Asymptotically Safe Gravity connecting continuum and Monte Carlo

Transcription

1 Asymptotically Safe Gravity connecting continuum and Monte Carlo Frank Saueressig Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen J. Biemans, A. Platania and F. Saueressig, arxiv: J. Biemans, A. Platania and F. Saueressig, arxiv: W. Houthoff, A. Kurov and F. Saueressig, arxiv: Lemaître Workshop Vatican Observatory, May 11th, 2017 p. 1/38

2 p. 2/38 Outline asymptotic safety: a primer gravitational renormalization group flows cosmic backgrounds backgrounds with topology S 1 S d a brief look at observables

3 Einstein s theory of classical general relativity R µν 1 2 g µν R+Λg µν }{{} dynamics of spacetime = 8πG N T µν }{{} matter strikingly successful in describing nature: highly successful cosmological models direct detection of gravitational waves p. 3/38

4 p. 4/38 Quantizing general relativity [G. t Hooft and M. Veldman, Ann. Inst. H. Poincare Phys. Theor. A20 (1974) 69] [M. H. Goroff and A. Sagnotti, Nucl. Phys. B266 (1986) 709] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] Einstein-Hilbert action S = 1 16πG d 4 x gr gives rise to a two-loop divergence: S div = 1 ǫ (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν.

5 p. 4/38 Quantizing general relativity [G. t Hooft and M. Veldman, Ann. Inst. H. Poincare Phys. Theor. A20 (1974) 69] [M. H. Goroff and A. Sagnotti, Nucl. Phys. B266 (1986) 709] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] Einstein-Hilbert action S = 1 16πG d 4 x gr gives rise to a two-loop divergence: S div = 1 ǫ (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν. divergence not of Einstein-Hilbert form: = cannot be absorbed into bare Newton s constant need to add new C 3 -interaction to bare action = introduce a new free parameter

6 p. 4/38 Quantizing general relativity [G. t Hooft and M. Veldman, Ann. Inst. H. Poincare Phys. Theor. A20 (1974) 69] [M. H. Goroff and A. Sagnotti, Nucl. Phys. B266 (1986) 709] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] Einstein-Hilbert action S = 1 16πG d 4 x gr gives rise to a two-loop divergence: S div = 1 ǫ (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν. divergence not of Einstein-Hilbert form: = cannot be absorbed into bare Newton s constant need to add new C 3 -interaction to bare action = introduce a new free parameter General Relativity is perturbatively non-renormalizable

7 p. 5/38 Perturbative quantization of General Relativity Dynamics of General Relativity governed by Einstein-Hilbert action S EH 1 = d 4 x g [ R+2Λ ] 16πG N Newton s constant G N has negative mass-dimension

8 p. 5/38 Perturbative quantization of General Relativity Dynamics of General Relativity governed by Einstein-Hilbert action S EH 1 = d 4 x g [ R+2Λ ] 16πG N Newton s constant G N has negative mass-dimension Wilsonian picture of perturbative renormalization: dimensionless coupling constant attracted to GFP (free theory) in UV introduce dimensionless coupling constants g 0.04 g k = k 2 G N, λ k Λk 2 GFP: flow governed by mass-dimension: k k g k =2g +O(g 2 ) k k λ k = 2λ+O(g) 0.02 GFP Λ

9 p. 5/38 Perturbative quantization of General Relativity Dynamics of General Relativity governed by Einstein-Hilbert action S EH 1 = d 4 x g [ R+2Λ ] 16πG N Newton s constant G N has negative mass-dimension Wilsonian picture of perturbative renormalization: dimensionless coupling constant attracted to GFP (free theory) in UV introduce dimensionless coupling constants g 0.04 g k = k 2 G N, λ k Λk 2 GFP: flow governed by mass-dimension: k k g k =2g +O(g 2 ) k k λ k = 2λ+O(g) 0.02 GFP Λ General Relativity is not asymptotically free

10 p. 6/38 General Relativity is non-renormalizable: conclusions a) Treat gravity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) predictivity breaks down at E 2 M 2 Pl

11 p. 6/38 General Relativity is non-renormalizable: conclusions a) Treat gravity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) predictivity breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory: introduces: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme

12 p. 6/38 General Relativity is non-renormalizable: conclusions a) Treat gravity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) predictivity breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory: introduces: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme c) Gravity makes sense as Quantum Field Theory: UV-completion beyond perturbation theory: UV-completion by relaxing symmetries: Asymptotic Safety Hořava-Lifshitz

13 p. 6/38 General Relativity is non-renormalizable: conclusions a) Treat gravity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) predictivity breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory: introduces: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme c) Gravity makes sense as Quantum Field Theory: UV-completion beyond perturbation theory: UV-completion by relaxing symmetries: Asymptotic Safety Hořava-Lifshitz

14 p. 7/38 The Asymptotic Safety package a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences

15 p. 7/38 The Asymptotic Safety package a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) NGFP has finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters required for predictivity

16 p. 7/38 The Asymptotic Safety package a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) NGFP has finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters required for predictivity c) connection to observable low-energy physics: tests of general relativity (cosmological signatures,...) compatibility with standard model of particle physics at 1 TeV (light fermions, Yukawa couplings, Higgs potential,...)

17 p. 7/38 The Asymptotic Safety package a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) NGFP has finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters required for predictivity c) connection to observable low-energy physics: tests of general relativity (cosmological signatures,...) compatibility with standard model of particle physics at 1 TeV (light fermions, Yukawa couplings, Higgs potential,...) d) structural demands: resolution of singularities: (black holes, Landau poles,...) background independence, unitarity,...

18 p. 8/38 Routes towards asymptotic safety: renormalization group starting point: path integral over metrics Z = Dĝe S[ĝ]+Ssource add k-dependent mass-term R k for fluctuations with p 2 k 2 : Z k = Dĝe S[ĝ]+Ssource hr k h functional renormalization group equation for effective average action Γ k k k Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] [C. Wetterich 93] [T. Morris 94] [M. Reuter 96] does not require to specify a fundamental action a priori integrate out quantum fluctuations shell-by-shell in momentum-space

19 p. 8/38 Routes towards asymptotic safety: renormalization group starting point: path integral over metrics Z = Dĝe S[ĝ]+Ssource add k-dependent mass-term R k for fluctuations with p 2 k 2 : Z k = Dĝe S[ĝ]+Ssource hr k h functional renormalization group equation for effective average action Γ k k k Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] [C. Wetterich 93] [T. Morris 94] [M. Reuter 96] does not require to specify a fundamental action a priori integrate out quantum fluctuations shell-by-shell in momentum-space Wilson s idea of the renormalization group

20 p. 9/38 Routes towards asymptotic safety: dynamical triangulations starting point: path integral over metrics Z = Dĝe S[ĝ] regularize partition sum by discretizing the geometry building blocks: D-simplices spacetimes arise from gluing together many building blocks

21 p. 9/38 Routes towards asymptotic safety: dynamical triangulations starting point: path integral over metrics Z = Dĝe S[ĝ] regularize partition sum by discretizing the geometry building blocks: D-simplices spacetimes arise from gluing together many building blocks Euclidean Dynamical Triangulations (EDT) Z = triangulations (detĝ) β e SEH [ĝ] Causal Dynamical Triangulations (CDT) Z = causal triangulations e SEH [ĝ]

22 results from the renormalization group p. 10/38

23 p. 11/38 Projecting the gravitational RG flow on the Einstein-Hilbert action Einstein-Hilbert truncation: Γ k retains two running couplings: G(k),Λ(k) Γ grav k = 1 16πG(k) d 4 x g[ R+2Λ(k)] β-functions for dimensionless couplings g k k 2 G(k), λ k Λ(k)k 2 k k g k =(η N +2)g k, k k λ k = (2 η N )λ k + g k 2π [ λ k η N is the anomalous dimension of Newton s constant ] 1 η 1 2λ N k

