A functional renormalization group equation for foliated space-times

Transcription

1 p. 1/4 A functional renormalization group equation for foliated space-times Frank Saueressig Group for Theoretical High Energy Physics (THEP) Institute of Physics E. Manrique, S. Rechenberger, F.S., Phys. Rev. Lett. 106 (2011) A. Contillo, S. Rechenberger, F.S., in preparation Workshop on Strongly-Interacting Field Theories Jena, Nov. 29th, 2012

2 p. 2/4 Outline Motivation for Quantum Gravity Foundations of Asymptotic Safety Functional Renormalization Group Equations: Part I metric construction Einstein-Hilbert results Functional Renormalization Group Equations: Part II ADM construction connection to Hořava-Lifshitz gravity Einstein-Hilbert results Conclusion and perspectives

3 p. 3/4 Classical General Relativity Based on Einsteins equations R µν 1 2 g µνr }{{} space-time curvature = Λg µν +8πG N T µν }{{} matter content Newton s constant: cosmological constant: G N = m3 kgs 2 Λ g m 3

4 p. 3/4 Classical General Relativity Based on Einsteins equations R µν 1 2 g µνr }{{} space-time curvature = Λg µν +8πG N T µν }{{} matter content Newton s constant: cosmological constant: G N = m3 kgs 2 Λ g m 3 describes gravity from sub-millimeter to cosmological scales [Adelberger, et. al., 2004]

5 p. 4/4 Motivations for Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum

6 p. 4/4 Motivations for Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity

7 p. 4/4 Motivations for Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity 3. cosmological observations: horizon problem: uniform CMB-temperature small positive cosmological constant

8 p. 4/4 Motivations for Quantum Gravity 1. internal consistency R µν 1 2 g µν R+Λg µν }{{} classical = 8πG N T µν }{{} quantum 2. singularities in solutions of Einstein equations black hole singularities Big Bang singularity 3. cosmological observations: horizon problem: uniform CMB-temperature small positive cosmological constant General Relativity is incomplete Quantum Gravity may give better answers to these puzzles

9 p. 5/4 General Relativity is perturbatively non-renormalizable conclusions: a) Treat General Relativity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) breaks down at E 2 M 2 Pl

10 p. 5/4 General Relativity is perturbatively non-renormalizable conclusions: a) Treat General Relativity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory (unifies all forces of nature): requires: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme

11 p. 5/4 General Relativity is perturbatively non-renormalizable conclusions: a) Treat General Relativity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory (unifies all forces of nature): requires: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme c) Gravity makes sense as Quantum Field Theory: non-perturbative UV-completion provided by non-trivial UV fixed point perturbative UV-completion by anisotropic QFT

12 p. 5/4 General Relativity is perturbatively non-renormalizable conclusions: a) Treat General Relativity as effective field theory: [J. Donoghue, gr-qc/ ] compute corrections in E 2 /M 2 Pl 1 (independent of UV-completion) breaks down at E 2 M 2 Pl b) UV-completion requires new physics: string theory (unifies all forces of nature): requires: supersymmetry, extra dimensions loop quantum gravity: keeps Einstein-Hilbert action as fundamental new quantization scheme c) Gravity makes sense as Quantum Field Theory: non-perturbative UV-completion provided by non-trivial UV fixed point perturbative UV-completion by anisotropic QFT

13 p. 6/4 basics of Asymptotic Safety renormalizing the non-renormalizable

14 p. 7/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations)

15 p. 7/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations) action = specific combination of interaction monomials build from field content compatible with symmetries

16 p. 7/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations) action = specific combination of interaction monomials build from field content compatible with symmetries theory space: space containing all actions coordinatized by coupling constants {g i }

17 p. 7/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations) action = specific combination of interaction monomials build from field content compatible with symmetries theory space: space containing all actions coordinatized by coupling constants {g i } renormalization group flow: describes change of action under integrating out quantum fluctuations for coupling constants = β-functions

