Quantum Gravitywhence, whither?

Claus Kiefer

Institut fur Theoretische PhysikUniversitat zu Koln

1

Contents

Why quantum gravity?

Steps towards quantum gravity

Covariant quantum gravity

Canonical quantum gravity

String theory

Black holes

(Quantum Cosmology)

Why quantum gravity?

◮ Unification of all interactions

◮ Singularity theorems

◮ Black holes◮ ‘Big Bang’

◮ Problem of time

◮ Absence of viable alternatives

Problem of time

◮ Absolute time in quantum theory:

i~∂ψ

∂t= Hψ

◮ Dynamical time in general relativity:

Rµν − 1

2gµνR+ Λgµν =

8πG

c4Tµν

QUANTUM GRAVITY?

Planck units

lP =

√

~G

c3≈ 1.62 × 10−33 cm

tP =lPc

=

√

~G

c5≈ 5.40 × 10−44 s

mP =~

lPc=

√

~c

G≈ 2.17 × 10−5 g ≈ 1.22 × 1019 GeV/c2

Max Planck (1899):Diese Grossen behalten ihre naturliche Bedeutung so lange bei, als dieGesetze der Gravitation, der Lichtfortpflanzung im Vacuum und die beidenHauptsatze der Warmetheorie in Gultigkeit bleiben, sie mussen also, von denverschiedensten Intelligenzen nach den verschiedensten Methodengemessen, sich immer wieder als die namlichen ergeben.

Structures in the Universe

Quantum region1 �1=2g 1 ��1=2g

mmpr

Atomi densityNu lear d

ensity

��3=2g

��1=2g��1g

mPWhite dw

arf StarsBla k holes

rrpr

��2g

Plan k holeProton

Mass in Hubblevolume whent � ��3=2g tPmassChandrasekhar

Hole withkBT � mpr 2

αg =Gm2

pr

~c=

(mpr

mP

)2

≈ 5.91 × 10−39

Steps towards quantum gravity

◮ Interaction of micro- and macroscopicsystems with an external gravitational field

◮ Quantum field theory on curvedbackgrounds (or in flat background, but innon-inertial systems)

◮ Full quantum gravity

Quantum systems in external gravitational fields

Neutron and atom interferometry

������

������

��������������������g � C D

BADete tors

Experiments:◮ Neutron interferometry in the field of the Earth

(Colella, Overhauser, and Werner (‘COW’) 1975)

◮ Neutron interferometry in accelerated systems(Bonse and Wroblewski 1983)

◮ Discrete neutron states in the field of the Earth(Nesvizhevsky et al. 2002)

◮ Neutron whispering gallery(Nesvizhevsky et al. 2009)

◮ Atom interferometry(e.g. Peters, Chung, Chu 2001: measurement of g with accuracy ∆g/g ∼ 10

−10)

Non-relativistic expansion of the Dirac equation yields

i~∂ψ

∂t≈ HFWψ

with

HFW = βmc2︸ ︷︷ ︸

rest mass

+β

2mp2

︸ ︷︷ ︸

kinetic energy

− β

8m3c2p4

︸ ︷︷ ︸

SR correction

+βm(a x)︸ ︷︷ ︸

COW

− ωL︸︷︷︸

Sagnac effect

− ωS︸︷︷︸

Mashhoon effect

+β

2mpa x

c2p +

β~

4mc2~Σ(a × p) + O

(1

c3

)

Black-hole radiation

Black holes radiate with a temperature proportional to ~

(‘Hawking temperature’):

TBH =~κ

2πkBc

Schwarzschild case:

TBH =~c3

8πkBGM

≈ 6.17 × 10−8

(M⊙

M

)

K

Black holes also have an entropy(‘Bekenstein–Hawking entropy’):

SBH = kBA

4l2P

Schwarzschild≈ 1.07 × 1077kB

(M

M⊙

)2

Analogous effect in flat spacetime

IV

III

IIBeschl.-

Horizont

IX

T

= constant

= constantτ ρ

Accelerated observer in the Minkowski vacuum experiencesthermal radiation with temperature

TDU =~a

2πkBc≈ 4.05 × 10−23 a

[cm

s2

]

K .

