Functional Renormalization Group Equations from a Differential Operator Perspective

Transcription

1 p. 1/4 Functional Renormalization Group Equations from a Differential Operator Perspective Frank Saueressig Group for Theoretical High Energy Physics (THEP) D. Benedetti, K. Groh, P. Machado, and F.S., arxiv: [hep-th] IHES, Bures-sur-Yvette, France February 10th, 2011

2 p. 2/4 Outline Goal: Quantum Gravity QFT from the renormalization group perspective Functional renormalization group equations Results: Perspectives: pre-industrial computations industrial revolution Summary and Outlook

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 s 2

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 s 2 good description for gravity from sub-millimeter to cosmological scales [Adelberger, et. al., 2004]

5 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 s 2 good description for gravity from sub-millimeter to cosmological scales [Adelberger, et. al., 2004]

6 p. 4/4 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 s 2 conceptual and phenomenological puzzles: gravity classical standard model as Quantum Field Theory? structure of space-time at short distances? tiny value of cosmological constant?

7 p. 4/4 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 s 2 conceptual and phenomenological puzzles: gravity classical standard model as Quantum Field Theory? structure of space-time at short distances? tiny value of cosmological constant? Theoretical guidance: Quantum Theory for Gravity

8 p. 5/4 Quantum Gravity 101: Quantizing General Relativity perturbative quantization of the Einstein-Hilbert action: GN has negative mass-dimension: infinite number of counterterms General Relativity is perturbatively non-renormalizable

9 p. 5/4 Quantum Gravity 101: Quantizing General Relativity perturbative quantization of the Einstein-Hilbert action: GN has negative mass-dimension: infinite number of counterterms General Relativity is perturbatively non-renormalizable conclusions: a) General Relativity is effective field theory: compute corrections in E 2 /MPl 2 1 (independent of UV-completion) breaks down at E 2 MPl 2

10 p. 5/4 Quantum Gravity 101: Quantizing General Relativity perturbative quantization of the Einstein-Hilbert action: GN has negative mass-dimension: infinite number of counterterms General Relativity is perturbatively non-renormalizable conclusions: a) General Relativity is effective field theory: compute corrections in E 2 /MPl 2 1 (independent of UV-completion) breaks down at E 2 MPl 2 b) UV-completion requires new physics: supersymmetry, extra dimensions, non-commutative geometry,... possibly: extension of QFT-framework

11 p. 5/4 Quantum Gravity 101: Quantizing General Relativity perturbative quantization of the Einstein-Hilbert action: GN has negative mass-dimension: infinite number of counterterms General Relativity is perturbatively non-renormalizable conclusions: a) General Relativity is effective field theory: compute corrections in E 2 /MPl 2 1 (independent of UV-completion) breaks down at E 2 MPl 2 b) UV-completion requires new physics: supersymmetry, extra dimensions, non-commutative geometry,... possibly: extension of QFT-framework c) Gravity makes sense as Quantum Field Theory: UV-completion requires going beyond perturbation theory

12 p. 6/4 Quantum Gravity 101: Quantizing General Relativity perturbative quantization of the Einstein-Hilbert action: GN has negative mass-dimension: infinite number of counterterms General Relativity is perturbatively non-renormalizable conclusions: a) General Relativity is effective field theory: compute corrections in E 2 /MPl 2 1 (independent of UV-completion) breaks down at E 2 MPl 2 b) UV-completion requires new physics: supersymmetry, extra dimensions, non-commutative geometry,... possibly: extension of QFT-framework c) Gravity makes sense as Quantum Field Theory: UV-completion requires going beyond perturbation theory

13 p. 7/4 Quantum Field Theory Wilsonian Renormalization Group Perspective

14 p. 8/4 Renormalization from the Wilsonian perspective theory specify a) field content (e.g. graviton) b) symmetries (e.g. coordinate transformations)

