Bachelor Thesis Holographic Dualities and …...University of Cyprus Department of Physics Bachelor Thesis Holographic Dualities and Emergent Gravity Panagiotis Charalambous Supervisor:

University of CyprusDepartment of Physics

Bachelor Thesis

Holographic Dualities

and

Emergent Gravity

Panagiotis Charalambous

2017-2018


Bachelor Thesis


and

Emergent Gravity


Supervisor: Prof. Nicolaos Toumbas

A thesis submitted in fulfillment of the requirements for the degree ofBachelor of Science in Physics

2017-2018


Bachelor Thesis


and

Emergent Gravity


Supervisor: Prof. Nicolaos Toumbas

A thesis submitted in fulfillment of the requirements for the degree ofBachelor of Science in Physics

2017-2018

Abstract

Gravity is the only fundamental force that has not yet been unified with the rest of the forcesof Nature, in a complete description of the world all the way from the microscopic quantumabsurdities to the macroscopic cosmological observations. What is more remarkable is thatgravity is the force more familiar to our mesoscopic (and macroscopic) realization and thefirst to be individually investigated. In fact, it is Newton’s early attempts to understandNature that made it possible not only to talk about gravity in a more systematic manner butto also quantify physics in a consistent framework. An impressive outcome of his analysis wasdifferential calculus. Indeed, the first differential equation ever written down was Newton’ssecond law of motion m~x = ~F (~x, t).

The most complete description for gravity at the classical level was obtained by Al-bert Einstein’s General Theory of Relativity. It is truly magnificent how Einstein’s purelyintellectual research provided humanity with such a groundbreaking interpretation of thegravitational force as a purely geometrical property. In accordance to this extraordinaryidea, Einstein managed to reshape the way we think of both space and time. But despite ofthe extensive research done for the sake of understanding gravity in the entire spectrum oflength scales, the result is always the same: Quantum Gravity is quirky.

There are, of course, good reasons for this peculiarity to be true. In particular, the factthat gravity is so different may suggest that gravity is an emergent/effective phenomenon.Even more counterintuitive than what we usually think of a theory of Quantum Gravity,the gravitational theory may arise from an equivalent description using a non-gravitationaltheory that lives in less dimensions! This bold claim is investigated in the present Thesis inthe content of holographic dualities, focusing on the Gauge-Gravity duality, the AdS/CFTcorrespondence, and its connection to quantum entanglement.

More details on the content of this script are presented in the Introduction where somehints on the emergent nature of gravity are discussed and the structure of the Thesis as well asthe notation and conventions followed throughout it are stated explicitly. In addition, someoriginal work on the entropic gravity proposal, an alternative interpretation of gravitationaleffects arising from purely thermodynamical laws, is presented at the last chapter whichhas also offered the author the achievement of accounting as a co-author for [1], currentlyawaiting to be published in the European Physical Journal C (Particles and Fields).

We promise to lead the reader throughout all the theoretical notions and ideas in asystematic way so that the tools developed will be well comprehended as to be applied atfurther personal interests. The reader may not have any background other than a typicalundergraduate knowledge on Quantum Mechanics and General Relativity, but even the mainaspects of General Relativity are reviewed in the Appendices. Having that said, the currentThesis is supposed to be self-consistent and pedagogical but if that proves not to be the casefor the curious reader, references herein can offer alternative descriptions.

Acknowledgments

I find it my obligation to express my gratitude to the Department of Physics of The Universityof Cyprus for its support throughout my introductory academic career. In particular, I wouldlike to thank Prof. Nicolaos Toumbas for supervising my Bachelor Thesis progress and foroffering me the opportunity to get enmeshed in original research, as well as Prof. StaurosTheodorakis without whom, I could not have exceeded my expectations and aim for anadvanced involvement beyond the preliminary Physics studies of a Bachelor degree.

“We can’t solve problems by using the same kind of thinking we usedwhen we created them.”

“Everything must be made as simple as possible, but not simpler.”– Albert Einstein

Contents

1 Introduction 31.1 Hints of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Why is Gravity different? . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Emergent Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I ORIGINS OF THE DUALITY 7

2 Black Holes Thermodynamics 92.1 What is a Black Hole? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Black Hole Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 ADM Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Applying the prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 The Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . . . 132.4.2 The AdS Black Brane Solution . . . . . . . . . . . . . . . . . . . . . 18

2.5 The Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 The AdS/CFT Correspondence 233.1 Gravity Side - Anti-de Sitter Spacetime . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.3 Boundary and Topology . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.4 Free Particles in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Gauge Side - Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Nother Currents for Conformal Symmetry . . . . . . . . . . . . . . . 353.2.3 The energy-momentum tensor in a CFT . . . . . . . . . . . . . . . . 353.2.4 Primary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.5 Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . 373.2.6 State-Operator Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 The original argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

i

ii CONTENTS

3.3.1 Large N Yang-Mills Theories . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Elements of String Theory . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Parameters matching and strong/weal coupling . . . . . . . . . . . . . . . . 523.5 Tests of the correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.1 Symmetries Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.2 Counting the degrees of freedom - The UV/IR Connection . . . . . . 533.5.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 The AdS/CFT Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

II MODERN REALIZATIONS OF THE DUALITY 71

4 Entropy and Entanglement in Quantum Systems 734.1 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Quantum Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 Reduced Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 Purifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Entropy in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 774.5 Quantum Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Two sided Black Holes 795.1 Eternal Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.1 A reminder on the Kruskal-Szekeres coordinates . . . . . . . . . . . . 795.2 Thermofield Double State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Holographic Duality Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 CFT side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Gravity Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 ER=EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Entanglement and Geometry 876.1 The Ryu-Takayanagi Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1.1 An example: The CFT vacuum state in Minkowski spacetime for aball-shaped region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.2 Evidence for Ryu-Takayanagi . . . . . . . . . . . . . . . . . . . . . . 936.2 Entanglement Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.1 First Law of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 976.2.2 Relative Entropy Inequalities . . . . . . . . . . . . . . . . . . . . . . 986.2.3 Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2.4 Strong Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3 Gravity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3.1 First Law of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 1066.3.2 Relative Entropy Inequalities . . . . . . . . . . . . . . . . . . . . . . 1126.3.3 Strong Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

CONTENTS iii

7 Entropic Gravity 115

7.1 Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.1.1 Why entropic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.1.2 Newtonian gravitational potential . . . . . . . . . . . . . . . . . . . . 118

7.1.3 General mass distributions . . . . . . . . . . . . . . . . . . . . . . . . 118

7.2 Einsteinian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2.1 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3 Test 1: Schwarzschild Black Hole . . . . . . . . . . . . . . . . . . . . . . . . 123

7.3.1 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3.2 Entropic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.4 Test 2: General Spherically Symmetric Mass Distribution . . . . . . . . . . . 128

7.4.1 Gravitational Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4.2 Entropic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4.3 Subtest 2: A statistical toy model . . . . . . . . . . . . . . . . . . . . 132

7.5 Conclusions/Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

III APPENDICES 135

Appendix A Gauge Field Theory 137

A.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.1.1 Nother’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.2 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 140

A.2.1 Nother’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A.3 Gauge field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.3.1 Abelian Gauge Theory - Maxwell Theory . . . . . . . . . . . . . . . . 144

A.3.2 Non-Abelian Gauge Theory - Yang Mills Theory . . . . . . . . . . . . 145

A.4 More on Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A.4.1 Global Vs Local Gauge Symmetry . . . . . . . . . . . . . . . . . . . . 148

A.4.2 Coupling of gauge fields to sources . . . . . . . . . . . . . . . . . . . 149

A.4.3 Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Appendix B Quantum Field Theory 153

B.1 General Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

B.1.1 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 153

B.1.2 Path Integral Quantization . . . . . . . . . . . . . . . . . . . . . . . . 154

B.2 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 156

B.2.1 Nother’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

B.2.2 Ward-Takahashi Identities . . . . . . . . . . . . . . . . . . . . . . . . 158

B.2.3 QFT realizations of symmetries . . . . . . . . . . . . . . . . . . . . . 159

B.3 Quantum gauge theories are haunted . . . . . . . . . . . . . . . . . . . . . . 160

B.3.1 The Faddeev-Popov Method . . . . . . . . . . . . . . . . . . . . . . . 160

B.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

iv CONTENTS

Appendix C General Relativity 167C.1 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167C.2 General Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . 168C.3 Einstein’s Field Equations and Geodesics . . . . . . . . . . . . . . . . . . . . 169C.4 Isometries and Killing Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 171

C.4.1 Constructing Maximally Symmetric Spacetimes . . . . . . . . . . . . 172C.5 Penrose Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C.6 Linearized Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Appendix D CFT in 2 Dimensions 179D.1 Holomorphic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179D.2 Ward-Takahashi Identities and Conformal Weights . . . . . . . . . . . . . . . 181D.3 Primary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183D.4 An example: The free massless scalar field theory . . . . . . . . . . . . . . . 183D.5 Central Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

D.5.1 Casimir Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190D.5.2 Weyl Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190D.5.3 Modular Invariance and Cardy’s formula . . . . . . . . . . . . . . . . 193D.5.4 c-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Appendix E Useful relations involving the generators of SU (N) 195

Appendix F Path Integral Formulation 197F.1 From Quantum Mechanics to Path Integrals . . . . . . . . . . . . . . . . . . 197F.2 From Quantum Field Theory to Path Integrals . . . . . . . . . . . . . . . . . 200F.3 Path integral representation of states . . . . . . . . . . . . . . . . . . . . . . 201F.4 Path integral representation of density matrices . . . . . . . . . . . . . . . . 202

Appendix G Entanglement structure of a CFT vacuum 205G.1 Domain of Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205G.2 Causal Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206G.3 Entanglement Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Appendix H Modular Hamiltonian 209H.1 Definition and Symmetry generator . . . . . . . . . . . . . . . . . . . . . . . 209H.2 Half-space subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210H.3 Ball-shaped subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Appendix I Newtonian Gravitational Potential in Higher Dimensions 215

Appendix J Post Newtonian Approximation 217

Appendix K Mean Value Theorem for Integrals 223

Appendix L Calculation of the Kretchmann invariant 225

CONTENTS v

References 227

vi CONTENTS

List of Figures

3.1 Universal cover CAdSD+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Penrose diagram of AdSD+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Poincare patch PAdSD+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Geodesics in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 Conformal map of Euclidean flat spacetime to Euclidean cylinder . . . . . . 393.6 Double line notation in large N Yang-Mills theories . . . . . . . . . . . . . . 433.7 Feynman rules for large N gauge theories . . . . . . . . . . . . . . . . . . . . 443.8 Planar and non-planar diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 443.9 String perturbation theory is a sum over topologies . . . . . . . . . . . . . . 513.10 Witten diagram with one 3-vertex interaction in the bulk . . . . . . . . . . . 673.11 Witten diagrams for the 4-point function in the bulk . . . . . . . . . . . . . 68

5.1 Penrose diagram for Lorentzian AdS-Schwarzschild black hole in AdSD . . . 815.2 Equivalent spatial slices under the isometry generator −i∂t in eternal black

hole geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Path integral representation of the thermofield double state . . . . . . . . . . 845.4 Gravity dual of the thermofield double state . . . . . . . . . . . . . . . . . . 85

6.1 Quantum subsystems are spatial subsets . . . . . . . . . . . . . . . . . . . . 886.2 Extremal area surfaces associated with boundary subsets in the Ryu-Takayanagi

formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Region A = SD−1

12

(R) associated with the ball-shaped subsystemA = BD−1(R)

according to the Ryu-Takayanagi formula . . . . . . . . . . . . . . . . . . . . 926.4 Entanglement entropy of the half-space subsystem with full quantum system

in the vacuum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.5 Subadditivity in the Ryu-Takayanagi formula . . . . . . . . . . . . . . . . . 1046.6 Strong subadditivity in the Ryu-Takayanagi formula . . . . . . . . . . . . . . 1056.7 AdS-Rindler wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1 Entropic force on a polymer immersed in a heat bath of temperature T . . . 1177.2 Space emergence after coarse graining . . . . . . . . . . . . . . . . . . . . . . 119

B.1 Quantum version of Nother’s theorem . . . . . . . . . . . . . . . . . . . . . . 157B.2 Ward-Takahashi Idenitities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B.3 Gauge Potential Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

vii

viii LIST OF FIGURES

C.1 Penrose diagram of (1 + 1)-dimensional flat spacetime . . . . . . . . . . . . . 175C.2 Penrose diagram of (D = d+ 1)-dimensional flat spacetime . . . . . . . . . . 176

F.1 Paths in discretized spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 198F.2 Pictorial depiction of path integral representations of vacuum states, partition

functions and density matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 204

G.1 Domain of dependence DA associated with a subsystem A . . . . . . . . . . 205G.2 Domains of dependence DA and DA associated with a subsystem A and its

complimentary A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

H.1 Domain of dependence DA associated with the half-space x1 > 0 at t = 0 . . 211

List of Tables

1.1 Fundamental forces of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Conformal coordinate transformations . . . . . . . . . . . . . . . . . . . . . 323.2 Conformal transformations generators . . . . . . . . . . . . . . . . . . . . . . 323.3 Nother Currents for Conformal Symmetry . . . . . . . . . . . . . . . . . . . 35

B.1 Feynman rules for ingoing and outgoing fundamental fields . . . . . . . . . . 166B.2 Feynman rules for propagators of fundamental fields . . . . . . . . . . . . . . 166

1

2 LIST OF TABLES

Chapter 1

Introduction

Gravity is the only fundamental force that has not yet been unified with rest of the forcesof Nature, in a complete description of the world all the way from the microscopic quantumabsurdities to the macroscopic cosmological observations. There are, of course, good reasonsfor this fact. In particular, the fact that gravity is so different suggests that gravity is anemergent concept. This bold claim is investigated in the present thesis in the content ofholographic dualities, focusing on Gauge-Gravity duality, the AdS/CFT correspondence andtheir connection to quantum entanglement. In addition, we review and present some originalwork on the entropic gravity proposal [1].

1.1 Hints of Holography

1.1.1 Why is Gravity different?

There only four fundamental forces in Nature1 giving rise to all the phenomena observedin the Universe: Electromagnetism, the Weak and the Strong Nuclear Forces and Gravity.Two of these forces (electromagnetism and gravity) have an infinite range, while the othertwo (the weak and the strong nuclear force) are felt only at short distances, of the orderof a few attometers to a few femptometers (atomic nucleus size). Electromagnetic forcescan be either attractive or repulsive, depending on the relative sign of the electrical chargesinteracting with each other. On the other hand, gravity can only be attractive becausethe gravitational charge (mass) is always positive. To be honest, all equations of Physicswork just as well with negative masses, but that would imply very different phenomena thanthe ones observed. Thus, it is almost experimentally proven that negative masses are notphysical. Whatever the case, what turns out to be more important is the mass square m2 ofa particle, since this is involved in the relativistic energy expression pµp

µ = −m2.But gravity is uniquely distinguished from the other three forces for many reasons. Firstly,

it is ridiculously weak! Compared to even to weakest of the other three forces, the weaknuclear force, which has a strength of the order of 10−6, gravity remains negligible with a

1Some theories suggest more fundamental forces.

3

4 CHAPTER 1. INTRODUCTION

Force Relative Strength Range Charge

Strong Nuclear Force 1 10−15 m Color (R, G, B)Weak Nuclear Force 10−6 10−18 m Isospin

Left-helicityElectromagnetism 1

137∞ Electric charge (±)

Gravity 10−39 ∞ Mass (+)

Table 1.1: Fundamental forces of Nature

relative strength of order 10−39. Some speculate that gravity is just as strong as the otherforces, but most of its flux escapes to other parallel universes.

Secondly, gravity is the only force that cannot be described by a Quantum Field Theory(QFT). All others forces are described by gauge theories in a fixed spacetime. Their basictheoretical structure is well understood and, in principle, everything is calculable. Anyattempt to apply this prescription of quantization to Einstein’s theory of general relativityyields infinities and cosmological singularities that cannot be removed. In other words, agravitational QFT is non-renormalizable and, therefore, unphysical. QFT is not the rightway to quantize gravity.

Lastly, gravity is a theory of space and time. This means that spacetime becomes dy-namical, in stark contrast to the fixed spacetime involved in the description of the otherthree forces. A quantum theory of gravity would, thus, involve quantum fluctuations ofspacetime itself. Finally, is spacetime continuous? Is it fundamental or does it emerge frommore fundamental objects?

1.1.2 Emergent Gravity

Regardless of the obscurity of the last question, there are hints in the general theory ofrelativity suggesting so. For example, in 1967, Sakharov argued that general relativity mayarise from ordinary condensed matter systems. In fact the general theory of relativity is verysimilar to hydrodynamics. Hydrodynamics is an effective field theory, and not a fundamentalone, that emerges from molecular dynamics.

Even though a quantum gravity theory has not yet been evidentially established, theforce carrier of the gravitational interactions, the graviton, must have spin-2. In addition, itmust be massless to account for the infinite range of the gravitational force. As a result, it isnatural to ask whether such a particle is elementary or a bound state of lower spin particles,such as photons, gluons or even electrons and quarks. If the answer is “yes”, then gravitycould be emergent.

There are a number of “no-go” theorems that forbid the existence of unphysical theoriesunder very natural assumptions. Eventually, loopholes in these theorems are accompaniedby potentially new physics. Two such no-go theorems were proven by Weinberg and Wittenin 1980 ([2]). The assumptions and restrictions of the theorems are surprisingly simple:

Theorem 1: A theory that allows the construction of a Lorentz covariant and conservedcurrent Jµ cannot contain massless particles with non-zero symmetry charge (Q =

1.2. THESIS STRUCTURE 5

d3~xJ0) of spin greater than 1

2.

Theorem 2: A theory that allows the construction of a Lorentz covariant and conservedenergy-momentum tensor T µν cannot contain massless particles of spin greater than1.

But gravity does contain a massless particle of spin greater than 1, the graviton withspin-2. Thus, it cannot emerge from renormalizable QFTs in a spacetime. But there is aloophole! The Weinberg-Witten theorems refer to theories defined in the same space theQFT lives. But what if gravity lives in a different spacetime?

For example, what if the gravitational theory lives in a spacetime with more dimensions?But then, how many are these extra dimensions? The answer comes from a well establishedand accepted principle: the holographic principle. The holographic principle provides aprescription for relating holographic dual theories, as we explain in Chapter 7, and revolvesaround the only known objects that are governed by both Quantum Mechanics and GeneralRelativity: black holes.

1.2 Thesis Structure

The present Thesis is organized as follows. In Part I the original ideas behind holographicdualities are discussed along with some theoretical aspects. In particular, Chapter 2 isconcerned with black hole thermodynamics and the holographic principle.

Chapter 3 is dedicated to the AdS/CFT correspondence. Unfortunately, it is impos-sible to present all aspects of the AdS/CFT correspondence due to the vast theoreticalbackground, but the main results are explained as part of an introduction to the subject.The chapter includes theoretical supplements on Anti-de Sitter geometry, Conformal FieldTheory as well as elements of String Theory.

Part II is devoted to modern realizations of gauge-gravity duality. In Chapter 4, wereview quantum entanglement and some aspects of quantum information theory. In Chapter5, we the connection between quantum entanglement and wormholes in the context of theAdS/CFT correspondence.

In Chapter 6, we present a connection between geometry and entanglement that exempli-fies the emergent nature of gravity. Lastly, in Chapter 7 we introduce the concept of entropicgravity, where the proposal is reviewed along with some tests and comments.

There are also a number of Appendices in Part III to assist the reader in gaining moreinsight on some either very general or very specific theoretical concepts and computationalresults related to the Thesis.

1.3 Notation and Conventions

Throughout this Thesis, the spacetime metric is taken to have signature (−1, . . . ,−1,+1, . . . ,+1),with the minus and positive signs corresponding to temporal and spatial dimensions respec-tively. The Minkowski metric is denoted by ηµν . For example, the flat 4-dimensional invariant

6 CHAPTER 1. INTRODUCTION

length is written asds2 = ηµνdx

µdxµ = −dt2 + d~x2 (1.1)

Spatial components of tensors and vectors are denoted by Latin indices. Greek indicesdenote spacetime components. Capital Greek letters indices are used when dealing withembedding spacetimes. Latin indices from the beginning of the alphabet, i.e. a, b, c, . . . ,are used to denote gauge field components. When performing some kind of transformationson fields, primed fields and indices represent the transformed quantities.

Spatial vectors are distinguished from spacetime vectors by an arrow. That is, ~x is aspatial vector while xµ is a spacetime vector. The inner product ~x · ~y is equal to the sum∑

i xiyi, while x · y is xµy

µ = gµνxµyν . Norms are frequently denoted without indices. For

example, the norm of spacetime vector Aµ is

A · A = AµAµ ≡ A2 (1.2)

The total number of dimensions (temporal + spatial) is denoted with the capital letterD. The number of temporal dimensions is denoted with q, while the number of spatialdimensions is denoted with p. Throughout most of the Thesis, the number of temporaldimensions is taken to be 1, in which case the number of spatial dimensions is writtenas d = D − 1. An exception is when considering the gravitational dual to a holographic(D = d+ 1)-dimensional QFT. The gravitational theory haw an additional spatial direction.That is, the gravitational theory lives in a (D + 1)-dimensional spacetime with D spatialdimensions.

Under these conventions, the operator in flat (D = d+ 1)-dimensional spacetime is,

= ∂2 = ∂µ∂µ = −∂2

t +∇2 (1.3)

which has an overall sign difference with the convention used in many field theory textbooks.The notation for covariant derivatives in curved spacetime is ∇µ, while for gauge covariantderivatives the symbol Dµ is used.

Part I

ORIGINS OF THE DUALITY

7

Chapter 2

Black Holes Thermodynamics

The recently deceased great physicist Stephen Hawking reshaped the way physicists lookat black holes. He managed to convince everyone that black holes are not really black, butthey emit radiation leading to their eventual evaporation. In this small chapter, some aspectsregarding the nature of black holes and their thermodynamic properties will be discussed.In addition, the famous holographic principle will be stated and commented upon.

2.1 What is a Black Hole?

First of all, what is a black hole? This seemingly trivial question has a misleading answer. Bydefinition, a black hole is a mass distribution in spacetime equipped with an event horizon,i.e. a surface beyond which no causal contact is possible with the exterior region. In otherwords, nothing, not even light, can escape the gravitational field of such distributions, andit is doomed to fall in the black hole never to return... However, Stephen Hawking provedthat some radiation can escape the black hole.

A general geometry,ds2 = gµνdx

µdxν (2.1)

may be identified with a black hole if an event horizon exists. The event horizon is a surfaceacross which the time coordinate flips to a space-like coordinate1. As a result, signals thatpass across the horizon must travel back in “time” in order to escape. Consequently, theevent horizon is defined by the equation

g00 = 0 (2.2)

Every black hole has a physical singularity, at which spacetime geometry cannot bedefined due to infinite curvature23. It is due to this singularity that a black hole cannot becompletely described in the context of general relativity.

1and one space-like coordinate flips to a time-like one in order to preserve the overall signature of themetric

2At a physical singularity, various curvature invariants, such as the Kretchmann invariant RµνκλRµνκλ,

diverge.3Coordinate singularities at which a metric component may diverge, can be removed by a suitable

9

10 CHAPTER 2. BLACK HOLES THERMODYNAMICS

2.2 Black Hole Mechanics

Black holes seem peculiar but in fact, they very simple objects. They are uniquely describedby three quantities: their mass M , their electric charge Q and their angular momentumJ . This statement is known as the no hair theorem: “regardless of their origin4, the onlyidentity that black holes carry are M , Q and J”. It is, thus, not surprising that black holesare governed by some laws. In particular, they are governed by four laws. These laws involvethe horizon area and its surface gravity.

2.2.1 ADM Energy

The mass M of a black hole is defined as the ADM energy or ADM mass ; an invariant energymeasured at infinity that accounts for all types of energy (matter, electromagnetic radiation,gravitation radiation). Since the geometry of spacetime contains information about thesources of energy inducing curvature, we interpret any possible form of energy as arisingfrom and giving rise to deviations of empty spacetime. In other words, given the metricgµν of a general geometry, which asymptotically approaches an empty spacetime with metric(0)g µν , the perturbation

hµν = gµν −(0)g µν (2.3)

contains all information about the total energy of the system relative to the energy of emptyspacetime. The perturbation hµν vanishes asymptotically. We only consider the case of

asymptotically flat (Minkowski) spacetime,(0)g µν = ηµν , even though the generalization to

other cases is straightforward. Expanding the metric as above, the Einstein equations canbe written as

(1)

Rµν −1

2ηµν

(1)

R = −8πG (Tµν + tµν) (2.4)

where(1)

Rµν is the part of the Ricci tensor linear in hµν ,

(1)

Rµν =1

2

(∂µ∂νh+hµν − ∂ρ∂µhρν − ∂ρ∂νhρµ

)(2.5)

Indices are raised and lowered with the unperturbed metric: ≡ ηµν∂µ∂ν , hρµ ≡ ηρσhσµ and

(1)

R ≡ ηµν(1)

Rµν . The “tensor” tµν contains the remaining terms not linear in hµν :

tµν =1

8πG

(Rµν −

1

2gµνR−Rlin.

µν +1

2ηµνR

lin.

)(2.6)

coordinate transformation. For example, because at the poles of a sphere the azimuthal angle cannot bedefined in a unique way, the inverse metric (in spherical coordinates) diverges. Transforming to Cartesiancoordinates these singularities are removed. The criteria of systematically classifying singularities is thefiniteness of coordinate invariant quantities.

4Physical black holes are formed from collapsing stars.

2.2. BLACK HOLE MECHANICS 11

This procedure is just rewriting the field equations. The crucial step is to realize that tµνbehaves as an energy-momentum “tensor”5 of the gravitation field itself. Consequently, thetotal energy-momentum “tensor” that takes into account matter and gravitation radiationis

τµν = ηµρηνσ (Tρσ + tρσ) (2.7)

There are a number of properties of τµν that support this claim, such as the linearizedBianchi identities

∂µ

((1)

R

µν

− 1

2ηµν

(1)

R

)= 0 (2.8)

and the local conservation law∂µτ

µν = 0 (2.9)

The total energy and momentum of the system are given by

P µ =

dd~x τ 0µ (2.10)

We consider the limit where some radial coordinate r becomes large. The ADM energyis given by the formula,

M ≡ EADM = limr→∞

1

16πG

dΩD−2r

D−2ni (∂jhij − ∂ihjj) (2.11)

where hµν = gµν − ηµν in Cartesian coordinates xi and ni = xir

. For more details, see thediscussion in Section 6, Chapter 7 of [3].

The area AH of the event horizon is given by

AH =

g00=0

dD−2x√g (2.12)

where gij is the induced metric on the surface g00 = 0.The surface gravity κ can be extracted from the near horizon geometry. Mathematically,

it is defined via the equationξν∇νξ

µ = κξµ (2.13)

where ξµ is a suitably normalized Killing vector (ξµξµ = −1 at infinity) that becomes null

(ξµξµ = 0) at the event horizon. Therefore, the event horizon is also a Killing horizon. At

the horizon this equation is equivalent to

∇µξ2 = κξµ

κ2 = −1

2∇µξν∇µξν

(2.14)

where we used the relation,

ξ[µ∇νξρ] = 0 on the Killing horizon (2.15)

5tµν is not really a tensor since it does not transform as a tensor under general coordinate transformations.


The surface gravity κ is the force needed to hold a unit mass at rest on the horizon (asmeasured at infinity).

As a result, the four laws of black mechanics are,

0th law: The surface gravity of a stationary black hole (J = 0) is constant.

1st law: Under a perturbation of a stationary black hole the change of its horizon areadA, its mass dM , its angular momentum dJ and its electric charge dQ satisfy

dM =κ

8πGdAH + ΩdJ + ΦdQ (2.16)

where κ is the surface gravity, Ω the angular velocity and Φ the electrostatic potential.

2nd law: The horizon area can only increase, dAH ≥ 0. For example, if two black holesof horizon areas A1 and A2 merge, the new black hole must have an area A3 ≥ A2 +A1.

3rd law: A black hole of vanishing surface gravity (κ = 0) cannot exist.

2.3 Black Hole Thermodynamics

The connection between black hole mechanics and thermodynamics becomes obvious aftersome natural identifications. To make these identifications explicit, we first write down thefour laws of thermodynamics

0th law: The temperature T of a system at thermal equilibrium is constant.

1st law: Under a process the energy E of a system changes in a manner dictated bythe equation

dE = TdS + dW (2.17)

where S is the entropy of the system and W the work done on the system by theenvironment.

2nd law: The entropy of a system can only increase, dS ≥ 0, with the equality holdingonly for closed and isolated systems.

3rd law: A system at absolute zero (T = 0) can never exist.

It is, thus, obvious that the two sets of laws become identical if the black hole is associatedwith a temperature TBH ∝ κ and an entropy SBH ∝ AH so that

TBHSBH =κAH8πG

(2.18)

2.4. APPLYING THE PRESCRIPTION 13

The precise relation between the entropy and the horizon area has been extracted by Beken-stein and Hawking and is called Bekenstein-Hawking entropy

SBH =AH4G

=AH

4lD−2P

⇒ TBH =κ

2π

(2.19)

where lP = G1

D−2 is the Plank length in units where ~ = c = 1.In fact, the temperature TBH is called Hawking temperature and is associated with ther-

mal radiation emmited by the black hole and reaching infinity. Of course, Hawking did notjust guessed the connection as was done here; he carefully counted the degrees of freedomand used the phenomenon of pair creation near the horizon to prove that a partner mayescape to infinity while the other partner falls into the black hole. It is these two resultsthat enable the identification of a black hole as a pure thermodynamic system. Actually,the knowledge of Hawking temperature suffices to determine the entropy via the 1st law ofthermodynamics:

dM = TBHdSBH + dW (2.20)

One way to study a QFT at finite temperature is to rotate to Euclidean space, t = −iτ ,and compactify the Euclidean time direction on a circle of period β: τ ∼ τ+β. Then the fieldvacuum amplitude becomes precisely the canonical partition function, with the temperaturebeing equal to the inverse period of the Euclidean time circle: T = 1/β. Thus,

〈1〉 =

φ(xi,τi+β)=φ0

φ(xi,τi)=φ0

Dφ e−βH[φ] = Tr e−βH = Zβ (2.21)

Similarly, the temperature of a black hole can be found by considering the near horizongeometry, which turns out to be flat in appropriate coordinates. Rotating to Euclidean sig-nature, enforces periodicity of the Euclidean time so as to avoid a conical singularity6. Goingback to Schwarzschild coordinates yields the Hawking temperature measured at infinity.

2.4 Applying the prescription

In this section we present a set of examples that will also turn out to be useful later on.

2.4.1 The Schwarzschild Black Hole

We consider first the simplest black hole. This is a stationary (J = 0), electrically neutral(Q = 0) black hole, called the Schwarzschild black hole, with metric

ds2 = −f(r)dt2 +dr2

f(r)+ r2dΩ2

D−2

f(r) = 1−(RS

r

)D−3 (2.22)

6The conical singularity can be avoided if we make an angle coordinate periodic with period 2π.


RS is the Schwarzschild radius. Classically, it corresponds to the radius of a star, such thatan object launched from the surface can barely escape when it travels at the speed of light.

Step 1: Event horizon and horizon area

The temporal component of the metric, g00 = −f(r), vanishes precisely at the Schwarzschildradius RS. Thus the horizon is a sphere with area7

AH = ΩD−2RD−2S (2.24)

Step 2: Mass of the black hole

Next we calculate the mass of the black hole, i.e. the ADM energy. In Cartesian coordinates(t, xi = rΩi), the metric becomes

ds2 = −f(r)dt2 +D−1∑i,j=1

[(1

f(r)− 1

)ninj + δij

]dxidxj (2.25)

with ni = xir

, and so

hij = gij − ηij =

(RS

r

)D−3

ninj +O(r−2(D−3)

)(2.26)

Using the relationsnini = 1

∂ir = ni

∂inj =δij − ninj

r

(2.27)

we find

∂jhij = −D − 3

RS

(RS

r

)D−2

∂jr ninj +

(RS

r

)D−3

(∂jni nj + ni∂jnj)

= −D − 3

RS

(RS

r

)D−2

ninjnj +

(RS

r

)D−3(0 +

D − 2

rni

)=RD−3S

rD−2ni

∂ihjj = −D − 3

RS

(RS

r

)D−2

∂ir

= −D − 3

RS

(RS

r

)D−2

ni

(2.28)

7The general area formula is (2.12), yielding in this case

AH =

g00=0

dD−2x√g

=

dΩD−2R

D−2S = ΩD−2R

D−2S

(2.23)


Then the ADM energy can be computed to be

EADM = limr→∞

1

16πG

dΩD−2r

D−2ni (∂jhij − ∂ihjj)

=(D − 2)RD−3

S

16πG

dΩD−2

=(D − 2) ΩD−2R

D−3S

16πG≡M

⇒ RS =

(16πGM

(D − 2) ΩD−2

) 1D−3

(2.29)

Step 3: Surface gravity

To calculate the surface gravity κ, we need to find a Killing vector ξµ that becomes null onthe event horizon. First we determine the most general Killing vector associated with theSchwarzschild solution. This procedure can be very tedious in some cases but, thankfully,there is a way to directly read off some of the Killing vectors. In general, if the componentsof the metric are independent of some particular coordinate xµ8 then ξµµ = δµµ is a Killingvector. This is true because the corresponding generator is Gµ = −iξµµ∂µ = −i∂µ. Indeed, ifthe metric does not depend on xµ, then the Euler-Lagrange equations for a particle movingalong a geodesic induce a conservation law

L = (−gµν xµxν)−12 = 1

⇒ ∂L

∂xµ= 0 if gµν is independent of xµ

⇒ d

dτ

(∂L

∂xµ

)= 0

⇒ gµµxµ = xµ = constant

(2.30)

So, for the Schwarzschild metric (2.22), which is independent of the time coordinate t andthe azimuthal angle θD−1 ≡ φ9, two Killing vectors can be immediately found to be

ξµt = δµ0 ⇒ Gt = −i∂tξµφ = δµD−1 ⇒ Gφ = −i∂φ

(2.32)

In addition, because the Schwarzschild solution is spherically symmetric, there are D(D−1)2

Killing vectors associated with the rotation generators. In Cartesian coordinates

ξµij = xiδµj − xjδ

µi ⇒ Gij ≡ Lij = −i (xi∂j − xj∂i) (2.33)

8The index µ represents a single constant value of the spacetime index µ = µ.9Recall that the line element of the unit sphere SD−2 is given by

dΩ2D−2 =

D−1∑i=2

(i−1∏n=2

sin2 θn

)dθ2i (2.31)

. Thus the metric components do not depend on the azimuthal angle θD−2.


Now ξµD−2,D−1 is precisely ξµφ = ∂φ = ∂xi∂φ∂i = xD−2∂D−1 − xD−1∂D−2, leaving a total of

D(D−1)2

+ 1 Killing vectors. This is D− 3 generators short from being maximally symmetric,but the Schwarzschild spacetime is not maximally symmetric.

Only the Killing vector that becomes null at the event horizon and is appropriatelynormalized at infinity is needed to find the surface gravity. The norms of the Killing vectors(ξµξ

µ = gµνξµξν) are

ξt µξµt = g00 = −f(r)

ξφµξµφ = gD−1,D−1 = r2

D−1∏n=2

sin2 θn

ξij µξµij = 0 (no sum)

(2.34)

On the event horizon (r = RS), both ξµt and ξµij become null. But the normalization condition,limr→∞ ξµξ

µ = −1, singles out ξµt . Finally we get

κ2 = −1

2∇µξt ν∇µξνt

= −1

2gνρg

µσ∇µξρt∇σξ

νt

= −1

2gνρg

µσΓρµ0Γνσ0

= −1

2

[g00g

11(Γ0

10

)2+ g11g

00(Γ1

00

)2]

= −1

2

[(−f 2(r)

)( f ′(r)2f(r)

)2

+

(− 1

f 2(r)

)(f(r)f ′(r)

2

)2] ∣∣∣∣

r=RS

=f ′2(RS)

4

(2.35)

⇒ κ =f ′(RS)

2=D − 3

2RS

(2.36)

As a result, the temperature of the Schwarzschild black hole is

TBH =κ

2π=D − 3

4πRS

(2.37)

Using the 1st law of thermodynamics (2.20), we can obtain the entropy of the black hole

dM = TBHdSBH

⇒ d

((D − 2) ΩD−2R

D−3S

16πG

)=D − 3

4πRS

dSBH

⇒ dSBH =(D − 2) ΩD−2

4GRD−2S dRS

⇒ SBH =ΩD−2R

D−2S

4G=AH4G

(2.38)


This is precisely the Bekenstein-Hawking entropy10.Alternatively, the QFT statement “the temperature is the period of the imaginary (Eu-

clidean) time” can be employed to make the calculation simpler, by extracting the Hawkingtemperature without direct calculation of the surface gravity.

Step 3 (alternative): Hawking Temperature

The idea is to investigate the near horizon geometry, r = RS(1 + ε), ε 1,

ds2 ' − (D − 3) εdt2 +1

D − 3

dr2

ε+R2

SdΩ2D−2 (2.39)

Defining the coordinate ρ measuring the proper distance from the event horizon,

ρ =

r

RS

dr′√f(r′)

' 2RS√D − 3

√ε (2.40)

and rescaling the time coordinate

ω =t (D − 3)

2RS

(2.41)

makes the local geometry Rindler× SD−2RS

:

ds2 ' −ρ2dω2 + dρ2 +R2SdΩ2

D−2 (2.42)

Next, we rotate to Euclidean time:

ω = −iωE⇒ ds2 ' ρ2dω2

E + dρ2 +R2SdΩ2

D−2

(2.43)

The Rindler part of the geometry has now turned into a 2-dimensional flat spacetime inpolar coordinates, with the Euclidean time playing the role of the polar angle. Avoiding theconical singularity requires periodicity of the Euclidean time ωE ∼ ωE + 2π. Going back toSchwarzschild coordinates (with Euclidean time tE) yields

tE ∼ tE +4πRS

D − 3(2.44)

The period of the imaginary time is precisely the inverse Hawking temperature:

T−1BH =

4πRS

D − 3⇒ TBH =

D − 3

4πRS

(2.45)

(which also implies that κ = D−32RS

).

10The “initial” condition for the entropy is SBH |RS=0 = 0; in other words, there is not any entropy whenthere is no black hole.


2.4.2 The AdS Black Brane Solution

Next we consider a black brane solution in Anti-de Sitter (AdS) spacetime11. The spacetimemetric in the so called Poincare coordinates (z, xµ)12 is

ds2 =b2

z2

(−f(z)dt2 + d~x2 +

dz2

f(z)

)f(z) = 1−

(z

z0

)D−1 (2.46)

We follow the same procedure as before.

Step 1: Event horizon and horizon area

The temporal component of the metric g00 vanishes is at z = z0. The event horizon hasplanar topology. The area is infinite. We may regulate it and use equation (2.12) (withgij|z,t=const. = b2

z2 δij) to find

AH =

g00=0

dD−2x√g

=

(b

z0

)D−2 RD−2

dD−2x

=

(b

z0

)D−2

VD−2

(2.47)

where VD−2 is the volume of the regulated (D − 2)-dimensional Euclidean space.

Step 2: Mass of the black hole

The ADM energy is tricky to define here. One needs to investigate asymptotically AdSspacetimes and extract an expression for the ADM energy in a similar way with asymp-totically flat spacetimes. However the fact that the AdS spacetime is not flat makes thecalculations much more tedious. Instead, the mass density of the black brane will be foundvia the Bekenstein-Hawking formula that relates the entropy with the horizon area, whichis a universal formula after all.

Step 3: Hawking Temperature

Near the horizon z = z0(1− ε), ε 113, the geometry becomes

ds2 ' −(D − 1) b2ε

z20

dt2 +b2

(D − 1) z20

dz2

ε+b2

z20

d~x2 (2.48)

11The AdS geometry is reviewed in detail in the next chapter.12These are defined to give rise to (3.13)13Near the horizon means between the event horizon and the boundary of AdS space. Thus, since the

boundary is at z = 0, z < z0.


Therefore, after defining the proper distance from the event horizon ρ,

ρ = b

z0

z

dz′

z′√f(r′)

' 2b√D − 1

√ε (2.49)

and rescaling the rest of the coordinates as follows

ω =t (D − 1)

2z0

yi =b

z0

xi

(2.50)

the metric takes the from of a Rindler× RD−2 geometry:

ds2 ' −ρ2dω2 + dρ2 + d~y2 (2.51)

The Euclidean time must be periodic in order to avoid a conical singularity

ω = −iωE⇒ ds2 ' ρ2dω2

E + dρ2 + d~y2

⇒ ωE ∼ ωE + 2π

(2.52)

The Hawking temperature is the inverse period of tE = it

tE ∼ tE +4πz0

D − 1

⇒ TBH =D − 1

4πz0

(2.53)

Back to Step 2!

As explained in step 2, the Bekenstein-Hawking entropy and the 1st law of thermodynamicsyield the mass density of the black brane

SBH =AH4G

=

(b

z0

)D−2VD−2

4G(2.54)

⇒ dM = TBHdSBH

=D − 1

4πz0

d

((b

z0

)D−2VD−2

4G

)

= −(D − 1) (D − 2)VD−2bD−2

16πG

dz0

zD0

(2.55)

⇒ M

VD−2

=(D − 1) (D − 2) bD−2

16πG

∞z0

dz′0z′D0

=(D − 2)

16πGb

(b

z0

)D−1 (2.56)


2.5 The Holographic Principle

The holographic principle is actually a very simple statement. It suggests a bound to thenumber of degrees of freedom of a system, taking into account gravitational interactions. Tounderstand the holographic bound, one needs to consider an isolated system of energy (ormass) E and entropy S, in asymptotically flat spacetime. If A is the area of a sphere thatencloses the system, called a holographic screen, then E must be smaller than the mass MA

of a black hole with area A. Otherwise the system would have collapsed to a black hole witharea greater than A or the holographic screen would not enclose the entire system. Next, anamount of energy E −MA and entropy S ′ is added to the system keeping the holographicscreen fixed, leading to the creation of a black hole of mass MA and horizon area A. The2nd law of thermodynamics ∆S ≥ 0 yields a bound for this process

∆S = Safter − Sbefore= SBH − (S + S ′) ≥ 0

⇒ S + S ′ ≤ SBH

⇒ S ≤ SBH

(2.57)

∴ S ≤ A

4G(2.58)

As a result, the maximum entropy inside a region of area A is precisely the Bekenstein-Hawking entropy of a black hole of the same area.

But what the holographic principle really implies is much more fundamental. It leads toan enormous reduction to the number of degrees of freedom needed to describe the system!Indeed, the maximum entropy of a system is a measure of the number of degrees of freedomassociated with the system. For example, consider the entropy corresponding to a completelyincoherent density matrix, of the form ρ = 1

N1 where N is the dimensionality of the Hilbert

space: S = −Tr (ρ log ρ) = Smax = logN . Consequently

logN ≤ A

4G(2.59)

In typical physical systems the number of simple degrees of freedom is of the order Ndof ∼logN . As a result the important bound associated with the holographic principle is

Ndof ≤A

4G=

A

4ld−1P

(2.60)

and translates to the statement:

Holographic Principle: In Quantum Gravity, a region of boundary area A can befully described by no more than A/4lD−1

P degrees of freedom, i.e. 1 degree of freedom perPlanck area.

However, in QFTs, the number of degrees of freedom scales with the volume V thatencloses the system. Therefore, a quantum theory of gravity living in a (D+ 1)-dimensional

2.5. THE HOLOGRAPHIC PRINCIPLE 21

spacetime MD+1, the bulk, could be emergent from a QFT living in a D-dimensional fixedspacetime BD such that BD = ∂MD+1, the boundary. The bulk may actually be higherdimensional as long the extra dimensions are compact. So, the exact argument is

Holographic Duality A quantum theory of gravity living in a (D+1+dc)-dimensionalspacetimeMD+1 ×Mdc, with Mdc a dc-dimensional compact manifold, could emerge froma QFT living in a D-dimensional fixed spacetimeBD such that BD = ∂MD+1.


Chapter 3

The AdS/CFT Correspondence

The AdSD+1/CFTD correspondence is a conjecture. Even though a rigorous, mathematicalproof is still lacking, the conjecture is believed to be true by the majority of the theoret-ical physics community. The correspondence is about a holographic duality : A (D + 1)-dimensional theory of quantum gravity, on asymptotically Anti-de Sitter spaces (AdSD+1)1

is dual to D-dimensional conformal field theory (CFTD) on the boundary [4]. The CFTis a local quantum field theory without gravity. In 1997 Juan Maldacena proposed a veryinteresting for dimensional example of a gauge/gravity duality between:

N = 4 Super Yang-Mills theory on 4-dimensional Minkowski spacetime, R3,1

Type IIB Superstring theory on AdS5 × S5

By now Maldacena’s conjecture has passed a large number of tests. Also, it has beenextended to other dimensions. In particular, string or M-theory on AdSD+1×M, whereMis a suitable compact manifold, is dual to a CFTD.

In this Thesis, we review basic aspects of the correspondence. We begin by describingproperties of the bulk geometry and then we introduce the boundary theory. We also presentMaldacena’s original argument, including the so called large N limit of an SU(N) Yang-Millstheory and elements of string theory in order to make the correspondence plausible. We alsodescribe some tests of the correspondence and present the general AdS/CFT dictionary.

3.1 Gravity Side - Anti-de Sitter Spacetime

After a quick look on Appendix C, in particular Section C.4, it is easy to introduce Anti-deSitter (AdS) spacetime. The definition of the spacetime, as well as the basic propertiesregarding its geometry and dynamics will be described in the following subsections. Otheruseful reviews on AdS spacetime include [5][6][7][8].

1times a compact manifold M, as in the original AdS5/CFT4 example of Maldacena

23

24 CHAPTER 3. THE ADS/CFT CORRESPONDENCE

3.1.1 Definition

AdSD+1 is a (D+1)-dimensional maximally symmetric spacetime with a negative cosmologicalconstant2. As discussed in C.4.1, AdSD+1 can be embedded spacetime in R

D,2 with metric

ds2 = −dX20 − dX2

D+1 +D∑i=1

dX2i (3.1)

via the equation

−X20 −X2

D+1 +D∑i=1

X2i = −b2 (3.2)

The positive constant b is the radius of curvature3 or just AdS radius. Since the spacisa hyperboloid embedded in flat spacetime, it is not compact. By construction, AdSD+1 isinvariant under SO (D, 2) transformations with generators

Lµν = −i (Xµ∂ν −Xν∂µ)

Lµ,D+1 = −i (Xµ∂D+1 +XD+1∂µ)

LD+1,D+1 = −2iXD+1∂D+1

(3.3)

from which only the generators Lµν of the SO (D, 1) group are Killing vectors4 satisfyingthe Killing equation.

With the metric of the spacetime given, it is easy to show that the scalar curvature andthe cosmological constant are given by,

R = −D (D + 1)

b2

Λ = −D (D − 1)

2b2

(3.4)

which are indeed constant and negative.

3.1.2 The metric

The form the AdS metric depends on the parameterization of the embedding coordinatesX0, Xi and XD+1. There are numerous ways to parameterize these; the most important onesare listed here.

2The metric is Lorentzian with signature (1, D)3not to be mistaken with the scalar curvature4with the right (maximum) number of generators

3.1. GRAVITY SIDE - ANTI-DE SITTER SPACETIME 25

Global coordinates

The global coordinates (τ, ρ,Ωi) are defined as follows

X0 = b cosh ρ sin τ

Xi = b sinh ρΩi

XD+1 = b cosh ρ cos τ

τ ∈ [0, 2π)

ρ ∈ R+

D∑i=1

Ω2i = 1

(3.5)

where Ωi are simply the embedding coordinates of the unit sphere SD−15,

Ω1 = cos θ2

Ωn =n∏

m=2

sin θm cos θn+1

ΩD =D∏

m=2

sin θm

(3.6)

In these coordinates, the metric takes the form

ds2 = b2(− cosh2 ρdτ 2 + dρ2 + sinh2 ρdΩ2

D−1

)(3.7)

with the metric on the unit sphere SD−1 given by

dΩ2D−1 =

D−1∑i=1

(i−1∏n=1

sin2 θn

)dθ2

i (3.8)

Another useful form of the AdSD+1 metric is obtained via the transformation r = b sinh ρ,t = bτ ,

ds2 = −f (r) dt2 +dr2

f (r)+ r2dΩ2

D−1

f (r) = 1 +r2

b2

(3.9)

From the definition of the temporal coordinate, τ ∈ [0, 2π), only closed timelike curvesexist, but one can unwrap the hyperboloid taking τ ∈ R. We end up with the universal coverof AdS, denoted as CAdSD+1 (Figure 3.1).

5The angles are denoted as θn with n = 2, . . . , D and not n = 1, . . . , D− 1 to make explicit the fact thatthere are D spatial coordinates: x1 = r, x2 = θ2, . . . , xD = θD.


Figure 3.1: The neck of the hyperboloid is close timelike curve. The universal cover CAdSD+1

is obtained by unwrapping the circle S1 and let the time coordinate take values in the entirerange of R.

𝜏

= 0

𝑆𝑑−1

→ 2

(Boundary)

Figure 3.2: Penrose diagram of AdSD+1.

A convenient form of the metric is obtained by defining sinh ρ = tan ρ, constraining ρ totake values in the interval

[0, π

2

),

ds2 =b2

cos2 ρ

(−dτ 2 + dρ2 + sin2 ρdΩ2

D−1

)(3.10)

As a result, the Penrose diagram (obtained by the conformal cover, i.e. omitting the overallpositive factor) is a cylinder as in figure 3.2.


1. AdS & Poincaré domain

Anti-de Sitter AdSd+1 (d ≥ 2) is the maximally symmetric solution to Einstein’sequations with a negative cosmological constant. It is defined by the relation

−X20 −X2

1 +∑d+1

i=2 X2i = −`2 , ` > 0 ,

where (X0, . . . , Xd+1) are Cartesian coordinates of M2,d.

Poincaré patch (t, z, xi), t ∈ R, z ∈ R>0 and xi ∈ R, i = 1, . . . , d− 1,

ds2 =`2

z2(−dt2 + dz2 + δijdxidxj

).

The region covered by this chart is the Poincaré fundamental domain, PAdSd+1.

Figure 3.3: The Poincare patch PAdSD+1. At each point in the Penrose diagram (left) livesa sphere SD−1.

Poincare coordinates

The Poincare coordinates (t, xi, u), (i = 1, . . . , D − 1) are defined as,

X0 =1

2u

(1 + u2

(b2 + ~x2 − t2

))Xi = uxi

XD =1

2u

(1− u2

(b2 − ~x2 + t2

))XD+1 = ut

t, xi ∈ Ru ∈ R+

(3.11)

with the AdSD+1 metric taking the form

ds2 = b2

(du2

u2+ u2 (dxµdx

µ)

)(3.12)

where xµ = (t, ~x). Equivalently, writing z = 1u, one ends up with the following form of the

metric

ds2 =b2

z2

(dz2 + dxµdx

µ)

(3.13)

In contrast to the global coordinates, the Poincare coordinates cover only a patch of thehyperboloid, the Poincare patch or the Poincare fundamental domain, denoted as PAdSD+1,as shown in figure 3.3.


Sausage coordinates

Lastly, the sausage coordinates (τ, ρ,Ωi) are defined as

X0 = b1 + ρ2

1− ρ2sin τ

Xi = b2ρ

1− ρ2Ωi

XD+1 = b1 + ρ2

1− ρ2cos τ

τ ∈ [0, 2π)

ρ ∈ [0, 1)

D∑i=1

Ω2i = 1

(3.14)

yielding the AdSD+1 metric in sausage coordinates

ds2 = −(

1 + ρ2

1− ρ2

)2

b2dτ 2 +4b2

1− ρ2

(dρ2 + ρ2dΩ2

D−1

)(3.15)

3.1.3 Boundary and Topology

It is important to understand the boundary behavior of AdS since, after all, the AdS/CFTcorrespondence concerns a CFT living at the boundary of the AdS spacetime. There areactually multiple conformally related boundaries for AdS. If read from CAdSD+1 in (3.10),the boundary at ρ→ π

2is conformally related to a cylinder R× SD−1,

ds2 ρ→π2−→ b2

cos2 ρds2

cylinder

ds2cylinder = −dτ 2 + dΩ2

D−1

(3.16)

If the boundary is read from PAdSD+1 in (3.13) at z → 0, it is conformally related to flatspacetime RD−1,1,

ds2 z→0−→ b2

z2ds2

flat

ds2flat = dxµdx

µ = −dt2 + d~x2(3.17)

The topology of the boundary is that of the sphere; it has genus g = 0. Since the entirespacetime is smooth, the entire AdS geometry has genus g = 0.