18 p. 7/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations) action = specific combination of interaction monomials build from field content compatible with symmetries theory space: space containing all actions coordinatized by coupling constants {g i } renormalization group flow: describes change of action under integrating out quantum fluctuations for coupling constants = β-functions physics at different scales captured by family of effective descriptions

19 p. 8/4 Fixed points of the RG flow Central ingredient in Wilsons picture of renormalization Definition: fixed point {g i } β-functions vanish (β g i ({g i }) gi =g i! = 0)

20 p. 8/4 Fixed points of the RG flow Central ingredient in Wilsons picture of renormalization Definition: fixed point {g i } β-functions vanish (β g i ({g i }) gi =g i! = 0) Properties: well-defined continuum limit trajectory captured by FP in UV has no unphysical UV divergences 2 classes of RG trajectories: relevant irrelevant = attracted to FP in UV = repelled from FP in UV predictivity: number of relevant directions = free parameters (determine experimentally)

21 p. 9/4 Weinbergs Asymptotic Safety scenario Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom (e.g. QCD)

22 p. 9/4 Weinbergs Asymptotic Safety scenario Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom (e.g. QCD) non-gaussian Fixed Point (NGFP) non-perturbatively renormalizable field theories fundamental theory: interacting asymptotic safety (e.g. gravity in d = 2+ɛ)

23 p. 10/4 Weinbergs Asymptotic Safety scenario Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom non-gaussian Fixed Point (NGFP) non-perturbatively renormalizable field theories fundamental theory: interacting asymptotic safety Weinberg s asymptotic safety conjecture (1979): gravity in d = 4 has non-gaussian UV fixed point

24 p. 11/4 Asymptotic Safety as viable theory for Quantum Gravity Requirements: a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences

25 p. 11/4 Asymptotic Safety as viable theory for Quantum Gravity Requirements: a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters ensures predictivity

26 p. 11/4 Asymptotic Safety as viable theory for Quantum Gravity Requirements: a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters ensures predictivity c) classical limit: RG-trajectories have part where GR is good approximation recover gravitational physics captured by General Relativity: (perihelion shift, gravitational lensing, nucleo-synthesis,...)

27 p. 11/4 Asymptotic Safety as viable theory for Quantum Gravity Requirements: a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters ensures predictivity c) classical limit: RG-trajectories have part where GR is good approximation recover gravitational physics captured by General Relativity: (perihelion shift, gravitational lensing, nucleo-synthesis,...) Quantum Einstein Gravity (QEG)

28 p. 12/4 Functional Renormalization Group Equations I covariant construction

29 p. 13/4 RG flows beyond perturbation theory Functional Renormalization Group Equations: Callan-Symanzik Equation Wegner-Houghton Equation Polchinski Equation Wetterich Equation

30 p. 13/4 RG flows beyond perturbation theory Functional Renormalization Group Equations: Callan-Symanzik Equation Wegner-Houghton Equation Polchinski Equation Wetterich Equation applicability: scalar field theory, QCD,..., Quantum Gravity condensed matter systems

31 flow equation for effective average action Γ k [C. Wetterich, Phys. Lett. B301 (1993) 90] p. 13/4 RG flows beyond perturbation theory Functional Renormalization Group Equations: Callan-Symanzik Equation Wegner-Houghton Equation Polchinski Equation Wetterich Equation applicability: scalar field theory, QCD,..., Quantum Gravity condensed matter systems For the purpose of studying gravity: adapted to gravity [M. Reuter, Phys. Rev. D 57 (1998) 971, hep-th/ ]

32 Theory space underlying the Functional Renormalization Group p. 14/4

33 p. 15/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d 4 x { 1 2 ( µχ) m2 χ 2 + interactions }

34 p. 15/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d 4 x { 1 2 ( µχ) m2 χ 2 + interactions } generating functional for connected Green functions W[J] = ln { Dχexp Sˆk[χ]+ d 4 xj χ }