15 p. 8/4 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. 8/4 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. 8/4 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. 8/4 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 Theory space underlying the Functional Renormalization Group p. 9/4

20 p. 10/4 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)

21 p. 10/4 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)

22 p. 11/4 Renormalization: asymptotic freedom and asymptotic safety Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom (example: QCD)

23 p. 11/4 Renormalization: asymptotic freedom and asymptotic safety Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom (example: QCD) non-gaussian Fixed Point (NGFP) non-perturbatively renormalizable field theories fundamental theory: interacting asymptotic safety

24 p. 11/4 Renormalization: asymptotic freedom and asymptotic safety Renormalization via UV fixed points = two classes of renormalizable QFTs Gaussian Fixed Point (GFP) perturbatively renormalizable field theories fundamental theory: free asymptotic freedom (example: QCD) non-gaussian Fixed Point (NGFP) non-perturbatively renormalizable field theories fundamental theory: interacting asymptotic safety Wilsonian picture: generalization of perturbative renormalization asymptotic safety as predictive as asymptotic freedom

25 p. 12/4 Examples: Asymptotically Safe Theories Theories with non-gaussian UV fixed point O(N)-sigma model (d = 2+ɛ) [Brézin, Zinn-Justin 76] critical exponents of Heisenberg ferromagnets Gross-Neveu model (d = 2+ɛ) [Gawedzki, Kupiainen 85] Grosse-Wulkenhaar model (non-commutative φ 4 -theory) [Grosse, Wulkenhaar 05; Disertori, Gurau, Magnen, Rivasseau 07] Gravity in 2+ɛ dimensions [Christensen, Duff; Gastmans, Kallosh, Truffin 78]

26 p. 12/4 Examples: Asymptotically Safe Theories Theories with non-gaussian UV fixed point O(N)-sigma model (d = 2+ɛ) [Brézin, Zinn-Justin 76] critical exponents of Heisenberg ferromagnets Gross-Neveu model (d = 2+ɛ) [Gawedzki, Kupiainen 85] Grosse-Wulkenhaar model (non-commutative φ 4 -theory) [Grosse, Wulkenhaar 05; Disertori, Gurau, Magnen, Rivasseau 07] Gravity in 2+ɛ dimensions [Christensen, Duff; Gastmans, Kallosh, Truffin 78] Weinberg s asymptotic safety conjecture (1979): gravity in d = 4 has non-gaussian UV fixed point

27 p. 13/4 Renormalizing gravity Wilsonian formulation: UV fixed points allow two classes of renormalizable Quantum Field Theories Gaussian Fixed Point perturbatively renormalizable field theories fundamental theory: free asymptotic freedom non-gaussian Fixed Point: 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

28 p. 14/4 Flows on Theory Space Functional Renormalization Group Equations

29 p. 15/4 RG flows beyond perturbation theory Functional Renormalization Group Equations: Callan-Symanzik Equation Wegner-Houghton Equation Polchinski Equation Wetterich Equation

30 p. 15/4 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. 15/4 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 p. 16/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d d x { 1 2 ( µχ) m2 χ 2 + interactions }

33 p. 16/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d d x { 1 2 ( µχ) m2 χ 2 + interactions } generating functional for connected Green functions W[J] = ln { Dχexp Sˆk[χ]+ d d xj χ }

34 p. 16/4 Effective action Γ in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d d x { 1 2 ( µχ) m2 χ 2 + interactions } generating functional for connected Green functions W[J] = ln { Dχexp Sˆk[χ]+ d d xj χ } effective action Γ[φ] gives 1PI correlation functions Γ[φ] = d d xjφ W[J] classical (expectation value) field φ = χ = δw[j] δj

35 p. 17/4 Effective average action Γ k in scalar field theory start: generic action Sˆk[χ] Sˆk[χ] = d d 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 d xj χ } k S[χ] = 1 2 d d p (2π) d R k(p 2 ) χ 2 discriminate between low/high-momentum modes R k (p 2 k 2 p 2 k 2 ) = 0 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