3.1.4 Free Particles in AdS

The dynamics of a particle in a fixed background metric are extracted from the geodesicequations (C.17) for massive particles and (C.18) for massless particles. As will be shown,


𝜏

= 0 → 2

Figure 3.4: The geodesics of particles moving in fixed AdS background. Massive particles(blue line) perform oscillatory motion, while massless particles (red line) move at straightlines with 45 slope.

the two types of particles behave extremely differently in spacetimes with negative curvaturesuch as AdS. Firstly, one needs to choose a coordinate system. It is more convenient to workwith global coordinates (τ, ρ,Ωi), with the metric given by (3.10)

g00 = − b2

cos2 ρ

g11 =b2

cos2 ρ

gij = b2 tan2 ρi−2∏m=1

sin2 θmδij , i, j = 2, . . . , d

(3.18)

Only radial motion will be considered, i.e. dθn = 0, for simplicity. So only g00 and g11 willbe needed. Consequently, only the following Christoffel symbols are required

Γ000 = Γ1

01 = 0

Γ001 = Γ1

00 = tan ρ(3.19)

Massive Particles

For a massive particle falling freely in AdS from ρ(τ0) = ρ06, the worldline is given by

ρ(τ) = | arcsin [sin ρ0 cos (τ − τ0)] | (3.20)

It is an oscillatory trajectory as indicated in figure 3.4.More importantly, the particle can never reach the boundary at ρ = π

2. In fact, its

trajectory is bounded by ρ ≤ ρ0.

6τ denotes the global time coordinate and not the proper time here


Massless Particles

It is obvious from the invariant length,

ds2 =b2

cos2 ρ

(−dτ 2 + dρ2

)(3.21)

that massless particles (ds2 = 0) follow straight lines with slope 45 in AdS. When theparticles reach the boundary, they are reflected to continue their motion (figure 3.4). Thisjourney to infinity and back lasts a finite time

∆τ[0→ π

2→ 0

]= π (3.22)

This is quite peculiar: An observer living in AdS cannot reach the boundary but he keepsseeing his reflection every ∆τ coming from this seemingly infinity! The AdS spacetime can,therefore, be understood as a perfectly reflecting cavity.

3.2 Gauge Side - Conformal Field Theory

The gauge side of the AdS/CFT correspondence is a Conformal Field Theory (CFT). A CFTis a conformally invariant QFT. This includes scale invariance. In other words, a necessarycondition to characterize a QFT as a CFT is a vanishing β-function 7; that is, the couplingconstants do not change with the energy scale. Conformal invariance is present essentiallywhenever dealing with massless fields at the classical level. This includes pure gauge theoriesas well as the free massless Klein-Gordon and Dirac fields. It is easy to understand why amass term breaks conformal symmetry. A mass parameter automatically introduces a lengthbreaking scale invariance. At the quantum level, however, things are always more fuzzy. InQED (Quantum ElectroDynamics) with massless electrons, for example, the one-loop beta-function is non-zero, thus violating conformal symmetry, even though the classical theory isconformally invariant. It is still not known if scale invariance is sufficient for a QFT to beconformally invariant in general, except in the case of 2 spacetime dimensions.

In this section, the group of conformal transformations, the conformal group, will beintroduced, along with the conservation laws associated with conformal symmetry at thequantum level. Next, some characteristic properties of general CFTs will be extracted,regarding the energy-momentum tensor, as well as some operator tools that apply only inCFTs. In addition, the “correspondence” between operators and states, the state-operatormapping, will be briefly reviewed. The central charge, an important quantity of a CFT, isintroduced in Appendix D.

Useful references for the material of this section, along with some aspects of CFT2 re-viewed in Appendix D, are [9], [10], as well as the lecture notes by Professor David Tong,http://www.damtp.cam.ac.uk/user/tong/string.html. (See Lecture 4 in particular).

7The β-function describes how the coupling constant g of the theory scales with the energy scale µ. Itis given by β (g) = µ ∂g∂µ . If there are more coupling constants, then there is a β-function for each couplingconstant.

3.2. GAUGE SIDE - CONFORMAL FIELD THEORY 31

3.2.1 Conformal Group

The conformal group is the group of coordinate transformations xµ → x′µ (x) that preservethe relative angle, uµυ

µ/√u2υ2, between two vectors uµ and υµ. This requires the metric to

transform as

gµν (x)Conformal Group−−−−−−−−−−→ g′µν (x′) =

∂xρ

∂x′µ∂xσ

∂x′νgρσ = Ω2 (x) gµν (3.23)

with Ω (x) a local function known as the scale factor. It must be positive to preserve thesignature of the metric and causality. We are interested in quantum field theories on a flatspacetime, so gµν = ηµν . Thus the Poincare group is a subgroup of the conformal group; itis the group of transformations with Ω2(x) = 1.

Under such transformations, the Jacobian is∣∣∣∣∂x′∂x

∣∣∣∣ =√− det

(g′µν(x

′))

=√− det (Ω2(x)ηµν)

= ΩD(x)

(3.24)

Since the conformal group is a continuous group, the investigation of the infinitesimaltransformations is sufficient to extract the entire finite elements of the group. Under generalinfinitesimal coordinate transformations, x′µ = xµ + εµ (x), the above metric transformationyields the following conformal Killing equations

∂µεν + ∂νεµ = 2σ (x) ηµν (3.25)

σ (x) =1

D∂µε

µ

Ω (x) = 1 + σ (x)(3.26)

where the function σ (x) is specified by contracting with ηµν the first equation of the groupabove. Some additional derivatives and contractions lead to the useful equations,

(ηµν+ (D − 2) ∂µ∂ν) (∂ · ε) = 0 (3.27)

(D − 1) (∂ · ε) = 0 (3.28)

∂µ∂νερ =1

D(−ηµν∂ρ + ηρµ∂ν + ηνρ∂µ) (∂ · ε) (3.29)

Equation (3.28) implies that

εµ = aµ + bµνxν + cµνρx

νxρ (3.30)

while equations (3.25) constrain the constants aµ, bµν and cµνρ, giving the most generalsolutions:

εµ = aµ + ωµνxν + λxµ + bµx

2 − 2 (b · x)xµ (3.31)


Transformation Infinitesimal Form Finite Form

Translations xµ → xµ + aµ xµ → xµ + aµ

Rotations xµ → xµ + ωµνxν xµ → Λµ

νxν

Dilatations xµ → xµ + λxµ xµ → λxµ

Special Conformal xµ → xµ + bµx2 − 2 (b · x)xµ xµ → xµ−x2bµ

1−2(b·x)+b2x2

Table 3.1: Conformal coordinate transformations

Tranformation Generator

Translations Pµ = −i∂µRotations Lµν = i (xµ∂ν − xν∂µ)

Dilatations D = −ixµ∂µSpecial Conformal Kµ = −i (2xµx

ν∂ν − x2∂µ)

Table 3.2: Conformal transformations generators

with ωµν = −ωνµ. These vectors that solve the conformal Killing equations are the conformalKilling vectors. It is already obvious that the Poincare transformations are also conformaltransformations: aµ parameterize translations and ωµν parameterize Lorentz transforma-tions. In addition, there are two other kinds of conformal transformations: dilatations,parameterized by λ and special conformal transformations parameterized by bµ.

The action of the generators of these 4 kinds of transformations on the fields φ(x) arefound by the standard prescription:

An infinitesimal coordinate transformation xµ → x′µ = xµ + εαδαxµ transforms a field

according toφ(x)→ φ′(x′) = φ(x) + εαδαF (φ(x)) (3.32)

where F (φ(x)) = φ′(x′).

The generators Gα are defined by 8,

φ(x)→ φ′(x) = e−iεαGαφ(x) = φ(x)− iεαGαφ(x) (3.33)

From these two statements, the generators act on the fields φ according to,

Gαφ (x) = −iδαxµ∂µφ (x) + iδαF (φ (x)) (3.34)

If the fields φ are scalar fields, then they are not affected by the transformation, i.e.φ′(x′) = φ(x) by definition, and the second term vanishes.

Following the above prescription, it is easy to write the generators in operator form actingon scalar fields (table 3.2).

8There is an overall minus sign in the usual definition of the generators Tα in Group Theory, eiεαTα =

e−iεαGα


Conformal Algebra

With the expressions for the generators at hand, we can extract the algebra

[D,Pµ] = iPµ

[D,Kµ] = −iKµ

[Kµ, Pν ] = 2i (ηµνD − Lµν)[Kρ, Lµν ] = i (ηρµKν − ηρνKµ)

[Pρ, Lµν ] = i (ηρµPν − ηρνPµ)

[Lµν , Lρσ] = −i (ηµρLνσ − ηµσLνρ − ηνρLµσ + ηνσLµρ)

(3.35)

From the above commutation relations, it is not clear what Lie algebra the generatorsgive rise to. What can be deduced though is the number of generators:

Number of translations aµ = D +

Number of rotations ωµν (ωµν = −ωνµ) = D(D−1)2

+Number of dilatations λ = 1 +

Number of special conformal transformations bµ = D

Number of generators = (D+2)(D+1)2

This number is exactly the same as the number of generators of a special orthogonal groupthat acts on a (D + 2)-dimensional vector space. Indeed, if the generators are combined inthe following way

Jµν ≡ Lµν

Jµ,D ≡1

2(Pµ +Kµ)

Jµ,D+1 ≡1

2(Pµ −Kµ)

JD,D+1 ≡ D

(3.36)

then they obey the algebra

[JMN , JPΣ] = −i (ηMPJNΣ − ηMΣJNP − ηNPJMΣ + ηNΣJMP ) (3.37)

with ηMN a flat spacetime metric that has +1 temporal dimension and +1 spatial dimension.(The capital Greek indices take values from 0 to D+ 1). For a D-dimensional flat spacetimewith p spatial dimensions and q temporal dimensions (p+ q = D), that is a spacetime withSO (q, p) isometry group, the conformal algebra (3.37) is precisely SO (q + 1, p+ 1). For thecase of q = 1, p = D − 1 we are interested in, the conformal group is SO (D, 2).

Representations of the Conformal Group

Although the generators of the conformal group were found by their action on scalar fields,this must be generalized to consider arbitrary fields9. For the case of the Poincare group,this requires one to consider fields of different spins. The procedure is as follows:

9The algebra itself does not depend on the nature of the fields but the representation of the generatorsdoes.


The Lorentz group leaves the point x = 0 invariant. The matrix representation Sµν ofthe Lorentz generators associated with a generic field Φ(x) is introduced by the actionat the invariant spacetime point10:

LµνΦ(0) = SµνΦ(0) (3.38)

The translation generators Pµ are used to translate infinitesimally away from the origin,according to

PµΦ(x) = −i∂µΦ(x) (3.39)

LµνΦ(x) = Lµνe−ixσPσΦ(0)

= e−ixσPσeix

σPσLµνe−ixσPσΦ(0)

= e−ixσPσ (Lµν − xµPν + xνPµ) Φ(0)

= e−ixσPσSµνΦ(0)− e−ixσPσ (xµPν + xνPµ) Φ(0)

= Sµνe−ixσPσΦ(0)− (xµPν + xνPµ) e−ix

σPσΦ(0)

= SµνΦ(x) + i (xµ∂ν − xν∂µ) Φ(x)

(3.40)

The formula,

e−ABeA = B + [B,A] +1

2![[B,A] , A] + . . . (3.41)

and the algebra (3.35) were used in the third line, while in the fifth line, the factthat the translation is infinitesimal was used to write the unit element as the entiretranslation element11.

We apply the same procedure for the conformal group by introducing two new quantum“numbers”12 ∆ = −i∆ and κµ associated with the dilatation D and special conformaltransformation Kµ generators respectively. Then one gets the following additional results

[∆, Sµν ] = 0

[∆, κµ] = κµ

[κµ, κν ] = 0

[κρ, Sµν ] = i (ηρµκν − ηρνκµ)

[Sµν , Sρσ] = −i (ηµρSνσ − ηµσSνρ − ηνρSµσ + ηνσSµρ)

(3.42)

DΦ(x) = −i (xµ∂µ + ∆) Φ(x)

KµΦ(x) =(κµ − xνSµν − i

(2xµ∆ + xµx

ν∂ν − x2∂µ))

Φ(x)(3.43)

The first equation in (3.42) says 13 that the matrix representing the dilatation generator (atthe point x = 0) is a number times the unit matrix. The fact that dilatations are scalings

10The spin is the part of the total angular momentum that remains after removing all contributions oforbital angular momentum. At the origin, the orbital angular momentum.

11e−ixσPσxµPν = xµPν +O

(x2)

= xµPν (1− ixσPσ) +O(x2)

= xµPνe−ixσPσ +O

(x2)

12They are not actually numbers but matrices associated with some quantum numbers, their eigenvalues.13According to Schur’s lemma


Conformal Transformation Nother Current(s)

Translations (xµ → xµ + aµ) jµνP = T µν (Energy-momentum tensor)Rotations (xµ → Λµ

νxν) jµνρL = T µνxρ − T µρxν

Dilatations (xµ → λxµ) jµD = T µνxν

Special Conformal Transformations (xµ → xµ−x2bµ

1−2(b·x)+b2x2 ) jµνK = T µρ(2xρxν − ηµνx2)

Table 3.3: Nother Currents for Conformal Symmetry.

implies that the group element e−iθ(λ)D associated with the transformation xµ → λxµ is“normally” real. This is why we set ∆ = −i∆. With this observation, the quantum number∆ is defined such that a dilatation acts on scalar fields according to

Φ(x)D−→ Φ(λx) = λ−∆Φ(x) (3.44)

For this reason, the number ∆ is called the scaling dimension of the field, and it is asfundamental as the spin in CFTs.

3.2.2 Nother Currents for Conformal Symmetry

Following the procedure described in Appendix A, Section A.2.1, it is not hard to derive theconserved currents associated with translations, dilatations and special conformal transfor-mations. The final symmetrized expressions are reported in the table below,

3.2.3 The energy-momentum tensor in a CFT

A CFT has many unique properties. One of these is the fact that the energy-momentumtensor is classically traceless. This comes from scale invariance alone. The energy-momentumtensor can be extracted by varying the action with respect to the metric tensor14,

T µν = − 2√−g

δS

δgµν(3.45)

Under conformal transformations (3.23) associated with a scale factor Ω, the metrictensor varies trivially

δgµν = 2gµνΩδΩ (3.46)

As a result, the variation of the action with respect to the metric gives

δgS =

dDx

δS

δgµνδgµν

=

dDx

(−√−g2

T µν)

(2gµνΩδΩ)

= −dDx√−g T µµΩδΩ = 0

(3.47)

14If the theory is defined in a fixed spacetime background, this is still applicable by letting the metric betnon-dynamical and then fixing it after the variation.


and for this to hold in general, the energy-momentum tensor must be traceless,

T µµ = 0 (3.48)

In a quantum CFT, this turns into an expectation value equation, and it is associatedwith a very important anomaly cancellation, the Weyl anomaly reviewed in Appendix D forD = 2.

3.2.4 Primary Operators

It was stated that scalar fields transform according to,

Φ(x)D−→ Φ(λx) = λ−∆Φ(x) (3.49)

under dilatations with ∆ the scaling dimension. At the quantum level, all field functionsO(x) become operators O(x), but we drop the “ ˆ ” sign for convenience.

The most important kind of local operators in a CFT are the primary operators, whichare annihilated by the generators of the special conformation transformations15,

KµO0(x) = 0 (3.50)

If the operators are annihilated by the Kµ corresponding to global special conformal trans-formations, then they are called quasi-primary operators. The above definition means thatthe corresponding conformal charge is zero16, κµ = 0, but they are still characterized by ascaling dimension ∆ and a spin s. The importance of primary operators comes from the factthat they can be used to build an entire representation of the conformal group. This can beseen from the conformal algebra (3.35), in particular the commutation relations involvingthe generator of the dilatations

[D,Pµ] = +iPµ

[D,Kµ] = −iKµ

(3.51)

These commutators imply that Pµ and Kµ act as creation and annihilation operators oneigenstates of D respectively, shifting the scaling dimension ∆ by steps of 1. In other words,primary operators are operators of minimum scaling dimension ∆0 and all higher ∆ = ∆0+noperators O∆ can be constructed by acting repeatedly with Pµ:

O∆ = Pµ1 . . . PµnO0 (3.52)

Operators created in such way from primary operators are called descendants, and, alongwith the initial primary operator, they form a conformal family.

15Annihilation of operators under the action of another operator implies the vanishing of acommutator,[Kµ,O0(x)] = 0. However, since in a CFT there is an equivalence between local operatorsand states, as we review in a later section, we express this property without writing the commutator explic-itly.

16This is trivially seen by computing the annihilation condition at x = 0, in which case KµO0(0) =κµO0(0) = 0⇒ κµ = 0.


For spinless quasi-primary operators φ(x), it can be shown that the annihilation conditionis equivalent to the transformation law

φ(x)Conformal Group−−−−−−−−−−→ φ′(x′) =

∣∣∣∣∂x′∂x

∣∣∣∣−∆D

φ(x) = Ω−∆(x)φ(x) (3.53)

3.2.5 Operator Product Expansion

There is a useful tool to approximate products of local operators. The product of two localoperators Oi(xi) and Oj(xj) inserted at nearby points can be approximated by a series ofoperators inserted at one of these points

Oi(xi)Oj(xj) =∑k

Ckij(|xi − xj|)Ok(xj) (3.54)

The coefficients Ckij depend on |xi − xj| only as a consequence of Poincare invariance. This

property is called the Operator Product Expansion (OPE ). The expansion above holds insidetime-ordered expectation values

〈Oi(xi)Oj(xj) . . .〉 =∑k

Ckij(|xi − xj|) 〈Ok(xj) . . .〉 (3.55)

with the ellipses representing arbitrary operator insertions at distances much larger than|xi − xj|. This ensures that the OPE is a good approximation in general QFTs. In CFTs,however, the OPEs turn out to be exact, with radius of convergence equal to the distance tothe nearest extra insertion.

OPEs become singular as xi → xj, and it is this singular behavior that gives rise tonon-trivial CFT quantities, including the central charge.

3.2.6 State-Operator Map

In this subsection we review the correspondence between local operators and states. Buthow could local operators be dual to states, including non-local ones? It turns out that isalways possible in a CFT due to conformal invariance. First though we need to understandthe idea of radial quantization.

Radial Quantization

The quantization scheme applied to fields can vary. For example, one may use canonicalquantization or path integral quantization. In all quantization schemes we must providea foliation of spacetime. This is essentially the procedure of dividing the D-dimensionalspacetime into (D − 1)-dimensional surfaces. For QFTs with Poincare invariance thesesurfaces can be chosen to be spatial slices of constant t. The Hilbert spaces on these surfacesmust be isomorphic. This is always true if these surfaces are related by a symmetry, e.g.time translation generated by the Hamiltonian.


The observables of a QFT are scattering amplitudes, i.e. sandwiches of “in” and “out”states on particular spatial slices characterized by some parameter σ. “In”, |in(σ)〉, and“out”, |out(σ)〉, states are constructed by inserting local operators on the “past” (< σ) andthe “future” (> σ) slices, respectively. The scattering amplitude can be eventually expressedas a correlation function of the fundamental fields of the theory.

Now the |out(σ′)〉 and |out(σ)〉 states are related by a unitary operator U(σ′, σ), accordingto |out(σ′)〉 = U †(σ′, σ) |out(σ)〉. So the scattering amplitude can be written as

〈out(σ′)|in(σ)〉 = 〈out(σ)|U(σ′, σ)|in(σ)〉 (3.56)

For slices of constant t, σ = t, we get

U(t′, t) = e−iH(t′−t) (3.57)

which is the usual evolution operator.In a CFT, it is more convenient to foliate spacetime by dividing it into slices related by

dilatations rather than time translations. This is achieved by using spherical (Euclidean)spheres SD−1(r) of different radii r centered at the origin for which the flat Euclidean space-time metric is

ds2E = dr2 + r2dΩ2

D−1

r ∈ R+ = [0,∞)(3.58)

Just like states on constant time slices are classified by their D-momenta pµ, the eigen-states of the translations generators,

Pµ |p〉 = pµ |p〉 (3.59)

states on these spheres are classified by their scaling dimension ∆ and their SO (D) spin l

D |∆, l〉 = i∆ |∆, l〉Lµν |∆, l〉 = Sµν |∆, l〉

(3.60)

To understand the role of the spheres SD−1(r) and find the evolution operator relating them,we perform the coordinate transformation τ = ln r. This brings the flat Euclidean spacetimemetric to the form

ds2E = e2τ

(dτ 2 + dΩ2

D−1

)= e2τds2

cylinder (3.61)

which is conformally equivalent to the cylinder

ds2cylinder =

(dτ 2 + dΩ2

D−1

)τ ∈ R = (−∞,+∞)

(3.62)

In other words this conformal transformation maps RD to R×SD−1. This allows to interpretτ as the natural “time” governing the evolution. Slices of constant τ are spheres of constantradius on the cylinder (figure 3.5). The evolution operator is, thus

U(τ ′, τ) = e−iD(τ ′−τ) (3.63)


𝑡𝐸 𝜏

RD R × 𝑆𝐷−1

Figure 3.5: Conformal map of Euclidean flat spacetime RD to Euclidean cylinder R× SD−1.Spatial slices (red shaded regions) map from planes to cylinders.

and it evolves eigenstates of the dilatations generator according to,

U †(τ, 0) |∆〉 = eiDτ |∆〉 = e−∆τ |∆〉 = r−∆ |∆〉 (3.64)

This particular foliation along with the above discussion is known as radial quantization.The radial quantization scheme then will involve not equal-time but rather equal-radiuscommutation/anti-commutation relations.

State-Operator Correspondence

In the above foliation, the center of the cylinder r = 0, the origin xµ = 0, corresponds to theinfinite “past” τ → −∞, while approaching the infinity r →∞ is equivalent to approachingthe infinite “future” τ → ∞. In every QFT, all physical states are constructed by theaction of some local operators on the vacuum state |0〉. The vacuum state is defined by itsinvariance under all global conformal transformations, i.e. its annihilation under the actionof all the conformal group generators.

If a local primary operator O∆ of scaling dimension ∆ is inserted at the origin xµ = 0 = r,then as can be seen by utilizing the conformal algebra, a new state |∆〉 is created with thesame scaling dimension

|∆〉 = O∆(0) |0〉 (3.65)

while if the same operator is inserted at some arbitrary point, then the new state is

|χ〉 ≡ O∆(x) |0〉 = eiP ·xO∆(0)e−iP ·x |0〉= eiP ·xO∆(0) |0〉= eiP ·x |∆〉

(3.66)


which is a superposition of states with different scaling dimensions. This is because thegenerators of translations Pµ and the generators of the special conformal transformations Kµ

act as creation and annihilation operators on the eigenstates of the dilatations respectively,as the conformal algebra implies.

Consequently, given a primary operator O∆ of scaling dimension ∆, a state |∆〉 with thesame scaling dimension, annihilated by the generators of the special conformal transforma-tions Kµ, can be constructed, by inserting the operator at the origin. For this reason, suchstates are called primary states. Equivalently, given a state |∆〉 with scaling dimension ∆that is annihilated by the generators of the special conformal transformations Kµ, a localprimary operator O∆ with the same scaling dimension can be constructed. The constructionof the associated operator is done by defining its correlators with other operators

〈O1(x1)O2(x2) . . .O∆(0)〉 = 〈0|O1(x1)O2(x2) . . . |∆〉 (3.67)

which does satisfy the usual transformation properties expected from conformal invariance,e.g. (3.53) for quasi-primary operator insertions, proving the correspondence.

The fact that all local operators can be constructed by acting with appropriate combi-nations of the generators of translations and special conformal transformations implies thatthere exist one associated state for each local operator. This one-to-one correspondenceof states with local operators is the famous state-operator correspondence or state-operatormapping. Notice that this correspondence involves the local operators of the theory and notthe non-local ones.

Unitarity Bounds

The state-operator mapping allows us to extract some constraints on the range of the scalingdimensions a local operator can have. A generic local operator is a descendant of a primaryoperator O∆ of scaling dimension ∆, corresponding to a primary state |∆〉 annihilated byKµ.

Since a unitary theory is a theory with positive probabilities, there are no negative-normstates

〈χ|χ〉 ≥ 0 (3.68)

So the state-operator correspondence provides a bound on the scaling dimension of primaryoperators and, hence, of any descendant. The bound is obtained by firstly realizing that thegenerators of translations Pµ and the generators of the special conformal transformationsKµ are related via hermitian conjugation in radial quantization. This can be seen by writing

the corresponding differential operators in cylindrical coordinates (τ,Ωi ≡ ni = xi

r),

Pµ = −i∂µ → −ie−τ (nµ∂τ + (δµν − nµΩν)∂nµ)

Kµ = −i(x2∂µ − 2xµx

ν∂ν)→ −ieτ (−nµ∂τ + (δµν − nµΩν)∂nµ)

(3.69)

Thus the two generators are related to each other by a τ -reflection. Time-reversal in Eu-clidean signature corresponds to hermitian conjugation in Lorentzian signature, proving the

3.3. THE ORIGINAL ARGUMENT 41

relation between the two operators. Consequently, (Pµ |∆〉)† = 〈∆|Kµ. Using this relationand the fact that unitarity requires a semi-positive norm, we get

〈∆|KµPν |∆〉 = 〈∆| ([Kµ, Pν ] + PνKµ) |∆〉= 2i 〈∆| (Dδµν − Lµν) |∆〉= 2∆ 〈∆|∆〉 − 2i 〈∆|Lµν |∆〉 ≥ 0

(3.70)

So, for states with zero spin

〈∆|KµPν |∆〉 = 2∆s=0δµν 〈∆|∆〉 ≥ 0 (3.71)

Since unitarity also requires 〈∆|∆〉 ≥ 0, the scaling dimension ∆s=0 of any scalar operatormust be positive or, for only the vacuum, zero:

∆s=0 ≥ 0 (3.72)

Repeating this procedure for twice “excited” states

〈∆|KµKνPρPσ|∆〉 ≥ 0 (3.73)

implies the stronger constraint for scalars

∆s=0 ≥D

2− 1 (3.74)

To ensure unitarity of the theory, one should keep applying the semi-positivity condition atall levels to find an absolute condition, but for scalar operators it turns out that the secondlevel constraint is sufficient.

For operators with spin, one would have to work in the corresponding spin representationof the rotation group. For s = 1

2the absolute constraint turns out to be

∆s= 12≥ D − 1

2(3.75)

while for operators with spin s ≥ 1 the absolute constraint reads

∆s≥1 ≥ D + s− 2 (3.76)

3.3 The original argument

In this section, some very fundamental observations that lead Maldacena to his proposal ofthe correspondence will be discussed.


3.3.1 Large N Yang-Mills Theories

Before going forward with this subsection, it is advised to make sure that Appendix A ongauge theories and Appendix B, Section B.4 are well comprehended.

The one side of the correspondence considers a gauge field theory; a Yang-Mills theorywhich is simply an SU (N) gauge theory with N the number of “colors”, i.e. the numberof gauge charges or kinds of particles the theory describes. Its Lagrangian density has thegeneral form,

LYM = −1

2Tr (FµνF

µν)

= −1

4F aµνF

µνa

F µν = ∂µAν − ∂νAµ + igYM [Aµ, Aν ]

AµSU(N)−−−−→ UAµU † − i

gYM∂µUU †

F µν SU(N)−−−−→ UF µνU †

U †U = UU † = 1 , detU = 1

(3.77)

Written solely in terms of the gauge fields Aµa , LYM contains products of two, three and fourgauge fields describing a kinetic term, a 3-vertex and a 4-vertex interaction respectively,

LYM =

Kinetic Term︷︸︸︷−2 Tr

(∂µAν∂

[µAν]) 3-vertex interaction︷︸︸︷

+2igYM Tr (∂µAν [Aµ, Aν ])

4-vertex interaction︷︸︸︷−1

2g2YM Tr (AµAν [Aµ, Aν ])

= −1

2∂µA

aν∂

µAνa +1

2∂µA

aν∂

νAµa − gYMf bca ∂µA

aνA

µbA

νb +

1

2g2YMf

eabf

cde AaµA

bνA

µcA

νd

(3.78)

with ∂[µAν] ≡ 12!

(∂µAν − ∂νAµ) the antisymmetrization operation. A rescaling of the gaugefields gYMA

µ → Aµ brings the Lagrangian density to a nicer form with respect to the couplingconstant,

LYM =1

g2YM

(−2 Tr

(∂µAν∂

[µAν])

+ 2iTr (∂µAν [Aµ, Aν ])− Tr (AµAν [Aµ, Aν ]))

(3.79)

The next step is to find the propagators of this theory. In momentum space17, theFeynman rules immediately imply,

〈AµaAνb 〉 (k) = Dµνab (k) =

i

k2δab

(−ηµν −

kµkνk2

)(3.80)

The trick now is to consider the matrix components Aµij = AµaTaij as the fundamental

fields of the theory. In terms of these, the propagator reads18,

17〈T Aµa(x)Aνb (y)〉 =

dDk(2π)D/2

〈AµaAνb 〉 (k)eik·(x−y)

18For more on the generators of the SU (N) group, see Appendix E.


−→ ij kl

Figure 3.6: Double line notation in large N Yang-Mills theories.

〈AµijAνkl〉 (k) = Dµνab (k)T aijT

bkl = Dµν

ab (k)

(δilδkj −

1

Nδijδkl

)(3.81)

Now it should be obvious why a large N limit is motivated. Taking the number of colorsN to be large, means that the second term in the propagator can be neglected, leaving onlya product of Kronecker-δ’s. This suggests a double line notation in the Feynman diagrams(figure 3.6).

In order to proceed with a more proper and meaningful description of the gauge theoryin the large N limit, the ’t Hooft coupling λ is introduced,

λ = g2YMN (3.82)

When talking about the large N limit, what is really meant is N 1 and g2YM 1 so

that the ’t Hooft coupling is of order of unity, λ ∼ 1; this is known as the ’t Hooft limit.Eventually, what is interesting is the way the coupling constant dependnce of the Feynmandiagrams scales with the geometry of the diagrams. In light of all of these redefinition andrescalings, the Feynman rules can now be translated to (figure 3.7):

Each propagator contributes a factor of g2YM = λ

N

Each 3-vertex contributes a factor of g−2YM = N

λ

Each 4-vertex contributes a factor of g−2YM = N

λ

Each index loop contributes a factor of N due to the summation over the index thattakes values from 1 to N

As a result, all vacuum-to-vacuum diagrams Dvac, i.e. diagrams with no external lines, areeventually triangulations of surfaces of different topology! Since the diagrams are polygons,the number of edges E (propagators), vertices V and faces F (index loops) can be combinedto construct a topological invariant quantity, the Euler characteristic χ = F −E + V whichin turn is written in terms of the genus g (the number of holes or handles) of the surface,

F − E + V = χ = 2− 2g (3.83)

Counting the contributions of the ’t Hooft coupling constant in each diagram with Epropagators, V vertices and F index loops, the value of Dvac is,

Dvac ∼(λ

N

)E (N

λ

)VNF

= NF−E+V λE−V

= N2−2gλE−V

(3.84)


∼ g2YM =

λ

N

∼ 1

g2YM

=N

λ

∼ 1

g2YM

=N

λ

∼ N

Figure 3.7: Feynman rules for large N gauge theories.

Figure 3.8: Planar (left) and non-planar (right) diagrams. The left diagram can be drawnon the plane without crossing line, while the right diagram needs to extend out of the planeto do so.

and the dependence on the number of colors N comes only from the topology of the diagram.Diagrams that can be drawn on a plane or sphere (genus-0 surfaces) without crossing linesare called planar diagrams, while diagrams that need to be drawn on a surface of g ≥ 1in order not to have crossing lines are just called non-planar diagrams (figure 3.8). Theeffective action of theory logZ, computed by the sum of all connected vacuum-to-vacuumdiagrams, is, thus, of the form,

logZ =∞∑g=0

N2−2gfg(λ) (3.85)

with fg(λ) the sum of all diagrams drawn-able on a genus-g surface. The large N limitis finally well comprehended: “keep only diagrams that can be drawn on minimum genussurfaces, i.e. planar diagrams”.


3.3.2 Elements of String Theory

String theory is a theory of quantum gravity. In this subsection, we will introduce somebasic aspects of string theory. The main focus will be on Bosonic String Theory ([10], [11]),describing bosonic strings propagating on a suitable background. Even though Maldacena’scorrespondence involves Superstring Theory ([12], [13]), some of the general ideas can beillustrated in the context of bosonic string theory. We will describe the action of a relativisticstring. In addition, some remarks regarding boundary conditions for open strings will bepresented in order to introduce Dp-branes. Finally, we will briefly describe string interactionsso as to motivate the AdS/CFT correspondence.

Action and Symmetries

In string theory point particles are extended to 1-dimensional objects, open and closedstrings. There is a distinction between two kinds of spacetimes when it comes to extendedobjects. There is the actual spacetime in which the objects propagate, and then there is thesurface produced by the motion of the extended object.

The trajectory Xµ of a point particle is determined by a worldline, parametrized by oneparameter, usually the proper time τ (of a clock following the particle): Xµ = Xµ(τ). Fora free particle, this worldline can be found by minimizing proper time, or extremizing theaction

SPP = −mds = −m

dτ

√−gµνXµXν (3.86)

where the dots denote derivatives with respect to proper time. The path integral quantizationof the relativistic particle will then involve an integral of the exponential of the action, andthe presence of the square root makes it difficult to perform explicit calculations. Fortunately,we can simplify matters via the use of Lagrange multipliers. In particular, we introduce aLagrange multiplier η and consider the action

S ′PP = −1

2

dτ[η−1gµνX

µXν −m2η]

(3.87)

The Lagrange multiplier can be integrated out using its equation of motion

− η−2gµνXµXν −m2 = 0⇒ η = m−1

√−gµνXµXν (3.88)

and then SPP is recovered.Similarly, a string moving in spacetime produces a two dimensional surface, the world-

sheet, parametrized by two parameters σa = (σ0, σ1), a = 1, 2. One parameter, σ1 ≡ σ, isspacelike, and specifies the position of a point along the string. The other parameter, σ0 ≡ τ ,is timelike and describes the time evolution of the particular point. So Xµ = Xµ(τ, σ).19.To obtain the worldsheet of a free relativistic string, we need to extremize an action. Thisaction is proportional to the area of the worldsheet. The procedure is known: Find the

19Even though the spatial parameter is σ, the argument of any function of the target space coordinateswill always mean the target position vector σa.


induced metric gab = gµν(X)∂aXµ∂bX

ν on the worldsheet, and compute an invariant surfaceintegral

SNG = −TdA = −T

d2σ

√−g (3.89)

where g = det (gab) and T the tension of the string, which is related to the Regge slope α′

according to

T =1

2πα′(3.90)

This parameterization has its origins in an earlier attempt to explain hadronic states, forwhich the experimental results indigated a linear relation between the square of the massand the spin with slope α′. From the Regge slope we obtain the characteristic string lengthls =√α′.

In any case, (3.89) is the famous Nambu-Goto action describing a relativistic stringpropagating in a background spacetime with metric gµν . Just like the point particle case,the square root can be eliminated via the introduction of Lagrange multipliers. The Lagrangemultipliers consist of a worldsheet metric γab(σ), and the action is the Polyakov 20 action

SPoly = −T2

d2σ√−γ γabgab = −T

2

d2σ√−γ γabgµν∂aXµ∂bX

ν (3.91)

Indeed, the equations of motion for the worldsheet metric components are,

gab = −1

2γabγ

cdgcd ⇒√−γ =

2√−g

γcdgcd(3.92)

which reproduce the Nambu-Goto action.

Symmetries of Polyakov action It is easy to find the symmetries of SPoly. First ofall, it is invariant under general coordinate transformations. In fact, when the propagationof the string is always taken to be in Minkowski spacetime gµν = ηµν , the local coordinatetransformations become global Poincare transformations. Secondly, it is invariant underredefining the parameters σa, i.e. reparameterizations of the worldsheet, and, finally, it isinvariant under Weyl transformations. Explicitly the symmetries are

Poincare

Xµ Poincare−−−−−→ X ′µ′(X) = Λµ′

µXµ + aµ

′(3.93)

Reparameterizations or Diffeomorphisms

σa = (τ, σ)Diff.−−−→ σ′a(σ) = (τ ′(τ, σ), σ′(τ, σ))

Xµ(σ)Diff.−−−→ X ′µ(σ′) = Xµ(σ)

γab(σ)Diff.−−−→ γ′a′b′(σ

′) = ∂a′σa∂b′σ

bγab(σ)

(3.94)

20Also known as Brink-Di Vecchia-Howe (-Deser-Zumino) (BDH) action.


Weyl Transformations

γab(σ)Weyl−−−→ γ′ab = Ω2(σ)γab (3.95)

The Polyakov action describes a 2-dimensional CFT21 with D scalar fields Xµ, µ =0, 1, . . . , D − 1. The last two of the above transformations are local gauge transformations,meaning that they come with redundancies that can be removed via gauge fixing. A sys-tematic way of gauge fixing is through the Fadeev-Popov method reviewed in Appendix B,Section B.3. The method gives rise to ghost fields and the vanishing of the Weyl anomalyeventually leads to fixing the number of spacetime dimensions. This fact is reviewed inAppendix D, Section D.5.2.

Boundary Conditions

With the action at hand, the equations of motion of the embedding fields Xµ can be ex-tracted. Due to the finite size of the string, there are surface terms that need to be removedvia suitable boundary conditions. Indeed, the variation with respect to the embedding fieldsyields

δXνSPoly = T

τf

τi

dτ

l

0

dσ ∂b(√−γ γabηµν∂aXµ

)δXν

− T τf

τi

dτ√−γ γa1ηµν∂aX

µδXν

∣∣∣∣σ=l

σ=0

− T l

0

dσ√−γ γa0ηµν∂aX

µ δXν

∣∣∣∣τ=τf

τ=τi︸︷︷︸=0

= T

τf

τi

dτ

l

0

dσ ∂b(√−γ γabηµν∂aXµ

)δXν

− T τf

τi

dτ√−γ ηµν∂σXµδXν

∣∣∣∣σ=l

σ=0

(3.96)

where the upper limit σ = l is the coordinate length of the string and δXν(τi, σ) =δXν(τf , σ) = 0 was used. Consequently, the equations of motion read

∂b(√−γ γabηµν∂aXµ

)= 0 (3.97)

and boundary conditions must be chosen so that the last term vanishes.The appropriate boundary conditions are:

Dirichlet Boundary Conditions: The endpoints are fixed at certain positions

δXµ(τ, 0) = δXµ(τ, l) = 0

⇒Xµ(τ, 0) = cµ = const. , Xµ(τ, l) = dµ = const.(3.98)

Neumann Boundary Conditions: The endpoints are free to move

∂σXµ(τ, 0) = ∂σXµ(τ, l) = 0 (3.99)21Conformal invariance in 2 dimensions means diff. and Weyl invariance as the Virasoro algebra suggests.


Periodic Boundary Conditions: The endpoints are connected,

Xµ(τ, 0) = Xµ(τ, l) , ∂σXµ(τ, 0) = ∂σXµ(τ, l)

⇔Xµ(τ, σ) = Xµ(τ, σ + l)(3.100)

and likewise for the worlsheet metric.

There is a clear distinction between open strings with disconnected endpoints and closedstrings with connected endpoints. Open strings can satisfy either Dirichlet or Neumannboundary conditions.

Strings Spectrum

The two types of strings, open and closed, give rise to different spectra. For the bosonicstring theory, the ground state22 is always a tachyon with negative mass square m2 < 0.This tachyonic state implies an instability. This state is absent in the superstring cases.

The first excited states of an open string comprise a massless spin-1 gauge boson, whilehigher excited states correspond to massive particles. The graviton appears in the closedstring spectrum. Indeed, the first excitation of the closed string is a massless spin-2 boson.

It is remarkable that no string theory can be consistent with only open strings becausestring interactions involve open strings joining to form closed strings. On the other hand,a string theory with only closed strings is a consistent theory. This means that there is nostring theory without closed strings and, hence, without gravity.

D-branes

The Dirichlet boundary conditions seem to violate Lorentz invariance. This makes sensebecause fixing the endpoints of the string at certain points breaks translational and rotationalsymmetry. But suppose that p+1 of the embedding fields Xµ, including X0, satisfy Neumannboundary conditions and the other D − p− 1 fields satisfy Dirichlet boundary conditions

∂aXm(τ, 0) = ∂aX

m(τ, l) = 0 , m = 0, . . . , p

XI(τ, 0) = cI , XI(τ, l) = dI , I = p+ 1, . . . , D − 1(3.101)

Then, the open strings end on hyperplanes extended in p+ 1 directions and the Lorentzgroup SO (1, D − 1) breaks into

SO (1, D − 1)→ SO (1, p)× SO (D − p− 1) (3.102)

These hyperplanes are the Dp-branes23. The worldvolume of a Dp-brane carries twoclasses of excitations:

22This is not the vacuum. The vacuum implies absence of any string. Here the ground state is the stringstate with the lowest energy.

23“D” for “Dirichlet” and “p” for the number of spatial directions


Rigid motions and deformations parameterized by the D−p−1 embedding coordinatesXI ≡ φI24 transverse to the worldvolume of the Dp-brane. These are scalar fields.

Internal excitations parameterized by an abelian (U (1)) gauge field Aα living on theworldvolume.

The action is the Dirac-Born-Infeld action

SDBI = −TDpdp+1x

√− det (gαβ + 2πl2sFαβ) (3.103)

with gαβ = gµν∂αXµ∂βX

ν the induced metric and Fαβ = ∂αAβ − ∂βAα the field strengthassociated with the gauge field Aα. TDp is the tension of the Dp-brane given in terms of thestring coupling constant25 gs and the string length ls by

TDp =1

(2π)p gslp+1s

(3.104)

The integration measure dp+1x is with respect to the worldvolume coordinates xm. The g−1s

dependence of the brane tension suggests that Dp-branes are non-perturbative objects instring theory.

In the case of a flat background metric gµν = ηµν , and in the static gauge, the inducedmetric takes the form

gαβ = ηµν∂αXµ∂βX

ν

= ηγδ ∂αXγ︸︷︷︸

δγα

∂βXδ︸︷︷︸

δδβ

+ηIJ∂αφI∂βφ

J

= ηαβ + ∂αφI∂βφI

(3.105)

Rescaling the scalar fields φI → 2πl2sφI and expanding the square root in the action up to

2nd order terms in derivatives gives

SDBI = −TDp(2πl2s)

2

2

dp+1x

(1

4FαβF

αβ +1

2∂αφI∂

αφI)

+ . . .

= − 1

2 (2π)p−2 gslp−3s

dp+1x

(1

4FαβF

αβ +1

2∂αφI∂

αφI)

+ . . .

(3.106)

which is just the free Lagrangian density of a U (1) gauge theory and D − p − 1 masslessscalar fields.

What is more interesting is the consideration of many parallel Dp-branes. For N parallelbranes, the worldvolume gauge theory is a U (1)N gauge theory. When the branes coincide,the theory attains U (N) gauge invariance. In this case, the gauge fields Aα and the scalarfields φI become matrices transforming in the adjoint representation of U (N). Thus for Ncoincident parallel Dp-branes the worldvolume theory is a non-abelian Yang-Mills theory inp+ 1 dimensions.

24The φI coordinates describe how the Dp-brane is embedded in the background spacetime25The string coupling constant will be introduced in a later subsection.


D3-branes

In ten dimensional supergravity, there is a unique solution corresponding to N coincidentDp-branes, which is asymptotically flat

ds2 = H−12 (r)dxmdx

m +H12 (r)

(dr2 + r2dΩ2

5

)H(r) = 1 +

(b

r

)D−p−3

bD−p−3 = Ngs (4π)5−p

2 Γ

(7− p

2

)lD−p−3s

(3.107)

where H(r) is the warp factor. In particular for p = 3, the D3-brane solution has aninteresting small r behavior. For r b, the metric becomes

ds2 ' r2

b2dxmdx

m +b2

r2dr2 + b2dΩ2

5

=b2

z2

(dz2 + dxmdx

m)

+ b2dΩ25

(3.108)

where z = b2

r. The is precisely the AdS5 × S5 metric, with the sphere and the AdS radii

given by

b = (4πNgs)14 ls (3.109)

This is why one side of the correspondence involves an AdS geometry.

Interactions

In QFT interaction terms are introduced by hand in the Lagrangian density. However,the Polyakov action is almost the most general action with diffeomorphism and local Weylinvariance. The only additional term respecting these symmetries is

χ =1

4π

d2σ√−γ R (3.110)

with R the Ricci scalar of the worldsheet metric and λ a parameter that will give rise to thestring coupling constant. The fact that the worldsheet is 2-dimensional makes the dynamicspurely topological. The term above is the Euler characteristic χ = 2− 2g (g is the genus ofthe surface, i.e. the number of holes or handles on the surface). The Euler characteristic ofany 2-dimensional surface is a topologically invariant quantity. As a result, the most generalaction consistent with diffeomorphisms and Weyl invariance is

Sstring = SPoly + λχ = SPoly + λ(2− 2g) (3.111)

All dynamics of string theory including the interactions are hidden in Sstring. In com-puting amplitudes, the path integral involves summing over all possible configurations ofthe embedding fields and all possible geometries of the worldsheet, including the ones with


Aሺ2 → 2ሻ =

+

+

+

+

. . .

Figure 3.9: String perturbation theory is a sum over topologies. In the above example, thescattering amplitude of two incoming and two outgoing closed strings is depicted. Eachadditional hole in the diagrams contributes a factor g2

s with gs the string coupling constant.

holes. This yields a weight for the various processes and any scattering amplitude A even-tually takes the form

A =∞∑g=0

g2g−2s Fg (3.112)

with the string coupling constant defined by

gs = e−λ (3.113)

and Fg is the genus g contribution.

Consequently, string perturbation theory is an expansion over topologies. For examplethe scattering amplitude of two incoming closed strings to two outgoing closed strings isdescribed in figure 3.9.

A simple comparison of the perturbative expansion of string theory and large N gaugefield theory suggests a connection or even an identification as long as the coupling constantsof the two theories are connected up to some constants according to

gs ∼1

N⇒ gs ∼ g2

YM (3.114)

This actually makes sense. The N coincided D3-branes give rise to a U(N) gauge theory,a Yang-Mills theory with coupling constant gYM . Open strings are attached on these D-branes and can collide to join their endpoint and escape the D-brane as a closed stringpropagating in the bulk. Each joining of endpoints contributes a factor gYM and since thereare two endpoints on each open string that join together, two such factor appear just as seenabove.


3.4 Parameters matching and strong/weal coupling

The two sides of the correspondence are characterized by some parameters. On the gravityside, these are the string coupling constant gs and the ratio b

ls, while, on the gauge side,

these are the Yang-Mills coupling constant gYM and the number of colors N .

The ratio bls

as seen from (3.107) is,

(b

ls

)D−p−3

= Ngs (4π)5−p

2 Γ

(7− p

2

)(3.115)

or, for D3-branes, and since D = 10,

b

ls= (4πNgs)

14 (3.116)

Recognizing the identification of the string coupling constant with the Yang-Mills cou-pling constant,

4πgs = g2YM (3.117)

the ratio can be expressed in terms of the gauge parameters,

b

ls=(Ng2

YM

) 14 (3.118)

Consequently, a large N limit makes the AdS radius and, hence, the curvature radiusmuch larger from the the string length and the string theory becomes weakly coupled. Thisis just the SUGRA limit; ignoring the size of the strings and the weak interactions amongthem approximates a Superstring theory with Supergravity.

It is also interesting to notice that the central charge of the boundary theory becomesvery large at the large N limit. Since the central charge counts the number of degrees offreedom, it is simply the number of colors,

c = N2 − 1 ∼ N2 (3.119)

The large N limit in such a CFT, thus, translates to a theory with numerous degrees offreedom, that is, a strongly coupled theory.

3.5 Tests of the correspondence

After the above introduction, it is time to put the conjecture through some tests. Thefollowing tests are very basic and some of them do not even concern the exact theoriesinvolved.

3.5. TESTS OF THE CORRESPONDENCE 53

3.5.1 Symmetries Matching

First of all, the symmetries of the two theories must match. This is a first step towardsfiguring out which exact theories are dual. For the gravity side in which quantum gravity isexpressed as a type IIB Superstring theory in AdS5×S5, the symmetries are of three types:

Isometry group of AdS5: SO (4, 2). This is explained in Chapter 6.

Isometry group of S5: SO (6).

32 Supersymmetries. This part will not be extended here. Let it just be said thata general supersymmetric theory comes with fermionic generators, the supercharges,whose algebra involves anti-commutation relations rather than commutators. Thenumber of supersymmetries is the number of these supercharges. In the gravity sidethey are manifested as Killing spinors in AdS5 × S5.

For the gauge side, the N = 4 Super Yang-Mills CFT as the following symmetries,

Global Conformal group in R3,1: SO (4, 2). This is explained in Chapter 4.

Global R-symmetry group: SU (4) ' S0 (6). This symmetry concerns the field contentof the supersymmetric theory. In particular the gauge theory contains 6 scalar fieldand 4 fermionic fields. The R-symmetry group is the group that transforms these fieldsinto another.

32 Supersymmetries. In the gauge theory, the supersymmetries are manifested assuperconformal algebra.

The symmetries of the two sides are in one-to-one correspondence. An interesting andgeneral observation is that gauge symmetries of the gravity theory correspond to globalsymmetries of the boundary theory. The above discussion can be found in more details in[14], but a less extended coverage is also made in [15]. In addition, a good introduction inSupersymmetry is [16], while an extended discussion on Supersymmetry and Supergravitycan be found in [17] and [18].

3.5.2 Counting the degrees of freedom - The UV/IR Connection

The first check was trivial and a fundamental one. But even more fundamental is the numberof degrees of freedom living on the two sides. The following analysis concerns general CFTsand gravity theories in AdS and does not depend on the details of the theories.

The procedure of counting the number of degrees of freedom is rather simple. Firstly,one introduces some cutoffs, and then the calculations are straightforward.

From the point of view of the CFT, the regulation of the theory is succeeded with thefollowing cutoffs:

IR cutoff: Place the system in a spatial box of side LIR

UV cutoff: Discretize space to have lattice spacing εUV


Therefore, there is one cell per “unit” volume εD−1UV , while there is a total number of

LD−1IR

εD−1UV

cells in the entire space. Introducing the central charge cCFT which counts the density ofthe degrees of freedom, i.e. the number of degrees of freedom per cell, yields that the totalnumber of degrees of freedom of the theory is,

NCFTd.o.f. = cCFT

(LIRεUV

)D−1

(3.120)

From the point of view of the gravity theory, the holographic principle says that thenumber of degrees of freedom within a region is equal to the maximum entropy and, accordingto the Bekenstein-Hawking entropy formula, this means,

NAdSd.o.f. =

A∂4G

=A∂

4lD−1P

(3.121)

where A∂ is the area of the boundary of the region and lP = G1/(D−1) is the Planck lengthin units where c = ~ = 1. For the case of AdSD+1, the total number of degrees of freedom isthe number of degrees of freedom within the boundary at z = 0 where the metric diverges.So, firstly, the following cutoffs must be introduced:

IR cutoff: Place the system in a spatial box of side LIR

“UV” cutoff26: Take the boundary to be at z = εUV → 0

The IR cutoff means that, RD−1

dD−1~x = LD−1IR (3.122)

while the “UV” cutoff says that the metric at the boundary for constant times is,

ds2 =

(b

εUV

)2

d~x2 (3.123)

The boundary area is, thus,

A∂ =

RD−1,z=εUV

dD−1~x√g =

(b

εUV

)D−1 RD−1

dD−1~x =

(bLIRεUV

)D−1

(3.124)

and the total number of degrees of freedom of the gravity theory is,

NAdSd.o.f. =

1

4

(b

lP

)D−1(LIRεUV

)D−1

(3.125)

For a duality to be taken seriously, the number of degrees of freedom must at least match.The two formulas (3.120) and (3.125) match with the identification,

1

4

(b

lP

)D−1

= cCFT (3.126)

26This is not an actual UV cutoff, i.e. a short distance cutoff. See the end of this subsection for morecomments.


