Copyright

by

Syed Asif Hassan

2015

The Dissertation Committee for Syed Asif Hassan certifies that this isthe approved version of the following dissertation:

Dynamical Refinement in Loop Quantum Gravity

Committee:

Richard A Matzner, Supervisor

Duane A Dicus

Daniel S Freed

Philip J Morrison

Steven Weinberg

Dynamical Refinement in Loop Quantum Gravity

by

Syed Asif Hassan, B.S.;M.S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the degree of

Doctor of Philosophy

The University of Texas at Austin

August, 2015

Acknowledgements

First I thank my advisor Richard Matzer for his warm words of encouragement, for

his patience and for allowing me the freedom to pursue a research topic that I am

passionate about. Thanks also to my committee for their time and effort, to Steven

Weinberg for keeping me on task about the physics, to Duane Dicus for his patience,

to Philip Morrison for his excellent course, and to Dan Freed also for excellent course-

work that profoundly affected my mathematical understanding of forms, bundles and

connections and that prompted me to learn about the beautiful theory of geometric

quantization.

Thanks to Sarah Biedenharn for her generous funding of the Biedenharn Fellow-

ship that provided me with summer support. Thanks also to the Texas Cosmology

Center for funding multiple conference trips and my semester-long visit to Marseille.

Thanks especially to Eichiro Komatsu for his flexibility and openmindedness about

Loop Quantum Gravity.

To Carlo Rovelli I am deeply thankful for his invitation to visit his research group

in Marseille (Centre de Physique Theorique at Luminy), for his hospitality and for

many conversations that helped me understand the foundations of spin foam models.

To Andy Randono, thank you for introducing me to Loop Quantum Gravity and

getting me excited about the field. To Francesca Vidotto, thank you for encouraging

me and for sharing the details of your work with me. To Tom Mainiero, Justin Feng,

Ed Wilson-Ewing, Wolfgang Wieland, Jacek Puchta, Joel Meyers, Hal Haggard, Jeff

Hazboun and Dan Carney, thank you for useful and entertaining conversations about

physics. If I misunderstood you, any errors in this manuscript are mine.

To Lindley Graham, thank you for keeping me company through many hours

of writing. To Robert D’Angelo and Michael Ritter, thank you for your love and

emotional support. To Bryan Dunkeld and Ash Neblett, thank you for giving me the

opportunity to work on my thesis surrounded by the beauty of nature. To Michael

Moore, Mark Baumann, Andrew Leavenworth, David Frank, Nora Bernstein, Kevin

Shores and Tanzeel Ansari, thank you for your friendship and encouragement.

Most importantly, thanks to my parents Carolyn Hassan and Dr. Syed Abdullah

Hassan for their love and support and for always encouraging me to reach my full

iv

potential. Mom and Abbi, this is for you.

v

Dynamical Refinement in Loop Quantum Gravity

Syed Asif Hassan, Ph.D.

The University of Texas at Austin, 2015

Supervisor: Richard Matzner

In Loop Quantum Gravity, a quantum state of the gravitational field has a semiclas-

sical interpretation as a three-dimensional lattice discretization of space. We explore

the possibility that the scale of the lattice is only as fine as it needs to be in order to

carry the dominant frequency excitations of the auxiliary fields living on the lattice,

by considering graph-changing transition amplitudes in the context of a pure gravity

quantum theory. We define regular graphs that correspond to closed spatial slices

of FLRW spacetime in a novel way, with coherent state labels that correspond to

physical observables. This correspondence is obtained using the novel concept of a

pseudoregular polyhedron which affords a dimensionless volume to surface area ratio

in terms of the number of faces of the polyhedron. We normalize these regular graph

states using a new method, employing a saddle point approximation based on the

valence of the nodes rather than the large-scale semiclassical limit to obtain a result

that holds in the quantum limit. Finally we employ the EPRL spin foam model to

obtain a transition amplitude between single-node graphs of arbitrary valence that

is valid in both the semiclassical and quantum regimes, using an improved method

of normalizing the amplitude. We find that if we fix the scale factor and the fiducial

volume of space the amplitude favors final states with infinitely large valence.

vi

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Chapter One: Dynamical Refinement . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Zero-Point Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Maximal Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Dynamical Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Minimal Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Dynamical Refinement in LQG . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter Two: Kinematic Variables . . . . . . . . . . . . . . . . . . . . . . . . 9

Metric Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Frame-field Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Lorentz Group: SL(2, C) vs. SO(3, 1), SU(2) vs. SO(3) . . . . . . . . . . 12

Full GR action with topological terms; Immirzi parameter . . . . . . . . . 13

Ashtekar variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

New variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter Three: LQG Hilbert Space, Operators . . . . . . . . . . . . . . . . . . 17

Cylindrical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Spin Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Spin Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Area Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Volume Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Chapter Four: LQG Coherent States . . . . . . . . . . . . . . . . . . . . . . . 21

Hall Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vii

Coherent Spin Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Twisted Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Semi-coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter Five: SL(2, C) Representation Theory . . . . . . . . . . . . . . . . . 26

Representations of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Matrix Representations of SL(2, C) . . . . . . . . . . . . . . . . . . . . . . 29

Unitary Representations of SL(2, C) . . . . . . . . . . . . . . . . . . . . . 30

Chapter Six: Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Covariant Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 34

Simplicity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

EPRL Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter Seven: FLRW Coherent State Labels and Normalization . . . . . . . 49

Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Coherent State Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Norm of an Isolated Holomorphic Link . . . . . . . . . . . . . . . . . . . . 56

Norm of an Isolated Semi-Coherent Node . . . . . . . . . . . . . . . . . . . 58

Norm of a Holomorphic Coherent Spin Network . . . . . . . . . . . . . . . 62

Chapter Eight: Quantum FRW Cosmology Transition Amplitude . . . . . . . 71

Transition Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter Nine: Conclusion and Future Work . . . . . . . . . . . . . . . . . . . 79

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Appendix A: Action Priciple for GR . . . . . . . . . . . . . . . . . . . . . . . 81

Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Notation, Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

GR Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Topological terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Remaining terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Appendix B: Pseudoregular Polyhedra . . . . . . . . . . . . . . . . . . . . . . 88

viii

Appendix C: Sub-leading Contributions to the Norm . . . . . . . . . . . . . . 94

Appendix D: An Alternative Normalization . . . . . . . . . . . . . . . . . . . 96

Coherent State Normalization . . . . . . . . . . . . . . . . . . . . . . . . . 96

Alternative Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

ix

List of Figures

1.1 A scalar wave packet refines the graph as it proceeds. . . . . . . . . . . . 4

6.1 2-complex and boundary spin network diagram elements. . . . . . . . . . 35

6.2 Example 2-complex interpolating between different dipole graph states. . 35

6.3 Slicing up a 2-complex at the faces to isolate each vertex. . . . . . . . . . 36

6.4 A vertex shown inside its dual 4-simplex with boundary graph. . . . . . . 37

6.5 2-complex with one vertex, interpolating between different daisy graph

states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.6 Gluing the single-vertex 4-cells at the faces to form a 2-complex. . . . . . 41

6.7 The relation between bulk and boundary holonomies. . . . . . . . . . . . 43

6.8 Group-averaged edge holonomies that appear in the vertex amplitude. . . 45

7.1 Examples of Daisy graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Geometric interpretation of the L=6 daisy graph. . . . . . . . . . . . . . 52

7.3 Geometric interpretation of the L=6 dipole graph. . . . . . . . . . . . . . 53

7.4 Norm of the Livine-Speziale coherent intertwiner of a regular tetrahedron. 62

7.5 Norm of the Livine-Speziale coherent intertwiner of a cube. . . . . . . . . 63

8.1 Normalized one-vertex amplitude for a single node graph with L = 6 and

t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.2 Normalized one-vertex amplitude for a single node graph with L = 8 and

t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.3 Normalized one-vertex amplitude for a single node graph with L = 12 and

t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.4 Normalized one-vertex amplitude for a single node graph with L = 20 and

t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.5 Normalized amplitude with Li = 6 and t = 1 as a function of refinement

L and fiducial volume V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.1 Degrees of useful forms and their relationships . . . . . . . . . . . . . . . 82

x

B.1 The face count obtained, as a function of the number of links L requested.

The straight line shown is (.99)L . . . . . . . . . . . . . . . . . . . . . . 89

B.2 Percent Error between the face count obtained and the number of links

requested, as a function of the number requested. . . . . . . . . . . . . . 90

B.3 Dimensionless volume to surface area ratio as a function of face count L,

for pseudo-regular polyhedra (curve), regular polyhedra (dots), or a sphere

(dashed line.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.4 Dimensionless volume to surface area ratio as a function of face count L,

for pseudo-regular polyhedra (curve), “nice” polyhedra (dots), or a sphere

(dashed line.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

D.1 Alternative normalized amplitude for a single node graph with L = 6 and

t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

D.2 Alternative normalized amplitude for a single node graph with L = 6 and

t = 1 (detail) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.3 Numerically determined peaks of the alternative normalized amplitude for

L = 6 and t = 1, with a circular curve fit. . . . . . . . . . . . . . . . . . 101

xi

Chapter One: Dynamical Refinement

1.1 Introduction

The calculation of the zero-point energy of a quantum field should depend on the

details of a quantum theory of gravity. Here we lay out a possible mechanism by

which Loop Quantum Gravity may affect the calculation of the zero-point energy of

a scalar field, in order to motivate the calculation of certain transition amplitudes.

The arguments presented here are not rigorous and are not meant as a proposed

solution but rather to show that the typical standard model calculation is not the

only possibility, to highlight the issue of dynamical refinement of the spin network

graph, and to motivate the current work and future avenues of research.

1.2 Zero-Point Energy

We begin with a simple calculation of the zero-point energy of a scalar field on an LQG

holomorphic coherent state[1]. Such a state is a graph consisting of nodes connected

by links labeled with spins (area eigenvalues), with extra labels needed to completely

specify a semiclassical 3d spatial geometry[2]. Recall that in this description nodes

correspond to flat polyhedra and links correspond to their faces. The additional labels

describe the curvature which is concentrated where the faces of adjacent polyhedra

(links shared by adjacent nodes) are glued together. These states are described in

detail in Chapter 4. In this model the scalar field lives at the nodes, much as in

discrete QFT on a fixed spacetime lattice; in LQG however the lattice is dynamical.

Note that the scalar field attains a single value at a node, regardless of how much

spatial volume that node represents.

We will restrict attention to regular graphs, in which the same number of links

meet at each node, and all links are labeled with the same eigenvalues. In this case

the lattice spacing l and the volume of each node V are related by

l = αV 1/3 (1.1)

1

where α = 1 for a cubic lattic and α = 2(

34π

)1/3 ≈ 1.24 as the number of links goes to

infinity (Appendix B). Though α doesn’t change much and is of order 1 we will leave

it in but consider the cubic lattice as our main example. Further, we will assume that

the semiclassical geometry is approximately flat (otherwise the subsequent analysis

in terms of plane waves does not work.) Note that this requirement might cause a

contradiction later if the resulting zero-point energy density is too large. We also

invoke near-flatness to side-step any issues about how the energy density is defined.

Now we may view the scalar field as a lattice of coupled harmonic oscillators,

and quantize the normal modes using annihilation and creation operators as per the

standard Fock space QFT construction. Each normal mode of frequency ωi then

contributes 12~ωi to the zero-point energy and the greatest frequency is ωmax = 2πc

2l

since the minimum wavelength is λmin = 2l. Each mode frequency is ωi = 2πc2Li,

where Vtot = L3 is the total volume of space (or a representative portion) and i =√i2x + i2y + i2z. Taking into account the density of states,

g(ω)dω = g(i)di =1

84πi2di =

π

2

(L

πc

)3

dω =Vtot

2π2c3ω2dω (1.2)

The zero-point energy is

Uzpe =

∫ ωmax

0

1

2~ω

Vtot2π2c3

ω2dω = Vtoth

32π3c3ω4max = Vtot

h

32π3c3π4c4

l4= Vtot

hπc

32α4V −4/3

(1.3)

so the zero-point energy density doesn’t depend on Vtot,

ρzpe =h

32π3c3ω4max =

hπc

32α4V −4/3. (1.4)

1.3 Maximal Refinement

The above calculation is fairly standard; the choice of ωmax corresponds to a choice of

momentum-space cutoff of a divergent integral. In the standard model when possible

one sets a cutoff, renormalizes, then sends the cutoff to infinity. It is a typical ex-

pectation that in a quantum theory of gravity there is an actual cutoff at the Planck

scale, so ωmax would be the Planck frequency, or equivalently we may take V to be

the Planck volume.

In LQG we have more control over how we model the discretization of space, so

we can think about assigning a particular V eigenvalue as the scale (which will be

2

roughly a Planck volume.) If we choose the smallest possible value for V as above,

we are choosing the case of maximal refinement. This choice is the one made in Loop

Quantum Cosmology for example, where it is assumed that the spin-network graph

is a cubical lattice and all of the links are minimal (j = 12).

For this choice, and a semiclassical coherent state corresponding to a flat geometry,

α = 1 and if A is the area of a side of each cube then V −4/3 = (A3/2)−4/3 = A−2.

Now for j = 12

we get A = 8πGγ~c3

√32

so finally

ρzpe ≈ 1098 g /cm 3 (1.5)

which is quite large (see below), hence we have a problem with our initial assumption

of near-flatness.

Also note that the zero-point energy density is constant through all cosmological

epochs, independent of the scale of the universe and its temperature. The predicted

dark energy density has this property as well, but unfortunately the experimental

predictions thus far indicate a much smaller value (by 127 orders of magnitude),

ΛDE ≈ 10−29 g /cm 3 (1.6)

So not only does the zero-point energy density not explain dark energy, in fact it

should wash it out completely, which indicates that this calculation of the zero-point

energy is probably wrong somehow.

1.4 Dynamical Refinement

Perhaps the flaw in the above argument is the choice of maximal refinement. One

heuristic argument against maximal refinement is that increasing refinement increases

the zero-point energy; therefore it is energetically favorable for the spin network graph

to coarsen as much as possible. What then sets the level of refinement of the graph?

Perhaps refinement of a spin network graph is dynamical.

One way to understand the idea of dynamical refinement is to visualize the fol-

lowing situation. Suppose we have a coherent spin network graph corresponding to

a flat spacetime with a large volume assigned to each node of the graph. Now put a

scalar field on the graph, refine a small section of the graph, and place a tight scalar

wave packet in the refined region (Figure 1.4, left.) Suppose the wave packet is trav-

eling along through spacetime; as it proceeds, the graph will be forced to dynamically

3

Figure 1.1: A scalar wave packet travels to the right, refining the graph as it proceeds.

refine in response so that it can carry the highest frequencies of the wave pulse. As

the pulse travels through the ambient coarse region it will leave a trail of refinement

in the graph. Thus it seems plausible that refinement is at least dynamical in that a

graph can become more refined. Note that Thiemann’s early proposal for the action

of the Hamiltonian constraint refines a graph. But just as concern has been expressed

that the inverse graph-coarsening behavior should also be present in the action of the

Hamiltonian constraint, one may wonder what happens in the wake of the wave pulse;

does the spatial spin network eventually relax back to a coarse graph? If the answer

is no, then one can imagine that in an early hot phase of the universe the graph would

maximally refine very quickly, and might remain quite refined (despite the expansion

of the universe) at late times.

1.5 Minimal Refinement

On the other hand, we may take the other extreme view and see what happens if we

assume that the spin network relaxes immediately to a state of minimal refinement.

That is, the graph is only as refined as it has to be to accurately portray the fields it

carries. In this picture the spin network graph may be refined in regions of tumultuous

activity such as our solar system; but in the vast empty depths of space, the graph may

be extremely coarse, only as refined as it needs to be to carry the Cosmic Microwave

Background radiation. This may seem to be a strange concept, but bear in mind it

is only offered here as the other extreme end of the refinement spectrum from the

4

former possibility of maximal refinement, which is in some sense just as strange. This

picture may also seem less strange if one thinks about numerical simulations of GR

which employ this sort of dynamical refinement of the simulation lattice or mesh in

regions of high curvature, but which employ a looser mesh in regions of low curvature.

Another way to think about this situation is by analogy with an image or film

that has undergone digital compression. A region of an image that is featureless is

represented as a large block of uniform color, whereas regions of the image with more

detail are represented with higher resolution. The level of detail in various regions of

the image determines the resolution there.

Now we estimate the zero point energy in this situation of minimal refinement.

Suppose space contains only scalar excitations of exactly one frequency ω, then this

frequency determines the granularity of space and hence the zero-point energy density

of the scalar field in that state |Γ;ω〉 is

ρzpe =h

32π3c3ω4. (1.7)

In the state |Γ;ω〉, Γ denotes the graph and all its coherent state labels that determine

a semiclassical geometry.

Now consider a thermal distribution of scalar excitations at a temperature T . Tak-

ing a very brutal approximation, ignore all of these except the dominant contribution

at the peak of the distribution where

ω =2π

bT (1.8)

for some constant b. Then the zero-point energy scales with temperature as

ρzpe =hπ

2b4c3T 4. (1.9)

Note that this scaling is the same as that of radiation, so that the zero-point energy

in this model is ineligible as a dark energy candidate. Just for comparison however,

plugging in the current CMB temperature T = 2.725 K yields a numerical value

which is 4 orders of magnitude smaller than the dark energy density,

ρzpe ≈ 10−33 g /cm 3, (1.10)

and small enough that the initial assumption of near-flatness is justified.

5

1.6 Dynamical Refinement in LQG

As stated at the outset, the aim here is not to provide a rigorous derivation of the

zero-point energy in LQG, but rather to show that its behavior may be radically

different from the standard model behavior depending upon one’s assumptions about

dynamical refinement of the spatial graph. Namely, rather than attaining a constant

energy density through all cosmological epochs, the zero-point energy density may in

fact scale the same way radiation does, or perhaps in some intermediary way that may

be discovered when a more rigorous calculation is performed. These considerations

provide motivation for exploring the issue of dynamical refinement in LQG.

First we review the status of the field to contextualize the present work and out-

line the first few chapters. Loop Quantum Gravity is a background-independent,

non-perturbative quantization of 4-dimensional General Relativity (for reviews, see

[3, 4, 5, 6, 7, 8, 9, 10].) In Chapters 2 and 3 we describe the kinematic variables

used in LQG and construct a useful Hilbert space of quantum states spanned by the

spin network basis. There are two approaches to the dynamics, one is the canonical

approach in which one attempts to proceed in a similar manner to canonical QFT

approaches, breaking manifest Lorentz invariance by working in a 3 + 1 split of the

spacetime manifold and studying the action of the Hamiltonian constraint as an evo-

lution operator as in [11, 12, 13, 14, 15, 16, 17]. We mention the canonical approach

here for completeness but do not detail it further. The second approach to obtaining

the quantum dynamics is the covariant one, which attempts to proceed along the lines

of the path integral formulation of QFT. This line of research has led to various spin

foam models culminating in the most recent version, the EPRL model [18, 19, 20, 21,

22, 23, 24, 25, 26]. These models are based on the Plebanski formulation of GR [27,

28, 29] in which the theory is written as a constrained topological theory. Since the

quantization of the topological theory is known, the challenge is to implement the

constraints in an appropriate way. In the EPRL model [22] the so-called simplicity

constraints are implemented weakly [30, 31, 32], and it reproduces a discretization of

GR in the classical (large-j) limit [33, 34, 35, 36]. The EPRL transition amplitude

can be written in terms of coherent states on the boundary of a spacetime region [37].

