Asymptotic Safety in the ADM formalism of gravity

Size: px

Start display at page:

Download "Asymptotic Safety in the ADM formalism of gravity"

Ethel Wright
5 years ago
Views:

1 Asymptotic Safety in the ADM formalism of gravity 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, in preparation FRG2017 Heidelberg, March 10th, 2017 p. 1/33

2 p. 2/33 What s the name of the game? Goal: a consistent quantum field theory describing gravity on all scales

3 p. 3/33 What s the name of the game? Goal: a consistent quantum field theory describing gravity Nature on all scales

4 p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences

5 p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) 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

6 p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) 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,...)

7 p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) 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: background independence, unitarity,... resolution of singularities: (black holes, Landau poles,...)

8 p. 5/33 The Asymptotic Safety sloganizer Gravity rules!

9 p. 5/33 The Asymptotic Safety sloganizer Gravity rules! Matter matters!

10 p. 5/33 The Asymptotic Safety sloganizer Gravity rules! Matter matters! Time matters!

11 p. 6/33 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

12 p. 6/33 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

13 p. 6/33 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

14 p. 7/33 Quantizing general relativity [M. H. Goroff and A. Sagnotti, Phys. Lett. B160 (1985) 81] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] quantizing the Einstein-Hilbert action S = 1 16πG d 4 x gr has a two-loop divergence = hallmark of perturbative non-renormalizability S div = 1 ǫ (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν.

15 Quantizing general relativity [M. H. Goroff and A. Sagnotti, Phys. Lett. B160 (1985) 81] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] quantizing the Einstein-Hilbert action S = 1 16πG d 4 x gr has a two-loop divergence = hallmark of perturbative non-renormalizability S div = 1 ǫ (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν. the Goroff-Sagnotti challenge mastered: Γ k = 1 16πG k [H. Gies, S. Lippholdt, B. Knorr, F.S., Phys. Rev. Lett. 116 (2016) ] d 4 x g ( R+2Λ k )+ σ k d 4 x gc µν ρσ C ρσ αβ C αβ µν +... possess a NGFP with two relevant directions: NGFP GS : λ = 0.193, g = 0.707, σ = θ 1,2 = 1.475±3.043i, θ 3 = p. 7/33

16 p. 8/33 renormalization group flows in the presence of a foliation

17 p. 9/33 Technical routes towards asymptotic safety starting point: path integral over metrics Z = Dĝe S[ĝ] FRGE t Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1 t R k ] linear split: ĝ µν = ḡ µν +h µν exponential split: ĝ µν = ḡ µρ [ e h ] ρ ν

18 Technical routes towards asymptotic safety starting point: path integral over metrics Z = Dĝe S[ĝ] FRGE t Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1 t R k ] linear split: ĝ µν = ḡ µν +h µν exponential split: ĝ µν = ḡ µρ [ e h ] ρ ν Euclidean Dynamical Triangulations (EDT) Z = triangulations (detĝ) β e SEH [ĝ] Causal Dynamical Triangulations (CDT) Z = causal triangulations e SEH [ĝ] p. 9/33

19 p. 10/33 A fundamental puzzle FRGE: existence of NGFP well-established extension to gravity-matter models possible Causal Dynamical Triangulations (CDT): phase diagram contains 2nd order phase transition = possible continuum limit Euclidean Dynamical Triangulations (EDT): phase transition is 1st order = existence of continuum limit questionable

20 p. 10/33 A fundamental puzzle FRGE: existence of NGFP well-established extension to gravity-matter models possible Causal Dynamical Triangulations (CDT): phase diagram contains 2nd order phase transition = possible continuum limit Euclidean Dynamical Triangulations (EDT): phase transition is 1st order = existence of continuum limit questionable Conclusions: measure in partition function can manifestly change physics foliation structure underlying CDT is important

21 p. 10/33 A fundamental puzzle FRGE: existence of NGFP well-established extension to gravity-matter models possible Causal Dynamical Triangulations (CDT): phase diagram contains 2nd order phase transition = possible continuum limit Euclidean Dynamical Triangulations (EDT): phase transition is 1st order = existence of continuum limit questionable Conclusions: measure in partition function can manifestly change physics foliation structure underlying CDT is important construct a FRGE tailored to CDT

22 p. 11/33 renormalization group flows in the ADM-formalism

23 p. 12/33 Introducing time 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