36 p. 18/4 Effective average action Γ k for scalars scale-dependent generating functional for connected Green functions { } W k [J] = ln Dχexp Sˆk[χ] k S[χ]+ d d xj χ Effective average action = modified Legendre-transform of W k [J] Γ k [φ] = d d xjφ W k [J] k S[φ]

37 p. 18/4 Effective average action Γ k for scalars scale-dependent generating functional for connected Green functions { } W k [J] = ln Dχexp Sˆk[χ] k S[χ]+ d d xj χ Effective average action = modified Legendre-transform of W k [J] Γ k [φ] = d d 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 ] upon specifying R k = vector field on theory space

38 p. 19/4 Effective average action Γ k for gravity start: generic diffeomorphism invariant action for metric S grav [g µν ] ˆk complication: diffeomorphisms = fix via background field method (g µν = ḡ µν +h µν ) adds gauge fixing term adds ghost action S gf [h;ḡ] S gh [h, ξ,ξ;ḡ] auxiliary background transformations diff. invariance of Γ k=0 [g = ḡ]

39 p. 19/4 Effective average action Γ k for gravity start: generic diffeomorphism invariant action for metric S grav [g µν ] ˆk complication: diffeomorphisms = fix via background field method (g µν = ḡ µν +h µν ) adds gauge fixing term adds ghost action S gf [h;ḡ] S gh [h, ξ,ξ;ḡ] auxiliary background transformations diff. invariance of Γ k=0 [g = ḡ] flow equation: analogous to scalar case [ ( k k Γ k [h, ξ,ξ;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

40 p. 19/4 Effective average action Γ k for gravity start: generic diffeomorphism invariant action for metric S grav [g µν ] ˆk complication: diffeomorphisms = fix via background field method (g µν = ḡ µν +h µν ) adds gauge fixing term adds ghost action S gf [h;ḡ] S gh [h, ξ,ξ;ḡ] auxiliary background transformations diff. invariance of Γ k=0 [g = ḡ] flow equation: analogous to scalar case [ ( k k Γ k [h, ξ,ξ;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 goal: study RG flow described by this equation

41 p. 20/4 Non-perturbative approximation: derivative expansion of Γ k caveat: FRGE cannot be solved exactly gravity: need non-perturbative approximation scheme

42 p. 20/4 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)

43 p. 20/4 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 Rk ) stability of results wrt extended truncation

44 p. 21/4 Pre-industrial computations probing theory space by hand

45 p. 22/4 Gravitational theory space: FKWC-basis for Γ grav k [g]. R 8... Einstein-Hilbert truncation R 7... R 6... R 5... R 4... R 3 C ρσ µν C κλ ρσ C µν κλ R R + 7 more R 2 C µνρσ C µνρσ R µν R µν R ½

46 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

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 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

48 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

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 Gaussian Fixed Point: at g = 0,λ = 0 free theory saddle point in the g-λ-plane

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 non-gaussian Fixed Point (η N = 2): at g > 0,λ > 0 interacting theory UV attractive in g k,λ k

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 Asymptotic safety: non-gaussian Fixed Point is UV completion for gravity

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

53 p. 26/4 Gravitational theory space: FKWC-basis for Γ grav k [g]. R 8... R 7... Einstein-Hilbert truncation polynomial f(r)-truncation R 2 +C 2 -truncation R 6... R 5... R 4... R 3 C µν ρσ C ρσ κλ C κλ µν R R + 7 more R 2 C µνρσ C µνρσ R µν R µν R ½

54 p. 27/4 Exploring the gravitational theory space Some key results... all computations confirm existence of NGFP strong evidence for asymptotic safety predictivity UV-criticial surface is finite dimensional (possibly 3 relevant parameters)

55 p. 27/4 Exploring the gravitational theory space Some key results... all computations confirm existence of NGFP strong evidence for asymptotic safety predictivity UV-criticial surface is finite dimensional (possibly 3 relevant parameters)... and open questions: existence of NGFP in extended truncations? (convergence of position, critical exponents,...) dimension of its UV-critical surface? (= number of parameters to be determined experimentally) universality class of the fundamental theory?