35 p. 15/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d 4 x { 1 2 ( µχ) m2 χ 2 + interactions } generating functional for connected Green functions W[J] = ln { Dχexp Sˆk[χ]+ d 4 xj χ } effective action Γ[φ] gives 1PI correlation functions Γ[φ] = d 4 xjφ W[J] classical (expectation value) field φ = χ = δw[j] δj

36 p. 16/4 Effective average action Γ k in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d 4 x { 1 2 ( µχ) m2 χ 2 + interactions } introduce scale-dependent mass term k S[χ] in W[J] W k [J] = ln { Dχexp Sˆk[χ] k S[χ]+ d 4 xj χ } k S[χ] = 1 2 d 4 p (2π) 4 R k (p 2 ) χ p 2 discriminate between low/high-momentum modes R k (p 2 0 p 2 k 2 ) = k 2 p 2 k 2 high momentum modes: integrated out low momentum modes: suppressed by mass term UV IR p 2 = ˆk 2 k 2 p 2 = 0

37 p. 17/4 Effective average action Γ k in scalar field theory scale-dependent generating functional for connected Green functions { } W k [J] = ln Dχexp Sˆk[χ] k S[χ]+ d 4 xj χ Effective average action = modified Legendre-transform of W k [J] Γ k [φ] = d 4 xjφ W k [J] k S[φ]

38 p. 17/4 Effective average action Γ k in scalar field theory scale-dependent generating functional for connected Green functions { } W k [J] = ln Dχexp Sˆk[χ] k S[χ]+ d 4 xj χ Effective average action = modified Legendre-transform of W k [J] Γ k [φ] = d 4 xjφ W k [J] k S[φ] k-dependence governed by Functional RG Equation (FRGE) k k Γ k [φ] = 1 2 Tr [ (δ 2 Γ k +R k ) 1k k R k ] Formally: exact equation no approximations in derivation independent of fundamental theory Sˆk enters as initial condition Limits: Γ k interpolates continuously between: k bare/classical action S k 0 = ordinary effective action Γ

39 p. 18/4 Covariant functional RG equation for gravity fundamental field: metric γ µν symmetry: general coordinate invariance δγ µν = L v (γ µν ) = v α α γ µν +γ αν µ v α +γ µα ν v α starting point: generic diffeomorphism invariant action S grav [γ µν ]

40 p. 18/4 Covariant functional RG equation for gravity fundamental field: metric γ µν symmetry: general coordinate invariance δγ µν = L v (γ µν ) = v α α γ µν +γ αν µ v α +γ µα ν v α starting point: generic diffeomorphism invariant action S grav [γ µν ] background covariance from background field formalism: γ µν = ḡ µν +h µν quantumtrafos : δ Q ḡ µν = 0, δ Q h µν = L v (ḡ µν +h µν ) backgroundtrafos : δ B ḡ µν = L v (ḡ µν ), δ B h µν = L v (h µν ) gauge-fixing breaks quantum gauge trafo (retains background gauge): S gf = 1 2α d 4 x ḡḡ µν F µ F ν, F µ = D µ h µν 1 2 D µ h, Jacobian captured by ghost term: S gh [h,c, C;ḡ]

41 p. 19/4 Covariant functional RG equation for gravity starting point: generic diffeomorphism invariant action S grav [g µν ] background covariance background field formalism g µν = ḡ µν +h µν ḡ provides structure for constructing k-dependent IR cutoff: k S[h;ḡ] = d 4 x ḡ{h µν R k [ḡ] µνρσ h ρσ +...} R k [ḡ] Z k k 2 R (0) = k-dependent mass term discriminates low/high- D 2 -eigenmodes R (0) (p 2 /k 2 0 p 2 k 2 ) = 1 p 2 k 2 UV p 2 = ˆk 2 k 2 high momentum modes: integrated out low momentum modes: suppressed by mass term IR p 2 = 0