24 p. 11/38 Projecting the gravitational RG flow on the Einstein-Hilbert action Einstein-Hilbert truncation: Γ k retains two running couplings: G(k),Λ(k) Γ grav k = 1 16πG(k) d 4 x g[ R+2Λ(k)] β-functions for dimensionless couplings g k k 2 G(k), λ k Λ(k)k 2 k k g k =(η N +2)g k, k k λ k = (2 η N )λ k + g k 2π [ λ k η N is the anomalous dimension of Newton s constant ] 1 η 1 2λ N k Gaussian fixed point at g = 0,λ = 0: saddle point in the g-λ-plane non-gaussian fixed point at g > 0,λ > 0: UV attractive in g k,λ k

25 p. 11/38 Projecting the gravitational RG flow on the Einstein-Hilbert action Einstein-Hilbert truncation: Γ k retains two running couplings: G(k),Λ(k) Γ grav k = 1 16πG(k) d 4 x g[ R+2Λ(k)] β-functions for dimensionless couplings g k k 2 G(k), λ k Λ(k)k 2 k k g k =(η N +2)g k, k k λ k = (2 η N )λ k + g k 2π [ λ k η N is the anomalous dimension of Newton s constant ] 1 η 1 2λ N k Gaussian fixed point at g = 0,λ = 0: saddle point in the g-λ-plane non-gaussian fixed point at g > 0,λ > 0: UV attractive in g k,λ k asymptotic safety: non-gaussian fixed point is UV completion for gravity

26 p. 12/38 Einstein-Hilbert-truncation: the phase diagram M. Reuter and F. Saueressig, Phys. Rev. D 65 (2002) g Type IIa Type Ia Type IIIa λ 0.25 Type IIIb Type Ib

27 p. 13/38 asymptotic safety: recent highlight results pure gravity: non-gaussian fixed point persists once 2-loop counterterm is included low number of relevant parameters ( 3) non-gaussian fixed point in d = 2 is a unitary CFT [H. Gies, B. Knorr, S. Lipholdt, and F. Saueressig, arxiv: ] [ R. Percacci and A. Codello, arxiv: ] [ D. Benedetti, P.F. Machado and F. Saueressig, arxiv: ] [ K. Falls, D. F. Litim, K. Nikolakopoulos and C. Rahmede, arxiv: ] [ A. Nink and M. Reuter, arxiv: ] black hole singularities weakened/removed by RG effects [talk by A. Platania]

28 p. 13/38 asymptotic safety: recent highlight results pure gravity: non-gaussian fixed point persists once 2-loop counterterm is included low number of relevant parameters ( 3) non-gaussian fixed point in d = 2 is a unitary CFT [H. Gies, B. Knorr, S. Lipholdt, and F. Saueressig, arxiv: ] [ R. Percacci and A. Codello, arxiv: ] [ D. Benedetti, P.F. Machado and F. Saueressig, arxiv: ] [ K. Falls, D. F. Litim, K. Nikolakopoulos and C. Rahmede, arxiv: ] black hole singularities weakened/removed by RG effects [ A. Nink and M. Reuter, arxiv: ] [talk by A. Platania] gravity coupled to matter: non-gaussian fixed point compatible with standard-model matter [ P. Dona, A. Eichhorn and R. Percacci, arxiv: ] [ J. Meibohm, J. M. Pawlowski and M. Reichert, arxiv: ] connection to masses of the standard model particles (Higgs mass) [ M. Shaposhnikov and C. Wetterich, arxiv: ] [ A. Eichhorn, A. Held and J. M. Pawlowski, arxiv: ] [talk by A. Eichhorn]

29 results from Causal Dynamical Triangulations p. 14/38

30 p. 15/38 The phase diagram of CDT on topology S 1 S 3 J. Ambjørn et. al., Phys. Rev. D 85 (2012)

31 p. 16/38 CDT correlation functions [J. Ambjørn, et. al., Phys. Rev. D 94 (2016) ] Best measured correlation functions build from spatial volumes V 3 : expectation value of volume profiles: V 3 (t) S 1 S 3, phase C S 1 T 3

32 p. 16/38 CDT correlation functions [J. Ambjørn, et. al., Phys. Rev. D 94 (2016) ] Best measured correlation functions build from spatial volumes V 3 : expectation value of volume profiles: V 3 (t) S 1 S 3, phase C S 1 T 3 Is there a connection between FRGE and CDT?