This is a result worth commenting on. First of all, the cutoff dependence of both theo-ries is the same, meaning that the above identification holds as it is in the continuum limitLIR →∞, εUV → 0, something already remarkable. In expansion of this, in general, a clas-sical theory arises from the corresponding quantum theory when the coupling constant (thecoefficient multiplying the action) is large27. Thus, the classical gravity theory in AdSD+1

is reliable when the AdS radius is large in Planck units because the coupling constant is

proportional to bD−1

G=(blP

)D−1

1. Therefore, a CFT has a classical gravity dual when

cCFT is large, that is when there is a large number of degrees of freedom per unit volumeor a large number of species/“colors”. This large number of species corresponds preciselyto the large N limit of an SU(N) gauge theory in which N is the number of colors of thetheory in the notion of QCD terminology.

Now it is time to resolve a misguidance. The “UV” cutoff in the gravity calculation isnot an actual UV cutoff, i.e. a short distance cutoff, but rather a long distance cutoff aswell. This happens because the proper distance from z = εUV to the boundary z = 0 isinfinite as already seen in Section 3.1.4 due the negative curvature of AdS spacetime. Thisis the famous UV/IR connection between the actual UV cutoff of the CFT and the actualIR cutoff of the AdS.

3.5.3 Correlation functions

Finally, the observables of the two theories must agree. Observables are expressed as cor-relation functions of local CFT operators. We will discuss some n-point functions of scalaroperators.

CFT Correlators

Conformal symmetry fully determines correlation functions up to n = 3. This is true whencomputing correlation functions involving quasi-primary operators. After all, most usefulcorrelators in a CFT are constructed in terms of them. The n-point function can in generalbe computed via a path integral

〈O1(x1) . . . On(xn)〉 =1

Z

Dφ O1(x1) . . . On(xn)e−S[φ] (3.127)

where φ denotes the fundamental fields of the CFT.Assuming that conformal invariance at the quantum level, i.e. invariance of both the

action and the path integral measure, then it is easy to see that

〈O1(x1) . . . On(xn)〉 = Ω∆1(x1) . . .Ω∆n(xn) 〈O1(x′1) . . . On(x′n)〉 (3.128)

27A large coupling constant suppresses higher loops contributions in a scattering process leaving onlythe tree-level amplitudes to dominate and the path integral is well approximated by its saddle point: theclassical solution.


Indeed, starting from 〈O1(x′1) . . . On(x′n)〉 and performing a conformal transformation x′ →x(x′), we get

〈O1(x′1) . . . On(x′n)〉 =1

Z

Dφ O1(x′1) . . . On(x′n)e−S[φ]

=1

Z ′

Dφ′ O′1(x1) . . . O′n(xn)e−S[φ′]

=1

Z

Dφ O′1(x1) . . . O′n(xn)e−S[φ]

= 〈O′1(x1) . . . O′n(xn)〉

=

∣∣∣∣ ∂x∂x′∣∣∣∣−

∆1D

x′=x′1

. . .

∣∣∣∣ ∂x∂x′∣∣∣∣−∆n

D

x′=x′n

〈O1(x1) . . . On(xn)〉

(3.129)

Inverting the transformation yields the required result

〈O1(x1) . . . On(xn)〉 =

∣∣∣∣∂x′∂x

∣∣∣∣∆1D

x=x1

. . .

∣∣∣∣∂x′∂x

∣∣∣∣∆nD

x=xn

〈φ1(x′1) . . . φn(x′n)〉

= Ω∆1(x1) . . .Ω∆n(xn) 〈O1(x′1) . . . On(x′n)〉(3.130)

AdS Correlators

According to the AdS/CFT dictionary (see Section 3.6), CFT operators are sourced bythe asymptotic values of bulk fields. For the case of scalar quasi-primary operators, thecorresponding bulk fields are also scalar. The dynamics of a free real scalar field in afixed background will, thus, be considered. Using the prescription that follows from theequivalence principle, ηµν → gµν , ∂µ → ∇µ,

RD,1

dD+1x →M dD+1x

√−g, the action for a

real scalar field propagating in a manifold M is

S =1

2

MdD+1x

√−g

(−gµν∇µφ∇νφ+m2φ2

)(3.131)

Now ∇µφ = ∂µφ since φ is a scalar field. As a result

S =1

2

MdD+1x

√−g

(−gµν∂µφ∂νφ+m2φ2

)=

1

2

MdD+1x

√−g φ

(+m2

)φ− 1

2

MdD+1x ∂µ

(√−ggµνφ∂νφ

)=

1

2

MdD+1x

√−g φ

(+m2

)φ− 1

2

∂M

dDy√−gφnµ∂µφ

(3.132)

where gab = ∂xµ

∂ya∂xν

∂ybgµν is the induced metric on the boundary ∂M parameterized by co-

ordinates ya, a = 0, . . . , D − 1. The coordinates ya are essentially the same as the bulkcoordinates xµ but with one coordinate that determines the position of the boundary beingfixed.


Now, in order for the field to satisfy the Klein-Gordon equation in curved spacetime,(−+m2

)φ = 0 (3.133)

where

=1√−g

∂µ(√−ggµν∂ν

)(3.134)

appropriate boundary conditions need to be specified because the variation of the actionincludes in general a surface integral28. For M = PAdSD+1, the boundary ∂M = R

D−1,1 isat z = 0. In Poincare coordinates (z, xµ) (µ = 0, . . . , D − 1), the metric is given by

gzz =b2

z2

gµν =b2

z2ηµν , µ, ν = 0, . . . , D − 1

(3.135)

So the surface term reads (gµν |z = b2

z2ηµν , nz =√gzz = z

b),

δφS = −RD−1,1

dDx√−gnz∂zφ δφ

∣∣∣∣z→∞z=ε

= −(b

ε

)D−1 RD−1,1

dDx ∂zφ δφ

∣∣∣∣z=ε

(3.136)

where ε is a cutoff that will eventually be taken to zero. We conclude that two types ofboundary conditions can be applied:

Dirichlet boundary conditions, with the field kept fixed at the boundary, φ(z, x)

∣∣∣∣z=0

= 0

Neumann boundary condition, ∂zφ(z, x)

∣∣∣∣z=0

= 0

The equation of motion reads(−(D) − ∂2

z +D − 1

z∂z +

m2b2

z2

)φ(z, x) = 0 (3.137)

where −(D) = ∂2t − ∇2

(D−1) is the D’Alambertian operator in D-dimensional Minkowski

space and ∇2(D−1) is the Laplacian operator on R

D−1.To solve this second order partial differential equation, we use the method of separation

of variables:

φ(z, x) = f(z)φ(x)

⇒−(D)φ(x)

φ(x)=

1

f(z)

(∂2z −

D − 1

z∂z −

m2b2

z2

)f(z) = −k2

(3.138)

28In Minkowski spacetime, this is not an issue because the boundary is at infinity where the fields decayto zero.


with the constant k2 being the norm of a vector kµ. As a result, the following equations areobtained

z2f ′′k (z)− (D − 1)zf ′k(z)− (k2z2 +m2b2)fk(z) = 0 (3.139)

(−(D) + k2)φk(x) = 0 (3.140)

The second one can be solved in terms of plane waves

φ(x) =eik·x

(2π)D(3.141)

In terms of fk(z), the most general solution takes the form

φ(x, z) =

dDk

(2π)Dfk(z)eik·x (3.142)

in accordance with translational invariance along the x-directions. The equation for fk(z)

takes a familiar form if we apply the transformation fk(z) = zD2 gk(z)

z2g′′(z) + zg′ −(

(kz)2 +D2

4+m2b2

)gk(z) = 0 (3.143)

This is the modified Bessel’s equation with general solution

gk(z) = akKν(kz) + bkIν(kz) (3.144)

Here ν is given by

ν ≡√D2

4+m2b2 =

D

2

√1 +

(2mb

D

)2

(3.145)

Regularity in the interior, z →∞, requires the rejection of Iν(kz). So bk = 0 and

fk = ck (kz)D2 Kν(kz) (3.146)

where we set ck ≡ akk−D

2 .The asymptotic behavior of Kν(kz) when z ' 0 gives

Kν(kz) '1

2Γ(ν)

(2

kz

)ν+

1

2Γ(−ν)

(2

kz

)−ν(3.147)

So near the boundary, the function fk(z) reads

fk(z) ' φ0(k)z∆− + φ1(k)z∆+ (3.148)

whereφ0(k) ≡ ck2

ν−1Γ(ν)k∆−

φ1(k) ≡ ck2−ν−1Γ(−ν)k∆+

(3.149)


The scaling exponents ∆± are given by the following expressions

∆± =D

2± ν =

D

2

1±

√1 +

(2mb

D

)2 (3.150)

The scaling exponents are real if the mass square satisfies the Breitenlohner-Freedman bound

m2b2 > −D2

4(3.151)

The full solution is, finally, written as

φ(z, x) =

dDk

(2π)DakKν(kz)e

ik·x (3.152)

Near the boundaryφ(z, x) ' z∆−φ0(x) + z∆+φ1(x) (3.153)

where

φ0(x) =

dDk

(2π)Dφ0(k)eik·x

φ1(x) =

dDk

(2π)Dφ1(k)eik·x

(3.154)

Quite remarkably, the divergent, non-renormalizable term29 produces a boundary fieldφ0(x) with conformal scaling dimension ∆−:

φ0(λx) = limz→0

z−∆−φ(z, λx)

= λ−∆− limz→0

(λ−1z

)−∆− φ(z, λx)

z=λz′= λ−∆− lim

z′→0z′−∆−φ(λz′, λx)

= λ−∆− limz′→0

z′−∆−φ(z′, x)

= λ−∆−φ0(x)

(3.155)

where the fact that φ(λz, λx) = φ(z, x) was used30. This result will be very important indeveloping the AdS/CFT dictionary, as described in the next section.

29The modes associated with φ0(k) are non-renormalizable (since ∆− < 0), but they are essential forspecifying the boundary conditions.

30This can be shown by redefining the dummy variable λk → k in the Fourier integral,

φ(λz, λx) =

dDk

(2π)DakKν(λkz)eiλk·x

k′=λk=

dDk′

(2π)Dak′Kν(k′z)eik

′·x = φ(z, x)

(3.156)


The boundary field φ0(x) acts as a source for a boundary primary operator with conformalscaling dimension ∆−. In a quantum field theory, the correlation functions can be computedusing propagators and vertices. So it is important to extract these propagators for thecurrent case. There are three kinds of propagators:

Bulk-to-Bulk Propagator

Boundary-to-Bulk Propagator

Boundary-to-Boundary Propagator

Bulk-to-Bulk Propagator The bulk-to-bulk propagator −iG∆(X;X ′), where X ≡(z, x), is defined by (

−+m2)G∆(X;X ′) =

1√−g

δD+1(X −X ′) (3.157)

In the presence of a source J(X), (−+m2)φ(X) = J(X), the general solution is given by

φ(X) =

MdD+1X ′ G∆(X;X ′)J(X ′) (3.158)

The δ-function is defined bydD+1X ′δD+1(X −X ′)f(X ′) = f(X). The subscript ∆ stands

for a scaling exponent. This is to distinguish between propagation of normalizable (∆ = ∆+)and non-normalizable (∆ = ∆−) modes, but the general expression applies for either one.

In terms of some initial bulk configuration φ(X ′) [8], the field can be written in terms ofthe bulk-to-bulk propagator as

φ(X) =

dD+1X ′

√−g(φ(X ′)LX′G∆(X;X ′)− LX′φ(X ′)G∆(X;X ′)

)(3.159)

where LX′ ≡ −X′ +m2.The bulk-to-bulk propagator takes the form

G∆(X;X ′) =

dDk

(2π)D(zz′)

D2 e−ik·(x−x

′)

(Θ(z− z′)Kν(kz)Iν(kz

′) + Θ(z′− z)Iν(kz)Kν(kz′)

)(3.160)

The above integral can be performed to express the propagator in terms of a hypergeometricfunction

G∆(X;X ′) =2C∆

ν

(ξ

2

)∆

2F1

(∆

2,∆ + 1

2; ν + 1; ξ2

)ξ =

2zz′

z2 + z′2 + (x− x′)2=

2zz′

(z − z′)2 + (x− x′)2 + 2zz′

(3.161)

where C∆ is a normalization constant

C∆ =Γ(∆)

πD2 Γ(ν)

(3.162)

and the scaling exponent ∆ satisfies ∆(D −∆) + m2 = 0. The choice of the normalizationconstant will be explained in the boundary-to-bulk propagator.


Boundary-to-Bulk Propagator The boundary-to-bulk propagator −iK∆(z, x;x′) isa solution of the free scalar equation of motion

(−+m2)K∆(z, x;x′) = 0 (3.163)

such that the general solution φ(z, x) is written as a convolution with source some boundaryconfiguration φ0(x′) or φ1(x′),

φ(z, x) =

∂M

dDx′ K∆(z, x;x′)φ0,1(x′) (3.164)

The boundary-to-bulk propagator can be extracted from the bulk-to-bulk propagatorafter taking the proper limit and properly exchanging the scaling exponents [19]

K∆±(z, x;x′) =ν

2∆∓bD−1limz′→0

√−gz′ z′∆± nz

′∂z′G∆∓(z, x; z′, x′)

=ν

2∆∓limz′→0

z′∆±−Dz′∂z′G∆∓(z, x; z′, x′)

=ν

2limz′→0

z′∆±−DG∆∓(z, x; z′, x′)

= C∆∓

(z

z2 + (x− x′)2

)∆∓

(3.165)

where gz = −(b2

z2

)Dis the determinant of the induced metric on a constant z surface and

nz =√gzz = z

bis the unit normal. To derive the above expression, we used the relation

z∂zG(z, x; z′, z′) = z′∂z′G(z, x; z′, z′) = ∆G(z, x; z′, x′) (3.166)

and the fact that the hypergeometric function is equal to 1 at the origin:

2F1

(∆

2,∆ + 1

2; ν + 1; 0

)= 1 (3.167)

The choice of the normalization constant C∆ was to enforce the relation

limz→0

z−∆∓K∆±(z, x;x′) = δD(x− x′) (3.168)

Boundary-to-Boundary Propagator Finally the boundary-to-boundary−iβ∆(x, x′)propagator can be extracted from the boundary-to-bulk propagator by taking the limit z → 0

β∆±(x;x′) = limz→0

z−∆∓K∆±(z, x;x′)

= C∆∓

1

|x− x′|2∆∓

(3.169)


A more intuitive derivation From all three propagators, the boundary-to-bulk prop-agator is the most important so it should be understood from a physical point of view interms of symmetries. Only K∆− ≡ K is useful since the φ1 boundary field does not con-tribute at the boundary due to the z∆+ factor that vanishes at z = 0. The idea is to considerthe equation of motion near the center z → ∞. The x-components of the metric can thenbe neglected since b2

z2ηµν → 0 and the defining equation for the boundary-to-bulk propagatorsimplifies to the equation of motion (3.143) with k = 0. So the solution is of the form

K∆−(z, 0; 0) = C∆+z∆+ , ∆+ =

D

2+

√D2

4+m2b2 (3.170)

The component multiplying z∆− can be neglected since this factor vanishes at the centerz →∞.

The full expression for the boundary-to-bulk propagator can be obtained by utilizing theisometries of AdSD+1. In particular, performing an inversion

z → z

z2 + x2(3.171)

followed by translation31

x→ x− x′ (3.172)

can lead to the required result

K∆−(z, x;x′) = C∆+

(z

z2 + (x− x′)2

)∆+

(3.173)

The next step is to determine the normalization constant C∆+ . For this purpose weimpose condition (3.168), or, equivalently

z−∆−

∂M

dDx K∆−(z, x;x′) = 1 (3.174)

Setting x′ = 0 and performing the integral, we obtain

z−∆−

∂M

dDx K∆−(z, x; 0) = C∆+z−D

Rd

dDx1

(z2 + x2)∆+

= C∆+z−DΩD−1

∞0

drrD−1

(z2 + r2)−∆+

(t= r2

z2) =

1

2C∆+ΩD−1

∞0

dttD2−1

(1 + t)−∆+

=1

2C∆+ΩD−1B

(D

2,∆+ −

D

2

)=

1

2C∆+

2πD2

Γ(D2

) Γ(D2

)Γ(∆+ − D

2

)Γ(∆+)

= C∆+

πD2 Γ(ν)

Γ(∆+)

(3.175)

31An inversion followed by a translation amounts to a special conformal transformation.


The beta-function was used to perform the integral

B(x, y) =

∞0

dttx−1

(1 + t)x+y=

Γ(x)Γ(y)

Γ(x+ y)(3.176)

The normalization condition, thus, sets

C∆+ =Γ(∆+)

πD2 Γ(ν)

(3.177)

as promised.The observables in the theory correspond to correlation functions of boundary opera-

tors, which can be calculated via Witten diagrams in the bulk. The latter break down toappropriate combinations of the boundary-to-bulk propagator.

1-point Functions

CFT Side Consider the 1-point function 〈φ(x)〉 of some quasi-primary operator withscaling dimension ∆. Applying (3.128) for dilatations x→ x′ = λx, we get

〈φ(x)〉 = λ∆ 〈φ(λx)〉 (3.178)

Writing the vacuum expectation value a function f(x), the result above suggests

f(x) = λ∆f(λx) (3.179)

By Taylor expanding32, f(x) =∑∞

n=0 cnxn, scale invariance allows only one power of x to

survive:∞∑n=0

cnxn =

∞∑n=0

cnλn+∆xn

⇒n = −∆

⇒f(x) = d∆x−∆

(3.181)

with d∆ = c−∆ a constant that depends on the field content. But now translational invarianceimplies that

〈φ(x)〉 = 〈φ(x+ a)〉⇒f(x) = f(x+ a)

⇒x−∆ = (x+ a)−∆

(3.182)

For this to hold for any translation aµ, the scaling dimension can only be zero. So the 1-pointfunction takes the form

〈φ(x)〉 = d0δ∆,0 (3.183)

32Note that f(x) is a multi-variable function, f(x) ≡ f(x0, . . . , xD−1), and the Taylor expansion shouldbe understood as such,

f(x) =

∞∑n=0

cµ1...µnxµ1 . . . xµn ≡

∞∑n=0

cnxn (3.180)


It is always possible to rescale one quasi-primary operator φ(x) = d∆φ(x) and get rid of theconstant coefficient to get

〈φ(x)〉 = δ∆,0 (3.184)

Only the vacuum is invariant under dilatations and, thus, only the unit operator, φ(x) = 1,gives a non-zero value.

AdS Side The vacuum expectation value of a scalar field in AdS vanishes and thevacuum state is normalized. So the 1-point functions of the two sides do match.

2-point Functions

CFT Side Consider the 2-point function 〈O1(x1)O2(x2)〉 of two quasi-primary opera-tors O1 and O2 of scaling dimensions ∆1 and ∆2 respectively. Applying (3.128) for dilatationsyields

〈O1(x1)O2(x2)〉 = λ∆1+∆2 〈O1(λx1)O2(λx2)〉 (3.185)

Poincare invariance enforces the 2-point function to be a function of |x1 − x2|:

〈φ1(x1)φ2(x2)〉 = f(|x1 − x2|) =∞∑n=0

cn|x1 − x2|n (3.186)

So we get

f(|x1 − x2|) = λ∆1+∆2f(|λx1 − λx2|) = λ∆1+∆2f(λ|x1 − x2|)

⇒∞∑n=0

cn|x1 − x2|n =∞∑n=0

cnλn+∆1+∆2 |x1 − x2|n

⇒n = −∆1 −∆2

(3.187)

⇒ 〈φ1(x1)φ2(x2)〉 =d12

|x1 − x2|∆1+∆2, d12 ≡ c−∆1−∆2 (3.188)

Applying now (3.128) for special conformal transformations

x′µi =xµi − x2

i bµ

γi, γi ≡ 1− 2b · xi + b2x2

i (3.189)

yields

〈φ1(x1)φ2(x2)〉 =1

γ∆11 γ∆2

2

〈φ1(x′1)φ2(x′2)〉 =1

γ∆11 γ∆2

2

d12

|x′1 − x′2|∆1+∆2(3.190)

Since

|x′1 − x′2| =|x1 − x2|√

γ1γ2

(3.191)


the 2-point function becomes

〈φ1(x1)φ2(x2)〉 =d12

γ∆11 γ∆2

2

(γ1γ2)∆1+∆2

2

|x1 − x2|∆1+∆2

=

(γ2

γ1

)∆1−∆22 d12

|x1 − x2|∆1+∆2

(3.192)

Now, for both (3.188) and (3.192) to hold, the scaling dimensions of the two quasi-primaryoperators must be equal. So

〈φ1(x1)φ2(x2)〉 =d12

|x1 − x2|2∆1δ∆1∆2 (3.193)

Finally, after the normalization of the 1-point function, we get

〈φ1(x1)φ2(x2)〉 =d12

|x1 − x2|2∆1δ∆1∆2 , d12 ≡

d12

d1d2

(3.194)

AdS Side In the largeN limit, the 2-point function is governed by tree level amplitudes.So it is just the boundary-to-boundary propagator

β∆−(x1;x2) = C∆+

1

|x1 − x2|2∆−(3.195)

with ∆− identified with ∆1. The 2-point functions turn out to be identical given that

C∆+ = d12 (3.196)

The asymptotic value of the bulk fields acts as a source for the dual conformal operators atthe boundary.

3-point Functions

CFT Side Applying (3.128) for the 3-point functions 〈O1(x1)O2(x2)O3(x3)〉, we get

〈O1(x1)O2(x2)O3(x3)〉 = λ∆1+∆2+∆3 〈O1(λx1)O2(λx2)O3(λx3)〉 (3.197)

In addition Poincare invariance implies that

〈O1(x1)O2(x2)O3(x3)〉 = f(|x1 − x2|, |x2 − x3|, |x3 − x1|) (3.198)

Therefore we must pick up only certain powers in the Taylor expansion

f(|x1 − x2|, |x2 − x3|, |x3 − x1|) = λ∆1+∆2+∆3f(λ|x1 − x2|, λ|x2 − x3|, λ|x3 − x1|) (3.199)

⇒∑

n1,n2,n3

cn1n2n3 |x1 − x2|n1|x2 − x3|n2|x3 − x1|n3 =

=∑

n1,n2,n3

cn1n2n3λ∆1+∆2+∆3+n1+n2+n3|x1 − x2|n1|x2 − x3|n2|x3 − x1|n3

(3.200)


⇒ n1 + n2 + n3 = −∆1 −∆2 −∆3 (3.201)

Next we apply (3.128) for special conformal transformations to get

〈O1(x1)O2(x2)O3(x3)〉 =1

γ∆11 γ∆2

2 γ∆33

〈O1(x′1)O2(x′2)O3(x′3)〉 (3.202)

⇒ f(|x1 − x2|, |x2 − x3|, |x3 − x1|) =1

γ∆11 γ∆2

2 γ∆33

f(|x′1 − x′2|, |x′2 − x′3|, |x′3 − x′1|)

=1

γ∆11 γ∆2

2 γ∆33

f(|x1 − x2|√

γ1γ2

,|x2 − x3|√

γ2γ3

,|x3 − x1|√

γ3γ1

)

(3.203)

⇒∑

n1,n2,n3

cn1n2n3|x1 − x2|n1 |x2 − x3|n2|x3 − x1|n3 =

=∑

n1,n2,n3

cn1n2n3

1

γ∆1+

n1+n32

1

1

γ∆2+

n1+n22

2

1

γ∆3+

n2+n32

3

|x1 − x2|n1|x2 − x3|n2|x3 − x1|n3

(3.204)

⇒

n1 + n3 = −2∆1

n1 + n2 = −2∆2

n2 + n3 = −2∆3

⇒

n1 = ∆3 −∆1 −∆2 = −(∆− 2∆3)n2 = ∆1 −∆2 −∆3 = −(∆− 2∆1)n3 = ∆2 −∆3 −∆1 = −(∆− 2∆2)

(3.205)

with ∆ ≡∑3

i=1 ∆i. This solution is also consistent with the constraint (3.201) obtained byscale invariance. The general 3-point function is, thus

〈O1(x1)O2(x2)O3(x3)〉 =d123

|x1 − x2|∆1+∆2−∆3 |x2 − x3|−∆1+∆2+∆3|x3 − x1|∆1−∆2+∆3

=d123

|x1 − x2|∆−2∆3|x2 − x3|∆−2∆1|x3 − x1|∆−2∆2

(3.206)

Implementing the normalization of the 1-point function, we finally obtain

〈O1(x1)O2(x2)O3(x3)〉 =d123

|x1 − x2|∆−2∆3|x2 − x3|∆−2∆1|x3 − x1|∆−2∆2

d123 ≡d123

d1d2d3

(3.207)

AdS Side To reproduce the 3-point functions, a 3-vertex interaction should be addedin the bulk action of 3 scalar fields,

S =1

2

MdD+1x

√−g

(3∑i=1

(−gµν∇µφi∇νφi +m2

iφ2i

)+ λφ1φ2φ3

)(3.208)

The leading contribution, then, is the Witten diagram with one vertex in the bulk (figure3.10). This interaction term is simply the product of 3 boundary-to-bulk propagators. The


Figure 3.10: Witten diagram with one 3-vertex interaction in the bulk.

correlator is then obtained by integrating over all possible positions of the bulk vertex:

〈φ1,0(x1)φ2,0(x2)φ3,0(x3)〉 = −λdDxdz

√−g K∆1(z, x;x1)K∆2(z, x;x2)K∆3(z, x;x3)

=λa1

|x1 − x2|∆−2∆3 |x2 − x3|∆−2∆1|x3 − x1|∆−2∆2

(3.209)with ∆ = ∆1 + ∆2 + ∆3, φi,0 the asymptotic behavior of the bulk field φi and a1 a numericalconstant equal to

a1 = −Γ(

∆2−∆1

)Γ(

∆2−∆2

)Γ(

∆2−∆3

)2π4Γ

(∆1 − D

2

)Γ(∆2 − D

2

)Γ(∆3 − D

2

)Γ

(∆−D

2

)(3.210)

The agreement with (3.206) is now obvious with the choice of the coupling constant λ, asimplied by the correspondence.

4-point Functions

CFT Side Finally, for 4-point functions 〈O1(x1)O2(x2)O3(x3)〉O4(x4), conformal in-variance still imposes some constraints, but these do not completely fix the correlators as inthe previous cases. This happens because for n-point functions with n > 3, it is possible toconstruct conformally invariant ratios, called anharmonic ratios or, cross ratios. For 4-pointfunctions, for example, there are two conformally invariant cross ratios:(

x12x34

x13x24

)2

≡ u ,

(x12x34

x23x14

)2

≡ υ (3.211)

where xij ≡ |xi − xj|. In terms of these, the general 4-point function takes the form

〈φ1(x1)φ2(x2)φ3(x3)〉φ4(x4) = f(u, υ)4∏i<j

x∆3−∆i−∆j

ij (3.212)

As a result, to summarize up, conformal symmetry completely determines correlationfunctions up to n = 3. For n > 3 the content and interactions of the particular theory must


Figure 3.11: Witten diagrams for the 4-point function in the bulk.

be taken into account. This is the main reason that if one wants to identify dual conformalfield theories, then one of the most fundamental tests to perform is to check whether 4-pointfunctions match. The details of the calculation for the particular theory will not be statedhere. Instead, the bulk result will be presented.

AdS Side Just like the 3-point function, the 4-point function of the gravitational theoryis computed by implementing the 4-vertex interactions. In this case, there are 3 differentWitten diagrams contributing to leading order in the large N limit (figure 3.11). It can beshown that the last two diagrams in the figure can be obtained from the first, which reads

D∆1∆2∆3∆4 ≡dDxdz

√−g K∆1(z, x;x1)K∆2(z, x;x2)K∆3(z, x;x3)K∆4(z, x;x4)

K∆(z, x;x′) =

(z

z2 + (x− x′)2

)∆ (3.213)

Perturbation theory can be used for the gravitational theory to show that

〈φ1,0(x1)φ2,0(x2)φ3,0(x3)φ4,0(x4)〉 =

(6

π

)2(16x2

24

(1

2s− 1

)D4433 +

64

9

x224

x213

1

sD3344

+16

3

x224

x213

D2233 − 14D4444 −46

9x213

D3344 −10

9x213

D2244 −8

3x613

D1144 + 64x224D4455

) (3.214)

which indeed reproduces the CFT expression.

3.6 The AdS/CFT Dictionary

This section is one of the most important of the chapter as the basic dictionary of theAdS/CFT correspondence will be presented. Each bulk field φ(x, z) corresponds to a bound-ary operator Oi(x) in the CFT. As Gubser, Klebanov, Polyakov [20] and Witten [7] show,the correspondence is based on an equality between partition functions:

Zgravity [φi,0; ∂M] =

⟨exp

(−∑i

φi,0(x)Oi(x)

)⟩CFT

[B = ∂M] (3.215)

3.6. THE ADS/CFT DICTIONARY 69

Here φi(x, z) are bulk fields with asymptotic values

φi(x, z) = zd−∆φi,0(x) + . . . (3.216)

Oi(x) are the corresponding CFT operators; M is the bulk manifold and B its boundary.The right hand side is the generating functional for CFT correlation functions where the

sources are given by the asymptotic values of the bulk fields⟨exp

(−∑i

φi,0(x)Oi(x)

)⟩CFT

(3.217)

(as seen in 3.2.2). The n-point correlation functions can be generated via

〈O1(x1) . . .On(xn)〉 = i−nδnZCFT [φ0]

δφ1,0(x1) . . . δφn,0(xn)

∣∣∣∣φ0=0

(3.218)

Thus the AdS/CFT dictionary is a mapping between the bulk fields and CFT operators.Each light field in the gravity theory corresponds to a local operator in CFT of low dimension.The spins of a bulk fields and its dual boundary operator are equal, while the mass of thebulk field fixes the scaling dimension of the CFT operator. In particular if the bulk field isa scalar field φ(x, z) with mass m, the dual CFT operator O(x) is also a scalar and has ascaling dimension

∆O =D

2

1 +

√1 +

(2bm

D

)2 (3.219)

For vector bulk fields Aµ(x, z), the dual CFT operators are spin-1 currents Jµ(x). If thebulk fields are massless, the CFT currents are conserved, i.e. ∂µJ

µ = 0 and have scalingdimension ∆J = D − 1. Consequently, gauge symmetries in the bulk correspond to globalsymmetries in the CFT as in Maldacena’s proposal.

Lastly, the bulk metric gµν(x, z), or the gravitational field, corresponds to the energy-momentum tensor Tµν(x) in the CFT. The energy-momentum tensor of the boundary theoryis conserved and traceless. In addition, it has a scaling dimension ∆T = D since the bulkgraviton is massless.

The above dictionary provides a non-perturbative, UV complete definition of quantumgravity.


Part II

MODERN REALIZATIONS OF THEDUALITY

71

Chapter 4

Entropy and Entanglement inQuantum Systems

It is remarkable how general relativity and the geometrical description of gravity are con-nected with thermodynamics, as illustrated via the properties of quantum black holes. To-gether with the holographic principle, this connection suggests that entropy and geometryshare something in common. In this chapters, we review aspects of quantum entanglementand entanglement entropy that will be useful for presenting the connection between gravityand thermodynamics. Further discussions can be found in [21].

4.1 Product Spaces

Consider Hilbert spaces H1, H2, . . . , HN with orthonormal bases |ψi1〉 , |ψi2〉 , . . . , |ψiN 〉 re-spectively. We can tensor these to form the product space

H = H1 ⊗H2 ⊗ · · · ⊗ HN (4.1)

The states|ψi1i2...iN 〉 = |ψi1〉 ⊗ |ψi2〉 ⊗ · · · ⊗ |ψiN 〉 ≡ |ψi1〉 |ψi2〉 . . . |ψiN 〉 (4.2)

form a basis for the product space. The most general state in the product Hilbert space canbe written as a linear superposition of this

|Ψ〉 =∑

i1,i2,...,iN

ci1,i2,...,iN |ψi1i2...iN 〉 (4.3)

where ci1,i2,...,iN are complex coefficients. When the linear combination above cannot bewritten as a product, then the degrees of freedom describing each initial Hilbert space factorare entangled.

4.2 Density Matrix

A generic mixed state can be described by a density matrix, which is a hermitian matrixwith non-negative eigenvalues and trace equal to unity. Choosing an orhonormal basis in

73

74 CHAPTER 4. ENTROPY AND ENTANGLEMENT IN QUANTUM SYSTEMS

which the density matrix is diagonal, we get

ρ =∑i

pi |ψi〉〈ψi| (4.4)

where pi ≥ 0 and sumipi = 1. We associate ρ to an ensemble of states |ψi〉 , pi, with pi theprobability for |ψi〉 to occur. So the probabilities pi are the eigenvalues of ρ and the states|ψi〉 are its eigenvectors :

ρ |ψi〉 =∑j

pj |ψj〉〈ψj|ψi〉 = pi |ψi〉 (4.5)

where orthonormality condition was used, 〈ψj|ψi〉 = δij. For a pure state |ψ〉, the densitymatrix is just a projection operator ρ = |ψ〉〈ψ| with one eigenvalue equal to unity and therest zero.

In terms of the density matrix, the mean value of an observable can be calculated bytracing the product of the density matrix and the matrix representing the observable

〈A〉 = Tr (ρA) (4.6)

Indeed, starting with the statistical definition of the mean value of the observable, we get

〈A〉 =∑i

pi 〈ψi|A|ψi〉

=∑i

pi 〈ψi|A

(∑n

|n〉〈n|

)|ψi〉

=∑n

∑i

pi 〈n|ψi〉〈ψi|A|n〉

=∑n

〈n|

(∑i

pi |ψi〉〈ψi|A

)|n〉

=∑n

〈n| ρA |n〉 = Tr (ρA)

(4.7)

In the above proof, an arbitrary orthonormal basis |n〉 and the completeness relation,∑n |n〉〈n| = 1, were used.Two basic properties of the density matrix are,

Tr ρ =∑i

pi = 1

Tr ρ2 =∑i

p2i ≤ 1

(4.8)

From these properties it is easy to distinguish between mixed and pure states. In particular,if Tr ρ2 = 1, then the density matrix corresponds to a pure state, while for mixed states wehave Tr ρ2 < 1. Of course, for pure states ρ2 = ρ.

4.3. QUANTUM SUBSYSTEMS 75

4.3 Quantum Subsystems

In general, a quantum system can be decomposed into a subsystem A and and its comple-mentary subsystem A. Accordingly, the Hilbert space decomposes as follows

H = HA ⊗HA (4.9)

There are two questions we would like to answer concerning this decomposition:

Given a pure state |Ψ〉 ∈ H of the full system, what is the density matrix ρA of thesubsystem A?

Given an ensemble |ψAi 〉 ∈ HA, pi of the subsystem A, is it possible to construct apure state |Ψ〉 ∈ H for the full quantum system?

4.3.1 Reduced Density Matrix

Let us answer the first question. The pure state |Ψ〉 ∈ H describing the full system can bewritten as a superposition of products of states |ψAi 〉 ∈ HA and |ψAi 〉 ∈ HA describing thesubsystem A and its complementary A:

|Ψ〉 =∑i,i

cii |ψAi 〉 ⊗ |ψAi 〉 (4.10)

The question can be rephrased in terms of expectation values. Given an operator OA thatacts only on the subsystem A, the task is to find the density matrix ρA so that

〈Ψ| OA ⊗ 1A |Ψ〉 = TrA (OAρA) (4.11)

Now

〈Ψ| OA ⊗ 1A |Ψ〉 = Tr ((OA ⊗ 1A) ρ) (4.12)

where ρ = |Ψ〉〈Ψ| is the density matrix for the entire system.


So using the relation above, we get

Tr ((OA ⊗ 1A) ρ) = Tr

(OA ⊗ 1A)

∑i,i

cii |ψAi 〉 ⊗ |ψAi 〉

∑j,j

cjj |ψAj 〉 ⊗ |ψAj 〉

†

= Tr

∑i,i

∑j,j

ciic∗jj (OA ⊗ 1A) |ψAi 〉 ⊗ |ψAi 〉〈ψ

Aj | ⊗ 〈ψ

Aj |

=∑i,i

∑j,j

ciic∗jj Tr

((OA ⊗ 1A) |ψAi 〉 ⊗ |ψAi 〉〈ψ

Aj | ⊗ 〈ψ

Aj |)

=∑i,i

∑j,j

ciic∗jj 〈ψ

Aj | ⊗ 〈ψ

Aj | (OA ⊗ 1A) |ψAi 〉 ⊗ |ψAi 〉

=∑i,i

∑j,j

ciic∗jj 〈ψ

Aj |ψ

Ai 〉〈ψ

Aj |OA|ψAi 〉

=∑i,j

∑i

ciic∗ji 〈ψ

Aj |OA|ψAi 〉

= TrA

(OA∑i,j

∑i

ciic∗ji |ψ

Ai 〉〈ψAj |

)≡ TrA (OAρA)

(4.13)

⇒ ρA =∑i,j

∑i

ciic∗ji |ψ

Ai 〉〈ψAj | ≡ TrA ρ (4.14)

The density matrix ρA of the subsystem is also called the reduced density matrix. Noticethat in (4.14), the partial trace over the subsystem A is taken.

Therefore, the subsystem A can be represented by an ensemble, described by the densitymatrix ρA.

4.3.2 Purifications

The second question concerns the construction of pure states given an ensemble for thesubsystem A, a procedure known as purification. There is no unique answer, but, in general,a pure state |Ψ〉 can be constructed in terms of the Schmidt decomposition

|Ψ〉 =∑i

√pi |ψAi 〉 ⊗ |ψBi 〉 (4.15)

where |ψBi 〉 is a set of orthonormal states spanning a Hilbert spaceHB of dimension greateror equal to the number of non-zero eigenvalues of ρA. The system B is the purifying system.

A useful purification is the thermofield double state (TFD), arising when considering a

canonical ensemble |Ei〉 , pi = e−βEiZ, and taking the system B to be a copy of the initial

system:

|TFD〉 =1√Z

∑i

e−βEi/2 |Ei〉 ⊗ |Ei〉 (4.16)

4.4. ENTROPY IN QUANTUM MECHANICS 77

The reduction to either of the two subsystems, as explained in the last subsection, gives thena thermal state1 with temperature β−1 in both cases.

4.4 Entropy in Quantum Mechanics

Entropy measures “ignorance”. For the microcanonical ensemble, |Ei〉 , pi = 1N, the en-

tropy is given by S = logN with N the number of states with energy E So, each statecontributes an amount

S =∑i=1

NSi

Si =1

NlogN = −pi log pi

(4.17)

For general ensembles |ψi〉 , pi, the entropy is given precisely by

S = −∑i

pi log pi (4.18)

and it is called the Von Neumann entropy. Remembering that the probabilities pi are theeigenvalues of the density matrix ρ, the Von Neumann entropy can be expressed by

S = −Tr (ρ log ρ) (4.19)

4.5 Quantum Entanglement

The last and most crucial part of this chapter regards Quantum Entanglement. Quantumentanglement is unavoidable when dividing a quantum system into subsystems. In terms ofthe reduced density matrix eigenvalues, the subsystem A is entangled with A if and only ifthe there is no pi = 1; in other words, entanglement arises whenever |Ψ〉 6= |ψA〉 |ψA〉 (ormore generally when it cannot be written as a product state). For example an entangledpair of two electrons is the antisymmetric state

|Ψ〉 =1√2

(|↑〉 ⊗ |↓〉 − |↓〉 ⊗ |↑〉) (4.20)

A measure of the entanglement of the subsystemA with its complement A is the entanglemententropy

Sentang. ≡ SA = −Tr (ρA log ρA) (4.21)

More generally, entanglement can be measured by the Renyi entropies

Sα =1

1− αlog

(∑i

pαi

)=

1

1− αlog Tr (ραA) (4.22)

1Canonical ensemble and thermal state are two names for exactly the same concept: A system at constanttemperature arising by weakly coupling the subsystem A with a heat bath (the remnant subsystem A).


with the exponent α typically being an integer. In the limit α → 1, the Renyi entropyS1 becomes the entanglement entropy. The usefulness of the Renyi entropies (4.22) overthe entanglement entropy (4.21) comes from the fact that they are easier to compute. Inaddition, the knowledge of the Renyi entropies Sα for all the integers α ∈ [1, D], withD = dim (HA) the dimension of the Hilbert space, is equivalent to the knowledge of theprobabilities pi characterizing the ensemble (and the size of quantum entanglement) sincethe characteristic polynomial of the density matrix can be expressed in terms of the tracesof the matrix powers:

D∏i=1

(λ− pi) = det (λ1A − ρA) = λD − λD−1 Tr (ρ) + . . . (4.23)

The set of probabilities pi is also known as the entanglement spectrum.

Chapter 5

Two sided Black Holes

Given the AdSD+1/CFTD conjecture, one can investigate additional dualities that follow.According to the conjecture, for any asymptotically AdSD+1 bulk geometry, there is a dualCFTD state at the boundary. For example, empty AdSD+1 corresponds to the CFTD vacuumstate on the sphere SD. There more asymptotically AdSD+1 solutions known. We recall fromchapter (Chapter 2), the case of the AdS-Schwarzschild black hole. In this chapter we presenta duality involving eternal AdS black holes ([22]).

5.1 Eternal Black Hole

The AdS-Schwarzschild black hole is described in global coordinates (t, r, θi) by the metric

ds2 = −g(r)dt2 +dr2

g(r)+ r2dΩ2

D−2

g(r) = 1 +r2

b2− wDM

rD−3

wD =(D − 1)16πG

(D − 2)ΩD−2

(5.1)

where wD is chosen so that M is the mass of the black hole ([23]). Another way to understandthe expression of wD is by taking the limit b→∞. In this limit, we get a Schwarzschild blackhole in flat space with wDM = RD−3

S ; RS is the Schwarzschild radius. The Penrose diagramof the geometry is extracted from the maximally extended geometry in Kruskal-Szekerescoordinates.

5.1.1 A reminder on the Kruskal-Szekeres coordinates

Given a spherically symmetric metric

ds2 = −g(r)dt2 +dr2

g(r)+ r2dΩ2

D−2 (5.2)

that describes a black hole with event horizon at rh (g(rh) = 0), the Kruskal-Szekerescoordinates are determined using the following prescription:

79

80 CHAPTER 5. TWO SIDED BLACK HOLES

Step 1: Define the tortoise coordinate r∗ via

dr∗ =dr

g(r)⇒ r∗(r) =

r

r0

dr′

g(r′)(5.3)

The metric takes the form

ds2 = g (r(r∗)) (−dt2 + dr2∗) + r2(r∗)dΩ2

D−2 (5.4)

Step 2: Define the lightlike coordinates u = t− r∗, υ = t+ r∗ transforming the metricto

ds2 = −g (r(r∗)) dudυ + r2(r∗)dΩ2D−2 (5.5)

Step 3: Define the Kruskal-Szekeres coordinates,

U = −e−ηu

V = eηυ(5.6)

with η a constant to be chosen such that the singularity at the horizon gets removed.The metric becomes

ds2 = −g (r(r∗))

η2e−2ηr∗dUdV + r2(r∗)dΩ2

D−2 (5.7)

Now η depends on the details of g(r). In general it is a function on the event horizonradius: η = f(rh). After this, the Kruskal-Szekeres coordinates satisfy the followingboundary conditions,

limr→∞

U V = −1

limr→rh

U V = 0

limr→0

U V = 1

(5.8)

In other words, the event horizon is at U V = 0 and the past and future singularitiesat U V = 1

Step 4: Finally, the Penrose diagram is obtained by bringing the infinities to finitepoints via the transformation,

U = tanU

V = tanV(5.9)

Unfortunately, the analytic form of the maximally extended AdS-Schwarzschild black hole isnot easy to obtain because the tortoise coordinate can be determined only up to an integralin terms of fundamental functions. The Lorentzian Penrose diagram of the geometry isshown in figure 5.1, with the vertical axes measuring time.

The Euclidean geometry is obtained by t = −itE. The Euclidean time is periodic withperiod the inverse temperature β of the black hole, i.e. tE ∈ S1( β

2π) = [0, β]. The topology is

5.2. THERMOFIELD DOUBLE STATE 81

FUTURE SINGULARITY

PAST SINGULARITY

LEFT

BO

UN

DA

RY

RIG

HT B

OU

ND

AR

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Figure 5.1: Penrose diagram for Lorentzian AdS-Schwarzschild black hole in AdSD. The redzig-zag lines are the positions of the future and past singularities. The left and right straightblue lines are the left and right boundary of the eternal black hole and the thin blue linesare the positions of the event horizons. At each point lives a sphere SD−1. The time axes isthe vertical axes. The spatial slice at which the two event horizons meet (the center) is theslice t = 0.

that of a solid ball: Bd+1. The heat capacity of a large enough AdS-Schwarzschild black holeis positive. In particular, at some critical temperature, a phase transition occurs, known asthe Hawking-Page phase transition ([24]). Large enough black holes can coexist in thermalequilibrium there radiation, without evaporating. These are the so called eternal black holes.It is these black holes that we examine here.

As can be seen from the Penrose diagram, the eternal black hole has a past and a futuresingularity. It also has two asymptotic regions, each with a boundary denoted by “L” (Left)and “R” (Right).

5.2 Thermofield Double State

The eternal black hole has two boundaries. The AdS/CFT conjecture instructs us to as-sociate a dual CFT on each boundary. But how can two disconnected CFTs give rise to aconnected bulk manifold? The answer is quantum entanglement.

In particular, there are two copies of the CFT, one for the left boundary and one forthe right boundary. As in section 4.3.2, we combine the two systems by taking the tensorproduct of the Hilbert spaces, and take the whole system to be in the thermofield doublestate:

|TFD〉 =1√Zβ

∑i

e−βEi/2 |Ei〉L ⊗ |Ei〉R (5.10)


Here |Ei〉L and |Ei〉R are the energy eigenstates of each CFT:

HL |Ei〉L = Ei |Ei〉LHR |Ei〉R = Ei |Ei〉R

(5.11)

The Hamiltonians HL and HR generate time translations on the L and R boundaries respec-tively.

The density matrix for the full system is

ρ = |TFD〉〈TFD| = 1

Zβ

∑i,i′

e−β(Ei+Ei′ )/2 |Ei〉L ⊗ |Ei〉R R 〈Ei′| ⊗ L 〈Ei′ | (5.12)

Each subsystem is decribed by a thermal density matrix associated with the canonical en-semble. Indeed taking a partial trace, the reduced density matrix for the left system is

ρL = TrR ρ

=∑j

R 〈Ej|

(1

Zβ

∑i,i′

e−β(Ei+Ei′ )/2 |Ei〉L ⊗ |Ei〉R R 〈Ei′| L 〈Ei′ |

)|Ej〉R

=1

Zβ

∑i

e−βEi |Ei〉L L 〈Ei|

=1

Zβe−βHL

(5.13)

Following the exact same steps for the R subsystem, we get

ρR = TrL ρ =1

Zβe−βHR (5.14)

It is important to understand that the two boundary theories are not coupled; the fullsystem Lagrangian does not contain products of operators associated with the left and rightCFTs. However, the the L and R degrees of freedom are entangled. So mixed correlationfunctions of the form,

〈TFD|OL ⊗OR|TFD〉 =1

Zβ

∑i,i′

e−β(Ei+Ei′ )/2L 〈Ei|OL|Ei′〉L R 〈Ei|OR|Ei′〉R (5.15)

do not generally vanish. For the special case of OR = 1R, the expectation values reduce tothermal expectation values on the corresponding boundary

〈TFD|OL ⊗ 1R|TFD〉 =1

ZβTrL (ρLOL) (5.16)

and similarly for OL = 1L.Next we ask “what is the Hamiltonian of the full system?” There are actually two

choices, H = HL−HR and H = HL +HR. The full quantum system is supposed to be dual

5.3. HOLOGRAPHIC DUALITY DERIVATION 83

FUTURE SINGULARITY

PAST SINGULARITY

LEFT

BO

UN

DA

RY

RIG

HT B

OU

ND

AR

Y

Figure 5.2: Equivalent spatial slices (green thin lines) under the isometry generator −i∂t ineternal black hole geometry. The intersect the vertical axes on the left and right boundariesat times tL and tR so that tL = −tR.

to the eternal black hole geometry. A standard result from the Kruskal-Szekeres analysisis that time in the L asymptotic region runs in the opposite direction than time in the Rasymptotic region. This suggests that the proper Hamiltonian is H = HL−HR

1 under whichthe thermofield double state is stationary:

H |TFD〉 =1

Zβ

∑i

e−βEi/2(

(HL |Ei〉L)⊗ |Ei〉R − |Ei〉L (HR |Ei〉R)

)= 0 (5.18)

⇒ |TFD(t)〉 = e−iHt |TFD〉 = |TFD〉 (5.19)

5.3 Holographic Duality Derivation

The spatial slices shown in figure 5.2 are equivalent under the isometry generator ξt = −i∂t2.The intersect the vertical axes on the left and right boundaries at times tL and tR so thattL = −tR.

Applying the AdS/CFT dictionary, we get

Zgravity[∂M = B] = ZCFT [B] (5.20)

In other words the path integral of gravity on M with boundary B is equal to the pathintegral of a CFT on B.

1More preciselyH = HL ⊗ 1R − 1L ⊗HR (5.17)

and similarly for H.2This is an isometry generator because the corresponding contra-variant Killing vector is simply ξµt = δ µt

since the eternal black hole metric does not depend explicitly on global time t.


ȁ𝑇𝐹𝐷 =

𝛽

2

Figure 5.3: Path integral representation of the thermofield double state. The red cutsrepresent states on the two boundaries.

5.3.1 CFT side

Several CFT observables can be expressed as Euclidean path integrals on the interval Iβ/2(of lenght β

2) times the sphere SD−1:

BE = Iβ/2 × SD−1 (5.21)

This space is shown in figure 5.3. The left cut represents a state on the L boundary andthe right cut a state on the R boundary. For example consider transition amplitudes of theform:

L 〈OL| ⊗ R 〈OR|TFD〉 (5.22)

We will show that

L 〈OL| ⊗ R 〈OR|TFD〉 = 〈OR|e−βH/2|O∗L〉 (5.23)

Indeed starting with the right hand side and inserting a complete basis of energy eigenstates,we get

〈OR|e−βH/2|O∗L〉 =∑i

〈OR|e−βH/2|Ei〉〈Ei|O†L〉

=∑i

e−βEi/2L 〈OL|Ei〉L R 〈OR|Ei〉R

= L 〈OL| ⊗ R 〈OR|∑i

e−βEi |Ei〉L ⊗ |Ei〉R

= L 〈OL| ⊗ R 〈OR|TFD〉

(5.24)

5.3.2 Gravity Side

Now consider the eternal black hole geometry. We cut this on a spatial slice of time reversalsymmetry, and glue to half of it half of the Euclidean AdS black hole geometry. See figure5.4. The arc then has coordinate length β

2. A Euclidean path integral on half the space,

determines a state for the Lorentzian geometry.

5.4. ER=EPR 85

FUTURE SINGULARITY

LEFT

BO

UN

DA

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RIG

HT B

OU

ND

AR

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𝑡

𝑡𝐸

Figure 5.4: Gravity dual of the thermofield double state. Half of the Euclidean eternal blackhole is glued with half of the Lorentzian eternal black hole. The spheres SD−1 denoted asred dots at the boundaries stand for the cuts in the path integral representation.

