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 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