33 p. 17/38 renormalization group flows in the presence of foliations

34 p. 18/38 Introducing a time-direction through the ADM formalism Preferred time -direction via foliation of space-time foliation structure M d+1 = R M d with y µ (t,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 ) g µν = N2 +N i N i N i N j σ ij

35 p. 19/38 FRGE for the ADM formalism fundamental fields: {N(t,x),N i (t,x),σ ij (t,x)} construction of FRGE uses background field method parameterizing fluctuations by linear split N = N + ˆN, N i = N i + ˆN i, σ ij = σ ij + ˆσ ij parameterizing fluctuations by exponential split N = N + ˆN, N i = N i + ˆN i, σ ij = σ il [ eˆσ ] l j preserves signature in spatial and time directions independently

36 p. 19/38 FRGE for the ADM formalism fundamental fields: {N(t,x),N i (t,x),σ ij (t,x)} construction of FRGE uses background field method parameterizing fluctuations by linear split N = N + ˆN, N i = N i + ˆN i, σ ij = σ ij + ˆσ ij parameterizing fluctuations by exponential split N = N + ˆN, N i = N i + ˆN i, σ ij = σ il [ eˆσ ] l j preserves signature in spatial and time directions independently Wetterich equation takes the usual form [ ( ) ] k k Γ k [h µν ;ḡ µν ] = 1 2 STr Γ (2) 1k k k +R k R k regulator breaks background Diff(M)-symmetry to foliation preserving diffeos

37 p. 20/38 renormalization group flows on cosmological backgrounds

38 p. 21/38 Off-shell flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime N = 1, N i = 0, σ ij = a 2 (t)δ ij. fluctuations: adapted to cosmological perturbation theory ˆN i =u i + i 1 B, i u i = 0 ˆσ ij =h ij ( σ ) 1 ij + i j ψ 1 + i j E + 1 i v j + j 1 v i.

39 p. 22/38 Gauge-fixing in the ADM formalism (flat space) 1 2 δ2 Γ grav k Index hh vv EE ψψ ψe matrix element of 32πG k δ 2 Γ grav k t 2 + 2Λ k 2 [ t 2 2Λ ] k Λ k (d 1) [ (d 2)( t 2 + ) (d 3)Λ ] k 1 2 (d 1)[ t 2 2Λ ] k ˆN ˆN 0 BB 0 uu uv 2 2 t Bψ 2(d 1) t ˆN ψ 2(d 1) [ ] Λ k ˆN E 2Λ k

40 p. 23/38 Off-shell flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime N = 1, N i = 0, σ ij = a 2 (t)δ ij. fluctuations: adapted to cosmological perturbation theory gauge-fixing ˆN i =u i + i 1 B, i u i = 0 ˆσ ij =h ij ( σ ) 1 ij + i j ψ 1 + i j E + 1 i v j + j 1 v i. F = t ˆN + i ˆNi 1 2 tˆσ + 2(d 1) d K ˆN, F i = t ˆNi i ˆN 1 2 iˆσ + jˆσ ji +(d 2) K ij ˆNj,

41 p. 24/38 Gauge-fixing in the ADM formalism (flat space) 1 2 δ2 Γ grav k +S gf Index hh vv EE ψψ ψe matrix element of 32πG k ( δ 2 Γ grav k t 2 + 2Λ k 2 [ t 2 + 2Λ ] k [ ] 2 t + Λ k 1 2 (d 1)(d 3) 4 [ 2 t + 2Λ k ] 1 2 (d 1)[ t 2 + 2Λ ] k +S gf) ˆN ˆN 2 t + BB 2 t + uu uv Bψ ˆN ψ ˆN E 2 [ 2 t + ] 2 t 2(d 1) t (d 1) [ t 2 + Λ ] k 2 t + 2Λ k

42 p. 25/38 RG flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime N = 1, N i = 0, σ ij = a 2 (t)δ ij. fluctuations: adapted to cosmological perturbation theory ˆN i =u i + i 1 B, i u i = 0 ˆσ ij =h ij ( σ ) 1 ij + i j ψ 1 + i j E + 1 i v j + j 1 v i. unique gauge-fixing of diffeomorphisms F = t ˆN + i ˆNi 1 2 tˆσ + 2(d 1) d K ˆN, F i = t ˆNi i ˆN 1 2 iˆσ + jˆσ ji +(d 2) K ij ˆNj, well-defined Hessian Γ (2) k with relativistic propagators