Now consider the intersection slice between the Euclidean and the Lorentzian geometries.This is divided into two (L and R) by the origin. Since time translations in Schwarzschildtime t correspond to rotations in the Euclidean geometry (as shown by the arrow), we caninterprete the path integral as a transition amplitude between data on the L slice and thedata on the R slice. The gravity data amount to insertion of CFT operators on the L andR boundaries. In all we end up with a transition amplitude of the form

〈φR|e−βH/2|phiL〉 → 〈OR|e−βH/2|O∗L〉 (5.25)

5.4 ER=EPR

On the CFT side, “mixed” CFT correlators of the form

〈TFD|OL ⊗OR|TFD〉 (5.26)

are non-zero due to quantum entanglement of CFTL and CFTR. On the gravity side, bulkcorrelators 〈φLφR〉, with φL and φR the asymptotic values of a bulk filed φ3 on the L andR boundary respectively, are non-zero because it is possible to draw Witten diagrams goingthrough the interior via a spacelike wormhole.

3A bulk field φ(z, x) gives rise to boundary values φL(x) and φR(x) that act as sources for OL(x) andOR(x) respectively:

φ(z, x) = zD−∆OLφL(x) + zD−∆ORφR(x) + . . . (5.27)


The notion of entanglement was historically introduced in the EPR experiment, a gedankenexperiment proposed by Einstein, Podolsky and Rosen. Wormholes were firstly discovered,quite remarkably, by Einstein and Rosen, also known as ER (Einstein-Rosen) bridges.

The duality between the CFT thermofield double state and the eternal black hole sug-gests, thus, a connection between quantum entanglement and wormholes, more famouslypresented as the “equality”

ER = EPR (5.28)

Chapter 6

Entanglement and Geometry

The AdS/CFT conjecture offers an identification of the entropy of the quantum system livingon the boundary of an asymptotically AdS spacetime with the entropy of the bulk boundedby the Bekenstein-Hawking formula according to the holographic principle. An even moreinsightful description of general holographic dualities comes when considering subsystemsof the boundary quantum system. In that case, the entropy of the subsystem is just theentanglement entropy. In this chapter, the connection between the entanglement entropy atthe boundary and the geometry of the bulk will be reviewed according to the Ryu-Takayanagiformula and its generalizations. In addition a number of entanglement constraints and theirgravity interpretation will be discussed to gain more insight on which theories theories canbe dual.

6.1 The Ryu-Takayanagi Formula

The proposal of Shinsei Ryu and Tadashi Takayanagi ([25]) states a conjecture similar tothe covariant entropy bound stated by the holographic principle, i.e. entropy is expressedin terms of a surface area. The configuration involves a holographic CFT living on a fixedspacetime geometry B with the full quantum system being at a state |Ψ〉. Breaking downthe system into a spatial subsystem A and its complement A is equivalent to taking a spatialslice ΣB of B and then choosing a subset A ⊂ ΣB. This subset may very well be a union ofdisconnected regions, not just a connected region (figure 6.1).

If the state |Ψ〉 is associated with a classical dual geometry MΨ, then B = ∂MΨ. Con-sequently, the spatial slice ΣB, as well as the subsets of the subsystems A and A correspondto some regions of the boundary ∂MΨ. The proposal states that the entropy S(A) of thesubsystem A, i.e. the entanglement entropy describing the entanglement of the fields in thesubsystem with the rest of the system, can be calculated by a “Bekenstein-Hawking”-likeformula,

S(A) =Area(A)

4G(6.1)

known as the Ryu-Takayanagi formula or holographic entanglement entropy formula.

87

88 CHAPTER 6. ENTANGLEMENT AND GEOMETRY

Figure 6.1: A spatial slice ΣB of the spacetime B in which the CFT lives. The subsystemA is a subset of the spatial slice and is generally constructed from disconnected regions, forexample A = A1 ∪ A2 ∪ A3 in the above figure.

The region A is a region that extends in the bulk and it is uniquely defined by threeconditions:

A has the same boundary as the region A enclosing the subsystem, i.e. ∂A = ∂A

A is homologous to A, i.e. A can be smoothly deformed and take the shape of Awithout discontinuous deformations that do not preserve the topology. These twoconditions say that the regions together make up the boundary of some d-dimensionalbulk spatial slice, a spacelike surface in MΨ.

A is a region of extremal and least area1.

For general bulk geometries that satisfy the null energy condition, the three conditionsthat determine the region A are equivalent to performing the following two steps:

Step 1: Minimize the area on a spatial slice ΣMΨbounded by ΣB

Step 2: Maximize the are over all possible spatial slices ΣMΨ

For static bulk geometries or geometries with time reflection symmetry about a spatial sliceending on ΣB, step 2 can be neglected.

1This does not mean minimum area necessarily; it means that if there are more than one surfaces thatextremize the area functional, then A is the one that gives the least area.

6.1. THE RYU-TAKAYANAGI FORMULA 89

Figure 6.2: The region A involved in the Ryu-Takayanagi formula has the same boundaryas the region A of the subsystem and the two regions together form the boundary of a bulkspatial slice. In addition A is such that the area of the region becomes extremal and as leastas possible.

6.1.1 An example: The CFT vacuum state in Minkowski space-time for a ball-shaped region

As an example of the validity of the Ryu-Takayanagi formula, the case of a CFTD in a vacuumstate in R

D−1,1 and a subsystem with the geometry of a ball of radius R, A = BD−1(R), willbe considered. The gravity dual of the vacuum state configuration is known to be PAdSD+1

with metric (equation (3.13)),

ds2 =b2

z2

(dz2 + dxµdx

µ), µ = 0, . . . , D − 1 (6.2)

The ball is defined to be centered at the origin, meaning that the boundary of the ballis at the locus

∑D−1i=1 (xi)2 = R2. In order to calculate the area of the region A, the area

functional must be extremized. The area functional of a (D − 1)-dimensional surface is ingeneral given by the integral,

Area =

dD−1y

√g (6.3)

with gab = gµν∂aXµ∂bX

ν (µ, ν = 0, . . . , D) the induced metric on the surface parameterizedby the coordinates ya. The “capital coordinates” Xµ(y) are the embedding coordinates thatdescribe how the surface is embedded in the entire geometry2. In the current case, the timecoordinate is set to some constant value since a bulk spatial slice is considered and the surfacecoordinates ya are taken to be the Poincare boundary coordinates xi (i = 1, . . . , D−1) letting

2The embedding coordinates are the actual coordinates used to parameterize the spacetime expressed asfunctions of the surface coordinates.


the surface being parameterized by xi = X i(~x), z = XD(~x) ≡ Z(~x). As a result, the inducedmetric and the area functional read,

gij = gµν∂Xµ

∂xi∂Xν

∂xj=b2

z2(δij + ∂iZ ∂jZ)

⇒ Area =

dD−1~x

(b

Z

)D−1√1 + ∂i Z∂iZ

(6.4)

The determinant of the induced metric above follows from the definition of the determinantthrough the Levi-Civita tensor,

g =1

(D − 1)!εi1...iD−1εj1...jD−1 gi1j1 . . . giD−1jD−1

=1

(D − 1)!

(b2

Z2

)D−1

εi1...iD−1εj1...jD−1 (δi1j1 + ∂i1Z ∂j1Z) . . .(δiD−1jD−1

+ ∂iD−1Z ∂jD−1

Z)

=1

(D − 1)!

(b2

Z2

)D−1

εi1...iD−1

(εi1...iD−1

+D−1∑n=1

ε jni1...in−1 in+1...iD−1

∂inZ ∂jnZ

)

=1

(D − 1)!

(b2

Z2

)D−1(

(D − 1)! + (D − 2)!D−1∑n=1

δ jnin∂inZ ∂jnZ

)

=

(b2

Z2

)D−1(

1 +1

D − 1

D−1∑n=1

∂inZ ∂inZ

)

=

(b2

Z2

)D−1(

1 +1

D − 1∂iZ ∂iZ

D−1∑n=1

1

)

=

(b2

Z2

)D−1

(1 + ∂iZ ∂iZ)

(6.5)Just some remarks on the above calculation:

Line 3: The fact that the Hessian matrix of Z(~x), the matrix with elements ∂iZ ∂jZ,is a symmetric matrix, ensures that no products of higher than order 1 will be appearafter antisymmetrization. It is simple to confirm this statement for a general symmetricmatrix A,

εij...Aij = εji...Aji = −εij...Aij ⇒ εij...Aij = 0 (6.6)

Line 4: The following useful relation for the general Levi-Civita tensor in D − 1 di-mensions was used,

εi1...ikik+1...iD−1εi1...ikjk+1...jD−1 = k!εik+1...iD−1

εjk+1...jD−1 (6.7)

⇒εi1...in−1inin+1...iD−1εi1...in−1jnin+1...iD−1 =

= (−1)D−1−n(−1)D−1−nεi1...in−1in+1...iD−1inεi1...in−1in+1...iD−1jn

= (D − 2)!δ jnin

(6.8)


Also, indices of the Levi-Civita tensor are raised and lowered through the flat spacemetric, the unity, so there is really no difference between upper and lower indices apartfrom the implied sum.

Line 6: The sum∑

n ∂inZ ∂inZ also contains a sum of the dummy indices in. Sincethey are dummy, they can always be redefined via, for example, in → i and drop outof the sum.

The determinant that was just computed is actually more general that it appears. It holdsfor any sum of the unit matrix and a symmetric matrix A,

det(1 + A) = 1 + Tr(A) (6.9)

The Euler-Lagrange equations for the “Lagrangian” LArea =√g that arise from making

the above functional extremal can then be solved to find Z(~x),

∂i∂LArea

∂(∂iZ)− ∂LArea

∂Z= 0 (6.10)

⇒(D − 1 + Z∂i∂iZ

)(1 + ∂jZ ∂jZ

)− Z∂iZ ∂jZ ∂i∂jZ = 0 (6.11)

Although the solution is not obvious due to the non-linearity of the differential equation, itturns out that the hemispheres,

Z2 + ~x2 = R2 (6.12)

satisfy the differential equation and have the proper boundary (at Z = 0, the boundary ofthe ball ~x2 = R2 is recovered). The region A is, thus, the hemisphere SD−1

12

(R) and the region

bounded by A is the half ball B12D−1(R) (figure 6.3). The integral can finally be calculated

with Z(~x) as above and integrating over the half ball of radius R,

Area(A) =

B

12D−1(R)

dD−1~x

(b

Z

)D−1√1 + ∂iZ ∂Z

= bD−1R

R

0

dr rD−2

dΩ

12D−2

1

(R2 − ~x2)D/2

=1

2bD−1RΩD−2

R

0

drrD−2

(R2 − r2)D/2

=bD−1ΩD−2

2R(D − 1)

rD−1

(R2 − r2)D−2

22F1

(1

2, 1;

D + 1

2;r2

R2

) ∣∣∣∣r=Rr=0

(6.13)

with dΩ12D−2 the angular integral measure for the half-sphere obtained by simply letting the

polar coordinate run from 0 to π2

instead of 0 to π and 2F1 (a, b; c;x) the hypergeometricfunction,

2F1 (a, b; c;x) =∞∑n=0

Γ(a+ n)Γ(b+ n)Γ(c)

Γ(a)Γ(b)Γ(c+ n)

1

n!xn (6.14)


Figure 6.3: Region A = SD−112

(R) associated with the ball-shaped subsystem A = BD−1(R)

according to the Ryu-Takayanagi formula.

a function investigated so much over the years that it can be regarded as fundamental as,say, the Bessel functions.

The radial integral is bound to be divergent at the upper limit corresponding to thethe boundary of AdS (r = R ⇔ z = 0) since proper distances that reach the boundary ofAdS spacetime are infinite. For this reason a “UV” cutoff3 εUV is introduced by setting theboundary at z = εUV → 0, or, equivalently, at r =

√R2 − ε2UV → R, therefore,

Area(A) =bD−1ΩD−2

2(D − 2)

(R

εUV

)D−2

(6.15)

where the asymptotic behavior of the hypergeometric function was taken into account,

limx→1

2F1 (a, b; c;x) =Γ (c) Γ (c− a− b)Γ (c− a) Γ (c− b)

(6.16)

⇒ limr→R

2F1

(1

2, 1;

D + 1

2;r2

R2

)=

Γ(D+1

2

)Γ(D2− 1)

Γ(D2

)Γ(D−1

2

)=

D−12

Γ(D−1

2

)Γ(D2− 1)(

D2− 1)

Γ(D2− 1)

Γ(D−1

2

)=D − 1

D − 2

(6.17)

through the defining property Γ(x+ 1) = xΓ(x) of the Γ-function. Consequently, the entan-glement entropy of the ball-shaped subsystem is predicted to be,

S(A) =bD−1ΩD−2

8G(D − 2)

(R

εUV

)D−2

, D > 2 (6.18)

This result holds for D > 2. For D = 2, the ball-shaped region of the subsystem becomesan interval, A = I2R. The extremal surface remains the same of course, A = S1

12

(R), but now

3Remember that this is actually an IR cutoff due to the infinite proper distance to the boundary.


the boundary of the interval is defined by the edges positions x = −R and x = R, while theextremal area region is simply the semicircle Z2 + x2 = R2. The integral of the extremalarea is evaluated to be,

Area(A) =

R

−Rdx

b

Z

√1 +

(∂Z

∂x

)2

= bR

R

−R

dx

R2 − x2

=b

2lnR + x

R− x

∣∣∣∣x=R

x=−R

(6.19)

and with the cutoff term z = εUV → 0 as before, which now translates to x = ±√R2 − ε2UV =

±R(

1− ε2UV2R2

), the entanglement entropy in D = 2 spacetime dimensions reads,

Area(A) = 2b lnL

εUV

⇒S(A) =b

2Gln

L

εUV, D = 2

(6.20)

with L = 2R the length of the subsystem interval. Alternatively, the fact that geodesics areby definition minimum length intervals can be used to gain the same result.

The corresponding result from CFT calculations in D = 2 dimensions gives for theentanglement entropy for a line interval of length L for the vacuum,

S(A) =cCFT

3ln

L

εUV(6.21)

with cCFT the central charge, only that now the CFT UV cutoff really is εUV . Comparingthe two expressions, the cutoff dependence is precisely the same and incorporates the UV/IRconnection, while the relation becomes exact by identifying,

cCFT =3

2

b

G(6.22)

which is the correct connection between the parameters of the two sides.

6.1.2 Evidence for Ryu-Takayanagi

The last example was actually the initial observation of Ryu and Takayanagi that lead themto propose the extension to any number of spatial dimensions. AD-dimensional example thatmotivates this generalization comes when considering a general QFT in RD−1,1 at its vacuumand taking the subsystem to be the half-space x1 > 0. The domain of dependence of thesubsystem is then the Rindler wedge (figure 6.4 in Appendix H) as the Rindler coordinates


Figure 6.4: Domain of dependence DA associated with the half-space x1 > 0 at t = 0. Thestraight blue lines are lines of constant Rindler time ω. The dashed lines are trajectories ofconstant ρ, i.e. constant acceleration. It is on such trajectories Rindler observers live. Inorder to find the entanglement entropy for the full system vacuum state in the half-spacesubsystem, the half-space is decomposed into infinitesimal slices of length dρ. For each suchslice, the entanglement entropy contribution is equal to the flat space entropy of the spatialslice.

suggest,

ds2 = −dt2 + d~x2 = −ρ2dω2 + dρ2 +D−1∑i=2

(dxi)2

t = ρ sinhω

x1 = ρ coshω

(6.23)

The entanglement entropy is the entropy of the subsystem. But first, it is essential tounderstand how the entropy of the vacuum state can be computed by QFT arguments. Sincea QFT at finite temperature T = β−1 is described by a canonical ensemble, the probabilityfor the quantum system to be in a state with energy Ei is given by

pi =1

Zβe−βEi (6.24)

and the entropy is generally given by,

S = −∑i

pi log pi = −β 〈E〉 − 〈logZβ〉 (6.25)


It is not hard to see that, for the vacuum,

〈Evac〉 ∼ VD−1TD (6.26)

at thermal equilibrium, where VD−1 is the volume of a spatial slice. To see this, it is useful torecall that the vacuum energy comes with two divergences, one IR divergence compensatingfor the infinite volume VD−1 of space and one UV divergence coming from very high energies,

Evac = VD−1

dD−1~p E~p

= VD−1ΩD−2

Λ

0

dp pD−2Ep

(6.27)

where an energy cutoff Λ4 was inserted and E~p =√~p2 +m2 is the relativistic energy for

a particle with mass m. The cutoff Λ is essentially the energy contribution of the degreesof freedom with the highest energy. The main contribution in the integral comes from veryvery high momenta, i.e. very high energies. At this regime, any mass terms can be neglectedand write Ep ' p, so the vacuum energy is given by,

Evac ' VD−1ΩD−2

Λ

0

dp pD−1 =ΩD−2

DVD−1ΛD (6.28)

If the quantum system is at equilibrium, the equipartition theorem applies so,

〈Λ〉 =1

2T (6.29)

with T the temperature, therefore,

〈Evac〉 'ΩD−2

2DDVD−1T

D ∼ VD−1TD (6.30)

Consequently, the entropy of a quantum system at temperature T at leading contributionis,

Svac 'ΩD−2

2DDVD−1T

D−1 (6.31)

For the Rindler subset, however, the temperature of the vacuum as measured by a Rindlerobserver is a local function of the ρ-coordinate,

T (ρ) =1

2πρ(6.32)

so, in order to apply (6.31), the spatial slice must be divided into many infinitesimal slicesof length dρ each carrying a temperature T (ρ) and having a difference,

dT (ρ) = − 1

2πρ2dρ (6.33)

4This should actually be√

Λ2 −m2 but, since Λ m,√

Λ2 −m2 ' Λ.


with the next slice at ρ+ dρ. Furthermore, the spatial volume of such an infinitesimal slicecan be calculated through the induced metric determinant on the slice which is just theRindler metric in absence of the time components. For the spatial slice bounded by theorigin and ρ, the volume of the region is,

VD−1(ρ) =

ρ

0

dρ′RD−2

dD−2x′√g =

ρ

0

dρ′RD−2

dD−2x′ = AD−2ρ (6.34)

where AD−2 is the surface area of a ρ = constant region. The entanglement entropy canthen be extracted by integrating the infinitesimal entropy dSentang(ρ) = dSvac over all ρ > 0.Due to a divergence at ρ→ 0, a UV cutoff ε should be inserted at small distances, thus,

dSentang(ρ) ' ΩD−2

2DD

(VD−1(ρ)dTD−1(ρ) + TD−1(ρ)dVD−1(ρ)

)= −(D − 2)ΩD−2AD−2

2D(4π)D−1

dρ

ρD−1

(6.35)

⇒ Sentang =

ρ→∞

ρ=ε

dSentag(ρ)

' −(D − 2)ΩD−2AD−2

2D(4π)D−1

∞ε

dρ

ρD−1

=ΩD−2

2D (4π)D−1

AD−2

εD−2∼ AD−2

εD−2

(6.36)

and the entanglement entropy of the half-space does indeed follow an area law. At this point,the cutoff ε must be receive an interpretation. It is supposed to represent a fundamentallength beyond which spacetime cannot be defined. In quantum gravity, this minimum lengthis the Planck length so εD−2 ∼ lD−2

P = G and the Ryu-Takayanagi formula immediatelymanifests itself (up to some constant factors),

Sentag ∼AD−2

G(6.37)

The surface area AD−2 appearing above is not the surface area of any black hole as in theBekenstein-Hawking formula, but rather an area of a region having the same properties thatdefine the region A in the Ryu-Takayanagi formula. Indeed the region with surface areaAD−2 does has the same boundary as the QFT subsystem (the plane t = 0, x1 > 0), itis homologous with the subsystem and it has a minimum extremal surface area (it has aconstant surface area which is always an extremal). To be honest, the region with surfacearea Ad−2 and ρ = ε→ 0 is the Rindler horizon so, eventually, it does has some resemblancewith the Bekenstein-Hawking formula in the sense that it relates entropy and horizon area.

It is also pedagogical to understand what the divergence at ρ → 0 means. It suggeststhat entanglement entropy diverges when approaching the boundary of the subsystem. Thisquantum entanglement, of course, is a monogamic pairing of a degree of freedom of thesubsystem with a degree of freedom of the rest of the quantum system. At the boundary ofthe subsystem, the two subsystems coincide and quantum entanglement is, thus, doomed todiverge since the closer the entangled pairs are, the greater the quantum entanglement.

6.2. ENTANGLEMENT CONSTRAINTS 97

6.2 Entanglement Constraints

The holographic entanglement entropy formula offers a connection between well defined andglobal quantities of both the gravity theory defined in the bulk and the gauge theory definedat the boundary. On the gravity side, an area functional needs to be computed, whileon the gauge side, the entanglement entropy of a subsystem must be evaluated which isequivalent to finding the matrix density through a path integral argument. Even thougha general quantum theory of gravity is not yet well established, the QFT living on theboundary has a very understandable theoretical structure and, in principle, everything canbe computed. Consequently, knowing the constraints that entanglement entropy is subjectedto will automatically yield gravitational constraints for the holographic dual living in thebulk. In this section, the entanglement constraints will be extracted in order to be translatedto bulk geometry constraints in the next section.

6.2.1 First Law of Entanglement

Evidentially, entanglement entropy is just the quantum version of the classical entropy ofthe full system. Perhaps the most well known law of thermodynamics is the first law. For aclosed and isolated system not subjected to any external forces, with the number of degreesof freedom conserved of course, the law states that

TdS = dE (6.38)

with T the temperature, S the entropy and E = 〈H〉 the mean energy of the system. Thefirst law of thermodynamics reaaltes configurations near equilibrium. The generalization ofthis classical law in quantum mechanics is written in terms of precisely the entanglemententropy.

To find this quantum version of the first law of thermodynamics, a general quantumsystem at state |Ψ(λ)〉 is considered. The parameter λ serves as a measure of the deviationfrom an unperturbed state. The unperturbed state5 is defined to be |Ψ(λ = 0)〉. As aresult, a subsystem A has a λ-dependent density matrix ρA(λ) and, thus, a λ-dependententanglement entropy6,

SA(λ) = −Tr (ρA(λ) log ρA(λ)) (6.39)

The quantum law is obtained by considering small perturbations λ → λ + δλ around theunperturbed state with λ = 0. For general small perturbations,

SA(λ+ δλ) = SA(λ) +dSA(λ′)

dλ′

∣∣∣∣λ′=λ

δλ+O((δλ)2

)⇒SA(λ+ δλ)− SA(λ)

δλ≡ δSA

δλ=dSAdλ

+O((δλ)2

) (6.40)

5The unperturbed state is usually chosen to be the vacuum state. This will also be the case for thetranslation of entanglement constraints to gravitational constraints in the next section.

6In the Ryu-Takayanagi formula, the subsystem region A was involved in the argument of the entangle-ment entropy as S(A) because it was a variable. In the current consideration, the subsystem region A iskept fixed, hence the index notation SA to avoid multiple variables in the argument. However it is alwaysimplied that SA(. . . ) ≡ S(A, . . . ).


The derivative can be easily calculated in terms of the reduced density matrix,

dSA(λ)

dλ= − d

dλTr (ρA(λ) log ρA(λ))

= −Tr

(dρA(λ)

dλlog ρA(λ)

)− Tr

ρA(λ)d

dλlog ρA(λ)︸︷︷︸

=dρA(λ)

dλρ−1A (λ)

= −Tr

(log ρA(λ)

dρA(λ)

dλ

)− Tr

(dρA(λ)

dλ

)︸︷︷︸= ddλ

Tr ρA(λ)=0

= −Tr

(log ρA(λ)

dρA(λ)

dλ

)

(6.41)

Now, for first order variations of the unperturbed state, the above equation reads as,

dSA(λ)

dλ

∣∣∣∣λ=0

=d

dλ〈HA〉 (6.42)

with,HA ≡ − log ρA(λ = 0) (6.43)

the so called modular Hamiltonian7 and 〈HA〉 = Tr (ρAHA). This is precisely the quantumgeneralization of the first law of thermodynamics; it is the first law of entanglement. Indeed,the special case of a thermal unperturbed state with ρA(λ = 0) = 1

Ze−H/T , the first law of

entanglement yields,

HA = − log

(1

Ze−H/T

)= logZ +

1

TH

⇒〈HA〉 = logZ 〈1〉+1

T〈H〉

⇒T dSAdλ

=d

dλ〈H〉

(6.44)

which is precisely (6.38). Apart from containing the first law of thermodynamics as a spe-cial case, the first law of entanglement relates configurations that are not necessarily nearequilibrium as the first law of thermodynamics requires, but rather, it relates configurationsthat are nearby some unperturbed state determined by the parameter λ.

6.2.2 Relative Entropy Inequalities

Besides the importance of the first law of entanglement (6.42), it only considers first orderperturbations. For finite perturbations, it turns into the inequality,

∆ 〈HA〉 −∆SA ≥ 0 (6.45)

7More details on the properties of the modular Hamiltonian can be found in Appendix H


This finite generalization is a natural consequence of properties of another importantquantity, the (quantum) relative entropy S(ρ||σ) between a mixed state with density matrixρ and a reference state8 with density matrix σ associated with the same Hilbert space. It isdefined by,

S(ρ||σ) ≡ Tr(ρ log ρ)− Tr(ρ log σ) (6.46)

The relative entropy measures how distinguishable the two states are. Indeed, for ρ = σ,S(σ||σ) = 0, while for ρ 6= σ, it can be proven that S(ρ||σ) > 0. The proof is straightforwardby decomposing the density marices according to the general matrix spectral decomposition,

A =∑n

an |n〉〈n| (6.47)

where |n〉 are the eigenvectors of the matrix A and an the associated eigenvalues9. Applyingthis for ρ and σ reads,

ρ =∑i

pi |ψi〉〈ψi| , σ =∑i

qi |ψi〉〈ψi| (6.48)

with pi and qi the probabilities associated with the two ensembles and |ψi〉 and |ψi〉; theeigenstates of the two density matrices. As a result, the relative entropy is,

S(ρ||σ) = Tr(ρ(log ρ− log σ)

)=∑i,j

pi

(log pj Tr

(|ψi〉〈ψi|ψj〉〈ψj|

)− log qj Tr

(|ψi〉〈ψi|ψj〉〈ψj|

))

=∑i

pi

(log pi −

∑j

log qj| 〈ψi|ψj〉 |2) (6.49)

where the orthonormality 〈ψi|ψj〉 = δij of the eigenstates was used.The quantities | 〈ψi|ψj〉 |2 ≡ Pij satisfy the following properties10,

Pij = | 〈ψi|ψj〉 |2 ∈ [0, 1]∑j

Pij =∑j

〈ψi|ψj〉〈ψj|ψi〉

= 〈ψi|

(∑j

|ψj〉〈ψj|

)︸︷︷︸

=1

|ψ〉i

= 〈ψi|ψi〉 = 1 =∑i

Pij

(6.50)

8The reference state can be either pure or mixed. There are no constraints on choosing the referencestate apart from convenience.

9By the way, the matrix |n〉〈n| is just the projection matrix along the n’th “direction”.10For the sake of some terminology, a square matrix P with non-negative real components Pij ≥ 0 with

each row and column summing up to 1 is called a doubly stochastic matrix.


where the last equality comes from the symmetric property Pij = Pji (or the explicit calcu-lation...). This means that the each column (as well as row) of the matrix with elements Pijcan be taken to be a set of statistical weights. Consequently, the sum over j in the secondterm can be realized as a mean value,

−∑j

log qjPij = 〈− log q〉i (6.51)

The Jensen’s inequality can now be applied,

〈− log q〉i ≥ − log (〈q〉i) = − log

(∑j

qjPij

)(6.52)

since the function f(x) = − log x is a convex function11. Jensen’s inequality is more familiarlyseen in statistical mechanics in the well known form 〈f(A)〉 ≥ f(〈A〉) with f a convexfunction, which is also why the variance of an observable var(x) = 〈x2〉 − 〈x〉2 is alwayspositive. Moving on with the proof of monotonicity, it is more convenient to temporarilydefine,

ri ≡∑j

qjPij (6.53)

to have things more tidy so,

S(ρ||σ) ≥∑i

pi (log pi − log ri)

= −∑i

pi logripi

= 〈− logr

p〉 ≥ − log 〈r

p〉

(6.54)

with another application of the Jensen’s inequality for the convex function − log x. Thequantity at the right side of the last inequality is zero since,

〈rp〉 =

∑i

piripi

=∑i

ri

=∑i,j

qj 〈ψi|ψj〉〈ψj|ψi〉

=∑i

〈ψi|

(∑j

qj |ψj〉〈ψj|

)|ψi〉

=∑i

〈ψi|σ|ψi〉

= Trσ = 1

(6.55)

11A function f is called convex if the line passing through two given points on the curve lies above thegraph of f . If the line lies below the graph, then f is called a concave function.


Consequently, the relative entropy is subjected to the positivity condition,

S(ρ||σ) ≥ 0 (6.56)

with equality holding only for ρ = σ. The generalized first law of entanglement is due toprecisely this positivity since the relative entropy can be written as,

S(ρ||σ) = Tr (ρ log ρ)− Tr (ρ log σ) + Tr (σ log σ)− Tr (σ log σ)

=[Tr(ρ (− log σ)

)− Tr

(σ (− log σ)

)]−[(− Tr (ρ log ρ)

)− (−Tr (σ log σ))

]=[

Tr (ρHσ)− Tr (σHσ)]−[Sρ − Sσ

]=[〈Hσ〉ρ − 〈Hσ〉σ

]−[Sρ − Sσ

]= ∆ 〈Hσ〉 −∆S

(6.57)where Hσ = − log σ is the modular hamiltonian associated with the reference state, Sρ andSσ are the entanglement entropies associated with the two states and 〈. . .〉ρ(〈. . .〉σ) meansmean value of “. . . ” using the density matrix ρ (σ).

Indeed, taking the reference state to be the unperturbed state |Ψ(λ = 0)〉, i.e. σ = ρA(λ =0), Hσ = HA and Sσ = SA(λ = 0), and the general mixed state to be the finitely perturbedstate |Ψ(λ)〉, i.e. ρ = ρA(λ) and Sρ = SA(λ), the positivity of relative (entanglement) entropy(6.56) is identical to (6.45).

Apart from being positive, the relative entropy is also a monotonic function of the sizeof the regions of the subsystems, that is, for two subsets A and B with B being bigger thanand including A,

S(ρA||σA) ≤ S(ρB||σB) , A ⊆ B (6.58)

The proof of monotonicity of relative entropy is a tedious one, but just by conception it isself-explanatory; it says that access to a larger subsystem offers more ease at distinguishingbetween two states of the full system.

A further investigation of the properties of relative entropy is to consider how it behavesunder small perturbations δρ around the reference state σ. Due to the first law of entangle-ment, the first order term vanishes leaving the leading contribution coming from the second


order term,

S(σ + δρ||σ) = Tr((σ + δρ) log(σ + δρ)

)− Tr

((σ + δρ) log σ

)= Tr

(σ log(σ + δρ)

)+ Tr

(δρ log(σ + δρ)

)− Tr

(σ log σ

)− Tr

(δρ log σ

)= Tr

(σ

[log σ +

d

dλlog(σ + λδρ)

∣∣∣∣λ=0

+1

2

d2

dλ2log(σ + λδρ)

∣∣∣∣λ=0

])+ Tr

(δρ

[log σ +

d

dλlog(σ + λδρ)

∣∣∣∣λ=0

])− Tr

(σ log σ

)− Tr

(δρ log σ

)= Tr

(δρ

d

dλlog(σ + λδρ)

∣∣∣∣λ=0

)+

1

2Tr

(σd2


∣∣∣∣λ=0

)=

1

2Tr

(δρ

d

dλlog(σ + λδρ)

∣∣∣∣λ=0

)≡ 〈δρ, δρ〉σ

(6.59)

where, the relation,

0 =d

dλTr(ρd

dλlog ρ

)= Tr

(δρ

d

dλlog(σ + λδρ)

∣∣∣∣λ=0

)+ Tr

(σd2


∣∣∣∣λ=0

) (6.60)

was used. The quantity 〈δρ, δρ〉σ defined in the last line is known as (quantum) Fisherinformation and positivity of relative entropy implies its positivity as well.

6.2.3 Subadditivity

At this point more disjoint subsystems will be introduced. As a result, some useful quantitiessimilar to the entanglement entropy come into play. Eventually, of course, everything isexpressed in terms of entanglement entropies. For starters, if the quantum system containstwo disjoint quantum subsystems A and B, then a measure of both the entanglement of thetwo systems and the correlations between them is the mutual information,

I(A;B) ≡ S(A) + S(B)− S(A ∪B) (6.61)

In a more quantum sense, mutual information bounds from above correlations betweenthe two subsystems. In particular, given two operators OA and OB acting on the Hilbertspaces HA and HB of the subsystems respectively, the inequality reads,

(〈OAOB〉 − 〈OA〉〈OB〉)2

2| 〈OA〉 |2| 〈OB〉 |2≤ I(A;B) (6.62)

Mutual information is a quantity that is always positive. This positivity follows from thepositivity of relative entropy (6.56) and is known as subadditivity. To prove it, simply the


case of the combined system A ∪ B is considered and the reference state associated with adensity matrix σ appearing in the definition (6.46) of relative entropy is taken to be the stateassociated with the density matrix σA∪B = ρA ⊗ ρB with ρA and ρB the density matricesassociated with the ensembles of the subsystems A and B respectively. As a result, therelative entropy for the combined subsystem reads,

S(ρA∪B||σA∪B) = TrA∪B

(ρA∪B

(log ρA∪B − log σA∪B

))= TrA∪B

(ρA∪B

(log ρA∪B − (log ρA ⊗ 1B + 1A ⊗ log ρB)

))= TrA∪B (ρA∪B log ρA∪B)

− TrA

log ρA TrB ρA∪B︸︷︷︸=ρA

− TrB

log ρB TrA ρA∪B︸︷︷︸=ρB

= −TrA (ρA log ρA)− TrB (ρB log ρB)−

(− TrA∪B (ρA∪B log ρA∪B)

)= S(A) + S(B)− S(A ∪B)︸︷︷︸

≡I(A;B)

≥ 0

(6.63)

∴ I(A;B) ≥ 0 (6.64)

In the above calculations, the matrix property log (ρA ⊗ ρB) = log ρA⊗ 1B + 1A⊗ log ρBwas used in the second line, while, in the third line, the fact that the two subsystemsare disjoint was taken into account. This fact means that the Hilbert spaces of the twosubsystems do not intersect and, hence, TrA∪B = TrA TrB = TrB TrA. In addition, theobservation that the subsystem B is the complimentary system of A with respect to thecombined system A∪B means that the reduced density matrix of the subsystem A is simplyρA ≡ TrA ρA∪A = TrB ρA∪B and vice versa for the reduced density matrix of B.

A nod to the Ryu-Takayanagi consistency

Apart from the significant consequences of the Ryu-Takayanagi formula, it must be consistentwith at least the fundamental properties of the entanglement entropy, such as subadditivity.At each subsystem corresponds a region of minimal-area surface so the regions A, B and

A ∪B are associated with the entanglement entropy of the subsystems A, B and A ∪ Brespectively. Consequently, subadditivity gets translated to the geometrical condition,

Area(A) + Area(B) = Area(A ∪ B) ≥ Area(A ∪B) (6.65)

This condition does hold by the definition of the surfaces involved. All surfaces are minimal-

area surfaces. In addition, the surface of the region A ∪B and the surface of the unionregion A ∪ B have the same boundary ∂(A ∪ B) = ∂A ∪ ∂B. However, the surface of the

region A ∪B has by definition the least possible area with boundary ∂(A ∪ B) so it is lessor equal to the extremal surface of the region A ∪ B. In particular, the equality takes place


Figure 6.5: Subadditivity constraint as is manifested in the geometrical description of the

holographic entanglement entropy formula (6.1). The regions A, B and A ∪B are drawn assemi-circles reflecting the fact that the minimal area regions corresponding to subsystems ofthe CFT vacuum are hemispheres.

when the two subsets are so far away from each other, that the the extremal area surface ofthe union is precisely the union of the two disconnected extremal surfaces for the individual

subsets, A ∪B = A ∪ B.

6.2.4 Strong Subadditivity

If the quantum system now contains three disjoint subsystems A, B and C, then it can beshown that entanglement entropy obeys a constraint that is stronger than subadditivity;strong subadditivity,

S(A ∪B) + S(B ∪ C) ≥ S(B) + S(A ∪B ∪ C) ≥ 0 (6.66)

or, in terms of mutual information,

I(A;B ∪ C) ≥ I(A;B) (6.67)

Strong subadditivity, just like subadditivity, also follows from the relative entropy but thistime from its monotonicity. Indeed, since the combined subsystem A∪B ∪C is larger thanand includes the combined subsystem A ∪B, the monotonicity (6.58) condition says,

S(ρA∪B∪C ||σA∪B∪C) ≥ S(ρA∪B||σA∪B) (6.68)

and following the exact same procedure as in the proof of subadditivity, it follows thatS(ρA∪B∪C ||σA∪B∪C) = I(A;B ∪ C) and S(ρA∪B||σA∪B) = I(A;B) proving (6.67).

What the condition says is that the mutual information between the subsystem A andthe combined subsystem B ∪ C can only be greater or equal than the mutual informationbetween the subsystem A and B. This makes sense since mutual information between twosystems measures how much information the two systems share in terms of entanglementand correlations. It sounds self-explanatory for the system A to share more stuff with thecombined system B ∪ C than with B alone with the equality holding only when A and Care completely uncorrelated.

6.3. GRAVITY CONSTRAINTS 105

Figure 6.6: Strong subadditivity constraint as is manifested in the geometrical description

of the holographic entanglement entropy formula (6.1). The regions B, A ∪B, B ∪ C, and˜A ∪B ∪ C are drawn as semi-circles reflecting the fact that the minimal area regions corre-

sponding to subsystems of the CFT vacuum are hemispheres.

Another nod to the Ryu-Takayanagi consistency

There has been a number of researches on whether the holographic entanglement formularespects strong subadditivity. It turns out, that strong subadditivity is preserved only forgeometries satisfying certain conditions. The exact conditions are still undetermined, but ithas been shown that the null energy condition is enough to ensure consistency.

“Local” Strong Subadditivity

Quite remarkably, strong subadditivity is implied by smaller subsets. For example, lettingthe subsets A and B as they are and considering the smaller subsets C1 and C2 of C yieldsthe following strong subadditivity conditions for the triples A,B,C1 and A,B ∪C1, C2,

S(A ∪B) + S(B ∪ C1) ≥ S(B) + S(A ∪B ∪ C1)

S(A ∪B ∪ C1) + S(B ∪ C1 ∪ C2) ≥ S(B ∪ C1) + S(A ∪B ∪ C1 ∪ C2)(6.69)

Adding them together gives,

S(A ∪B) + S(B ∪ C1 ∪ C2) ≥ S(B) + S(A ∪B ∪ C1 ∪ C2) (6.70)

This says that if strong subadditivity holds for the triples A,B,C1 and A,B,C2, then italso holds for A,B,C1∪C2. As a result, breaking down the subsets A and C into infinites-imal parts Ai and Ci, the full strong subadditivity constraint for A,B,C = ∪Ai, B,∪Cican be recovered by the “local” strong subadditivity constraints for Ai, B, Ci.

6.3 Gravity Constraints

Finally, the entanglement constraints described in the last section can now be translated intogravity constraints through the holographic entanglement entropy formula (6.1). The setup


is similar to the one stated in the subsection regarding the first law of entanglement. TheCFT is taken to be in a quantum state |Ψ(λ)〉 parametrized by one parameter λ carryingperturbations around the vacuum state defined to be |0〉 = |Ψ(λ = 0)〉. The gravity dual is,thus, the geometry of the manifoldMΨ(λ) ≡M(λ) which is asymptotically AdS with λ = 0corresponding to pure vacuum AdS, the dual to the CFT vacuum state.

The metric of a general asymptotically AdSD+1 spacetime can be expressed in Fefferman-Graham coordinates as,

ds2 =b2

z2

(dz2 + Γµν(x, z)dx

µdxν)

Γµν(x, z) = ηµν +O(zD) (6.71)

As a result, the first order perturbation around pure AdSD+1 is,

ds2 =b2

z2

(dz2 + (x, z)dxµdx

µ + zDHµν(x, z)dxµdxν

)limλ→0

Hµν(x, z) = 0(6.72)

where the tensor Hµν(x, z) also depends on the parameter λ, i.e. Hµν(x, z) ≡ Hµν(x, z, λ),but the argument is omitted for simplifying the notation.

6.3.1 First Law of Entanglement

The subset A to which an entanglement entropy will be assigned is taken to be a ball ofradius R centered at the boundary position ~x0, i.e. A = BD(R, ~x0) ≡ B. For this case,the unperturbed reduced density matrix ρB is, by definition, thermal with respect to themodular Hamiltonian HB which takes the form12,

ρB =1

Ze−HB

HB =π

R

B

dD−1~x(R2 − (~x− ~x0)2

)T00(~x)

(6.73)

The difficult part turns out to be the extraction of the minimal area surface neededto apply the Ryu-Takayanagi formula for the computation of the entanglement entropy.Fortunately, there is a simple observation that makes things much easier. Firstly, the areafunctional can be thought as a functional of both the bulk metric gµν and the embeddingcoordinates Xµ, Area = Area [g,X]. As a result, its variation is of the form,

δArea =δArea

δgµνδgµν +

δArea

δXµδXµ (6.74)

The observation now is that the unperturbed surface (the hemisphere z2+(xi−xi0)2 = R2, i.e.

A = B12D(R, ~x0) ≡ B, for the case of A = BD(R, ~x0)) extremizes the area functional for the

unperturbed metric (pure AdSD+1 for the CFTD vacuum). What this means is that the first

12See Appendix H for more details


order variation of the area functional with respect to the embedding coordinates vanishessince it is an extremal. Consequently, the procedure of finding the first order perturbedextremal area functional is to keep the embedding coordinates fixed and vary with respectto the spacetime metric as it is written in the induced metric on the surface, therefore,

d

dλS(B) =

1

4G

d

dλArea(B)

=1

4G

d

dλ

B

dD−1~x√g

=1

8G

B

dD−1~x√g gij

d

dλgij

(6.75)

where the famous result for the determinant of any matrix A,

d

dλdetA = detATr

(A−1 d

dλA

)(6.76)

was used in the last line13. From the metric (6.72), the induced metric on the surface of theregion B is,

gij =b2

Z2

(δij + ∂iZ∂jZ + ZDHij (~x, Z)

)=

(0)

g ij +(1)

g ij

(6.78)

where(0)

g ij is the unperturbed metric and(1)

g ij = b2

Z2ZDHij is the perturbation. In order to

make the notation and the calculations more easy on the eye, the boundary coordinates xi

may be temporarily replaced by the shifted boundary coordinates xi − xi0 → xi. This shifthas no consequences since both the derivatives and the metric remain the same. Now, thefact that the function Z(~x) is fixed and has been found to be,

Z(~x) =√R2 − ~x2 ⇒ ∂iZ = −xi

Z(6.79)

allows the exact calculation of the inverse of the induced metric,

(0)

gij =Z2

b2

(δij − xixj

R2

)(6.80)

13This result can easily be proven from the identity,

log(detA) = Tr(logA)

⇒ d

dλlog(detA) =

d

dλTr(logA) = Tr

(d

dλlogA

)⇒ 1

detA

d

dλdetA = Tr

(A−1 d

dλA

)⇒ d

dλdetA = detATr

(A−1 d

dλA

)(6.77)


In addition, the zeroth order term in the determinant takes the form,

(0)

g =

(b2

Z2

)D−1R2

Z2(6.81)

As a result, the varied entanglement entropy of the ball-shaped subsystem becomes,

d

dλS(B) =

1

8G

B

dD−1~x

[(b

Z

)D−1R

Z

] [Z2

b2

(δij − xixj

R2

)]× d

dλ

[b2

Z2

(δij +

xixj

Z2+ ZdHij

)]=bD−1

8GR

B

dD−1~x

(R2 d

dλH i

i(~x, z)− xixj ddλHij(~x, z)

) (6.82)

and the first law of entanglement (6.42) reads,

bD−1

8πG

B

dD−1~x

(R2 d

dλH i

i(~x, z)− xixj ddλHij(~x, z)

)=

B

dd−1~x(R2 − ~x2

) d

dλ〈T00(~x)〉

(6.83)

Inifinitesimal Balls

For starters, the case of infinitesimal balls should be considered. At this limit, the radius Rof the ball becomes so small that the integrated quantities Hij(~x, z) and T00(~x) take theirconstant values at the center of the boundary ball so,

B

dD−1~x(R2 − ~x2

) d

dλ〈T00(~x)〉 =

d

dλ〈T00(~x0)〉

R

0

dr rD−2

dΩD−2

(R2 − ~x2

)=

d

dλ〈T00(~x0)〉

( R

0

dr rD−2(R2 − r2

))(dΩD−2

)=

2RD+1ΩD−2

D2 − 1

d

dλ〈T00(~x0)〉

(6.84)B

dD−1~x R2 d

dλH i

i(~x, z) =d

dλH i

i(~x0, z = 0)

R

0

dr rD−2

dΩ

12D−2 R

2

=d

dλH i

i(~x0, z = 0)R2

( R

0

dr rD−2

)(dΩ

12D−2

)︸︷︷︸

=ΩD−2

2

=RD+1ΩD−2

2(D − 1)

d

dλH i

i(~x0, z = 0)

(6.85)


B

dD−1~x xixjd

dλHij(~x, z) =

d

dλHij(~x0, z = 0)

R

0

dr rD−2

dΩ

12D−2 x

ixj

=d

dλHij(~x0, z = 0)

( R

0

dr rD−2 r2

)(dΩ

12D−2ΩiΩj

)︸︷︷︸

=ΩD−22(D−1)

δij

=Rd+1ΩD−2

2(D − 1)(D + 1)

d

dλH i

i(~x0, z = 0)

(6.86)

where dΩ12D−2 is the angular integral measure for the half ball which is just the same as the

full ball but with the polar angle θ2 running from 0 to π2

instead of 0 to π. As a result, thefirst law of entanglement for infinitesimal balls is,

D bD−1

32πG

d

dλH i

i(~x0, z = 0) =d

dλ〈T00(~x0)〉 (6.87)

At this point, the equation is only considering the special time instant t = 0, that is, thequantum observer sits still at the origin with D-velocity uµ = (1,~0) so the above equationcan also be written as,

D bD−1

32πG

d

dλuµuν(Hµν − ηµνHρ

ρ)

∣∣∣∣z=0

=d

dλuµuν 〈Tµν〉 (6.88)

with the argument ~x0 dropped for convenience14. This is a tensor equation and must holdfor all reference frames with the observer moving at any timelike D-velocity. Consequently,

D bD−1

32πG

d

dλ(Hµν − ηµνHρ

ρ)

∣∣∣∣z=0

=d

dλ〈Tµν〉 (6.89)

To make the result final, some basic properties of the energy-momentum tensor need to bereminded. For general QFTs, the energy-momentum tensor is conserved due to translationalinvariance, while for general CFTs it is also traceless due to scaling symmetry,

∂µ 〈T µν〉 = 〈T ρρ〉 = 0 (6.90)

which yield,∂µH

µν∣∣z=0

= Hρρ

∣∣z=0

= 0 (6.91)

and the final result,D bD−1

32πG

d

dλHµν

∣∣∣∣z=0

=d

dλ〈Tµν〉 (6.92)

The result suggests a direct connection between the asymptotic metric and the expecta-tion value of the CFT energy-momentum tensor. This is a standard result in AdS/CFT.

14To be more insightful, the first law of entanglement for infinitesimal balls can be expressed for any ballcentered at any point, so the constant position vector ~x0 of the center of the ball can be regarded as a generalpoint ~x in the final result. In other words, the fact that the center ~x0 can be arbitrarily chosen and give thesame results allows the generalization to any point ~x.


General Balls

Moving on to balls of finite radius, equation (6.83) can be written in terms of only the metricperturbation using the last result (6.92) for infinitesimal balls,

≡δSgravB︷︸︸︷B

dD−1~x(R2δH i

i(~x, z)− xixjδHij(~x, z))

=

≡δEgravB︷︸︸︷D

4

B

dD−1~x(R2 − ~x2

)δH i

i(~x, z = 0) (6.93)

The above constraint is a non-local constraint relating the metric perturbation in the interiorwith the asymptotic metric perturbation. To be more accurate, there is an infinite numberof such constraints, one for each pair of radii and centers (R, ~x0) of the boundary ball-shapedsubsystem. Associating each such constraint to a bulk point (~x = ~x0, z = R) at the tip ofthe extremal surface, it seems that there is one constraint for each bulk point. This infinitenumber of non-local constraints suggests a single local constraint.

This is indeed the case and the conversion of the non-local constraint to a local equationinvolves the generalized Stokes theorem as expressed in the language of differential forms.The details are of no real importance. What needs to be understood is that there exists adifferential form χB for each choice of the ball B with three basic properties,

B

χB = δEgravB (6.94)

B

χB = δSgravB (6.95)

dχB = 2ξ0B(x, z)δE00(x, z)volΣ (6.96)

where ξµB is the bulk Killing vector,

ξB = ξµB∂µ = −2π

Rt(z∂z + xi∂i

)+π

R

(R2 − z2 − t2 − ~x2

)∂t (6.97)

defined in the half-ball shaped Cauchy region Σ, between B and B that generates evolutionwithin the AdS-Rindler patch formed by the union of all the extremal surfaces associatedwith the ball shaped spatial slices within the domain of dependence DB of the boundarysubsystem (figure 6.7) and, thus, satisfies the Killing equation ∇µξBν +∇νξBµ = 0, vanishesat z2 − t2 + ~x2 = R2 and reduces to the boundary Killing vector ζµB governing evolutionwithin DB at the boundary of AdS,

ζB = ζµB∂µ = −2π

Rtxi∂i +

π

R

(R2 − t2 − ~x2

)∂t (6.98)

δE00(x, z) is the 00-component of the Einstein field equation linearized about AdS,

δE00(x, z) ∝ zd(∂2z +

D + 1

z∂zH

ii + ∂j∂

jH ii − ∂i∂jHij

)(6.99)


Figure 6.7: AdS-Rindler wedge. The wedge is formed by the union of the extremal surfaces(hemispheres) associated with spatial slices of the domain of dependence of the ball shapedregion in the boundary CFT. The boundary isometry generator ζB = ζµB∂µ generates trans-lations along the rigid blue lines, while the bulk isometry generator ξB = ξµB∂µ generatestranslations along the dotted red lines (original figure found in [21]).

and volΣ is the volume form on the region Σ,

(volΣ)µ =√gεµµ1...µd−1

dxµ1 ∧ · · · ∧ dxµd−1 (6.100)

As a result, the non-local constraints can be written as,

δEgravB = δSgravB

⇔B

χB =

B

χB

⇒∂Σ

χB = 0

(Stokes Theorem)⇒

Σ

dχB = 0

⇒

Σ

ξ0B(x, z)δE00(x, z)volΣ = 0

(6.101)

It can be shown that the integral vanishes for all possible half-ball regions Σ if and onlyif δE00(x, z) = 0. Again, this holds in the reference frame of the observer with D-velocityuµ = (1,~0) and for it to hold in any reference frame it must also hold for the other componentswith boundary indices as well,

δEµν = 0 (6.102)

In addition, the remaining components of the linearized Einstein field equations,

δEzν = δEzz = 0 (6.103)


come from the fact that they hold everywhere given that they hold at the boundary z = 0due to the gauge freedom diffeomorphisms offer. Evidentially, these components are preciselythe conditions (6.91).

The result is remarkable: Given that a CFT is holographic, then the first law of entan-glement for ball-shaped regions in the CFT is precisely equivalent to the linearized Einsteinequations in the gravity dual theory!

6.3.2 Relative Entropy Inequalities

The relative entropy inequalities to be translated are the positivity (6.56) and the mono-tonicity (6.58) conditions. Like already argued, the relative entropy offers a generalizationof the first law of entanglement by considering finite perturbations. For finite perturbations,the infinitesimally perturbed δSgravB and δEgrav

B defined above are replaced by the finitelyperturbed ∆SgravB and ∆Egrav

B such that the relative entropy is expressed as the difference∆Egrav

B − ∆SgravB . This quantity now needs to be positive and monotonic with increasingsystem size.