56 p. 28/4 Industrial Revolution The Universal RG Machine

57 p. 29/4 Perimeter Institute, Nov. 09: What if one could track the flow of 20, 100,... couplings???

58 p. 30/4 Obstructions All previous constructions rely on special background ḡ: Interaction monomials become indistinguishable example: curvature 2 -terms on background sphere d 4 x ḡ R µν Rµν = 1 4 d 4 x ḡ R 2, d 4 x ḡ R µναβ Rµναβ = 1 6 d 4 x ḡ R 2 Insufficient to probe theory space in full generality

59 p. 30/4 Obstructions All previous constructions rely on special background ḡ: Interaction monomials become indistinguishable example: curvature 2 -terms on background sphere d 4 x ḡ R µν Rµν = 1 4 d 4 x ḡ R 2, d 4 x ḡ R µναβ Rµναβ = 1 6 d 4 x ḡ R 2 Insufficient to probe theory space in full generality Working with general background ḡ: Tr contains very complicated operator structures not computable by standard methods

60 p. 31/4 The universal RG machine: blueprint goal: systematic derivative expansion of k k Γ k [h,ξ, ξ;ḡ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ]

61 p. 31/4 The universal RG machine: blueprint goal: systematic derivative expansion of k k Γ k [h,ξ, ξ;ḡ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] 4-step implementation: 1. expand Γ k [h;ḡ] to second order in fluctuations 2. reduce operator structure of Γ (2) k to standard form 3. perturbative inversion of [Γ (2) k +R k] 4. evaluation of operator traces via off-diagonal heat-kernel

62 p. 31/4 The universal RG machine: blueprint goal: systematic derivative expansion of k k Γ k [h,ξ, ξ;ḡ] = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] 4-step implementation: 1. expand Γ k [h;ḡ] to second order in fluctuations 2. reduce operator structure of Γ (2) k to standard form 3. perturbative inversion of [Γ (2) k +R k] 4. evaluation of operator traces via off-diagonal heat-kernel virtues: lifts technical requirement of having simple backgrounds ḡ each step can be handled by a computer algebra software no numerical integrations

63 p. 32/4 Step 1: expand Γ k [h; φ] to second order Notation: φ: multiplet of background fields (e.g. {ḡµν,āµ, h } ) h: multiplet of fluctuations around φ (e.g. {h µν,h µ,h})

64 p. 32/4 Step 1: expand Γ k [h; φ] to second order Notation: φ: multiplet of background fields (e.g. {ḡµν,āµ, h } ) h: multiplet of fluctuations around φ (e.g. {h µν,h µ,h}) Γ (2) k : results from Taylor expansion: Γ k [h, φ] = d d x ḡ h i [ Γ (2) k [ φ]] ij hj +... [ Γ (2) k [ φ]] ij is operator-valued matrix in field space

65 p. 32/4 Step 1: expand Γ k [h; φ] to second order Notation: φ: multiplet of background fields (e.g. {ḡµν,āµ, h } ) h: multiplet of fluctuations around φ (e.g. {h µν,h µ,h}) Γ (2) k : results from Taylor expansion: Γ k [h, φ] = d d x ḡ h i [ Γ (2) k [ φ]] ij hj +... [ Γ (2) k [ φ]] ij is operator-valued matrix in field space Example: free massive scalar field h in curved space-time ḡ µν : Γ k [h,ḡ] = 1 d d x ḡh [ D 2 +m 2 2 k] h (2) Γ k [ḡ]: Laplace-type differential operator Γ (2) k [ḡ] = D 2 +m 2 k

