How to define a continuum limit in CDT?

Transcription

1 How to define a continuum limit in CDT? Andrzej Görlich Niels Bohr Institute, University of Copenhagen Kraków, May 8th, 2015 Work done in collaboration with: J. Ambjørn, J. Jurkiewicz, A. Kreienbühl, R. Loll and J. Studnicki

2 Outline Causal Dynamical Triangulations in Four Dimensions Phase diagram Renormalization group flow Anisotropic scaling and Hořava-Lifshitz gravity Conclusions

3 Gravitational path integral What is Causal Dynamical Triangulations? Causal Dynamical Triangulations (CDT) is a background independent approach to quantum gravity. CDT provides a lattice regularization of the formal gravitational path integral via a sum over causal triangulations. D[g]e iseh [g] continuous e SR [T ] T discrete

4 Regge action in four dimensions The Einstein-Hilbert action has a natural realization on piecewise linear geometries called Regge action S E [g] = 1 G dt d D x g(r 2Λ) S R [T ] = K 0 N 0 + K N + (N 1 6N 0 ) N 0 number of vertices N number of simplices N 1 number of simplices of type {1, } K 0 K bare coupling constants (G, Λ, α = a t /a s )

5 CDT in a nutshell To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The partition function of quantum gravity, defined as a formal integral over all geometries, is written as a nonperturbative sum over all causal triangulations. Due to global proper-time foliation, the Wick rotation is well defined (a t ia t ). Using Monte Carlo techniques we can approximate expectation values of observables. 2D 3D D

6 CDT in a nutshell To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The partition function of quantum gravity, defined as a formal integral over all geometries, is written as a nonperturbative sum over all causal triangulations. Due to global proper-time foliation, the Wick rotation is well defined (a t ia t ). Using Monte Carlo techniques we can approximate expectation values of observables. time

7 CDT in a nutshell To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The partition function of quantum gravity, defined as a formal integral over all geometries, is written as a nonperturbative sum over all causal triangulations. Due to global proper-time foliation, the Wick rotation is well defined (a t ia t ). Using Monte Carlo techniques we can approximate expectation values of observables. D[g]e iseh [g] T e isr [T ] T e SR [T ]

8 CDT in a nutshell To make sense of the gravitational path integral one uses the standard method of regularization - discretization. The partition function of quantum gravity, defined as a formal integral over all geometries, is written as a nonperturbative sum over all causal triangulations. Due to global proper-time foliation, the Wick rotation is well defined (a t ia t ). Using Monte Carlo techniques we can approximate expectation values of observables. O[T ] K 1 K i=1 O[T (i) ] Example observable is N(i), the number of tetrahedra building slice i, i = 1,..., T.

9 A glimpse at the phase diagram 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) Two coupling constants Tuned cosmological constant C A B Triple point K 0

10 A glimpse at the phase diagram 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) C A B Triple point K 0

11 A glimpse at the phase diagram 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) C A B Triple point K 0

12 A glimpse at the phase diagram 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) C A B Triple point K 0

13 A glimpse at the phase diagram 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) C A 0 B Second order phase transition line K 0 Triple point

14 De Sitter phase - properties Phase C, the so called De Sitter phase, is physically most interesting.

15 De Sitter phase - properties Phase C, the so called De Sitter phase, is physically most interesting. A four-dimensional background geometry emerges dynamically. The background geometry corresponds to Euclidean de Sitter space (S ), a classical vacuum solution.

16 De Sitter phase - properties Phase C, the so called De Sitter phase, is physically most interesting. A four-dimensional background geometry emerges dynamically. The background geometry corresponds to Euclidean de Sitter space (S ), a classical vacuum solution. It is also possible to study quantum fluctuations around it.

17 De Sitter phase - properties Phase C, the so called De Sitter phase, is physically most interesting. A four-dimensional background geometry emerges dynamically. The background geometry corresponds to Euclidean de Sitter space (S ), a classical vacuum solution. It is also possible to study quantum fluctuations around it. Consider the three-volume N(i) defined as the number of tetrahedra building slice i, i = 1,..., T. N = i N(i) Total four-volume

18 De Sitter phase - background geometry In phase C the time translation symmetry is spontaneously broken and the three-volume profile N(i) is bell-shaped. N(i) center of volume i N(i) N(i) Average profile Snapshot configuration

19 De Sitter phase - background geometry In phase C the time translation symmetry is spontaneously broken and the three-volume profile N(i) is bell-shaped. The average volume N(i) is with high accuracy given by formula ( ) i N(i) = H cos 3 W a classical vacuum solution. N(i) i N(i) δn(i) Average profile Quantum fluctuations