In Chapter 4 we define holomorphic coherent states obtained via geometric quantiza-

tion or the heat kernel method. In Chapter 5 we set up the SL(2, C) representation

theory needed to implement the simplicity constraints in Chapter 6, where we define

6

the EPRL spin foam model which is used to calculate transition amplitudes in the

remainder of the work. Modifications of the EPRL model not detailed in the present

work include [38, 39] in which an auxiliary fermion field is included, and [40, 41,

42, 43, 44] in which the Lorentz group is replaced with a q-deformed Lorentz group

resulting in a theory that reproduces discrete GR with a cosmological constant. The

issue of coarse graining a graph is explored in [45], though from a different perspective

than the present work.

In the present work we pose a question the answer to which is at least partially

accessible with the current LQG candidate theory for the dynamics, the EPRL spin

foam model. We wish to sidestep the issue of the inclusion of auxiliary scalar or

other fields, as most current efforts in the field focus on pure GR, so we need to

ask a question which is answerable in that context. If we recall the scenario (Figure

1.4) described earlier in which a scalar wave packet leaves a trail of refinement in

an otherwise coarse graph, the question may be posed as to whether or not the

graph will return to a state of coarseness after the wave packet has gone. One may

simplify the question further by doing away with the scalar field that induced the

refinement, considering an empty spacetime described by an initially refined graph

and calculating the transition amplitude as a function of the refinement of the final

state graph. In particular, a generalization of the dipole cosmology model (in which

space is modelled as two tetrahedra glued together [46]) using regular inital and final

graphs with arbitrary numbers of nodes and links has been considered[47]. In that

work the focus was to establish that the Friedmann equation is recovered in the

classical limit regardless of the refinement of the graph in the case where the initial

and final states have the same graph structure, so the overall normalization of the

amplitude was not relevant. In the present work we set up the calculation in more

detail, defining and normalizing the initial/final states with different numbers of nodes

and links. We work out some details for the case of a general regular graph, but the

main cases of interest are the dipole graph with two nodes connected by an arbitrary

number of links and an n × n × n cubic lattice of nodes. We explicitly compute

the amplitude for the case in which the initial and final states are a single self-glued

node of arbitrary valence (the 6-valent case is studied in [48, 49],) corresponding to

a polyhedral region of space with opposing sides identified. In Chapter 7 we define

and normalize the boundary states, and in Chapter 8 we calculate the transition

amplitude. We mainly restrict attention to calculations that are tractable without

7

invoking the large-j (classical) limit, and employ some new calculation techniques that

should be of general interest to practitioners in the field (most notably, employing a

saddle point approximation based on valence rather than area.)

8

Chapter Two: Kinematic Variables

A typical place to begin describing a physical theory is with an articulation of the

kinematics, that is the terms used to specify the state of a physical system, before

moving on to describe the dynamics of the system, the way in which those kinematical

elements evolve or are related at different times. In Newtonian mechanics for example,

one choice of kinematic variables could be the positions and velocities of all the

particles in a system, and the dynamics would then be captured by the equation

F = ma. In this sense a particular kinematic framework is a choice of language for

expressing the dynamics, so the separation between the two is not completely crisp

in a theory that is still in flux; clearly the language must adapt to better express

the behavior of the system. This has certainly been the case in the history of Loop

Quantum Gravity. The choice of LQG kinematic variables has evolved for various

reasons as the theory has developed, and some of the original reasons for certain

choices have been supplanted by other motivations. Perhaps one can expect - even

hope - that the description that follows will seem outdated several years from now.

Given that the exploration of the dynamics provided by the EPRL spin foam model

has really only just begun, it is to be expected that even the language in which it is

expressed - and the deeper significance of the choices made therein - will change as

the depths of the theory are more fully plumbed.

It can be argued that the most significant accomplishment of the Loop Quantum

Gravity program so far is a radical reformulation of General Relativity in terms of new

kinematic variables that ostensibly facilitate a quantization of the theory. Certainly

an appreciation for the elegance and utility of these variables provides necessary

motivation for any researcher in the field. The objective of this chapter is to convey

to the reader an intuitive sense of why this is so, providing a minimum of historical

and theoretical context without excessive conceptual clutter.

2.1 Metric Variables

The kinematics of classical GR were originally articulated in terms of the space-

time metric gµν and its derivatives, packaged as the Riemannian curvature or as the

9

Christoffel symbols, which were then used to formulate the dynamics of the the-

ory: Einstein’s field equations for the metric itself and the geodesic equation for test

particles moving in curved spacetime. Expressed using these variables, the (Einstein-

Hilbert) action is

S =

∫d4x√− det g R (2.1)

where R is the Ricci scalar obtained from the curvature tensor.

These 4-d variables are ill-suited to the canonical quantization program used to

develop the Quantum Field Theory of the Standard Model, in which a Hamiltonian

approach is taken and Lorentz covariance is explicitly broken at the outset (then

recovered at the end). So following the typical QFT program a so-called ADM split

is taken, where spacetime is foliated (sliced into 3-d spatial leaves at equal coordinate

times for some observer) and the basic kinematic variables in each leaf are the 3-d

spatial metric gij and its conjugate momenta πij, which is related to the extrinsic

curvature of the leaf.

2.2 Frame-field Variables

Now these variables, the 3-metric and its momentum, are still undesirable because

they lack parallel structure with the description of the other fields of the standard

model, and because they do not allow for the inclusion of fermionic fields. One of the

basic ingredients of LQG is the idea that the gravitational field is to be treated not

as a Spin-2 field but rather as a Spin-1 field with an internal Lorentz symmetry, a key

distinction that recovers some parallel structure with the other fields of the Standard

Model.

In the geometric language of fiber bundles, every field of the Standard Model is

described either by a fiber bundle or a corresponding connection. A fiber bundle is

a manifold N equipped with a projection π : N →M to a submanifold M (which in

QFT is spacetime), and the fiber over a point m ∈ M is π−1m. A simple example

is the manifold F × M where F is the typical fiber, a manifold of possible field

configurations at each point in M , and the fiber over any point is F . A section of

the bundle is a choice of a point in the fiber over each point of M , that is a global

choice of the field configuration everywhere in spacetime. Now to take spacetime

10

derivatives of the field one needs an additional structure, a connection A, which in

effect specifies how adjacent fibers are glued to each other. In a QFT the fibers are

also G-torsors for some internal symmetry group G (typically a Lie group), that is

there is an action of G defined on the elements of each fiber. Because of this fact,

the connection A takes values in the Lie algebra of G. The definition of a QFT also

includes a choice of unitary representation for G and the spaces upon which it acts.

For example, quantum electrodynamics can be defined as a fiber bundle where the

typical fiber F is a space of 4-component spinors (electrons), the group G is U(1)

represented as multiplication by a unit complex number eıθ, and the connection A is

the photon field. The theories of the electroweak and strong forces (other Yang-Mills

fields) are similarly defined, using different symmetry groups G and representations

thereof.

In this language, GR can be defined in terms of a frame field or tetrad field eIµ,

which is a section of the frame bundle. This field may be viewed as a map from the

tangent space at each spacetime point to a standard internal Minkoski space, and is a

one-form field that takes values in the internal Minkowski vector space. The internal

symmetry group G which acts on each fiber is the Lorentz group (more on this in the

next subsection), and a section of the connection bundle is ωIJµ (a one-form field that

takes values in the Lorentz algebra). The internal Minkowski space comes equipped

with a flat metric ηIJ which allows one to reconstruct the metric from the frame field,

gµν = eIµeJνηIJ .

Two benefits of taking the frame field as fundamental are immediately apparent.

First, the metric can be seen to be a composite field rather than a fundamental

field. Second, one can now incorporate fermions into GR by introducing the standard

gamma matrices as a fixed structure defined on the internal Minkoski space. One also

finds that the cumbersome factors of√− det g in the action can now be swapped for

factors of | det e| which are easier to manipulate, and furthermore the rest of the action

can now be expressed as a simple wedge product of forms. Removing the orientation

factor sgn(det e) and using the fixed Minkowski space antisymmetric tensor ε, the

(Palitini) action is

S =

∫εIJKL e

I ∧ eJ ∧ FKL (2.2)

where F is the curvature obtained from the connection ω.

11

Note that these structures closely parallel those of the Yang-Mills fields of the stan-

dard model, with some important differences. In a Yang-Mills field theory the fibers

correspond to fermionic (Spin- 12) fields and the connections correspond to bosonic

(Spin-1) fields, while in GR the fibers correspond to a bosonic (Spin-1) field and the

connection also corresponds to a bosonic (Spin-1) field. Further, in a Yang-Mills field

theory one expects to have two observable quanta, the fermion and its gauge field,

while in GR one might expect from the classical theory to have only one observable

quantum, the graviton. This reduction of degrees of freedom is perhaps related to the

appearance of topological terms that appear in the GR action in its most complete

form (see Appendix A.)

2.3 Lorentz Group: SL(2, C) vs. SO(3, 1), SU(2) vs.

SO(3)

The symmetry group G of classical GR is SO(3, 1), but it is standard in LQG to

employ instead its double cover SL(2, C). Similarly, when taking a 3 + 1 foliation of

spacetime the symmetry group on the spacelike leaves in classical GR is the familiar

3-d rotation group SO(3), but in LQG one uses instead its double cover SU(2).

The reason for this choice is purely pragmatic, that the representations of SU(2)

and SL(2, C) are well-understood and thus easier to deal with. Locally a group

and its double cover look the same; that is, they have the same Lie alegbra. The

global topological structure of these groups may affect the predictions of the theory

in some way, but for the sake of simplicity those issues are deferred for later study.

In this sense LQG may be understood as a toy model that informs us about part

of the behavior of the full theory, or LQG may turn out to be correct as it stands;

experimental results will be the final arbiter. Throughout this work, we will refer to

SL(2, C) as the Lorentz group.

12

2.4 Full GR action with topological terms;

Immirzi parameter

Now it is usual in a physical theory to adapt the kinematic variables to best suit the

solution of a particular problem; that is, the dynamics inform the definition of the

kinematics. So to understand the choice of variables in LQG one must revisit the

classical GR action and restore terms which have been implicitly dropped. This is a

typical situation in a QFT obtained from a classical thoery. Broadly, there are two

kinds of terms which can be ignored classically: terms that are topological (that can

be expressed as a total derivative, and hence do not affect the equations of motion),

and terms that vanish because of some equation of motion. Both of these types of

terms do not affect the classical dynamics, but may affect the quantum dynamics.

Topological terms affect the overall value of the action, which is irrelevant in a

classical theory since the dynamics are only dependent on stationary points of the

action. In a quantum theory, however, since the action appears in the path integral

as a phase, the value itself can be important. That said, the effect of topological

terms on the quantum dynamics in GR is an issue for later study; in this work such

terms will be ignored.

Terms that vanish due to an equation of motion are important in a QFT because

the equations of motion are true for expectation values only, and do not hold in gen-

eral. That is, the quantum dynamics are influenced by trajectories that are classically

forbidden. The full GR action contains a term of this type, and including it in the

theory is crucial for the definition of the kinematic variables of LQG.

The construction of an action for a QFT may be systematized as follows: First,

one decides which fundamental fields are to be considered. Next, one includes every

possible combination of those fields that transforms as a scalar with respect to all the

internal gauge symmetries and as a spacetime volume form, with an undetermined

multiplicative constant for each term.1 Following this procedure, the full GR action

is built from the frame field e, the curvature F and covariant derivative D arising

from the connection ω, and the internal Minkoski tensors ε and η. It consists of six

terms.

1In the Standard Model one may also disqualify any terms which are not UV renormalizeable,however this consideration is irrelevant for our purposes since UV renormalization is not expectedto be necessary.

13

LGR = α1L1 + α2L2 + α3L3 + α4L4 + α5L5 + α6L6 (2.3)

L1 = εIJKL FIJ ∧ FKL L2 = ηIK ηJL F

IJ ∧ FKL (2.4)

L3 = εIJKL eI ∧ eJ ∧ FKL L4 = ηIK ηJL e

I ∧ eJ ∧ FKL (2.5)

L5 = εIJKL eI ∧ eJ ∧ eK ∧ eL L6 = ηIJ De

I ∧DeJ (2.6)

This full action for GR is considered in detail in Appendix A; here we highlight

the main points.

Three terms in the action are topological: L1 and L2 are respectively known as

the Euler and Pontryagin invariants, while the combination (L4−L6) is known as the

Nieh-Yan invariant. Setting aside these invariants, one may then drop L1, L2, and

either L4 or L6 (with a suitable redefinition of the α’s). Further, the Euler-Lagrange

equations yield the equation of motion DeI = 0, the torsion is zero2, so in a classical

second-order formalism one may drop L6. This leaves only the familiar terms L3 and

L5, where the latter is the cosmological constant term.

As discussed earlier, in a quantum theory one should consider all six terms but

for the sake of preliminary simplicity in LQG one drops the topological terms. The

standard choice is to drop the Euler term L1, the Pontryagin term L2 and the torsion

term L6. The new term to be considered is the Immirzi term L4, and it has a crucial

impact on the choice of kinematic variables.3 Dropping the cosmological constant

term (again for simplicity), one obtains the (Holst) action for LQG,

LLQG = α

(εIJKL e

I ∧ eJ ∧ FKL − 1

γηIK ηJL e

I ∧ eJ ∧ FKL

)(2.7)

where γ is known as the Immirzi-Barbero parameter. This is the action that serves

as the starting point for our quantum theory.

2However, it is worth noting that if the action includes fermionic fields then the torsion is notzero. This issue is sometimes overlooked in classical GR because classical actions do not typicallyinclude fundamental fermionic fields.

3Conceptually it may be helpful to remember that one could just as well have kept the torsionterm L6 instead of the Immirzi term L4. The quantum effects of the Immirzi term are thus dueto contribitions from connections that are not torsion-free. Note also that the torsion term has thestructure of a “kinetic” term for the frame field.

14

2.5 Ashtekar variables

The Holst action for GR can be re-expressed as a theory of a(n) (anti)self-dual con-

nection as follows: First, complexify spacetime and hence the frame field and the

connection. The internal symmetry group is now SL(2, C) ⊗ C, which splits into

SL(2, C)⊗C = SL(2, C)⊕ SL(2, C). The corresponding connection then splits into

self-dual and antiself-dual parts. Expressing the action in terms of spinorial variables,

each of the terms splits into a self-dual and an antiself-dual part. For the special choice

of Immirzi parameter γ = ±ı, two terms cancel and two terms combine, leaving only

one (anti)self-dual term remaining in the action. The beauty of this reformulation

of GR is that the 3 + 1 Hamiltonian decomposition takes a very simple form which

was initially thought to solve some of the problems arising during an attempt at

quantization.4 However, the initial step of complexifying spacetime leads to the new

problem of dealing with imposition of reality constraints on the quantum theory, a

problem which has not (yet) been solved. For this reason, (anti)self-dual connections

are not the kinematic variables of LQG. Rather, the Immirzi parameter is taken to

be real-valued but certain features of the construction are retained.

2.6 New variables

The so-called “new variables” used in LQG are a 3-d connection Aim (m is a spatial

index, i is an internal su(2) index) and the densitized inverse triad Emi , defined on

the spacelike leaves of a 3+1 foliation of spacetime. The connection A is built out of

the connection ω, the extrinsic curvature K, and the Immirzi parameter γ,

A = ω + γK, (2.8)

as in the (anti)self-dual construction outlined earlier. The Immirzi parameter is

taken here to be real-valued, however. Moreover in the (anti)self-dual case the 3-d

4More specifically: the Hamiltonian decomposition of standard GR contains square roots ofthe determinant of the metric which are nonpolynomial and hence require an infinite number ofoperator-ordering choices during quantization, so the theory loses predictive power. In the self-dualHamiltonian decomposition these square root factors were initially absorbed into a redefinition ofcertain Lagrange multipliers, leaving only polynomial terms to be quantized. However, from a moremodern point of view this redefinition is mathematically unsound. In essence the problem was beingswept under the rug rather than solved.

15

connection A is the pullback to the leaves of a 4-d connection, while here it is not.

The Poisson bracket of E with K immediately gives the bracket of E with A,

Emi , K

jn = 8πG~ δmn δ

ji =⇒ Em

i , Ajn = 8πG~γ δmn δ

ji . (2.9)

The presence of γ in this Poisson bracket is one of the ways that the Immirzi parameter

enters into observable predictions of the theory. In this work we follow the typical

convention in the LQG literature and refer to the connection A as the Ashtekar

connection.

16

Chapter Three: LQG Hilbert Space, Operators

Loop Quantum Gravity is so named for the discovery of a Hilbert space of states

that satisfies the Gauss (gauge) constraint and the diffeomorphism constraint. In the

following sections we will construct these states and define physical operators that

are diagonal on them.

3.1 Cylindrical Functions

A reasonable starting point is to assert that in GR a quantum state is a functional

of the connection. It is more convenient to work instead with holonomies, which are

path-ordered exponentials of the integral of the connection along a path. Given a

path1 γ from point p to point q in a manifold with connection A, the holonomy is a

group element Uγ (in our case an element of SU(2)) that parallel transports vectors

in the tangent space at p to vectors in the tangent space at q.

Uγ = P exp

∫γ

A (3.1)

Now the space of functionals of holonomies along all possible paths is the same as

the space of functionals of the connection. If we choose a particular graph Γ that

consists of multiple curves, a cylindrical function is a function of the holonomies along

the curves that compose Γ. Henceforth we will refer to these curves as links, and the

endpoints of curves as nodes. The valence of a node is the number of link endpoints at

that node. A suitable choice of inner product turns this space of cylindrical functions

into a Hilbert space (which is not yet the one we want.)

3.2 Spin Networks

Now to get a better handle on the space of cylindrical functions we may apply the

Peter-Weyl theorem, which says that a function of an SU(2) group element U may be

expanded as a sum over the matrix elements of the irreducible representations of U .2

1Not to be confused with the Immirzi parameter γ.2For an intuitive explanation of why this is so, see Chapter 5.

17

We may therefore work with a basis of functions of explicit matrix representations of

the holonomies, for some representation labels. Now we are in a position to solve the

Gauss constraint by enforcing guage invariance.

Note that a local guage transformation acts independently at each node, so we

must enforce gauge invariance at each node separately. At a given node the links that

meet there each correspond to one index of a holonomy in a particular SU(2) repre-

sentation. The coefficient that tensors all those indices together must transform as

an SU(2) scalar. To find such a scalar, we look at the Clebsch-Gordan decomposition

of the tensor product of the incoming representations and choose one of the scalars.

Such a choice is called an intertwiner. A guage invariant state is thus completely

determined by a choice of graph Γ together with a choice of representation jl for each

link l and a choice of intertwiner in for each node n, and such a state is called a spin

network.

In the definition of a cylindrical function the valence of a node could be any

number, even one. Clearly a spin network can only have nodes of valence 2 or more.

Since the choice of intertwiner is trivial for a 2-valent node, typically only graphs

with nodes of valence 3 or more are considered. Further, a link with spin label j = 0

is considered to be trivial hence only spin labels j = 12

or higher are allowed.

3.3 Spin Knots

So far a spin network is defined in terms of curves drawn in a manifold, so that a

slight deviation in any curve defines a completely different state. The solution of

the diffeomorphism constraint is achieved simply by identifying any two states for

which the graphs are topologically the same (and are colored with the same spins

and intertwiners.) In fact this strategy enforces a stronger constraint than strict

diffeomorphism invariance, which would preserve the angles between links meeting

at a node. However these extra parameters at each node would clutter the notation

unneccessarily, so we follow the standard convention and ignore them. Such states

are known as spin knots, though often in the literature these states are also referred

to as spin networks.

