Universality classes of Quantum Gravity Frank Saueressig Research - - PowerPoint PPT Presentation

Universality classes of Quantum Gravity

Frank Saueressig

Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen

• A. Contillo, G. D’Odorico, E. Manrique, S. Rechenberger, M. Schutten

arXiv:1102.5012, arXiv:1212.5114, arXiv:1309.7273, arXiv:1406.4366 Non-Perturbative Methods in Quantum Field Theory Balatonfüred, October 8-10, 2014

– p. 1/29

Quantum Gravity within Quantum Field Theory

Requirements: a) well-defined behavior at high energy

• RG-fixed point controlling the UV-behavior of the theory
• ensures the absence of UV-divergences

– p. 2/29

Quantum Gravity within Quantum Field Theory

Requirements: a) well-defined behavior at high energy

• RG-fixed point controlling the UV-behavior of the theory
• ensures the absence of UV-divergences

b) predictivity

• fixed point has finite-dimensional UV-critical surface SUV
• fixing the position of a trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

– p. 2/29

Quantum Gravity within Quantum Field Theory

Requirements: a) well-defined behavior at high energy

• RG-fixed point controlling the UV-behavior of the theory
• ensures the absence of UV-divergences

b) predictivity

• fixed point has finite-dimensional UV-critical surface SUV
• fixing the position of a trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

c) classical limit

• reconcile quantum theory with the experimental success of GR
• RG-trajectories have part where GR is good approximation

– p. 2/29

Quantum Gravity within Quantum Field Theory

Requirements: a) well-defined behavior at high energy

• RG-fixed point controlling the UV-behavior of the theory
• ensures the absence of UV-divergences

b) predictivity

• fixed point has finite-dimensional UV-critical surface SUV
• fixing the position of a trajectory in SUV

⇐ ⇒ experimental determination of relevant parameters

c) classical limit

• reconcile quantum theory with the experimental success of GR
• RG-trajectories have part where GR is good approximation

d) question of unitarity

• information loss in black holes?

– p. 2/29

Proposals for UV fixed points (incomplete. . .)

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN

– p. 3/29

Proposals for UV fixed points (incomplete. . .)

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling

– p. 3/29

Proposals for UV fixed points (incomplete. . .)

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• Lorentz-invariant, non-perturbatively renormalizable

– p. 3/29

Proposals for UV fixed points (incomplete. . .)

• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: Einstein-Hilbert action
• perturbation theory in GN
• isotropic Gaussian Fixed Point (GFP)
• fundamental theory: higher-derivative gravity
• perturbation theory in higher-derivative coupling
• non-Gaussian Fixed Point (NGFP)
• fundamental theory: interacting
• Lorentz-invariant, non-perturbatively renormalizable
• anisotropic Gaussian Fixed Point (aGFP)
• fundamental theory: Hoˇ

rava-Lifshitz gravity

• Lorentz-violating, perturbatively renormalizable

– p. 3/29

The phase diagram of Asymptotic Safety

• M. Reuter and F. Saueressig, Phys. Rev. D 65 (2002) 065016 [hep-th/0110054]

−0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.75 −0.5 −0.25 0.25 0.5 0.75 1 λ

g

Type IIIa Type Ia Type IIa Type Ib Type IIIb

– p. 4/29

The phase diagram of Causal Dynamical Triangulations

• J. Ambjørn, J. Jurkiewicz, R. Loll; D. Benedetti, J. Cooperman, . . .