20 Continuum limit in lattice field theory Consider a field theory on a lattice with a bare mass m 0 and a bare coupling constant λ 0. Correspondingly, the renormalized mass m R and renormalized coupling constant λ R are fixed by an experiment. Given an observable O for which we can calculate a correlation length ξ, we have O(x n )O(x m ) e n m /ξ = e m R x n x m, x n = n a, where a is the lattice spacing, m R = (ξ a) 1 In order to define a continuum limit where a 0 we need a divergent correlation length ξ (expressed in lattice units). This happens in the vicinity of a second-order phase transition 1 ξ(λ 0 ) λ 0 λ 0 ν

21 Continuum limit in lattice field theory Consider a field theory on a lattice with a bare mass m 0 and a bare coupling constant λ 0. Correspondingly, the renormalized mass m R and renormalized coupling constant λ R are fixed by an experiment. Given an observable O for which we can calculate a correlation length ξ, we have O(x n )O(x m ) e n m /ξ = e m R x n x m, x n = n a, This defines a path in where a is the lattice spacing, (m 0, λ 0 ) which leads to an UV fixed point λ m R = (ξ a) if it exists - such that m R stays constant In order to define a continuum limit where a 0 we need a divergent correlation length ξ (expressed in lattice units). This happens in the vicinity of a second-order phase transition ξ(λ 0 ) 1 λ 0 λ 0 ν

22 De Sitter phase - scaling For different total volumes N, N(i) scales as a genuine four-dimensional Universe. ( ) N(i) = 3 ω N3/ cos 3 i ωn 1/ ( ) δn(i) = γn 1/2 i F ωn 1/ k 0k 80k 120k 160k k 0k 80k 120k 160k Fit N(i) v(t) i t

23 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω ) k 0k 80k 120k 160k Fit v(t) t

24 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω ) Expressed in lattice units. Directly measured. v(t) k 0k 80k 120k 160k Fit Physical quantities. Independent of cut-off a t

25 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i Proper time N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω ) k 0k 80k 120k 160k Fit v(t) t

26 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i Physical volume N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω ) k 0k 80k 120k 160k Fit v(t) t

27 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i Quantum fluctuations ) N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω k 0k 80k 120k 160k Fit v(t) t

28 De Sitter phase - scaling Scaling the lattice cut-off a with Hausdorff dimension d H =, a N 1/, gives physical quantities t and v(t). i t = N 1 / i Width, shape parameter N(i) v(t) = N 3 / N(i) = 3 ( t ω cos3 ω ( δn(i) δv(t) = N 3 / δn(i) = γn 1/ t ) F ω ) v(t) k 0k 80k 120k 160k Fit Amplitude of fluctuations. Note N 1/! t

29 Lines of constant physics Taking a continuum limit is achieved by sending the lattice spacing a 0, while keeping physical quantities fixed. We assume that the scaling holds everywhere in phase C, and that the three-volume profile v(t) as well as its fluctuations δv(t) are physical quantities. The width parameter ω and fluctuations γ are functions of the coupling constants, but do not depend on N, ω = ω(k 0, ), γ = γ(k 0, ) This leads us to the conditions for a path of constant physics: V = N a v(t) = 3 ( ) ω cos3 t ω δv(t) = γn 1/ F ( t ω ) fixed N + ω = const γ N 1/ = const

30 Lines of constant physics Taking a continuum limit is achieved by sending the lattice spacing a 0, while keeping physical quantities fixed. We assume that the scaling holds everywhere in phase C, and that the three-volume profile v(t) as well as its fluctuations δv(t) are physical quantities. The width parameter ω and fluctuations γ are functions of the coupling constants, but do not depend on N, Fixed total volume ω = ω(k 0, ), γ = γ(k 0, ) Fixed shape This leads us to the conditions Fixedfor amplitude a path of ofconstant fluctuations physics: (Newton s constant) N + ) fixed ω = const V = N a v(t) = 3 ( ) ω cos3 t ω δv(t) = γn 1/ F ( t ω γ N 1/ = const

31 Renormalization group flow To approach the continuum limit, we follow a path in the coupling constants plane (K 0, ) = (K 0 (N ), (N )), parametrized by N or a, such that physical properties are fixed. While the lattice spacing a goes to zero, the lattice size (number of simplices) has to grow to infinity (a N 1/ ). From the conditions, it follows that the fluctuation parameter γ has to approach infinity in the continuum limit. A second-order (or higher) phase transition is needed. Conditions for the line of constant physics ω(k 0 (N ), (N )) = const, γ(k 0 (N ), (N )) N 1/ N