18

3.4 Area Operator

We now define an operator which is diagonal in the spin network basis, the area oper-

ator. First observe that since the densitized inverse triad E is canonically conjugate

to the connection A, canonical quantization indicates that as an operator it becomes

i8πG~γ ddA

where G is the gravitation constant and γ is the Immirzi parameter. A

spin network depends on the connection only through the holonomy of each link, so

we consider first the action of E on a 2-d surface that intersects the graph Γ at a

single point on one link. The action of E on a more generic 2-d surface that inter-

sects multiple links (or the same link at multiple points) may then be easily obtained

by breaking up the surface into smaller surfaces that satisfy the single-intersection

criterion and summing the results. Further, the action on a surface that intersects

the graph at a node may also be defined, though we omit the details as we will not

need this result.

Schematically, the action of E at the point where the surface intersects a link with

holonomy U = Pe∫A (the path-ordered exponential of the connection along the link)

is

EiU = EiPe∫A = i8πG~γ

d

dAiPe

∫Aiτi = i8πG~γ τi Pe

∫Aiτi = i8πG~γ τi U (3.2)

where τi are basis elements for the Lie Algebra su(2) in the representation j. Noting

that τiτi = j(j + 1), one may then define and regularize an operator A =

√|E2| so

that A U = 8πG~γ√j(j + 1) U . The physical significance of this operator is that it

yields the area associated to the surface pierced by the link. Since the area operator

A only depends on the representation label j of the link, it is also well-defined on

spin knots.

The fact that the area operator has a discrete spectrum has great physical signif-

icance. It indicates that quantum space itself is discrete on the smallest scales. One

application is that one may explicitly count the quantum states available to a surface

with a given area and hence associate an entropy to that surface, for example the

horizon of a black hole.

19

3.5 Volume Operator

One may also define and regularize a volume operator V ; we will not describe its

construction in detail here3, but rather state some of its properties. The volume

operator acts only at the nodes of a spin network (or knot) state and has a discrete

eigenvalue spectrum. It is not diagonal in the intertwiner basis, but one may easily

find a suitable basis in which it is diagonal and label spin networks with volume

eigenvalues rather than intertwiners at the nodes.

The area and volume operators together provide a rough physical interpretation

of a spin network state: each node corresponds to a discrete nugget of 3-d space of

a certain volume, and each adjoining link corresponds to a 2-d planar surface of a

certain area that bounds the volume. Note that the volume operator annihilates any

node of valence less than four, which supports this physical interpretation because

one cannot bound a volume with less than four planar surfaces. Note also that despite

this loose physical interpretation a spin network does not correspond to a classical

Regge geometry; such a state is delta-function peaked in the Area operator which

is related to the E operator, hence the state is completely spread in the canonically

conjugate connection operator A and the holonomy operator U and as such contains

no information about curvature. In the following chapter we will construct coherent

states that are peaked in both canonically conjugate operators and correspond in the

semi-classical limit to a discrete classical geometry that bears a resememblance to a

Regge geometry.

3There are several proposals in the literature, e.g. [3, 32, 31, 50], some of which pertain to thespin network basis described here and some of which rely on coherent states. There is a simpleconstruction given in [3] for the case of a four-valent node, but for nodes of higher valence theappropriate choice of volume operator is less clear as there are many choices which asymptote tothe classical volume in an appropriate limit.

20

Chapter Four: LQG Coherent States

In the case of the quantum simple harmonic oscillator one may employ states that

are diagonal in either position or momentum operators, but there also exist (Segal-

Bargmann) coherent states which are gaussian-peaked at particular position and mo-

mentum values, with a relative spread that goes to zero in the ~→ 0 (classical) limit.

These states are considered semi-classical in this sense, and may be used to provide

a quantum description of a system with particular classically observable properties.

These coherent states may be obtained either via heat-kernel methods (in which a

delta function state is allowed to spread, evolving according to a heat equation ob-

tained by defining a Laplacian on the space), or by geometric quantization (in which

the symplectic manifold is complexified then foliated diagonally into Lagrangian sub-

manifolds upon which states are defined.)

4.1 Hall Transform

It turns out that the same strategies work in the case of the Hilbert space corre-

sponding to a single link. That is, in the previous section we began with functionals

of holonomies, that is functionals on multiple copies of a group (one per link.) Sup-

pose we consider just one link, then we can consider the tangent manifold of the

group, which may be endowed with a symplectic structure, an appropriate Laplacian,

and all the other structures needed to proceed by either the heat-kernel or geomet-

ric quantization routes to construct (Hall) coherent states in the same way that the

Segal-Bargmann states were constructed.

These coherent link states are labeled by an SU(2) group element (holonomy) Ul

and an su(2) algebra element Ll which correspond to the expectation values of the

holonomy operator U and the flux operator L. Now these labels may be used to

construct an SL(2, C) group element via a left polar decomposition,

Hl = Ul exp(it

El8πG~γ

)(4.1)

in the non-unitary representation of SL(2, C) obtained by complexifying SU(2) rep-

21

resentations. This is equivalent to the right polar decomposition

Hl = exp(it

E ′l8πG~γ

)Ul (4.2)

with the identification

E ′l = UlElU−1l . (4.3)

The difference is that El is the flux as seen by the source node, whereas E ′l is the flux

as seen by the target node [50]. In our cases of interest this distinction is immaterial

as El commutes with Ul.

The positive real number t in (4.1) is known as the heat kernel time, and governs

the relative spread of the states1, and is typically chosen such that 0 < t < 1. A

coherent state on one link may be explicitly written as

ΨHl(hl) = Kt(Hlh−1l ) (4.4)

Kt(g) =∑j

(2j + 1)e−t2j(j+1)Tr[Dj(g)] (4.5)

where Kt is the heat kernel on SU(2), and Dj(g) is the Wigner representation matrix

of g in the respresentation j (or since Hl is complex, the analytical continuation

thereof.)

4.2 Coherent Spin Networks

Having constructed coherent states for each link of a graph, the next step is to make

this state gauge invariant. If we apply an SU(2) gauge transformation at each node

un, then each holonomy transforms as

hl → u−1tl hlusl (4.6)

So we group average over all un to obtain a gauge invariant state:

ΨHl(hl) =

∫SU(2)N−1

dun⊗l

ΨHl(u−1tlhlusl) (4.7)

which we call a coherent spin network. Note that N is the number of nodes, and we

need only integrate over N − 1 group elements since there is an overall symmetry

that may be used to trivialize one un.

1Note that different conventions for t exist in the literature. We follow the choice of [49] here.

22

This construction imposes gauge invariance strongly at all but one node, which

we can see schematically as follows:

|ΨGI〉 =

∫du u|Ψ〉 (4.8)

u′|ΨGI〉 = u′∫du u|Ψ〉 =

∫du u|Ψ〉 = |ΨGI〉. (4.9)

The translation-invariance of the Haar measure du allows the externally acting trans-

formation u′ to be absorbed into the integral by a change of variable so that the state

|ΨGI〉 is explicitly gauge-invariant.2 The constraint is thus not imposed uniformly at

all the nodes, which may be cause for concern; in Appendix D we explore the effect

of imposing gauge invariance strongly at every node.

It is important to note that we have not shown the action of the volume operator

on a coherent spin network. Indeed, there is not even consensus in the literature

as to the most appropriate definition of the volume operator for nodes of valence

higher than four. For the purposes of this work, we assume a definition of the volume

operator (for example as defined in [51]) such that its action on a node of a coherent

spin network reproduces the classical volume of the correspoding polyhedron.

4.3 Twisted Geometries

Now while the Ul, Ll labels are convenient for discussing the peakedness properties

of the coherent states, and the Hl labels are convenient for explicitly writing out

an expression for the coherent states for use in calculations, there is a third way of

expressing the coherent state variables that provides a nice physical interpretation.

In particular, any SL(2, C) element Hl may be decomposed as

Hl = ntle−i(ξl+iηl)

σ32 n−1sl (4.10)

where each n is an SU(2) group element that may be identified with a vector n by

applying that transformation to a fixed reference vector z,

n = nz. (4.11)

2In a weak imposition of gauge invariance, on the other hand, we would have 〈ΨGI|u′|ΨGI〉 =〈ΨGI|ΨGI〉 but u′|ΨGI〉 6= |ΨGI〉.

23

These state labels offer a useful geometric interpretation of a coherent state. As

in the case of a normal spin network, each node corresponds to a nugget of volume

enclosed by surfaces, one for each link emanating from it. With these labels, we may

assign an area Al = 8πγGη to each such face as well as a normal vector ~n. Due

to a theorem of Minkowski, the specification of the areas and normals to the faces

defines a unique convex polyhedron for each node. Moreover, the normals as seen

from each side give some curvature information. The rest is provided by ξ, which

in a particular gauge is related to the extrinsic curvature. This geometrical picture,

dubbed twisted geometry, is similar to Regge geometry but somewhat more relaxed.

First, adjoining faces from adjacent polyhedra need not match shapes. Second, there

is the aforementioned twisting; the factor e−iξlσ32 is a rotation about the normal vector

so the faces are attached with a relative twist of the angle (−ξl/2).

4.4 Semi-coherent States

It is convenient to take a limit of the above holomorphic, fully coherent states, to

obtain a set of semi-coherent states (or Perelomov coherent states). These states are

still peaked on the normal vectors with some minimal spread, but are sharply peaked

on the j parameter (areas) and fully spread on the conjugate intrinsic curvature. To

see how these states arise naturally, take (4.10) and insert two resolutions of unity∑m |m〉〈m|,

Hl =∑m1,m2

nt,l|m1〉〈m1|e−i(ξl+iηl)σ32 |m2〉〈m2|n−1s,l (4.12)

=∑m

nt,l|m〉e−i(ξl+iηl)m〈m|n−1s,l (4.13)

Then for large area, η 1 so the dominant term in the sum is m = j,

Hl ≈ nt,l|j〉e−i(ξl+iηl)j〈j|n−1s,l (4.14)

and just as the highest weight state |j〉 corresponds to the vector z, the state |n〉 =

n|j〉 corresponds to the vector n = nz.

That |n〉 has minimal spread in ~L follows from the fact that |j〉 does:

〈j|~L|j〉 = jz; 〈j|L2|j〉 = j(j + 1); σ2L = j(j + 1)− j2 = j. (4.15)

24

The states |n〉 are the Perelomov or semi-coherent states. In many situations they

are easier to work with than holomorphic coherent states, and thus are useful in the

large area approximation. We can also use them as a check on calculations using

the holomorphic coherent states, to verify that the same results are recovered in the

appropriate limit.

In particular, these link states have a simple inner product so it is easier to calcu-

late the norm of a gauge invariant semi-coherent spin network state. We may define

such a gauge invariant graph state (the Livine-Speziale coherent intertwiner[24]) in

the same way as we did with the holomorphic coherent link states, by tensoring the

states then group averaging at the nodes:

|nl〉 =

∫SU(2)N−1

dun

(⊗l

un(l)|nl〉)

(4.16)

Note that because the link states factor into source and target Perelomov states in

(4.14), the un integrals factor into an integral per node. We will explicitly compute

the norm of these states later on in Section 7.4.

25

Chapter Five: SL(2, C) Representation Theory

One particular choice of unitary representation for SL(2, C) plays a key role in the

construction of the EPRL spin foam amplitude; specifically, the simplicity constraints

are imposed by projecting SL(2, C) representations into an SU(2) representation on

a boundary graph. The unitary representation theory of noncompact groups such as

SL(2, C) is likely to be unfamiliar to the reader; here we describe its basic structure

and properties. A noncompact group does not have unitary matrix representations,

but rather unitary representations are elements of a Hilbert space which is built out

of matrix representations of a compact subgroup (the so-called little group). In our

case the noncompact group is SL(2, C) and the little group is SU(2); representations

of SL(2, C) will thus be built out of the more familiar representations of SU(2).

5.1 Representations of SU(2)

Representations of SU(2) are labeled by non-negative half-integers j, and may be

explicitly represented as (2j+ 1)× (2j+ 1) matrices acting on elements of a (2j+ 1)-

dimensional vector space. One may choose a basis |j,m〉 for the vector space, where j

is fixed for a particular representation and m is a half-integer such that −j ≤ m ≤ j

and (m + j) is an integer. The representation ρj of an element u of SU(2) or an

element X of the lie algebra su(2) is then given by its action on the basis vectors

|j,m〉, or equivalently by its matrix elements. A standard shorthand notation for the

matrix elements of u ∈ SU(2) in the representation j is

Djm1m2

(u) = 〈j,m1|ρj(u)|j,m2〉. (5.1)

The action of the SU(2) generators of the Lie algebra Jx, Jy and Jz may be conve-

niently expressed using J± = Jx ± ıJy:

Jz|j,m〉 = m|j,m〉 J±|j,m〉 =√

(j ±m+ 1)(j ∓m) |j,m± 1〉, (5.2)

where we employ the typical physicist’s abuse of notation and write for example Jx

rather than ρj(Jx) for the representation. Note that for the Casimir of the group J2,

26

the above relations imply that

J2 = J2x + J2

y + J2z = J2

z + 12(J+J− + J−J+) J2|j,m〉 = j(j + 1)|j,m〉. (5.3)

The Wigner D matrices have a simple orthogonality relation,∫dh Dj′(h)pqD

j(h)mn =1

2j + 1δjj′δpmδqn, (5.4)

and the explicit matrix elements may be realized as sines and cosines in a particular

parameterization of the group elements. It is not hard to accept then that they form

a good basis for “Fourier transforming” arbitrary functions on the group. In fact,

the Peter-Weyl theorem states that any reasonable function on the group may be

decomposed as a linear combination of matrix elements of all the representations.

This is also called the Plancherel decomposition. Even some distributions may be so

expressed, such as the delta function which we will come to in (5.16).

The vector space that carries the representation may be realized as a finite-

dimensional space of homogeneous polynomials of degree 2j. These are functions fj(z)

of a normalized complex 2-component spinor z = (z1, z2), with 〈z, z〉 = |z1|2 + |z2|2 =

1, that behave under a scaling λz = (λz1, λz2) by a complex factor λ as

f(λz) = λ2jf(z). (5.5)

The functions may be explicitly expanded as

fj(z1, z2) =

j∑m=−j

cmzj+m1 zj−m2 (5.6)

for some coefficients cm. The homogeneity property is apparent from this expression.

Note that the matrix

u(z) =

[z1 −z2z2 z1

](5.7)

is a representation of SU(2), hence one may view the functions fj(z) as functions

fj(u(z)) on SU(2). If one then coordinatizes SU(2) one may express each fj as a

linear combination of products of sines and cosines of the coordinates, and it becomes

evident that the set of all fj forms a suitable basis for the Fourier decomposition of any

function on SU(2); this is another way to think about the Plancherel decomposition

which we employ in the construction of the LQG spin network basis. These spaces

27

of homogeneous functions are also interesting because suitable generalizations will

afford both nonunitary and unitary representations of SL(2, C) as well.

Another useful expression for the Wigner D matrices is given in terms of the

parameterization

u(z) =

[a b

c d

], (5.8)

Djmn(u) =

∑l

√(j +m)!(j −m)!(j + n)!(j − n)!

(j −m− l)!(j + n− l)!(m− n+ l)!l!aj+n−ldj−m−lbm−n+lcl. (5.9)

This form is useful for obtaining the character χj(u) = Trj[u] = Djmm(u) of the group

element u in the j representation. First diagonalize g, obtaining g = hgdh−1 where

gd is diagonal. Then using the cyclic property of the trace and the faithfulness of the

representation,

Trj[u] = Trj[hudh−1] = Trj[udh

−1h] = Trj[ud]. (5.10)

Thus without loss of generality we may take b = c = 0, so that only the l = 0 term

survives in the trace:

χj(u) =∑m

∑l

(j +m)!(j −m)!

(j −m− l)!(j +m− l)!(l)!l!aj+m−ldj−m−lblcl (5.11)

=∑m

aj+mdj−m. (5.12)

Now since u ∈ SU(2), Det[u] = 1, so if a = λ is one eigenvalue then d = λ−1 is the

other. Summing the geometric series, we obtain the identity

χj(u) =λ2j+1 − λ−(2j+1)

λ− λ−1. (5.13)

Further, using the invariance of the trace we may solve the equation

λ+ λ−1 = Tr 12[u] (5.14)

to obtain

λ = x+√x2 − 1; λ−1 = x−

√x2 − 1; x =

1

2Tr 1

2[u] ≥ 0

λ = x−√x2 − 1; λ−1 = x+

√x2 − 1; x < 0

(5.15)

28

where the sign choice is to ensure that |λ| ≥ |λ−1|. Note that the formula (5.13) holds

even when u is the identity and λ = 1 by taking an appropriate limit1.

The character appears in the explicit expression for the delta function on SU(2),

δ(u) =∑j

djχj(u), (5.16)

which we will use in the definition of the spin foam amplitude. That (5.16) works

as a delta function may be directly verified by writing out an integral and using the

orthogonality relation (5.4),∫dh Dj(h)mnδ(uh

−1) =∑j′

dj′

∫dh Dj(h)mnD

j′(uh−1)pp (5.17)

=∑j′

dj′

∫dh Dj(h)mnD

j′(u)pqDj′(h−1)qp (5.18)

=∑j′

dj′Dj′(u)pq

∫dh Dj′(h)pqD

j(h)mn (5.19)

=∑j′

dj′Dj′(u)pq

1

dj′δjj′δpmδqn (5.20)

= Dj(u)mn (5.21)

which is precisely what a delta function does.

5.2 Matrix Representations of SL(2, C)

Representations of SL(2, C) in matrix form are easily obtainable, but they are unsuit-

able for quantum applications because they are not unitary. However, 2 × 2 matrix

representations will be useful later on as they appear in the description of the unitary

representations in the next section.

The generators Ji of rotations and Ki of boosts have 2× 2 representations

Ji =ı

2σi, Ki =

1

2σi, (5.22)

where σi are the standard Hermitian Pauli matrices

σx =

[0 1

1 0

], σy =

[0 −ıı 0

], σz =

[1 0

0 −1

]. (5.23)

1In contrast with the trace of the identity for the unitary SL(2,C) representations, which as wewill see later diverges.

29

Matrix representations of group elements are obtained by exponentiation, and act on

a vector space of complex 2-component spinors.

Note that these representations are used to label holomorphic coherent states.

As these SL(2, C) representations are complexified versions of the previous SU(2)

representations, functions on SU(2) group elements may be analytically continued to

SL(2, C) group elements.

5.3 Unitary Representations of SL(2, C)

The principal series of unitary irreducible representations of SL(2, C), labeled by

a half-integer k and a real number p, are given as operators acting on an infinite-

dimensional Hilbert space Hχ = H(k,p). Here we mostly follow Ruhl [52], using

the conventions of [53] which are standard in the Spin Foam literature.2 For some

formulas it is convenient to label the representations instead using complex numbers

n1 and n2, where

n1 = k + ip; n2 = −k + ip. (5.24)

First we define the space H(k,p) that carries the group representation and provide a

convenient basis in which to work. Next we show the action of the rotation and boost

generators on a basis element of H(k,p).