– p. 5/29

Once upon a time there was a . . . puzzle

FRGE and Dynamical Triangulations investigate the same path integral continuum functional renormalization group (FRGE):

• covariant computation, Euclidean signature
• non-Gaussian fixed point (NGFP)
• classical general relativity recovered at ℓ ≈ 10ℓPl

– p. 6/29

Once upon a time there was a . . . puzzle

FRGE and Dynamical Triangulations investigate the same path integral continuum functional renormalization group (FRGE):

• covariant computation, Euclidean signature
• non-Gaussian fixed point (NGFP)
• classical general relativity recovered at ℓ ≈ 10ℓPl

Monte Carlo Simulation of gravitational partition sum

• Causal Dynamical Triangulations (CDT)
• second order phase transition line
• “classical universes” at ℓ ≈ 10ℓPl
• Euclidean Dynamical Triangulations (EDT)
• no second order phase transition line
• no “classical universes”

– p. 6/29

Once upon a time there was a . . . puzzle

FRGE and Dynamical Triangulations investigate the same path integral continuum functional renormalization group (FRGE):

• covariant computation, Euclidean signature
• non-Gaussian fixed point (NGFP)
• classical general relativity recovered at ℓ ≈ 10ℓPl

Monte Carlo Simulation of gravitational partition sum

• Causal Dynamical Triangulations (CDT)
• second order phase transition line
• “classical universes” at ℓ ≈ 10ℓPl
• Euclidean Dynamical Triangulations (EDT)
• no second order phase transition line
• no “classical universes”

How does this fit together?

– p. 6/29

Functional Renormalization Group Equation for foliated spacetimes

– p. 7/29

Foliation structure via ADM-decomposition

Preferred “time”-direction via foliation of space-time

• foliation structure Md+1 = S1 × Md with yµ → (τ, xa):

ds2 = N2dt2 + σij

• dxi + Nidt

dxj + Njdt

• fundamental fields: gµν → (N, Ni, σij)

gµν =

 N2 + NiNi

Nj Ni σij

 

– p. 8/29

Foliation structure via ADM-decomposition

Preferred “time”-direction via foliation of space-time

• foliation structure Md+1 = S1 × Md with yµ → (τ, xa):

ds2 = ǫN2dt2 + σij

• dxi + Nidt

dxj + Njdt

• fundamental fields: gµν → (N, Ni, σij)

gµν =

 ǫN2 + NiNi

Nj Ni σij

  Allows to include signature parameter ǫ = ±1

– p. 9/29

Foliated functional renormalization group equation

Flow equation: formally the same as in covariant construction

k∂kΓk[h, hi, hij; ¯ σij] = 1

2 STr

• Γ(2)

k

+ Rk

−1

k∂kRk

• covariant: M4

STr ≈

• fields
• d4y√¯

g

• foliated: S1 × M3

STr ≈ √ǫ

• component fields
• KK−modes
• d3x

√ ¯ σ

• structure resembles: quantum field theory at finite temperature!

– p. 10/29

Foliated functional renormalization group equation

Flow equation: formally the same as in covariant construction

k∂kΓk[h, hi, hij; ¯ σij] = 1

2 STr

• Γ(2)

k

+ Rk

−1

k∂kRk

• covariant: M4

STr ≈

• fields
• d4y√¯

g

• foliated: S1 × M3

STr ≈ √ǫ

• component fields
• KK−modes
• d3x

√ ¯ σ

• structure resembles: quantum field theory at finite temperature!

Advantages of the foliated flow equation:

• ǫ-dependence: keep track of signature effects
• structure: same as in Causal Dynamical Triangulations

– p. 10/29

Comparison: phase diagrams for ADM-variables

ΓADM

k

= √ǫ 16πGk

• dτd3xN√σ
• ǫ−1

KijKij − K2 − R(3) + 2Λk

• + Sgf + Sgh

0.4 0.2 0.0 0.2 0.4 Λ 0.2 0.4 0.6 0.8 1.0 g

covariant computation

0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Λ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 g 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Λ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 g

Euclidean ǫ = 1 Lorentzian: ǫ = −1

– p. 11/29

It’s all about choosing a gauge:

covariant formulation:

gµν = ¯ gµν + hµν

perform covariant gauge-fixing (e.g., harmonic gauge)

Fµ = ¯ Dνhµν − 1

2 ¯

Dµhν ν = 0 .