66 p. 33/4 Step 2: simplify operator structure of Γ (2) k generically, Γ (2) k has non-laplacian part: [ Γ (2) k Examples: ] ij = K( )δ ij ½ i }{{} kin. terms + D(D µ ) + M(R,D µ ) }{{}}{{} uncontracted derivatives background curvature D = (1 α)d µ D ν, D = D µ D ν D α D β M = R µν D µ D ν

67 p. 33/4 Step 2: simplify operator structure of Γ (2) k generically, Γ (2) k has non-laplacian part: [ Γ (2) k Examples: ] ij = K( )δ ij ½ i }{{} kin. terms + D(D µ ) + M(R,D µ ) }{{}}{{} uncontracted derivatives background curvature D = (1 α)d µ D ν, D = D µ D ν D α D β M = R µν D µ D ν D 0 Γ (2) k is non-minimal differential operator: obstructs evaluation of traces via standard heat-kernel formulae

68 p. 34/4 Step 2: simplify operator structure of Γ (2) k generically, Γ (2) k has non-laplacian part: [ ] Γ (2) ij k = K( )δ ij ½ i + D(D µ ) + M(R,D µ ) }{{}}{{}}{{} kin. terms uncontracted derivatives background curvature D 0 Γ (2) k is non-minimal differential operator general solution: Transverse decomposition of fluctuation fields [York, 75; Reuter, Lauscher 02] vector: h µ = h T µ +D µh, D µ h T µ = 0 graviton: h µν = h T µν +D µξ ν +D ν ξ µ 1 2 g µνd α ξ α g µνh D µ h T µν = 0, g µν h T µν = 0, g µν h µν = h

69 p. 34/4 Step 2: simplify operator structure of Γ (2) k generically, Γ (2) k has non-laplacian part: [ ] Γ (2) ij k = K( )δ ij ½ i + D(D µ ) + M(R,D µ ) }{{}}{{}}{{} kin. terms uncontracted derivatives background curvature D 0 Γ (2) k is non-minimal differential operator general solution: Transverse decomposition of fluctuation fields [York, 75; Reuter, Lauscher 02] vector: h µ = h T µ +D µh, D µ h T µ = 0 graviton: h µν = h T µν +D µξ ν +D ν ξ µ 1 2 g µνd α ξ α g µνh D µ h T µν = 0, g µν h T µν = 0, g µν h µν = h Implemented via non-local projection operators: [Π L ] µ ν = D µ 1 D ν, [Π L h] µ = D µ h

70 p. 34/4 Step 2: simplify operator structure of Γ (2) k generically, Γ (2) k has non-laplacian part: [ ] Γ (2) ij k = K( )δ ij ½ i + D(D µ ) + M(R,D µ ) }{{}}{{}}{{} kin. terms uncontracted derivatives background curvature D 0 Γ (2) k is non-minimal differential operator general solution: Transverse decomposition of fluctuation fields [York, 75; Reuter, Lauscher 02] vector: h µ = h T µ +D µh, D µ h T µ = 0 graviton: h µν = h T µν +D µξ ν +D ν ξ µ 1 2 g µνd α ξ α g µνh D µ h T µν = 0, g µν h T µν = 0, g µν h µν = h Implemented via non-local projection operators: [Π L ] µ ν = D µ 1 D ν, [Π L h] µ = D µ h Removes D-part from [Γ (2) k ]ij

71 p. 35/4 Step 3: Perturbative inversion of choose R k to regulate kinetic terms: [ Γ (2) k +R k ] ij K( ) P( ), following P k = +R k

72 p. 35/4 Step 3: Perturbative inversion of choose R k to regulate kinetic terms: [ Γ (2) k +R k ] ij K( ) P( ), following P k = +R k [ Γ (2) k +R k] ij operator-valued matrix in field space: [ ] Γ (2) ij k +R k = P 1½ 1 +M 1 M M P 2 ½ 2 +M 2 inversion: via formula for block matrices