H(k,p) is a space of functions f(z) of a complex 2-component spinor z = (z1, z2)

which are homogenous in z, that is under scaling of the argument λz = (λz1, λz2) by

a complex factor λ they behave as

f(λz) = λ−1+ıp+kλ−1+ıp−kf(z) = λn1−1λn2−1f(z). (5.25)

This property is useful because it allows one to restrict attention to special values

of z, then extend the result to any z by homogeneity. For example, we may set

ξ = z/√〈z, z〉 so that ξ is normalized, then as in (5.7) ξ may be mapped to an

element u ∈ SU(2) via a j = 1/2 matrix representation. This affords an explicit

Plancherel decomposition of f(ξ) in terms of SU(2) representation matrix elements,

f(ξ)(k,p) = cmj fjm(ξ)(k,p); f jm(ξ)(k,p) =

√2j + 1

πDjkm(ξ), (5.26)

2Ruhl [52] labels these representations by χ = (m, ρ) for an integer m and real number ρ, whichcorrespond to our labels χ = (k, p) via k = − 1

2m and p = 12ρ. In other formulas there the “magnetic”

index q is our m.

30

where the “Fourier coefficients” cmj are given by

cmj =

∫f jm(u)(k,p)f(u)(k,p)du (5.27)

and we have used the identification f(u) = f(u(ξ)) = f(ξ) and the SU(2) Haar

measure du. The functions f(ξ)(k,p) can then be extended back to z =√〈z, z〉 ξ by

homogeneity,

f(z)(k,p) = cmj fjm(z)(k,p); f jm(z)(k,p) =

√2j + 1

π〈z, z〉−1+ıp−j Dj

km(z), (5.28)

using also the fact that Djkm(ξ) = Dj

km(z/√〈z, z〉) = 〈z, z〉−j Dj

km(z) by homogeneity

of the SU(2) representation. Note that the first index of Djkm(z) is fixed to the

representation label k. This is because the scaling ξ = z/√〈z, z〉 does not “use up”

all of the information provided by the homogeneity condition. Explicitly, scaling by

a complex phase gives, using (5.9),

f jm(eiωξ)(k,p) =

√2j + 1

πDjkm(eiωξ) = eikω

√2j + 1

πDjkm(ξ) (5.29)

precisely as required.

The functions f jm(z)(k,p) are called the canonical basis, written |(k, p); j,m〉 or sim-

ply |j,m〉. One can explicitly see that H(k,p) is a direct sum of SU(2) representations

Hj with j ≥ k,

H(k,p) =∞⊕j=k

Hj. (5.30)

We will use this fact later to solve the simplicity constraints of the spin foam model.

The representation T(k,p)a of a ∈ SL(2, C) acts on a function f(z) ∈ H(k,p) by

T (k,p)a f(z) = f(zT

( 12)

a ) (5.31)

where T( 12)

a is the 2 × 2 nonunitary matrix representation of a discussed earlier and

z is treated as a row vector.

31

The action of the SL(2, C) generators on the canonical basis elements is:

Jz|j,m〉 =m|j,m〉 (5.32)

J±|j,m〉 =√

(j ±m+ 1)(j ∓m) |j,m± 1〉 (5.33)

Kz|j,m〉 =− γ(j)√

(j2 −m2)|j − 1,m〉 − β(j)m|j,m〉

+ γ(j+1)

√(j + 1)2 −m2|j + 1,m〉

(5.34)

K±|j,m〉 =∓ γ(j)√

(j ∓m− 1)(j ∓m)|j − 1,m± 1〉

− β(j)√

(j ±m+ 1)(j ∓m)|j,m± 1〉

∓ γ(j+1)

√(j ±m+ 1)(j ±m+ 2)|j + 1,m± 1〉,

(5.35)

where

β(j) =kp

j(j + 1)γ(j) =

ı

j

√(j2 − k2)(j2 + p2)

4j2 − 1. (5.36)

Note that the rotation generators J respect the SU(2) subspaces Hj, while the boost

generators K do not.

To define the trace of an operator, we need to develop a notion of Fourier transform

of functions on the group. Given an integrable function x(a) for a ∈ SL(2, C), we

define an operator T χx by

T χx =

∫T χa da (5.37)

where da is the SL(2, C) Haar measure and the action of T χa is given by (5.31). If we

define multiplication of functions by convolution,

x1 · x2(a) =

∫x1(a1)x2(a

−11 a)da1 (5.38)

then these functions form an algebra (with formal unit element, the delta function)

isomorphic to the algebra of operators T χx ,

T χx1·x2 = T χx1Tχx2. (5.39)

We also define the adjoint of a function by

x(a)† = x(a−1) (5.40)

so that

(T χx )† = T χx†

(5.41)

32

The operators T χx are integral operators on the space of L2 functions on SU(2)

spanned by the canonical basis functions f jm(u)χ,

T χx f(u1) =

∫Kx(u1, u2|χ)f(u2)du2. (5.42)

We call either the operator T χx or the kernel Kx(u1, u2|χ) the Fourier transform of

the function x(a).

The trace of an operator is defined as a distribution on the group,

Tr(T χx ) =

∫x(a)

λn1λn2 + λ−n1λ−n2

|λ− λ−1|2da (5.43)

where λ is any solution of

λ+ λ−1 = Tr[a] (5.44)

and a is a 2 × 2 nonunitary matrix representation of the SL(2,C) group element.

Despite the apparently simple form of this expression, it may be divergent in general

so the functions x(a) must be chosen carefully. For example, if a is the identity

matrix, then λ = 1 and the expression diverges. The divergences arise because of

the tower of SU(2) representations contained in a given representation χ = (k, p);

the trace in the finite dimensional sense involves a sum over all indeces of the matrix

elements, and since the index j ranges from k to infinity the sum may diverge. In

practice we will instead project to just the lowest of the SU(2) representations and

take traces there where everything is manifestly finite.

33

Chapter Six: Spin Foams

Having articulated the kinematics of the theory, including how to model semi-classical

states, we now turn to the dynamics. One strategy considered in the literature is

a Hamiltonian approach, which we will not discuss here. Recently, progress has

been made in describing the dynamics in a covariant setting in a way which allows

computations to be performed[22]. This is the Spin Foam approach described in this

chapter.

6.1 Covariant Transition Amplitudes

First we must define what is meant by a transition amplitude in a covariant setting[54,

55]. In standard Quantum Field Theory, a scattering amplitude is calculated from

ingoing to outgoing states by taking the field states to be plane waves asymptotically

at future and past infinity. In a covariant setting this kind of a setup is ill-defined

since distances (times) have to do with the field configuration itself. What we can

do however is take some 4-d region of spacetime that in some sense encloses the

“interaction region”, and then set up some known spatial state on the 3-d boundary

of that region. In the case of a QFT that would be a plane wave state. In the case

of GR, we choose a known classical solution to Einstein’s equations, then set up an

equivalent coherent state (in the semi-classical limit) on the 3-d boundary.

Just as in the case of a Feynman diagram, there are many ways to choose a bulk

configuration of spacetime such that it agrees with the chosen boundary graph and

coloring (spin and intertwiner labels.) Such a configuration is called a 2-complex, and

consists of 0-d vertices connected by oriented 1-d edges, which border oriented 2-d

faces. See Figure 6.1 for a lexicon of diagram elements. Figure 6.2 shows a possible

2-complex whose boundary graph consists of two disconnected components, an initial

dipole graph and a final dipole graph. Slicing a 2-complex with a 3-d surface (for

example the boundary of the interaction region), an edge yields a node and a face

yields a link of a boundary spin network. As such, edges are colored with intertwiners

and faces are colored with spins.

One may also imagine slicing up the 2-complex into cells (Figure 6.1) such that

34

element name represents lives in intersecting with boundary

node 3D volume 3D boundary

link 2D area 3D boundary

vertex 4D volume 4D bulk

edge 3D volume 4D bulk

face 2D area 4D bulk

Figure 6.1: 2-complex and boundary spin network diagram elements.

Figure 6.2: Example 2-complex interpolating between different dipole graph states.

35

Figure 6.3: Slicing up a 2-complex at the faces to isolate each vertex, shown from theperspective of a single face which lies in the plane of the diagram. Shaded regions donot lie in the plane of the diagram.

36

Figure 6.4: A vertex shown inside its dual 4-simplex with boundary graph (top,) andits boundary graph alone (bottom.)

37

Figure 6.5: 2-complex with one vertex, interpolating between different daisy graphstates.

exactly one vertex is inside each 4-d cell, with a particular boundary spin network

associated with each vertex (Figure 6.4.) This idea is useful for enumerating all

possible 2-complexes consistent with a given boundary graph.

At any rate, the overall “transition amplitude” for a given boundary state is

obtained via a sum over histories, again analagous to the Feynman diagram sum,

expressed as a sum over the amplitudes for all the possible 2-complexes[56]. In this

work we will only consider the leading order term in this sum that corresponds to the

simplest possible 2-complex, one with a single vertex (specifically diagrams like the

one shown in Figure 6.5.)

6.2 Simplicity Constraints

The next task is to define an amplitude for a given 2-complex. This choice is the

essence of a particular spin foam model. Early models were based on the observation

38

that given the following “BF” action:∫B ∧ F (6.1)

where B is an arbitrary 2-form and F is the curvature associated with its connection,

one equation of motion is F = 0. Thus there is no curvature and the only degrees of

freedom are topological, that is how the space is connected to itself. These degrees

of freedom are entirely articulated using the 2-complexes defined earlier.

Now the GR action including the Immirzi parameter is∫ (e ∧ e+

1

γ? e ∧ e

)∧ F (6.2)

which is similar except that B has more structure. One may thus think of GR as a

BF theory plus constraints that force B to factor appropriately. These are called the

simplicity constraints, and one way to express them is

~K + γ~L = 0. (6.3)

We impose this form of the simplicity constraints weakly as follows. Using the SL(2,C)

representation detailed earlier, we can look up the action of the operators ~K and ~L

on a canonical basis state (5.32) and impose the condition that the matrix element

of the constraint for a canonical basis element vanishes,

〈k, p; jm| ~K + γ~L|k, p; jm〉 = 0 (6.4)

which yields the equation

− kp

j(j + 1)m+ γm = 0 (6.5)

Note that so far this equation does not ensure that the constraint vanishes when

sandwiched with a general state that is a superposition of basis elements, as there

may be cross terms. We will however be mainly concerned with coherent states that,

in the large j limit, are strongly peaked on m = j so we argue that the cross terms

arising from the ladder operators K+, K−, L+, L− are negligible. The trivial solution

m = 0 may be discarded, so we have

kp = γj(j + 1). (6.6)

39

There is an infinite set of solutions to this equation, but the typical choice is

p = γ(j + 1); k = j (6.7)

which corresponds to projecting into the lowest j representation in the set of repre-

sentations since j ≥ k. This choice of projection reduces in the large j limit to

p = γj; k = j. (6.8)

The latter choice is most often made in the literature. An alternate pathway to this

projection is to consider the so-called “master constraint” version of the simpicity

constraint

〈| ~K + γ~L|2〉 = 0 (6.9)

and use the Casimirs of the group to obtain an equation that relates k, p, j. One must

still invoke the large k, j limit in this case, and one arrives at the same solution (6.8).

It is important to note that in any case the existing solution to the simplicity con-

straints has the classical limit already built in, so to make reliable predictions about

the deep quantum regime one would need to refine the way the theory implements

the constraints.

Thus the simplicity constraints define a map Y † from SL(2,C) states to SU(2)

states. Recall that SL(2,C) representations were presented as a Hilbert space of

states constructed out of an infinite tower of SU(2) representations. The map Y †

projects to the bottom SU(2) space in the tower. Similarly, the map Y injects an

SU(2) representation into an SL(2,C) representation in the obvious way. Note that

while Y †Y is the identity operator on SU(2), Y Y † is a projection in the SL(2,C)

representation space.

6.3 EPRL Model

Let’s first look at the amplitude for B-F theory as a starting point, then slightly

modify it. If we assume locality, it’s reasonable to start with a form for the partition

function like so:

Z =

∫SU(2)

dhvf∏f

δ(hf )∏v

Av(hvf ) (6.10)

There is an amplitude for each face and an amplitude for each vertex.

40

Figure 6.6: Gluing the single-vertex 4-cells at the faces to form a 2-complex. Theproduct hf = hv1fhv2fhv3f = 1 since the path exactly retraces itself to its startingpoint.

41

Recalling our earlier definition of a 2-complex, each vertex is encapsulated in a

4d cell with a 3d surface that slices through the 2-complex, allowing one to define a

boundary graph on that surface in which links correspond to faces of the 2-complex

and nodes correspond to edges (Figure 6.4.) The holonomies hvf are along the links

of the boundary graph, and the labels keep track of the vertex to whose boundary

graph they belong and the face in which they lie. In the face amplitude, δ(hf ), hf

is the oriented ordered product of the holonomies that all lie in the face but belong

to the boundary graph of different vertices (Figure 6.6.) Explicitly, suppose a given

interior face is bounded by N edges which intersect at vertices v1, v2, ...vN then the

holonomy in the face amplitude is the product hf = hv1fhv2f ...hvNf (assuming the

orientations coincide, otherwise any or all of the holonomies will appear as inverses

instead.) From Figure 6.6 it is clear that the holonomy hf must be trivial since the

overall path retraces itself exactly back to the starting point. The face amplitude

δ(hf ) in the partition function reflects this geometric fact.1

If the face intersects the boundary of the 2-complex at a link with holonomy hl as

in Figure 6.7 then hf = hv1fhv2f ...hvNfhl, again with the caveat about orientations.

The appearance of δ(hf ) in the partition function is to guarantee the composition

law of arbitrary 2-complexes with boundary, as it ensures that they will glue together

correctly and the partition function of the composition will retain its form. The

face amplitude is in a sense just book-keeping, as it guarantees internal consistency

and identifies the external boundary state variables with internal bulk variables that

appear in the vertex amplitude. In our cases of interest we will be dealing with the

simplest possible 2-complex, with only one vertex in the bulk, so that (as in Figure 6.7,

bottom) our initial/final states correspond exactly to the boundary graph of that one

vertex (which has two disconnected components, the initial state and the final state.)

In this case the face amplitude is just a product of delta functions like δ(hvf1h−1l1

)

that make the correspondence explicit. After integrating out the variables hvf , the

partition function is then only a function of the boundary variables hl plugged into

the vertex amplitude, which is the transition amplitude associated to our initial/final

boundary states.

The vertex amplitude thus contains the essence of the dynamics of the theory.

1In the EPRL model, the face amplitude is an SU(2) delta function since boundary states involveholonomies of the SU(2) Ashtekar connection, while in other models it may be an SL(2,C) deltafunction of SL(2,C) holonomies.

42

Figure 6.7: Two examples of the relation between vertex boundary graph holonomiesand holonomies on the boundary of the 2-complex.

43

One possibility is to take the boundary spin network, which is a function of boundary

holonomies, and evaluate it at the identity for each group element. This is effec-

tively the trace of the intertwiners. This can be expressed in a variety of equivalent

notations,

〈Ψ|W 〉 =

∫dhlΨ(hl)Av(hl) = Av(Ψ) = Ψ(1); Av(hl) =

∏l

δ(hl) (6.11)

This is the Ooguri quantization of BF theory, and it is no surprise that it is a flat

topological theory; the vertex amplitude is just a product of delta functions imposing

flatness.

To obtain General Relativity, we must do something to this vertex amplitude

that takes into account the simplicity constraints. The key modification of the EPRL

model is to implement the map Y to inject SU(2) boundary spin network states

into SL(2,C) states, enforce SL(2,C) gauge invariance, then evaluate at the identity.

Schematically, the theory is defined by

〈Ψ|W 〉 =

∫dhlΨ(hl)Av(hl) = Av(Ψ) = (fY ·Ψ)(1) (6.12)

where the map fY stands for the injection Y followed by group averaging. Explicitly,

Av(hl) =

∫SL(2,C)N−1

dgn∏l

δ(h−1l Y †g−1tl 1gslY ). (6.13)

The group averaging followed by the nontrivial projection Y into SU(2) rescues the

theory from flatness. Another way to understand the vertex amplitude is to view gsl

and gtl as the SL(2,C) holonomies along the edges from the vertex to the boundary

node as in Figure 6.8. Then the product of these g’s around a face is precisely what

is needed to capture curvature, as each 2D face is dual to a 2D “hinge” as in a

Regge geometry, so their appearance in the vertex amplitude is natural. That they

are integrated over reflects a smearing over all possible internal geometries for the

vertex 4-cell, and the Y map ensures that only relevant ones (satisfying the simplicity

constraints) contribute.

Since the group variables gn always appear together in the form gslg−1tl , by changes

of variable it is possible to eliminate the dependence of the integrand on exactly

one group integration and due to the infinite volume of SL(2,C) the integral over

this variable will diverge. It is a simple matter to regularize this divergence by

44

Figure 6.8: Group-averaged edge holonomies that appear in the vertex amplitude.

45

dropping one vertex integral (and setting the corresponding gn to 1.) Recall the

parallel situation when imposing gauge invariance on SU(2) coherent spin networks

in (4.7); we dropped one redundant gauge integral there as well.

This form of the amplitude is somewhat opaque for concrete calculations, but it

may be translated into other more useful forms. In the group basis,

Av(hl) =

∫SL(2,C)N−1

dgn∏l

K(hl, g−1tl gsl) (6.14)

K(h, g) =∑j

∫SU(2)

dk d2jχj(hk)χγj,j(kg). (6.15)

Note that dj = 2j+1 by definition, and χj denotes the character (trace) of the SU(2)

representation. The character χγj,j of an SL(2,C) group element is understood to

be the SU(2) trace after projecting with the Y map described earlier. To see how

this form relates to (6.13), observe that the integration over k with one factor of dj

glues the two traces together into one via the orthogonality relation of the SU(2)

representations (5.4), and the sum over j of the characters with the other factor of

dj is an SU(2) delta function (5.16). The “evaluation at the identity” happens in the

kernel K,

K(hl, gsl1g−1tl ) = K(hl, gslg

−1tl ). (6.16)

In our application we will specify the boundary states using gauge-invariant holo-

morphic coherent states, so it is useful to express the amplitude directly in terms of

them. Recall (4.4, 4.5, 4.7),

ΨHl(hl) = Kt(Hlh−1l ) (6.17)

Kt(g) =∑j

(2j + 1)e−t2j(j+1)Tr[Dj(g)] (6.18)

ΨHl(hl) =

∫SU(2)N−1

dun⊗l

ΨHl(u−1tlhlusl) (6.19)

46

so putting the pieces together we obtain

〈ΨHl|W 〉 =

∫dhlΨHl(hl)Av(hl) (6.20)

=

∫dhl

∫dun

∫dgn

∏l

ΨHl(u−1tlhlusl)δ(h

−1l Y †g−1tl gslY ) (6.21)

=

∫dun

∫dgn

∏l

ΨHl(u−1tlY †g−1tl gslY usl) (6.22)

=

∫dun

∫dgn

∏l

ΨHl(Y†u−1tl g

−1tl gsluslY ) (6.23)

=

∫dgn

∏l

ΨHl(Y†g−1tl gslY ) (6.24)

=

∫dgn

∏l

Kt(HlY†g−1sl gtlY ) (6.25)

=

∫dgn

∏l

∑jl

(2jl + 1)e−t2jl(jl+1)Trjl [HlY

†g−1sl gtlY ] (6.26)

and finally the vertex amplitude is

〈ΨHl|W 〉 =

∫dgn

∏l

∑jl

(2jl + 1)e−t2jl(jl+1)Tr[Djl(Hl)D

(jl,γjl)jl

(g−1sl gtl)] (6.27)

Note that in the simplification we pushed the un elements through Y and used the

translation invariance of the dgn measure to absorb them. For this to be possible,

the omitted un and gn integrals are chosen to match.