foliated formulation with ADM-fields gµν → {N, Ni, σij}

N = ¯ N + h , Ni = ¯ Ni + hi , σij = ¯ σij + hij

perform temporal gauge-fixing (non-covariant):

h = 0 , hi = 0

• fluctuations in the metric on the spatial slice only

– p. 12/29

It’s all about choosing a gauge:

covariant formulation:

gµν = ¯ gµν + hµν

perform covariant gauge-fixing (e.g., harmonic gauge)

Fµ = ¯ Dνhµν − 1

2 ¯

Dµhν ν = 0 .

foliated formulation with ADM-fields gµν → {N, Ni, σij}

N = ¯ N + h , Ni = ¯ Ni + hi , σij = ¯ σij + hij

perform temporal gauge-fixing (non-covariant):

h = 0 , hi = 0

• fluctuations in the metric on the spatial slice only

ADM fields in temporal gauge No fluctuations in stacking spatial slices!

– p. 12/29

Symmetries conserved by the foliated FRGE

fundamental fields: ˜

N(τ, x), ˜ Ni(τ, x), ˜ σij(τ, x)

symmetry: general coordinate invariance inherited from γµν:

δγµν = Lv(γµν) , vα = (f(τ, x) , ζa(τ, x))

induces

δ ˜ N = f∂τ ˜ N + ζk∂k ˜ N + ˜ N∂τf − ˜ N ˜ Ni∂if , δ ˜ Ni = ˜ Ni∂τ f + ˜ Nk ˜ Nk∂if + ˜ σki∂τ ζk + ˜ Nk∂iζk + f∂τ ˜ Ni + ζk∂k ˜ Ni + ǫ ˜ N2∂if δ˜ σij = f∂τ ˜ σij + ζk∂k˜ σij + ˜ Nj∂if + ˜ Ni∂jf + ˜ σjk∂iζk + ˜ σik∂jζk

• Non-linearity of ADM-decomposition: symmetry realized non-linearly

– p. 13/29

Symmetries conserved by the foliated FRGE

fundamental fields: ˜

N(τ, x), ˜ Ni(τ, x), ˜ σij(τ, x)

symmetry: general coordinate invariance inherited from γµν:

δγµν = Lv(γµν) , vα = (f(τ, x) , ζa(τ, x))

induces

δ ˜ N = f∂τ ˜ N + ζk∂k ˜ N + ˜ N∂τf − ˜ N ˜ Ni∂if , δ ˜ Ni = ˜ Ni∂τ f + ˜ Nk ˜ Nk∂if + ˜ σki∂τ ζk + ˜ Nk∂iζk + f∂τ ˜ Ni + ζk∂k ˜ Ni + ǫ ˜ N2∂if δ˜ σij = f∂τ ˜ σij + ζk∂k˜ σij + ˜ Nj∂if + ˜ Ni∂jf + ˜ σjk∂iζk + ˜ σik∂jζk

• Non-linearity of ADM-decomposition: symmetry realized non-linearly
• in ADM it is impossible to combine:
• linear background field method
• regulator ∆kS quadratic in fluctuation fields
• background Diff(M)-symmetry

– p. 13/29

Symmetries conserved by the foliated FRGE

background symmetry respected by FRGE:

• subgroup of linear transformations

δ ˜ N = f∂τ ˜ N + ζk∂k ˜ N + ˜ N∂τ f −✘✘✘

✘

˜ N ˜ Ni∂if , δ ˜ Ni = ˜ Ni∂τ f +✘✘✘

✘

˜ Nk ˜ Nk∂if + ˜ σki∂τ ζk + ˜ Nk∂iζk + f∂τ ˜ Ni + ζk∂k ˜ Ni +✘✘✘ ǫ ˜ N2∂if δ˜ σij = f∂τ ˜ σij + ζk∂k˜ σij + ˜ Nj∂if + ˜ Ni∂jf + ˜ σjk∂iζk + ˜ σik∂jζk