42 p. 20/4 Covariant functional RG equation for gravity starting point: generic diffeomorphism invariant action S grav [γ µν ] background covariance background field formalism γ µν = ḡ µν +h µν add: k-dependent IR cutoff: k S[h;ḡ] = d 4 x ḡ{h µν R k [ḡ] µνρσ h ρσ +...} exact RG equation for Γ k : [ ( ) ] k k Γ k [h;ḡ] = 1 2 STr Γ (2) 1k k k +R k R k Γ (2) k = Hessian with respect to fluctuation fields extra ḡ-dependence necessary for formulating exact equation

43 p. 21/4 Non-perturbative approximation: derivative expansion of Γ k caveat: FRGE cannot be solved exactly gravity: need non-perturbative approximation scheme

44 p. 21/4 Non-perturbative approximation: derivative expansion of Γ k caveat: FRGE cannot be solved exactly gravity: need non-perturbative approximation scheme expand Γ k in derivatives and truncate series: Γ k [Φ] = N i=1 ū i (k)o i [Φ] = substitute into FRGE = projection of flow gives β-functions for running couplings k k ū i (k) = β i (ū i ;k)

45 p. 21/4 Non-perturbative approximation: derivative expansion of Γ k caveat: FRGE cannot be solved exactly gravity: need non-perturbative approximation scheme expand Γ k in derivatives and truncate series: Γ k [Φ] = N i=1 ū i (k)o i [Φ] = substitute into FRGE = projection of flow gives β-functions for running couplings k k ū i (k) = β i (ū i ;k) testing the reliability: within a given truncation: cutoff-scheme dependence of physical quantities (= vary R k ) stability of results within extended truncations

46 p. 22/4 Letting things flow The Einstein-Hilbert truncation

47 p. 23/4 The Einstein-Hilbert truncation: setup Einstein-Hilbert truncation: two running couplings: G(k), Λ(k) 1 Γ k = d 4 x g[ R+2Λ(k)]+S gf +S gh 16πG(k) project flow onto G-Λ plane

48 p. 23/4 The Einstein-Hilbert truncation: setup Einstein-Hilbert truncation: two running couplings: G(k), Λ(k) 1 Γ k = d 4 x g[ R+2Λ(k)]+S gf +S gh 16πG(k) project flow onto G-Λ plane explicit β-functions for dimensionless couplings g k := k 2 G(k), λ k := Λ(k)k 2 Particular choice of R k (optimized cutoff) k k g k =(η N +2)g k, k k λ k = (2 η N )λ k g k 2π [ λ k ] 1 η 1 2λ N k anomalous dimension of Newton s constant: B 1 = 1 3π [ 5 ] 1 1 2λ 9 1 (1 2λ) 2 7 η N = gb 1 1 gb 2, B 2 = 1 12π [ 5 ] 1 1 2λ +6 1 (1 2λ) 2

49 p. 24/4 Einstein-Hilbert truncation: Fixed Point structure β-functions for 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 ] 1 η 1 2λ N k microscopic theory fixed points of the β-functions β g (g,λ ) = 0, β λ (g,λ ) = 0

50 p. 24/4 Einstein-Hilbert truncation: Fixed Point structure β-functions for 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 ] 1 η 1 2λ N k microscopic theory fixed points of the β-functions β g (g,λ ) = 0, β λ (g,λ ) = 0 Gaussian Fixed Point: at g = 0,λ = 0 free theory saddle point in the g-λ-plane

51 p. 24/4 Einstein-Hilbert truncation: Fixed Point structure β-functions for 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 ] 1 η 1 2λ N k microscopic theory fixed points of the β-functions β g (g,λ ) = 0, β λ (g,λ ) = 0 Gaussian Fixed Point: at g = 0,λ = 0 free theory saddle point in the g-λ-plane non-gaussian Fixed Point (η N = 2): at g > 0,λ > 0 interacting theory UV attractive in g k,λ k

52 p. 24/4 Einstein-Hilbert truncation: Fixed Point structure β-functions for 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 ] 1 η 1 2λ N k microscopic theory fixed points of the β-functions β g (g,λ ) = 0, β λ (g,λ ) = 0 Gaussian Fixed Point: at g = 0,λ = 0 free theory saddle point in the g-λ-plane non-gaussian Fixed Point (η N = 2): at g > 0,λ > 0 interacting theory UV attractive in g k,λ k Asymptotic safety: non-gaussian Fixed Point is UV completion for gravity