(‘Davies–Unruh temperature’)

Is thermodynamics more fundamental than gravity?

Main Approaches to Quantum Gravity

No question about quantum gravity is more difficultthan the question, “What is the question?”(John Wheeler 1984)

◮ Quantum general relativity

◮ Covariant approaches (perturbation theory, path integrals,. . . )

◮ Canonical approaches (geometrodynamics, connectiondynamics, loop dynamics, . . . )

◮ String theory◮ Fundamental discrete approaches (quantum

topology, causal sets, group field theory, . . . ); have partiallygrown out of the other approaches

Covariant quantum gravity

Perturbation theory:

gµν = gµν +

√

32πG

c4fµν

◮ gµν : classical background

◮ Perturbation theory with respect to fµν

(Feynman rules)

◮ ‘Particle’ of quantum gravity: graviton(massless spin-2 particle)

Perturbative non-renormalizability

Effective field theory

Concrete predictions possible at low energies(even in non-renormalizable theory)

Example:Quantum gravitational correction to the Newtonian potential

V (r) = −Gm1m2

r

1 + 3G(m1 +m2)

rc2︸ ︷︷ ︸

GR−correction

+41

10π

G~

r2c3︸ ︷︷ ︸

QG−correction

(Bjerrum–Bohr et al. 2003)

Analogy: Chiral perturbation theory (small pion mass)

Asymptotic Safety

Weinberg (1979): A theory is called asymptotically safe if allessential coupling parameters gi of the theory approach fork → ∞ a non-trivial fix point

Preliminary results:

◮ Effective gravitational constant vanishes for k → ∞◮ Effective gravitational constant increases with distance

(simulation of Dark Matter?)◮ Small positive cosmological constant as an infrared effect

(Dark Energy?)◮ Spacetime appears two-dimensional on smallest scales

(H. Hamber et al., M. Reuter et al.)

The sigma model

Non-linear σ model: N -component field φa satisfying∑

a φ2a = 1

◮ is non-renormalizable for D > 2

◮ exhibits a non-trivial UV fixed point at some coupling gc(‘phase transition’)

◮ an expansion in D − 2 and use of renormalization-group (RG)techniques gives information about the behaviour in the vicinityof the non-trivial fixed point

Example: superfluid HeliumThe specific heat exponent α was measured in a space shuttleexperiment (Lipa et al. 2003): α = −0.0127(3), which is in excellentagreement with three calculations in the N = 2 non-linear σ-model:

◮ α = −0.01126(10) (4-loop result; Kleinert 2000);

◮ α = −0.0146(8) (lattice Monte Carlo estimate; Campostrini et al. 2001);

◮ α = −0.0125(39) (lattice variational RG prediction; cited in Hamber 2009)

Path integrals

Z[g] =

∫

Dgµν(x) eiS[gµν(x)]/~

In addition: sum over all topologies?

◮ Euclidean path integrals(e.g. for Hartle–Hawking proposal or Regge calculus)

◮ Lorentzian path integrals(e.g. for dynamical triangulation)

Dynamical triangulation

◮ makes use of Lorentzian path integrals◮ edge lengths of simplices remain fixed; sum is performed

over all possible combinations with equilateral simplices◮ Monte-Carlo simulations

t

t+1

(4,1) (3,2)

Preliminary results:◮ Hausdorff dimension H = 3.10 ± 0.15

◮ Spacetime two-dimensional on smallest scales(cf. asymptotic-safety approach)

◮ positive cosmological constant needed

◮ continuum limit?

(Ambjørn, Loll, Jurkiewicz from 1998 on)

Canonical quantum gravity

Central equations are constraints:

HΨ = 0

Different canonical approaches:

◮ Geometrodynamics –metric and extrinsic curvature

◮ Connection dynamics –connection (Ai

a) and coloured electric field (Eai )

◮ Loop dynamics –flux of Ea

i and holonomy

Hamilton–Jacobi equation

Hamilton–Jacobi equation −→ guess a wave equation

Vacuum case:

16πGGabcdδS

δhab

δS

δhcd−

√h

16πG( (3)R− 2Λ) = 0 ,

DaδS

δhab= 0

(Peres 1962)

Find wave equation that yields the Hamilton–Jacobi equation inthe semiclassical limit:

Ansatz : Ψ[hab] = C[hab] exp

(i

~S[hab]

)

The dynamical gravitational variable is the three-metric hab! It isthe argument of the wave functional.