• foliation-preserving diffeomorphisms: Diff(M, Σ) ⊂ Diff(M)

δγµν = Lv(γµν) , vα = (f(τ) , ζa(τ, x))

– p. 14/29

Symmetries conserved by the foliated FRGE

background symmetry respected by FRGE:

• subgroup of linear transformations

δ ˜ N = f∂τ ˜ N + ζk∂k ˜ N + ˜ N∂τ f −✘✘✘

✘

˜ N ˜ Ni∂if , δ ˜ Ni = ˜ Ni∂τ f +✘✘✘

✘

˜ Nk ˜ Nk∂if + ˜ σki∂τ ζk + ˜ Nk∂iζk + f∂τ ˜ Ni + ζk∂k ˜ Ni +✘✘✘ ǫ ˜ N2∂if δ˜ σij = f∂τ ˜ σij + ζk∂k˜ σij + ˜ Nj∂if + ˜ Ni∂jf + ˜ σjk∂iζk + ˜ σik∂jζk

• foliation-preserving diffeomorphisms: Diff(M, Σ) ⊂ Diff(M)

δγµν = Lv(γµν) , vα = (f(τ) , ζa(τ, x))

symmetry group of Hoˇ rava-Lifshitz gravity

– p. 14/29

Wetterich Equation for projectable Hoˇ rava-Lifshitz gravity

[E. Manrique, S. Rechenberger, F.S., arXiv:1102.5012] [S. Rechenberger, F.S., arXiv:1212.5114] [A. Contillo, S. Rechenberger, F.S., arXiv:1309.7273] [G. D’Odorico, M. Schutten, F.S., arXiv:1406.4366] [M. Baggio, J. de Boer and K. Holsheimer, arXiv:1112.6416] [D. Benedetti, F. Guarnieri, arXiv:1311.6253]

– p. 15/29

projective Hoˇ rava-Lifshitz gravity in a nutshell

P . Hoˇ rava, Phys. Rev. D79 (2009) 084008, arXiv:0901.3775

central idea: find a perturbatively renormalizable quantum theory of gravity fundamental fields:

{N(τ), Ni(τ, x), σij(τ, x)}

symmetry: Diff(M, Σ) ⊂ Diff(M)

• spatial higher-derivative terms make theory power-counting renormalizable
• anisotropic dispersion relation breaks Lorentz-invariance

– p. 16/29

projective Hoˇ rava-Lifshitz gravity in a nutshell

P . Hoˇ rava, Phys. Rev. D79 (2009) 084008, arXiv:0901.3775

central idea: find a perturbatively renormalizable quantum theory of gravity fundamental fields:

{N(τ), Ni(τ, x), σij(τ, x)}

symmetry: Diff(M, Σ) ⊂ Diff(M)

• spatial higher-derivative terms make theory power-counting renormalizable
• anisotropic dispersion relation breaks Lorentz-invariance

Can construct the effective average action for projectable HL-gravity

• scale-dependence governed by functional renormalization group equation

k∂kΓk[φ, ¯ φ] = 1

2 STr

• Γ(2)

k

+ Rk

−1

k∂kRk

• Complication: anisotropic models have two correlation lengths

– p. 16/29

RG-flows of Hoˇ rava-Lifshitz gravity in the IR

• A. Contillo, S. Rechenberger, F.S., JHEP 1312 (2013) 017

RG-flow of anisotropic λ-R truncation

Γgrav

k

[N, Ni, σij] = 1 16πGk

• dτd3xN√g
• KijKij − λkK2 −(3) R + 2Λk
• – p. 17/29

RG-flows of Hoˇ rava-Lifshitz gravity in the IR

• A. Contillo, S. Rechenberger, F.S., JHEP 1312 (2013) 017

RG-flow of anisotropic λ-R truncation

Γgrav

k

[N, Ni, σij] = 1 16πGk

• dτd3xN√g
• KijKij − λkK2 −(3) R + 2Λk
• Fixed points of the beta functions:
• Wheeler-de Witt metric ⇒ line of GFPs