As usual, the infinitesimal case should be considered. At this regime, the leading termin the relative entropy is the second order term which turned out to be the quantum Fisherinformation,

〈δρA, δρA〉λ=0 =1

2

d2

dλ2S(ρA(λ = 0) + δρA||ρA(λ = 0)

)=

1

2

d2

dλ2(δEgrav

B − δSgravB )∣∣λ=0

(6.104)

On the gravity side, the quantity dual to the quantum Fisher information turns out to bea certain perturbative gravitational energy, the canonical energy associated with the Killingvector ξµB. This energy is the conserved charge associated with Rindler-time translationswithing the AdS-Rindler wedge,

Ecanonical =

Σ

ξµBTµν (volΣ)ν (6.105)

The energy-momentum tensor can in turn be broken down into a matter part Tmatterµν con-taining contributions from matter fields, collectively denoted as φ, and a gravitational partT gravµν defined from the second order linearized Einstein’s field equation; it is the expressionquadratic in the first order perturbation of the metric that acts as a source for the secondorder perturbation of the metric,

(2)

Rµν −1

2

((0)g µν

(2)

R +(2)g µν

(0)

R

)= −8πGT gravµν (

(1)g

(1)g ) (6.106)

where the oversets (... ) denote the perturbative order of the quantity. Consequently, relativeentropy inequalities for infinitesimal perturbations around the vacuum for ball-shaped regionsimply that the canonical energy for the AdS-Rindler wedges associated with all possibleboundary balls must be positive and increase with the size of the ball.


For general asymptotically AdS spacetimes, there is, in general, no isometry group and,thus, no Killing vectors making the definition of a canonical energy seemingly ambiguous. Forball-shaped boundary subsystems, however, it is always possible to find a “pseudo-Killing“vector ξµ that satisfies the Killing equation at the boundary, vanishes at the extremal surfaceB and reduces to the conformal Killing vector ζµB at the boundary. This vector ξµ is alsoa generator of a diffeomorphism, which is a real symmetry of the gravitational theory. Asa result, the notion of the canonical energy can be introduced as the conserved chargeassociated with this generator, only this time the integral is performed over a spatial sliceof the entanglement wedge. Despite the large number of choices of possible vectors ξµ, theenergy EB defined in this way turns out to be independent of ξ and it is exactly the gravitydual to the relative entropy.

6.3.3 Strong Subadditivity

Finally, the strong subadditivity condition needs to receive a dual gravitational interpreta-tion. Wall ([26]) suggests that the null energy condition is sufficient, but the minimum set ofconditions may be weaker yet still unknown, apart from the D = 2 case. For 2-dimensionalholographic CFTs, the minimal area functional is replaced by minimal length geodesics withendpoints the boundary points defining the subsystem line interval. Evidentially ([27],[28]),geometries dual to the vacuum state of Lorentz invariant field theories corresponding to RG-flows between UV and IR CFTs or non-vacuum states corresponding to small perturbationsin the vacuum need to satisfy precisely the following version of the null energy conditionaveraged over a spatial geodesic,

B

gµνkµkν ≥ 0 (6.107)

for strong subadditivity to hold on the boundary theory. The vector kµ is a null vectordefined along B with some proper normalization.


Chapter 7

Entropic Gravity

In the last chapter, it was seen how the classical laws of gravity, Einstein’s field equations, inthe bulk geometry naturally rise through entanglement on the boundary system. It seems asthe gravity could be an emergent property rising from quantities such as entropy which dealwith the microscopic degrees of freedom. Verlinde took this literally and precisely suggestedthat gravity is entropic. It is useful to note that in this and only this chapter, namely onlyin the first section, units will be restored and the theory will be presented in D = 3 + 1spacetime dimensions.

7.1 Newtonian Gravity

Verlinde’s idea ([29]) was to apply classical statistical mechanics on the holographic principlestarting by uniformly distributing N degrees of freedom on a holographic screen AH witheach degree of freedom occupying a Planck area,

N =AHl2P

=AHc

3

G~(7.1)

According to the holographic principle, these N degrees of freedom are sufficient to fullydescribe what is happening in the region bounded by the holographic screen AH .

Next, classical statistical mechanics get into the game by taking into account the equipar-tition theorem,

E =1

2kBNT ⇒ T =

2E

kBN=

2G~EAHc3kB

(7.2)

The equipartition theorem is actually a definition of the temperature T in terms of the totalenergy E and the number of degrees of freedom N of the system.

Since any force is measured through the acceleration a, a prescription to insert theacceleration needs to be identified. For uniformly accelerated observers in the vacuum,it is known that the vacuum is realized as a heat bath at temperature,

T =~a

2πckB(7.3)

115

116 CHAPTER 7. ENTROPIC GRAVITY

an effect known as the Unruh effect. This result can be immediately extracted by remem-bering that a uniformly accelerated observer is a Rindler observer who uses the Rindlercoordinates (ω, ρ) instead of the Minkowski coordinates (t = ρ sinhω, x = ρ coshω). In hisreference frame, the Rindler observer stationary at the Rindler position ρ = ρ measuresa temperature T = ~

2πρckB. According to the Minkowski observer, the Rindler observer is

uniformly accelerating with acceleration a = 1ρ

reproducing the Unruh effect.

Everything else is straight forward. The energy of the system E is just the mass E = Mc2

enclosed within the holographic screen taken to be sphere of radius r and, thus, of areaAH = 4πr2, so the force F = ma acting on a test particle with mass m lying just outsidethe holographic screen is simply,

F = ma = m2πckBT

~=

2mπckB~

2G~Mc2

4πr2c3kB= G

mM

r2(7.4)

which is just Newton’s universal law of gravity. So the “statistical” force is the gravitationalforce.

7.1.1 Why entropic?

The “statistical” force considered above can be identified as an entropic force by taking acloser look at how the test particle affects the information content of the holographic screen.It was argued by Bekenstein himself in his derivation of the black hole entropy that in orderfor the second law of thermodynamics to be valid when lowering a test particle of mass mnear the horizon, the particle should be considered as part of the black hole when it is oneCompton wavelength λC = ~

mcaway from the horizon in which case the information content

is increased by definition by one bit of information.This argument is applied in the current case by replacing the black hole with a spherically

symmetric distribution of mass M and replacing the horizon with the holographic screen ofarea AH . Then when the test particle is at distance ∆x = λC = ~

mcaway from the holographic

screen, it should change the entropy of the system by one bit, postulated to be given by1,

∆S = 2πkB = 2πkB∆x

λC(7.5)

The change entropy was written in terms of the distance ∆x above to take more generaldistance later on. For now, the above writing is motivated by the fact that the change inentropy must be proportional to the mass m since, if more particles are considered nearthe holographic screen, both the change in entropy caused by each individual particle andtheir mass are additive. An effective force should then be felt by the particle due to themicroscopic tendency to increase the entropy,

F∆x = T∆S (7.6)

with T the temperature of the system. This is precisely the entropic force,

Fentropy = T∆S

∆x(7.7)

1The factor 2π is picked to ensure that the entropic force will recover Newton’s second law of motionF = ma.

7.1. NEWTONIAN GRAVITY 117

Figure 7.1: Entropic force on a polymer immersed in a heat bath of temperature T . Oneend of the polymer molecule is fixed while thee other end is pulled by an observer with aforce F . The tendency to maximize entropy and return to a configuration with a more coiledshape induces an effective macroscopic force, the entropic force Fentropy, resisting the furtherdistortion from the equilibrium state.

An intuitive example: A polymer molecule in a heat bath

A more physical intuition regarding the effect of an entropic force is obtained by consideringthe elasticity of a polymer molecule. If the molecule is heat bathed, its observed configurationis more likely to be a random coiling. This comes from the fact that there are more con-figurations associated with a short molecule, i.e. more compact coiling, than configurationassociated with a long, stretched molecule and the tendency to maximize entropy immedi-ately implies the randomly coiled shape. It is precisely this tendency that gets translated toan effective macroscopic force, the entropic force.

This force can be realized as a resistance by pulling one end of the polymer (figure7.1). The polymer tends to return to its equilibrium state, the configuration with maximumentropy, i.e. heavily coiled, compact shape, and, hence, an elastic force appears to resist thepull. The entropic force Fentropy will then be the force F needed to keep the polymer at fixedlength x.

Since the polymer is heat bathed in a big tank of temperature T , the system is subjectedto the canonical ensemble. In the canonical ensemble the force F is introduced in thepartition function according to the external work Fx done to keep the length fixed,

Z(T, F ) =

dEdx Ω(E, x)e

−E+FxkBT (7.8)

with Ω(E, x) the number of configurations having energy E and length x. Consequently, theforce F is obtained from the saddle point equations,

1

T=∂S

∂E(7.9)


F

T=∂S

∂x(7.10)

with S = S(E, x) = kBΩ(E, x) the entropy of the configuration with energy E and lengthx. The entropic force is, thus,

Fentropy = T∂S

∂x(7.11)

7.1.2 Newtonian gravitational potential

The observation that the Newtonian gravitational force has arisen from a purely entropicforce is intriguing. But in order to properly understand what the statement “gravity isentropic” means, a one-to-one relation between the statistical quantities S (entropy) and T(temperature) and the gravitational quantities m (mass) and Φ (Newtonian gravitationalpotential) must be established.

The temperature is clearly associated with the acceleration according to the Unruh effect,so the gradient of the entropy ∆S

∆xmust be associated with the mass. Interestingly enough,

when the test particle of mass m has fully merged with the holographic screen, its informationgets encoded in a number n of bits of the same type as the those living on the screen, somc2 = 1

2nkBT , and combining the Unruh effect (7.3) and the change in entropy (7.5), the

entropy change is directly related to the change of the Newtonian gravitational potential2,

∆S

n= −kB

∆Φ

2c2(7.12)

Quite remarkably, the Newtonian gravitational potential Φ encodes the reduction of theentropy per bit. In particular, since ∆S ≥ 0 according to the 2nd thermodynamic law, thechange of the gravitational potential along one process must be negative, i.e. the force isattractive.

7.1.3 General mass distributions

Holographic screens are equipotential surfaces

Up until now, the holographic screen was taken to be a sphere surrounding a sphericallysymmetric mass distribution. To deal with general mass distributions, a prescription forchoosing holographic screens needs to be identified. The identification becomes natural byremembering that space emerges macroscopically after coarse graining.

Coarse graining is the procedure of taking a microscopic region and then blowing it upby increasing the size of this region. To make it more explicit, the initial microscopic regionis discretized to a lattice. This can be regarded as the idea that there is no notion of smallerregion than a minimum region; for spacetime this is understood as the Planck scale, i.e.spacetime is fundamentally made out of elementary lattice sites of lattice spacing a0 = lP .This elementary lattice site can be labeled as an order n = 0 part of the coarse graining

2The Newtonian gravitational potential is introduced from the acceleration which is given by the gradientof the potential, i.e. ~a = −~∇Φ in general, while for constant variations ∆x along one direction only, a = −∆Φ

∆x .

7.1. NEWTONIAN GRAVITY 119

Figure 7.2: Coarse graining is the procedure of taking a fundamental lattice cite of themicroscopic space and gradually expanding it while averaging over the smaller sites (redshaded regions). Eventually the continuous macroscopic space is recovered after infinitelyblowing the elementary lattice cell.

procedure. The next order n = 1 is obtained by doubling the lattice spacing, a1 = 2a0 andthen replacing the multiple elementary lattice sites of order n = 0 with a single lattice site oforder n = 1 by averaging the lattice variables over the (n−1 = 0)-order lattice sites. Movingfurther ahead, the general order n is obtained by doubling the lattice spacing an = 2an−1

and then replacing the multiple (n − 1)-order lattice sites with a single n-order lattice byaveraging the lattice variables over the the lattice sites of lower order n − 1. Eventually,the macroscopic spacetime is recovered by taking n→∞ (figure 7.2). This coarse grainingprocedure is known as the Kadanoff-Wilson renormalization group approach.

Now, since entropy is a measure of ignorance of the macroscopic observer regarding themicroscopic information, it governs the coarse graining procedure. Bigger surfaces thatcontain more microscopic information with less accuracy mean bigger entropy. And as seenin the last subsection, entropy is directly related to the Newtonian gravitational potentialhence, the Newtonian gravitational potential is a natural measure of coarse graining on theholographic screens. Holographic screens that enclose smaller regions correspond to lessentropy and higher potential. This allows to identify the holographic direction, the directionof increasing size of the holographic screens, with the gradient of the Newtonian gravitationalpotential −~∇Φ and the holographic screens are identified as equipotential surfaces.


Entropic derivation of the Poisson equation

This procedure of determining holographic screens can now be applied to extract the entropicforce associated with a general mass distribution. The holographic screen S is, thus, chosen tobe an equipotential surface with Newtonian gravitational potential energy Φ0. The analysisis similar with the case of the spherically symmetric distribution. The mass distribution istaken to be entirely contained inside the region bounded by the holographic screen. The onlydifference is that the temperature is a local function, rather than constant. In particular,the temperature can be analogously defined through a local Unruh effect by introducing ascalar function Φ whose dynamics are still unknown,

T =~~∇Φ · n2πkBc

(7.13)

where n is a unit vector normal to the holographic screen pointing outwards, hence thechange of sign. Since temperature is a local function, the equipartition theorem takes anintegral form,

E =1

2kB

SdN T (7.14)

where dN is the number of bits contained in an infinitesimal surface dA of the screen. Underthe same assumption of uniform bits distribution along the holographic screen, dN must begiven by,

dN =N

AdA =

c3

G~dA (7.15)

so the equipartition theorem rearranges to (with E = Mc2),

4πGM =

SdA~∇Φ · n (7.16)

which is just Gauss’ law for gravitation. Expressing the mass in terms of the mass densityρ(~r) of the distribution and applying Gauss’ theorem, the Poisson equation is immediatelyderived,

∇2Φ(~r) = 4πGρ(~r) (7.17)

This function Φ that was introduced in the definition of the temperature can thus onlybe identified with the Newtonian gravitational potential.

7.2 Einsteinian Gravity

A very similar procedure can give rise to the generally relativistic gravitational force. Forthis, a proper “Einsteinian” gravitational potential needs to be identified. This is known tobe expressed in terms of a global time like Killing vector ξµ3 as,

φ =1

2log(−ξ2

)(7.18)

3In the notation used before in the current script, this Killing vector would be more consistently denotedas ξµt with t the local time coordinate but, for the sake of notational simplicity, it is omitted here.

7.2. EINSTEINIAN GRAVITY 121

Just a remark here. It is implied that such a Killing vector exists. This is not self-explanatory in general but it holds for all static background geometries; in particular, itscontra-variant form is precisely ξµ = δ µ0 . In what follows, such a static background isassumed to be the case. As is commonly done in general relativity, the speed of light,Planck’s constant and the Boltzmann constant are set to the unit, c = ~ = kB = 1, butNewton’s gravitational constant G is left as it is.

To obtain some intuition into why the above φ is a generalization of the Newtoniangravitational potential Φ, it is reminded that the redshift factor is precisely eφ. This comesfrom the explicit form of the Killing vector which yields,

eφ =√−ξµξµ =

√−g00 (7.19)

This is indeed the redshift factor that connects the local time coordinate, the proper timeτ , with the time x0 measured by an observer at infinity (or the boundary if the spacetime iscompact),

ds2 = −g00(dx0)2 = −dτ 2 ⇒ dτ =√−g00dx

0 = eφdx0 (7.20)

The amount of redshift is directly related to the gravitational field strength; the strongerthe gravitational field, the greater the redshift. To be more accurate, it is the gravitationalpotential energy that says how big the redshift will be, hence the above connection. Anotherway to see that φ is a generalization of the Newtonian gravitation potential Φ is to take itsNewtonian limit. At this regime, g00 = −(1 + 2Φ) +O(Φ2), so,

φ =1

2log(−g00)

=1

2log(1 + 2Φ +O(Φ2))

= Φ +O(Φ2)

(7.21)

The equipotential surfaces are, thus, translated to surfaces of equal redshift. Followingthe previous discussion, the holographic screens chosen to measure the gravitational forceare precisely these surfaces. This is a natural way of choosing the holographic screens sincesurface of equal redshift are surfaces of equal time x0 or, more appropriately, surfaces ofequal rate of time dx0.

The next step is to identify a generally relativistic acceleration. This is just the 4-acceleration aµ expressed in terms of the 4-velocity uµ as4 aµ = uν∇νu

µ. It is easy to expressthese quantities using the Killing vector ξµ,

uµ ≡ dxµ

dτ= e−φ

dxµ

dx0= e−φδ µ0 = e−φξµ (7.22)

4Just a reminder. the D-velocity and D-acceleration are defined by uµ = dxµ

dτ and aµ = duµ

dτ . In a local

inertia reference frame, the acceleration can also be written as aµ = dxν

dτ∂uµ

∂xν = uν∂νuµ and since it is a

vector, the equivalence principle suggesting ∂ν → ∇ν yields the result aµ = uν∇νuµ.


aµ = e−φξν∇ν

(e−φξµ

)= e−2φξν∇νξ

µ − e−2φξµ ξν∇νφ︸︷︷︸=0

= −e−2φξν∇µξν

=1

2e−2φ∇µ

(ξ2)

= −∇µφ

(7.23)

The above calculation involves the fact that ξµ satisfies the Killing equation∇µξν+∇νξµ = 0.The zero value of the under-braced quantity comes from the definition of φ,

e2φ =− ξ2 ∇ν×=⇒ 2e2φ∇νφ = −2ξρ∇νξρ

⇒ ξν∇νφ =− e−2φξνξρ∇νξρ

=e−2φξνξρ∇ρξνν↔ρ= e−2φξρξν∇νξρ = −ξν∇νφ

⇒ ξν∇νφ =0

(7.24)

The “local” Unruh effect allows, thus, to relate the Einsteinian gravitational potentialwith the locally measured temperature,

T0 =1

2π∇µφn

µ (7.25)

where nµ is a unit vector normal to the holographic screen S pointing outwards. Thetemperature measured at infinity is related to this by a redshift factor,

T = eφT0 =1

2πeφ∇µφn

µ (7.26)

Verlinde’s postulation ∆S = 2πm~ ∆x now translates to,

∇µS = −2πmnµ (7.27)

and the entropic force,

F µentropy = T∇µS = −meφ∇µφ = F µ

gravity (7.28)

is precisely the gravitational force measured from the asymptotic observer.

7.2.1 Einstein’s Field Equations

Finally, Verlinde offers an entropic derivation of Einstein’s field equations. The proceduregoes as in the non-relativistic case. The degrees of freedom are assumed to be uniformlydistributed on the holographic screen S,

dN =dA

G(7.29)

7.3. TEST 1: SCHWARZSCHILD BLACK HOLE 123

and the equipartition theorem reads,

M =1

2

SdNT =

1

4πG

SdAµ eφ∇µφ (7.30)

where dAµ = dAnµ. This is known to be the generally relativistic generalization of Gauss’law for gravity. In particular, M is Komar’s definition of mass enclosed inside a volume Vwith boundary ∂V = S in a static curved spacetime. It is originally derived from Einstein’sfield equations but the question is whether the procedure can be done backwards, i.e. areEinstein’s field equations the only equations that give rise to the Komar mass?

The answer turns out to be yes, but it includes a little more knowledge on differentialforms and Komar’s derivation. In particular, the Komar mass can be rewritten in terms ofthe Killing field as,

M =1

8πG

Sdxµ ∧ dxνεµνρσ∇ρξσ (7.31)

After applying the generalized Stokes theorem, using the fact that ∇ν∇νξµ = −Rµνξν and

expressing the mass as a volume integral over energy-momentum tensor components, theresult is,

V

dV nµξν(Tµν −

1

2Tgµν

)= − 1

8πG

V

dV nµξνRµν (7.32)

where V is the three dimensional volume bounded by the holographic screen S = ∂V and nµ

is its normal unit vector. It can be shown that the above integral must hold for any choiceof the normal nµ and the Killing vector ξµ and Einstein’s field equations are recovered,

Rµν = −8πG

(Tµν −

1

2Tgµν

)⇔Rµν −

1

2Rgµν = −8πGTµν

(7.33)

7.3 Test 1: Schwarzschild Black Hole

Besides the impressive claims of the entropic gravity proposal, it must be put through ex-tensive tests before becoming plausible. The difficulty comes to finding systems with welldefined equations of state. The only generally known such case in general relativity is theblack hole whose entropy is given by the Bekenstein-Hawking formula, its event horizon areain Planck units (divided by 4). The insertion of a test particle of mass m would then deformthe shape of the event horizon through a computable way and the shift in the horizon areawill translate to shift in the entropy of the system.

After this discussion it is only natural to consider the simplest possible unperturbedblack hole to test the conjecture, i.e. a static uncharged black hole in asymptotically flat


spacetime; the Schwarzschild black hole,


f(r)+ r2dΩ2

D−2

f(r) = 1−(RS

r

)D−3

RS =

(16πGM

(D − 2) ΩD−2

) 1D−3

(7.34)

with RS the Schwarzschild radius and M the mass of the black hole.

7.3.1 Gravitational Force

The first thing to do is calculate the gravitational force as predicted by general relativity.This result will then put the entropic force into test.

In order to calculate the gravitational force, an observer must be defined. The observeris taken to be a static observer, i.e. a Schwarzschild observer, at some fixed distance r = r.A test particle of mass m is left to freely fall from some initial position r0 following a radialtrajectory (θi = const.). Without loss of generality, r0 can be taken to be on the positive axisxd. Expanding the geodesic equations (C.17), the equations of motion for proper velocityand acceleration in the r-direction measured by the static observer read,(

dr

dτ

)2

= RD−3S

(1

rD−3− 1

rD−30

)d2r

dτ 2= −D − 3

2RS

(RS

r

)D−2(7.35)

while the acceleration measured by a Minkowski observer living at infinity is,

d2r

dt2= −D − 3

2RS

(RS

r

)D−2

[

1−(RS

r

)D−3]− 3

(dr

dt

)2[

1−(RS

r

)D−3]−1 (7.36)

The gravitational force is then just Fgravity = m d2rdτ2 for the Schwarzschild observer. For the

calculation of the entropic force, the approximation of a large black hole with mass M mand near horizon dropping r0 = RS(1 + ε), ε 1 will be needed. At this regime, the nearhorizon geometry takes the form (Section 2.4.1),

ds2 ' −(D − 3)εdt2 +1

D − 3

dr2

ε+R2

SdΩ2D−2 (7.37)

which can be brought to a Rindler× SD−2(RS) geometry by defining the proper distance ρ


from the event horizon and a rescaled time ω,

ρ =

r

RS

dr′√f (r′)

' 2RS√D − 3

√ε

ω =D − 3

2RS

t

⇒ ds2 ' −ρ2dω2 + dρ2 +R2SdΩ2

D−2

(7.38)

Then a Schwarzschild observer at position r = r is also a Rindler observer at position,

ρ = ρ =2RS√D − 3

√r −RS

RS

(7.39)

and measures a temperature,

T = T =TBH√−g00

'√D − 3

4πRS

√RS

r −RS

' 1

2πρ(7.40)

Consequently, the equation of motion for the test mass dropped from ρ0 = 2RS√D−3

√r0−RSRS

is,d2ρ

dω2= −ρ+

2

ρ

(dρ

dω

)2

(7.41)

This can be obtained from the fact that a freely falling observer, a Minkowski observer, seesthe particle stationary at ρ0 = xd = ρ coshω so ρ(ω) = ρ0

coshωyielding the above result after

differentiating. Alternatively, the geodesic equations can be applied.From the last result, the initial5 proper acceleration measured by the static observer is

just,d2ρ

dτ 2R

= −ρ0

ρ2(7.42)

where dτR = ρdω is the proper time used by the Rindler observer stationary at ρ. Inparticular, if the test particle is dropped from the observer’s hand, i.e. ρ0 = ρ, then theinitial gravitational force is measured to be,

Fgravity = md2ρ

dτ 2R

= −mρ

(7.43)

7.3.2 Entropic Force

Now that the reference result (7.43) has been stated, the entropic force,

Fentropy = TdS

dρ(7.44)

5The fact that only the initial time instant is considered gets rid of any velocity dependent terms.


needs to be computed and check the consistency of the proposal. To do this, the back-reaction of the test particle on the background geometry has to be taken into account.Unfortunately, perturbative corrections on the geometry are only available for backgroundMinkowski spacetime. Fortunately, the near horizon geometry is precisely Minkowskian inproper coordinates. Indeed, resuming the analysis from the Rindler × SD−2(RS) geometry,the following “flat” coordinates can be defined,

tM = ρ sinhω

xd = ρ coshω(7.45)

to give an almost flat spacetime,

ds2 ' −dt2M + dx2d +R2

SdΩ2D−2 (7.46)

There is not a direct way to define the rest xi, i = 1, . . . , d− 1 flat coordinates from thesphere but an interesting observation allows to approximate the spherical surface as a plane.The observation is that the Kretchmann invariant RµνκλR

µνκλ scales with a negative powerof the mass M of the black hole at the horizon6,

RµνκλRµνκλ

∣∣RS

=(D − 1)(D − 2)2(D − 3)

R4S

(RS

r

)2(D−1) ∣∣∣∣r=RS

=(D − 1)(D − 2)2(D − 3)

R4S

(7.47)

This means that if the black hole is extremely large, then RµνκλRµνκλ

∣∣RS' 0 and the horizon

is approximately flat allowing to approximate its spherical shape with a plane employed withflat coordinates xi, i = 1, . . . , d− 1 so,

ds2 ' −dt2M +d∑i=1

dx2i (7.48)

Well this is precisely a metric that can be perturbed by the insertion of a small testparticle with mass m M . As described in Appendix J, the first order in m correction isgiven by,

ds2 ' −(1 + 2Φ)dt2M +

(1− 2

D − 3Φ

) d∑i=1

dx2i (7.49)

with Φ the Newtonian gravitational potential of the test particle (placed at initial positionxd = ρ0 = ρ),

Φ = − 8πGm

(D − 2) ΩD−2

((xd − ρ)2 +

d−1∑i=1

x2i

)−D−32

(7.50)

6The calculation of the Kretchmann invariant was done via the MATHEMATICA software and wasproven by induction. For more details, everything is written in Appendix L.


The Newtonian gravitational potential in D = d+ 1 dimensions is reviewed in Appendix I.Going back to Rindler coordinates at initial Rindler times ω ' 0, xd ' ρ, the perturbed

metric reads,

ds2 ' − (1 + 2Φ) ρ2dω2 +

(1− 2

D − 3Φ

)dρ2 +R2

S (1− 2Φ) dΩ2D−2 (7.51)

From this metric, the area of the event horizon (at ρ = 0 and constant Rindler time; g isthe determinant of the induced metric gij = gij|ρ=0, i, j = 2, ..., D − 1),

ABH =

dθ1dθ2...dθD−2

√g

'dθ1dθ2...dθD−2 (sin θ1)D−3 (sin θ2)D−2 ... sin θD−3

[R2S

(1− 2

D − 3Φ

)]D−22

' ΩD−2RD−2S − D − 2

D − 3RD−2S

SD−2(RS)

dΩD−2Φ

(7.52)

has shifted by,δABH = ABH − ΩD−2R

D−2S

' −D − 2

D − 3RD−2S

SD−2(RS)

ΦdΩD−2

= −D − 2

D − 3RD−2S ΦCΩD−2

=8πGm

D − 3RD−2S (RS + ρ)−(D−3)

(7.53)

where the mean value theorem for integrals over a sphere SD−2(RS) of radius RS and centerC = (xd = −RS, Xi = 0) that does not coincide with ρ = 0 was applied (equation (K.1) inAppendix K). Therefore, the change in entropy is,

δS =δABH

4G' 2πm

D − 3RD−2S (RS + ρ)−(D−3) (7.54)

and the entropic force can be calculated to be,

Fentropy = TdS

dρ

= Td(δS)

dρ

' − 1

2πρRD−2S

2πm

D − 3(D − 3) (RS + ρ)−(D−2)

' −mρ

(7.55)

which is precisely the gravitational force (7.43) calculated before. The entropic gravityconjecture has passed the first test!


7.4 Test 2: General Spherically Symmetric Mass Dis-

tribution

The second test consists of a test particle moving in the gravitational field of a generalspherically symmetric star with mass M and radius R0 > RS. The geometry outside thedistribution is of course the same as that of a Schwarzschild black hole. For this analysis,it is more convenient to work in isotropic coordinates (t, R(r), θi) related to the standardSchwarzschild coordinates by the transformation7,

r = R

[1 +

1

4

(R

RS

)D−3] 2D−3

(7.58)

⇒ ds2 = −

1− 14

(RRS

)D−3

1 + 14

(RRS

)D−3

2

dt2 +

[1 +

1

4

(R

RS

)D−3] 4D−3

(dR2 +R2dΩ2D−2) (7.59)

7.4.1 Gravitational Force

In order to proceed with calculations, the approximation of being very far away from theSchwarzschild radius, R RS, will be considered. According to this, the geometry lookslike,

ds2 ' −

[1−

(RS

R

)D−3]dt2 +

[1 +

1

D − 3

(RS

R

)D−3]

(dR2 +R2dΩ2D−2) (7.60)

or, in Cartesian coordinates (t, ~x) with ~x2 = R2 and recognizing that(RSR

)D−3= −2Φ(R)

with Φ(R) the Newtonian gravitational potential induced by the mass distribution,

ds2 ' −(1 + 2Φ)dt2 +

(1− 2

D − 3Φ

)(dR2 +R2dΩ2

D−2)

= −(1 + 2Φ)dt2 +

(1− 2

D − 3Φ

)d~x2

(7.61)

At this regime, the gravitational force takes the Newtonian expression,

Fgravity(R) = −mg(R) (7.62)

7The isotropic coordinates are defined so that the spatial part of the metric looks like a function timesthe flat space metric,

ds2 = −A(x)dt2 +B(x)d~x2 = −A(x(R))dt2 + B(R,Ω)(dR2 + dΩ2D−2) (7.56)

Applying this for the Schwarzschild solution implies that the isotropic coordinate R(r) must satisfy thefollowing differential equation,

r√f(r)R′(r) = R (7.57)

whose integration immediately yields the expression stated.

7.4. TEST 2: GENERAL SPHERICALLY SYMMETRIC MASS DISTRIBUTION 129

with g(R) the magnitude of the acceleration due to gravity felt by an observer at radialposition R,

g(R) =dΦ(R)

dR=

8πGM(D − 3)

(D − 2)ΩD−2

1

RD−2(7.63)

7.4.2 Entropic Force

For the entropic force, an expression for temperature measured by an observer stationaryat some position R = R0 needs to be found. To do this, a small patch around the observeris considered. The observer’s position can be taken to lay on the positive xd axes so theposition vector of the observer is ~R0 = (0, . . . , 0, R0). The small patch is then understood asthe region with |xd−R0|, |xi| R0. Defining the new coordinates Xd = xd−R0 and Xi = xi,i = 1, . . . , d − 1 measuring displacement from the position of the observer, the geometryaround the observer becomes approximately flat. This can be seen by the expansion of theNewtonian gravitational potential around the location of the observer,

Φ(~x) = −1

2

(RS

|~x|

)D−3

= −1

2RD−3S

((xd −R0 +R0)2 +

d−1∑i=1

x2i

)−D−32

= −1

2

(RS

R0

)D−3(

1 + 2Xd

R0

+d∑i=1

(Xi

R0

)2)−D−3

2

= −1

2

(RS

R0

)D−3(1− (D − 3)

Xd

R0

)+O

((X

R0

)2)

= Φ(R0) + g(R0)Xd +O

((X

R0

)2)

(7.64)

with g(R0) = Φ′(R0) the acceleration due to gravity at the location of the observer,

g(R0) =(D − 3)8πGM

(D − 2)ΩD−2

1

RD−20

(7.65)

Taking the equipotential surface Φ(R0) as the reference point for measuring energies, themetric in the new coordinates becomes,

ds2 ' − (1 + 2g(R0)Xd) dt2 + d ~X2 (7.66)

This patch is approximately flat because the Kretchmann invariant there reads,

RµνκλRµνκλ

∣∣∣∣R0

=(D − 1)(D − 2)2(D − 3)

R4S

(RS

R0

)2(D−1)

(7.67)


so for D > 1, it approaches zero.To extract the temperature, it is useful to recall the Rindler metric,

ds2 = −ρ2dω2 + dρ2 +d−1∑i=1

dx2i (7.68)

The static Rindler observer at ρ = ρ is seen by the Minkowski observer to have uniformacceleration a = ρ−1 along the xd-direction. A change of coordinates Xd = ρ − ρ andXi = xi, i = 1, . . . , d− 1 for small Xd brings the Rindler metric in the form,

ds2 ' −ρ2

(1 +

2

ρXd

)dω2 + d ~X2

= −(

1 +2

ρXd

)dτR + d ~X2

(7.69)

which is just the metric of the original case with the acceleration due to gravity identifiedwith the acceleration of the Rindler observer,

g(R0) =1

ρ(7.70)

The temperature measured by the observer sitting still at R0 RS is, thus,

T (R0) =1

2πρ=g(R0)

2π(7.71)

This is just the local Unruh effect originally used by Verlinde along with the postulationthat 1 bit of information is added in the holographic screen, the spherical surface of radiusR0 intersecting the observer’s position, when a test particle of mass m approaches the screenat distance equal to its Compton wavelength λC = m−1,

dS

dX0

= −2πm (7.72)

to prove that,

mg(R0) = T (R0)dS

dX0

(7.73)

with X0 the position of the test particle with respect to the observer.It is interesting to see, though, under what assumption this holds in the current case

of the spherically symmetric star. The entropy of such distribution cannot be calculatedvia the Bekenstein-Hawking formula because the event horizon seen by the observer is at adistance,

ρ =1

g(R0)=

RS

2(D − 3)

(R0

RS

)D−2

(7.74)

which extends far away from the patch surrounding the observer approximated as flat.

7.4. TEST 2: GENERAL SPHERICALLY SYMMETRIC MASS DISTRIBUTION 131

Beside this difficulty, the insertion of a test particle of m at position xd = x0 wouldback-react on the metric (7.61) up to first order in its mass according to,

ds2 ' −(1 + 2Φ′)dt2 +

(1− 2

D − 3Φ′)d~x2

= −(1 + 2Φ′)dt2 +

(1− 2

D − 3Φ′)

(dR2 +R2dΩ2D−2)

(7.75)

with Φ′ = Φ+φ the total gravitational potential caused by the mass distribution contributingthe Φ part and the test particle contributing the φ part,

Φ(~x) = − 8πGM

(D − 2)ΩD−2

(d∑i=1

x2i

)−D−32

φ(~x) = − 8πGM

(D − 2)ΩD−2

(d−1∑i=1

x2i + (xd − x0)2

)−D−32

(7.76)

The shift in the holographic screen area at R = R0 can then be calculated to be (primedquantities are the quantities after the insertion of the test particle),

δAH = A′H − AH

= RD−20

SD−2(R0)

dΩD−2

((1−−D − 2

D − 3Φ′)− (1−−D − 2

D − 3Φ)

)= −D − 2

D − 3RD−2

0

SD−2(R0)

dΩD−2φ

= −D − 2

D − 3RD−2

0 φCΩD−2

=8πGm

D − 3RD−2

0 x−(D−3)0

=8πGm

D − 3RD−2

0 (R0 +X0)−(D−3)

(7.77)

where the mean theorem for integrals over a sphere SD−2(R0) of radius R0 and center C =(xd = 0, xi = 0) was applied. The derivative of this shift with respect to the position of thetest particle for positions very close to the holographic screen,

d(δAH)

dX0

= −8πGm (7.78)

is then related to the shift in entropy according to the Bekenstein-Hawking formula,

d(δS)

dX0

=1

4G

d(δAH)

dX0

(7.79)

In other words, the postulation that dSdX0

= −2πm and the equipartition theorem can begenerally combined only if the entropy of the distribution is maximum.


7.4.3 Subtest 2: A statistical toy model

The last test showed that for the entropic force associated with a general spherically sym-metric mass distribution to be the gravitational force, the change in entropy encoded in theholographic screen must be related to the change in its area through the maximum entropyformula (7.79). It is only natural, thus, to ask whether a general spherically symmetric starcan admit such ignorance as well as have the equipartition theorem M = 1

2NT in power

with N the number of degrees of freedom. Taking the temperature to be as extracted above,the equipartition theorem says that,

M =1

2Ng(R0)

2π= 2GMN

D − 3

D − 2

1

ΩD−2RD−20

⇒ N = 2D − 2

D − 3

AH4G∼ AH

4G

(7.80)

in agreement with the holographic principle.The first law of thermodynamics dM = TdS now implies that,

dM =2M

NdS ⇒ N

Mmax

M

dM ′

M ′ = S − Smax (7.81)

⇒ S = Smax −N

2lnMmax

M=AH4G

(1− D − 2

D − 3lnMmax

M

)(7.82)

with Mmax the mass of the distribution corresponding to the maximum entropy Smax = AH4G

.For M > Mmax the distribution extends outside the holographic screen and a new largerholographic screen needs to be chosen; Mmax is the mass of a black hole of event horizonradius R0,

Mmax =(D − 2)ΩD−2R

D−30

16πG=

(D − 2)(ΩD−2A

D−3H

) 1D−2

16πG(7.83)

The above equation suggests that for thermal equilibrium to be established, and, hence,for the equipartition theorem to be applicable, the mass of the distribution cannot but bean appreciable fraction of the maximum mass,

M ≥Mmaxe−D−3D−2 (7.84)

The insertion of a test mass near the holographic screen then disturbs its area by δAHwhich also translates to a variation of the maximum mass δMmax. These two are related by,

δMmax =(D − 2)Ω

1D−2

D−2

16πG

D − 3

D − 2A

D−3D−2−1

H δAH

⇒ δMmax

Mmax

=D − 3

D − 2

δAHAH

(7.85)

and the associated change in entropy can be read to be,

δS =δAH4G

(1− D − 2

D − 3lnMmax

M

)− D − 2

D − 3

AH4G

δMmax

Mmax

= −D − 2

D − 3

δAH4G

lnMmax

M6= δAH

4G

(7.86)

7.5. CONCLUSIONS/COMMENTS 133

Consequently, the change in entropy cannot be given by δAH4G

unless M = Mmax8 in which

case the mass distribution has already collapsed into a black hole or M = MmaxeD−3D−2 > Mmax

in which case the mass distribution extends beyond the holographic screen and the analysisdoes not make sense.

7.5 Conclusions/Comments

In the light of the last result, for gravity to be entropic under Verlinde’s assumptions, thesystem is either never in a thermal equilibrium or the postulation dS

dX0= −2πm is invalid.

Despite the fact that a general spherically symmetric distribution seems to disprove Ver-linde’s original assumptions, it is interesting to ask why the first test involving a Schwarzschildblack hole is compatible with the notion of entropic gravity. In fact, even the general chargedand rotating black hole turns out to satisfy the proposal. In remembrance of the true natureof black holes (Chapter 2), things could not be otherwise. After all, black holes are identifiedas purely thermodynamical systems and gravitational processes cannot but be entropic.

To summarize up, if Verlinde’s assumption that the equipartition theorem holds and thepostulation that one bit of information is added to the system when the probe mass is oneCompton wavelength away from the holographic screen are left unperturbed, then gravityis entropic only when it is supposed to be entropic, i.e. when dealing with black holes atthe semi-classical level. If gravity is really entropic for any distribution, then either theequipartition theorem fails to apply or the postulation regarding the entropy change mustbe modified, otherwise gravity is simply not entropic.

8In that case, the above analysis simplifies to

S = Smax =AH4G⇒ δS =

δAH4G

(7.87)

or, equivalently, the same analysis can be done by also varying M = Mmax to give an extra term cancelingthe logarithm-depended term.


Part III

APPENDICES

135

Appendix A

Gauge Field Theory

This appendix is dedicated to the notion of gauge invariance in field theories. It is mainlyconstructed through a combination of the author’s personal notes and a hint of his own under-standing on the subject. If the reader finds this insufficient, a well written and pedagogicalscript is [30]. The worrying reader who seeks the fundamental aspects of the theoreticalbackground can take a look into the more demanding, but also more precise, books [31] and[32], while a complete script on elements of Group Theory applied in Physics is [33].

A.1 General Formalism

The idea of the field is merely the extension of the finite numbered degrees of freedomqn,i (t) (generalized coordinates)1 to the infinite numbered degrees of freedom φ (~x, t). Thisis equivalent to identifying,

n, i→ xi , q → φ⇒ qn,i (t)→ φ (~x, t) (A.1)

A.1.1 Nother’s Theorem

Nother’s theorem is a powerful statement. It suggests that for every continuous symmetryα of the action, that is any transformation with infinitesimal form,

φ→ φ′ = φ+ ε δαφ (A.2)

of its component fields that leave it invariant, there exists a conserved charge Qa. Here,this statement will be proven in its exact form and the conserved charges associated withthe most common continuous symmetry transformations, such as Poincare and global gaugetransformations, will be derived. The symbol α is supposed to represent the symmetrytransformation itself; it’s not really a mathematical object, it’s just a way of giving a nameto the transformations. The constants ε are some infinitesimal parameters that characterize

1n = 1, . . . , N counts the number ofN particles, and i = 1, . . . , d counts the number of spatial componentsof the vector ~qn. There are, thus, Nd degrees of freedom in such a configuration.

137

138 APPENDIX A. GAUGE FIELD THEORY

the symmetry transformations. For example, if the symmetry is rotation around some axes,then ε would be the infinitesimal angle of rotation.

The procedure of proving the theorem is similar to deriving the Euler-Lagrange equations,since the statement δαS = 0, which means the action is invariant under the symmetrytransformation, is similar to Hamilton’s principle of stationary action but not to be confusedwith it; these are two separate statements. This infinitesimal statement δαS = 0 is valid dueto the continuity of the symmetry. Before directly considering this variance of the action, afirst question to be answered is whether the Lagrangian density is unique. In other words,does there exist a different Lagrangian density L that gives the same action as L ? Theanswer is yes. It is easy to prove that,

L = L + ∂µAµ (φ) (A.3)

does the trick. Indeed the action just changes by some surface terms and since the fieldφ vanishes at the boundary, so do the functions Aµ (φ) and the action remains the same.Consequently, the statement δαS = 0 is equivalent to,

ε δαL = ε ∂µAµα (A.4)

The nature of the transformation is what determines the finite factors Aµα.Now, momentarily forgetting what a symmetry transformation means, a general trans-

formation ε δαφ which is not necessarily a symmetry changes the Lagrangian density by,

ε δαL =∂L

∂φε δαφ+

∂L

∂ (∂µφ)ε δα (∂µφ)

= ε

(∂µ

(∂L

∂ (∂µφ)

)δαφ+

∂L

∂ (∂µφ)∂µ (δαφ)

)= ε ∂µ

(∂L

∂ (∂µφ)δαφ

) (A.5)

where the Euler-Lagrange equation of motion (??) and the fact that the change of thederivative is the same as the derivative of the change were used to go to the second line.Remembering that ε δαφ is a symmetry means that (A.26) should match to (A.25) whichproves Nother’s theorem,

∂µ

(Aµα −

∂L

∂ (∂µφ)δαφ

)= ∂µj

µα = 0 (A.6)

with jµα the conserved current. Putted explicitly, the quantity,

Qα =

dd~x j0 =

dd~x

(A0α −

∂L

∂ (∂tφ)δαφ

)(A.7)

is not only globally conserved, i.e. Qα = 0, but also locally conserved as the existence of acontinuity equation implies. For more details regarding the classical Nother’s theorem, [30]should be insightful.

A.1. GENERAL FORMALISM 139

Translation Symmetry

As a first example, the symmetry under spacetime displacements xµ → xµ+aµ is considered.Infinitesimally, for δaµx

µ = δaµ ≡ εµ small, this changes the scalar field2 φ (~x, t) ≡ φ (xµ) as,

ενδaνφ = φ (xµ + δaxµ)− φ (xµ) = ∂νφ δa

ν (A.8)

Since the Lagrangian density is also a scalar,

εµδaµL = ∂µL δaµ

⇒ ενAµa ν = L δµν δaν (A.9)

so there are d+ 1 conserved currents, one for each coordinate translation,

jµa ν ≡ T µν = L δµν −∂L

∂ (∂µφ)∂νφ (A.10)

which is, by definition, the energy-momentum tensor. In terms of its components, theHamiltonian density is,

H ≡P0 = T 00 = η00T 00 = L − π∂tφ (A.11)

and the momentum density is,

P i = T 0i = ηijT 0j = −π∂iφ (A.12)

so the total energy (the Hamiltonian) and the total momentum are,

H =

dd~xH =

dd~x (L − π∂tφ) (A.13)

P i =

dd~xP i = −

dd~x π∂iφ (A.14)

The energy-momentum tensor defined in (A.31) has an important subtlety. Although itis clearly a symmetric tensor for the case of scalar fields, the more general case of arbitraryfields Φ, which transform as Φ (xµ)→ Φ′ (x′µ) = Φ (xµ) + aµδaµF with F (Φ (xµ)) = Φ′ (x′µ)the function relating the new field Φ′ evaluated at the transformed coordinates x′µ to theold field Φ evaluated at the old coordinates xµ,

T µν = L δµν −∂L

∂ (∂µΦ)∂νΦ +

∂L

∂ (∂µΦ)δaνF (A.15)

does not ensure the symmetric property. Fortunately, there is a method borrowed fromgeneral relativity that ensures this property. The idea is to forget that the spacetime is flat

2Only the case of scalar fields is considered here. Of course, not all fields are scalar. Non-scalar fieldsmay have additional terms in their variation under the action of a transformation. In general, under atransformation the generic field Φ (xµ) changes as Φ (xµ)→ Φ′ (x′µ) ≡ F (Φ (xµ)) which means that δαΦ =∂µΦδαx

µ − δαF


and include the invariant measure in the integral of the action for a general metric tensorgµν ,

S =

dDx√−g L (A.16)

with g ≡ det (gµν) the determinant of the metric tensor. The energy-momentum tensor isthen extracted by variating the action with the metric itself,

T µν = − 2√−g

δS

δgµν(A.17)

and remembering the actual form of the metric after performing the variation. This is validbecause translations are coordinate transformations that yield a variation in the generalmetric as well.

Lorentz Symmetry

For the second example, the symmetry under Lorentz transformations (boosts and rotations)will be considered. An infinitesimal Lorentz transformation is of the form,

Λµν = δµν + δωµν (A.18)

with δωµν an antisymmetric infinitesimal parameter which will eventually drop out, so δΛxµ =

δωµνxν . Following the same procedure as before, it easy to confirm that the conserved

currents, expressed with only contra-variant indices, are,

Mρµν = xνT ρµ − xµT ρν (A.19)

with Mρµν = −Mρνµ. As a result the d(d+1)2

associated angular momentum densities are,

M µν = M0µν = xνT 0µ − xµT 0ν (A.20)

which yield the d boosts generators (center of mass),

Ki =

dd~xM 0i =

dd~x

(xiH − tP i

)(A.21)

and the d(d−1)2

rotations generators (angular momenta),

J i =1

2

dd~x εijkM

jk =

dd~x εijkx

jPk (A.22)

A.2 Symmetries and Conservation Laws

Perhaps the most remarkable recognition of the Lagrangian formalism is its capability totranslate the meaning of a symmetry, which becomes mathematically transparent in theframework of Group Theory, into observables, the conserved charges. Especially nowadays,when symmetries have come to define theories rather than the other way around, it mustbecome self-explanatory how to construct a Lagrangian with specific symmetries.

A.2. SYMMETRIES AND CONSERVATION LAWS 141

A.2.1 Nother’s Theorem

Nother’s theorem is a powerful statement. It suggests that for every continuous symmetryα of the action, that is any transformation with infinitesimal form,

φ→ φ′ = φ+ ε δαφ (A.23)

of its component fields that leave it invariant, there exists a conserved charge Qa. Here,this statement will be proven in its exact form and the conserved charges associated withthe most common continuous symmetry transformations, such as Poincare and global gaugetransformations, will be derived. The symbol α is supposed to represent the symmetrytransformation itself; it’s not really a mathematical object, it’s just a way of giving a nameto the transformations. The constants ε are some infinitesimal parameters that characterizethe symmetry transformations. For example, if the symmetry is rotation around some axes,then ε would be the infinitesimal angle of rotation.

The procedure of proving the theorem is similar to deriving the Euler-Lagrange equations,since the statement δαS = 0, which means the action is invariant under the symmetrytransformation, is similar to Hamilton’s principle of stationary action but not to be confusedwith it; these are two separate statements. This infinitesimal statement δαS = 0 is valid dueto the continuity of the symmetry. Before directly considering this variance of the action, afirst question to be answered is whether the Lagrangian density is unique. In other words,does there exist a different Lagrangian density L that gives the same action as L ? Theanswer is yes. It is easy to prove that,

L = L + ∂µAµ (φ) (A.24)

does the trick. Indeed the action just changes by some surface terms and since the fieldφ vanishes at the boundary, so do the functions Aµ (φ) and the action remains the same.Consequently, the statement δαS = 0 is equivalent to,

ε δαL = ε ∂µAµα (A.25)

The nature of the transformation is what determines the finite factors Aµα.Now, momentarily forgetting what a symmetry transformation means, a general trans-

formation ε δαφ which is not necessarily a symmetry changes the Lagrangian density by,

ε δαL =∂L

∂φε δαφ+

∂L

∂ (∂µφ)ε δα (∂µφ)

= ε

(∂µ

(∂L

∂ (∂µφ)

)δαφ+

∂L

∂ (∂µφ)∂µ (δαφ)

)= ε ∂µ

(∂L

∂ (∂µφ)δαφ

) (A.26)

where the Euler-Lagrange equation of motion (??) and the fact that the change of thederivative is the same as the derivative of the change were used to go to the second line.


Remembering that ε δαφ is a symmetry means that (A.26) should match to (A.25) whichproves Nother’s theorem,

∂µ

(Aµα −

∂L

∂ (∂µφ)δαφ

)= ∂µj

µα = 0 (A.27)

with jµα the conserved current. Putted explicitly, the quantity,

Qα =

dd~x j0 =

dd~x

(A0α −

∂L

∂ (∂tφ)δαφ

)(A.28)

is not only globally conserved, i.e. Qα = 0, but also locally conserved as the existence of acontinuity equation implies. For more details regarding the classical Nother’s theorem, [30]should be insightful.

Translation Symmetry

As a first example, the symmetry under spacetime displacements xµ → xµ+aµ is considered.Infinitesimally, for δaµx

µ = δaµ ≡ εµ small, this changes the scalar field3 φ (~x, t) ≡ φ (xµ) as,

ενδaνφ = φ (xµ + δaxµ)− φ (xµ) = ∂νφ δa

ν (A.29)

Since the Lagrangian density is also a scalar,

εµδaµL = ∂µL δaµ

⇒ ενAµa ν = L δµν δaν (A.30)

so there are d+ 1 conserved currents, one for each coordinate translation,

jµa ν ≡ T µν = L δµν −∂L

∂ (∂µφ)∂νφ (A.31)

which is, by definition, the energy-momentum tensor. In terms of its components, theHamiltonian density is,

H ≡P0 = T 00 = η00T 00 = L − π∂tφ (A.32)

and the momentum density is,

P i = T 0i = ηijT 0j = −π∂iφ (A.33)

so the total energy (the Hamiltonian) and the total momentum are,

H =

dd~xH =

dd~x (L − π∂tφ) (A.34)

3Only the case of scalar fields is considered here. Of course, not all fields are scalar. Non-scalar fieldsmay have additional terms in their variation under the action of a transformation. In general, under atransformation the generic field Φ (xµ) changes as Φ (xµ)→ Φ′ (x′µ) ≡ F (Φ (xµ)) which means that δαΦ =∂µΦδαx

µ − δαF

A.2. SYMMETRIES AND CONSERVATION LAWS 143

P i =

dd~xP i = −

dd~x π∂iφ (A.35)

The energy-momentum tensor defined in (A.31) has an important subtlety. Although itis clearly a symmetric tensor for the case of scalar fields, the more general case of arbitraryfields Φ, which transform as Φ (xµ)→ Φ′ (x′µ) = Φ (xµ) + aµδaµF with F (Φ (xµ)) = Φ′ (x′µ)the function relating the new field Φ′ evaluated at the transformed coordinates x′µ to theold field Φ evaluated at the old coordinates xµ,

T µν = L δµν −∂L

∂ (∂µΦ)∂νΦ +

∂L

∂ (∂µΦ)δaνF (A.36)

does not ensure the symmetric property. Fortunately, there is a method borrowed fromgeneral relativity that ensures this property. The idea is to forget that the spacetime is flatand include the invariant measure in the integral of the action for a general metric tensorgµν ,

S =

dDx√−g L (A.37)

with g ≡ det (gµν) the determinant of the metric tensor. The energy-momentum tensor isthen extracted by variating the action with the metric itself,

T µν = − 2√−g

δS

δgµν(A.38)

and remembering the actual form of the metric after performing the variation. This is validbecause translations are coordinate transformations that yield a variation in the generalmetric as well.

