A new spinfoam vertex for quantum gravity by Livine and Speziale

In this paper the authors introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the 15j symbol, in a particular basis constructed using SU(2) coherent states. This basis makes the geometric interpretation of the variables transparent – they are the vectors normal to the triangles within each tetrahedron. They study the condition under which these states can be considered semiclassical, and show that the semiclassical ones dominate the evaluation of quantum correlations. Finally, they describe how the constraints reducing BF to gravity can be directly written in terms of the new variables, and how the semiclassicality of the states might improve understanding the correct way to implement  the constraints.

Introduction
The spinfoam formalism for loop quantum gravity is a covariant approach to  the definition of the dynamics of quantum General Relativity. It provides transition  amplitudes between spin network states. The most studied example in the literature is the
Barrett–Crane model. This model has interesting aspects, such as the inclusion  of Regge calculus in a precise way, but it can not be considered a complete proposal. In  particular, recent developments on the semiclassical limit show that it does not give the full correct dynamics for the free graviton propagator. In this paper the authors introduce a  new model that can be taken as the starting point for the definition of a better behaved  dynamics.

Most spinfoam models, including Barrett-Crane, are based on BF theory, a topological theory whose relevance for 4d quantum gravity has long been conjectured, and has been  exploited in a number of ways. In constructing a specific model for quantum gravity, there
is a key difficulty of quantum BF theory that has to be overcome: not all the variables  describing a classical geometry turn out to commute. This fact has two important  consequences.

The first is that there is in general no classical geometry associated to the spin network in the boundary of the spinfoam. This leads immediately to the problem of finding semiclassical quantum states that approximate a given classical geometry, in the sense in which wave packets or coherent states approximate classical configurations in ordinary quantum theory. This is the problem of defining ‘coherent states’ for LQG.

The second consequence concerns the definition of the dynamics. This is typically obtained by constraining the BF theory, a mechanism well understood classically, but still unsettled at the quantum level. The constraints involve non-commuting variables, and the way to
properly impose them is still an open issue. In particular, it can be argued that the specific procedure leading to the BC model imposes them too strongly, a fact which also complicates a proper match with the states of the canonical theory (LQG) living on the  boundary. These key difficulties are present for both lorentzian and euclidean signatures.

Consider a definition of the partition function of SU(2) BF theory on a Regge triangulation, where the dynamical variables entering the sum have a clear semiclassical interpretation: they are the normal vectors n associated to triangles t within a tetrahedron t.  The classical geometry of this discrete manifold (areas, angles, volumes, etc.) can be  described in a transparent way in these variables, provided they satisfy a constraint for each  tetrahedron of the triangulation. This constraint is the closure condition, that says that the  sum of the four normals associated to each tetrahedron must vanish. If this condition is  satisfied, this new choice of variables positively addresses the issues described above.

First of all, from the boundary point of view, each tetrahedron corresponds to a node of the  boundary spin network. The states associated to the new variables are linear superpositions of the conventional ones, with the property of minimizing the uncertainty of the non commuting operators: they thus provide a solution to the problem of  finding coherent states. Upon satisfying the closure condition, the states are coherent and carry a given classical geometry.

Secondly, as far as the dynamics is concerned, the constraints reducing BF theory to GR are functions of the full bivectorial structure of the B field, this structure is related to the vectors n in a precise way.

A key point of this approach is the closure condition. This is crucial for the semiclassical interpretation of the states. This is not satisfied by all the states entering the partition function. Yet one of the main results of this paper is to show that quantum correlations are
dominated by the semiclassical states in the large spin limit. The key to this mechanism lies in the fact that the coherent states introduced are not normalized, and the configurations maximizing the norm are the ones satisfying the closure condition – the configurations
satisfying the closure condition are exponentially dominating. The results obtained are valid for SU(2) BF theory, but can be straightforwardly extended to Spin(4), and the same logic
applied to any Lie group, including noncompact cases which are of interest for lorentzian signatures.

The new vertex amplitude
The vertex amplitude that characterizes this spinfoam model can be constructed using the conventional procedure for the spinfoam quantization of SU(2) BF theory.