˜ G∗ = 0 , ˜ Λ∗ = 0 , λ = λ∗

• ne IR attractive, one IR repulsive, one marginal direction
• NGFP:

˜ G∗ = 0.49 , ˜ Λ∗ = 0.17 , λ∗ = 0.44

• three UV-attractive eigen-directions
• imprint of Asymptotic Safety

– p. 17/29

RG-flows of Hoˇ rava-Lifshitz gravity in the IR

• A. Contillo, S. Rechenberger, F.S., JHEP 1312 (2013) 017

RG-flow of anisotropic λ-R truncation

Γgrav

k

[N, Ni, σij] = 1 16πGk

• dτd3xN√g
• KijKij − λkK2 −(3) R + 2Λk
• Fixed points of the beta functions:
• Wheeler-de Witt metric ⇒ line of GFPs

˜ G∗ = 0 , ˜ Λ∗ = 0 , λ = λ∗

• ne IR attractive, one IR repulsive, one marginal direction
• NGFP:

˜ G∗ = 0.49 , ˜ Λ∗ = 0.17 , λ∗ = 0.44

• three UV-attractive eigen-directions
• imprint of Asymptotic Safety

anisotropic GFP providing UV-limit of HL-gravity not in truncation

– p. 17/29

Hoˇ rava-Lifshitz gravity: recovering general relativity in the IR

0.0 0.2 0.4 0.6 G

• 0.5

0.0 0.5

• 0.0

0.5 1.0 1.5 2.0 Λ

– p. 18/29

Scale-dependence of dimensionful couplings

1 1 2 3 4 5 t 104 0.001 0.01 G 1 1 2 3 4 5 t 0.01 0.1 1 10 100 1000

• 1

1 2 3 4 5 t 0.4 0.6 0.8 1. 1.2 Λ

GFP governs IR-behavior of HL-gravity small value of cosmological constant makes λ compatible with experiments

– p. 19/29

projectable Hoˇ rava-Lifshitz gravity zooming into the anisotropic gaussian fixed point

• G. D’Odorico, F. Saueressig, M. Schutten, PRL in press

– p. 20/29

Zooming into the aGFP in D = 3 + 1

Compute matter-induced gravitational β-functions

Γk = ΓHL

k

+ SLM

for two wave-function renormalizations and 8 potential couplings

ΓHL

k

= 1 16πGk

• dtd3x√σ

KijKij − λk K2 − g7R ∆x R − g8Rij ∆x Rij + . . . SLM = 1 2

• dtd3x√σ [φ (∆t + (∆x)z) φ]

key ingredient: anisotropic Laplace operator

D = ∆t + (∆x)z ∆t = −√σ−1 ∂t √σ∂t , ∆x = −σij(t, x)DiDj

– p. 21/29

Zooming into the aGFP in D = 3 + 1

Compute matter-induced gravitational β-functions

Γk = ΓHL

k

+ SLM

for two wave-function renormalizations and 8 potential couplings

ΓHL

k

= 1 16πGk

• dtd3x√σ

KijKij − λk K2 − g7R ∆x R − g8Rij ∆x Rij + . . . SLM = 1 2

• dtd3x√σ

φ (∆t + (∆x)z) φ

Gravitational propagators in flat space: σij = δij + √16πGk hij

[Gs=2(ω, p)] ∝ ω2 − g8 p 6 [Gs=0(ω, p)] ∝ 1

3 − λk

ω2 − 1

3 − λk

−1 8

9 g7 + 1 3g8

• p 6

– p. 22/29

Heat kernel expansion of anisotropic operators

FRGE computations use heat-kernel expansion of Laplacian ∆ ≡ −gµνDµDν

Tre−s∆ ≃ 1 (4πs)d/2

• ddx√g
• n≥0

sn a2n ≃ 1 (4πs)d/2

• ddx√g

1 + s

6 R + . . .

– p. 23/29

Heat kernel expansion of anisotropic operators

FRGE computations use heat-kernel expansion of Laplacian ∆ ≡ −gµνDµDν

Tre−s∆ ≃ 1 (4πs)d/2

• ddx√g
• n≥0

sn a2n ≃ 1 (4πs)d/2

• ddx√g

1 + s

6 R + . . .

Heat kernel expansion of anisotropic operators

D ≡ ∆t + (∆x)z

• compute from off-diagonal heat-kernel techniques
• G. D’Odorico, F. Saueressig, M. Schutten, arXiv:1406.4366, PRL in press