Lorentz Symmetry

For the second example, the symmetry under Lorentz transformations (boosts and rotations)will be considered. An infinitesimal Lorentz transformation is of the form,

Λµν = δµν + δωµν (A.39)

with δωµν an antisymmetric infinitesimal parameter which will eventually drop out, so δΛxµ =

δωµνxν . Following the same procedure as before, it easy to confirm that the conserved

currents, expressed with only contra-variant indices, are,

Mρµν = xνT ρµ − xµT ρν (A.40)

with Mρµν = −Mρνµ. As a result the d(d+1)2

associated angular momentum densities are,

M µν = M0µν = xνT 0µ − xµT 0ν (A.41)

which yield the d boosts generators (center of mass),

Ki =

dd~xM 0i =

dd~x

(xiH − tP i

)(A.42)

and the d(d−1)2

rotations generators (angular momenta),

J i =1

2

dd~x εijkM

jk =

dd~x εijkx

jPk (A.43)


A.3 Gauge field theories

One important aspect of physical theories is the gauge field, a field that determines physicalquantities up to some gauge transformation that leaves observables invariant. An example of

a gauge field is the familiar “electromagnetic potential Aµ =(φ, ~A

)” which determines the

observables “electric field strength ~E” and “magnetic field strength ~B”. These observablesare invariant under the gauge transformation A′µ = Aµ + ∂µΛ with Λ = Λ (~x, t) a general,good behaving, function of spacetime.

A.3.1 Abelian Gauge Theory - Maxwell Theory

The last example regarding the electromagnetic potential gives rise to pure gauge theoriesthat are invariant under U (1) gauge transformations. The U (1) group has a single gen-erator which reflects the fact that the group is abelian, i.e. its group elements commutewith each other. It makes its presence in Quantum Electrodynamics (QED) that describeselectromagnetic interactions. For a U (1) gauge theory, the gauge field Aµ has the gaugetransformation,

AµU(1)−−→ Aµ + ∂µΛ (A.44)

where Λ appears in the exponent of group elements

U (~x, t) = e−ieΛ(~x,t) ∈ U (1) (A.45)

with −e the electron charge. When dealing with gauge fields, the action of the free theoryis built from the field strength tensor F µν . For abelian gauge theories, this is,

F µν = ∂µAν − ∂νAµ ≡ ∂[µAν] (A.46)

or, written in terms of the electromagnetic field,

Ei = F 0i , Bi = − 1

2√d− 2

εijkFjk (A.47)

where the notation “[µ1µ2 . . . ]” translates to “fully antisymmetrize with respect to indicesµ1, µ2, ...”. In the language of differential forms, this is written as F ≡ dA and translates as“the differential form F of rank p+ 1 is the exterior derivative of the differential form A ofrank p”. The field strength tensor is defined as above to ensure that a gauge transformationacts on it as,

F µν U(1)−−→ UF µνU−1 = F µν (A.48)

By the definition of the field strength tensor, it obeys the Bianchi identities,

∂[ρFµν] = ∂ρFµν + ∂νFρµ + ∂µFνρ = 0 (A.49)

which yield two of the four Maxwell equations, the Gauss law for magnetism and the Maxwell-Faraday equation,

~∇ · ~B = 0 ,∂ ~B

∂t+ ~∇× ~E = 0 (A.50)

A.3. GAUGE FIELD THEORIES 145

The other two Maxwell equations arise from the free Lagrangian density of a pure U(1)gauge field,

LMaxwell = −1

4FµνF

µν (A.51)

whose Euler-Lagrange equations of motion are,

∂µFµν = 0 (A.52)

or, in terms of the electromagnetic fields,

~∇ · ~E = 0 , ~∇× ~B − ∂ ~E

∂t= 0 (A.53)

The Lagrangian density LMaxwell is formed by the Lorentz invariant inner product of thefield strength tensor with itself to manifest special relativity.

Conjugate Momenta For LMaxwell, the conjugate momenta are,

πµ =∂LMaxwell

∂ (∂tAµ)= F0µ (A.54)

The fact that π0 = 0 is of great importance as it gets rid of negative norm states in quantumfield theory.

Hamiltonian The Hamiltonian of the free abelian gauge field is,

H = −dd~x

(−1

4FµνF

µν − πµ∂tAµ)

=1

2

dd~x

(E2 +B2

)(A.55)

where in the second equality, the case of the electromagnetism was taken into account toshow that it is indeed the familiar formula for the electromagnetic energy.

A.3.2 Non-Abelian Gauge Theory - Yang Mills Theory

When someone starts considering non-abelian gauge fields, things get more complicated.Under the action of a general gauge group transformation, the gauge field Aµ = AµaT

a (withT a the generators of the algebra and Aµa the true, physical gauge fields) transform as,

AµGauge Group−−−−−−−→ UAµU−1 − i

g∂µUU−1 (A.56)

where the notion of a coupling constant g is straightforwardly introduced for the first time inthis textbook. The above transformation is actually not necessary for a pure gauge theory.Its necessity comes from coupling the matter fields with the gauge fields or by foreseeing thatthe gauge fields are bosons. To be more precise, the gauge field transforms in the adjointrepresentation of the gauge group. This means that the gauge field Aµ is a matrix withthe row and column indices identified with the gauge group indices, while the generators T a


have components (T a)bc = −ifabc with fabc the structure constants of the algebra. This willbe expanded in the next section.

The field strength tensor of such a theory is,

F µν = ∂µAν − ∂νAµ + ig[Aµ, Aν ] (A.57)

or, in terms of components (F µν = F µνa T a),

F µνa = ∂µAνa − ∂νAµa − gf bc

a AµbAνc (A.58)

This definition ensures again a nice gauge transformation,

F µν Gauge Group−−−−−−−→ UF µνU−1 (A.59)

As a nod to the mathematics of differential forms, this would be written as F = dA+A∧Awith ∧ the wedge product. For more details regarding the language of differential forms, [13]contains a well stated description.

Due to this property, a Lagrangian density that is invariant under gauge transformationscan be constructed by tracing the field strength tensor with respect to the gauge groupindices, Tr (F µν) = F µν

a Tr (T a), thanks to the cyclic propetry of the trace. If the gauge groupis a semi simple Lie group, which is always the case in physical theories, then some GroupTheory analysis proves that the generators are traceless meaning that a gauge invariantLagrangian density must contain at minimum order the term Tr (F µνF ρσ). Higher orderterms would describe interactions, but this is a free field theory, so they are not consideredhere. To additionally ensure Lorentz invariance, the spacetime indices µ, ν are contracted,so, taking into account the normalization of the group generators Tr

(T aT b

)= 1

2δab, a free

gauge and Lorentz invariant Lagrangian density is,

Lgauge = −1

2Tr (FµνF

µν)

= −1

4F aµνF

µνa

(A.60)

The Euler-Lagrange equations of motion are, thus,

∂µFµνc = −gfabcF µν

a Abµ (A.61)

A more useful notation that will prove natural is the gauge covariant derivative (Dµ)ab thatcarries both Lorentz and gauge indices,

Dµ ≡ ∂µ1 + igAµ (A.62)

or, in terms of components,(Dµ)ab ≡ ∂µδ

ab − gfabcAcµ (A.63)

according to which, (A.61) is more familiarly written as,

DµFµν = 0 (A.64)

A.3. GAUGE FIELD THEORIES 147

The gauge covariant derivative Dµ is essentially the generalization of the spacetime deriva-tive ∂µ to a more appropriate derivative with a “tensor-nature” in the context of gaugetransformations, such that things look more like the abelian case. Indeed, the field strengthtensor may now be written as,

Fµν = DµAν −DνAµ (A.65)

This is the same as Einstein’s equivalence principle which says that to go from SpecialRelativity to General Relativity, replace ∂µ → ∇µ and ηµν → gµν . It is a “gauge equivalenceprinciple” which says that to go from an abelian gauge theory to a non-abelian gauge theory,replace ∂µ → Dµ. For example, this prescription predicts that the Bianchi identities (A.49)should generalize to,

D[ρFµν] = DρFµν +DνFρµ +DµFνρ = 0 (A.66)

which is indeed the case. This gauge covariant derivative is then natural, because it trans-forms as,

DµGauge Group−−−−−−−→ UDµU

−1 (A.67)

It is nice to also realize that the analysis of the previous paragraphs is completely generaland unbiased, while it reproduces the LMaxwell along with the correct properties of the gaugefield for Gauge Group = U (1).

The most common example of a non-abelian gauge group is SU (N). Then, the puregauge theory is called Yang-Mills theory. It is the second group one needs to familiarize within order to construct the Standard Model, particularly SU (2) and SU (3). A Yang-Millstheory is, therefore, summarized by the following equations,

LYM = −1

2Tr (FµνF

µν)

= −1

4F aµνF

µνa

F µν = ∂µAν − ∂νAµ + ig[Aµ, Aν ]

AµSU(N)−−−−→ UAµU † − i

g∂µUU †

F µν SU(N)−−−−→ UF µνU †

U †U = UU † = 1 , detU = 1

(A.68)

Conjugate Momenta Having established a theory, the rest are straightforward. Theconjugate momenta of the gauge fields Aµa for LYM are,

πaµ =∂LYM

∂tAµa

= F a0µ (A.69)


Hamiltonian The Hamiltonian of the free non-abelian gauge fields is,

H = −dd~x

(−1

4F aµνF

µνa − πaµ∂tAµa

)=

dd~xTr

([Dt

~A+ ~DA0]2

+1

d− 2

[~D × ~A

]2)

=1

2

dd~x

([(Dt)

ab~Aa +

(~D)abA0a

]2

+1

d− 2

[(~D)ab× ~Aa

]2) (A.70)

where in the second line, the square power means sum over Lorentz indices, while in thethird line, the square power means sum over Lorentz indices and the gauge index b. Thefirst and second square brackets contain something similar to the electric field strength andthe magnetic field strength respectively, in agreement with the “gauge equivalence principle”.Once again, the Hamiltonian is positive definite which will yield the bosonic nature of thegauge fields.

Lastly, a good observation is the absence of any mass term in the Lagrangian density; eventhough the Lagrangian density contains products of the gauge fields in the form of fabcA

bµA

cν ,

the antisymmetrization of the structure constants do not allow products of the same gaugefield of the form AaµA

µa . The interpretaion of this observation is that the gauge fields are

massless. If a mass term was to be present, then gauge invariance would automatically beviolated.

A.4 More on Gauge Theories

Gauge theories are so fundamental in the world of particle physics, that they deserve abetter interpretation for the sake of intuition and understanding. Particularly, since theaction is invariant under the action of a gauge group, gauge transformations are symmetrytransformations. With Nother’s theorem in hand, this would naively mean that there shouldexist some conserved charge associated with the gauge symmetry, so where is it? Beyondthis, only the cases of pure gauge theories were introduced. These would describe a universein which no fermions exist. The next step would be, thus, to see how one introduces thematter fields in the gauge theory (or how one introduces the gauge fields in the mattertheory) and how the two kind of fields couple to each other.

A.4.1 Global Vs Local Gauge Symmetry

The thing to pay attention to is that the elements U (~x, t) of the gauge group are localquantities for a gauge theory, leading to the distinguishment between global gauge symmetry(also known as gauge symmetry of the first kind) and local gauge symmetry (also known asgauge symmetry of the second kind). The first one is a real symmetry with its associatedconserved currents while the second one is a redundant symmetry; it does not come with anyconservation laws and it needs to be fixed via gauge fixing (determine the function Λ (~x, t)for the case of electromagnetism). Examples of gauge fixings in electromagnetism are the

A.4. MORE ON GAUGE THEORIES 149

Coulomb gauge condition ~∇ · ~A′ = 0 and the Lorentz gauge condition ∂µA′µ = 0 which yield

∇2Λ = −~∇ · ~A and Λ = −∂µAµ respectively. This redundant symmetry is what allows theexistence of negative norm states. Eventually, these drop out after gauge fixing and do notlive in the physical Hilbert space. Of course, any gauge theory that is invariant under localgauge transformations of a gauge group is also invariant under global gauge transformationsof the same group, thus they do come with the associated conserved currents. These giverise to conservation of the number of particles involved in a process.

A.4.2 Coupling of gauge fields to sources

It is time to get more realistic by firstly considering the case of sources, that is, apart fromwhatever would happen in the vacuum, there exist some sources that produce or consumegauge bosons. This could be a bulb producing photons, or a venta black material consumingthem or, perhaps, some kind of particle that has the ability to absorb the boson and thenmaybe emit another one. In any case, this source is described by an external current jµa ,where the gauge index is carefully kept explicit. As a result, an additional term in the actionneeds to be introduced,

Lgauge−source = Lgauge − 2 Tr (jµAµ)

= −1

4F aµνF

µνa − jµaAaµ

(A.71)

which yields the equations of motion,

DµFµν = jν (A.72)

or, in terms of components,

∂µFµνa = jνa + igf bcaF

µνb Acµ (A.73)

These do reproduce the Maxwell equations with sources for the case of U (1) gauge theory.

A.4.3 Minimal Coupling

The last discussion regarding coupling to sources was merely an introduction rather thanreal physics. For what could that source be? The answer is other fields! For example thephoton couples to electrically charged particles; this is the statement that electrically chargedparticles feel electromagnetic forces. So, for the case of electromagnetism, the sources areparticles with non-zero electric charge. An analogous statement is issued for the other inter-actions. Gluons (gauge fields of the strong nuclear interactions) couple to colored particles,the quarks. In fact, the non-abelian nature of the SU (3) color gauge group gets translatedto “gluons also have color” (particularly, they are double-colored) meaning that the gluonsthemselves may act as sources by interacting with each other. Indeed the terms in (A.68)that do not contain derivatives of the gluon fields, describe precisely this procedure of “self-coupling”. This “self-coupling” property is not shared by the photon since it does not havean electric charge. Another example is the Higgs particle, the interaction with which is


responsible for the existence of mass, while itself has a non-zero mass allowing it to coupleto itself.

In all its beauty, this fundamental coupling is made precisely by the “gauge equivalenceprinciple” prescription. Eventually, what is left is to prove that the gauge fields of any gaugetheory must transform according the adjoint representation of the gauge group (equation(A.56)) which automatically makes the prescription meaningful in the sence that the gaugecovariant derivative transforms as in (A.67) keeping everything nice and tide. There is onlyone very natural requirement: the matter fields transform according to the fundamentalrepresentation of the gauge group under gauge transformation,

ψGauge Group−−−−−−−→ Uψ (A.74)

with ψ representing any kind of matter field (leptons and quarks). The above requirementsays that a gauge transformation shifts the matter fields by an overall phase factor (this istrue for the cases of unitary group which evidentally are the only ones invloved in gaugetheories). For the case of electromagnetism, this phase factor is just the Aharanov-Bohmphase.

The normal derivative, however, does not transform in such a manner for a local gaugesymmetry due to the position dependency of the group elements U = U (~x, t),

∂µψGauge Group−−−−−−−→ ∂µ (Uψ) = ∂µUψ + U∂µψ 6= U∂µψ (A.75)

This is where the gauge covariant derivative Dµ gets involved. It is defined so that,

DµψGauge Group−−−−−−−→ UDµψ (A.76)

and, because of (A.74), it means that Dµ transforms according to the adjoint representationof the gauge group,

DµGauge Group−−−−−−−→ UDµU

−1 (A.77)

It is known from classical electromagnetism that the coupling of matter is done via theminimal coupling prescription,

pµ → pµ + qAµ (A.78)

where q is the electric charge of the particle. Translated in a quantum mechanical laguagewith the classical momentum replaced with its operator pµ = −i∂µ, the prescription reads(with the indices lowered),

∂µ → ∂µ + iqAµ ≡ Dµ (A.79)

which is precisely (A.62) with the electric charge q identified as the coupling constant ofthe theory. With the past as a guide, it make sence to consider a similar gauge covariantderivative for the case of non-abelian gauge groups,

Dµ = ∂µ + igAµ (A.80)

A direct substitution in the definition (A.76) yields the transformation law for the gaugefield itself,

AµGauge Group−−−−−−−→ UAµU−1 − i

g∂µUU−1 (A.81)

A.4. MORE ON GAUGE THEORIES 151

Therefore, the coupling of matter fields ψ to gauge fields Aµ is achived by the general minimalcoupling prescription,

∂µ → Dµ = ∂µ + igAµ (A.82)

given that the matter fields and gauge fields transform according to the fundamental andadjoint representations of the gauge group respectively.

For example, a gauge theory that contains a charged real scalar field is described by theLagrangian density,

L = Lgauge + LKG

= −1

2Tr (FµνF

µν)− 1

2DµφDµφ+

1

2m2φ2

(A.83)

while a gauge theory that contains a charged fermion is,

L = Lgauge + LDirac

= −1

2Tr (FµνF

µν)− ψ(i /D −m

)ψ

(A.84)


Appendix B

Quantum Field Theory

After the familiarization with classical field theories, especially gauge theories, in AppendixA, the passage to quantum field theories is smooth and straightforward. In this appendix, itwill become clear how to deal with quantum field theories in terms of quantizing the classicaltheory and being able to do meaningful calculations in the context of Feynman diagrams. Avery good and famous introductory book on QFT is [34]. [31] and [32] are more fundamentalon the subject but they are less trivial.

B.1 General Prescription

The term “quantization” immediately implies a quantum mechanical approach to a physicalproblem, meaning that starting from a classical theory some kind of quantum rules areintroduced to come up with a quantum theory. There are two different main approachesthat give the same result: canonical quantization and path integral quantization. The firstone offers more intuition regarding what is really happening, mainly because it stronglydepends on the theory itself, while the second one is more systematic and manifests thequantum nature of the theory more effectively.

B.1.1 Canonical Quantization

The first and most fundamental prescription for quantizing a classical theory is the canonicalquantization which only involves two simple steps:

Step 1: Promote everything to operators,

φ (t, ~x)→ φ (t, ~x) , π (t, ~x)→ π (t, ~x) (B.1)

Step 2: Apply equal-time canonical commutation/anti-commutation relations,[φ (t, ~x) , π (t, ~y)

]±

= iδd (~x− ~y) (B.2)

with [A,B]− ≡ [A,B] and [A,B]+ ≡ A,B.

153

154 APPENDIX B. QUANTUM FIELD THEORY

Some important explanations need to be made now. When saying “promote everything tooperators”, what is really meant is “promote the degree of freedoms (the fields) and theirconjugate momenta to operators”. To appropriately do this, one firstly needs to solve theclassical equations of motion and then promote the parameters involved in the solution tooperators. These parameters are simply the integration constants that are determined bythe initial and boundary conditions. It is most common to express the classical solutions asseries of Fourier modes, i.e. superpositions of free wave solutions, and then, using step 2,it turns out that the Fourier coefficients become raising and lowering operators, collectivelycalled ladder operators. In the end, canonical quantization is synonymous to “learning howto climb”. In completion, the choice between commutation or anti-commutation relations instep 2 is a matter of whether the degrees of freedom are bosonic or fermionic respectively.Only one will work for each type fields, unless the field does not contain any informationregarding the spin, like the Schrodinger field.

Eventually, what need to be calculated to extract predictions, are vacuum expectationvalues (VEVs) of time-ordered field products,

G(n) (x1, . . . , xn) ≡ 〈φ (x1) . . . φ (xn)〉

= 〈0|Tφ (x1) . . . φ (xn)

|0〉

(B.3)

where the variables x represent the spacetime position xµ. These are Green’s funtions involv-ing n fields being created at the corresponding spacetime coordinates, hence the notation;they are more commonly referred to as n-point functions. Particularly, the 2-point functionis precisely the propagator describing a particle propagating from xµ1 to xµ2 . From here on,the notation of collectively writing the spacetime position xµ as x when within the argumentof a function will be adopted. Space and time will be explicitly reported only when it isessential to differentiate them as in the step 2 above.

B.1.2 Path Integral Quantization

The second and most useful prescription for quantizing a classical theory is the path integralquantization. This involves the notion of Feynman’s Path Integral which is an alternativeformulation of quantum mechanics. For example, in one space dimension for simplicity,the probability amplitude K (xi, ti;xf , tf ) = 〈xf , tf |xi, ti〉 (also known as propagator in thelanguage of quantum mechanics, even though it is a Green function to be mathematicallyprecise) for a particle to propagate from an initial position xi at time ti to a final positionxf at time tf ,

K (xi, ti;xf , tf ) =

x(tf)=xf

x(ti)=xi

Dx eiS[x] (B.4)

In the above symbolic notation, x(tf)=xf

x(ti)=xiDx means “integrate over all possible paths

x (t) with boundary conditions x (ti) = xi, x (tf ) = xf”. Each such path is weighted by

the exponent of the action S [x] = tftidt L [x, x] for the corresponding path. As a result,

B.1. GENERAL PRESCRIPTION 155

expectation values are given by the path integral,

〈x (t1) . . . x (tn)〉 = 〈xf , tf |x (t1) . . . x (tn) |xi, ti〉

=

x(tf)=xf

x(ti)=xi

Dx x (t1) . . . x (tn) eiS[x](B.5)

The path integral formulation is properly reviewed in Appendix ??.

Similar formulation holds in a QFT. The Green’s functions (B.3) are written as,

G(n) (x1, . . . , xn) = NDφ φ (x1) . . . φ (xn) eiS[φ] (B.6)

with N a normalization constant, namely,

N−1 = 〈1〉 =

Dφ eiS[φ] (B.7)

There is an immediate way to generate these Green’s functions from one single generatingfunctional. The idea is to consider a source function J (x) and the generating functional,

Z [J ] =

Dφ exp

iS [φ] + i

dd+1x J (x)φ (x)

(B.8)

from which the normalization constant is related to Z[0] according to,

Z[0] = N−1 (B.9)

Then, the n-point functions are directly obtained by n functional derivatives of the generatingfunctional,

G(n) (x1, . . . , xn) =i−nδnZ [J ]

δJ (x1) . . . δJ (xn)

∣∣∣∣J=0

(B.10)

A second generating functional defined by,

W [J ] = −i lnZ [J ] (B.11)

generates another type of Green’s function,

G(n) (x1, . . . , xn) =i1−nδnW [J ]

δJ (x1) . . . δJ (xn)

∣∣∣∣J=0

(B.12)

This second type of Green’s functions turns out to be more appropriate for calculations dueto their correspondence to the so called connected Feynman diagrams.


B.2 Symmetries and Conservation Laws

The quantum world is peculiar and classical intuition is not applicable for most of the cases.One such case is the subject of symmetries. For a classical field theory, Nother’s theoremis an exact argument. However, in a quantum field theory things tend to fluctuate aroundclassical results and the quantum version of Nother’s theorem needs to be extracted seper-ately. Fortunately, the path integral formulation offers a simple way to do this. In additionto Nother’s theorem, an entire new kind of “conservation laws” arises in QFT, the Ward-Takahashi identities. These take into account the operator nature of the degrees of freedomand constrain the operator products with the currents according to their transformationlaws.

B.2.1 Nother’s Theorem

The starting point is the partition function,

Z =

Dφ eiS[φ] (B.13)

which is nothing more that mean value of unity in the vaccum.From this, it becomes clear that for a transformation φ→ φ′ to be a symmetry transfor-

mation, which means at least leave the partition function unchanged, two things need to beinvariant:

The action functional S [φ′] = S [φ]

The path integral measure Dφ′ = Dφ

so a symmetry of the classical field theory (the first condition in the list) is not necessarily asymmetry of the quantum field theory unless the path integral measure is also invariant. Tobe more accurate, it is the product Dφ eiS[φ] that needs to be invariant, but such subtletiesare taken into account through the following proof.

Under a general infinitesimal transformation, the field φ varies according to,

φ (x)→ φ′ (x) = φ (x) + ε δαφ (x) (B.14)

If this is a symmetry transformation, then the change of variables,

φ′ (x) = φ (x) + ε (x) δαφ (x) (B.15)

is not a symmetry and as a result neither the action nor the path integral measure areinvariant. However, the total change of the path integral has to be proportional to firstderivatives of the function ε (x) to leading order in ε to recover the fact that (B.14) is asymmetry,

Z →Dφ′ eiS[φ′] =

Dφ eiS[φ]

(1 +

dd+1x jµα∂µε

)(B.16)

B.2. SYMMETRIES AND CONSERVATION LAWS 157

𝑂1(𝑥1)

×

𝑂2(𝑥2)

×

𝑂3(𝑥3)

×

𝑂4(𝑥4)

×

𝜖 ≠ 0

𝑂5(𝑥5)

×

Figure B.1: Quantum version of Nother’s theorem. The operators Oi(xi) are inserted in theregion within which ε = 0.

The current jµα may have contributions from both the variations of the action and the pathintegral measure.

Now, since Z is eventually an actual integral over φ, (B.15) is simply a redefinition ofthe dummy integration variable φ leaving the actual value of Z unchanged, therefore,

Dφ eiS[φ]

dd+1x jµα∂µε =

Dφ eiS[φ]

dd+1x ∂µj

µαε = 0 (B.17)

and for this to hold for any ε, the current jµα needs to obey the quantum version of Nother’stheorem,

Dφ eiS[φ]∂µjµα = 〈∂µjµα〉 = 0 (B.18)

which says that the VEV of the conserved current vanishes. This is precisely Ehrenfest’stheorem: mean values of quantum theory follow the classical equations of motion.

The concept of a symmetry, however, is stronger than simply demanding invariance ofthe partition function. It also suggests invariance of the correlation functions,

〈O1 (x1) . . .On (xn)〉 =1

Z

Dφ eiS[φ]O1 (x1) . . .On (xn) (B.19)

which involves insertions of local operators Oi (xi) at spacetime point xµi . The case of n-point functions fall in this category but (B.19) is more general since it takes into accountthe potentially different transformation rules for the local operators. The operators Oi are,of course, constructed from the field φ, but they may transform in a different way than φitself,

Oi → O′i = Oi + ε δαOi (B.20)

The first step is to again promote ε → ε (x). Choosing ε to be non-zero in a region thatdoes not contain any of the spacetime points on which the local operators live as in figure.... (meaning that δαOi = 0), the above procedure gives,

〈∂µjµα . . .〉 = 0 (B.21)


𝑂1(𝑥1)

×

𝑂2(𝑥2)

×

𝑂3(𝑥3)

×

𝑂4(𝑥4)

×

𝜖 ≠ 0

𝑂5(𝑥5)

×

Figure B.2: Ward-Takahashi Idenitities. The operators Oi(xi) are inserted in the regionwithin which ε = 0 but with one operator O1(x1) inserted within the region with ε 6= 0.

This result suggests the operator equation,

∂µjµα = 0 (B.22)

Such statements are meant to be realized as being present in time-ordered correlation func-tions so the above “suggestion” is really just a notation to keep from writing the “. . . ”insertions over and over again.

B.2.2 Ward-Takahashi Identities

The last discussion regarding local operators insertions lead to the operator equation (B.22)for the case of non-zero ε in a region with no insertions. If ε is non-zero in a region thatcoincides with the an operator insertion, say O1 (x1), as in figure ..., then the correlationfunction in the new variables φ′ is,

1

Z

Dφ eiS[φ]

(1 +

dd+1x jµα∂µε

)(O1 + ε δαO1) . . .On (xn) (B.23)

Invariance of the correlation function yields the Ward-Takahashi identitity,

ε

dd+1x 〈∂µjµαO1 (x1) . . .〉 = 〈δαO1 (x1) . . .〉 (B.24)

or, in the operator equation form,

ε

dd+1x ∂µjµαO1 (x1) = δαO1 (x1) (B.25)

The integratalεdd+1x is performed over the region where ε does not vanish.

The Ward-Takahashi identity associated with a symmetry transformation puts con-straints on operator products of the Nother current with other general local operators interms of the symmetry transformation laws of the local operators themselves and vice versa.

B.2. SYMMETRIES AND CONSERVATION LAWS 159

B.2.3 QFT realizations of symmetries

In order to properly close the section of how symmetries are realized in a QFT, one needsto distinguish between three cases of symmetries that may appear:

Symmetry a la Wigner

Symmetry a la Nambu-Glodstone

Anomalous Symmetry

Each one of these symmetries is realized in the action functional, i.e. it is a symmetryof the classical theory, but may or may not be a symmetry of the quantum theory. Inthis subsection, all of them will be reviewed in very basic lines without paying too muchattention.

Symmetry a la Wigner

This kind of symmetry is also known as exact symmetry and is what people usually havein mind when talking about symmetries, although in physics great care should be taken toavoid misconceptions and ambiguities. It is also a symmetry in the quantum sense meaningthat the path integral measure is also invariant. Exact symmetries transformations are thosethat leave the vacuum |0〉 of the theory invariant,

Qα |0〉 = 0 (B.26)

with Qα the operator of the conserved charged associated with the symmetry. As a result,particles form degenerate multiplets, referred to as symmetry multiplets, with each multipletcontaining particles of the same mass.

Symmetry a la Nambu-Glodstone

A symmetry a la Nambu-Glodstone is more commonly known as spontaneously broken sym-metry. As in the case of exact symmetries, spontaneously broken symmetries are also sym-metries of the quantum theory. The only difference is that such symmetry transformationsdo no leave the vacuum of the theory invariant,

Qα |0〉 6= 0 (B.27)

leading to no symmetry multiplets. A very famous theorem was found by Goldstone in 1961([35]). The Goldstone theorem states that for each charge Qα of the spontaneously brokesymmetry, there exists a massless scalar particle, the Goldstone boson.

A symmetry may be partially spontaneously broken in which cases the theory has bothexact and spontaneously broken symmetry. If the symmetry group is G, then the symmetrymay be spontaneously broken to a subrgroup H ∈ G. This subgroup H is an exact symmetrygroup, while the remaining symmetry group, the coset G/H, is a spontaneously brokensymmetry group.


Anomalous Symmetry

As the name may suggest, this is not a realized symmetry of the QFT. The path integralmeasure is not invariant. Even if the integration measure is invariant, such not realized sym-metries may rise from specifying some boundary conditions in an attempt of regularization.Even though the anomaly of the symmetry may become manifested by such attempts ofregularization, it turns out that it is independent of the particular choice of the regulariza-tion.

B.3 Quantum gauge theories are haunted

After becoming familiar with the prescription that gives rise to QFTs for the simple cases, itis time to see how gauge fields should be treated quantum mechanically. The first problemone comes against to is in the path integral. As already mentioned, the path integral is doneover all possible configurations of the degrees of freedoms, the fields, for all kinds of fields.For a gauge theory with gauge fields Aµa and some other non-gauge fields φ, the correspondingpath integral would be,

Z =

DADφ eiSgauge[A,φ] (B.28)

The path integral measure DA is now meant to be also over all components Aµa , i.e. DA ≡∏dµ=0

∏Ngena=1 DAµa with Ngen the number of generators of the gauge group. And the difficulty

comes from the fact that integration must be made over all physically distinct configurationsnot related by gauge transformations. But the gauge fields are specified up to some non-physical gauge transformations. Subsequently, the above path integral is not valid becauseit takes into account all possible gauge fields including the infinite choices of the gaugeconditions. For the path integral to be meaningful, a prior gauge fixing must take place toseparate physically distinct gauge trajectories from a whole bunch of gauge orbits (figureB.3). There is a general way to do this through path integrals, the Faddeev-Popov method.

B.3.1 The Faddeev-Popov Method

The idea is simple and perhaps naive. Since the path integral (B.28) involves all possiblegauge orbits, then it sounds effective to leave the integral as it is and simply divide out thosenon-physically distinct gauge orbits through the volume V ol of the gauge action on fieldspace,

Z =1

V ol

DADφ eiS[A,φ] (B.29)

Some explanatory comments should be made here. The gauge fields live in a spaceA =

Aaµ (x)

, the “gauge potential space” (this is figure B.3).

In this space, there are multiple gauge orbits which describe the path a gauge field takesafter a gauge transformation (the hard red lines in the figure). Gauge fixing is realized astaking a slice in this gauge potential space (the dashed green line in the figure).

The next step is to split the integral over the gauge fields into two pieces: one for thephysically distinct gauge configurations and one for the gauge transformations. Such change

B.3. QUANTUM GAUGE THEORIES ARE HAUNTED 161

GAUGE ORBITS GAUGE

FIXING

GAUGE POTENTIAL SPACE

Figure B.3: Gauge Potential Space. The red arrows are possible gauge orbits, while the bluedashed arrow is a particular gauge fixing. The gauge fixing is taken to be “good” meaningthat it intersects every gauge orbit only once, i.e. there is only one gauge transformationthat takes a gauge orbits to lie on the gauge fixed trajectory.

of coordinates is expected to come with some kind of Jacobian factor because, after all,the path integral is still an integral. A certain way to perform this splitting is through aδ-function (a δ-functional to be more precise) that plays the role of gauge fixing. For thesake of this, the following notation is adopted:

A gauge transformated gauge field is represented as,

AµGauge Group−−−−−−−→ Aζµ ≡ U ζAµ

(U ζ)−1 − i

g∂µU

ζ(U ζ)−1

(B.30)

The symbol ζ is meant to represent the parameters that characterize the elements ofthe gauge group. More explicitly,

U ζ = exp (igζaTa) = exp (igζ) (B.31)

After gauge fixing, the gauge field is fixed to Aµ called a fiducial gauge field through

a gauge fixing condition F [A] = Fa[A]T a = 0.

In terms of this notation, the δ-functional can be inputted from a resolution of 1,

∆FP [A]

Dζ δ

[F [Aζ ]

]= 1 (B.32)

The factor ∆FP [A] is called the Faddeev-Popov determinant and corresponds to a Jacobianfactor in the same way that 1 =

dpx δp (x) = det (f ′)

dpx δp (f (x)) with f ′ evaluated at

the roots of f (x). Therefore,

∆FP [A] = det

(∂Fa[A

ζ ]

∂ζb

∣∣∣∣ζ=0

)(B.33)


with the determinant being taken over the gauge indices a and b.Even though the whole discussion is rather ambiguous, the final results will be sensible

and well defined. Just a reminder regarding the notation of the arguments of the functionalFa[A]: What is really meant is Fa[A

bµ]. In addition, the lower gauge indices and the upper

gauge indices are exactly the same, θa = θa. Their position is only such that a sum is impliedin the “contraction of lower and upper indices” sense.

Before proceeding, there are some important assumptions and observations. Firstly, thegauge fixing is supposed to be “good” meaning that the gauge trajectory passes only oncethrough each physically distrinct configuration; in other words, for every point in the gaugepotential space Aaµ ∈ A there exist one and only one gauge transformation U ζ0 that takes

Aµ to Aζ0µ = Aµ preventing potential discrete ambiguities. Secondly, the measure Dζ isgauge invariant because it is the Haar measure of a Lie group being unaffected by left andright actions, Dζ = D (ζ ′ζ) = D (ζζ ′). Lastly, the Faddeev-Popov determinant is also gaugeinvariant,

∆−1FP [Aζ ] =

Dζ ′ δ

[F [Aζζ

′]]

=

D(ζ−1ζ ′′

)δ[F [Aζζ

−1ζ′′ ]]

=

Dζ ′′ δ

[F [Aζ

′′]]

= ∆−1FP [A]

(B.34)

In the above proof, a change of “coordinates” ζ ′ = ζ−1ζ ′′ was made in the second line andgauge invariance of the measure was taken into account in the third line.

Inserting the unity in the path integral gives,

Z =1

V ol

DζDφDA ∆FP [A]δ

[F [Aζ ]

]eiS[φ,A]

=1

V ol

DζDφ ∆FP [Aζ ]eiS[φ,Aζ ]

=1

V ol

(Dζ)

Dφ ∆FP [A]eiS[φ,A]

=

Dφ ∆FP [A]eiS[φ,A]

(B.35)

where in the third line, gauge invariance of both the action functional and the Faddeev-Popov determinant were used, while

Dζ is precisely the volume V ol of the gauge potential

space.

Faddeev-Popov determinant computation

The last thing to do is compute ∆FP . The inverse of the Faddeev-Popov determinant isevaluated by (B.33). For an infinetismal gauge transformation U ζ = eigζ

aTa = 1 + igζaTa,the gauge fields get shifted by (see (A.56)),

AaµGauge Group−−−−−−−→ Aζ aµ = Aaµ + ∂µζ

a + gfabcAbµζ

c = Aaµ + (Dµ)ab ζb (B.36)

B.3. QUANTUM GAUGE THEORIES ARE HAUNTED 163

so the Faddeev-Popov determinant is,

∆FP [A] = det

(∂Fa[Aµ +Dµζ]

∂ζb

∣∣∣∣ζ=0

)(B.37)

At this point, the gauge fixing condition must be specified. It would be more proper topick a condition that realizes Lorentz invariance, like the Lorentz condition ∂µA

µa = 0. More

generally, it could be a Lorentz-Feynman-Landau gauge condition,

∂µAµa (x) = εa (x)⇒ Fa[A] = ∂µA

µa (x)− εa (x) (B.38)

for some given function εa (x), resulting in,

∆FP [A] = det (∂µDµ) (B.39)

Ghost fields

The expression (B.39) of the Faddeev-Popov determinant is a meaningful result, but thereis a better way to engulf it in the path integral. It is based on a well-known result fromfermionic path integral calculations,

DcDc exp

(i

dd+1x cMc

)= det (M) (B.40)

with c (x) and c (x) two Grassman variables, i.e. anticommuting fields, related by hermitianconjugation (c = c†) with dim (M) components each. Consequently, (B.39) may equivalentlybe written as,

∆FP [A] =

DcDc exp

(i

dd+1x c∂µDµc

)=

DcDc exp

(−i

dd+1x ∂µcDµc

) (B.41)

where the anticommuting “spinors” now have Ngen components. These newly inserted anti-commuting fields ca and ca are called ghost fields.

Plugging this result in the original path integral yields,

Z =

DφDcDc eiSeff [φ,A,c,c] (B.42)

with Seff [φ, A, c, c] = S[φ, A] + Sghosts[c, c] an effective action functional that includes theghost action,

Sghosts[c, c] = −dd+1x ∂µcDµc (B.43)

If it was quantize a gauge theory, these ghost fields would violate the spin-statisticstheorem, hence their characterization as “non-physical” fields. For example, in Yang-Mills


theories the ghost fields ca (x) have spin-0, even though they obey anti-commutation rela-tions. The ghost fields transform in the same representation as the gauge fields, the adjointrepresentation, so their action may be written explicitly in terms of components,

Sghosts[c, c] = −dd+1x

(∂µca∂µc

a + gfabcAbµcc)

(B.44)

They carry the same gauge charges as the gauge fields but not any spinor or Lorentz indices.Of course, a quantum theory that violates the spin-statistics theorem is unacceptable butthere is a loop hole in the theorem; the theorem is violated unless the ghost fields createnegative norm states. This does not sound much better, but it is actually essential becausethe negative norm states created by ghost fields cancel out the negative norm states createdby the longitudinal polarizations of the gauge fields restoring the unitarity of the entiretheory. As a result, these spooky fields are some kind of negative degrees of freedom cancelingout those unphysical degrees of freedom that were not taken care of. This statement willbecome explicit in the next chapter when the contribution of ghost fields to the centralcharge, a quantity that measures the degrees of freedom of the theory, will be calculated tobe negative.

B.4 Interactions

After the theory has been specified, one needs to make sense out of it in terms of meaningfulcalculations capable of experimental testing. Nature does not come with free field theories.It is dynamical and considers interactions among particles carrying charges. These chargesare gauge charges and interactions are carried out by the gauge fields. For example, theelectric charge is a U (1) gauge charge and electromagnetic interactions are carried out bythe U (1) gauge field, the photon.

There are two main questions that need to be answered:

What to measure?

What to compute?

The first question is an experimental one. It is well known that in order to investigatethe microscopic world of particle physics, large energy scales nedd to be achieved, hence theuse of accelerators. In general, particle physics experiments are restricted by technologicalcapabilities and even more fundamental are the restrictions subsequent to the Heisenberguncertainty principle. Eventually, only statistical quantities can be measured. In particular,there are two principal experimental observables of particle physics: scattering cross-sectionsσ(1 + 2→ 1′ + 2′ + · · ·+ n′) and decay widths Γ(1→ 1′ + 2′ + · · ·+ n′).

Similar to Fermi’s golden rule,

Γ(α→ β) =2π

~| 〈β|VI |α〉 |2 ρf (B.45)

which is effective in non-relativistic quantum mechanics, there are two golden rules in rela-tivistic quantum mechanics that express the differential scattering cross-section dσ and thepartial decay rate dΓ in terms of theoretical quantities,

B.4. INTERACTIONS 165

Partial decay rate for 1→ 1′ + 2′ + · · ·+ n′

dΓ =1

2E1

|M|2dLips1n (B.46)

Differential scattering cross-section for 1 + 2→ 1′ + 2′ + · · ·+ n′

dσ =1

uα

1

2E1

1

2E2

|M|2dLips2n (B.47)

where uα is the relative velocity of the incoming particles,

uα =

√(p1 · p2)2 −m2

1m22

E1E2

(B.48)

dLipsmn is the Lorentz invariant phase space for interactions of the form 1 + 2 + · · ·+m→1′ + 2′ + . . . n′,

dLipsmn = (2π)d+1 δd+1

(m∑j=1

pi −n∑j=1

p′i

)n∏i=1

dd~p ′i

(2π)d 2E ′i(B.49)

andM is the Feynman invariant amplitude 1. Its calculation is done via Feynman diagramswhich synonymous to “QFT perturbation theory”.

As in non-relativistic quantum mechanics, interactions are carried out by some potentialenergies in the Hamiltonian. The full Hamiltonian is decomposed into a free Hamiltonianpart H0 and an interaction Hamiltonian part Hint. The H0 part is simply the part in thefree Lagrangian density, more commonly referred to as the kinetic Lagrangian density, whichinvolves exactly two fields. Higher order products of fields fall in Hint. In the Feynmandiagrams, one needs to carefully count lines and vertices. Each line corresponds to thepropagation of the field from an initial point to a final point in spacetime. Vertices are thosepoints in spacetime that 3 or more fields meet. The number and kinds of fields that maymeet is determined by the corresponding terms in the Lagrangian density. For example,LYM contains products of 3 fields and 4 fields. These give rise to 3-vertices and 4-vertices.

The Feynman diagrams are simply correlation functions of fields. Thanks to Wick’stheorem, all correlation functions may be calculated by only knowing the 2-point functions.Following the Feynman rules 2, the quantity −iM may be calculated. The Feynman rulesin general are,

Each vertex contributes a factor of i times the interaction term in the Lagrangiandensity with all the fields removed

Each line (propagator) contributes a factor of i times the inverse of the kinetic op-erator in the momentum space. For the Klein-Gordon field with equation of motion(p2 −m2)φ in momentum space, this is just i (p2 −m2)

−1

1|M|2 is the Feynman invariant amplitude averaged over unmeasured particle spins2The Feynman rules in momentum space to be more precise


Fundamental Field Feynman Diagram Feynman Rule

Ingoing scalar (φ) 1Outgoing scalar (φ) 1

Ingoing anti-scalar (φ∗) 1Outgoing anti-scalar(φ∗) 1

Ingoing fermion (u) uOutgoing fermion (u) u

Ingoing anti-fermion (υ) υOutgoing anti-fermion (υ) υ

Ingoing abelian gauge boson (γ) εµ

Outgoing abelian gauge boson (γ) εµ∗

Ingoing non-abelian gauge boson (g) εµ

Outgoing non-abelian gauge boson (g) εµ∗

Table B.1: Feynman rules for ingoing and outgoing fundamental fields. u and υ are thespinors for fermions and anti-fermions and εµ is the polarization vector of the gauge boson.The full circles at the one of the ends of the diagram indicates where an interaction vertexis present, while no circle on the left or right means propagation from the infinite past or tothe infinite future respectively.

The propagators for all cases of fundamental fields are demonstrated in table B.2. Thediagram contribution is then obtained by multiplying everything and integrating over theLorentz invariant phase space dLips. When saying everything, it is not only meant thevertex and propagators contributions but also all ingoing and outgoing states. These statesmay be fermionic or bosonic and they are characterized by their D-momentum. The contri-bution of each such ingoing and outgoing fundamental field along with its Feynman diagramrepresentation is read from table B.1.

Fundamental Field Propagator Feynman Diagram Feynman Rule

Scalar (〈φφ〉 (p)) i(p2 −m2)−1

Anti-scalar (〈φ∗φ∗〉 (p)) i(p2 −m2)−1

Fermion (〈ψψ〉 (p)) i(/p−m)−1

Anti-Fermion (〈ψψ〉 (p)) i(/p+m)−1

Abelian Gauge Boson (〈AµAν〉 (k)) ik2

(−ηµν + (1− ξ)kµkν

k2

)Non-Abelian Gauge Boson (〈AµaAνb 〉 (k)) i

k2

(−ηµν + (1− ξ)kµkν

k2

)δab

Table B.2: Feynman rules for propagators of fundamental fields. p is the D-momentum ofthe fundamental field. For gauge fields, k is the wavenumber D-vector. In the gauge fieldspropagators, the parameter ξ fixes the gauge. For Feynman and Lorentz gauges, xi = 1,x = 0 respectively.

Appendix C

General Relativity

In this appendix, some basic elements of General Relativity will be reviewed in order to makesure that the dynamics of Anti-de Sitter spacetime and black holes will be well understood.For the sake of this, some general aspects of tensor analysis will be stated as well as theirapplications in gravity. In addition, the subject of isometries will be analyzed in order to getfamiliar with commonly used concepts in the language of gravity physics. Two of the mostcomplete textbooks on General Relativity are [3] and [36].

C.1 Tensor Analysis

All notions of General Relativity are expressed in terms of tensors. Tensors are objects thatlive on a differential manifold, in other words a curved continuous space. They are char-acterized by the way they transform under diffeomorphisms (coordinate transformations).They carry two types of indices:

Upper or contra-variant indices

Lower or covariant indices

Tensor are classified by their rank which simply indicates how many upper and howmany lower indices the tensor carries. The rank is written as (n,m) with the first integern counting the number of contra-variant indices the tensor carries, while the second integerm counts the number of covariant indices. For example, a (0, 0) tensor is simply a scalar,a (1, 0) tensor is a (contra-variant) vector, such as the spacetime position xµ, and a (2, 0),like the energy-momentum tensor T µν or the inverse metric tensor gµν , is a... tensor1.Although it more transparent to write expressions with the tensor indices explicitly shown,it is sometimes more convenient to consider the notion of a tensor without the indices anddefine some special kinds of mathematical operations such as the wedge product. This is thelanguage of differential forms. The symbols for these “indices-less” tensors is the symbol ofthe tensor itself in bold mode. For example g is the metric tensor gµν .

1The only tensors that have been baptized are those with ranks (0, 0) (scalars), (1, 0) (contra-variantvectors or just vectors) and (0, 1) (covariant vectors or just 1-forms). All tensors of other ranks are plaintensors. (2, 0), (1, 1) and (0, 2) tensors are matrices

167

168 APPENDIX C. GENERAL RELATIVITY

One can construct higher rank tensors by multiplying tensors of lower ranks. This mul-tiplication is a tensor product and its symbol is ⊗. Provided two tensors A and B of ranks(n1,m1) and (n2,m2), the tensor,

C = A⊗B (C.1)

is a rank-(n1 + n2,m1 +m2) tensor with components,

Ci1...in1k1...kn2j1...jm1 l1...lm2

= Ai1...in1j1...jm1

Bk1...kn2l1...lm2

(C.2)

The indices are raised or lowered through the metric tensor. For example Aµ = gµνAν .

Geometrically, the metric tensor is nothing more but a matrix whose components are theinner products of the basis vectors. It describes how much “not-orthogonal” the axes are. Adiagonal metric means that all axes are orthogonal, while non-diagonal terms measure theangle between the axes.

Tensors need not only refer to objects that transform under some spacetime coordinatetransformations. They may very well refer to gauge transformations or any other kind oftransformations. It should now be clear that the gauge fields Aµa and the field strengthtensor F a

µν encountered in Appendix A are tensors of ranks (0, 1) and (1, 0) respectivelywhen considering their transformations under the gauge group. In particular, the metric ofthe gauge group that is used to raise and lower the gauge indices is just the unity matrixgab = δab

2. This is why not any care was taken in positioning the gauge indices; tensors ofgauge rank (n,m) transform in the same way as tensors of gauge rank (m + k, n − k) withk an integer counting how many times the gauge index was raised or lowered. On the otherhand, the gauge fields and the field strength tensor transform in a different manner undercoordinate transformations specified by the indices µ and ν. The gauge fields Aµa transformas tensors of Lorentz rank (1, 0) (vectors), while the field strength tensor F a

µν transforms asa tensor of Lorentz rank (0, 2). The spacetime (Lorentz) indices are raised and lowered withthe spacetime metric gµν (= ηµν for Minkowski spacetime).

C.2 General Coordinate Transformations

The gauge group for General Relativity is the group of general coordinate transformations,also known as diffeomorphisms. Under diffeomorphisms,

xµDiff.−−−→ x′µ(x) (C.4)

a tensor of general rank (n,m) transforms as,

T ν1...νnµ1...µm

Diff.−−−→ T′ ν′1...ν′nµ′1...µ

′m

=∂x′ν

′1

∂xν1. . .

∂x′ν′n

∂xνn∂xµ1

∂x′µ′1. . .

∂xµm

∂x′µ′mT ν1...νnµ1...µm

(C.5)

2The axes are the group generators Ta and the inner product is twice the trace,

gab = 2 Tr (TaTb) = δab (C.3)

C.3. EINSTEIN’S FIELD EQUATIONS AND GEODESICS 169

As in the case of gauge field theories, the partial derivative ∂µ is not a tensor 3 leading tothe need of defining a tensor derivative ∇µ. The action of this covariant derivative dependson the position of the indices of the tensor on which it acts. For vectors the action of thecovariant derivative reads,

∇µVν = ∂µV

ν + ΓνµρVρ

∇µVν = ∂µVν − ΓρµνVρ(C.7)

with Γρµν the Christoffel symbols expressed in terms of the metric as,

Γρµν =1

2gρσ (∂µgνσ + ∂νgσµ − ∂σgµν) (C.8)

All higher order tensors are “tensorly” differentiated in the same manner but by adding moreChristoffel symbols reconciling for the extra indices. For example,

∇µTρσκλ = ∂µT

ρσκλ + ΓρµνT

νσκλ − ΓνµσT

ρνκλ − ΓνµκT

ρσνλ − ΓνµλT

ρσκν (C.9)

Even though the Christoffel symbols carry spacetime indices, they are not tensors becausethey do not transform as such,

ΓρµνDiff.−−−→ Γ′ρ

′

µ′ν′ =∂x′ρ

′

∂xρ∂xµ

∂x′µ′∂xν

∂x′ν′Γρµν +

∂x′ρ′

∂xµ∂2xµ

∂x′µ′∂xν′(C.10)

The Christoffel symbols are actually the gauge fields of the group of diffeomorphisms.The field strength tensor is, thus, the Riemann tensor, defined by,

RρσµνV

σ ≡ [∇µ,∇ν ]Vρ (C.11)

Expressed in terms of the gauge fields, this is,

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓ

λνσ − ΓρνλΓ

λµσ (C.12)

C.3 Einstein’s Field Equations and Geodesics

Finally, the procedure of making sense of the above mathematical review is through theEinstein’s field equations,

Rµν −1

2Rgµν + Λgµν = −8πGTµν (C.13)

3Although ∂µDiff.−−−−→ ∂′µ′ = ∂xµ

∂x′µ′ ∂µ, the quantity ∂µVν , for example, with Vν a (covariant) vector, is nota tensor because,

∂µVνDiff.−−−−→ ∂′µ′V ′ν′ =

∂xµ

∂x′µ′ ∂µ∂xν

∂x′ν′ Vν

=∂xµ

∂x′µ′

∂xν

∂x′ν′ ∂µVν +∂2xν

∂x′µ′∂x′ν′ Vν 6=∂xµ

∂x′µ′

∂xν

∂x′ν′ ∂µVν

(C.6)


with Rµν = Rρµρν the Ricci tensor, R = gµνRµν the scalar curvature, Lambda the cosmolog-

ical constant and Tµν the energy-momentum tensor. The left hand side contains only termsof the metric tensor with up to second derivatives and it describes the curved spacetime,while the right hand side is associated with the presence of matter and radiation. The aboveequations arise from the Einstein-Hilbert action,

SEH =

dDx

√|g| (R− 2Λ) + Smatter (C.14)

with the energy-momentum tensor, calculated by varying Smatter with respect to the metric,

T µν = −2δSmatterδgµν

(C.15)

Einstein’s field equations (C.13) are solved to extract the metric components gµν .In addition, the motion of particle in a fixed spacetime background is described by the

Lagrangian,L =

√−gµν xµxxν = 1 (C.16)

with xµ the trajectory of the particle and the dots being derivatives in respect with theproper time τ . The above Lagrangian gives rise to the geodesic equations,

xρ + Γρµν xµxν = 0 (C.17)

For a massless particle, proper time cannot be defined. Even so, massless particles do followgeodesics parametrized by an affine parameter λ 4,

d2xρ

dλ2+ Γρµν

dxµ

dλ

dxν

dλ= 0 (C.18)

The geodesic path is a “straight line in a curved spacetime”, reflecting the fact that gravity,as described by general relativity, is not a real force but an apparent one arising from thenon-linear, but free, motion in curved spacetime.