Finally, we must address the issue of normalization. The convention in the liter-

ature [46, 47, 49] is to normalize the amplitude as

〈ΨHl|W 〉〈ΨHl|ΨHl〉

(6.28)

but this is at odds with basic intuition about quantum mechanics; states are rays

in a Hilbert space, and probability amplitudes are insensitive to scaling of a state

by a real parameter, for example the norm. The above expression (6.28) clearly is

not invariant under a real scaling of the boundary state |ΨHl〉. Put another way,

it is not the amplitude which is to be normalized but rather the state that must be

normalized for it to make sense quantum-mechanically. Thus a more logical choice of

normalization is〈ΨHl|W 〉√〈ΨHl|ΨHl〉

(6.29)

47

and we shall proceed with this definition, keeping in mind the other one for comparison

with results from the literature.

48

Chapter Seven: FLRW Coherent State Labels and

Normalization

We now begin to set up the calculation of a transition amplitude in which the initial

and final states are quantum coherent states that correspond to FLRW spacetime

slices in the classical limit. This calculation refines and expands upon prior work in

which the goal was simply to establish that the quantum dynamics match the classical

dynamics at the lowest order approximation [46, 44, 47, 48, 49]. Here we will correct

some errors in the literature, and also relax certain approximations. Our main goal

however is to examine the dependence of the transition amplitude on the refinement

of the coherent state graph, an issue which has not yet been treated in the literature.

We begin therefore with an articulation of the specfication of the relevant coherent

states, with an emphasis on the physically relevant parameters.

7.1 Regular Graphs

In order to construct a coherent state that corresponds to a spatial slice of a clas-

sical FLRW spacetime with a particular choice of granularity, first we choose a reg-

ular graph[47]. A regular graph is defined by the requirements of homogeneity and

isotropy; every node has the same valence (number of links,) and each node’s links

are uniformly distributed around the node. Moreover, every link has the same area

eigenvalue label, and the corresponding paths in the FLRW manifold are geodesic

and all have the same proper length. Only graphs with a single connected compo-

nent are considered. We do not address the issue of whether or not such graphs exist

for general values of the number of nodes N and links per node L, but rather focus

on a few concrete examples. We also restrict attention to the cases where L ≥ 4 so

that the classical volume of each node is nonvanishing.

The requirements that the graph be homogeneous and isotropic and have only

one connected component constrain the link structure of the graph, specifically the

source/target relations. If the graph contains a link with the same source and target

node then by isotropy all the links attached to that node also have the same source

49

and target and by homogeneity every node looks like this. The requirement that

there only be one connected component implies that there is only one possible graph,

the one with a single node (N = 1) with all the links self-glued. All other regular

graphs will have no self-glued links.

Now suppose the graph contains two nodes that are connected by two different

links, then there is a path that goes from node n1 to node n2 and back to n1 without

retracing itself. By isotropy all the links leaving n1 also connect back to n1 in this

way via some intermediate node. If the intermediate node is the same for any two

pairs of links, then all the links connect to the same intermediate node and there

is one possible graph (again invoking homogeneity and connectedness), the one with

two nodes (N = 2). Otherwise the intermediate nodes are all distinct and moreover

by homegeneity they all have the same link structure as well. We do not exhaustively

pursue the possibilities here but rather we exhibit an example, the 2× 2 cubic lattice

(N = 8) with opposite sides joined.

Aside from these three (possibly degenerate) cases, we consider the general case in

which each pair of nodes of a regular graph are connected by exactly one link. More

precisely, graphs with N ≥ 3 where the inverse source and target maps s−1(n) and

t−1(n) have the property that given two distinct nodes n1 and n2 the set s−1(n1) ∩t−1(n2) is either empty or contains exactly one link.

A node is taken to correspond classically to a regular polyhedron with L faces

if such a polyhedron exists, or a pseudoregular polyhedron (Appendix B) with L

faces for more general values of L. Certain geometric information is needed for the

coherent state labels detailed in the next section, namely the relation between the

normal coordinate distance h from the center of each polyhedron to its faces and the

quantum observables (the volume per node VN and the area per face AL.) This is

of crucial importance because our objective is to compare different choices of graph

which correspond to the same physical space, which we take to mean that the quantum

observables must match. The principal observable of interest is the total volume of

spacetime V = NVN .

For a pseudoregular polyhedron in flat space, each face is the base of a pyramid

of height hL, base area AL, and volume VL = 13hLAL. The volume of the polyhedron

is VN = LVL, so

hL =3VLAL

=3VNLAL

. (7.1)

50

Figure 7.1: Examples of Daisy graphs, with L=6 (left) and L=20 (right.)

Furthermore, we have (B.1,B.2)

α(L) ≡ V1/3N

(LAL)1/2=

1

(36π)1/6

( (L− 2)2

L(L− 1)

)1/6(7.2)

so that

AL = (36π)1/3( (L− 1)

L2(L− 2)2

)1/3(VN

)2/3(7.3)

and

hL =( 3

4π

)1/3( (L− 2)2

L(L− 1)

)1/3(VN

)1/3. (7.4)

We now apply these considerations to some concrete examples:

• k = 0, N = 1, L ≥ 6 and even (Figures 7.1, 7.2)

The classical space is flat (k = 0) and topologically a torus. The so-called

“daisy” graph consists of one node of valence L, where L ≥ 6 and is even.

Links in one hemisphere are taken to be outgoing and the rest incoming, and

opposite links are connected. Note we choose to count the paths leading from

the node as L for compatibility with the other cases. As with the other cases,

the total number of links in the graph is Ltot = NL/2 = L/2.

The geometric quantities in terms of the volume of space V are

AL = (36π)1/3( (L− 1)

L2(L− 2)2

)1/3V 2/3 (7.5)

51

Figure 7.2: Geometric interpretation of the L=6 daisy graph. The node represents acube, and the dashed link corresponds to the dashed faces that are identified.

and

hL =1

2

( 6

π

)1/3( (L− 2)2

L(L− 1)

)1/3V 1/3. (7.6)

The L = 6 case (self-glued cube) is treated in [48, 49].

• k = 0, N = 2, L ≥ 4 and even (Figure 7.3)

The classical space is flat (k = 0) and topologically a torus. The so-called

“dipole” graph consists of two nodes of valence L, where L ≥ 4 and is even.

Links in one hemisphere are taken to be outgoing and the rest incoming, and

each outgoing link connects to the opposite ingoing link of the other node. In

some contexts it is easier to instead orient the links so that for all links one node

is the source and the other is the target. In the special case L = 4 one of the

tetrahedra is flipped so that the normals correspond in an analagous manner to

the other cases. The total number of links in the graph is Ltot = NL/2 = L

The geometric quantities in terms of the volume of space V are

AL = (9π)1/3( (L− 1)

L2(L− 2)2

)1/3V 2/3 (7.7)

and

hL =1

2

( 3

π

)1/3( (L− 2)2

L(L− 1)

)1/3V 1/3. (7.8)

The L = 4 case (two glued tetrahedra) is treated in [46].

52

Figure 7.3: Geometric interpretation of the L=6 dipole graph. Each node representsa cube, and each patterned link corresponds to the patterned faces that are identified.

• k = 0, N = n3 where n ∈ Z, n ≥ 2, L = 6

The classical space is flat (k = 0) and topologically a torus. Nodes are arranged

in an n×n cubic lattice and each node has valence six. Links in one hemisphere

are taken to be outgoing and the rest incoming, and are connected in the obvious

way with opposite sides of the lattice connected. Note that the case n = 2 must

be treated separately as discussed previously. The case n = 1 is a special case

of the first type already described. The total number of links in the graph is

Ltot = NL/2 = 3n3.

The geometric quantities in terms of the volume of space V are

AL =(VN

)2/3(7.9)

and

hL =1

2

(VN

)1/3. (7.10)

53

7.2 Coherent State Labels

Following [48, 49], we use the holonomy-flux parameterization (4.1) for the coherent

state labels. Each link carries an SL(2, C) group element Hl, with

Hl = Ul exp(it

El8πG~γ

)(7.11)

where Ul is the holonomy along the link and El is the flux of the inverse densitized

triad through the corresponding surface. The holonomy (3.1) is the path ordered

integral of the exponential of the connection along the path γl associated with the

link,

Ul = P exp

∫γl

A; Aa = Aiaτi; τi = − i2σi (7.12)

and the ~σ are the standard Pauli matrices,

σ1 =

(0 1

1 0

); σ2 =

(0 −ii 0

); σ3 =

(1 0

0 −1

); (7.13)

In all three classes of regular graph considered in the previous section, space is flat

(k = 0) so the curves γl are straight lines. The gluing of spatial cells corresponding to

each node is such that the coordinate system used in each cell can be oriented identi-

cally and the tangent to each curve remains unchanged across cell boundaries. This

tangent vector nl = ˆns(l) is the same as the normal to the surface of the polyhedron

corresponding to the source node, and opposite to the normal of the surface of the

polyhedron corresponding to the target node, nt(l) = −ns(l).The connection A defined in (2.8) is

Aia = ωia + γKia, (7.14)

and since space is flat the spin connection ω vanishes, so we need only to compute

the extrinsic curvature K.

The familiar FLRW line element is

ds2 = −dt2 + a(t)2(dx2 + dy2 + dz2) (7.15)

so the corresponding spacetime metric is

gµν = diag (−1, a(t)2, a(t)2, a(t)2) (7.16)

54

and writing

gµν = ηIJeIµeJν ; ηIJ = diag (−1, 1, 1, 1) (7.17)

the tetrad eIµ is

eIµ = diag (1, a(t), a(t), a(t)). (7.18)

The spatial three-metric hab and triad eia are therefore

hab = a(t)2δab; eia = a(t)δia; δ = diag (1, 1, 1) (7.19)

and their inverses are

hab = a(t)−2δab; eai = a(t)−1δai . (7.20)

A standard 3 + 1 split of the spacetime manifold yields the general form of the line

element in terms of the three-metric hab, the lapse N , and the shift Na,

ds2 = −N2dt2 + hab(dxa +Nadt)(dxb +N bdt) (7.21)

which by comparison with (7.15) implies that N = 1 and Na = (0, 0, 0). Hence the

extrinsic curvature is

Kab = 12L∂thab = a(t)a(t)δab; Ki

a = ηijebjKab = a(t)δia, (7.22)

the connection is

Aia = γa(t)δia (7.23)

and the holonomy of each link is obtained by integrating A along a path of coordinate

length 2hL in the direction nl,

Ul = exp(−ihLγa(t) nl · ~σ). (7.24)

Now integrating the hodge dual of the inverse densitized triad over the surface Sl

dual to a link using a suitable smearing function, we obtain

El = −iALa(t)2nl · ~σ. (7.25)

Note that both factors in the polar decomposition of Hl are exponentials of some

(complex) number times nl · ~σ so they commute, i.e. the left and right polar de-

compositions are identical. We may then combine the exponentials and write the

coherent state labels in the form

Hl = exp(− i2z nl · ~σ) (7.26)

55

where the complex number z is given by

z = hLγa+ i2ALa

2t

8πG~γ(7.27)

and the hL and AL parameters are given in the previous section for various choices

of regular graph.

7.3 Norm of an Isolated Holomorphic Link

We tackle the normalization of the coherent states in three steps. First, we com-

pute the normalization of a single isolated holomorphic coherent link state. Second,

we compute the normalization of a spin network with Perelomov semi-coherent link

states. Finally, we compute the normalization of a coherent spin network (with holo-

morphic link states) and show the relation with the first two results. The first two

steps do not depend on the specific choice of regular graph, whereas the full normal-

ization result does. We also provide a suggestion as to how to generalize the result

to any regular graph aside from the specific cases laid out above.

For the normalization of a single isolated holomorphic coherent link state we follow

[49]. Recall the definition of such a state (4.4, 4.5),

Ψtg(h) =

∑j

(2j + 1)e−t2j(j+1)Trj(gh

−1) (7.28)

where we have used the shorthand Trj(gh−1) = Tr[Dj(gh−1)]. We recollect some

useful facts about the Wigner D representation matrices. The faithfulness of the

representation ensures that

D(g)mn = D(g†)nm; D(g1g2) = D(g1)D(g2). (7.29)

Note that in our coherent states the argument h ∈ SU(2) is unitary (h† = h−1),

whereas the label g = Hl ∈ SL(2, C) is not unitary. The orthogonality relation for

the representation matrices is (5.4)∫dh Dj′(h)pqD

j(h)mn =1

2j + 1δjj′δpmδqn. (7.30)

56

The trace in any representation Trj(g) can be written in terms of the trace in the

fundamental representation Tr1/2(g) via the identity (5.13, 7.31),

λ = x+√x2 − 1; λ−1 = x−

√x2 − 1; x =

1

2Tr 1

2[g] ≥ 0

λ = x−√x2 − 1; λ−1 = x+

√x2 − 1; x < 0

(7.31)

Trj(g) = χj(g) =λ2j+1 − λ−(2j+1)

λ− λ−1=λ2j+1 − λ−(2j+1)

2√x2 − 1

. (7.32)

Now we write out the inner product of two link states and simplify,

〈Ψtg|Ψt

g′〉 =

∫dh Ψt

g(h)Ψtg′(h); f t(j) = (2j + 1)e−

t2j(j+1) (7.33)

=∑jj′

f t(j)f t(j′)

∫dh TrDj(gh−1)TrDj′(g′h−1) (7.34)

=∑jj′

f t(j)f t(j′)

∫dh TrDj(hg†)TrDj′(g′h−1) (7.35)

=∑jj′

f t(j)f t(j′)

∫dh Dj(h)mnD

j(g†)nmDj′(g′)pqD

j′(h−1)qp (7.36)

=∑jj′

f t(j)f t(j′)Dj(g†)nmDj′(g′)pq

∫dh Dj′(h)pqD

j(h)mn (7.37)

=∑jj′

f t(j)f t(j′)Dj(g†)nmDj′(g′)pq

( 1

2j + 1δjj′δpmδqn

)(7.38)

=∑j

(2j + 1)e−tj(j+1)Dj(g†)nmDj′(g′)mn (7.39)

=∑j

f 2t(j)Trj(g†g′) = Ψ2t

g†g′(1) (7.40)

=∞∑

2j=0

(2j + 1)e−tj(j+1) λ2j+1 − λ−(2j+1)

2√x2 − 1

; n = 2j + 1 (7.41)

=1

2√x2 − 1

∞∑n=1

ne−t4(n2−1)(λn − λ−n) (7.42)

=et4

2√x2 − 1

∞∑n=−∞

ne−t4n2

λn (7.43)

If we assume λ > 0, we may write λn = exp(n lnλ) without the ambiguity of a choice

57

of branch to define ln(−1). Then approximating the sum as an integral1,

〈Ψtg|Ψt

g′〉 ≈et4

2√x2 − 1

∫ ∞−∞

dn ne−t4n2+n lnλ (7.44)

= 2

√π

t

et4

√x2 − 1

lnλ

te(lnλ)

2/t (7.45)

Now to compute the norm ‖ΨtHl‖2 we set g = g′ = Hl using (7.26):

Hl = exp(− i2z nl · ~σ); g†g′ = H†lHl = exp(Im(z)nl · ~σ) (7.46)

x = cosh(Im(z));√x2 − 1 = sinh(Im(z)); λ = eIm(z) (7.47)

Now we put these values in the sum, approximate it as an integral, and achieve our

result2.

‖ΨtHl‖2 ≈ 2

√πet4

t3/2Im(z)

sinh(Im(z))exp

(Im(z)2

t

)(7.48)

7.4 Norm of an Isolated Semi-Coherent Node

When a holomorphic link is placed in a spin network and the group averaging proce-

dure is employed at the nodes, the traces of the group elements complicate consider-

ably. Therefore we first compute the effect of the group averaging on the norm when

the links carry Perelomov semi-coherent states. Not only is the norm of a Perelomov

state trivial to compute, but the link states factor into source and target states (4.14)

so the overall structure of the graph is irrelevant for the computation and we may

focus on the norm of a single node without regard to how the nodes connect to other

nodes. We expect that in the large j limit the norm of a holomorphic coherent state

will factorize into a norm of this type for the nodes times a norm for the links as

computed in the previous section, and we will see in fact it does even without the

large j approximation.

1In the present context, the assumption λ > 0 is valid but we shall see later that the groupaveraging at the nodes can introduce sub-leading terms with λ < 0. For a discussion of sub-leadingterms originating from the approximation of the sum as an integral and from considering λ < 0contributions, see appendix C

2Note that equation (B.25) of [49] is off by a factor of 2

58

At a particular node, we have the Livine-Speziale coherent intertwiner (4.16)

|nl〉 =

∫dg(⊗

l

g|nl〉)

(7.49)

where the nl refer to the links connected to this particular node. The inner product

of this state with itself is

〈nl|nl〉 =

∫dg′∫dg∏l

〈nl|(g′)†g|nl〉 (7.50)

but the Haar measure dg on the group is invariant under left/right translations and

inversions, so a simple change of variable (g′)†g → g leaves the dg integration measure

unchanged and makes the dg′ integral trivial, so that

〈nl|nl〉 =

∫dg∏l

〈nl|g|nl〉. (7.51)

Now this norm can be computed explicitly using computational methods, for example

using the normal vectors to the faces of a regular polyhedron with L faces and using a

particular representation j, and we will show these results later. However, it is better

to have a method for quickly obtaining the norm using an approximation that works

for a node with any number of links L that are distributed isotropically. To this end,

we rewrite the norm in a form such that the product of link factors becomes a sum

over links and hence scales as L.

〈nl|nl〉 =

∫dg exp

(L( 1

L

L∑l=1

ln〈nl|g|nl〉))

(7.52)

The norm has now been cast in a form that is appropriate for a saddle point ap-

proximation to the dg integral for large L. A useful explicit parameterization for the

group elements [24] is given by

g = cos γ1 + i sin γu · ~σ, γ ∈ [0, π], u ∈ S2 (7.53)

with

u = (sinα cos β, sinα sin β, cosα), α ∈ [0, π], β ∈ [0, 2π]. (7.54)

Introducing a vector ~p = sin γu we have cos γ = ±√

1− ~p2 where the sign ambiguity

must be taken into account by defining pη = η√

1− ~p2 for η = ±1. The integral is

then taken over two unit three-balls Bη such that |~p| ≤ 1, one for each value of η,∫SU(2)

dg =1

2π2

∑η=±1

∫Bη

d3~p√1− ~p2

. (7.55)

59

The group element takes the form

g = pη1 + i~p · ~σ, (7.56)

and the expression in the norm integral takes a convenient form as well,

〈n|g|n〉 = (pη1 + i~p · n)2j (7.57)

hence

〈nl|nl〉 =1

2π2

∑η=±1

∫Bη

d3~p√1− ~p2

exp

(L( 1

L

∑l

ln(pη + i~p · n)2j))

(7.58)

=1

2π2

∑η=±1

∫Bη

d3~p√1− ~p2

exp

(2jL

( 1

L

∑l

ln(pη + i~p · n)))

(7.59)

So in fact the saddle point approximation is valid in the large 2jL ≈ djL regime. This

fact illuminates the nature of the classical limit; it is associated with large surface

area of the polyhedron and hence physically large distance scales, but from another

perspective it is also associated with there being a large number of surface states

available.

We now perform the saddle point approximation on the integral

〈nl|nl〉 =1

2π2

∑η=±1

∫Bη

d3~p√1− ~p2

exp(

2jL S(η, ~p))

(7.60)

S(η, ~p) =1

L

∑l

ln(pη + i~p · n) (7.61)

Note that S(η, ~p) ≤ 0, and the maximum value S(η, ~p) = 0 is attained when ~p = 0

for η = ±1, so there are two critical points,3 c = 2. We now compute the Hessian of

3In fact this is an over-counting of the number of critical points as we will see later whenconsidering fully coherent spin network states. Once η is fixed for one node, to leading order all theother nodes must use the same η. Then since one of the integration variables is dropped, its valueis fixed and the orientation of the rest of the integration must match it.