53 p. 25/4 Einstein-Hilbert-truncation: the phase diagram g Type IIa Type Ia Type IIIa λ 0.25 Type IIIb Type Ib

54 p. 26/4 Einstein-Hilbert truncation: Stability properties Ref. g λ g λ θ ±iθ gauge R k BMS ± 1.78i geometric II, opt RS ± 3.15i harmonic I, sharp LR ± 3.84i harmonic I, exp ± 4.08i Landau I, exp L ± 4.31i Landau I, opt CPR ± 3.04i harmonic I, opt ± 1.27i harmonic II, opt ± 2.07i harmonic III, opt BMS: Benedetti, Machado, Saueressig, RS: Reuter, Saueressig, LR: Lauscher, Reuter, L: Litim, CPR: Codello, Percacci, Rahmede, 2009.

55 p. 27/4 Exploring the gravitational theory space Some key results... evidence for asymptotic safety non-gaussian fixed point provides UV completion of gravity finite dimensional UV-critical surface possibly: 3 relevant parameters perturbative counterterms: gravity + matter: asymptotic safety survives 1-loop counterterm

56 p. 27/4 Exploring the gravitational theory space Some key results... evidence for asymptotic safety non-gaussian fixed point provides UV completion of gravity finite dimensional UV-critical surface possibly: 3 relevant parameters perturbative counterterms: gravity + matter: asymptotic safety survives 1-loop counterterm... and current research lines: construction of fixed functionals in f(r)-gravity? Γ grav k [g] = [D. Benedetti, F. Caravelli, JHEP 06 (2012) 017] [M. Demmel, F. S., O. Zanusso, JHEP 11 (2012) 131] [J. Dietz, T. Morris, arxiv: ] d d x gf k (R)

57 p. 27/4 Exploring the gravitational theory space Some key results... evidence for asymptotic safety non-gaussian fixed point provides UV completion of gravity finite dimensional UV-critical surface possibly: 3 relevant parameters perturbative counterterms: gravity + matter: asymptotic safety survives 1-loop counterterm... and current research lines: construction of fixed functionals in f(r)-gravity? Γ grav k [g] = [D. Benedetti, F. Caravelli, JHEP 06 (2012) 017] [M. Demmel, F. S., O. Zanusso, JHEP 11 (2012) 131] [J. Dietz, T. Morris, arxiv: ] d d x gf k (R) How does a time -direction affect Asymptotic Safety?

58 p. 28/4 Functional Renormalization Group Equations II foliated construction

59 p. 29/4 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 ) g µν = N2 +N i N i N i N j σ ij

60 p. 30/4 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 = ɛn 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 = ɛn2 +N i N i N i N j σ ij Allows to include signature parameter ɛ = ±1

61 p. 31/4 Foliated FRGE: full diffeomorphism symmetry fundamental fields: {Ñ(τ,x), Ñ i (τ,x), σ ij (τ,x) } symmetry: general coordinate invariance inherited from γ µν : δγ µν = L v (γ µν ), v α = (f(τ,x), ζ a (τ,x)) induces δñ = f τñ +ζk k Ñ +Ñ τf ÑÑi i f, δñi = Ñi τ f +ÑkÑk i f + σ ki τ ζ k +Ñk i ζ k +f τ Ñ i +ζ k k Ñ i +ɛñ2 i f δ σ ij = f τ σ ij +ζ k k σ ij +Ñj i f +Ñi j f + σ jk i ζ k + σ ik j ζ k

62 p. 32/4 Foliated FRGE: full diffeomorphism symmetry fundamental fields: {Ñ(τ,x), Ñ i (τ,x), σ ij (τ,x) } symmetry: general coordinate invariance inherited from γ µν : δγ µν = L v (γ µν ), v α = (f(τ,x), ζ a (τ,x)) induces δñ = f τñ +ζk k Ñ +Ñ τf ÑÑi i f, δñi = Ñi τ f +ÑkÑk i f + σ ki τ ζ k +Ñk i ζ k +f τ Ñ i +ζ k k Ñ i +ɛñ2 i f δ σ ij = f τ σ ij +ζ k k σ ij +Ñj i f +Ñi j f + σ jk i ζ k + σ ik j ζ k symmetry realized non-linearly

