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 S UV ◦ fixing the position of a trajectory in S UV ◦ ⇐ ⇒ 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 S UV ◦ fixing the position of a trajectory in S UV ◦ ⇐ ⇒ 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 S UV ◦ fixing the position of a trajectory in S UV ◦ ⇐ ⇒ 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 G N ◦ – p. 3/29
Proposals for UV fixed points (incomplete . . . ) isotropic Gaussian Fixed Point (GFP) • fundamental theory: Einstein-Hilbert action ◦ perturbation theory in G N ◦ 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 G N ◦ 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 G N ◦ 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] g 1 0.75 0.5 Type IIa 0.25 Type Ia Type IIIa λ −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 −0.25 Type IIIb −0.5 Type Ib −0.75 – 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 ≈ 10 ℓ Pl ◦ – 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 ≈ 10 ℓ Pl ◦ Monte Carlo Simulation of gravitational partition sum Causal Dynamical Triangulations (CDT) • second order phase transition line ◦ “classical universes” at ℓ ≈ 10 ℓ Pl ◦ 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 ≈ 10 ℓ Pl ◦ Monte Carlo Simulation of gravitational partition sum Causal Dynamical Triangulations (CDT) • second order phase transition line ◦ “classical universes” at ℓ ≈ 10 ℓ Pl ◦ 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 M d +1 = S 1 × M d with y µ �→ ( τ, x a ) : • ds 2 = N 2 dt 2 + σ ij dx i + N i dt � � dx j + N j dt � � fundamental fields: g µν �→ ( N, N i , σ ij ) • N 2 + N i N i N j g µν = N i σ ij – p. 8/29
Foliation structure via ADM-decomposition Preferred “time”-direction via foliation of space-time foliation structure M d +1 = S 1 × M d with y µ �→ ( τ, x a ) : • ds 2 = ǫN 2 dt 2 + σ ij dx i + N i dt � � dx j + N j dt � � fundamental fields: g µν �→ ( N, N i , σ ij ) • ǫN 2 + N i N i N j g µν = N i σ ij Allows to include signature parameter ǫ = ± 1 – p. 9/29
Foliated functional renormalization group equation Flow equation: formally the same as in covariant construction �� � � − 1 Γ (2) σ ij ] = 1 k∂ k Γ k [ h, h i , h ij ; ¯ 2 STr + R k k∂ k R k k covariant: M 4 • d 4 y √ ¯ � � STr ≈ g fields foliated: S 1 × M 3 • √ STr ≈ √ ǫ � � � d 3 x σ ¯ component fields KK − modes 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 �� � � − 1 Γ (2) σ ij ] = 1 k∂ k Γ k [ h, h i , h ij ; ¯ 2 STr + R k k∂ k R k k covariant: M 4 • d 4 y √ ¯ � � STr ≈ g fields foliated: S 1 × M 3 • √ STr ≈ √ ǫ � � � d 3 x σ ¯ component fields KK − modes 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 √ ǫ dτd 3 xN √ σ � � K ij K ij − K 2 � − R (3) + 2Λ k � + S gf + S gh Γ ADM ǫ − 1 � = k 16 πG k g 1.0 0.8 0.6 0.4 0.2 Λ � 0.4 � 0.2 0.0 0.2 0.4 covariant computation g g 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 Λ Λ � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 0.3 0.4 � 0.3 � 0.2 � 0.1 0.0 0.1 0.2 0.3 0.4 Euclidean ǫ = 1 Lorentzian: ǫ = − 1 – p. 11/29
It’s all about choosing a gauge: covariant formulation: g µν = ¯ g µν + h µν perform covariant gauge-fixing (e.g., harmonic gauge) D µ h ν ν = 0 . F µ = ¯ D ν h µν − 1 2 ¯ foliated formulation with ADM-fields g µν �→ { N, N i , σ ij } N = ¯ N i = ¯ N + h , N i + h i , σ ij = ¯ σ ij + h ij perform temporal gauge-fixing (non-covariant): h = 0 , h i = 0 fluctuations in the metric on the spatial slice only • – p. 12/29
It’s all about choosing a gauge: covariant formulation: g µν = ¯ g µν + h µν perform covariant gauge-fixing (e.g., harmonic gauge) D µ h ν ν = 0 . F µ = ¯ D ν h µν − 1 2 ¯ foliated formulation with ADM-fields g µν �→ { N, N i , σ ij } N = ¯ N i = ¯ N + h , N i + h i , σ ij = ¯ σ ij + h ij perform temporal gauge-fixing (non-covariant): h = 0 , h i = 0 fluctuations in the metric on the spatial slice only • ADM fields in temporal gauge No fluctuations in stacking spatial slices! – p. 12/29
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