60

S(η, ~p) at ~p = 0:

S(η, ~p) =1

L

∑l

ln f(η, ~p, n) f = pη + i~p · n (7.62)

∂ipη = −η2pipη

f |crit = pη|crit = η (7.63)

∂if = −η2pipη

+ i ni ∂if |crit = i ni (7.64)

∂i∂jf = −η2δijpη− η4pipj

p3η∂i∂jf |crit = −η δij (7.65)

∂iS =1

L

∑l

(∂iff

)(7.66)

Hij = ∂i∂jS|crit =1

L

∑l

(∂i∂jff− (∂if)(∂jf)

f 2

)∣∣∣crit (7.67)

Hij =1

L

∑l

(−η δijη− (i ni)(i nj)

η2

)(7.68)

=1

L

∑l

(−δij + ninj) (7.69)

=1

L

(− Lδij +

1

3Lδij

)= −2

3δij (7.70)√

det(−H) =(2

3

)3/2(7.71)

where we used the fact that∑

l ninj =∑

l13δij since by symmetry

∑l ninj ∝

∑l δij

and the proportionality factor is fixed by tracing. The final result is

〈nl|nl〉 ≈ c( 2π

2jL

)3/2 1

2π2

(1√

1− ~p2exp

(2jL S(η, ~p)

) 1√det(−H)

)∣∣∣∣∣crit (7.72)

= 2( 2π

2jL

)3/2 1

2π2

(3

2

)3/2(7.73)

=1√π

( 3

2jL

)3/2(7.74)

This approximation to the norm of the Livine-Speziale coherent intertwiner agrees

well with the exact norm even for low values of j and L. The exact norm computed

directly in the case of a regular tetrahedron (Figure 7.4) and a cube (Figure 7.5) are

plotted against the approximate norm for comparison.

61

Figure 7.4: Norm of the Livine-Speziale coherent intertwiner of a regular tetrahedron(dots) for various values of j compared to the approximate norm (curve)

7.5 Norm of a Holomorphic Coherent Spin

Network

Now we turn to the full case of the norm of a holomorphic coherent spin network

state, the definition of which we recall as (4.7)

ΨHl(hl) =

∫SU(2)N−1

dun⊗l

ΨHl(u−1tlhlusl). (7.75)

Note that the group averaging is only over N−1 nodes since one of these is redundant.

Later we will see explicitly in equation (7.106) why this is true. For the simplest graph

with N = 1 there is no averaging at all, and the norm of the graph state is just the

product of L/2 single link norms given by (7.48),

〈ΨtHl|Ψ

tHl〉 = ‖Ψt

Hl‖L (7.76)

where the right hand side is a function of Im(z) and t only (which are the same for

all links.)

62

Figure 7.5: Norm of the Livine-Speziale coherent intertwiner of a cube (dots) forvarious values of j compared to the approximate norm (curve)

For graphs with N ≥ 2 we proceed with the implicit understanding that the last

integral duN is omitted and uN = 1. Following the same steps as in (7.33 - 7.40), the

norm may be expressed as

〈ΨtHl|Ψ

tHl〉 =

∫dun

∫du′n

∏l

(∑j

f 2t(j)Trj(u−1t(l)Hlus(l)u

′s(l)H

†l (u′t(l))

−1))

(7.77)

=

∫dun

∏l

(∑j

f 2t(j)Trj(u−1t(l)Hlus(l)H

†l ))

(7.78)

after performing the change of variable un → un(u′n)−1 and the du′n integrals, which

are then trivial. As in (7.45) we approximate the sum for each link factor as an

integral. There are Ltot such factors, where Ltot = NL/2 is the number of links

overall.

‖ΨtHl‖

2 ≈

(2√πet4

t3/2

)Ltot ∫dun

(∏l

lnλl√x2l − 1

)exp

(2LIm(z)

tS(un)

)(7.79)

63

S(un) =1

2LIm(z)

∑l

(lnλl)2 (7.80)

λl = xl +√x2l − 1; λ−1l = xl −

√x2l − 1; xl =

1

2Tr 1

2[u−1t(l)Hlus(l)H

†l ] ≥ 0

λl = xl −√x2l − 1; λ−1l = xl +

√x2l − 1; xl < 0

(7.81)

Note the holomorphic coherent link state is a superposition of states corresponding to

different representations of spin j, but for Im(z) 1 the j = Im(z)/t term dominates.

Hence our choice of large parameter 2LIm(z)/t ≈ djL that justifies the saddle point

approximation tracks with the choice in the case where we considered a single node

(7.60), namely 2jL ≈ djL. The classical limit is thus not just associated with large

scales, but also with a large total number of surface states per node. We will also see

that the Hessian takes a simple form with this definition of S. As before, we write

un = pηn1 + i~pn · ~σ; u−1n = pηn1− i~pn · ~σ; pηn = ηn√

1− ~p2n (7.82)∫SU(2)N−1

dun =

(1

2π2

)N−1 ∑ηn∈±1

∫Bηn

d3 ~pn√1− ~p2n

. (7.83)

so that we cast the norm as

‖ΨtHl‖

2 ≈

(2√πet4

t3/2

)Ltot(

1

2π2

)N−1

∑ηn

∫d3 ~pn√1− ~p2n

(∏l

lnλl√x2l − 1

)exp

(2LIm(z)

tS(ηn, ~pn)

)(7.84)

where, as explained in Appendix C, we drop terms with λ < 0 since they are sub-

leading. To evaluate the norm, first we obtain an expression for x.

Hl = cos( 12z)1− i sin( 1

2z)nl · ~σ; H†l = cos( 1

2z)1 + i sin( 1

2z)nl · ~σ (7.85)

xl =1

2Tr[(pηt(l)1− i~pt(l) · ~σ)(cos( 1

2z)1− i sin( 1

2z)nl · ~σ)

· (pηs(l)1 + i~ps(l) · ~σ)(cos( 12z)1 + i sin( 1

2z)nl · ~σ)] (7.86)

We simplify using the familiar identities

Tr[1] = 2; Tr[σi] = 0; Tr[σiσj] = 2δij; Tr[σiσjσk] = 2iεijk;

Tr[σiσjσkσp] = 2(δijδkp − δikδjp + δipδjk)(7.87)

64

2xl = Tr[(pηt(l)1− i~pt(l) · ~σ)(pηs(l)1 + i~ps(l) · ~σ)] cos( 12z) cos( 1

2z)

− i Tr[(pηt(l)1− i~pt(l) · ~σ)(nl · ~σ)(pηs(l)1 + i~ps(l) · ~σ)] sin( 12z) cos( 1

2z)

+ i Tr[(pηt(l)1− i~pt(l) · ~σ)(pηs(l)1 + i~ps(l) · ~σ)(nl · ~σ)] cos( 12z) sin( 1

2z)

+ Tr[(pηt(l)1− i~pt(l) · ~σ)(nl · ~σ)(pηs(l)1 + i~ps(l) · ~σ)(nl · ~σ)] sin( 12z) sin( 1

2z)

(7.88)

xl =(pηs(l)pηt(l) + ~ps(l) · ~pt(l)

)cos( 1

2z) cos( 1

2z)

+(− pηs(l)(~pt(l) · nl) + (~ps(l) · nl)pηt(l) + pis(l)p

jt(l)n

kl εijk

)sin( 1

2z) cos( 1

2z)

+(

pηs(l)(~pt(l) · nl)− (~ps(l) · nl)pηt(l) + pis(l)pjt(l)n

kl εijk

)cos( 1

2z) sin( 1

2z)

+(pηs(l)pηt(l) − ~ps(l) · ~pt(l) + 2(~ps(l) · nl)(~pt(l) · nl)

)sin( 1

2z) sin( 1

2z)

(7.89)

xl =(pηs(l)pηt(l) + (~ps(l) · nl)(~pt(l) · nl)

)cosh(Im(z))

+((~ps(l) · ~pt(l))− (~ps(l) · nl)(~pt(l) · nl)

)cos(Re(z))

+ i((~ps(l) · nl)pηt(l) − pηs(l)(~pt(l) · nl)

)sinh(Im(z))

+(pis(l)p

jt(l)n

kl εijk

)sin(Re(z))

(7.90)

This expression attains a simple form when ~pn = 0, leading to

xl|~pn=0 = ηs(l)ηt(l) cosh(Im(z)) (7.91)√x2l − 1|~pn=0 = sinh(Im(z)) (7.92)

λl|~pn=0 = ηs(l)ηt(l)eIm(z) (7.93)

so the requirement λ > 0 implies that all ηn are equal and there are only two global

choices ηn = η = ±1. Further, fixing uN = 1 fixes the global orientation to ηn = 1.

Then

lnλl|~pn=0 = Im(z). (7.94)

S|~pn=0 =Ltot

2LIm(z) (7.95)

Now to perform a saddle point approximation of the integral we show that ~pn = 0

65

corresponds to the critical points of S.

∂niS(ηn, ~pn) =∂

∂pinS(ηn, ~pn) =

1

2LIm(z)

∑l

∂ni(lnλl)2 (7.96)

=1

2LIm(z)

∑l

2 lnλlλl

∂niλl (7.97)

=1

LIm(z)

∑l

lnλlλl

(1 +

xl√x2l − 1

)∂nixl (7.98)

=1

LIm(z)

∑l

lnλl√x2l − 1

∂nixl (7.99)

∂is(l)xl =

(−pis(l)pηt(l)pηs(l)

+ nil(~pt(l) · nl))

cosh(Im(z))

+(pit(l) − nil(~pt(l) · nl)

)cos(Re(z))

+ i

(nilpηt(l) +

pis(l)(~pt(l) · nl)pηs(l)

)sinh(Im(z))

+(pjt(l)n

kl εijk

)sin(Re(z))

(7.100)

∂it(l)xl =

(−pηs(l)p

it(l)

pηt(l)+ (~ps(l) · nl)nil)

)cosh(Im(z))

+(pis(l) − (~ps(l) · nl)nil

)cos(Re(z))

− i(

(~ps(l) · nl)pit(l)pηt(l)

+ pηs(l)nil

)sinh(Im(z))

−(pjs(l)n

kl εijk

)sin(Re(z))

(7.101)

∂inxl = δns(l)∂is(l)xl + δnt(l)∂

it(l)xl (7.102)

∂inxl|~pn=0 = i(ηt(l)δns(l) − ηs(l)δnt(l))nil sinh(Im(z)) (7.103)

∂inS|~pn=0 =i

L

∑l

(ηt(l)δns(l) − ηs(l)δnt(l))nil (7.104)

=i

L

( ∑l∈s−1(n)

ηt(l)nil −

∑l∈t−1(n)

ηs(l)nil

)= 0 (7.105)

This critical point condition is related to the closure relation at node n, as in [24].

For the leading order terms we have ηs(l) = ηt(l) = 1 so the source and target terms

are opposite in sign, which simply reflects the fact that nl is outgoing at the source

node and incoming at the target node. If we make the identifications nsl = nl and

66

ntl = −nl so that each nnl is outward pointing from node n, then (7.104) becomes

∂inS|~pn=0 =i

L

( ∑l∈s−1(l)

nisl +∑

l∈t−1(l)

nitl

)= 0. (7.106)

This is the closure relation at node n, or equivalently the condition of isotropic dis-

tribution of outgoing normals from node n. Note that the closure relation for the

N = 1 state is automatic, so it makes sense that we did not need to group average to

enforce this condition. We can also see that for N = 2 enforcing closure at one node

automatically guarantees closure at the other node as the equations are identical (up

to an overall minus sign), so only one group average is needed. It is straightforward

to see that in general closure only needs to be enforced at N − 1 nodes, so we need

only N − 1 group integrals thus justifying our construction.

The critical points of S are given by ~pn = 0, ηn = η = 1∀n. Note that the fixed

group element uN also corresponds to ~pN = 0, ηN = 1 so it does not require special

treatment when evaluating at the critical point.

Now we compute the Hessian matrix, continuing from (7.99).

∂ni∂mjS =1

LIm(z)

∑l

(1

x2l − 1− xl lnλl

(x2l − 1)3/2

)(∂nixl)(∂mjxl)

+lnλl√x2l − 1

∂ni∂mjxl

(7.107)

∂ni∂mjS|~pn=0 =1

L

∑l

(1

Im(z)− cosh(Im(z))

sinh(Im(z))

)(∂nixl)(∂mjxl)|~pn=0

sinh2(Im(z))

+∂ni∂mjxl|~pn=0

sinh(Im(z))

(7.108)

where it is understood throughout that 1 ≤ n,m ≤ N − 1. The derivatives of xl we

67

need are

∂is(l)∂js(l)xl = −

(δij +

pis(l)pjs(l)

p2ηs(l)

)pηt(l)pηs(l)

cosh(Im(z))

+ i

(δij +

pis(l)pjs(l)

p2ηs(l)

)(~pt(l) · nl)pηs(l)

sinh(Im(z))

(7.109)

∂it(l)∂jt(l)xl = −

(δij +

pit(l)pjt(l)

p2ηt(l)

)pηs(l)pηt(l)

cosh(Im(z))

− i(δij +

pit(l)pjt(l)

p2ηt(l)

)(~ps(l) · nl)pηt(l)

sinh(Im(z))

(7.110)

∂is(l)∂jt(l)xl =

(pis(l)p

jt(l)

pηs(l)pηt(l)+ niln

jl

)cosh(Im(z)) +

(δij − niln

jl

)cos(Re(z))

− i(nilp

jt(l)

pηt(l)−pis(l)n

jl

pηs(l)

)sinh(Im(z)) +

(nkl εijk

)sin(Re(z))

(7.111)

∂it(l)∂js(l)xl =

(pit(l)p

js(l)

pηt(l)pηs(l)+ niln

jl )

)cosh(Im(z)) +

(δij − niln

jl

)cos(Re(z))

− i(pit(l)n

jl

pηt(l)−nilp

js(l)

pηs(l)

)sinh(Im(z))−

(nkl εijk

)sin(Re(z))

(7.112)

which reduce when evaluated at the critical point ~pn = 0.

∂is(l)∂js(l)xl|~pn=0 = ∂it(l)∂

jt(l)xl|~pn=0 = −ηs(l)ηt(l)δij cosh(Im(z)) (7.113)

∂is(l)∂jt(l)xl|~pn=0 = niln

jl cosh(Im(z)) +

(δij − niln

jl

)cos(Re(z))

+(nkl εijk

)sin(Re(z))

(7.114)

∂it(l)∂js(l)xl|~pn=0 = niln

jl cosh(Im(z)) +

(δij − niln

jl

)cos(Re(z))

−(nkl εijk

)sin(Re(z))

(7.115)

∂ni∂mjxl = δns(l)δms(l)∂is(l)∂

js(l)xl + δnt(l)δmt(l)∂

it(l)∂

jt(l)xl

+ δns(l)δmt(l)∂is(l)∂

jt(l)xl + δnt(l)δms(l)∂

it(l)∂

js(l)xl

(7.116)

∂in∂jmxl|~pn=0 = −

(δns(l) + δnt(l)

)ηs(l)ηt(l)δnmδ

ij cosh(Im(z))

+(δns(l)δmt(l) + δnt(l)δms(l)

)·(niln

jl cosh(Im(z)) +

(δij − niln

jl

)cos(Re(z))

)+(δns(l)δmt(l) − δnt(l)δms(l)

)(nkl εijk

)sin(Re(z))

(7.117)

68

Now putting (7.103 - 7.94) and (7.117) into (7.108) and specializing to the leading

order term where ηn = 1, we have

Hnimj = ∂ni∂mjS|~pn=0 (7.118)

H ijnm =

1

L

∑l

(δns(l) − δnt(l)

)(δms(l) − δmt(l)

)niln

jl

(cosh(Im(z))

sinh(Im(z))− 1

Im(z)

)−(δns(l) + δnt(l)

)δnmδ

ij cosh(Im(z))

sinh(Im(z))

+(δns(l)δmt(l) + δnt(l)δms(l)

)·(niln

jl

cosh(Im(z))

sinh(Im(z))+(δij − niln

jl

) cos(Re(z))

sinh(Im(z))

)+(δns(l)δmt(l) − δnt(l)δms(l)

)(nkl εijk

) sin(Re(z))

sinh(Im(z))

(7.119)

−H ijnm =

1

L

∑l

(δns(l) + δnt(l)

)δnm

((δij − niln

jl

)cosh(Im(z))

sinh(Im(z))+

nilnjl

Im(z)

)−(δns(l)δmt(l) + δnt(l)δms(l)

)((δij − niln

jl

) cos(Re(z))

sinh(Im(z))+

nilnjl

Im(z)

)−(δns(l)δmt(l) − δnt(l)δms(l)

)(nkl εijk

) sin(Re(z))

sinh(Im(z))

(7.120)

Note that the Hessian is insensitive to the orientation of any link. Flipping a link

reverses the source/target relations and also flips the sign of nl. The first two terms

are unchanged under both of these reversals, while in the third term the two rever-

sals cancel one another. Note also that the Hessian matrix is symmetric under the

interchange ni ↔ mj.In the limit Im(z) 1, the Hessian is approximately

−H ijnm ≈

1

L

∑l

(δns(l) + δnt(l)

)δnm(δij − niln

jl

)(7.121)

=1

L

∑l

(δns(l) + δnt(l)

)δnm(δij − 1

3δij)

(7.122)

H ijnm ≈ −

2

3δnmδ

ij (7.123)

det[−H]−12 ≈

(3

2

)3(N−1)/2

(7.124)

69

where we used the fact that∑

l nilnjl = 1

3Lδij since the sum over links includes all

outgoing and incoming normals for the node n. This is precisely the Hessian of the

single isolated node (7.70), duplicated for each node.

Let us examine the form of the norm in general. Applying the results of the saddle

point approximation, the expression (7.84) reduces to

‖ΨtHl‖

2 ≈ ‖ΨtHl‖2Ltot

(1

2√π

(t

LIm(z)

)3/2)N−1

det[−H]−12 (7.125)

so we see that as expected the norm factors into Ltot link factors and N − 1 node

factors. Moreover all of the dependence on Re(z) is contained in the Hessian, and in

the limit Im(z) 1 with the identification Im(z)/t = j the node factor corresponds

precisely to the single-node norm computed using Perelomov semi-coherent states

(except for the factor of 2 that came from over-counting the allowed orientations.)

For the regular graph with N = 2,

−H ij11 =

1

L

∑l

(δij − niln

jl

)cosh(Im(z))

sinh(Im(z))+

nilnjl

Im(z)(7.126)

=(δij − 1

3δij)cosh(Im(z))

sinh(Im(z))+

13δij

Im(z)(7.127)

=2

3δij(

cosh(Im(z))

sinh(Im(z))+

1

2Im(z)

)(7.128)

det[−H]−12 =

(3

2

)3/2(cosh(Im(z))

sinh(Im(z))+

1

2Im(z)

)−3/2(7.129)

‖ΨtHl‖

2 ≈ ‖ΨtHl‖2L 1

2√π

(3t

2LIm(z)

)3/2(cosh(Im(z))

sinh(Im(z))+

1

2Im(z)

)−3/2. (7.130)

Computing the determinant explicitly by hand in more general cases is unwieldy

but is easily done using analytic computation software to write out the Hessian (7.120)

given a set of source/target relations for a graph of interest, then insert the determi-

nant in (7.125).

70

Chapter Eight: Quantum FRW Cosmology

Transition Amplitude

Now that we have specified our initial and final coherent states and calculated their

norm, we are prepared to calculate the relevant transition amplitudes.