Quantum geometrodynamics

Vacuum case:

HΨ ≡(

−2κ~2Gabcd

δ2

δhabδhcd− (2κ)−1

√h(

(3)R− 2Λ))

Ψ = 0,

κ = 8πG

Wheeler–DeWitt equation

DaΨ ≡ −2∇b~

i

δΨ

δhab= 0

quantum diffeomorphism (momentum) constraints

Problem of time

◮ no external time present; spacetime has disappeared!

◮ local intrinsic time can be defined through localhyperbolic structure of Wheeler–DeWitt equation(‘wave equation’)

◮ related problem: Hilbert-space problem –which inner product, if any, to choose between wavefunctionals?

◮ Schrodinger inner product?◮ Klein–Gordon inner product?

◮ Problem of observables

The semiclassical approximation

Wheeler–DeWitt equation and momentum constraints in thepresence of matter (e.g. a scalar field):

{

− 1

2m2P

Gabcdδ2

δhabδhcd− 2m2

P

√h (3)R+ Hm

⊥

}

|Ψ[hab]〉 = 0 ,

{

−2

ihabDc

δ

δhbc+ Hm

a

}

|Ψ[hab]〉 = 0

(bra and ket notation refers to non-gravitational fields)

Make comparison with a quantum-mechanical model:

− 1

2M

∂2

∂Q2↔ − 1

2m2P

Gabcd

δ2

δhabδhcd

,

V (Q) ↔ −2m2P

√h (3)R ,

h(q,Q) ↔ Hm⊥ ,

Ψ(q,Q) ↔ |Ψ[hab]〉 .

Recovery of time

Ansatz:|Ψ[hab]〉 = C[hab]e

im2

PS[hab]|ψ[hab]〉

One evaluates |ψ[hab]〉 along a solution of the classicalEinstein equations, hab(x, t), corresponding to a solution,S[hab], of the Hamilton–Jacobi equations;this solution for hab(x, t) is obtained from

hab = NGabcdδS

δhcd+ 2D(aN b)

compare q =1

m

∂S

∂q

∂

∂t|ψ(t)〉 =

∫

d3x hab(x, t)δ

δhab(x)|ψ[hab]〉

This leads to a functional Schrodinger equation for quantizedmatter fields in the chosen external classical gravitational field:

i~∂

∂t|ψ(t)〉 = Hm|ψ(t)〉

Hm ≡∫

d3x{

N(x)Hm⊥(x) +Na(x)Hm

a (x)}

Hm: matter-field Hamiltonian in the Schrodinger picture,parametrically depending on the (generally non-static) metriccoefficients of the curved space–time background.

WKB time t controls the dynamics in this approximation

Quantum gravitational corrections

Next order in the Born–Oppenheimer approximation gives

Hm → Hm +1

m2P

(various terms)

(C. K. and T. P. Singh (1991); A. O. Barvinsky and C. K. (1998))

Simple example: Quantum gravitational correction to the traceanomaly in de Sitter space:

δǫ ≈ − 2G~2H6

dS

3(1440)2π3c8

(C. K. 1996)

Path Integral satisfies Constraints

◮ Quantum mechanics: path integral satisfies

Schrodinger equation

◮ Quantum gravity: path integral satisfies

Wheeler–DeWitt equation and diffeomorphism

constraints

A. O. Barvinsky (1998): direct check in the one-loopapproximation that the quantum-gravitational path integralsatisfies the constraints−→ connection between covariant and canonical approach

The full path integral with the Einstein–Hilbert action (if definedrigorously) should be equivalent to the constraint equations ofcanonical quantum gravity

Ashtekar’s new variables

◮ new momentum variable: densitized version of triad,

Eai (x) :=

√

h(x)eai (x) ;

◮ new configuration variable: ‘connection’ ,

GAia(x) := Γi

a(x) + βKia(x)

{Aia(x), E

bj (y)} = 8πβδi

jδbaδ(x, y)

Loop quantum gravity

◮ new configuration variable: holonomy,

U [A,α] := P exp(G

∫

αA)

;

◮ new momentum variable: densitized triad flux

Ei[S] :=∫

Sdσa E

ai

S

S

P1P2

P3

4P

Quantization of area:

A(S)ΨS [A] = 8πβl2P∑

P∈S∩S

√

jP (jP + 1)ΨS[A]

(cf. review talk by C. Fleischhack)

String theory

◮ Inclusion of gravity unavoidable(E. Witten: “String theory forces general relativity upon us.”)