73 p. 35/4 Step 3: Perturbative inversion of choose R k to regulate kinetic terms: [ Γ (2) k +R k ] ij K( ) P( ), following P k = +R k [ Γ (2) k +R k] ij operator-valued matrix in field space: [ ] Γ (2) ij k +R k = P 1½ 1 +M 1 M M P 2 ½ 2 +M 2 inversion: via formula for block matrices Perturbative expansion of inverse matrix elements in M: [Γ (2) +R k ] 1 11 = 1 P 1 1 P 1 M 1 1 P Two classes of traces: pure Laplacian and Laplacian + vertex insertions truncation expansion terminates at finite order:

74 p. 36/4 Step 4a: evaluate operator traces without insertions M Uses: standard heat-kernel expansion of Laplace-operators Laplace transform W( ) W(s) Tr i [W( )] = 0 [ Tr i e s ] 1 = (4πs) 2 ds W(s)Tr i [ e s ] d 4 x g [ tr i a 0 +str i a 2 +s 2 tr i a 4 + ],

75 p. 36/4 Step 4a: evaluate operator traces without insertions M Uses: standard heat-kernel expansion of Laplace-operators Laplace transform W( ) W(s) Tr i [W( )] = 0 [ Tr i e s ] 1 = (4πs) 2 ds W(s)Tr i [ e s ] d 4 x g [ tr i a 0 +str i a 2 +s 2 tr i a 4 + ], Coefficients for unconstrained fields: well known in math-literature tr 0 a 0 = 1, tr 0 a 2 = 1 6 R,...

76 p. 36/4 Step 4a: evaluate operator traces without insertions M Uses: standard heat-kernel expansion of Laplace-operators Laplace transform W( ) W(s) Tr i [W( )] = 0 [ Tr i e s ] 1 = (4πs) 2 ds W(s)Tr i [ e s ] d 4 x g [ tr i a 0 +str i a 2 +s 2 tr i a 4 + ], Coefficients for unconstrained fields: well known in math-literature tr 0 a 0 = 1, tr 0 a 2 = 1 6 R,... Coefficients for transverse fields: construct via projection operators [ Tr 1T e s ] [ Tr 1 e s ] Π T [ =Tr 1 e s ] +Tr [ 0 Dµ 1 D µ e s ]

77 p. 37/4 Step 4a: evaluate operator traces without insertions M Heat-kernel expansion for constrained fields on spherical topology Tr i [ e s ] = 1 (4πs) 2 d 4 x g [ c 1 +c 2 R+c 3 R 3 +c 4 R µν R µν +c 5 R µναβ R µναβ] T 2T c c c c c

78 p. 37/4 Step 4a: evaluate operator traces without insertions M Heat-kernel expansion for constrained fields on spherical topology Tr i [ e s ] = 1 (4πs) 2 d 4 x g [ c 1 +c 2 R+c 3 R 3 +c 4 R µν R µν +c 5 R µναβ R µναβ] T 2T c c c c c Straightforward to evaluate operator traces without vertices M

79 p. 38/4 Step 4b: Evaluate the operator traces including insertions M 1. use commutators to bring trace argument into standard form: contracted cov. derivatives: = collected into a single function W( ) remainder: = matrix-valued insertion O 2. Laplace transform W( ) W(s) Tr[W( )O] = 0 ds W(s) x e s O x

80 p. 38/4 Step 4b: Evaluate the operator traces including insertions M 1. use commutators to bring trace argument into standard form: contracted cov. derivatives: = collected into a single function W( ) remainder: = matrix-valued insertion O 2. Laplace transform W( ) W(s) Tr[W( )O] = 0 ds W(s) x e s O x 3. evaluate trace using off-diagonal Heat-kernel (act O on H) x Oe s x = x O x x e s x = d 4 x ḡtr i [ OH(s,x,x ) ] x=x H(s,x,x ) := x e s x = 1 σ(x,x ) (4πs) 2 e 2s s n A 2n (x,x ) n=0 A 2n (x,x ): heat-coefficients at non-coincident point 2σ(x,x ): geodesic distance between x,x