Recall that there is a vector space

spinfoam vertex equ0
associated with each edge of the spinfoam. Here the half-integer (spin) j labels the irreducible representations of SU(2), and F is the number of faces around the  edge. If the spinfoam is defined on a Regge triangulation, F = 4 for every edge. The spinfoam quantization assigns to each edge an integral over the group, that is evaluated by
inserting in Ho the resolution of identity

spinfoam vertex equ1

The labels i are called intertwiners, and the sums run over all half-integer values allowed by the Clebsch-Gordan conditions. Taking into account the combinatorial structure of  the Regge triangulation (four triangles t in every tetrahedron t, five tetrahedra in every
4-simplex s), one ends up with the partition function,

spinfoam vertex equ2

where dj = 2j+ 1 is the dimension of the irrep j, and the vertex amplitude is As(j, i)= {15j}, a well known object from the recoupling theory of SU(2).

This partition function endows each 4-simplex with 15 quantum numbers, the ten irrep labels j and the five intertwiner labels i. Using LQG operators associated with the boundary of the 4-simplex, the ten jtcan be interpreted as areas of the ten triangles in the 4-simplex while the five it give 3d dihedral angles between triangles (one – out of the  possible six – for each tetrahedron). On the other hand, the complete characterization of  a classical geometry on the 3d boundary requires 20 parameters, such as the ten areas plus
two dihedral angles for each tetrahedron. Therefore the quantum numbers of the partition function are not enough to characterize a classical geometry. To overcome these difficulties, this paper writes the same partition function in different variables, which will endow each 4-simplex with enough geometric information.

The key observation is that SU(2) coherent states |j, n>, here n is a unit vector on the two- sphere S2, provide a (overcomplete) basis for the irreps. By group averaging the tensor  product of F coherent states, we obtain a vector

spinfoam vertex equ2a

in Ho. Here j and n denote the collections of all j’s and n’s. The set of all these vectors when varying the n’s forms an overcomplete basis in Ho. Using this basis the  resolution of the identity in Ho can be written as

spinfoam vertex equ3a
This formula can be used in the edge integration. The combinatorics is the same as before.  In particular, it assigns to a triangle t a normal n for each tetrahedron sharing t. Within a  single 4-simplex, there are two tetrahedra sharing a triangle, denoted u(t)and d(t).  Furthermore, it assigns two group integrals to each tetrahedron, one for each 4-simplex  sharing it. Taking also into account the dj factors and using the conventional spinfoam  procedure, gives

spinfoam vertex equ4
where the new vertex is

spinfoam vertex equ5

The new vertex gives a quantization of BF theory in terms of the variables j and n. Together,  these represent the full bivector Bi(x) discretized on triangles belonging to tetrahedra. Taking the B field constant on each tetrahedron, the discretization and quantization procedures can  be schematically summarized as follows,

spinfoam vertex equ6

In the quantum theory, the field Bi (x) is represented by a vector jn associated to each triangle in a given tetrahedron. The set of all these vectors can be used to describe a  classical discrete geometry.

On the other hand, the conventional quantization of BF theory differs in the quantization  step, which reads

spinfoam vertex equ7

where Jt are SU(2) generators associated to each triangle. Consequently the variables entering the partition function are irrep labels j and intertwiner labels i. The first variables  are related to discretization of the modulus of B, and thus to the area of triangles. The  intertwiner labels are related to one angle for each tetrahedron. Therefore in these  variables a classical geometry is hidden, and this in turn makes it hard to implement the  dynamics.

This vertex can be used directly for SU(2) BF theory, and straightforwardly generalized  to the Spin(4) case, exploiting the homomorphism Spin(4) = SU(2) ×SU(2).

spinfoam vertex equ8
The Barrett Crane vertex can be obtained in its integral representation as,

spinfoam vertex equ9

To understand the geometry of this new vertex amplitude, it is crucial to study its asymptotic behaviour in the large spin limit. The large spin limit is dominated by values of the group elements such that the factors of the integrand in

spinfoam vertex equ5

are close to one, namely such that
spinfoam vertex equ10

This has a compelling geometric interpretation. Recall that u(t) and d(t) are the normals to the same triangle as as seen from the two tetrahedra sharing it. They are changed   into one another by a gravitational holonomy. This is in contrast with the BC model, which fixes u(t) = d(t) thus not allowing any gravitational parallel transport between tetrahedra. This is another way of seeing the well known problem that the Barrett Crane model has not enough  degrees of freedom.