◮ Gauge invariance, supersymmetry, higher dimensions◮ Unification of all interactions◮ Perturbation theory probably finite at all orders, but sum

diverges◮ Only three fundamental constants: ~, c, ls◮ Branes as central objects◮ Dualities connect different string theories

Space and time in string theory I

Z =

∫

DXDh e−S/~

(X: embedding; h: metric on worldsheet)

++ ...Action S still contains background fields (spacetime metric,dilaton, . . . )

Absence of quantum anomalies −→◮ Background fields obey Einstein equations up to O(l2s );

can be derived from an effective action◮ Constraint on the number of spacetime dimensions:

10 resp. 11

Generalized uncertainty relation:

∆x ≥ ~

∆p+l2s~

∆p

Space and time in string theory II

String theory contains general relativity −→ the abovearguments apply: Wheeler–DeWitt equation shouldapproximately be valid away from the Planck scale

‘Problem of time’ is the same here; new insight is obtained forthe concept of space

◮ Matrix models: finite number of degrees of freedomconnected with a system of D0-branes for the descriptionof M-theory; fundamental scale is the 11-dimensionalPlanck scale

◮ AdS/CFT correspondence: non-perturbative string theoryin a background spacetime that is asymptotically anti-deSitter (AdS) is dual to a conformal field theory (CFT)defined in a flat spacetime of one less dimension(Maldacena 1998).

AdS/CFT correspondence

Often considered as a mostly background-independentdefinition of string theory (background metric enters onlythrough boundary conditions at infinity).

Realization of the “holographic principle”:laws including gravity in d = 3 are equivalent tolaws excluding gravity in d = 2.

In a sense, space has here vanished, too!

Black holes

Microscopic explanation of entropy?

SBH = kBA

4l2P

◮ Loop quantum gravity: microscopic degrees of freedomare the spin networks; SBH only follows for a specificchoice of β: β = 0.237532 . . .

◮ String theory: microscopic degrees of freedom are the‘D-branes’; SBH only follows for special (extremal ornear-extremal) black holes

◮ Quantum geometrodynamics: e.g. S ∝ A in the LTB model

Problem of information loss

◮ Final phase of evaporation?◮ Fate of information is a consequence of the fundamental

theory (unitary or non-unitary)◮ Problem does not arise in the semiclassical approximation

(thermal character of Hawking radiation follows fromdecoherence)

◮ Empirical problems:◮ Are there primordial black holes?◮ Can black holes be generated in accelerators?

Primordial black holes

Primordial Black Holes could form from density fluctuations inthe early Universe (with masses from 1 g on); black holes withan initial mass of M0 ≈ 5 × 1014 Gramm would evaporate“today” −→ typical spectrum of Gamma rays

Fermi Gamma-ray Space Telescope; Launch: June 2008

Generation of mini black holes at the LHC?

CMS detector

Only possible if space has more than three dimensions

Why Quantum Cosmology?

Gell-Mann and Hartle 1990:Quantum mechanics is best and most fundamentallyunderstood in the framework of quantum cosmology.

◮ Quantum theory is universally valid:Application to the Universe as a whole as the only closedquantum system in the strict sense

◮ Need quantum theory of gravity, since gravity dominateson large scales

John Wheeler 1968:These considerations reveal that the concepts of spacetimeand time itself are not primary but secondary ideas in thestructure of physical theory. These concepts are valid in theclassical approximation. However, they have neither meaningnor application under circumstances whenquantum-geometrodynamical effects become important.. . . There is no spacetime, there is no time, there is no before,there is no after. The question what happens “next” is withoutmeaning.

Literature: C. K., Quantum Gravity (Second edition, Oxford 2007)