8.1 Transition Amplitude

We consider the transition amplitude corresponding to a boundary state that is the

tensor product of coherent states on disjoint graphs, corresponding to the initial and

final states,

|Ψ〉 = |Ψi〉 ⊗ |Ψf〉. (8.1)

We take the lowest order term in the vertex expansion, corresponding to the 2-

complex with a single vertex. In this case1 the amplitude factorizes into a product of

the amplitudes for each component of the boundary state,

〈Ψ|W 〉 = 〈Ψi|W 〉〈Ψf |W 〉. (8.2)

We will therefore just study one factor. In previous investigations in the literature

[46, 47] the focus was on reproducing the classical dynamics. Here we are interested

in the graph dependence of the amplitude. Namely, given a fixed initial state we

wish to compare the probability of transitioning to various physically equivalent final

states as a function of the number of nodes and links of the graph. We define two

states to be physically equivalent if the total fiducial volume of space is the same

for both. This may or may not be the appropriate criterion, but it is a serviceable

starting point.

Our initial/final states are defined in the previous chapter, and the transition

amplitude is given by (6.27)

〈ΨHl|W 〉 =

∫dgn

∏l

∑jl

(2jl + 1)e−t2jl(jl+1)Tr[Djl(Hl)D

(jl,γjl)jl

(g−1sl gtl)] (8.3)

1For a general 2-complex the amplitude does not factorize, which may be cause to be suspiciousof the lowest order term as possibly degenerate.

71

The normalized amplitude is given by (6.29)

〈ΨHl|W 〉√〈ΨHl|ΨHl〉

(8.4)

We consider the concrete example of initial and final graphs with N = 1 and

L ≥ 6 (Figure 6.5,) a case where the transition amplitude is tractable for all values

of z (even the deep quantum regime.) Recall that L is the valence of the node, so

the total number of links is still Ltot = NL/2 = L/2. In this case there are no

group integrations, and for every link we have gsl = gtl = 1. The amplitude therefore

simplifies dramatically to

〈ΨHl|W 〉 =∏l

∑jl

(2jl + 1)e−t2jl(jl+1)Trjl [Hl] (8.5)

Proceeding as we did for the calculation of the norm, we use (5.13, 7.31) for the trace

and approximate the sum as an integral as in (7.45) to obtain

〈ΨHl|W 〉 ≈∏l

2√π

et/8

(t/2)3/2lnλ√x2 − 1

e2(lnλ)2/t (8.6)

where

Hl = exp(− i2z nl · ~σ) = cos( 1

2z)1− i sin( 1

2z)nl · ~σ (8.7)

x =1

2Tr 1

2[Hl] = cos( 1

2z) (8.8)

√x2 − 1 = i sin( 1

2z) (8.9)

λ = x+√x2 − 1 = exp( 1

2iz) (8.10)

lnλ = 12iz. (8.11)

Note that in accordance with the discussion in Appendix C, the branch cut in

lnλ = 12iz is tied to the infinite series of Fourier transform integrals. That is, the

dominant term in the series picks out the branch where −π < Im(lnλ) ≤ π, and when

Im(lnλ) deviates significantly from zero the next-to-leading order term becomes more

comparable to the leading order term. At Im(lnλ) = ±π the two dominant terms

are of the same order, but they both contain a suppressing Gaussian factor. The net

effect of this is that the above expression for the amplitude is valid only for Re(z)

near zero (mod 4π,) the amplitude is suppressed as Re(z) moves away from zero, and

72

Figure 8.1: Normalized one-vertex amplitude for a single node graph with L = 6 andt = 1

the amplitude is periodic in Re(z) with period 4π. With these caveats, the amplitude

is approximately

〈ΨHl|W 〉 ≈

(2√π

et8

(t/2)3/2

12z

sin( 12z)

exp

(−z

2

2t

))L/2

. (8.12)

For the sake of brevity here we do not repeat the correction terms shown in Appendix

C but in the plot shown we include several of them, the main effect of which is to

exhibit the periodicity in Re(z).

The norm is given by (7.76, 7.48)

〈ΨtHl|Ψ

tHl〉 ≈

(2√πet4

t3/2Im(z)

sinh(Im(z))exp

(Im(z)2

t

))L/2

(8.13)

73

Figure 8.2: Normalized one-vertex amplitude for a single node graph with L = 8 andt = 1

The normalized amplitude is then given by (6.29)

〈ΨHl|W 〉√〈ΨHl|ΨHl〉

(8.14)

which is plotted in Figures 8.1 to 8.4 for t = 1 and various values of L. The amplitude

is sharply peaked on Re(z) = 0, which corresponds to the classical Friedmann equation

with no matter or cosmological constant, a = 0. The amplitude grows quickly with

Im(z), but does not vanish at Im(z) = 0 so there does not appear to be singularity

resolution as one might have hoped. Smaller values of t result in a sharper peak.

Larger values of L also sharpen the peak and enhance the growth of the amplitude

for large Im(z).

Unlike previous results in the literature, our amplitude does not asymptote to a

constant value for large Im(z), which is a result of a different normalization. It is also

worth noting that using (6.28) instead for the normalization yields an unsatisfactory

74

Figure 8.3: Normalized one-vertex amplitude for a single node graph with L = 12and t = 1

result; the amplitude is in that case very small everywhere except for a sharp peak

at Re(z) = Im(z) = 0, which is unphysical.

Now recall that the transition amplitude at the one-vertex level is just a prod-

uct of the initial and final amplitudes, each of which is of the above form. Both of

these factors must have vanishing Re(z), i.e. ai = af = 0, otherwise the amplitude

is suppressed. However the transition amplitude is large for initial/final states with

different values of the scale factor (ai 6= af ,) a situation which is difficult to interpret

physically2. Part of the difficulty comes from the covariant setting; there is no speci-

fication of how “far away” in time the initial and final states are from each other. In

fact the state itself carries information about a and a, and the amplitude correlates

them to determine a differential equation. It would be nice if the amplitude also guar-

2Note that the situation is no better if the amplitude asymptotes to a constant for large Im(z)as in [46], as it appears then that the transition probability is equal from a given initial scale factorto any final scale factor.

75

Figure 8.4: Normalized one-vertex amplitude for a single node graph with L = 20and t = 1

anteed that the initial and final states were consistent with each other, but perhaps

that is too much to hope for at the one-vertex level. This is a well-known problem

associated with the factorization of the amplitude that is special to the one-vertex

amplitude; it may be that the lowest order 2-complex just doesn’t capture enough of

the dynamics for the amplitude to be physically meaningful.

That said, we will still try to see if we can say anything about the graph refinement

by looking at this amplitude. Suppose we fix an initial scale factor ai and a fidicial

total volume of space V along with the initial number of links Li, then we will study

the behavior of the amplitude as a function of Lf while holding V and af = ai fixed

(and also ai = af = 0 so that the amplitude is not suppressed.) Recall from (7.27,

76

7.5) that

z = hLγa+ i2ALa

2t

8πG~γ(8.15)

AL = (36π)1/3( (L− 1)

L2(L− 2)2

)1/3V 2/3 (8.16)

so we may write

Im(z) = 2(36π)1/3a2t( (L− 1)

L2(L− 2)2

)1/3V 2/3; V 2/3 =

V 2/3

8πG~γ. (8.17)

Here we take a to be dimensionless, and we pair V with the dimensionful constant to

form the dimensionless volume V which is now essentially in Planck units. Observe

that the L dependence is roughly 1/L for large enough L. We set ai = af = 1 and

Vi = Vf = V and plot the amplitude as a function of Lf and V for some choice of Li.

The result (Figure 8.5) shows that the amplitude favors large Lf regardless of V . For

the reasons already discussed, this result is probably not to be taken too seriously; we

could swap the V dependence for af , and already the interpretation of a transition

from ai to af is murky. It is useful however as an illustration of a possible way to

set up a calculation to address the issue of refinement. Perhaps the question itself

needs to be posed in a different way, and a better understanding of how transition

amplitudes convey physical information (especially beyond the one-vertex level) will

provide a new perspective.

77

Figure 8.5: Normalized amplitude with Li = 6 and t = 1 as a function of refinementL and fiducial volume V .

78

Chapter Nine: Conclusion and Future Work

In this work we have constructed a large class of coherent states with a semiclassical

interpretation in terms of psuedo-regular polyhedra. We computed the normalization

of these states, in a couple of simple cases explicitly, without resorting to the large-

scale (semi-classical) limit. In doing so, we introduced the technique of performing a

saddle point approximation using the valence of the nodes as an expansion parameter

rather than the area eigenvalue, a novel method that arises naturally from considering

graphs with nodes of arbitrary valence but works well even for relatively low valence

and so may be of general use. We uncovered some standing issues with the way

the normalization of states is performed in the literature, and chose a convention

which seems to be in line with standard quantum thinking and produces a sensible

normalized amplitude when applied. We explicitly computed a transition amplitude

for the simplest class of graph, with a single self-glued node of arbitrary valence, that

does not employ the standard large-distance approximation hence is valid in the deep

quantum regime. We also propose an alternate form of the normalization for which

the amplitude displays interesting quantum behavior (See Appendix D.)

9.1 Future Work

One obvious direction for future work is to carry out the computation of the nor-

malized amplitude for the other two classes of graphs defined and normalized here.

This investigation should shed some more light on the appropriateness of the choice

of normalization made in the current work. The class of dipole diagrams (N = 2)

would be an interesting next step, though it would only say a limited amount about

granularity since N is only stepped up by one. To really give an answer as to whether

the amplitude favors higher or lower N one would need to investigate a class of dia-

grams that admits a range of values for N , for example the cubic lattice graphs. The

result of this amplitude might show an indication of whether the zero point energy

density is large or small in pure quantum gravity.

Another point of future interest is the implementation of gauge invariance, which

is imposed strongly in the current theory described in this work; LQG is sometimes

79

criticized for this fact. Recently operators built out of coherent states have been pro-

posed in [50] that might be used to impose the gauge invariance weakly. Alternately,

it might be possible to impose the gauge invariance more directly in the context of

geometric quantization.

Further, the EPRL spin foam model as currently defined seems to contain the

classical limit in its implementation of the simplicity constraints. Moreover, in the

present work crucial use was made of the SU(2) character formula to avoid taking

the classical limit. There may be a way to implement the simplicity constraints in a

more covariant way that comes naturally from the underlying mathematical structure

of SL(2,C), so that the SL(2,C) character comes into the amplitude calculation in

the same way the SU(2) character did with the normalization. Since the SL(2,C)

character is a distribution on the group, it must be integrated over some “smearing”

function and the simplicity constraints might fill this role. Concretely, we might seek

to impose the simplicity constraints as a “group averaging” procedure similar to the

way gauge invariance was imposed at the nodes,

|Ψ〉SC =

∫ds exp

(~s · ( ~K + γ~L)

)|Ψ〉. (9.1)

In fact such an operator T χy(a) has a nice Fourier transform, namely

y(a) =

∫ds δ(S−1a); S = exp

(~s · ( ~K + γ~L)

)(9.2)

and it may serve as precisely the smearing function needed to make the relevant

SL(2,C) traces finite. The measure ds here is chosen as an appropriate invariant

measure on the coset space associated to the decomposition of a general SL(2,C)

group element

a = Su (9.3)

where S is an exponentiation of the constraint as defined earlier, and u is an SU(2)

element. Thus the simplicity constraints act to select out a particular SU(2) subgroup.

The appropriate definition of the measure, the question of the uniqueness of the

decomposition (9.3), and how to define a finite amplitude using these structures are

interesting objects for future study. The hope is that this line of investigation may

produce a new definition of the spin foam amplitude that is tractable at small scales.

80

Appendix A: Action Priciple for GR

A.1 Mathematical Framework

Here we derive the appropriate boundary terms and the equations of motion from

the most general form of the GR Lagrangian L. We will work in the geometrical

language of bundles, connections, and exterior derivatives. First we describe the

general procedure for any QFT, then apply it to GR.

We begin with a base space M (spacetime) and a bundle over M with typical

fiber F . For simplicity we take the bundle to be topologically trivial so that the total

space is F ×M . Now we have an exterior derivative d on the base space M and an

exterior variation δ on the field configuration space F . Both of these are De Rham,

that is d2 = δ2 = 0. We define an exterior derivative D = d + δ on the total space

M × F , and require that it is also De Rham. Then we have

D2 = (d+ δ)2 = d2 + dδ + δd+ δ2 = dδ + δd = 0; ⇒ dδ = −δd, (A.1)

that is d and δ anticommute. This relation greatly simplifies the calculations later

on.

The physical theory is defined by a choice of Lagrangian L which is a density of

weight one on M and a 0-form (a scalar function) on F . A density of weight one is

a top form (in our case a 4-form) tensored with a section of the orientation bundle,

but for the sake of simplicity we will ignore the latter and treat it as a top form.

The boundary term θ is a next-to-top form (a 3-form) on M and a 1-form on F ,

and is determined (up to total derivatives) by the requirement that δL+ dθ must be

linear over functions, that is it contains no mixed partials dδ or δd.1 Further, the

requirement δL + dθ = 0 yields the field equations (the Euler-Lagrange equations.)

Finally the symplectic form Ω, a next-to-top form (3-form) on M and a 2-form on F , is

1This requirement replaces the traditional step in most physics textbook treatments where theboundary terms are assumed to vanish outside some compact region of spacetime, but is moregeneral. Also note that using the property dδ = −δd replaces the step where one integrates byparts.

81

obtained from the boundary term θ via Ω = δθ.2 Figure A.1 shows diagrammatically

how these objects are related.

Figure A.1: Degrees of useful forms and their relationships

degree in F

deg

ree

inM

0 1 2 · · ·4 L δ−→ δL+ dθ = 0

↑ d3 θ

δ−→ Ω...

In what follows, when taking the exterior variation of a section of a bundle we

omit the evaluation map, writing simply δe for example.

A.2 Notation, Identities

First we establish notational conventions for the following sections and some useful

identities. We mostly employ an index-free notation using the symbols ∧ and ∧to indicate contraction of internal Lorentz indices with ε and η respectively, so for

example F ∧F = εIJKL FIJ ∧ FKL and e∧e = ηIJ e

I ∧ eJ . The trace operator tr is

also used to indicate contraction of initial and final indices with η. Indeces may be

reintroduced as needed for clarification.3

Using this notation, the covariant derivative of e (the torsion) is

De = de+ ω∧e. (A.2)

Sometimes it will be more convenient to work with a Hodge star operator ? on the

internal Lorentz indices rather than using the ∧ notation. This operator is defined

on a Lorentz rank 2 tensor T by

(?T )KL = 12ε KLIJ T IJ (A.3)

2Note that Ω here has the correct structure to be projected to a leaf of a spacetime foliation toobtain the symplectic form presented in tradition textbook treatments, but with the advantage thatit has been completely disentangled from the choice of foliation.

3As in [57], we employ an abstract index notation throughout this appendix; that is, indices donot refer to a basis but rather indicate the kind of object and specify where contractions are takingplace.

82

The factor of 12

is chosen so that (using standard ε identites)

? ? T = T (A) (A.4)

where the symmetric and antisymmetric parts of T are defined as

(T (A))IJ = T [IJ ] = 12

(T IJ − T JI

)(A.5)

(T (S))IJ = T (IJ) = 12

(T IJ + T JI

)(A.6)

so that

T = T (A) + T (S). (A.7)

We also introduce a wedge bracket notation,

[ω∧T ] = ω∧T − T ∧ω (A.8)

which is useful when writing the covariant derivative of T ,

DT = dT + [ω∧T ]. (A.9)

Taking these conventions together, one may show (again using standard ε iden-

tites) that for T = T (A),

?[ω∧ ? T ] = [ω∧T ] (A.10)

which leads to the identity

D ? T = ?DT. (A.11)

Finally, recall that the curvature F is given by

F = dω + ω∧ω, (A.12)

and the second Bianchi identity is

DF = d(ω∧ω) + ω∧dω +ω∧(ω∧ω)− dω∧ω −

(ω∧ω)∧ω = 0 (A.13)

so the previous identity implies that we also have

D ? F = ?DF = 0. (A.14)

83

A.3 GR Action

The basic objects out of which GR is built are the spacetime manifold M and a tetrad

bundle e over M with connection ω. As stated in the main text (2.3), we will begin

with the “kitchen sink” Lagrangian that contains all possible terms one can write

down that are 4-forms on M and 0-forms on F (i.e. taking local gauge invariance

into account).

LGR = α1L1 + α2L2 + α3L3 + α4L4 + α5L5 + α6L6 (A.15)

L1 = F ∧F L2 = − trF ∧F (A.16)

L3 = e∧F ∧e = −2 tr (?(e ∧ e)∧F ) L4 = e∧F ∧e = tr ((e ∧ e)∧F ) (A.17)

L5 = e∧e∧e∧e L6 = De∧De (A.18)

A.4 Topological terms

A topological term in the Lagrangian is one that does not affect the equations of

motion. The GR action contains two such terms that are total derivatives, that is

they can be written in the form Li = d(· · · ). First note that

δLi = δd(· · · ) = −dδ(· · · ), (A.19)

so an appropriate choice of θi to make δLi + dθi linear over functions is

θi = δ(· · · ). (A.20)

Now since δLi + dθi = 0 there is no contribution to the equations of motion. There

is also no contribution to the symplectic form,

Ωi = δθi = δ2(· · · ) = 0. (A.21)

Note the power and elegance of the exterior variation notation in the above calcula-

tions.

84

It remains to show that the Lagrangian contains three topological terms. First

the Pontryagin term,

L2 = − trF ∧F

= − tr ((dω + ω∧ω)∧(dω + ω∧ω))

= − tr (dω∧dω + ω∧ω∧dω + dω∧ω∧ω +((((((

ω∧ω∧ω∧ω)

= − tr (d(ω∧dω) + 2 ω∧ω∧dω)

= − tr

(d(ω∧dω) +

2

3d(ω∧ω∧ω)

)= −d

(tr

(ω∧dω +

2

3ω∧ω∧ω

))= −d

(tr

(ω∧F − 1

3ω∧ω∧ω

))θ2 = −δ

(tr

(ω∧F − 1

3ω∧ω∧ω

)). (A.22)

The term inside the trace in the last line is known as the Chern-Simons form.

Second the Nieh-Yan term,

L6 = De∧De

= (de+ e∧ω)∧(de+ ω∧e)

= de∧de+ e∧ω∧de+ de∧ω∧e+ e∧ω∧ω∧e

= d(e∧de) + d(e∧ω∧e) + e∧dω∧e+ e∧ω∧ω∧e

= d(e∧De) + e∧F ∧e

L6 − L4 = d(e∧De)

θ46 = δ(e∧De). (A.23)

Third the Euler term,

δL1 = 2F ∧δF = −2F ∧dδω + 2F ∧δ(ω∧ω)

= −2d(F ∧δω) + 2dF ∧δω + 2F ∧(δω∧ω)− 2F ∧(ω∧δω)

= −2d(F ∧δω)− 4 tr (?dF ∧δω + ?F ∧δω∧ω − ?F ∧ω∧δω)

= −2d(F ∧δω)− 4 tr (D ? F ∧δω)

θ1 = 2F ∧δω = −4 tr(?F ∧δω)

δL1 + dθ1 = 0. (A.24)

85

Note that the Euler term does contribute to the symplectic form, unlike the other

two topological terms above.

