Diffeomorphism invariant coordinate system on a CDT dynamical geometry with a toroidal topology

Transcription

1 1 / 32 Diffeomorphism invariant coordinate system on a CDT dynamical geometry with a toroidal topology Jerzy Jurkiewicz Marian Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland Quantum Gravity in Paris, March 2017

2 2 / 32 Outline Outline Causal Dynamical Triangulations Causal Dynamical Triangulations for S 3 X [0, 1]. Causal Dynamical Triangulations for T 4.

3 2 / 32 Outline Outline Causal Dynamical Triangulations Causal Dynamical Triangulations for S 3 X [0, 1]. Causal Dynamical Triangulations for T 4.

4 2 / 32 Outline Outline Causal Dynamical Triangulations Causal Dynamical Triangulations for S 3 X [0, 1]. Causal Dynamical Triangulations for T 4.

5 Introduction Causal Dynamical Triangulations CDT is based on the idea of Feynman path integral. The purpose is to calculate the quantum amplitude between the initial and final geometry. Two-dimensional example where initial and final states are represented by closed circular spatial World states. This is the analogue of a Feynman trajectory. The amplitude is a sum (integral) over a class of admissible trajectories (space-times). 3 / 32

6 Introduction Postulates The main tool will be numerical simulations. We assume that it is legitimate to consider only discretized (triangulated) geometries. We assume that it is possible to perform a Wick rotation to imaginary time. The basic assumption is causality, which means that the initial spatial topology is preserved in time evolution. We consider a class of geometries admitting a global time foliation. Each spatial slice has a fixed topology. The assumption of causality reduces drastically a class of admissible space-times included in the Feynman integral, permitting to avoid a number of problems related to the instability of the amplitude resulting from a baby-universe formation. 4 / 32

7 5 / 32 Introduction Triangulation of 3D spatial geometric states. CDT was studied for 2D (where it can be solved analytically) and 3D (where it was studied numerically). Let me come directly to the case of a 4D system, which is the most interesting from a physical point of view. In all cases discretization permits to label foliations by a discrete time variable assigned to vertices of a system. At each discrete time t we construct a 3D geometry by gluing together regular tetrahedra (3-simplices) with a universal edge length a s. We assume that 3D geometries have a simplicial manifold structure with a closed topology. In our earlier results we assumed that the spatial topology is S 3. Recently we extended our study to the case of a T 3 topology.

8 6 / 32 Introduction Connecting 3D states to obtain a 4D space-time. Elementary tetrahedra are glued along triangular faces. For a fixed topology a number of different inequivalent manifolds typically grows exponentially with the number of tetrahedra. Spatial states at time t have to be connected to states at t ± 1. Each tetrahedron becomes a base of {4, 1} and {1, 4} simplices with 4 vertices at time t and one at t ± 1. To close the manifold structure we need {3, 2} and {2, 3} simplices (3 vertices at t and 2 at t ± 1). The time edges have also a universal length a t = αa s.

9 7 / 32 Introduction Direct and dual lattice structure Simplices and subsimplices: In 4D each 4-simplex has 5 vertices (0-simplices), 10 edges (links - 1-simplices), 10 triangles (hinges - 2-simplices) and 5 faces (tetrahedra - 3-simplices). On a dual lattice each dual vertex is connected to 5 neighbouring dual vertices. Each d-simplex on a direct lattice can be assigned a corresponding (4-d)-simplex on a dual lattice. Each d-simplex has a 4-d topological ball surrounding it with a boundary being a (3-d)-sphere. We can measure ergodic distance between vertices both on a direct and on a dual lattice. We usually choose a dual lattice.

10 8 / 32 Introduction Simplex types Two basic types of simplices form a space-type manifold structure. Each space-time trajectory is characterized by a set of global numbers: N {41} 4, N {3,2} 4 corresponding to the number of {4, 1} and {3, 2} simplices. N 0 is a number of vertices. All other numbers of this type can be expressed as linear combinations of the three global numbers. Formulation is background independent, diffeomorphism invariant and nonperturbative. t+1 t

11 9 / 32 Introduction Hilbert-Einstein action. The quantum amplitude is represented as a weighted sum over all simplicial manifolds (trajectories) satisfying the boundary conditions. The trajectories will be weighted by a factor e S HE, where S HE is the discretized version of Hilbert-Einstein action S[g] = 1/G Curvature(g) + λ Volume(g) The curvature contribution can be expressed by a sum of deficit angles around the d 2-dimensional hinges (triangles). The simplicity of a construction leads to a discretized action parametrized by three dimensionless coupling constants κ 4 - bare cosmological constant, κ 0 related to the inverse Newton constant and depending on the ratio a t /a s. S HE = (κ )N 0 + κ 4 (N {4,1} 4 + N {3,2} 4 ) + N {4,1} 4