63 p. 32/4 Foliated FRGE: full diffeomorphism symmetry fundamental fields: {Ñ(τ,x), Ñ i (τ,x), σ ij (τ,x) } symmetry: general coordinate invariance inherited from γ µν : δγ µν = L v (γ µν ), v α = (f(τ,x), ζ a (τ,x)) induces δñ = f τñ +ζk k Ñ +Ñ τf ÑÑi i f, δñi = Ñi τ f +ÑkÑk i f + σ ki τ ζ k +Ñk i ζ k +f τ Ñ i +ζ k k Ñ i +ɛñ2 i f δ σ ij = f τ σ ij +ζ k k σ ij +Ñj i f +Ñi j f + σ jk i ζ k + σ ik j ζ k symmetry realized non-linearly impossible to construct regulator k S quadratic in fluctuation fields perserving Diff(M) as background symmetry

64 p. 33/4 Foliated FRGE: projectable Hořava-Lifshitz fundamental fields: {Ñ(τ), Ñ i (τ,x), σ ij (τ,x) } symmetry: foliation-preserving diffeomorphisms Diff(M): δγ µν = L v (γ µν ), v α = (f(τ), ζ a (τ,x)) induces δñ = f τñ +ζk k Ñ +Ñ τf ÑÑi i f, δñi = Ñi τ f + Ñ k Ñ k i f + σ ki τ ζ k +Ñk i ζ k +f τ Ñ i +ζ k k Ñ i + ɛñ2 i f δ σ ij = f τ σ ij +ζ k k σ ij +Ñj i f +Ñi j f + σ jk i ζ k + σ ik j ζ k symmetry realized linearly construct regulator k S: quadratic in fluctuation fields allows foliation-preserving diffeomorphisms as background symmetry

65 Embedding of QEG in Hořava-Lifshitz gravity p. 34/4

66 p. 35/4 Foliated FRGE: construction fundamental fields: {Ñ(τ), Ñ i (τ,x), σ ij (τ,x) } symmetry: foliation-preserving diffeomorphisms Diff(M) starting point: action invariant under foliation-preserving diffeomorphisms S grav [Ñ,Ñi, σ ij ]

67 p. 35/4 Foliated FRGE: construction fundamental fields: {Ñ(τ), Ñ i (τ,x), σ ij (τ,x) } symmetry: foliation-preserving diffeomorphisms Diff(M) starting point: action invariant under foliation-preserving diffeomorphisms S grav [Ñ,Ñi, σ ij ] gauge-fixing: background field formalism for { Ñ,Ñi, σ ij } : Ñ = N +h, Ñ i = N i +h i, σ ij = σ ij +h ij impose temporal gauge: h = 0,h i = 0 S gf = 1 2α ɛ dτ d 3 x σ N 1 { h 2 + σ ij h i h j } ghost action: S gh = ɛ dτ d 3 x σ { C τ C + C i τ C i}

68 p. 36/4 Foliated FRGE: construction fundamental fields: symmetry: {Ñ(τ), Ñ i (τ,x), σ ij (τ,x) } foliation-preserving diffeomorphisms Diff(M) starting point: action invariant under foliation-preserving diffeomorphisms S grav [Ñ,Ñi, σ ij ] gauge-fixing: background field formalism for { Ñ,Ñi, σ ij } : Ñ = N +h, Ñ i = N i +h i, σ ij = σ ij +h ij k-dependent IR-cutoff k S k S[h; σ] = ɛ dτ d 3 x σ { h ij R k [ σ]h ij +... } depends on spatial Laplacian: = σ ij Di Dj is positive definite respects foliation-preserving diffeomorphisms explore RG-flows in Hořava-Lifshitz gravity

69 p. 37/4 Foliated functional renormalization group equation Flow equation: formally the same as in covariant construction k k Γ k [h,h i,h ij ; σ ij ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] covariant: M 4 STr fields d 4 y ḡ foliated: S 1 M 3 STr ɛ component fields KK modes d 3 x σ structure resembles: quantum field theory at finite temperature!