What these two sets of equations of motion say is very well stated by John Wheeler:“Spacetime tells matter how to move (geodesics) ; matter tells spacetime how to curve”(Einstein’s Field Equations).

A last comment here regards invariant quantities, quantities that all observers measurethe same regardless of the coordinate system. Invariant quantities are basically all scalarquantities, that is quantities with no spacetime indices or with all spacetime indices con-tracted. Such is the scalar curvature R, as well as the length element,

ds2 = gµνdxµdxν = −dτ 2 (C.19)

The length element ds2 measure the spacetime distance between infinitesimal separatedevents. If ds2 < 0, then the events are time-like separated (well defined proper time: massivesignals travelling slower than light). If ds2 = 0, the events are light-like separated (zeroproper time: massless signals travelling at the speed of light), while if ds2 > 0, the events arespace-like separated (imaginary proper time: tachyonic signals travelling faster than light).Tachyons are unphysical, thus, only events that are either time-like or light-like separatedcan be in causal contact, i.e. interchange information and define an observation.

4An affine parameter is one that satisfies gµνdxµ

dλdxν

dλ = 0

C.4. ISOMETRIES AND KILLING VECTORS 171

C.4 Isometries and Killing Vectors

In Minkowski spacetime, it is said that the spacetime has an SO (p, q) isometry group,the Poincae group. In general, an isometry transformation is a diffeomorphism that leavesdistances invariant. This means that the new metric g′µν in the old coordinates xµ equalsthe old metric tensor gµν in the old coordinates as well 5,

gµν(x)Isometry Group−−−−−−−−−→ g′µν(x

′) , such that g′µν(x) = gµν(x) (C.20)

Infinitesimally for x′µ = xµ + ξµ, the general coordinate transformation of the metric,

gµν(x)Diff.−−−→ g′µ′ν′(x

′) =∂xρ

∂x′µ′∂xσ

∂x′ν′gρσ(x) (C.21)

yields,g′µν(x) = gµν(x)−∇µξν −∇νξµ (C.22)

The generators Ga of the isometry group are constructed from the vector fields ξµ, calledKilling vector fields or just Killing vectors6 according to,

Ga = −iξµa∂µ (C.23)

with the index a counting the number of the Killing vectors.To be more conceptual, an isometry along one direction means that moving each point

of an object (e.g. a sphere) the same distance along the direction of the Killing vectors willnot distort distances on the object (e.g. the radius). In order to find these Killing vectorsξµ, one needs to solve the Killing vector equation,

∇µξν +∇νξµ = 0 (C.24)

The identification of generators of a symmetry group with conserved charges suggests thatfor each Killing vector there exists a charge that is conserved along the geodesic path. Themore the solutions of the Killing equation, the bigger the symmetry group. In particular,maximally symmetric spacetimes are those with the maximum number of Killing vectors, i.e.D(D+1)

2. In these cases, the Riemann tensor receives a nice expression,

Rρσµν =R

D (D − 1)(gρµgσν − gρνgσµ) (C.25)

As already said, the solution to Einstein’s field equations (C.13) is the metric tensor. Tosolve the field equations, two quantities need to be specified: the energy-momentum tensorand the cosmological constant. Although seemingly ambiguous, the cases of empty spacewith Tµν = 0, called Einstein’s spacetimes, are already quite rich in content. Apart from

5Remember that the metric components are precisely the inner products of the axes unit vectors. As aresult, they calculate the distances

6The “Killing” is actually after Wilhelm Killing and not some property that has killing/annihilatingeffects when acting on some quantity.


describing flat spacetime (when the cosmological constant vanishes), they also capture thedynamics of black holes, such as a single black hole propagating in empty space or evencollision of black holes accompanied by gravitational radiation. The field equations in thesecases can be written as,

Rµν =2

D − 2Λgµν (C.26)

where the scalar curvature was expressed as a function of the cosmological constant bycontracting the original field equations with the inverse metric to get,

R =2D

D − 2Λ (C.27)

The two arguments (maximum number of Killing vectors and Riemann tensor givenby (C.25)) are equivalent. For example, in Minkowski spacetime, the Lorentz group withgenerators,

Lµν = −i (xµ∂ν − xν∂µ) (C.28)

that satisfy the Killing equation, form the isometry group SO (D − 1, 1) which has preciselyD(D+1)

2generators making flat spacetime a maximally symmetric spacetime. Equivalently,

R = 0 and (C.25) is ensured7.

C.4.1 Constructing Maximally Symmetric Spacetimes

Maximally symmetric spacetimes are extremely interesting so it is of great importance toconstruct them at will. Fortunately, there is a straightforward way to do this. The keyidea is actually already stated: flat spacetimes are always maximally symmetric. This isnot restricted to Minkowski spacetime, but also includes flat spacetimes with a metric ofsignature (p, q) that carry p space dimensions and q time dimensions8, i.e. the manifold isRp,q with metric,

(ηMN) = diag

−1, . . . ,−1︸︷︷︸q times

, 1, . . . , 1︸︷︷︸p times

(C.29)

from which a generally not flat spacetime is constructed by a submanifold of Rp,q definedas the locus of points placed at some fixed distance from the origin,

ηMNXMXN = b2 (C.30)

The submanifold is then a maximally symmetric spacetime with isometry group SO (p, q)by construction, that describes a (p+q−1)-dimensional universe. This method of construct-ing maximally symmetric spacetimes is sometimes referred to as the embedding techniquebecause it is precisely a submanifold embedded in the manifold R

p,q. The submanifold is the

7Not all the Poincare group is the isometry group. The translations generators Pµ do not satisfy theKilling vector equation and are, thus, not Killing vectors.

8but with only the case of q = 1 time dimension being of current interest. Nevertheless, the procedureis completely general, perhaps, some ingenious idea may come in need of extra time dimensions.

C.5. PENROSE DIAGRAMS 173

embedded spacetime, while Rp,q is the embedding spacetime. Regardless of the signature ofthe metric of the embedding spacetime, the signature of the metric of the embedded space-time may be whatever one desires depending on the locus condition (C.30), but the isometrygroup of the embedded spacetime will be the same as the isometry group of the embeddingspacetime by construction.

C.5 Penrose Diagrams

Apart from the theoretical fundamentals of general relativity and its elegant mathematicalconstruction, it is essential to have a way of visualizing spacetimes. But how can infinite,higher dimensional spacetimes be depicted on a 2-dimensional sheet of paper? The answeris through Penrose diagrams9. To obtain them, a number of coordinate transformations isperformed to map the infinite ranges of the original coordinates to some finite ranges andmake the geometry free of any potential coordinate singularities. To be more precise, thePenrose diagram of a spacetime geometry is the actual drawing of the spacetime up to apositive scale factor. This means that the geometry drawn is a conformally related with theexact geometry.

Eventually, a Penrose diagram is useful because it captures the complete causal structureof the spacetime it describes since a scale factor Ω2 relating two spacetimes,

ds2 = Ω2ds2 (C.31)

conserves causality. The scale factor is usually the way of removing global factors thatbecome divergent at some points of the range of the new coordinates. Events in causalcontact are those events that fall within the light cone of each other and these light conesare bounded by null trajectories, i.e. light rays. To make causality even more obvious, it ispreferred to keep the notion of a light cone as an actual cone in the Penrose diagram. Inother words, in the new finitely ranged coordinates, the null geodesics are to be straight linesforming propagating at an angle of 45. The finite range is most often obtained by definingthe new coordinates to be angle-like through a transformation that involves trigonometricfunctions.

As an example, the case of a (1+1)-dimensional Minkowski spacetime can be considered,

ds2 = −dt+dx2

t ∈ R = (−∞,+∞)

x ∈ R = (−∞,+∞)

(C.32)

The new angle-like spatial coordinate u and time coordinate υ are defined through the

9Named after the mathematical physicist Roger Penrose. It is more proper to refer to them as Penrose-Carter diagrams in remembrance of both Roger Penrose and Brandon Carter who firstly employed thediagrams.


following coordinate transformations10,

tan(u± υ) = x± t (C.33)

As a result, the new coordinates are defined in the range (−π2,+π

2) so that the above coor-

dinate transformation is a one-to-one map11. Consequently, the flat spacetime metric takesthe form,

ds2 = (sec(u+ υ) sec(u− υ))2 (−dυ2 + du2)

υ ∈ (−π2,+

π

2)

u ∈ (−π2,+

π

2)

(C.34)

and overall factor can be conformally omitted and draw the metric,

ds2 = Ω2ds2 = −dυ2 + du2

υ ∈ (−π2,+

π

2)

u ∈ (−π2,+

π

2)

(C.35)

with the scale factor being identified with Ω = sec(u+ υ) sec(u− υ). This conformal trans-formation removes the poles u = ±υ. The final metric turns out to have the exact same formas the Minkowskian metric, but with coordinates restricted to the finite range (−π

2,+π

2). To

properly draw the spacetime on a sheet of paper, the edges must first be identified. Thespacelike infinities x → ±∞ get mapped to u → ±π

2and υ = 0. The future and past

timelike infinities t → ±∞ get mapped to u = 0 and υ → ±π2. These four points are the

corners of the Penrose diagram of the infinite Minkowski spacetime in figure C.1. Light rays,of course, still propagate at straight lines u = ±υ at an angle of 45 and the are used toconnect these four points and draw the entire flat spacetime as a diamond. Trajectories ofconstant position x = x and constant time t = t are the lines with equations,

2x = tan(u+ υ) + tan(u− υ) = sec(u+ υ) sec(u− υ) sin 2u

2t = tan(u+ υ)− tan(u− υ) = sec(u+ υ) sec(u− υ) sin 2υ(C.36)

The same procedure can be applied in the (D = d+1)-dimensional flat spacetime writtenin spherical coordinates,

ds2 = −dt2 + dr2 + r2dΩ2D−2

t ∈ R = (−∞,+∞)

r ∈ R+ = [0,∞)

θ2 ∈ (0, π)

θn ∈ (0, 2π) , n = 3, . . . , d

(C.37)

10The general idea is to first the introduce light cone coordinates u± = t± x. The next step depends onthe exact form of the metric and it usually is to do the trigonometric transformation u = tanu+, υ = tanu−as here.

11Surely, for y = tanx, the angle x can take values in the entire range R, but then again, the inversetransformation x = arctan y makes sense only for x in the range (−π2 ,+

π2 ). Otherwise, there are multiple

x’s that give the same y due to the periodicity of the tan function and the mapping is not one-to-one.

C.5. PENROSE DIAGRAMS 175

Figure C.1: Penrose diagram of (1 + 1)-dimensional flat spacetime. The horizontal axesis the u-axes and the vertical axes is the υ-axes. The 4 vertices of the diamond are thepoints (υ, u) = (0,±π

2) (spacelike infinities) and (υ, u) = (±π

2, 0) (timelike infinities). The

red and green lines are trajectories of constant Minkowski positions x = x and time t = trespectively.

The θ-coordinates are already defined in a finite range, so only the t and r coordinates needbe altered. Again, the new coordinates are defined by,

tan(u± υ) = r ± t (C.38)

but now the fact that the r coordinate takes values in half positive region [0,∞) means thatthe spatial coordinate u now ranges from 0 to π

2, rather than from −π

2to π

2. The conformally

related metric is, thus,

ds2 = Ω2ds2 = −dυ2 + du2 + sin2 2udΩ2D−2

υ ∈ (−π2,+

π

2)

u ∈ [0,+π

2)

θ2 ∈ (0, π)

θn ∈ (0, 2π) , n = 3, . . . , d

(C.39)

with Ω = sec(u+υ) sec(u−υ). The spacetime infinity r →∞ gets mapped to the point withυ = 0 and u = π

2and the timelike infinities t→ ±∞ get mapped to the points with υ = ±π

2

and u = 0, while the constant t and constant r trajectories are as already described in the2-dimensional case. The Penrose diagram is, thus, the half-diamond in figure C.2 with eachpoint being identified as a sphere SD−2.


Figure C.2: Penrose diagram of (D = d + 1)-dimensional flat spacetime. The horizontalaxes is the u-axes and the vertical axes is the υ-axes. The 3 vertices of the half-diamondare the points (υ, u) = (0, π

2) (spacelike infinity) and (υ, u) = (±π

2, 0) (timelike infinities).

The red and green lines are trajectories of constant Minkowski radius r = r and time t = trespectively. At each point of the diagram lives a sphere Sd−1.

C.6 Linearized Gravity

Lastly, a small review on the linearized Einstein’s field equation will prove useful later. The

idea is to take a known unperturbed metric(0)gµν and perturb it by adding a small variation

(1)gµν ≡ hµν ,

gµν =(0)gµν + hµν (C.40)

Under such deformations, the inverse metric becomes,

gµν =(0)

gµν +(1)

gµν +O(h2)

(1)

gµν = −(0)

gµρhρσ(0)

gσν(C.41)

and, the new Christoffel symbols are,

Γρµν =(0)

Γρµν +(1)

Γρµν +O(h2)

(1)

Γρµν =1

2

(0)

gρσ (∇µhσν +∇νhσµ −∇σhµν)

(C.42)

where covariant derivatives computed using the unperturbed metric(0)gµν . Doing the calcula-

C.6. LINEARIZED GRAVITY 177

tions for the Ricci tensor it turns out that12,

Rµν =(0)

Rµν +(1)

Rµν +O(h2)

(1)

Rµν = ∇ν

(1)

Γρρµ −∇ρ

(1)

Γρµν

(C.44)

Putting everything together in Einstein’s field equations,

Rµν −1

2gµνR = −8πGTµν (C.45)

the first order field equations are the linearized Einstein’s field equations,

hµν +∇µ∇νh−∇µ∇ρhρν −∇ν∇ρhρµ

+(0)gµν

(∇ρ∇σhρσ −h+ hρσ

(0)

Rρσ

)− hµν

(0)

R = −16πG(1)

Tµν(C.46)

where ≡ ∇µ∇µ, h ≡(0)

gµνhµν and(1)

Tµν is the first order perturbation in the energy-momentum tensor13.

12There is a simple way to extract the perturbed Ricci tensor without doing the calculations explicitly.

The trick is to recognize that(1)

Γρµν is a tensor (in contrast to(0)

Γρµν), and work in a locally flat coordinatesystem for which Γρµν = 0 to get,

(1)

Rµν = ∂ν(1)

Γρρµ − ∂ρ(1)

Γρµν (C.43)

Then the tensor nature of(1)

Γρµν and(1)

Rµν allows to apply the equivalence principle and generalize to anycoordinate system by the substitution ∂µ → ∇µ.

13The perturbation in the metric is sourced by a perturbation in the energy-momentum tensor(1)

Tµν . Itis within this variation that a small parameter characterizing the size of the perturbation is hidden. Forexample, this parameter may be the mass m of a small probe backreacting on the geometry of a black holewith mass M m or the speed υ 1 of a slowly moving particle.


Appendix D

CFT in 2 Dimensions

In the main text (Section 3.2), the conformal group and its algebra in D > 2 dimensions areanalyzed as well as what constraints they employ on correlation functions. But the mostcharacteristic quantity associated with a CFT is the central charge cCFT , a quantity thatmeasures the degrees of freedom of the theory. The central charge is involved in almostevery aspect of a quantum CFT from quantum anomalies to computations of entropies.In this appendix, an attempt to smoothly introduce the notion of the central charge willbe performed. To achieve this, the special case of D = 2 needs to be considered due tosignificant simplifications, although the tools employed are of general, but less trivial, use inhigher dimensional CFTs.

D.1 Holomorphic Coordinates

In D = 2, the metric is a 2×2 matrix, (ηµν) = diag (−1, 1). It is more convenient to work inEuclidean signature by defining the Euclidean time, x2 ≡ ix0 to simplify the analysis evenmore. In this case, the Lorentz index runs from 1 to 2 instead of from 0 to 1 and the metrictensor is the unit matrix. The conformal Killing equations (3.25) then read explicitly,

∂1ε1 = ∂2ε

2 , ∂1ε2 = −∂2ε

1 (D.1)

These are just the Cauchy-Riemann equations encountered in an introductory course onComplex Analysis. There are infinite solutions of these two equations which can be groupedtogether to from a complex function,

f(x1, x2) = ε1(x1, x2) + iε2(x1, x2) (D.2)

that has the important property of holomorphism. To understand more properly what aholomorphic function is, it is elegant to introduce the complex coordinates1,

z = x1 + ix2 , z = x1 − ix2 (D.3)

1In Lorentzian signature, these complex coordinates are simply the light cone coordinates x± = t± x1.

179

180 APPENDIX D. CFT IN 2 DIMENSIONS

which will be called holomorphic coordinates. In these coordinates, the metric tensor be-comes,

ds2 = (dx1)2 + (dx2)2 = dzdz (D.4)

gzz = gzz = 0

gzz = gzz =1

2

(D.5)

while the holomorphic derivatives read,

∂z =∂1 − i∂2

2≡ ∂ , ∂z =

∂1 + i∂2

2≡ ∂ (D.6)

From the form of the metric it is obvious a general coordinate transformation of the formz → z′ = f(z), z → z′ = f(z)2 conformally transforms the metric,

ds2 = dzdz → ds′2 =

∣∣∣∣dfdz∣∣∣∣2dz′dz′ (D.7)

with the scale factor identified with the derivative of the holomorphic function f(z). Holo-morphy, eventually, means that the function f(z) does not depend on the complex conjugateholomorphic coordinate z, i.e. ∂f(z) = 0 3. The conformal group in 2 dimensions is thusthe set of all holomorphic coordinate transformations.

Tracelessness of the energy-momentum tensor T µµ = 0 now translates to Tzz = Tzz = 0,while its conservation, ∂µT

µν reads,

∂Tzz = 0⇒ Tzz = Tzz(z) ≡ T (z)

∂Tzz = 0⇒ Tzz = Tzz(z) ≡ T (z)(D.8)

In other words, T (z) is a holomorphic function, while T (z) is an anti-holomorphic function.Even though the actual space is R2 it is elegant to treat the complex coordinates z, z ∈ C asindependent variable and extend space to be C2. This is useful because it allows to applyvarious theorems from Complex Analysis.

Translations, rotations and dilatations on holomorphic coordinates are then realized as,

Translations: z → z + a

Rotations: z → ζz, |ζ| = 1

Dilatations: z → ζz, Re ζ 6= 1

while the conserved currents associated general holomorphic transformations δz = ε(z) andδz = ε(z), i.e. conformal transformations, are found to be,

δz = ε(z): jz = 0, j z = T (z)ε(z)

δz = ε(z): jz = T (z)ε(z), j z = 02It should be enough to just write z′ = f∗(z) = f(z) but the notation f(z) ≡ f(z) makes explicit the

fact that everything associated with z is “barred”.3Or, in Lorentzian signature, that the function f is “left-moving”, i.e. ∂+f = 0.

D.2. WARD-TAKAHASHI IDENTITIES AND CONFORMAL WEIGHTS 181

D.2 Ward-Takahashi Identities and Conformal Weights

The Ward-Takahashi identity (B.25) is written in a more useful form for conformal transfor-mations in 2 dimensions by defining the conserved current in the proof with an additional− 1

2πfactor4,

− 1

2π

ε

d2x ∂µjµαO(x1) = δαO(x1) (D.9)

Working in holomorphic coordinates (x1, x2) → (z, z) and applying the Stokes theoremexpressed in the content of Complex Analysis, the Ward-Takahashi identity offers a trans-formation rule for a general operator O(w, w) inserted at position (x1

1, x21)→ (w, w),

d2x∂µj

µ(x) =

∂ε

dljµ(x)nµ

=

∂ε

(j1dx2 − j2dx

1)

= −i∂ε

(jzdz − jzdz)

(D.10)

⇒ δO(w, w) =i

2π

(∂ε

dz jzO1(w, w)−∂ε

dz jzO1(w, w)

)(D.11)

The contour integral vanishes unless the contour encloses simple poles of the integratedfunction. The fact that the currents associated with the two independent conformal trans-formations δz = ε(z) and δz = ε(z) are holomorphic and anti-holomorphic respectivelymeans that the contour integral will pick up the residues5,

δz = ε(z)⇒ δO(w, w) = −Resz→wε(z)T (z)O(w, w) (D.13)

δz = ε(z)⇒ δO(w, w) = −Resz→w

ε(z)T (z)O(w, w)

(D.14)

The above Ward-Takahashi identities say that if the OPE of the general operator Owith the energy-momentum tensor is given, in particular the OPEs TO and TO, then thetransformation rule for the operator under conformal transformations is immediately known.

Interpreting the result the other way around, if the transformation rule for the operatorunder conformal transformations is given, then part of the TO and TO is immediatelyknown.

4The ε in the lower limit of the integral is the region ε 6= 0 explained in section 3.3 and is not be confusedwith the infinitesimal conformal Killing vector ε(x).

5As a reminder from Complex Analysis,

dzf(z) = 2iπRes

z→wf(z) (D.12)

with w being a simple pole of the complex function f(z).


Translations

Considering the special case of infinitesimal translations ζ → z + ε and z → z + ε, with εand ε two constants, the operator transforms as,

O(z, z)→ O(z − ε, z − ε) = O(z, z)− ε∂O(z, z)− ε∂O(z, z) + . . .

⇒δO(z, z) = −ε∂O(z, z)− ε∂O(z, z)(D.15)

so the Ward-Takahashi identities read,

Resz→wT (z)O(w, w) = ∂O(w, w)

Resz→w

T (z)O(w, w)

= ∂O(w, w)

(D.16)

and the OPEs with T and T have the form,

T (z)O(w, w) = · · ·+ ∂O(w, w)

z − w+ . . .

T (z)O(w, w) = · · ·+ ∂O(w, w)

z − w+ . . .

(D.17)

Rotations and Dilatations

Likewise before, under infinitesimal rotations and dilatations z → z + εz and z → z +εz, the transformation rule for a general local operator O will give the term in the OPEwith energy-momentum tensor that has a second order pole. The problem is that, unlikethe case of translations, the transformation rules under rotations and dilatations are notgenerally known. For operators which are eigenstates of dilatations and rotations, however,the procedure can be carried out. Such operators are characterized by a conformal weight6

(h, h) which describes their transformation rule under δz = εz and δz = εz via,

δO(z, z) = −ε(hO(z, z) + z∂O(z, z)

)− ε(hO(z, z) + z∂O(z, z)

)(D.18)

The terms involving derivatives come from the usual Taylor expansion and are common forany operator, not just eigenstates of dilatations and rotations.

The conformal weights are real numbers and they are actually more familiar than theylook. In particular, the spin s, the eigenvalue under rotation, and the scaling dimension ∆,the eigenvalue under dilatations, are expressed in terms of the conformal weights as,

s = h− h , ∆ = h+ h (D.19)

This is motivated by the fact that the rotations and dilatations generators are given by,

L = −i(x1∂2 − x2∂ − 1) = z∂ − z∂ , i∆ = xµ∂µ = z∂ + z∂ (D.20)

Consequently, the TO and TO OPEs take the general form,

T (z)O(w, w) = · · ·+ hO(w, w)

(z − w)2+∂O(w, w)

z − w+ . . .

T (z)O(w, w) = · · ·+ hO(w, w)

(z − w)2+∂O(w, w)

z − w+ . . .

(D.21)

6The notation h instead of h is used to avoid mistaking h with the complex conjugate of h.

D.3. PRIMARY OPERATORS 183

D.3 Primary Operators

The primary operators are generally defined by their annihilation under the action of thespecial conformal transformations generator Kµ which meant that primary operators cor-respond to the lowest scaling dimension eigenstates of the dilatations generator. To avoidconsidering explicitly the action of the special conformal transformations generators, theequivalent definition (3.53) regarding the transformation law will be used. In holomorphiccoordinates (z, z), the transformation law is written as,

O(z, z)Conformal Group−−−−−−−−−−→ O′(z′, z′) =

(∂z′

∂z

)−h(∂z′

∂z

)−hO(z, z) (D.22)

In fact, another equivalent definition of primary operators is the truncation of their OPEswith the energy-momentum tensor at the square inverse power,

T (z)O(w, w) = hO(w, w)

(z − w)2+∂O(w, w)

z − w+ . . .

T (z)O(w, w) = +hO(w, w)

(z − w)2+∂O(w, w)

z − w+ . . .

(D.23)

For general small conformal transformations,

δz = ε(z) = ε(w) + (z − w)∂ε(w) + . . .

δz = ε(z) = ε(w) + (z − w)∂ε(w) + . . .(D.24)

the Ward identities following by the new alternative definition of primary operators say thatprimary operators transform infinitesimally under conformal transformations according to,

δO(w, w) = −h∂ε(w)− ε(w)∂O(w, w)− h∂ε(w)− ε(w)∂O(w, w) (D.25)

and, since the conformal group is a continuous group, these can be integrated up to finiteconformal transformations z → z′ = f(z) and z → z′ = f(z) to yield precisely the definition(D.22).

D.4 An example: The free massless scalar field theory

An example to see how the above formulation applies in a CFT2 is the free massless scalarfield theory with Euclidean action,

SE =T

2

d2x ∂µX

I∂µXI =1

2T

dz∂XI ∂XI (D.26)

where the capital index I = 1, . . . , NX counts the number NX of the non-interactive masslessscalar fields XI whose classical equation of motion is,

δSEδXI(x)

= −2TXI(x) = 0⇒ XI(x) = 0⇔ ∂∂XI(z, z) = 0 (D.27)


and T a constant to restore proper units. The operator is, of course, equal to,

= ∂21 + ∂2

2 = ∂∂ (D.28)

in Euclidean signature. Raising and lowering the degrees of freedom index I is done viaa flat metric gIJ = δIJ

7. The above Euclidean action is actually the Euclidean Polyakovaction, the action functional of bosonic string theory, in which case the constant T is thetension of the string8. This is good to have in mind to justify some steps that have a naturalgeometrical interpretation such as the idea of metric for the degree of freedom index.

The propagator can then be found through either canonical quantization or using thefollowing path integral trick. The path integral of a total derivative is zero so,

0 =

DX δ

δXI(x)

(e−SEXJ(x′)

)=

DX e−SE

(− δSEδXI(x)

XJ(x′ +δXJ(x′)

δXI(x)

)=

DX e−SE

(2TxX

I(x)XJ(x′ + δIJδ2(x− x′))

= 2T 〈xXI(x)XJ(x′)〉+ δIJδ2(x− x′)= 2Tx 〈XI(x)XJ(x′)〉+ δIJδ2(x− x′)

(D.29)

⇒ x 〈XI(x)XJ(x′)〉 = − 1

2TδIJδ2(x− x′) (D.30)

where the index “x” in the x operator is to make it explicit that it acts on functions of x,such as XI(x), and not on functions of x′, such as XJ(x′). Now, since9,

x ln |x− x′|2 = 4πδ2(x− x′) (D.32)

the differential equation can be solved to give,

〈XI(x)XJ(x′)〉 = − 1

8πTln |x− x′|2δIJ (D.33)

or, in complex coordinates,

〈XI(z, z)XJ(w, w)〉 = − 1

8πTln |z − w|2δIJ (D.34)

7In Lorentzian signature, this degrees of freedom metric will be the Minkowski metric gIJ = ηIJ . Thereis no reason to consider this specific metric but it allows the analysis to focus on the main subject insteadof non-trivial generalizations which eventually give the same results.

8By the way NX in the bosonic string theory action is the number of spacetime dimensions.9To prove this, one simply needs to show that x ln |x− x′|2 vanishes for x 6= x′, while integrating with

respect to x′ over a region that includes the point xµ, the result should be 4π,

d2x′x ln |x− x′|2 = 4π (D.31)

D.4. AN EXAMPLE: THE FREE MASSLESS SCALAR FIELD THEORY 185

This can be written as an operator equation,

XI(z, z)XJ(w, w) = − 1

8πTln |z − w|2δIJ + . . . (D.35)

with “. . . ” representing non-singular terms. The equation of motion ∂∂XI(z, z) = ∂∂XI(z, z) =0 now says that ∂XI(z, z) is holomorphic function, while ∂XI(z, z) is an anti-holomorphicfunction so the general solution is of the form,

XI(z, z) = XI(z) + XI(z) (D.36)

allowing to write the operator equation for the propagator as the two equations,

XI(z)XJ(w) = − 1

8πTln(z − w)δIJ + . . .

X(z)X(w) = − 1

8πTln(z − w)δIJ + . . .

(D.37)

The logarithm means that the fundamental fields XI(z) and XI(z) do not transform nicelyunder conformal transformations so they are not the really the objects of interest in thistheory. The operators ∂XI(z) and ∂XI(z), however, do have a nice looking OPE,

∂XI(z)∂XJ(w) = − 1

8πT

1

(z − w)2+ . . .

∂XI(z)∂XJ(w) = − 1

8πT

1

(z − w)2+ . . .

(D.38)

The energy-momentum tensor can thus be found from the usual formula predicted byNother’s theorem or from the trick borrowed by general relativity to10,

T (z) = −4πT ∂XI(z)∂XI(z)

T (z) = −4πT ∂XI(z)∂XI(z)(D.39)

but this holds in the classical theory. If the same expression was promoted directly to anoperator in the quantum theory, there are bound to be divergences because it is as expressedas a product of two operators at the same position (UV divergence). This is avoided bynormal ordering everything, i.e. putting all annihilation operators to the right. Without anyreference to ladder operators, the normal ordering of the energy-momentum tensor, and anygeneral product of two local operators at the same point, is defined as,

T (z) = −4πT : ∂XI(z)∂XI(z) :≡ −4πT limw→z

(∂XI(z)∂XI(w)− 〈∂XI(z)∂XI(w)〉

)(D.40)

ensuring that the vacuum infinities are set to zero, i.e. quantities such as energy and momen-tum are measured with respect to the vacuum. Under this construction, any OPE can becomputed by also employing Wick’s theorem, i.e. by summing over all possible contractions

10Since a factor 12π was introduced in the conserved current in the Ward-Takahashi identities, the same

factor must be divided for the energy-momentum tensor to reconcile for it.


of pairs of non-ordered operators and then normal ordering them where “contraction” means“replace the pair with the propagator”,

XI(z)XJ(w) = − 1

8πTln(z − w)δIJ + . . .

XI(z)XJ(w) = − 1

8πTln(z − w)δIJ + . . .

(D.41)

or, the more useful contraction,

∂XI(z)∂XJ(w) = − 1

8πT

1

(z − w)2δIJ + . . .

∂XI(z)∂XJ(w) = − 1

8πT

1

(z − w)2δIJ + . . .

(D.42)

For example, the T∂X OPE is,

T (z)∂XI(w) = −4πTδJK : ∂XJ(z)∂XK(z) : ∂XI(w)

= −4πTδJK

(: ∂XJ(z)∂XK(z)∂XI(w) : + : ∂XJ(z)∂XK(z)∂XI(w) :

)= −4πTδJK

(: ∂XK(z) : δIJ+ : ∂XJ(z) : δIK

)(− 1

8πT

1

(z − w)2+ . . .

)=

∂XI(z)

(z − w)2+ . . .

=∂XI(w)

(z − w)2+∂2X(w)

z − w+ . . .

(D.43)where, in the last line, ∂X(z) was Taylor expanded around z = w to have only operators atz = w on the right-hand side while the fact that : ∂X := ∂X was used. From this example,by the way, it is obvious the the operator ∂XI(z) is a primary operator with conformalweights (h∂X , h∂X) = (1, 0)11. Similarly, ∂X(z) is also a primary operator with conformalweights (h∂X , h∂X) = (0, 1).

Another operator that appears frequently in interactions is the exponential operator: eik·X(z,z) :. To find its OPE with the energy-momentum tensor, it is more useful to first

11The h∂X = 0 is extracted from the fact that T (z)∂XI(z) = 0 or, equivalently, from the fact that ∂XI

is a holomorphic function as the equations of motions suggest.

D.5. CENTRAL CHARGE 187

compute,

∂XI(z) : eik·X(w) : =∞∑n=0

in

n!∂XI(z) :

(kJX

J(w))n

:

=∞∑n=0

in

n!n :(kJX

J(w))n−1

: kK∂XI(z)XK(w)

= kK∂

(− 1

8πTln(z − w)δIK

) ∞∑n=0

in

(n− 1)!:(kJX

J(w))n−1

: + . . .

= − ikI

8πT

1

z − w

∞∑m=0

im

m!:(kJX

J(w))m

: + . . .

= − ikI

8πT

: eik·X(w) :

z − w+ · · · ≡ ∂XI(z) : eik·X(w) :

(D.44)

⇒ T (z) : eik·X(w) : = −4πT : ∂XI(z)∂XI(z) :: eik·X(w) :

= −4πTδIJ

(: ∂XI(z)∂XJ(z) : eik·X(w) :: + : ∂XI(z)∂XJ(z) : eik·X(w) ::

+ : ∂XI(z)∂XJ(z) : eik·X(w) ::

)= −4πTδIJ

(− i

8πT

1

z − w

[kI : ∂XJ(z) : eik·X(w) :: +kJ : ∂XI(z) : eik·X(w) ::

]− kIkJ

64π2T 2

: eik·X(w) :

(z − w)2

)+ . . .

=k2

16πT

: eik·X(w) :

(z − w)2+∂ : eik·X(w) :

z − w(D.45)

The same holds for the T (z) : eik·X(w) : OPE so the operator : eik·X(z,z) : is a primary operatorwith conformal weights (heikX , heikX ) = ( k2

16πT, k2

16πT).

D.5 Central Charge

The initial observation is that the quantum energy-momentum tensor has a scaling dimen-sion ∆T = 2 since the energy is obtained by integrating the energy-momentum tensor overthe 2-dimensional space and scale invariance says that energy is scale-independent. In Ddimensions, this would be ∆T = D. In addition, the fact that the energy-momentum tensoris rank-2 symmetric tensor, like the metric, means that it has a spin s = 2. ConsequentlyT (z) and T (z) have conformal weights (hT , hT ) = (2, 0) and (hT , hT ) = (0, 2) in any CFT2.


These mean the TT and T T OPEs in any CFT2 have the form,

T (z)T (w) = · · ·+ 2T (w)

(z − w)2+∂T (w)

z − w+ . . .

T (z)T (w) = · · ·+ 2 ¯T (w)

(z − w)2+∂T (w)

z − w+ . . .

(D.46)

The next task is to find all the singular terms in the above OPEs. This can be explicitlydone in D = 2 dimensions after three basic observations:

The scaling dimension of TT and T T products is,

∆TT = ∆T T = D2 = 4 (D.47)

This means that all terms in the TT OPE must be of the form,

T (z)T (w) =∞∑

n=−∞

On(w)

(z − w)n(D.48)

with On some general operator of scaling dimensions ∆On = δTT −n = D2−n = 4−nand similarly for T T .

For unitary CFTs in any number of dimensions, it is proven in the main text that theconformal weights of any operator On are positive12, hOn , hOn ≥ 0,

∆On ≥ 0⇒ n ≥ D2 = 4 (D.49)

so the TT and T T OPEs truncate at the fourth inverse power,

T (z)T (w) =c1

(z − w)4+

c2

(z − w)3+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

T (z)T (w) =c1

(z − w)4+

c2

(z − w)3+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

(D.50)

Since everything commutes with everything,

T (z)T (w) = T (w)T (z)

=c1

(w − z)4+

c2

(w − z)3+

2T (z)

(w − z)2+∂T (z)

w − z+ . . .

=c1

(z − w)4+

−c2

(z − w)3+

2T (w) + 2(z − w)∂T (w)

(w − z)2− ∂T (w)

z − w+ . . .

=c1

(z − w)4+

−c2

(z − w)3+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

(D.51)

so c2 = 0. Similarly, T (z)T (w) = T (w)T (z) yields c2 = 0.

12To be more precise, it is proven that the scaling dimension is always positive, but along the full levelconstraints and the positivity of the spin magnitude it implies that the conformal weights h = ∆+s

2 and

h = ∆−s2 are also always positive.


Therefore, the most general form for TT and T T OPEs in any CFT2 is,

T (z)T (w) =c

2(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

T (z)T (w) =c

2(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

(D.52)

where the constants c1 ≡ c2

and c1 ≡ c2

were redefined. These constants c and c are preciselythe (left-moving and right-moving) central charges of the particular theory.

The Ward-Takahashi identities (D.13) and (D.14) can now be applied for the energy-momentum tensor to extract its transformation rule. The TT OPE truncates at the inversefourth power and the residue picks up the coefficient of the Taylor expansion that multipliesthe inverse first power, so the general small conformal transformation is needed up to thethird power,

δz = ε(z) = ε(w) + ∂ε(w)(z − w) +1

2∂2ε(w)(z − w)2 +

1

6∂3ε(w)(z − w)3 + . . . (D.53)

⇒ δT (w) = −Resz→wε(z)T (z)T (w)

= Resz→w

(ε(w) + ∂ε(w)(z − w) +

1

2∂2ε(w)(z − w)2 +

1

6∂3ε(w)(z − w)3

)(

c

2(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w

)+ . . .

= −ε(w)∂T (w)− 2∂ε(w)T (w)− c

12∂3ε(w)

(D.54)

Doing infinite successive such infinitesimal transformations, it can be proven that underfinite conformal transformations z → z′(z), the T (z) component of the energy-momentumtensor obeys the following transformation rule,

T (z)Conformal Group−−−−−−−−−−→ T ′(z′) =

(∂z′

∂z

)2 (T (z)− c

12S(z′, z)

)(D.55)

where S(z′, z) is the Schwarzian,

S(z′, z) =

(∂3z′

∂z3

)(∂z′

∂z

)−1

− 3

2

(∂2z′

∂z2

)2(∂z′

∂z

)−2

(D.56)

Working in the same manner, the transformation rule for the other component T (z) of theenergy-momentum tensor under finite conformal transformations z → z′() is analogous,

T (z)Conformal Group−−−−−−−−−−→ T ′(z′) =

(∂z′

∂z

)2(T (z)− c

12S(z′, z)

)(D.57)

Apart from other important properties, they measure the number of degrees of freedomof the theory. As an example to motivate this claim, in the free massless scalar field theory


seen before, the TT OPE is,

T (z)T (w) = 16π2T 2δIJδKL : ∂XI(z)∂XJ(z) :: ∂XK(w)∂XL(w) :

= 16π2T 2δIJδKL

(: ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) : + : ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) :

+ : ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) : + : ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) :

+ : ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) : + : ∂XI(z)∂XJ(z)∂XK(w)∂XL(w) :

)= 16π2T 2δIJδKL

(− 1

8πT

1

(z − w)2

[δIK : ∂XJ(z)∂XL(w) : +δIL : ∂XJ(z)∂XK(w) :

+ δJK : ∂XI(z)∂XL(w) : +δJL : ∂XI(z)∂XK(w) :

]+

1

64π2T 2

1

(z − w)4

[δIKδJL+ δILδJK

])+ . . .

=NX

2(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

(D.58)with the last line being obtained after multiple sums of Kronecker δ’s and Taylor expansionsof ∂X(z)’s around z = w on the right-hand side. The central charge of this theory is finallyread to be precisely the number of the fundamental fields,

cX = NX (D.59)

D.5.1 Casimir Energy

The extra term − c12

(∂z′

∂z

)2S(z′, z) in (D.55) does not depend on T (z). This means that it

will be the same evaluated on all states. In other words, it only affects the constant term inthe energy, the so called zero mode, i.e. it is the Casimir energy. The central charge, thus,is directly related to the Casimir effect.

D.5.2 Weyl Anomaly

In classical CFT, it was proven the energy-momentum tensor is traceless, T µµ = 0. Inquantum CFT, however, this statement is fuzzy. In particular, the statement comes withan anomaly, the Weyl anomaly, arising from contact terms. To see this, the property oftracelessness should be forgot and look how the TzzTww OPE behaves. Quantum CFTmight cheat on obeying classical constraint, but it cannot violate symmetries so the energy-momentum tensor is still conserved,

∂Tzz = −∂Tzz (D.60)

⇒ ∂zTzz(z, z)∂wTww = ∂zTzz(z, z)∂wTww(w, w)

= ∂z∂w

(c

2(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w+ . . .

)(D.61)


The last line is an anti-holomorphic derivative of a holomorphic quantity and one wouldnaively say that it vanishes. This is certainly true ∀z 6= w, but not for z → w a contactterm appears. This can be seen from the fact that,

x ln |x− x′|2 = 4πδ2(x− x′) (D.62)

that was used in the computation of the propagator for the free massless scalar field theoryin Section ??. In holomorphic coordinates, this translates to,

∂z∂z ln |z − w|2 = 2πδ(z − w, z − w)⇒ ∂z1

z − w= 2πδ(z − w, z − w) (D.63)

allowing to rewrite inverse fourth power term in the OPE as,

∂z∂w1

(z − w)4= ∂z∂w

(1

6∂2z∂w

1

z − w

)= ∂z∂w

(π3∂z∂wδ(z − w, z − w)

)(D.64)

so the OPE can be regarded as differential equation with the following OPE solution,

⇒ Tzz(z, z)Tww(w, w) =cπ

6δ(z − w, z − w) (D.65)

It is this contact term, needed to ensure conservation of energy and momentum, thatis responsible for the Weyl anomaly. Under a general conformal transformation, the metricshifts infinitesimally as δgµν = 2ΩgµνδΩ so the shift in the expectation value of the trace ofthe energy-momentum tensor is,

δ 〈T µµ(x)〉 = δ

DΦ e−SET µµ(x)

=1

4π

DΦ e−SE

(T µµ(x)

d2x′√gδgρσTρσ(x′)

)= − 1

2π

DΦ e−SE

d2x′√gΩ(x′)δΩ(x′)T µµ(x)T ρρ(x

′)

=c

6

DΦ e−SE

d2x′√gΩ(x′)δΩ(x′)xδ

2(x− x′)

=c

6xΩ(x)δΩ(x) = − c

12δR(x)

(D.66)

Some remarks on the above calculations. In the fourth line, the OPE for the components ofthe energy-momentum tensor in holomorphic coordinates were used to write,

T µµ(x)T ρρ(x′) = 16TzzTww

=cπ

38∂z∂wδ(z − w, z − w)

= −cπ3xδ

2(x− x′)

(D.67)

In addition, the fact that the Ricci scalar for metric of the form gµν(x) = Ω2(x)δµν can beexpressed in terms of the scale factor as,

R(x) = −2(ΩxΩ(x)− ∂µΩ∂µΩ⇒ δR(x) = −2xΩ(x)δΩ(x) (D.68)


was used in the last line. The final result for finite shifts is precisely the Weyl anomaly,

〈T µµ(x)〉 = − c

12R(x) (D.69)

The Weyl anomaly is present in higher dimensional CFTs as well but with additionalcentral charges. For quantum CFT2’s to be consistent in fixed, curved backgrounds, itis sufficient to require that c = c, but for dynamical spacetimes the only way to ensureconsistency is to demand that the central charge vanishes. Such is the case of string theorywhere a vanishing central charge yields a specific critical number of spacetime dimensions.

An aside: Critical Dimensions in String Theories

A brief nod to the critical dimensions of string theories is considered fun to realize. It is afamous statement that bosonic string theory lives in a (D = 26)-dimensional spacetime whilesuperstring theory lives in a (D = 10)-dimensional spacetime. These critical dimensions arenecessary for the theory to be consistent. The way these dimensions are extracted fromthe theory content comes from the fact that they are 2-dimensional CFTs, so they posses acentral charge c which comes along with the Weyl anomaly,

〈T aa〉 = − c

12R (D.70)

In usual QFTs defined in Minkowski spacetime, the Ricci scalar always vanishes and theanomaly is relieved. However, in string theory, R is the Ricci scalar on the target spacewhich can have any geometry, so, for the theory to be consistent, i.e. 〈T aa〉 = 0, the onlyquantity that can be set to zero is the central charge itself.

Bosonic String Theory In the bosonic string theory with action SPoly, the bosonicfields Xµ contribute a factor of cbosons = D in the central charge. At the same time, theprocedure of gauge fixing the path integral through the Fadeev-Popov method yields ghostfields, known as bc-ghosts, whose central charge turns out to be cghost = −26. Therefore,consistency of the theory requires,

c = cbosons + cghost

= D − 26 = 0

⇒D = 26

(D.71)

Supersting Theory In superstring theory, there are D bosonic fields and also thesame number of fermionic fields due to supersymmetry. Each bosonic degree of freedomcontributes a factor of 1, while each fermionic degree of freedom contributes a factor of 1

2,

thus, cbosons = D, cfermions = D2

. In addition to the ghost fields associated with conformalsymmetry with cghosts = −26, there are additional ghost fields associated with the localgauge invariance of supersymmetry transformations, the βγ-ghosts. These new ghost fields


give an extra cSUSY = +11, from which the critical dimension can be found,

c = cbosons + cfermions + cghosts + cSUSY

=3D

2− 15 = 0

⇒D = 10

(D.72)

Eventually, all these extra dimensions need to be compactified into submanifolds livingin the extended 4-dimensional spacetime to restore what is seen in Nature.

D.5.3 Modular Invariance and Cardy’s formula

Apart from the Casimir effect and the Weyl anomaly, the central charge is also related tothe density of high energy scales. A way to see this is by studying CFT2 on a Euclideantorus, that is, x1 ∈ [0, 2π) and x2 = x0

E ∈ [0, β) with β−1 the temperature. The partitionfunction of a theory with periodic Euclidean time is related to the free energy of the theoryvia,

Z[β] = Tr e−βH = e−βF (D.73)

At very low temperatures β → ∞, the free energy is dominated by the lowest energystate, the vacuum with the Casimir energy. For the case of a cylinder, the Casimir energyis E = − c

12so,

Z [β]β→∞−→ e

cβ12 (D.74)

The fact that the space is Euclidean allows to interchange the notion of x0E as Euclidean

time and x1 as spatial coordinate and say that x1 will be the Euclidean time and x0E will be the

spatial coordinate. But it is not enough to just decide how to identify the coordinates. Theyalso need to be appropriately ranged. In particular, the spatial coordinate needs to be definedin the [0, 2π) to be identified as an actual spatial coordinate on the torus. Remarkably,conformal invariance allows to do just that by rescaling the coordinates according to,

xµ → 2π

βxµ ⇒ x0

E →2π

βx0E ∈ [0, 2π) , x1 → 2π

βx1 ∈ [0,

4π2

β] (D.75)

Since the CFT partition function is invariant under conformal transformations, the partitionfunction in the new coordinates must equal the partition function in the old coordinates,

Z

[4π2

β

]= Z [β] (D.76)

This is known as modular invariance and it offers a connection between high-temperatureand low-temperature partition functions. In particular, writing β′ = 4π2

βsays that the very

high temperature β′ → 0 behavior of the partition function for the cylinder is,

Z [β′]β′→0−→ e

cπ2

3β (D.77)


But the very high temperature limit of the partition function is sampling all states in thetheory dominated by the high energy states. Consequently, this computation is measuringthe number of high energy states. It was shown by John Cardy in 1986 that the entropy ofa general CFT2 is given by,

S = 2π

√cR

6

(E − c

24R

)(D.78)

where E and R are the total energy and radius of the system. The shift − c24

in the totalenergy within the brackets is related precisely to the Casimir effect. In his honor, this formulais known as Cardy’s formula.

D.5.4 c-theorem

Lastly, the very important c-theorem will be reviewed here. The c-theorem was proved byAlexander Zamolodchikov in 1988 and it gives further weight to the connection betweenthe central charge and the degrees of freedom. The theorem is considering RenormalizationGroup (RG) flows

The idea to realize what this means is to stand back and look at the space of all theoriesand RG flows between them. CFTs are the fixed points of the RG. If one perturbes a CFTby adding an extra term to the action,

S → S + α

d2xO(x) (D.79)

for some coefficient α and some local operator O(x) of th theory, then these perturbationsfall in one of the following three classes depending on the scaling dimension of O,

Relevant Deformations (∆O < 2): The UV physics are still described by the originalCFT but the IR physics is altered. RG generates flows away from the original CFTtowards a new CFT, a new fixed point.

Marginal Deformations (∆O = 2): The deformed theory defines a new CFT.

Irrelevant Deformations (∆O > 2): The IR physics are still described by the originalCFT but the UV physics is altered.

The c-theorem now says that there exists a positive real function c(gi, µ) depending on thecoupling constants gi of the QFT and the energy scale µ which monotonically decreasesalong RG flows, while at fixed points it coincides with the central charge of the CFT.

In order to complete the discussion on (D = 2)-dimensional CFTs, the conformal algebrashould be re-evaluated by introducing the Virasoro algebra and proceed with quantizationschemes, namely radial quantization. However, such details are not really needed for under-standing the research done in this script and will be omitted. If the reader wishes to get incontact with all these details, a good suggestion is to check out [10].

Appendix E

Useful relations involving thegenerators of SU (N)

In this appendix, some useful relations regarding the SU (N) generator should be reviewed.First of all, there are N2 − 1 generators T a satisfying the commutation relations,[

T a, T b]

= if cabTc (E.1)

It doesn’t really matter whether the group index is an upper or lower index since the groupmetric is just the unity, but they are carefully positioned to imply summation. Due to thefact that SU (N) is a semi simple Lie group, the generators are traceless,

Tr (T a) = 0 (E.2)

and they are normalized according to,

Tr(T aT b

)=

1

2δab (E.3)

For the fundamental representation of any Lie group, there exists a Casimir operator1,the quadratic operator TaT

a which, according to Schur’s lemma must be proportional to theunit matrix, TaT

a = C1. For SU (N), the normalization condition implies,

C =N2 − 1

2N(E.4)

Lastly, any N × N complex matrix M can be written as a complex linear combinationthe the unit matrix the SU (N) generators T a in the fundamental representation,

M = M01 +MaTa (E.5)

for proper complex coefficients M0 and Ma. These coefficients can be easily extracted to be,

M0 =1

NTr(M) , Ma = 2 Tr (Ta) (E.6)

1A Casimir operator is an operator that commutes with all the generators of the group.