A.5 Remaining terms

Three terms remain. The easiest to tackle is

L5 = e∧e∧e∧e

δL5 = δe∧e∧e∧e− e∧δe∧e∧e+ e∧e∧δe∧e− e∧e∧e∧δe

= −4 e∧e∧e∧δe = −8 e∧ ? (e ∧ e)∧δe

θ5 = 0 (A.25)

So this term contributes to the equations of motion but not to the symplectic form.

Next,

L3 = e∧F ∧e = −2 tr (?(e ∧ e)∧F ) = −2 tr ((e ∧ e)∧ ? F )

δL3 = −2 tr (δ(e ∧ e)∧ ? F )− 2 tr (?(e ∧ e)∧δF )

= −4 tr ((δe ∧ e)∧ ? F ) + 2 tr (?(e ∧ e)∧dδω)− 2 tr (?(e ∧ e)∧δ(ω∧ω))

= −4e∧ ? F ∧δe+ 2d tr (?(e ∧ e)∧δω)− 2 tr (d ? (e ∧ e)∧δω)

− 2 tr ([ω∧ ? (e ∧ e)]∧δω)

= 2d tr (?(e ∧ e)∧δω)− 2 tr (D ? (e ∧ e)∧δω)− 4e∧ ? F ∧δe

θ3 = −2 tr (?(e ∧ e)∧δω) = e∧δω∧e (A.26)

δL3 + dθ3 = −2 tr (?D(e ∧ e)∧δω)− 4e∧ ? F ∧δe (A.27)

Finally, following essentially the same steps,

L4 = e∧F ∧e = − tr ((e ∧ e)∧F )

δL4 = δe∧F ∧e− e∧δF ∧e− e∧F ∧δe

= d tr ((e ∧ e)∧δω)− tr (D(e ∧ e)∧δω)− 2e∧F ∧δe

θ4 = − tr ((e ∧ e)∧δω) = e∧δω∧e (A.28)

δL4 + dθ4 = − tr (D(e ∧ e)∧δω)− 2e∧F ∧δe (A.29)

86

A.6 Field equations

Now we reorganize the GR action,

LGR = α1L1 + α2L2 + α3L3 + (α4 + α6)L4 + α5L5 + α6(L6 − L4) (A.30)

to obtain the equations of motion.

δL+ dθ = α3(δL3 + dθ3) + (α4 + α6)(δL3 + dθ3) + α5(δL5 + dθ5) = 0 (A.31)

δL+ dθ = α3 (−2 tr (?D(e ∧ e)∧δω)− 4e∧ ? F ∧δe)

+ (α4 + α6) (− tr (D(e ∧ e)∧δω)− 2e∧F ∧δe)

+ α5 (−8e∧ ? (e ∧ e)∧δe)

= − tr ((2α3 ? D(e ∧ e) + (α4 + α6)D(e ∧ e)) ∧δω)

− 2e∧ (2α3 ? F + (α4 + α6)F + 4α5 ? (e ∧ e)) ∧δe = 0

2α3 ? D(e ∧ e) + (α4 + α6)D(e ∧ e) = 0 (A.32)

e∧ (2α3 ? F + (α4 + α6)F + 4α5 ? (e ∧ e)) = 0 (A.33)

Some remarks are in order. For the traditional (Einstein-Hilbert) GR action with

nonzero cosmological constant, α1 = α2 = α4 = α6 = 0 so eq. A.32 implies

?D(e ∧ e) = 0 ⇒ e ∧De = 0 ⇒ De = 0 (A.34)

which is the usual torsion-free connection condition. If we also assume that the tetrad

e is invertible and define

RIµ = F IJ

µν eνJ , R = RI

µeνI , (A.35)

the second field equation eq. A.33 may be recast as

−4α3

(RIµ − 1

2eIµR

)+ 12α5e

Iµ = 0 (A.36)

from which we may set α5 = −13α3Λ where Λ is the standard cosmological constant.

Another interesting special case is the Holst action (2.7) described in the main

text.

87

Appendix B: Pseudoregular Polyhedra

We define by construction a type of polyhedron which is approximately regular, and

explore its properties. Take the number of faces L to be large (L 1), and let each

face have the same area and be such that a circle is a good approximation to it. Now

each face subtends a solid angle

Ωface = 4π/L

while the solid angle subtended by a cone with vertex angle 2θ is

Ωcone = 2π(1− cos θ) ≈ πθ2

so that, for large L, each face corresponds roughly to a cone with vertex angle 2θ =

2 arccos(1 − 2L

) ≈ 4/√L. We now want to pack these cones into a sphere in a

systematic arrangement that should allow us to construct each of the normals. All

cones are packed in pairs, corresponding to diametrically opposed faces (thus the

closure constraint is automatically satisfied). In effect we will only pack half the

sphere. The first cone is aligned with the z axis, and subsequent cones are arranged

in circular layers around the first cone. Each layer may be pictured as the region

between two large nested bounding cones, which will be packed with small face cones.

Take the first cone to be layer zero, then the nth layer is bounded by an outer cone

of vertex angle (2n + 1)2θ and an inner cone of vertex angle (2n − 1)2θ. The solid

angle subtended by each layer is then

Ωn = 2π(

cos(2n− 1)θ − cos(2n+ 1)θ)

= 4π sin θ sin 2nθ ≈ 4πθ sin 2nθ

so the number of cones that fits in each layer is (except for the last middle layer, in

which case the opposite cone sits in the same layer so the count must be halved)

Cn = Ωn/Ωcone =2 sin θ sin 2nθ

(1− cos θ)≈ 4πθ sin 2nθ

πθ2=

4 sin 2nθ

θ

(note that the approximate expressions shown above are merely shown for interest

and the exact expressions are used in the numerical results described subsequently.)

We may thus construct the normal to each face, labeled by a pair of integers (n,m)

88

Figure B.1: The face count obtained, as a function of the number of links L requested.The straight line shown is (.99)L

by first rotating the unit vector z around the y axis by an angle 2nθlayer into the

correct layer, where

θlayer =π

2Int(nmax)

then rotating the resulting vector around the z axis by an angle 2πm/Cn to position

it within that layer, where 0 ≤ m ≤ (Cn − 1) and 1 ≤ n ≤ π4θ

. The opposite face’s

normal may be obtained by following the same rotations starting with the unit vector

−z. Note that when n is at its maximum, each face and its opposite face may both

sit in the same layer, so to avoid double counting we may need to divide Cnby two. If

we round nmaxat .7, and correct for double counting when 2nmaxis even, the number

of faces constructed using the above procedure as a function of the value of L used in

the construction is shown in Figure B.1. The percent error between the two is shown

in Figure B.2.

Note that for L > 200 the percent error is under 5%. Also note that throughout

the range of L there are many specific values of L that one may choose to make the

error almost zero. We could choose these specific values of L to construct a set of

89

Figure B.2: Percent Error between the face count obtained and the number of linksrequested, as a function of the number requested.

pseudoregular polyhedra for use as boundary states for our spin foam calculations.

We could alternately choose a nominal value of L for the purposes of the construction,

then ignore it and use the actual number of faces constructed. However, there is a

simple consistency condition which allows us to choose certain preferred values of

L. One object of the construction is to produce polyhedra with directly opposing

faces, which we have enforced by hand; but if instead of aborting the cone packing

procedure at the halfway point we continue all the way to the other side of the sphere

(packing cones singly instead of in opposing pairs), if we can only pack a single cone

in the final layer, diametrically opposed to the starting cone, then the construction

procedure closes (for even L) and while we will still enforce the symmetry by hand it

is at least justified a posteriori. This condition allows us to computationally produce

a list of preferred polyhedra for arbitrarily large L (that is, for L > 200 we may not

be able to produce a polyhedron at L but we can produce a polyhedron within 5% of

L that satisfies the condition of equality between input and output L, which should

be close to the condition that the polyhedron has only opposing faces). We may

choose to only use such polyhedra if an explicit expression for the normals is needed.

90

Note that in general for such polyhedra the input L does not equal the output L, in

fact the output L is smaller, which accounts for the fact that we have used a conical

approximation for the solid angle subtended by each face, which is an underestimate.

In practice, we do not need the normal vectors explicitly in our calculations in the

main text, so the above construction just serves as a demonstration of the existence

of these polyhedra. The main issue of interest is the surface area to volume ratio,

which is crucial so that one may identify two different graphs as representative of the

same semiclassical spatial geometry. For this to be true, we want to keep the volume

of the node fixed while varying the number of links (faces of the polyhedron dual to

the node). We therefore need to eliminate the area labels in favor of the volume and

number of links. We can do so by defining the dimensionless quantity

α ≡ (Volume)13

(Surface Area)12

(B.1)

which for our pseudo-regular polyhedra may be calculated (using the approximation

that each face subtends a cone)

αasymptotic =1

(36π)1/6

( (L− 2)2

L(L− 1)

)1/6(B.2)

and is plotted in Figure B.3, along with the same ratio calculated for regular polyhedra

(dots). Figure B.4 shows the same thing including more polyhedra that are not regular

but have a high degree of symmetry.

Note that the curve agrees reasonably well with the dots even for small L, and

asymptotes to the ratio for a sphere as one might expect. We will therefore use

this approximate formula for α in all regimes. This gives us our desired relationship

between area and volume as a function of the number of links.

91

Figure B.3: Dimensionless volume to surface area ratio as a function of face count L,for pseudo-regular polyhedra (curve), regular polyhedra (dots), or a sphere (dashedline.)

92

Figure B.4: Dimensionless volume to surface area ratio as a function of face count L,for pseudo-regular polyhedra (curve), “nice” polyhedra (dots), or a sphere (dashedline.)

93

Appendix C: Sub-leading Contributions to the

Norm

Here we investigate sub-leading contributions to the norm to justify neglecting them

as done in the main body. We begin with (7.43),

〈Ψtg|Ψt

g′〉 =et4

2√x2 − 1

∞∑n=−∞

ne−t4n2

λn (C.1)

and observe that the summand f(n) = ne−t4n2λn is a Schwartz function since it drops

off faster than any inverse power of n as n → ∞. Thus the Poisson summation

formula applies, namely ∑n∈Z

f(n) =∑k∈Z

f(k) (C.2)

where f is the Fourier transform of f ,

f(k) =

∫ ∞−∞

dnf(n)e−2πikn =4√π

t3/2(lnλ− 2πik)e

(lnλ−2πik)2

t . (C.3)

Then the norm is

〈Ψtg|Ψt

g′〉 =2√π

t3/2et4

√x2 − 1

∑k∈Z

(lnλ− 2πik)e(lnλ−2πik)2

t . (C.4)

For real λ > 0 the leading term in the sum is k = 0, which reproduces (7.45). To

estimate the remaining terms, we sum them in pairs to obtain

〈Ψtg|Ψt

g′〉 =2√π

t3/2et4 lnλ√x2 − 1

e(lnλ)2

t

(1 +

∑k≥1

2

(cosαk −

2πk

lnλsinαk

)e−4π2k2

t

)(C.5)

αk =4πk lnλ

t(C.6)

and observe that the oscillatory factor is at most 1 so the sub-leading terms are

suppressed by at least e−4π2k2

t hence we neglect them.

Now for λ < 01 a choice of branch is required to define lnλ, but any such choice is

equivalent to lnλ = ln |λ| − πi via a suitable redefinition of k. Then the norm (C.4)

1A similar situation arises in the computation of the amplitude, where λ is complex.

94

becomes

〈Ψtg|Ψt

g′〉 =2√π

t3/2et4

√x2 − 1

∑k∈Z

(ln |λ| − 2πi(k + 12))e

(ln |λ|−2πi(k+12))2

t , (C.7)

from which we see that even the leading terms are suppressed relative to the λ > 0

case. Summing the series pairwise,

〈Ψtg|Ψt

g′〉 =2√π

t3/2et4 ln |λ|√x2 − 1

e(ln |λ|)2

t

(∑k≥0

2

(cosαk+ 1

2− 2π(k + 1

2)

ln |λ|sinαk+ 1

2

)e−4π2(k+

12)2

t

)(C.8)

αk =4πk ln |λ|

t(C.9)

the leading term is suppressed by at least e−π2

t thus we neglect contributions to the

norm from λ < 0 configurations.

Moreover, a detailed examination of the critical points associated with λ < 0

shows that the closure condition is not satisfied, so these subleading configurations

are also nonphysical and are further suppressed when the coherent state labels are

chosen appropriately.

95

Appendix D: An Alternative Normalization

Here we work out the details of an alternative to the normalization scheme used in the

main body of the thesis that has an interesting impact on the normalized amplitude.

This is not simply a choice of convention, the normalization is different because the

states considered are different. When the gauge-invariant coherent states were defined

in (4.7), we dropped one group integral as redundant since we see later that it is not

needed to enforce closure at the last node after closure has been enforced at all the

others. However it does have an impact on the normalization of the state, and it does

no harm to the closure relation to include it. We can still drop an integral from the

amplitude provided we do so after it “eats” the extra SU(2) gauge integral, either

by putting a delta function in the amplitude, or taking the perspective that dividing

by the infinite volume of SL(2,C) is just part of the normalization of the amplitude.

In this appendix we consider the transition amplitude for these states, for which the

gauge invariance is imposed more strongly.

D.1 Coherent State Normalization

The extra gauge integral adds one to the relevant exponent in (7.125),

‖ΨtHl‖

2 ≈ ‖ΨtHl‖2Ltot

(1

2√π

(t

LIm(z)

)3/2)N

det[−H]−12 (D.1)

and though the calculation of the Hessian matrix (7.120) is unchanged, it is now

N × N rather than (N − 1) × (N − 1) so its determinant is different. We explicitly

compute it for the two simplest cases.

96

For the regular graph with just one node,

−H ij =2

L

∑l

(δij − niln

jl

)cosh(Im(z))− cos(Re(z))

sinh(Im(z))(D.2)

= 2(δij − 1

6δij)cosh(Im(z))− cos(Re(z))

sinh(Im(z))(D.3)

=5

3δij

cosh(Im(z))− cos(Re(z))

sinh(Im(z))(D.4)

det[−H] =

(5

3

cosh(Im(z))− cos(Re(z))

sinh(Im(z))

)3

(D.5)

where in the second step we used the fact that∑

l nilnjl = 1

6Lδij. Recall that this

sum taken over all outgoing normals would be 13Lδij, but here each link connects

to the same node and the target normals do not appear in the sum. Moreover,

source/target paired normals are opposite to each other, but since (−nil)(−njl ) = niln

jl

they contribute equally to the sum. Therefore summing over only one member of each

pair produces half the value. Note that for Im(z) 1, the Hessian tends to (−5/3)δij

unlike the other cases.

For the next simplest regular graph with just two nodes, any orientation of links

will do and we choose s(l) = 1, t(l) = 2∀l. Then

−H ij11 = −H ij

22 =1

L

∑l

(δij − niln

jl

)cosh(Im(z))

sinh(Im(z))+

nilnjl

Im(z)(D.6)

=(δij − 1

3δij)cosh(Im(z))

sinh(Im(z))+

13δij

Im(z)(D.7)

= 13δij(

2cosh(Im(z))

sinh(Im(z))+

1

Im(z)

)(D.8)

−H ij12 =

1

L

∑l

−((δij − niln

jl

) cos(Re(z))

sinh(Im(z))+

nilnjl

Im(z)

)(D.9)

−(nkl εijk

) sin(Re(z))

sinh(Im(z))(D.10)

= − 13δij(

2cos(Re(z))

sinh(Im(z))+

1

Im(z)

)(D.11)

= −H ij21 (D.12)

where we used the fact that∑

l nkl = 0 and

∑l n

ilnjl = 1

3Lδij since in this case the

sum over links includes all outgoing and incoming normals.

97

Since the Hessian takes a block or partitioned form,

H =

[H11 H12

H21 H22

](D.13)

its determinant is given by

det[H] = det[H11] det[H/H11] (D.14)

where H/H11 is the Schur complement of H11 in H. It is simple to compute since the

blocks are all diagonal.

H/H11 = H22 −H21H−111 H12 (D.15)

= − 13δij

((2

cosh(Im(z))

sinh(Im(z))+

1

Im(z)

)(D.16)

−(

2cosh(Im(z))

sinh(Im(z))+

1

Im(z)

)−1(2

cos(Re(z))

sinh(Im(z))+

1

Im(z)

)2)

(D.17)

det[−H] = ( 13)6

((2

cosh(Im(z))

sinh(Im(z))+

1

Im(z)

)2

−(

2cos(Re(z))

sinh(Im(z))+

1

Im(z)

)2)3

(D.18)

=

(2

3

)6(cosh(Im(z)) + cos(Re(z))

sinh(Im(z))+

1

Im(z)

)3

·(

cosh(Im(z))− cos(Re(z))

sinh(Im(z))

)3(D.19)

In both cases we ended up with a factor of (cosh(Im(z)) − cos(Re(z))), which

vanishes when Re(z) = Im(z) = 0. This factor appears in the Hessian, which is

inverted in the norm then inverted again in the normalized amplitude, so it suppresses

the amplitude at the origin as we will see next.

D.2 Alternative Amplitude

The extra gauge integral has no effect on the single-node amplitude other than

through the norm of the states. The new normalized amplitude is plotted in Fig-

ures D.1,D.2. At large Im(z) the amplitude looks the same as in the main text,

but a detail view of the origin shows a bifurcation of the peak that avoids the point

98

Figure D.1: Alternative normalized amplitude for a single node graph with L = 6and t = 1

Re(z) = Im(z) = 0. Aside from being a nice example of a nontrivial relation between

a and a, it has an interesting interpretation. If we view the peak as a kind of phase

space trajectory and the system is at a point on the trajectory near the origin with

Re(z) < 0, this corresponds to a < 0 so the system is traveling in the direction of

decreasing a (decreasing Im(z).) When it reaches a = 0 it still has a negative value

of a so it travels through the singularity and out the other side into a region of neg-

ative Im(z), which could be interpreted as a state with oppositely oriented volume.

The other branch (Re(z) > 0) circulates in the opposite direction towards increasing

Im(z).

99

Figure D.2: Alternative normalized amplitude for a single node graph with L = 6and t = 1 (detail)

It is possible to carry the analysis forward in more detail. While on the scale of

Figure D.2 it is not readily apparent, the peak of the amplitude in the bifurcated

region very closely follows a circle. This trajectory joins discontinuously to the linear

solution Re(z) = 0 for Im(z) greater than the radius of the circle. Figure D.3 shows

the numerically determined peak of the amplitude at various values of Im(z), and the

fit to a circle. Further numerical investigation shows that the radius of the circle R

scales with t and L approximately as R2 ∝ t/L. The equation for the circle leads to

a differential equation of the form

C1a2 + C2a

4 = R2 = C3t/L (D.20)

100

Figure D.3: Numerically determined peaks of the alternative normalized amplitudefor L = 6 and t = 1, with a circular curve fit.

101

where the constants C1, C2 are shown in (7.27) and C3 may be determined numerically.

Written another way, (a

a

)2

=C3t

C1L

1

a2− C2

C1

a2. (D.21)

The differential equation has oscillatory solutions for a that may be explicitly given in

terms of Jacobi elliptic functions. So while the singularity is technically not avoided,

the system behaves classically at large Im(z) but below a certain value of Im(z) it

suddenly transitions to an oscillatory solution which exhibits a kind of “quantum

bounce” at a = 0 instead of sticking at a crunch.

Whether or not this normalization of the amplitude is physical is a matter for

future work, specifically how it impacts the amplitude in more complicated cases.

As stated at the outset, these states result from imposing the gauge invariance more

strongly, which may not be the right thing to do, but nonetheless the interesting

behavior of the transition amplitude warrants the given description; perhaps other

more physically relevant transition amplitudes may produce similar results.

102

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