Tre−sD ≃ s− 1

2 (1+d/z)

(4π)(d+1)/2

• dtddx√σ
• s

6

• e1 K2 + e2 KijKij

+

• n≥0

sn/z bn a2n

• e1 = d − z + 3

d + 2 Γ( d

2z )

z Γ( d

2 )

, e2 = − d + 2z d + 2 Γ( d

2z )

z Γ( d

2 )

– p. 23/29

Heat kernel coefficients for anisotropic operators

d = 2 d = 3 z = 1 z = 2 z = 3 z = 1 z = 2 z = 3 z = 4 b0 1

√π 2

Γ( 4

3 )

1

4 3√π Γ( 7 4 ) 2 3 4 3√π Γ( 11 8 )

b1 1 1 1 1

2 √π Γ( 5 4 ) 2 √π Γ( 7 6 ) 2 √π Γ( 9 8 )

b2 1 1

1 √π Γ( 3 4 ) 1 √π Γ( 5 6 ) 1 √π Γ( 7 8 )

b3 1 −2 1 − 2

√π Γ( 5 4 )

− 1

2

−

1 2√π Γ( 5 8 )

b4 1 6 1 − 4

√π Γ( 7 4 ) 9 2√π Γ( 7 6 ) 2 √π Γ( 11 8 )

• z = 1: reproduces standard heat-kernel
• z = 2, d = 2: reproduces

[M. Baggio, J. de Boer and K. Holsheimer, arXiv:1112.6416]

• d even: zero coefficients in heat kernel expansion

– p. 24/29

matter-induced RG flows in d = z = 3

20000 40000 60000 80000 100000 t 0.5 0.4 0.3 0.2 0.1 Gt 1 2 3 4 5 t

• 2. 1081
• 1. 1081
• 1. 1081
• 2. 1081

Λt13 20000 40000 60000 80000 100000 t 0.14 0.16 0.18 0.20 0.22 0.24 g7t 20000 40000 60000 80000 100000 t 0.02 0.04 0.06 0.08 0.10 0.12 0.14 g8t

UV attractive anisotropic GFP

G∗ = 0, λ∗ = 1/3, g∗

7 = 5π 84 ,

g∗

8 = π 42

– p. 25/29

Tracing the anisotropic Gaussian fixed point in z 40 30 20 10 10 20 G

1 d

1

Λ

z1 zd

– p. 26/29

Conclusions

– p. 27/29

Asymptotic Safety and Hoˇ rava-Lifshitz gravity live in same space

– p. 28/29

Conclusions

many proposals for quantum gravity within QFT:

• Asymptotic Safety
• Causal Dynamical Triangulations
• Hoˇ

rava-Lifshitz gravity

• Shape Dynamics

differences:

• field content

(metric, vielbein, ADM-variables, . . . )

• symmetry group

(diffeomorphisms, foliation preserving diff.) ??? Which formulations describe the same Universality Class ???

– p. 29/29

Conclusions

many proposals for quantum gravity within QFT:

• Asymptotic Safety
• Causal Dynamical Triangulations
• Hoˇ

rava-Lifshitz gravity

• Shape Dynamics

differences:

• field content

(metric, vielbein, ADM-variables, . . . )

• symmetry group

(diffeomorphisms, foliation preserving diff.) ??? Which formulations describe the same Universality Class ??? RG techniques crucial for solving this question

– p. 29/29