Coherent intertwiners
This section focuses on the building blocks of the new vertex, namely the states |j,n>  associated to the tetrahedra. Before studying the details of the mathematical  structure of these states, let us discuss their physical meaning. In the canonical picture, a  tetrahedron is dual to a 4-valent node in a spin network. More in general, when the edge
of the spinfoam is on the boundary, there is a one–to–one correspondence between the number of faces F around it, and the valence V of the node of the boundary spin network. Given a discrete atom of space dual to the node with V faces, its classical geometry can be entirely determined using the V areas and 2(V-3) angles between them. Alternatively, one can use the (non–unit) normal vectors n associated with the faces, constrained to close, namely to satisfy sum(ni) = 0.

On the other hand, the conventional basis used in gives quantum numbers for V areas but only V-3 angles. This is immediate from the fact that one associates SU(2) generators Ji with each face, and only V- 3 of the possible scalar products Ji · Jk commute among each
other. Thus half the classical angles are missing. To solve this problem, it is argued that the states |j,n> carry enough information to describe a classical geometry for the discrete atom of space dual to the node. In particular, we interpret the vectors j, n precisely with the
meaning of normal vectors n to the triangle, and we read the geometry off them.

This requires the vectors to satisfy:

spinfoam vertex equ11
This is a closure condition, if it holds, all classical geometric observables can be parametrized as O({j,n}).

To describe the mathematical details, recall basic properties of the SU(2) coherent states. A coherent state is obtained via the group action on the highest weight state,

spinfoam vertex equ12

where g(n) is a group element rotating the north pole z = (0,0,1) to the unit vector n (and such that its rotation axis is orthogonal to z). These coherent states are semiclassical in the sense that they localize the direction n of the angular momentum.

This localization property is preserved by the tensor product of irreps: in the the large spin limit we have for instance <Ji·Jk> ~ jijk ni·nk for any i and k. More in general any operator O({Ji })on the tensor product of coherent states satisfies:

spinfoam vertex equ13

with vanishing relative uncertainty.

So far, the n’s are completely free parameters. If the closure condition holds, then the states admit a semiclassical interpretation.

The partition function is the tensor product of coherent states projected on the invariant subspace Ho, in order to implement gauge invariance. This is achieved by group averaging. The resulting states written as a linear combination of the conventional basis of intertwiners are

spinfoam vertex equ14
We call these states coherent intertwiners. Projecting onto Ho does not force the V-simplex to close identically, but it does imply that closed simplices dominate the dynamics.
Four-valent case: the coherent tetrahedron

spinfoam vertex equ14

shows the construction of coherent intertwiners for nodes of generic valence. In the 4-valent case, whose dual geometric picture is a tetrahedron, is of particular interest as it enters the construction of vertex amplitudes for spinfoam models. The coherent tetrahedron can be decomposed in the conventional basis of virtual links, which are fixed by choosing to add J1 and J2 first.

Introducing the shorthand notation |i> = |ji,ni>, in the 4-valent case we have,

spinfoam vertex equ31

To study the asymptotics, it is convenient to introduce an auxiliary unit vector n, writing the character in the basis of coherent states,

spinfoam vertex equ32

This gives,

spinfoam vertex equ34

with,

spinfoam vertex equ35
The asymptotics of these coefficients are dominated by g and h close to the identity. Expanding S around p= q = 0 and denoting r = (p,q), we have

spinfoam vertex equ36

where N is the following 6-dimensional vector,

spinfoam vertex equ37

The Hessian matrix has the following structure,

spinfoam vertex equ38

 

where,

spinfoam vertex equ39-40

The asymptotics are given by

spinfoam vertex equ41

where det H can expressed in term of 3 × 3 determinants as,

For fixed ji, ni, this integral i represents the probability of the eigenstate j12 as a function of the ni’s. For closed configurations, we expect this to be a Gaussian peaked on the semiclassical value computed from the ni’s. Let’s consider for simplicity the equilateral case. In this case, we expect j12 to be peaked around j such that

spinfoam vertex equ41a
The diagram below shows that for large spin this is approximated by the Gaussian

spinfoam vertex equ42
where N(jo)is the normalization.

spinfoam vertex fig1

Analogous results hold for arbitrary closed configurations. This shows in a concrete way in which sense |j,n> represents a semiclassical state for a quantum tetrahedron, and more in general for a V-simplex dual to a node of valence V.