12 10 / 32 Introduction Monte Carlo simulations Starting from a very simple (small) initial configuration we perform a random walk in a configuration space. It is based on a set of 7 elementary moves (selected Pachner moves), which Preserve topology (and foliation). Constitute the ergodic set in the configuration space. Are performed with a probability determined by the detailed balance condition determined by the H-E action. Measurements are performed on a large but finite set of statistically independent configurations of space-time. In this approach a vacuum state is a result of a balance between the H-E action and the entropy of configurations quite distant from a naive classical solution of GR.

13 11 / 32 CDT results Phase diagram CDT amplitude is defined only for κ 4 > κ crit 4 (κ 0, ). The number of configurations with a fixed number N 4 of 4-simplices grows exponentially with N 4. For κ 4 κ crit 4 we approach an infinite volume limit where the average number of simplices becomes large (infinite). This is the limit we want to study. In numerical simulations (for practical reason) we replace it by a growing (but finite) sequence of N 4 and study the scaling properties as a function of N 4. Physical properties of the system depend in effect on two bare coupling constants, κ 0 and.

14 12 / 32 CDT results Phase diagram in a spherical case The phase diagram presented below shows results obtained by studying a sequence of systems with a spherical topology in spatial direction and periodic in (imaginary) time. The different phases appear as a result of a subtle balance between the H-E action and the entropy of configurations. Physically interesting are the regions near phase transitions, where a continuum limit can be studied (provided the phase transitions are higher order).

15 CDT results Phases We find a surprisingly reach and physically very interesting phase structure C A 0.2 Bifurcation 0 B κ 0 Quadruple point 13 / 32

16 14 / 32 CDT results Phase C - de Sitter phase It is not a simple problem to define physical observables in the background independent setup. The simplest to measure is a time distribution of spatial volume. Characteristic behaviour of a trajectory in phase C has a bump and a cut-off scale stalk resulting from periodic boundary conditions in imaginary time.

17 CDT results Distribution averaged over configurations and scaled. Averaging over trajectories one obtains a probability distribution of spatial volume in the blob. The dependence on N 4 can be scaled as a function of τ = (t t 0 )/N 1/d H 4, where d H is a Hausdorff dimension. t 0 is the center of volume necessary to freeze the translational symmetry in time. The solid line on the plot corresponds to d H = 4 and P(τ) cos 3 (ατ) v t 15 / 32

18 16 / 32 CDT results The effective mini-superspace action in the de Sitter phase. The plot presented above suggests that it represents a semiclassical solution obtained as a minimum of some effective action. This can be determined measuring the volume-volume covariance matrix. Volume as a function of time is related to the effective scale factor V 3 (τ) a 3 (τ). Inverting the measured covariance matrix we obtain a matrix of second derivatives of the effective action and in effect we can determine the effective action itself. Our result is (in a continuum version N 4 ): S eff = 1 ( T ( ) ) 1 dv3 (τ) 2 dτ + V 1/3 Γ 9V 3 (τ) dτ 3 (τ) λ 2 V 3 (τ) 0

19 17 / 32 CDT results Remarks The form is a well-known mini-superspace action by Hartle and Hawking. Here λ 2 is a Lagrange multiplier to enforce τ 0 dτv 3 (τ) = V 4 and Γ is the (dimensionless) effective Newton s constant. Observe that V 3 is a collective coordinate measuring the spatial volume at a given τ. It is averaged over all geometric realizations. The observed semi-classical distribution corresponds to a minimum of the effective action (in Euclidean time). There is no conformal factor instability, which is cured by the entropy of states.

20 18 / 32 CDT results Other phases Phases A and B seem to lack a physical interpretation. In phase A we loose a causal connection between neighbouring times. In phase B we observe a spotanous compactification of time. A transition between A and C is first order. Recently we discovered a new bifurcation phase with some fascinating properties, we do not (yet) completely understand. The phase transition to the C phase is higher order (probably second order) Phase transition to B phase is also higher order. A quadruple fixed point is a candidate for a non-gaussian fixed point of Quantum Gravity.