32 Renormalization group flow To approach the continuum limit, we follow a path in the coupling constants plane (K 0, ) = (K 0 (N ), (N )), parametrized by N or a, such that physical properties are fixed. While the lattice spacing a goes to zero, the lattice size (number of simplices) has to grow to infinity (a N 1/ ). From the conditions, it follows that the fluctuation parameter γ has to approach infinity in the continuum limit. A second-order (or higher) phase transition is needed. Conditions for the line of constant physics ω(k 0 (N ), (N )) = const, γ(k 0 (N ), (N )) N 1/ N

33 Renormalization group flow Follow the constant physics line: K 0 (N ), (N ) ω contour plot ω(k 0, ) = const 0.8 γ(k 0, ) N 1/ V = N a V = const a N 1/ Δ Phase C Phase A Second order phase transition line K0 Phase B Triple point

34 Renormalization group flow Follow the constant physics line: K 0 (N ), (N ) ω contour plot ω(k 0, ) = const γ(k 0, ) N 1/ V = N a V = const a N 1/ Δ K0

35 Renormalization group flow Follow the constant physics line: K 0 (N ), (N ) γ contour plot ω(k 0, ) = const γ(k 0, ) N 1/ V = N a V = const a N 1/ Δ K0

36 Renormalization group flow Follow the constant physics line: K 0 (N ), (N ) γ contour plot ω(k 0, ) = const γ(k 0, ) N 1/ V = N a V = const a N 1/ Δ K0

37 Renormalization group flow Follow the constant physics line: K 0 (N ), (N ) γ contour plot ω(k 0, ) = const γ(k 0, ) N 1/ V = N a V = const a N 1/ Δ K0

38 Alternative approaches The presented recipe is not the only one possible. As the total four-volume V = N a is constant, it seems natural to require that δv(t) v(t) = δn(i) N(i) = const. This fixes the product ω(k 0, ) γ(k 0, ) N 1/, but doesn t fix the path in coupling constant plane. Previously, we have chosen additional condition ω(k 0, ) = const, but alternatively one could choose e.g. (by scaling the time coordinate) γ(k 0, ) = const

39 Alternative approaches The presented recipe ω(k 0 is, ) not the γ(konly 0, ) one Npossible. 1/ As the total four-volume V = N a ω γ contour is plot constant, it seems natural to require that δv(t) v(t) = δn(i) 0.8 N(i) = const. 0.6 This fixes the product Δ 0. ω(k 0, ) γ(k 0, ) N 1/, 1. but doesn t fix the path in coupling constant plane. 1 Previously, we 0.2 have chosen additional condition ω(k 0, ) = const, but alternatively one could choose e.g. (by scaling the time coordinate) 0.0 γ(k 0, ) = const K

40 Anisotropic scaling P. Hořava,,,Quantum Gravity at a Lifshitz Point, Phys.Rev.D79, (2009) Hořava proposed a non-relativistic power-counting renormalizable theory of gravitation, which reduces to General Relativity at large distances, and might provide a UV completion of Einstein s theory. So far, we assumed isotropic scaling between time and space. However, the CDT framework is richer, and might be viewed as a discretization not only of General Relativity but also Hořava-Lifshitz gravity (HLG). Common features of HLG and CDT are global time foliation (which breaks full four-dimensional diffeomorphism invariance) and unitarity. They show similar scale dependence of the spectral dimension and phase structure. The UV fixed point corresponds to an anisotropic Lifshitz scaling of space and time.

41 Hořava-Lifshitz gravity Hořava action Hořava modified the Einstein-Hilbert action by adding terms of higher order in the spatial derivatives of the metric, S H [g µν ] = 1 dtd D x gn { K ij K ij λk 2 V[g ij ] } G V[g ij ] = ωc ij C ij + + γr 2Λ K ij is the extrinsic curvature of the foliation leaves and C ij is the Cotton tensor. For λ = 1 and V = R 2Λ the Einstein action is recovered. Due to higher order spatial derivatives in V, in the UV limit there is an anisotropic Lifshitz scaling between space and time with dynamical critical exponent z: x bx, t b z t

42 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter

43 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter

44 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter dφ > 0, dt dg dt > 0

45 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter

46 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter Φ = 0, [g] = 0

47 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter

48 Lifshitz scalar phase diagram Hořava gravity is a generalization of Lifshitz model to gravity. The theory of Lifshitz scalar describes a tricritical phenomena, evinces an anisotropic behavior and spatially modulated phases. Identification of average geometry [g] with Lifshitz order parameter Φ > 0, [g] > 0

49 Modification of the RG flow We can view the B-C line in CDT as a second-order UV phase transition line for the HLG action. In this interpretation an anisotropy between space and time develops as one moves along flow lines. In such case, fixing the shape of the Universe is not equivalent to fixing the ω parameter, and the lines of constant physics will be modified. Although, the curves of constant ω eventually turn away from the triple point, it is also possible for flow lines to end up in the triple point. The triple point itself may represent the isotropic scaling.

