Causal RG equation for Quantum Einstein Gravity

Transcription

1 Causal RG equation for Quantum Einstein Gravity Stefan Rechenberger Uni Mainz arxiv: v1 [hep-th] with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

2 Outline 1 Motivation 2 Causal Dynamical Triangulations and Horava Gravity 3 ADM decomposition 4 Causal RG equation 5 Results 6 Conclusion Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

3 Motivation Classical GR reaches its limits close to space-time singularities Black Holes Big Bang Solution probably lies within a theory of Quantum Gravity different approaches to a theory of QG String Theory Loop Quantum Gravity Causal Dynamical Triangulations Horava Gravity Asymptotic Safety... which one is correct? Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

4 lack of experimental data nobody helps us to decide which is the best approach what to do? Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

5 lack of experimental data nobody helps us to decide which is the best approach what to do? start crying and wait for data OR Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

6 lack of experimental data nobody helps us to decide which is the best approach what to do? start crying and wait for data OR compare different approaches Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

7 Causal Dynamical Triangulations (arxiv: v1 [hep-th]) discretization of gravitational path integral Dg µν e isgrav by summing over piecewise flat geometries modeling space-time geometries by gluing together simplices (higher dimensional generalizations of triangles) (a) (b) (arxiv:hep-th/ v1) curvature introduced via deficit angles pos./neg. deficit angles correspond to pos./neg. curved space y y δ x x δ (arxiv:hep-th/ v3) (a) (b) causal structure (equal-sided instead of equilateral triangles) Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

8 Horava Gravity (arxiv: v2 [hep-th]) different scaling of space and time x bx, t b z t Lorentz invariance is broken in UV but reestablished in IR foliation-preserving diffeomorphism invariance infinitesimal generators: δx i = ζ i (t,x), δt = f(t) Horava-like action: 1 S H = d 3 xdtn g 16πG N [ ] K ij K ij λk 2 γr (3) +2Λ+V(g ij ) maybe connection to CDT due to global time foliation (arxiv: v2 [hep-th]) Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

9 ADM decomposition time function... t(x µ ) µ t... future-dir. timelike VF x μ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

10 ADM decomposition Σ t2 time function... t(x µ ) µ t... future-dir. timelike VF Σ ti... spatial slices Σ t1 x μ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

11 ADM decomposition y a y a Σ t2 Σ t1 time function... t(x µ ) µ t... future-dir. timelike VF Σ ti... spatial slices coordinates y a on Σ ti x μ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

12 ADM decomposition γ(t) P 2 ya y a P 1 Σ t2 Σ t1 time function... t(x µ ) µ t... future-dir. timelike VF Σ ti... spatial slices coordinates y a on Σ ti parametrized curve γ(t) y a (P 1 ) = y a (P 2 ) x μ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

13 ADM decomposition γ(t) P 2 μ t Σ t2 t μ ea μ P 1 Σ t1 time function... t(x µ ) µ t... future-dir. timelike VF Σ ti... spatial slices coordinates y a on Σ ti parametrized curve γ(t) y a (P 1 ) = y a (P 2 ) ( ) t µ = γ µ t, y eµ a a = ( γ µ y a )t x μ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

14 ADM decomposition γ(t) time function... t(x µ ) µ t... future-dir. timelike VF Σ ti... spatial slices x μ P 2 μ t Σ t2 t μ ea μ P 1 Σ t1 coordinates y a on Σ ti parametrized curve γ(t) y a (P 1 ) = y a (P 2 ) ( ) t µ = γ µ t, y eµ a a = t µ = N 2 µ t +N a e a µ lapse function... N shift vector... N a ( γ µ y a )t Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

15 ADM decomposition change of coordinates: x µ (τ,y a ) old coordinates: ds 2 = γ µν dx µ dx ν new coordinates: ds 2 = N 2 dτ 2 +σ ij ( dy i +N i dτ )( dy j +N j dτ ) with induced spatial metric: σ ij = γ µν e µ i e ν j ( N γ µν = 2 +N i N i ) N j N i σ ij Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

16 ADM decomposition change of coordinates: x µ (τ,y a ) old coordinates: ds 2 = γ µν dx µ dx ν new coordinates: ds 2 = ǫn 2 dτ 2 +σ ij ( dy i +N i dτ )( dy j +N j dτ ) with induced spatial metric: σ ij = γ µν e µ i e ν j ( ǫn γ µν = 2 +N i N i ) N j N i σ ij Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

17 Causal RG equation Starting point: Einstein Hilbert action 1 S EH = d D x γ( R +2Λ) 16πG N G N... Newton constant D... space-time dimension (D = d +1) γ... metric R... curvature scalar of space-time Λ... cosmological constant g Λ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