43 p. 26/38 beta functions for the gravity-matter system β g =(2+η)g, β λ =(η 2)λ+ g 4π anomalous dimension of Newton s constant: [ ( λ λ )( B det (λ) 1 η) ] 6 8 η = 16πgB 1 (λ) (4π) 2 +16πgB 2 (λ) and B 1 (λ) (1 2λ) 5 6(1 2λ) λ 3B det (λ) λ+196λ2 24B det (λ) 2 B 2 (λ) = (1 2λ) 5 36(1 2λ) λ 12B det (λ) λ+196λ2 144B det (λ) 2

44 p. 27/38 Einstein-Hilbert-truncation on cosmological background J. Biemans, A. Platania and F. Saueressig, arxiv: g A O C Λ

45 p. 28/38 Phase diagram robust when adding standard model matter J. Biemans, A. Platania and F. Saueressig, arxiv:

46 p. 29/38 renormalization group flows on CDT backgrounds S 1 S d

47 p. 30/38 Off-shell flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: S 1 S d N = 1, N i = 0, σ ij = σ ij (r) S d. fluctuations: adapted to the background sphere ˆN i =u i + D i 1 B, Di u i = 0 ˆσ ij =h ij + D i 1 v j + D j 1 v i + ( D i D j + 1 d σ ij ) ψ + 1 d σ ijh. unique gauge-fixing of diffeomorphisms S gf 1 = dtd d y σ [ F 2 +F i σ ij ] F j 32πG k F = t ˆN + D i ˆN i 1 2 tˆσ, F i = t ˆN i D i ˆN 1 2 D iˆσ + D jˆσ ji

48 p. 31/38 Phase diagrams on S 1 S 3 Type I, linear g Type I, exponential g A 1.0 A O C Λ O C Λ Type II, linear Type II, exponential g g A A B O C Λ O C Λ

49 p. 32/38 Phase diagrams on S 1 S 3 : Type II, linear g A 0.5 B O C Λ

50 p. 33/38 IR completion of the flow 10 6 G ln k ln k flow approaches quasi-fixed point C: dimensionful Newton s constant: lim k 0 G k = 0 dimensionful cosmological constant: lim k 0 Λ k = 0 transition to IR-phase at Hubble-horizon scales

51 p. 34/38 FRGE meets CDT relating the phase diagrams

52 p. 35/38 Background geometries from foliated flows Determine volume profiles V 3 (t) from solving lim k 0 δγ k δh µν = 0

53 p. 35/38 Background geometries from foliated flows Determine volume profiles V 3 (t) from solving lim k 0 δγ k δh µν = 0 V 3 t Type IIIa 5 Type IIa t independent of the background where the RG flow was computed peak determines Λ 0

54 p. 36/38 Background geometries from foliated flows Determine volume profiles V 3 (t): V3 t t independent of the background where the RG flow was computed peak determines Λ 0 Correlation functions provide map between CDT and FRGE results volume-volume fluctuations (δv 3 (t))(δv 3 (t )) determine G 0 CDT FRGE S 1 S 3 Phase C Type IIIa S 1 T 3 Type IIa

55 Background geometries from foliated flows Determine volume profiles V 3 (t): V3 t t independent of the background where the RG flow was computed peak determines Λ 0 Correlation functions provide map between CDT and FRGE results volume-volume fluctuations (δv 3 (t))(δv 3 (t )) determine G 0 CDT FRGE S 1 S 3 Phase C Type IIIa S 1 T 3 Type IIa First systematic relation between CDT and FRGE phases p. 36/38

56 conclusions p. 37/38

57 p. 38/38 conclusions on asymptotic safety gravity rules

58 p. 38/38 conclusions on asymptotic safety gravity rules matter matters

59 p. 38/38 conclusions on asymptotic safety gravity rules matter matters time is essential