81 p. 39/4 Step 4b: Evaluate the operator traces including insertions M 3. evaluate trace using off-diagonal Heat-kernel (act O on H) x Oe s x = x O x x e s x = d 4 x [ ḡtr i OH(s,x,x ) ] x=x H(s,x,x ) := x e s x = 1 σ(x,x ) (4πs) 2 e 2s s n A 2n (x,x ) n=0 properties of H(s,x,x ) in the coincidence limit: A 2n (x,x) derivatives of A 2n σ(x,x) = 0, σ ;µ = 0 σ ;µν (x,x) = g µν (x) standard heat-kernel coefficients additional powers of curvatures vanish in coincidence limit non-vanishing

82 p. 39/4 Step 4b: Evaluate the operator traces including insertions M 3. evaluate trace using off-diagonal Heat-kernel (act O on H) x Oe s x = x O x x e s x = d 4 x [ ḡtr i OH(s,x,x ) ] x=x H(s,x,x ) := x e s x = 1 σ(x,x ) (4πs) 2 e 2s s n A 2n (x,x ) n=0 properties of H(s,x,x ) in the coincidence limit: A 2n (x,x) derivatives of A 2n σ(x,x) = 0, σ ;µ = 0 σ ;µν (x,x) = g µν (x) standard heat-kernel coefficients additional powers of curvatures vanish in coincidence limit non-vanishing example: O = R µν D µ D ν Tr[W( )O] = 1 32π 2 0 ds 1 s 3 W(s) d 4 x gr+o(r 2 )

83 p. 40/4 The algorithm on the computer mathematical input off-diagonal heat-kernel coefficients for unconstrained fields = heat-kernel coefficients for transverse fields

84 p. 40/4 The algorithm on the computer mathematical input off-diagonal heat-kernel coefficients for unconstrained fields = heat-kernel coefficients for transverse fields manipulation of traces including vertices commutation rules for covariant derivatives canonicalize trace-arguments including M apply off-diagonal heat-kernel

85 p. 40/4 The algorithm on the computer mathematical input off-diagonal heat-kernel coefficients for unconstrained fields = heat-kernel coefficients for transverse fields manipulation of traces including vertices commutation rules for covariant derivatives canonicalize trace-arguments including M apply off-diagonal heat-kernel bookkeeping add all terms occurring in the expansion

86 p. 40/4 The algorithm on the computer mathematical input off-diagonal heat-kernel coefficients for unconstrained fields = heat-kernel coefficients for transverse fields manipulation of traces including vertices commutation rules for covariant derivatives canonicalize trace-arguments including M apply off-diagonal heat-kernel bookkeeping add all terms occurring in the expansion implementation: in progress

87 p. 41/4 Summary: The Universal RG Machine Algorithmic expansion of FRGE: allows: systematic exploration of RG flows on theory space applicable to gauge theory, gravity,... realization on a computer algebra system

88 p. 41/4 Summary: The Universal RG Machine Algorithmic expansion of FRGE: allows: systematic exploration of RG flows on theory space applicable to gauge theory, gravity,... realization on a computer algebra system springboard for unveiling many physics features encoded by RG flow

89 p. 41/4 Summary: The Universal RG Machine Algorithmic expansion of FRGE: allows: systematic exploration of RG flows on theory space applicable to gauge theory, gravity,... realization on a computer algebra system springboard for unveiling many physics features encoded by RG flow first applications to gravity: EH-truncation: background-independence of β-functions ghost-wavefunction renormalization Confirms NGFP in full agreement with asymptotic safety