18 Causal RG equation Starting point: Einstein Hilbert action 1 S EH = d D x γ( R +2Λ) 16πG N G N... Newton constant D... space-time dimension (D = d +1) γ... metric R... curvature scalar of space-time Λ... cosmological constant g Λ S ADM = ǫ 16πG N dτ d d xn [ σ {ǫ 1 K ij σ ik σ jl σ ij σ kl] } K kl R +2Λ Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

19 gauge fixing background field method first try: γ µν = ḡ µν +h µν, S k h µν R µνρσ k h ρσ but split nonlinear in N,N i,σ ij cutoff non-quadratic in N,N i,σ ij Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

20 gauge fixing background field method first try: γ µν = ḡ µν +h µν, S k h µν R µνρσ k h ρσ but split nonlinear in N,N i,σ ij cutoff non-quadratic in N,N i,σ ij ( ) ǫ 0 second try: special background ḡ µν = N = 1, N 0 σ i = 0 ij gauge freedom can be fixed with N = 0 = N i (arxiv:hep-th/ v2) purely spatial fluctuations h ij ghost action: S gh = ǫ dτ d d x σ { C τ C + C i τ C i} Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

21 Fourier decomposition in time direction time direction circle S 1 with interval length T φ(τ,x) = φ n (x)e 2πinτ/T φ n (x) = 1 T n= T 0 dτφ(τ,x)e 2πinτ/T T 0 T 0 dτ φ(x,τ) 2 = T dτ φ(x,τ) 2 τφ(x,τ) = T n= n= φ n (x)φ n (x), φ n (x) ( ) 2πn 2φ T n (x) Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

22 regulator and simplifications regulator: purely spatial R k ( ) with = σ ij Di Dj R k ( ) is given by implementing: +R k for simplicity optimized cutoff: R k = k 2( 1 ) ( ) k θ 1 2 k 2 further simplification: transverse-traceless decomposition h ij (x) + field redefinition no Jacobians { } hij T(x),ξT i (x),σ(x),h(x) Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

23 analytic structure k k Γ k = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] = T TT +T 0 Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

24 analytic structure k k Γ k = 1 2 STr [ ( Γ (2) k +R k ) 1k k R k ] = T TT +T 0 T TT = ǫk d d 2T (4π) d/2 n d d x σ [ q 1,0 d/2 (w 2T) + Rk 2( )] 1 6 q1,0 d/2 1 (w 2T) d2 3d+4 d(d 1) q2,0 d/2 (w 2T) T 0 = + d 2 2dλ k Rk 2 ǫk d (4π) d/2 n { d d x σ [ q 1,0 d/2 (w 0)+ 1 6 Rk 2 q 1,0 d/2 1 (w 0) q 2, 1 d/2 (w 0) q 1,0 d/2 (w 0)+ ( ) 3 2ǫ m2 n 2 4(d 3) d 2 λ k q 2,0 d/2 (w 0) }] Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

25 analytic structure Kaluza-Klein mass: m = 2π Tk carry out sums analytically: n n n 1 n 2 +x 2 = 1 n 2 +x 2 = q 1,0 d/2 (w 2T) n ǫ m2 n 2 2λ k π x tanh(πx), x2 > 0 (hyperbolic functions) π i x tan(πx), x2 < 0 (trigonometric functions) Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

26 flow equation k k g k = β g (g,λ;m), k k λ k = β λ (g,λ;m) m = 2π Tk analytic structure depends on signature and on λ ǫ λ < λ (1) < 0 λ (1) < λ < λ (2) = 1/2 λ (2) < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

27 Gaußian fixed point scenario m = 2π Tk T = const k k m k = m k m k = 0 in this limit all trigonometric functions diverge ǫ λ < λ (1) < 0 λ (1) < λ < λ (2) = 1/2 λ (2) < λ +1 hyperbolic mixture trigonometric 1 trigonometric mixture hyperbolic NGFP only for g < 0 in Euclidean signature! Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

28 floating point scenario m = const. T 1 k if m > 5/2 trigonometric terms stay finite for λ (1) < λ < λ (2) Example: m = 2π T = 1 k Non-Gaußian Fixed Point in Euclidean and Lorentzian signature ǫ g λ g λ θ 1, ± 3.31i ± 3.10i Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

29 Fixed Points depending on m g Λ Fixed-Point values depending on m Euclidean FP... blue Lorentzian FP... red both FPs converge for increasing Kaluza-Klein mass Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

30 Trajectories g Λ covariant calculation g 0.35 g Λ Λ Euclidean Lorentzian Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

31 Conclusion FP for Euclidean and Lorentzian signature characteristics are similar also similar to covariant formulation time circle collapses toward UV signature does NOT matter in UV formulation prepares ground for comparison to other theories Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26

32 Thank you for your attention! Questions? Stefan Rechenberger (Uni Mainz) Causal RGE for QEG / 26