50 Conclusions 1. In Causal Dynamical Triangulations a four-dimensional background geometry emerges dynamically. It corresponds to the Euclidean de Sitter space, i.e. classical solution of the minisuperspace model. 2. We have presented the most direct way of addressing the renormalization group flow in CDT via defining lines of constant physics. 3. There is, however, no unique prescription. Other aspects might influence the outcome: Inclusion of the asymmetry parameter α = at /a s which dependends on K 0 and. Anisotropic scenarios á la Hořava-Lifshitz Gravity. Different way of defining constant physics. Different scaling relations.. Existence of the new bifurcation phase should be also taken into account, while considering the coupling constant flow.

51 Thank you for your attention!

52 Lifshitz phase diagram and CDT 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) C A B Triple point K 0

53 Lifshitz phase diagram and CDT 0.8 S[T ] = K 0 N 0 + K N + (N 1 6N 0 ) Φ > 0, [g] > 0 C A dφ dt > 0, dg dt > B Φ = 0, [g] = 0 Triple point K 0

54 Effective action Simulations show that inside phase C, there exists an effective action for three-volume N(i), P({N(i)}) = 1 Z e S[N] S[N] = 1 ( (N(i + 1) N(i)) 2 Γ N(i + 1) + N(i) i ) + µn(i) 1/3 λn(i) It agrees with the discretization of the minisuperspace action S[v] = 1 v 2 G v + v 1/3 λv dt obtained from the Einstein-Hilbert action by,,freezing all degrees of freedom except the scale factor. Classical trajectory corresponds to Euclidean de Sitter space, v(t) = 3 ( t ) ω cos3 ω The form of S[v] gives the previously described scaling.

55 Effective action Simulations show that inside phase C, there exists an effective action for three-volume N(i), P({N(i)}) = 1 Z e S[N] S[N] = 1 ( (N(i + 1) N(i)) 2 Γ N(i + 1) + N(i) i ) + µn(i) 1/3 λn(i) It agrees with the discretization of the minisuperspace action S[v] = 1 v 2 Couples only adjacent G v + v 1/3 λv slices dt obtained from the Einstein-Hilbert action by,,freezing all degrees of freedom except the scale factor. Classical trajectory corresponds to Euclidean de Sitter space, v(t) = 3 ( t ) ω cos3 ω The form of S[v] gives the previously described scaling.

56 Effective transfer matrix The effective action suggests existence of an effective transfer matrix M labeled by the scale factor P({N(i)}) = 1 Z n M m = N e 1 Γ N(1) M N(2) N(2) M N(3) N(T ) M N(1) }{{} [ e S[N] ] (n m) 2 n+m +µ( n+m 2 ) 1/3 λ n+m 2 Potential term Product of matrix elements log n M n n

57 Effective transfer matrix The effective action suggests existence of an effective transfer matrix M labeled by the scale factor P({N(i)}) = 1 Z n M m = N e 1 Γ N(1) M N(2) N(2) M N(3) N(T ) M N(1) }{{} [ e S[N] ] (n m) 2 n+m +µ( n+m 2 ) 1/3 λ n+m 2 Kinetic term, gaussian Product of matrix elements c = 2000 c = 2800 c = n M c n n

58 New bifurcation phase? In phase C the cross-diagonals of the transfer matrix n M s n are given by a centered Gaussian function. In the bifurcation region the cross-diagonals split into a sum of two shifted Gaussians

59 New bifurcation phase? Recent results show between phase B and foregoing phase C, there is a new region with different behavior of the transfer matrix. In this region, the scaling arguments seem not to be valid in the whole range of volumes C A B K 0 Triple point

60 New bifurcation phase? Recent results show between phase B and foregoing phase C, there is a new region with different behavior of the transfer matrix. In this region, the scaling arguments seem not to be valid in the whole range of volumes C A 0.2 Bifurcation 0 B K 0 Quadruple point