Towards the quantum gravity amplitude

How can  the coherent state presented here can be used to define the dynamics of quantum GR, starting from the spinfoam model?Spinfoam quantization of GR usually relies on reformulating GR as a constrained BF theory with an action of the following type

spinfoam vertex equ43

ω is a connection valued on a given Lie algebra (for euclidean signature, su(2) or spin(4)), and Bµ. is a bivector field (or two-form) with values in the same Lie algebra. The term C(B) includes polynomial constraints, reducing topological BF to GR. Typically, it gives the set of second class constraints expressing B in terms of the tetrad field (or vierbein)  and leading to GR in the first order formalism.

The BF theory can be quantized by discretizing the spacetime manifold with a Regge triangulation (or more generally a cellular decomposition), and then evaluating the partition function, where the variables are the representations j and intertwiners i . The natural extension of this procedure to quantize GR with action would be to discretize the constraints C(B) and include them in the computation of the discretized path integral.

 

The partition function for the quantum BF theory in terms of coherent states offers a natural way to impose the constraints on average. The key is that the partition function provides a quantization of BF theory where the B field is represented not through generators of the group, but as representation and
intertwiner labels.

The vertex is dominated in the large spin limit by semiclassical states
satisfying the closure condition for each tetrahedron and the relation between adjacent tetrahedra in the same 4-simplex. On these states the variables j and n give classical values with an (almost minimal) uncertainty decreasing as the jt’s increase.

If we use the j and n to construct a Regge geometry, we expect that the role of the constraints is to generate deficit angles when we glue together various 4-simplices, thus allowing the geometry to be curved.

Conclusions

The standard intertwiner basis which leads to BF theory with the vertex amplitude given by the {15j} symbol does not appear to be the most suitable one to study the semiclassical geometry of BF theory. Furthermore, it also makes it hard to understand the quantum
structure of the constraints reducing BF to GR.

This paper considered a basis constructed out of SU(2) coherent states. It defined nonnormalized coherent intertwiners, and studied their norm as a function of the geometric configuration. For each configuration, the norm is an integral over SU(2) that can be solved exactly.  Evaluation of the leading order of this integral in the large spin limit proves a very accurate approximation even for small spins, and shows very neatly that the norm is exponentially maximized
by the states admitting a semiclassical interpretation, namely the ones whose quantum numbers can be interpreted as vectors j, n describing the classical discrete geometry of a V -simplex. Due to this result, the semiclassical states will dominate the
evaluation of quantum correlations. Using these coherent intertwiners w the BF partition function can be rewritten with a new
vertex amplitude, where the discrete Bt(τ ) variables are interpreted in terms of the vectors j,n. This reformulation of the BF spinfoam amplitudes  improves the geometric interpretation of the theory.

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Numerical work in sage math 14: 1+1 dimensional Causal Dynamical Triangulation

In this post, a theory of quantum gravity called Causal Dynamical Triangulation (CDT) is explored. The 1+1 dimensional universe, the simplest case of an empty universe of one spatial and one temporal dimension is simulated in sagemath. This post explains CDT in general and presents and explains the results of the 1+1 dimensional simulations.

Understanding gravity at the fundamental level is key to a deeper understanding of the workings of the universe. The problem of unifying Einstein’s theory of General Relativity with Quantum Field Theory is an unsolved problem at the heart of understanding how gravity works at the fundamental level. Various attempts have been made so far at solving the problem. Such attempts include String Theory, Loop Quantum Gravity, Horava-Lifshitz gravity, Causal Dynamical Triangulation as well as others.