24 p. 13/33 FRGE for the ADM-formalism fundamental fields: {N(t,x),N i (t,x),σ ij (t,x)} N = N + ˆN, N i = N i + ˆN i, σ ij = σ ij + ˆσ ij symmetry: general coordinate invariance inherited from γ µν : δγ µν = L v (γ µν ), v α = (f(t,x), ζ a (t,x)) induces δn = f t N +ζ k k N +N t f NN i i f, δn i = N i t f +N k N k i f +σ ki t ζ k +N k i ζ k +f t Ñ i +ζ k k N i +N 2 i f δσ ij = f t σ ij +ζ k k σ ij +N j i f +N i j f +σ jk i ζ k +σ ik j ζ k Non-linearity of ADM-decomposition: symmetry realized non-linearly = background Diff(M)-symmetry broken to foliation preserving diffeos [ ( ) ] k k Γ k [h µν ;ḡ µν ] = 1 2 STr Γ (2) 1k k k +R k R k

25 p. 14/33 renormalization group flows on cosmological backgrounds

26 p. 15/33 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-Robertson-Walker 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.

27 p. 15/33 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-Robertson-Walker 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,

28 p. 16/33 Gauge-fixing in the ADM formalism (flat space) δ 2 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

29 p. 17/33 Gauge-fixing in the ADM formalism (flat space) δ 2 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

30 p. 18/33 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-Robertson-Walker 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

31 p. 19/33 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

32 p. 20/33 Einstein-Hilbert-truncation on cosmological background J. Biemans, A. Platania and F. Saueressig, arxiv:

33 adding minimally coupled matter fields p. 21/33

34 p. 22/33 Minimally coupled matter fields in ADM Supplement Einstein-Hilbert action by N S minimally coupled scalar fields S scalar = 1 2 N S i=1 dtd d xn σ [ φ i 0 φ i] N V abelian gauge fields (gauge-fixed) S vector = 1 4 N V i= ξ N V i=1 dtd d xn σ [ g µν g αβ Fµα i ] Fi νβ dtd d x N σ [ N ḡ µν Dµ A i ] 2 V ν + i=1 dtd d x N σ [ Ci 0 C i] N D Dirac spinors S fermion =i N D i=1 dtd d xn σ [ ψi / ψ i].

35 p. 23/33 beta functions for the gravity-matter system β g =(2+η)g, β λ =(η 2)λ+ g 4π [ ( λ λ )( B det (λ) 1 η) ] 8+dλ 6 anomalous dimension of Newton s constant: 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) d g B 2 (λ) = (1 2λ) 5 36(1 2λ) λ 12B det (λ) λ+196λ2 144B det (λ) 2 Type I regulator: d g = N S N V N D, d λ = N S +2N V 4N D

36 Gravity-matter fixed points are quite common d g = N S N V N D, d λ = N S +2N V 4N D d g d blue: 1 NGFP black: 0 NGFPs red: 3 NGFP green: 2 NGFPs p. 24/33

37 most of them are UV fixed points d λ d g green: UV-FP: real exponents black: 0 NGFPs blue: UV-FP: complex exponents rest: saddle/ir fixed points p. 25/33

38 multi-fixed point systems exist 5 d λ d g filled symbols: 3 fixed point systems open symbols: 2 fixed point systems p. 26/33

39 p. 27/33 most matter sectors are in a good region! model N S N D N V g λ θ 1 θ 2 pure gravity ± 5.38 i Standard Model (SM) SM, dark matter (dm) SM, 3 ν SM, 3 ν, dm, axion MSSM SU(5) GUT SO(10) GUT

40 p. 28/33 renormalization group flows on CDT backgrounds S 1 S d

41 p. 29/33 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 cosmological perturbation theory 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

42 p. 29/33 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 cosmological perturbation theory 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

43 p. 30/33 Phase diagrams on S 1 S 3 Type I, linear g Type I, exponential g A A O C Λ O C Λ Type II, linear Type II, exponential g g A 1.0 A 0.5 B 0.5 O Λ O C Λ

44 p. 31/33 IR completion of the flow: Type II - linear 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!!!

45 outlook p. 32/33

46 p. 33/33 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) correlators for volume fluctuations: (δv 3 (t))(δv 3 (t )) S 1 S 3 S 1 T 3 now within reach of the FRGE based on the ADM-formalism

47 p. 33/33 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) correlators for volume fluctuations: (δv 3 (t))(δv 3 (t )) S 1 S 3 S 1 T 3 now within reach of the FRGE based on the ADM-formalism Thank you!

Asymptotically Safe Gravity connecting continuum and Monte Carlo

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