70 p. 37/4 Foliated functional renormalization group equation Flow equation: formally the same as in covariant construction k k Γ k [h,h i,h ij ; σ ij ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] covariant: M 4 STr fields d 4 y ḡ foliated: S 1 M 3 STr ɛ component fields KK modes d 3 x σ structure resembles: quantum field theory at finite temperature! Advantages of the foliated flow equation: limits: same as covariant equation ɛ-dependence: keep track of signature effects structure: same as Monte-Carlo simulations following CDT

71 p. 38/4 signature-dependent renormalization group flows Einstein-Hilbert truncation

72 p. 39/4 ADM-decomposed Einstein-Hilbert truncation: setup ADM-decomposed Einstein-Hilbert truncation: running couplings: G k,λ k Γ ADM k = ɛ 16πG k dτd 3 xn σ {ɛ 1 K ij [ σ ik σ jl σ ij σ kl] K kl R (3) +2Λ k } +S gf +S gh K ij : extrinsic curvature R (d) : intrinsic curvature ɛ = ±1: signature parameter

73 p. 39/4 ADM-decomposed Einstein-Hilbert truncation: setup ADM-decomposed Einstein-Hilbert truncation: running couplings: G k,λ k Γ ADM k = ɛ 16πG k dτd 3 xn σ {ɛ 1 K ij [ σ ik σ jl σ ij σ kl] K kl R (3) +2Λ k } +S gf +S gh K ij : extrinsic curvature R (d) : intrinsic curvature ɛ = ±1: signature parameter Structure of the flow equation T TT = ɛk 3 d 2T (4π) 3/2 n k k Γ k = T TT +T 0 d 3 x σ [ ( )] q 1,0 3/2 (w 2T)+ R 1 k 2 6 q1,0 1/2 (w 2T) 2 3 q2,0 3/2 (w 2T)

74 p. 39/4 ADM-decomposed Einstein-Hilbert truncation: setup ADM-decomposed Einstein-Hilbert truncation: running couplings: G k,λ k Γ ADM k = ɛ 16πG k dτd 3 xn σ {ɛ 1 K ij [ σ ik σ jl σ ij σ kl] K kl R (3) +2Λ k } +S gf +S gh K ij : extrinsic curvature R (d) : intrinsic curvature ɛ = ±1: signature parameter Structure of the flow equation T TT = ɛk 3 d 2T (4π) 3/2 n k k Γ k = T TT +T 0 d 3 x σ [ ( )] q 1,0 3/2 (w 2T)+ R 1 k 2 6 q1,0 1/2 (w 2T) 2 3 q2,0 3/2 (w 2T) β-functions depend parametrically on m = 2π Tk : k k g k = β g (g,λ;m), k k λ k = β λ (g,λ;m)

75 p. 40/4 Analyticity properties of β-functions Kaluza-Klein sums: carry out analytically: n q 1,0 d/2 (w 2T) n ɛ m2 n 2 2λ k Summation: depends on signature ɛ: n n 1 n 2 +x 2 = 1 n 2 +x 2 = π x tanh(πx), x2 > 0 (hyperbolic functions) π ix tan(πx), x2 < 0 (trigonometric functions)

76 p. 40/4 Analyticity properties of β-functions Kaluza-Klein sums: carry out analytically: n q 1,0 d/2 (w 2T) n ɛ m2 n 2 2λ k Summation: depends on signature ɛ: n n 1 n 2 +x 2 = 1 n 2 +x 2 = π x tanh(πx), x2 > 0 (hyperbolic functions) π ix tan(πx), x2 < 0 (trigonometric functions) analytic structure of β-functions: determined by ɛ, λ: ɛ λ < λ (1) < 0 λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic

77 p. 41/4 results: compactification limit T 0,m d time -circle collapses = β-functions independent of signature ɛ dimensionless couplings λ k, g (d) k = g (D) m k 2π, τ = λ (g (d) ) 2/(d 2) β-functions give non-gaussian fixed point: g (d) λ τ θ 1,2 τ LF ± 3.22i θ LF 1, ± 4.63i ± 3.04i ± 6.26i ± 5.15i ± 7.86i ± 7.14i ± 9.41i ± 9.05i [comparison: P. Fischer, D. Litim, hep-th/ ]

78 p. 42/4 results: decompactification limit T,m 0 trigonometric terms in β-functions diverge: ɛ λ < λ (1) λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic no Non-Gaussian fixed point!

79 p. 42/4 results: decompactification limit T,m 0 trigonometric terms in β-functions diverge: ɛ λ < λ (1) λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic no Non-Gaussian fixed point! can be traced to the conformal factor problem artifact of the truncation

80 p. 43/4 results: NGFP for Kaluza-Klein mass T k 1 lim k m k = m 0 (sequel :m = 2π) β-functions well-defined in all regions: ɛ λ < λ (1) λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic

81 p. 43/4 results: NGFP for Kaluza-Klein mass T k 1 lim k m k = m 0 (sequel :m = 2π) β-functions well-defined in all regions: ɛ λ < λ (1) λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic Obtain: NGFP for both signatures: ɛ g λ g λ θ 1, ± 3.27i ± 3.16i

82 p. 43/4 results: NGFP for Kaluza-Klein mass T k 1 lim k m k = m 0 (sequel :m = 2π) β-functions well-defined in all regions: ɛ λ < λ (1) λ (1) < λ < 1/2 1/2 < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic Obtain: NGFP for both signatures: ɛ g λ g λ θ 1, ± 3.27i ± 3.16i stability coefficients: almost the same!

83 p. 44/4 result: signature dependence of NGFP g Λ 0.4 for m finite NGFPs separate: ɛ = +1: Euclidean signature (blue) ɛ = 1: Lorentzian signature (magenta)

84 p. 45/4 result: phase diagrams g Λ covariant computation g 0.35 g Λ Λ Euclidean Lorentzian

85 p. 46/4 including anisotropy preliminary results

86 p. 47/4 Probing the RG-flow of Hořava-Lifshitz gravity include anisotropy parameter λ: Γ HL k = 1 16πG k dτd 3 xn [ σ {K ij σ ik σ jl λ k σ ij σ kl] } K kl R (3) +2Λ k G Λ 2.0 NGFP known from Einstein-Hilbert still present no evidence for diffeomorphism invariant surface as IR-attractor

87 Conclusions p. 48/4

88 p. 49/4 Conclusion and Perspectives novel causal functional renormalization group equation symmetries: foliation-preserving diffeomorphism applications: RG flows of Euclidean and Lorentzian signature metrics analytic complement to causal dynamical triangulations Hořava-type gravitational theories

89 p. 49/4 Conclusion and Perspectives novel causal functional renormalization group equation symmetries: foliation-preserving diffeomorphism applications: RG flows of Euclidean and Lorentzian signature metrics analytic complement to causal dynamical triangulations Hořava-type gravitational theories Asymptotic Safety ADM-decomposed Einstein-Hilbert action: Euclidean and Lorentzian signature: similar non-gaussian fixed points phase portraits identical to covariant computation

90 p. 49/4 Conclusion and Perspectives novel causal functional renormalization group equation symmetries: foliation-preserving diffeomorphism applications: RG flows of Euclidean and Lorentzian signature metrics analytic complement to causal dynamical triangulations Hořava-type gravitational theories Asymptotic Safety ADM-decomposed Einstein-Hilbert action: Euclidean and Lorentzian signature: similar non-gaussian fixed points phase portraits identical to covariant computation gravity in UV signature does not affect asymptotic safety