195

196APPENDIX E. USEFUL RELATIONS INVOLVING THE GENERATORS OF SU (N)

thanks to the tracelessness of the generators. In terms of these coefficients, the matrixcomponents of M can be written explicitly as,

Mij =1

NMkkδij + 2Mlk(Ta)kl(T

a)ij (E.7)

or, equivalently,

δilδjkMlk =

(1

Nδijδkl + 2(Ta)kl(T

a)ij

)Mlk (E.8)

and for this to be true for any complex matrix M , the generators must satisfy the followuseful relation,

(Ta)ij(Ta)kl =

1

2

(δilδjk −

1

Nδijδkl

)(E.9)

Appendix F

Path Integral Formulation

In this appendix, the path integral formulation of quantum mechanics and quantum fieldtheory will be derived. In addition, the path integral of the most used notions in the script,states and density matrices, will be explained in a way applicable to any case.

F.1 From Quantum Mechanics to Path Integrals

The derivation for quantum mechanics starts by considering the probability amplitudeK(~xf , tf ; ~xi, ti) for a single particle localized at position ~xi = ~x(ti) at time instant ti tobe found localized at position ~xf = ~x(tf ) at some later time instant xf ,

K(~xf , tf ; ~xi, ti) = 〈~xf , tf |~xi, ti〉 (F.1)

the well known propagator.These initial and final states |~xi, ti〉 and |~xf , tf〉 are not the position eigenstates usually

encountered in the Schrodinger picture; they are eigenstates of the position operator ~x(t) inthe Heisenberg picture,

~x(t) |~x, t〉 = ~x(t) |~x, t〉 (F.2)

The two pictures are connected via the time evolution operator U(tf , ti) according to theprescription,

~x(t) = U †(t, 0)~xU(t, 0) , |~x, t〉 = U †(t, 0) |~x(t = 0)〉 (F.3)

where the time independent operator ~x and the time dependent position eigenstate |~x, t = 0〉 ≡|~x〉 are in the Schrodinger picture. There is usually a misconception here. The eigenstates|~x, t〉 are not the time evolved position eigenstates |~x(t)〉 = U(t, 0) |~x(t = 0)〉 but just the,

also time-dependent, eigenstates of the time-dependent Heisenberg position operator ~x(t),hence the U † evolution. Nevertheless, the position operator in the Heisenberg picture is aHermitian operator, so these peculiar eigenstates form a complete set of states,

dd~x(t) |~x, t〉〈~x, t| = 1 (F.4)

with d the number of spatial dimensions. The trick now is to discretize time and divide the

197

198 APPENDIX F. PATH INTEGRAL FORMULATION

𝑡0 = 𝑡𝑖

𝑡𝑁 = 𝑡𝑓

𝑡1 = 𝑡𝑖 + 𝛥𝑡

𝑡2 = 𝑡𝑖 + 2𝛥𝑡

𝑡3 = 𝑡𝑖 + 3𝛥𝑡

𝑡4 = 𝑡𝑖 + 4𝛥𝑡

𝑡𝑁−2 = 𝑡𝑖 + (𝑁 − 2)𝛥𝑡

𝑡𝑁−2 = 𝑡𝑖 + (𝑁 − 1)𝛥𝑡

.

.

.

𝑥𝑖

𝑥𝑓

Figure F.1: Paths in discretized spacetime. The general path (red line) contributes to thepath integral but it is weighted by the exponential of the action for that particular path.

time interval [ti, tf ] into N discrete time intervals [tn−1, tn] of constant step,

∆t = tn − tn−1 =tf − tiN

(F.5)

with (t0, ~x0 ≡ (ti, ~xi) and (tN , ~xN) ≡ (tf , ~xf ) and use the completeness of the eigenstates|~xn, tn〉 for each time instant tn,

K(~xf , tf ; ~xi, ti) =

dd~x1(t1) . . . dd~xN−1(tN−1) 〈~xf , tf |~xN−1, tN−1〉 . . . 〈~x1, t1|~xi, ti〉

=

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

〈~xm, tm|~xm−1, tm−1〉

)

=

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

〈~xm|U†(tm, tm−1)|~xm−1〉

) (F.6)

The continuous limit will be restored at the end of the calculation by taking N → ∞ or,equivalently, ∆t → 0, but it is important to have in mind that ∆t is supposed to be aninfinitesimal quantity.

The time evolution operator is in general impossible to be found explicitly due the factthat the differential equation is obeys (the Schrodinger equation) is an operator equationexcept for some special cases. One of such special case is the case of time independentHamiltonians for which the time evolution operator is takes the familiar form U(tf , ti) =

F.1. FROM QUANTUM MECHANICS TO PATH INTEGRALS 199

e−iH(tf−ti). Then the propagator becomes,

K(~xf , tf ; ~xi, ti) =

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

〈~xm|e−iH(tm−tm−1)|~xm−1〉

)

=

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

〈~xm|e−iH∆t|~xm−1〉

) (F.7)

The Hamiltonian operator now is an operator that depends on the position ~x and the con-jugate momentum ~p, H = H(~x, ~p). To exploit this fact, the completeness of the momentumoperator eigenstates1 |~p〉 is used in each inner product to write,

K(~xf , tf ; ~xi, ti) =

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

dd~pm 〈~xm|e−iH(~x,~p)∆t|~pm〉〈~pm|~xm−1〉

)(F.8)

Due to the fundamental commutation relation [xi, pj] = iδij, the Hamiltonian can al-

ways be rearranged to have all x’s on the right and all p’s on the left so 〈~xn|H|~pm〉 =H(~xn, ~pm) 〈~xn|~pm〉, therefore,

K(~xf , tf ; ~xi, ti) =

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

dd~pme

−iH(~xm,~pm)∆t 〈~xm|~pm〉〈~pm|~xm−1〉

)

=

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

dd~pme

−iH(~xm,~pm)∆t ei~xm·~pm

(2π)d2

e−i~xm−1·~pm

(2π)d2

)

=

(N−1∏n=1

dd~xn(tn)

)(N∏m=1

dd~pm

(2π)dexp

−i(H(~xm, ~pm)∆t− (~xm − ~xm−1) · ~pm

))(F.9)

The last step to complete the derivation is to perform the p-integrals. This is possible becausethe p dependence in the classical Hamiltonian H(~x, ~p) = T (~p)+V (~x) comes from the kinetic

term T (~p) = ~p2

2mand, thus, all p-integrals are Gaussian integrals. As a result,

dd~pm exp

−i(H(~xm, ~pm)∆t− (~xm − ~xm−1) · ~pm

)=

= e−iV (~xm)∆t

dd~pm exp

−i∆t

2m~p2m + i (~xm − ~xm−1) · ~pm

= e−iV (~xm)∆t

(d∏i=1

∞−∞

dpm,i exp

−i∆t

2mp2m,i + i (xm,i − xm−1,i) pm,i

)

= e−iV (~xm)∆t

(d∏i=1

√2πm

i∆tei∆t 1

2m(xm,i−xm−1,i

∆t

)2)

=

(d∏i=1

√2πm

i∆t

)exp

i∆t

(1

2m

(~xm − ~xm−1

∆t

)2

− V (~xm)

)(F.10)

1These are the actual eigenstates in the Schrodinger picture and, more accurately, at t = 0.


⇒ K(~xf , tf ; ~xi, ti) =

(2πm

i∆t

)Nd2

(N−1∏n=1

dd~xn(tn)

)×(

N∏m=1

exp

i∆t

(1

2m

(~xm − ~xm−1

∆t

)2

− V (~xm)

))

=

(2πm

i∆t

)Nd2

(N−1∏n=1

dd~xn(tn)

)×

exp

i

N∑m=1

∆t

(1

2m

(~xm − ~xm−1

∆t

)2

− V (~xm)

)(F.11)

Now it is finally time to restore continuity of space and time by taking ∆t → 0, N → ∞.At this regime, the quantity ~xm−~xm−1

∆tbecomes the time derivative of the position, i.e. the

velocity, and the sum turns into an integral. The integral is precisely the classical action ofthe system,

K(~xf , tf ; ~xi, ti) =

~x(tf )=~xf

~x(ti)=~xi

D~x exp

i

tf

ti

dt

(1

2m~x2 − V (~x)

)=

~x(tf )=~xf

~x(ti)=~xi

D~xeiS[~x]

(F.12)

with the integration measure defined by,

D~x ≡ lim∆t→0 , N→∞

(2iπm

∆t

)Nd2

(N−1∏n=1

dd~xn(tn)

)(F.13)

This result is remarkably simple. It says that the propagator is evaluated by adding upall the possible paths ~x(t), hence the name “path integral”, starting from ~xi at time instantti and ending to ~xf at time instant tf with each path weighted by the exponent of the action.The weight ensures that the leading contribution comes from the trajectory that extremizesthe action functional, i.e. the classical path. This is due to the oscillatory nature of theweight.

F.2 From Quantum Field Theory to Path Integrals

The derivation for the case of a quantum field theory with Lagrangian density L [φ, ∂φ]involving one field φ follows the same steps. The only difference is that both time and spaceneed to be discretized at the beginning. Eventually, the result will be almost identical. Theexplicit derivation will not be presented here since the line of thought has already beenintroduced right above. Eventually, the result will be,

〈φf (~x), tf |φi(~x), ti〉 = N φ(tf ,~x)=φf (~x)

φ(ti,~x)=φi(~x)

DφeiS[φ] (F.14)

F.3. PATH INTEGRAL REPRESENTATION OF STATES 201

with N a normalization factor.

This now is not a “path” integral, but rather a “configuration” integral because thepropagator is obtained by adding up all possible configurations φ(t, ~x), starting from φi(~x)at time instant ti and ending to φf (~x) at time instant tf with each configuration weightedby the exponent of the action functional. Nevertheless, the integral is also called a “pathintegral“ regardless of this misconception.

F.3 Path integral representation of states

The path integral formulation offers more than a way of simply evaluate the propagator. Italso allows the construction of the vacuum state from the action functional alone from whichthe wavefunctional Ψ[φ(x)] = 〈φ(~x), t|Ψ〉 of any state call also be acquired with the help ofmore details regarding the theory. This happens because the general state |Ψ〉 can alwaysbe written as a superposition of energy eigenstates,

|Ψ(t)〉 =∑n

cne−iEnt |En〉 (F.15)

and each energy eigenstate is obtained by acting with appropriate combinations of someladder operators, determined by the content of the theory, on the vacuum state |0〉 ≡ |E0〉,defined as the state with the lowest possible energy E0,

|En〉 = A†n |0〉 (F.16)

with A†n representing that combination.

The analysis becomes much more transparent by performing a Wick rotation to analyti-cally continue to Euclidean times τ = it,

|Ψ(τ)〉 =∑n

cne−Enτ |En〉 = e−E0τ

∑n

cne−(En−E0)τ |En〉 (F.17)

Consequently, when dealing with large Euclidean times τ →∞, all excitations decay leavingonly the vacuum state,

limτ→∞

e−τH |Ψ〉 = |0〉 (F.18)

As a result, the wave functional of the ground state Ψ0[φ(x)] is obtained by integratingover all possible configuration coming from the infinite past and ending on φ(~x) at timeinstant zero,

〈φ(x)|0〉 = N φ(0,~x)=φ(~x)

Dφ(τ < 0)e−SE [φ] (F.19)

with SE the Euclidean action functional.


F.4 Path integral representation of density matrices

In addition, there is a direct connection between components of the density matrix ρ associ-ated with a general state and the Euclidean path integral. In particular, they are identicalup to a normalization factor. This follows by considering the case of a canonical ensemble at

temperature β−1 for which the density matrix is thermal ρβ = 1Zβe−βH with Zβ = Tr

(e−βH

)the partition function. The expectation value of any observable O is then written as,

〈O〉β = Tr(ρβO

)=

1

ZβTr(e−βHO

)(F.20)

Things become very interesting when considering a 2-point function, that isO = O1(t)O2(t′),coming from the cyclic property of the trace which allows the introduction of unity in termsof the time evolution operator U(t, 0) = e−iHt

〈O1(t)O2(t′)〉β =1

ZβTr(e−βHO1(t)1O2(t′)

)=

1

ZβTr(e−βHO1(t)eβHe−βHO2(t′)

)=

1

ZβTr(e−βHO2(t′)e−βHO1(t)eβH

)=

1

ZβTr(e−βHO2(t′)O1(t− iβ)

)= 〈O2(t′)O1(t− iβ)〉β

(F.21)

known as the KMS2 relation. In quantum field theory, the propagator is just the product oftwo fundamental fields φ1 = φ(t, ~x) and φ2 = φ(t′, ~x′) and the product of two fundamentalfields is symmetric (anti-symmetric) for bosons (fermions) so the above relation is interpretedas a periodicity condition saying that the propagator is periodic (anti-periodic) with respectto the Euclidean time with period β,

〈φ(τ, ~x)φ(τ ′, ~x′)〉β = ±〈φ(τ + β, ~x)φ(τ ′, ~x′)〉β (F.22)

with “+” for bosons and “−” for fermions.The partition function Zβ can now be expressed as a path integral by working in the

basis of position eigenstates |~x〉 which means to write the trace as an integral,

Zβ = Tr(e−βH

)=

dd~x′ 〈~x′|e−βH |~x′〉

=

dd~x′ K(~x′, β; ~x′, 0)

=

dd~x′

~x(β)=~x′

~x(0)=~x′D~x e−SE [~x]

(F.23)

2“KMS”=“Kubo-Martin-Schwinger”

F.4. PATH INTEGRAL REPRESENTATION OF DENSITY MATRICES 203

This is just a path integral over all possible periodic paths with Euclidean period β so it canbe equivalently written as,

Zβ =

~x(0)=~x(β)

D~x e−SE [~x] (F.24)

or, in the case of a quantum field theory,

Zβ =

φ(0,~x)=φ(β,~x)

Dφ e−SE [φ] (F.25)

As a result, the thermal density matrix components can also be calculated via a pathintegral,

〈~xi|ρβ|~xj〉 =1

Zβ〈~xi|e−βH |~xj〉

=1

Zβ

~x(β)=~xi

~x(0)=~xj

D~x e−SE [~x] (for Quantum Mechanics)

〈φi(~x)|ρβ|φj(~x)〉 =1

Zβ〈φi(~x)|e−βH |φj(~x)〉

=1

Zβ

φ(β,~x)=φi(~x)

φ(0,~x)=φj(~x)

Dφ e−SE [φ] (for Quantum Field Theory)

(F.26)

Apart from the density matrix of a full system, the density matrix ρA,β of a subsystemA, the reduced density matrix, can also be represented with a path integral. After all, it isjust a density matrix and it was shown just above how to compute it with a path integral.The only difference now is that the path integral must be performed with respect to thesubsystem fields φA(t, ~x), while the fields φA(t, ~x) need to be traced out. This tracing outwas already seen in the case of the partition function to be path represented by a pathintegral over all possible periodic configurations so,

〈φAi (~x)|ρA,β|φAj (~x)〉 =1

Zβ

φA(0,~x)=φA(β,~x)

DφA φA(β,~x)=φAi (~x)

φA(0,~x)=φAj (~x)

DφA e−SE [φ] (F.27)

All of the above path representations are pictorially depicted in figure F.2.


ሺ0, 𝑥Ԧሻ = ሺ𝑥Ԧሻ

ሺ𝛽, 𝑥Ԧሻ = 𝑗ሺ𝑥Ԧሻ

ሺ0, 𝑥Ԧሻ = 𝑖ሺ𝑥Ԧሻ

Aሺ𝛽, 𝑥Ԧሻ = 𝑖

Aሺ𝑥Ԧሻ

Aሺ0, 𝑥Ԧሻ = 𝑗Aሺ𝑥Ԧሻ

𝛽

𝛽

𝛢ҧ

𝛢

= ሺ𝑥Ԧሻȁ0ۦ

= 𝑍𝛽 ർ𝑖ሺ𝑥Ԧሻቚ𝜌𝛽ቚ𝑗ሺ𝑥Ԧሻ

= 𝑍𝛽

= 𝑍𝛽 ർ𝑖Aሺ𝑥Ԧሻቚ𝜌𝐴,𝛽ቚ𝑗

Aሺ𝑥Ԧሻ

Figure F.2: Pictorial depiction of path integral representations of vacuum states, partitionfunctions and density matrices

Appendix G

Entanglement structure of a CFTvacuum

In this appendix, the basic notion for considering subsystems of a CFT in the vacuumstate will be discussed. The setup is straightforward: The full quantum system living ina spacetime B is in its vacuum state |0〉 and a subsystem A is considered. Choosing asubsystem means to take a spatial slice ΣB at some constant time instant t = t0 and thendraw a subset A on that spatial slice, as in figure 6.1. Essentially, the only review here is onhow some special regions in both the boundary and the dual bulk geometries.

G.1 Domain of Dependence

The first such region is associated with the boundary spacetime. This is the set DA ofall points p for which a causal trajectory (time-like or light-like) passing through p mustnecessarily intersect the subset A, known as domain of dependence associated with A1, as infigure G.1.

𝐴

𝛴𝛣

𝐷𝐴

Figure G.1: Domain of dependence DA associated with a subsystem A

1Sometimes, this is also referred to as causal development region or causal diamond associated with A

205

206 APPENDIX G. ENTANGLEMENT STRUCTURE OF A CFT VACUUM

𝐴 𝐴ҧ

𝐷𝐴

𝐷𝐴ҧ

𝐷𝐴ҧ

𝐷𝐴

Figure G.2: Domains of dependence DA and DA associated with a subsystem A and itscomplimentary A.

Classically, the domain of dependence is DA is that region within which initial dataspecified on the subset A are enough to fully specify the solutions of the field equations.Quantum mechanically, evolution of local operators defined in A determine all operators inthe entire domain DA. This means that a field theory defined in DA is a complete quantumsystem on its own.

It is interesting to notice that the domain of dependence DA of the complementary systemA does not cover the rest of the boundary spacetime (figure G.2). In other words, the densitymatrices ρA and ρA associated with the two subsystems do not contain all the informationabout the full system. The missing information is encrypted in the entanglement detailsbetween A and A, as represented by the following “relation”2,

|Ψ〉 → ρA + ρA + (entanglement information) (G.1)

G.2 Causal Wedge

The next two regions are defined in the bulk and are only valid for holographic boundarytheories with a dual bulk geometry. The first one is the set of all bulk points that are incausal contact with the domain of dependence DA of the subsystem A, called the causalwedge associated with DA. This the region within which an observer living in DA can sendand receive signals.

The causal wedge associated with the domain of dependence DA of the complementarysubsystem does not intersect the causal wedge associated with DA. Holographically, thismeans that changes made withing the causal wedge associated with DA do not alter the

2This is not an equation, it is way of representing how the information contained in the state |Ψ〉 of thefull system can be broken down into the information contained in the subsystems (which is stored in theensembles or, equivalently, the density matrices) plus the information contained in the entanglement of thetwo subsystems.

G.3. ENTANGLEMENT WEDGE 207

reduced density matrix ρA associated with DA. An example of the shape of the causalwedge is the case of the CFT vacuum state and choosing the subsystem A to be a ball-shaped region. Then the causal wedge is precisely the the AdS-Rindler wedge.

G.3 Entanglement Wedge

Lastly, there is another important region that extend in the bulk geometry associated withthe holographic CFT. This is the entanglement wedge defined as the region bounded by thedomain of dependence DA associated with a subsystem A and the region A, the extremalbulk area surface with the same boundary as A involved in the Ryu-Takayanagi formula.

The entanglement wedge is just the union of all regions A′ associated with all spatialslices A′ of the domain of dependence DA according to the Ruy-Takayanagi prescription.For example, the entanglement wedge for a ball-shaped subsystem with radius R, A = Bd−1

R

with the full system in its vacuum state is an AdS-Rindler-like wedge.

208 APPENDIX G. ENTANGLEMENT STRUCTURE OF A CFT VACUUM

Appendix H

Modular Hamiltonian

In this appendix, the notion of the modular Hamiltonian will be better understood in termsits definition and the symmetry it is associated with. In addition, the modular Hamiltonianwill be extracted for two special cases used in the main part of this script, namely for thehalf-space and the ball-shaped regions.

H.1 Definition and Symmetry generator

The modular Hamiltonian is an alternative way of saying that the density matrix is a her-mitian operator with positive eigenvalues. These two properties, allow to write the densitymatrix ρ as an exponential of a hermitian operator Hmod,

ρ = e−Hmod (H.1)

which defines the modular Hamiltonian Hmod. In general, modular Hamiltonians are non-local operators and is most cases they are extremely difficult to extract. Its importance,however, comes a very simple observation: the density matrix is thermal in terms of Hmod

with temperature β = 1. This might look like a peculiar statement since a thermal densityneeds to have the partition function Zβ for normalization issues. This does not really mattersince a more “Hamiltonian-like” operator H can be expressed directly through the modularHamiltonian according to,

Hmod = βH + logZβ (H.2)

where a more natural temperature β is allowed to be present and the normalization of thedensity matrix (Tr ρ = 1) is ensured by the partition function,

Zβ = Tr(e−βH

)(H.3)

This is just a rewriting of the modular Hamiltonian with some numerical coefficients (βand Zβ) on the right. As already seen in appendix F, a quantum field theory at imaginarytimes turns into a field theory at finite temperature with the propagator becoming a thermaldensity matrix at a temperature β−1 with β giving the periodicity of the Euclidean time;

209

210 APPENDIX H. MODULAR HAMILTONIAN

the Hamiltonian of the quantum field theory is precisely the Hamiltonian appearing in thedensity matrix exponent.

Now the thought is done backwards. Given that a density matrix can be written as athermal one for some properly chosen operator Hmod, then Hmod is an actual Hamiltonianin the sense that it is associated with a full quantum system described by a quantum fieldtheory. For the case of a subsystem A, this full quantum system is the domain of dependenceDA as already argued in appendix G.

Since Hmod is an actual Hamiltonian, it generates translations along some kind of timedirection. This “time” is the actual time describing the evolution of the quantum systemonly when the entire system is taken into account, but when a subsystem is considered,then this is a fictitious time, call it the “modular time”, describing evolution only withinthe domain of dependence associated with the subsystem. This modular time ranges from−∞ to +∞ like usual time but the infinite past and infinite future correspond to the pastand future peaks of the domain of dependence respectively. In other words, it is the time anobserver restricted within the domain of dependence realizes.

Formally, the symmetry group associated with this symmetry is called the modular groupbut this is not a uniquely defined group, not even an actual group as a matter of fact. Aswill be demonstrated in the following two examples, the modular group can be the set ofboosts or special conformal transformations or even something else. These are not full groupsbecause their algebra is in general not closed, i.e. some group transformations are missing.

H.2 Half-space subsystems

The first example to be considered is the case of the full quantum system living in (D = d+1)-dimensional Minkowski spacetime,

ds2 = −dt2 + d~x2 , ~x = (x1, . . . , xd) (H.4)

and taking the subsystem A to be the half-space region x1 > 0 at time instant t = 0. Thedomain of dependence associated with this subsystem is depicted in figure H.1 and it isjust the Rindler wedge which is familiarly introduced in an undergraduate course on generalrelativity.

To work in the domain of dependence of the half-space, proper coordinates need to beintroduced which are restricted within the Rindler wedge. These coordinates are no otherthan the Rindler coordinates,

t = ρ sinhω

x1 = ρ coshω

ω ∈ [−∞,∞]

ρ > 0

(H.5)

in terms of which the invariant length reads,

ds2 = −ρ2dω2 + dρ2 +d∑i=2

(dxi)2

(H.6)

H.2. HALF-SPACE SUBSYSTEMS 211

𝑡

𝑥1

𝜔 = 𝑡 = 0

Figure H.1: Domain of dependence DA associated with the half-space x1 > 0 at t = 0. Thestraight blue lines are lines of constant Rindler time ω. ω-translations are generated by theRindler Hamiltonian Hω and these are just boosts with respect to the stationary Minkowskiobserver. The dashed line is a trajectory of constant ρ, i.e. constant acceleration. It is onthis trajectory a Rindler observer lives.

The modular Hamiltonian will now be the generator of ω-translations. These look likerotations along the t-axes to the Minkowski observer and, as should be known from theLorentz symmetry discussion, rotations around time are boosts. To make it explicit, it ismore transparent to see what happens in Euclidean times tE = it and ωE = iω. Then, thecoordinate transformation takes the form of a Cartesian-to-polar coordinates transformationin 2 dimensions,

tE = ρ sinωE

x1 = ρ cosωE(H.7)

and evolution along the Euclidean Rindler time now means to actually rotate the tE = ωE = 0spatial slice by an angle ωE. These rotations are generated by the 2-dimensional rotationelements RωE ,

RωE

tEx1

xi

=

cosωE sinωE 0− sinωE cosωE 0

0 0 1(d−1)×(d−1)

tEx1

xi

(H.8)

In Lorentzian times, the above transformation reads,

Rω

tx1

xi

=

coshω − sinhω 0− sinhω coshω 0

0 0 1(d−1)×(d−1)

tx1

xi

(H.9)

This is just a boost along the x1 direction with rapidity ω so the modular Hamiltonian


associated with half-space is just the boost generator (A.42) along x1 at t = 0,

HHSmod = 2πHω + Z2π = 2π

t=0,x1>0

dd~x x1T00(~x) + Z2π (H.10)

The modular Hamiltonian above was written as in the expansion (H.2) with 2π theinverse temperature β a Rindler observer measures1 and Hω the Rindler Hamiltonian (theboost generator). The partition function Z2π is written just to ensure the normalization ofthe density matrix (Tr ρ = 1) and it is given by,

Z2π = Tr(e−2πHω

)(H.11)

An alternative extraction of the Rindler Hamiltonian is obtained by considering theKilling vector ζµω associated with translations along the Rindler time ω. This is trivial tofind since the metric in Rindler coordinates does not depend on the Rindler time. Followingthe relative discussion in Step 3 of Section 7.4.1, the contra-variant Killing vector is just,

ζµω = δµω (H.12)

and the corresponding generator is precisely the generator of Rindler time translations,

Gω = −iζµω∂µ = −i∂ω (H.13)

This allows the extraction of the Rindler Hamiltonian in terms of the energy-momentumtensor Tµν(x) measured in Minkowski spacetime. In particular, since the Killing vector ζµωis an actual vector field, coordinate transformations are easily translated via the definingtransformation of a general vector,

ζ ′µ′

ω =∂x′µ

′

∂xµζµω (H.14)

where ζ ′µ′

ω is the same Killing vector expressed in the new x′µ′coordinates. For the case where

the new coordinates are the Minkowski coordinates x′µ′= (t, ~x) and the old coordinates are

the Rindler coordinates xµ = (ω, ρ, xi), i = 2, . . . , d, the Killing vector reads,

ζ ′µ′

ω =∂x′µ

′

∂xµδµω

=∂x′µ

′

∂ω

=∂t

∂ωδµ′

t +∂x1

∂ωδµ′

1

= x1δµ′

t + tδµ′

1

(H.15)

and the corresponding generator is precisely the boost generator K1 along the x1-direction,

G′ω = −iζ ′µ′ω ∂′µ′ = −i(x1∂t + t∂1

)= K1 (H.16)

1This comes from the fact that the Euclidean Rindler time ωE needs to be periodic with period 2π toavoid a conical singularity. Euclidean Rindler time is just the polar angle in polar coordinates.

H.3. BALL-SHAPED SUBSYSTEMS 213

The relevant conserved charge, the Rindler Hamiltonian by definition, is then associatedwith the energy-momentum tensor according to (A.42),

Hω =

dd~x

(x1T00(x) + tT01(x)

)(H.17)

and at the time instant ω = t = 0, this is precisely the Rindler Hamiltonian reported in theprevious discussion.

H.3 Ball-shaped subsystems

There is another special case of the subset A for which the modular Hamiltonian can beexactly extracted. It is the case of ball-shaped subsystems A = Bd(R, ~x0) of radius Rcentered at the point ~x0. So, somehow, a proper coordinate transformation must be foundto map the Minkowski spacetime to the spacetime of a d-dimensional ball.

Fortunately, such a transformation exists; it is a conformal transformation that mapsthe half-space region x1 > 0 into the region of the ball. In particular, the coordinatetransformation,

yµ =xµ − x2bµ

1− 2b · x+ b2x2+ 2R2bµ

bµ =1

2Rδ µ

1

(H.18)

maps the Minkowski spacetime parametrized by xµ into the “locus”2,

y2 = R2 1− x1

R+ x2

4R2

1 + x1

R+ x2

4R2

(H.19)

with metric,ds2 = ηµνdx

µdxν

= Ω2ηµνdyµdyν

(H.20)

where the scale factor Ω is given by,

Ω = 1− 2b · x+ b2x2

=(1− 2b · y + b2y2

)−1 (H.21)

The above transformation has some basic properties that justify its importance. Forstarters the spatial slice t ≡ x0 = 0 is the spatial slice y0 = 0. In addition, at t = 0, ~y2 ≤ R2

given that x1 ≥ 0. In other words, the ball Bd(R) centered at the origin3 is just a morecomplex form of the half-space so the temperatures are the same.

2x2 is not the coordinate xi with i = 2; it is the norm x2 = xµxµ. On the other hand, x1 is the actual

coordinate xi with i = 1.3For ~x0 6= ~0 and t0 6= 0, simply replace xi, i = 1, . . . , d with xi − xi0 and t with t− t0.


Taking a closer look at the transformation (H.18) it is just a conformal transformationxµ → x′µ = eibνK

νxµ followed by a translation x′µ → yµ = eiR

2bνP νx′µ. Consequently, thegenerators of yµ-translations are,

Gµ =1

2R

(Kµ +R2Pµ

)(H.22)

Here seems to come a problem though. The coordinate y0 cannot be directly identi-fied with the modular time due to multiple discontinuities coming from the poles of thedenominator, namely, when t = ±tb, tb =

√~x2 + 4Rx1 + 4R2,

y0 =t

1 + x1

R+ ~x2−t2

4R2

(H.23)

In order to properly constraint the y0 coordinate in the range (−∞,∞) and have it to becontinuous, the Minkowski time needs to be limited in the range (−tb,+tb). This shouldbe expected since these two points in time correspond to the “infinite” past and “infinite”future, the peaks, of the domain of dependence of the ball.

Consequently, the modular time can be identified with y0 and the modular Hamiltonianis ontained by the generator G0 of y0-translations,

Hmod = 2π1

2R

(Kt +R2Pt

)+ logZ2π

=π

R

Bd(R)

dd~x(R2 − ~x2

)T00(~x) + logZ2π

(H.24)

In the above equation, the “ball”-Hamiltonian is just G0 and the inverse temperature isβ = 2π.

Appendix I

Newtonian Gravitational Potential inHigher Dimensions

In this appendix the Newtonian gravitational potential will be extracted for the case of aflat (D = d+ 1)-dimensional spacetime with d spatial dimensions.

In general, the Newtonian gravitational potential is what remains from the Newtonianlimit of Einstein field equations, that is, the limit hµν 1 in the perturbation expansion ofthe full metric gµν = ηµν+hµν for a static source where ηµν is the Minkowski spacetime metric.A simple argument regarding the geodesic equations of motion suggests that h00 = −2φwith φ being the Newtonian gravitational potential. The next step would be to see what thecoefficient κ in Einstein’s field equations,

Rµν −1

2gµνR = κTµν (I.1)

should be equal to in order to give the proper Newtonian limit for a pressureless ideal liquid.When talking about a theory of gravitational interactions, however, it is natural to definethis constant κ to be the coupling constant that is present in the Einstein-Hilbert action,

S =1

16πG

dDx√−gR ≡ 1

2κ

dDx√−gR (I.2)

which is the same as saying that the “D-dimensional Newton’s constant” G is defined sothat Einstein’s field equations are,

Rµν −1

2gµνR = 8πGTµν (I.3)

Contracting with the inverse of the metric allows to rewrite the Ricci scalar as a function ofthe trace of the energy-momentum tensor yielding the equivalent form of the Einstein fieldequations,

Rµν = 8πG

(Tµν −

1

D − 2gµνT

)(I.4)

The advantage of (I.4) over (I.3) becomes clear when applying the Newtonian limit. The00-component of the Ricci tensor up to first order in the perturbation is actually straight

215

216APPENDIX I. NEWTONIANGRAVITATIONAL POTENTIAL IN HIGHER DIMENSIONS

forward to calculate since the products of the Christoffel symbols are O (h2) and ∂thµν = 0,

R00 ' ∂iΓi00 ' −

1

2∇2h00 (I.5)

The energy-momentum tensor itself is trivial. For a pressureless ideal liquid with massdensity ρ,

Tµν = ρδµ0δν0 ⇒ T = g00ρ = −ρ+ O (h) (I.6)

and the 00-component of Einstein’s field equations gives rise to the Poisson equation,

∇2φ =8πG (D − 3)

D − 2ρ (I.7)

From this, the Newtonian gravitational potential of a point particle with mass M at position~r0 can be calculated to be,

φ = − 8πGM

(D − 2) ΩD−2

1

| ~r − ~r0 |D−3(I.8)

This result becomes more transparent from the fact that,

∇2 1

| ~r − ~r0 |d−2= − (d− 2) Ωd−1δ

d (~r − ~r0) (I.9)

easily proven by integrating both sides and applying Gauss’s theorem on the left side.

Appendix J

Post Newtonian Approximation

In this appendix the method of post Newtonian approximation in general relativity willbe explained and extract the main results used in Section 7.2.1. The post Newtonian ap-proximation is nothing more than an analytical method for founding corrections of the flatspacetime metric in D = d+ 1 dimensions due to the presence of some mass distribution.

The idea is that a probe moving in the gravitational field of the source of typical massM at a typical distance r would have a typical speed of order,

υ2 ∼ GM

rD−3(J.1)

As a result, one is allowed to write the following perturbative expansions of the metrictensor gµν , the affine connection Γρµν and the Ricci Tensor Rµν ,

Metric Tensor

((N)gµν ∼ υN

)

g00 = −1 +(2)g00 +

(4)g00 + . . .

gi0 =(3)gi0 +

(5)gi0 + . . .

gij = δij +(2)gij +

(4)gij + . . .

(J.2)

217

218 APPENDIX J. POST NEWTONIAN APPROXIMATION

Affine Connection

((N)

Γρµν ∼ υN

r

)

Γ000 =

(3)

Γ000 +

(5)

Γ000 + . . .

Γ0i0 =

(2)

Γ0i0 +

(4)

Γ0i0 + . . .

Γ0ij =

(3)

Γ0ij +

(5)

Γ0ij + . . .

Γi00 =(2)

Γi00 +(4)

Γi00 + . . .

Γij0 =(3)

Γij0 +(5)

Γij0 + . . .

Γijk =(2)

Γijk +(4)

Γijk + . . .

(J.3)

Ricci Tensor

((N)

Rµν ∼ υN

r2

)R00 =

(2)

R00 +(4)

R00 + . . .

Ri0 =(3)

Ri0 +(5)

Ri0 + . . .

Rij =(2)

Rij +(4)

Rij + . . .

(J.4)

From here until the end of this appendix, everything will be expressed with lower indices,but still, a repeated index will imply a sum over that index. The task now is to find the first

order perturbations(2)g00 and

(2)gij. Subsequently, it is sufficient to express the

(2)

R00 and(2)

Rij interms of these. The first step is to find the inverse metric first order corrections,

g00 = −1 +(2)

g00 + . . .

gij = δij +(2)

gij + . . .

(J.5)

which is easy to do since gµρgρµ = δµν , so,

(2)

g00 = −(2)g00

(2)

gij = −(2)gij

(J.6)

The second step is to express(2)

Γ0i0,

(2)

Γi00 and(2)

Γijk in terms of the metric corrections. Fromthe expression of the affine connection components in terms of the metric components, this

219

turns out be,(2)

Γ0i0 =

(2)

Γi00 = −1

2∂i

(2)g00

(2)

Γijk =1

2

(∂k

(2)gij + ∂j

(2)gik − ∂i

(2)gjk

) (J.7)

The first order corrections to the Ricci tensor components are, thus,

(2)

R00 = −∂i(2)

Γi00

=1

2∇2 (2)g00

(2)

Rij = ∂j

(2)

Γ0i0 + ∂j

(2)

Γkik − ∂k(2)

Γkij

= −1

2∂i∂j

(2)g00 +

1

2∂i∂j

(2)gkk −

1

2∂j∂k

(2)gik

− 1

2∂i∂k

(2)gjk +

1

2∇2 (2)gij

(J.8)

Working in harmonic coordinates, that is applying the harmonic coordinates conditions,

gµνΓρµν = 0 (J.9)

implies (second order term for ρ = i),

1

2∂i

(2)g00 + ∂j

(2)gij −

1

2∂i

(2)gjj = 0

⇒∂i∂j(2)gjk + ∂j∂k

(2)gij − ∂i∂k

(2)gjj + ∂i∂k

(2)g00 = 0

(J.10)

and the Ricci tensor components reduce to,

(2)

R00 =1

2∇2 (2)g00

(2)

Rij =1

2∇2 (2)gij

(J.11)

Now, Einstein’s Field equations (I.4) can be applied. From their interpretation as energydensity, momentum density and momentum flux, the T 00, T i0 and T ij components of the

energy-momentum tensor may be expanded as

((N)

T µν ∼ MυN

rD−1

),

T 00 =(0)

T 00 +(2)

T 00 + . . .

T i0 =(1)

T i0 +(3)

T i0 + . . .

T ij =(2)

T ij +(4)

T ij + . . .

(J.12)


As a result, the tensor,

Sµν ≡ Tµν −1

D − 2Tgµν (J.13)

has the perturbative expansion,

S00 =(0)

S00 +(2)

S00 + . . .

Si0 =(1)

Si0 +(3)

Si0 + . . .

Sij =(2)

Sij +(4)

Sij + . . .

(J.14)

It is crucial to realize that the perturbative Einstein’s field equations are,

(N)

Rµν = −8πG(N−2)

Sµν (J.15)

so what is needed for the first order corrections are the quantities(0)

S00 and(0)

Sij which in termsof only the fully covariant energy-momentum tensor components are simply,

(0)

S00 =D − 3

D − 2

(0)

T 00

(0)

Sij =1

D − 2δij

(0)

T 00

(J.16)

Putting together (J.11) and (J.16) yields,

∇2 (2)g00 = −16πG (D − 3)

D − 2

(0)

T 00

∇2 (2)gij = − 16πG

D − 2δij

(0)

T 00

(J.17)

and since(0)

T 00 is just the rest-mass density, the Poisson equation (I.7) for the Newtoniangravitational potential implies the final result,

(2)g00 = −2φ

(2)gij = − 2

D − 3φδij

(J.18)

An example to check this result, is the Schwarzschild solution,


f(r)+ r2dΩ2

D−2

f(r) = 1−(RS

r

)D−3 (J.19)

221

with RS the Schwarzschild radius which is related to mass M of the Schwarzschild blackhole,

RS =

(16πGM

(D − 2) ΩD−2

) 1D−3

(J.20)

The final results (J.18) are expressed in harmonic coordinates, so the Schwarzschild coor-dinates (t, r, θi), i = 2, . . . , D−1 need to be transformed to them. If the harmonic coordinatesare chosen to be of the form (t,Xi = R(r)Ωi), i = 1, . . . , D−1 then the harmonic coordinateconditions (J.9) imply that the coordinate R(r) must solve the differential equation,

R′′(r) +

(f ′ (r)

f (r)+D − 2

r

)R′ (r)− D − 2

r2f (r)R (r) = 0 (J.21)

In terms of the harmonic coordinates the Schwarzschild metric is,

ds2 = −fdt2 +r2

R2d ~X2 +

[1

fR2R′2− r2

R4

](~X · d ~X

)2

= −fdt2 +

[(1

fR′2− r2

R2

)ΩiΩj +

r2

R2δij

]dXidXj

(J.22)

⇒ g00 = −f , gij =

(1

fR′2− r2

R2

)ΩiΩj +

r2

R2δij (J.23)

The solution of the differential equation is written in terms of (RS/r)D−3 (which is linear

in the mass and, hence, the Newtonian gravitational potential) as,

R (r) = r 2F1

(− 1

D − 3,− 1

D − 3;− 2

D − 3;

(RS

r

)D−3)

(J.24)

properly normalized to give flat spacetime in the limit r →∞.

The next step is to see what the ∼(RSr

)D−3terms give,

1

fR′2=

[1−

(RS

r

)D−3]−1

×

[2F1

(− 1

D − 3,− 1

D − 3;− 2

D − 3;

(RS

r

)D−3)

+

(RS

r

)D−3

2F1

(D − 4

D − 3,D − 4

D − 3;D − 5

D − 3;

(RS

r

)D−3)]−2

= 1 +1

D − 3

(RS

r

)D−3

+O

(RS

r

)2(D−3)

(J.25)


and,

r2

R2=

[2F1

(− 1

D − 3,− 1

D − 3;− 2

D − 3;

(RS

r

)D−3)]−2

= 1 +1

D − 3

(RS

r

)D−3

+O

(RS

r

)2(D−3) (J.26)

where the expansion of the hypergeometric function,

2F1 (a, b; c;x) =∞∑n=0

(a)(n) (b)(n)

(c)(n)

xn

n!

= 1 +ab

cx+O

x2 (J.27)

was used, so, finally,

gij =

(1 +

1

D − 3

(RS

r

)D−3)δij +O

(RS

r

)2(D−3)

=

(1− 2

D − 3φ

)δij +O

(RS

r

)2(D−3) (J.28)

while the 00-component is, imperturbably,

g00 = −

(1−

(RS

r

)D−3)

= − (1 + 2φ) (J.29)

with φ the Newtonian gravitational potential,

φ = − 8πGM

(D − 2) ΩD−2

1

rD−3= −2

(RS

r

)D−3

(J.30)

extracted in Appendix E. These are precisely what the post Newtonian approximation (J.18)predicts in first order to the Newtonian gravitational potential.

Appendix K

Mean Value Theorem for Integrals

In this appendix, the mean value theorem for integrals will proved in d spatial dimensionsfor the sake of applying it in Section 7.3.2.

To begin with, Green’s theorem is applied to the following integral over the volume of ad-dimensional ball of radius R, denoted as Bd(R),

Bd(R)

ddx(ψ∇2φ− φ∇2ψ

)=

Sd−1(R)

d ~Ad−1 · (ψ∇φ− φ∇ψ) (K.1)

where d ~Ad−1 = Rd−1dΩd−1r is the elementary vector normal to the sphere Sd−1(R) = ∂Bd(R)of radius R. Choosing the function ψ to be,

ψ (r) = − 1

(d− 2) Ωd−1

(1

rd−2− 1

Rd−2

)(K.2)

such that ψ (R) = 0 and ∇2ψ = δd (~r) and assuming that ∇2φ = 0 inside the ball, equation(K.1) reduces to the mean value theorem for integrals in d spatial dimensions,

Sd−1(R)

dΩd−1φ = Ωd−1φC (K.3)

with φC being the value of the function φ (~r) at the center of the sphere Sd−1R . If ∇2φ 6= 0

then, the integral includes an extra term,Sd−1(R)

dΩd−1φ = Ωd−1φC +1

d− 2

Bd(R)

drdΩd−1

(r − rd−1

Rd−2

)∇2φ (K.4)

For example, the mean value theorem applied for the case of,

φ = − 8πGM

(D − 2) ΩD−2

1

(r2 + z20 − 2z0r cos θ2)

(D−3)/2(K.5)

where z0 is a positive constant and θ2 is the angle between the position vector and the “z”axis (the axis number 1) gives the result,

Sd−1(R)

dΩd−1φ = − 8πGM

(D − 2)

1

zD−30

(K.6)

223

224 APPENDIX K. MEAN VALUE THEOREM FOR INTEGRALS

where the center of the sphere was chosen to be at the origin and R < z0.The explicit calculation involves the integrals,

Kn =

π

0

dθsinn θ

(R2 + z20 − 2z0R cos θ)

n/2=

1

zn0

√π

Γ(n+1

2

)Γ(n+2

2

)In =

π

0

dθ sinn θ =√π

Γ(n+1

2

)Γ(n+2

2

) (K.7)

and the result is,

Sd−1(R)

dΩd−1φ = − 8πGM

(D − 2) ΩD−2

KD−3

(D−4∏n=1

In

)2π

= − 8πGM

(D − 2) ΩD−2

1

zD−30

2π(D−1)/2

Γ(D−1

2

)= − 8πGM

(D − 2) ΩD−2

1

zD−30

ΩD−2

= − 8πGM

(D − 2)

1

zD−30

(K.8)

in full agreement with what the theorem predicts.

Appendix L

Calculation of the Kretchmanninvariant

In this appendix, the MATHEMATICA code behind the calculation of the Kretchmann in-

variant RµνκλRµνκλ for the Schwarzschild metric in D = d + 1 dimensions will be reported.

For starters, the code for evaluating the Kretchmann invariant in any number of dimension

is,

Kretchmann[d Integer]:=Kretchmann[d Integer]:=Kretchmann[d Integer]:=

Module[g, ginv,Γ,Riemann,RiemannCo,RiemannContra,Module[g, ginv,Γ,Riemann,RiemannCo,RiemannContra,Module[g, ginv,Γ,Riemann,RiemannCo,RiemannContra,

g = Module[f, gii,g = Module[f, gii,g = Module[f, gii,

f :=(1− (R/x[2])∧(d− 3));f :=(1− (R/x[2])∧(d− 3));f :=(1− (R/x[2])∧(d− 3));

gii[i ]:=If[i == 3, 1,Product[(Sin[x[p]])∧2, p, 3, i− 1]];gii[i ]:=If[i == 3, 1,Product[(Sin[x[p]])∧2, p, 3, i− 1]];gii[i ]:=If[i == 3, 1,Product[(Sin[x[p]])∧2, p, 3, i− 1]];

Table[Table[Table[

Which[Which[Which[

i == j == 1, f,i == j == 1, f,i == j == 1, f,

i == j == 2, 1/f,i == j == 2, 1/f,i == j == 2, 1/f,

i == j, (x[2])∧2 ∗ gii[i],i == j, (x[2])∧2 ∗ gii[i],i == j, (x[2])∧2 ∗ gii[i],

i 6= j, 0i 6= j, 0i 6= j, 0

],],],

i, 1, d, j, 1, d]i, 1, d, j, 1, d]i, 1, d, j, 1, d]

225

226 APPENDIX L. CALCULATION OF THE KRETCHMANN INVARIANT

]//Simplify;]//Simplify;]//Simplify;

ginv = Inverse[g]//Simplify;ginv = Inverse[g]//Simplify;ginv = Inverse[g]//Simplify;

Γ = Table[Table[(1/2) ∗ Sum[ginv[[ρ, σ]]∗Γ = Table[Table[(1/2) ∗ Sum[ginv[[ρ, σ]]∗Γ = Table[Table[(1/2) ∗ Sum[ginv[[ρ, σ]]∗

(D[g[[σ, ν]], x[µ]] +D[g[[σ, µ]], x[ν]]−D[g[[µ, ν]], x[σ]]),(D[g[[σ, ν]], x[µ]] +D[g[[σ, µ]], x[ν]]−D[g[[µ, ν]], x[σ]]),(D[g[[σ, ν]], x[µ]] +D[g[[σ, µ]], x[ν]]−D[g[[µ, ν]], x[σ]]),

σ, 1, d], µ, 1, d, ν, 1, d], ρ, 1, d]//Simplify;σ, 1, d], µ, 1, d, ν, 1, d], ρ, 1, d]//Simplify;σ, 1, d], µ, 1, d, ν, 1, d], ρ, 1, d]//Simplify;

Riemann = Table[Table[(D[Γ[[ρ]][[ν, σ]], x[µ]]−D[Γ[[ρ]][[µ, σ]], x[ν]]+Riemann = Table[Table[(D[Γ[[ρ]][[ν, σ]], x[µ]]−D[Γ[[ρ]][[µ, σ]], x[ν]]+Riemann = Table[Table[(D[Γ[[ρ]][[ν, σ]], x[µ]]−D[Γ[[ρ]][[µ, σ]], x[ν]]+

Sum[(Γ[[ρ]][[µ, λ]] ∗ Γ[[λ]][[ν, σ]]− Γ[[ρ]][[ν, λ]] ∗ Γ[[λ]][[µ, σ]]),Sum[(Γ[[ρ]][[µ, λ]] ∗ Γ[[λ]][[ν, σ]]− Γ[[ρ]][[ν, λ]] ∗ Γ[[λ]][[µ, σ]]),Sum[(Γ[[ρ]][[µ, λ]] ∗ Γ[[λ]][[ν, σ]]− Γ[[ρ]][[ν, λ]] ∗ Γ[[λ]][[µ, σ]]),

λ, 1, d]), σ, 1, d, µ, 1, d, ν, 1, d], ρ, 1, d];λ, 1, d]), σ, 1, d, µ, 1, d, ν, 1, d], ρ, 1, d];λ, 1, d]), σ, 1, d, µ, 1, d, ν, 1, d], ρ, 1, d];

RiemannCo = Table[Sum[g[[ρ, κ]] ∗ Riemann[[κ]][[σ, µ, ν]],RiemannCo = Table[Sum[g[[ρ, κ]] ∗ Riemann[[κ]][[σ, µ, ν]],RiemannCo = Table[Sum[g[[ρ, κ]] ∗ Riemann[[κ]][[σ, µ, ν]],

κ, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];κ, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];κ, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];

RiemannContra = Table[Sum[ginv[[σ, κ]] ∗ ginv[[µ, λ]] ∗ ginv[[ν, ε]] ∗ Riemann[[ρ]][[κ, λ, ε]],RiemannContra = Table[Sum[ginv[[σ, κ]] ∗ ginv[[µ, λ]] ∗ ginv[[ν, ε]] ∗ Riemann[[ρ]][[κ, λ, ε]],RiemannContra = Table[Sum[ginv[[σ, κ]] ∗ ginv[[µ, λ]] ∗ ginv[[ν, ε]] ∗ Riemann[[ρ]][[κ, λ, ε]],

κ, 1, d, λ, 1, d, ε, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];κ, 1, d, λ, 1, d, ε, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];κ, 1, d, λ, 1, d, ε, 1, d], ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d];

Sum[RiemannCo[[ρ, σ, µ, ν]] ∗ RiemannContra[[ρ, σ, µ, ν]],Sum[RiemannCo[[ρ, σ, µ, ν]] ∗ RiemannContra[[ρ, σ, µ, ν]],Sum[RiemannCo[[ρ, σ, µ, ν]] ∗ RiemannContra[[ρ, σ, µ, ν]],

ρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d]//Simplifyρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d]//Simplifyρ, 1, d, σ, 1, d, µ, 1, d, ν, 1, d]//Simplify

with the following symbolic identifications,

“ d ”= D (Number of spacetime dimensions1)

“ g[[µ, ν]] ”= gµν

“ R ”= RS (Schwarzschild radius)

“ x[µ] ”= xµ (x[2] = r)

“ ginv[[µ, ν]] ”= gµν

“ Γ[[ρ]][[µ, ν]] ”= Γρµν

“ Riemann[[ρ]][[σ, µ, ν]] ”= Rρσµν

1The symbol “D” in MATHEMATICA is reserved for the derivative.

227

“ RiemanCo[[ρ, σ, µ, ν]] ”= Rρσµν

“ RiemanContra[[ρ, σ, µ, ν]] ”= Rρσµν

It is important to realize that the final result,

RµνκλRµνκλ =

(D − 1)(D − 2)2(D − 3)

R4S

(RS

r

)2(D−1)

(L.1)

was extracted through induction. To justify this, the following new function in MATHE-

MATICA is defined,

KretchmannTest[d Integer]:=KretchmannTest[d Integer]:=KretchmannTest[d Integer]:=

(d−1)(d−2)2(d−3)R4

(Rx[2]

)2(d−1)

//Simplify(d−1)(d−2)2(d−3)

R4

(Rx[2]

)2(d−1)

//Simplify(d−1)(d−2)2(d−3)R4

(Rx[2]

)2(d−1)

//Simplify

and the following tests for D = 4, 5, . . . , 9 show the correctness of the expression,

Kretchmann[4]Kretchmann[4]Kretchmann[4]

KretchmannTest[4]KretchmannTest[4]KretchmannTest[4]

12R2

x[2]6

12R2

x[2]6



72R4

x[2]8

72R4

x[2]8



240R6

x[2]10

240R6

x[2]10



600R8

x[2]12

600R8

x[2]12



1260R10

x[2]14

1260R10

x[2]14



2352R12

x[2]16

2352R12

x[2]16

228 APPENDIX L. CALCULATION OF THE KRETCHMANN INVARIANT

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