21 19 / 32 Space-time torus Toroidal topology Using spherical spatial topology has many advantages, but also disadvantages. The advantage is its relative simplicity. It is also a disadvantage: there are very few observables one can observe. Time plays a special role: it is used to foliate the space-time and is not a dynamical quantity. This is a variable closest to the semi-classical IR limit of the theory. Our numerical algorithm works independently of the choice of the spatial topology, it is only a matter of defining the initial configuration. Choosing this to be T 3 may be useful to address the non-trivial question: how a classical system of coordinates may at all emerge in a dynamical geometric setup.

22 20 / 32 Space-time torus First observations We use the initial topology based on a triangulation of a hypercube with 4 hypercubes in each direction and let it evolve using MC algorithm. We find that the toroidal system has the same phase structure as the spherical one although precise quantitative measurements are not finished. We concentrated on a study of the properties of the system in the range corresponding to phase C in one special point in the parameter space, the same where we made most of our measurements for the spherical system. We measured a distribution of the spatial volume as a function of imaginary time and determined the effective action using the covariance matrix of volume fluctuations.

23 21 / 32 Space-time torus Effective action for spatial volume fluctuations S eff = 1 Γ T 0 dτ ( 1 9V 3 (τ) ( dv3 (τ) dτ ) ) λ 2 V 3 (τ) There are small finite-volume corrections in a form of a volume-dependent potential with a negative power. The semi-classical minimum of this action with periodic boundary conditions is a constant. No curvature term in this case V 1/3 3 (τ).

24 22 / 32 Space-time torus Coordinates in the spatial directions On the initial hyper-cubic configuration of a torus we choose a 3D boundary separating elementary cells in the x direction. The boundary is represented as a thicker line. We let the geometry evolve. The presence of a boundary does not influence geometric updates. We add a condition that the boundary keeps a minimal volume during simulations (after each Pachner move).

25 23 / 32 Space-time torus Position of a minimal boundary A minimal boundary tries to follow a valley between volume mountains produced by geometric fluctuations.

26 24 / 32 Space-time torus Structure of a boundary between elementary cells The definition is not unique. For the initial configuration we define it as a 3D hypersurface with 4-simplices in the positive and negative direction sharing the face belonging to the boundary. This way we obtain a set of simplices lying to the left or to the right of the boundary. Simplices which have a face at a boundary (in a positive direction) are assigned a coordinate x = 0. Those in a negative direction are assigned a coordinate x = 0.

27 25 / 32 Space-time torus Determination of a distance from the boundary. Moving on a dual lattice we determine sets of simplices at a distance x = 1, 2,... from the initial set at x = 0 until we determine coordinate x of all simplices. We repeat the same moving in the negative direction measuring distances x = 1, 2,.... Following this procedure we assign to each simplex in the system two coordinates x and x. Notice that if the geometry would be that of a perfect torus x + x = L, where L would be a period in the x direction.

28 26 / 32 Space-time torus Distribution in x + x The distribution of the number of simplices labelled by x and x averaged over many configurations for a system with N {41} 4 = is presented below. The plot on the right shows that L is not a constant, but has a distribution similar to a normal distribution centered at some N 4 dependent value x+x'

29 27 / 32 Space-time torus Distribution in x x Below we show an averaged distribution in x x. Position of a boundary at x x = 0 is exactly equidistant from a boundary in positive and negative directions. It shows how the torus looks like in the spatial direction volume x-x'

30 28 / 32 Space-time torus Effective action in x x Preliminary results for the effective action in the spatial direction (inverse of the volume-volume covariance matrix) are very similar to that in the time direction, producing a pseudo-diagonal second derivative structure.

31 29 / 32 Space-time torus Final comments Similar definition of coordinates can be made in all spatial directions, which gives for a particular trajectory a distribution of a four-volume (number of simplices) as a function of semi-classical coordinates in spatial and time directions ( g(x, y, z, t)). We show the effect averaged over y, z, t. Preliminary results indicate that it may be possible to determine the effective action parametrized by semi-classical spatial coordinates, similar to that in the time direction. There are many numerical experiments necessary to determine the scaling properties.

32 30 / 32 Space-time torus Shifted boundary The definition of a boundary is not unique. We may start with a different position and compare distributions of volume obtained this way. The plot on the next transparency presents a distribution of simplices located on boundaries defined by a parallel shift of the initial boundary definition in the original hypercubic initial configuration by one, two and three units. In all cases the Monte Carlo evolution of a geometry is exactly the same, so the plots may be viewed as an effect of a translation in the x direction by one effective unit. The red plot on the right corresponds to a distribution of a distance between two periodically shifted boundaries shown before.

33 31 / 32 Space-time torus Plot of the shifted distributions d

34 32 / 32 Space-time torus Thank you