Causal Dynamical Triangulation is an to quantum gravity that recovers classical spacetime at large scales by enforcing causality at small scales. CDT combines quantum physics with general relativity in a Feynman sum over-geometries and converts the sum into a discrete statistical physics problem. I solve this problem using a Monte Carlo simulation to compute the spatial fluctuations of an empty universe with one space and one time dimensions. The results compare favourably with theory and provide an accessible but detailed introduction to quantum gravity via a simulation that runs on a computer.

In order to use the CDT approach, the Einstein-Hilbert action of General Relativity and the path integral approach to Quantum Field Theory are both important. I’ll begin by introducing both concepts as well as the metric and the Einstein Field equations. In this post I attempt, at least briefly, to explain CDT in general and explain what I have found with my simulation.

 Quantum gravity

Theories of quantum gravity attempt to unify quantum theory with general relativity, the theory of classical gravity as spacetime curvature. Superstring theory tries to unify gravity with the electromagnetic and weak and strong nuclear interactions, but it requires supersymmetry and higher dimensions, which are as yet unobserved. It proposes that elementary particles are vibrational modes of strings of the Planck length and classical spacetime is a coherent oscillation of graviton modes.

Loop quantum gravity does not attempt to unify gravity with the other forces, but it directly merges quantum theory and general relativity to conclude that space is granular at the Planck scale. It proposes that space is a superposition of networks of quantised loops and spacetime is a discrete history of such networks.

Causal Dynamical Triangulation is a conservative approach to quantum gravity that constructs spacetime from triangular-like building blocks by gluing their time-like edges in the same direction. The microscopic causality inherent in the resulting spacetime foliation ensures macroscopic space and time as we know it. Despite the discrete foundation, CDT does not necessarily imply that spacetime itself is discrete. It merely grows a combinatorial spacetime from the building blocks according to a propagator fashioned from a sum-over-histories superposition. Dynamically generating a classical universe from quantum fluctuations has been accomplished.CDT recovers classical gravity at large scales, but predicts that the number of dimensions drops continuously from 4 to 2 at small scales. Other approaches to quantum gravity have also predicted similar dimensional reductions from 4 to 2 near the Planck scale.

 Classical gravity

In the theory of relativity, space  and time  are on equal footing and mix when boosting between reference frames in relative motion. In the flat Minkowski spacetime of special relativity, the invariant proper space dr  and time ds  between two nearby events follow from the generalised pythagorean theorem.

2dcdtequ1which involves the difference in the squares of the relative space dx and time dt  between the events in some reference frame. This difference prevents causality violation, because light cones defined by dr = 0 or dt =+/-dx partition spacetime into invariant sets of past and future at each event. Free particles  follow straight trajectories or world-lines x[t] of stationary proper time. In the curved spacetime of general relativity, the separation between two events follows from

2dcdtequ2where the metric g  encodes the spacetime geometry. Free particles follow curved world lines of stationary proper time. The vacuum gravitational field equations can be derived from the Einstein-Hilbert scalar action

2dcdtequ3

where the Gaussian curvature is half the Ricci scalar curvature, K = R/2, and Lambda is the cosmological constant. By diagonalizing the metric, the invariant area element

2dcdtequ4

where g = det g.  Demanding that the action be stationary with respect to the metric, dS/dg = 0, implies the gravitational field equations

2dcdtequ5

where the Ricci tensor curvature is the variation of the Ricci scalar with the metric, R = dR/dg. The Einstein curvature G is proportional to the cosmological constant, which can be interpreted as the vacuum energy density.

Quantum mechanics

In classical mechanics, the action

2dcdtequ6

is the cumulative difference between a particle’s kinetic energy T and potential energy V[x]. A particle of mass m follows a worldline x[t] of stationary action. Demanding that the action be stationary with respect to the worldline, dS/dx = 0, implies Newton’s equation

2dcdtequ7

In quantum mechanics, a particle follows all worldlines. Along each worldline, it accumulates a complex amplitude whose modulus is unity and whose phase is the classical action S[x] in units of the quantum of action h. The Feynman propagator, or amplitude to transition from place a to place b, is the sum-over-histories superposition

2dcdtequ8

where Dx denotes the integration measure. The corresponding probability:

2dcdtequ9

is the absolute square of the amplitude. If the wave function 2dcdtequ9ais the amplitude to be at a place b, and the kinetic energy

2dcdtequ9b

then the path integral between infinitesimally separated places implies the nonrelativistic Schrodinger wave equation

2dcdtequ10

In quantum gravity, the corresponding sum is over all spacetime geometries and the quantum phase of each geometry is the Einstein-Hilbert action. The probability amplitude to transition from one spatial geometry to another is

2dcdtequ11If

2dcdtequ11ais the probability amplitude of a particular spatial geometry, then this path integral implies the timeless Wheeler-DeWitt equation:

2dcdtequ12

2dcdtfig3

 Gauss–Bonnet theorem

In 1 + 1 = 2 dimensions, the Gauss–Bonnet theorem  dramatically simplifies the Einstein-Hilbert action by relating the total curvature of an orientable closed surface to a topological invariant. The curvature of a polyhedron is concentrated at its corners where the deficit angle

2dcdtequ13and the total curvature

2dcdtequ14

where Kv is the discrete Gaussian curvature at vertex v, and av is the area closer to that vertex than any other. The Gaussian curvature of a circle of radius r is the reciprocal of its radius 1/r. The Gaussian curvature at a point on a surface is the product of the corresponding minimum and maximum sectional curvatures. Hence, the total curvature of a sphere

2dcdtequ15

like the topologically equivalent cube. More generally,

2dcdtequ16

where X is the surface’s Euler characteristic and the genus G is its number of holes. For a sphere G = 0, and for a torus G = 1. Total curvature can only change discretely and only by changing the number of holes.

Wick rotation

The sum over histories path integral is difficult to evaluate because of the oscillatory nature of its integrand. A Wick rotation converts this difficult problem in Minkowski spacetime to a simpler one in Euclidean space by introducing an imaginary time coordinate. For example,

2dcdtequ17

One motivation for the Wick rotation comes from complex analysis, where the integral of an analytic function f[z] around a closed curve in the complex plane always vanishes. If the function also decreases sufficiently rapidly at infinity, then around the contour C,

2dcdtequ18

and so

2dcdtequ19

which effectively rotates the integration 90o in the complex plane. Multiplying a complex number by similarly rotates the number 90o in the complex plane. The Wick rotation maps the line element by

2dcdtequ20

and the area element by

2dcdtequ21

This gives the  Einstein-Hilbert action by

2dcdtequ22

and the the probability amplitude by

2dcdtequ23

This Wick rotation converts the Feynman phase factor to the Boltzmann weight thereby connecting quantum and statistical mechanics, where toroidal boundary conditions and a positive cosmological constant Lambda > 0 ensure the negativity of the Euclidean action Se < 0.

Regge calculus

Triangulation can simplify the continuum to the discrete. The equilateral Euclidean triangle has height

2dcdtequ24

and area

2dcdtequ25

fig4

Regge calculus approximates curved spacetime by flat Minkowski triangles (or simplexes in higher dimensions) whose edge lengths ‘ may ultimately be shrunk to zero to recover a continuum theory. Curvature is concentrated in the deficit angles at the vertices.Dynamical Triangulation uses only equilateral triangles, but incorporates both positive and negative curvature by suitably gluing the triangles together. Causal Dynamical Triangulations uses only dynamical triangulations with a sliced or foliated structure with the time-like edges glued in the same direction

2dcdtfig6

Local light cones match to enforce global causality and preclude closed time-like curves. Regge calculus and the Gauss–Bonnet theorem dramatically simplifies the Wick-rotated Euclidean action from

2dcdtequ28

to

2dcdtequ29

where Nis the number of triangles. Assuming periodic boundary conditions in space and time, so the model universe is toroidal with genus G =1 the Euler characteristic X =2 (G-1) and the action

2dcdtequ30

where the rescaled cosmological constant

2dcdtequ31

The integral in amplitude becomes the sums

2dcdtequ32

where the number of triangles is twice the number of vertices, N = 2Nv,

The Simulation

Monte Carlo analysis

To analyse the statistical physics system defined by the  effective partition function, take a random walk through triangulation space sampling different geometries. The 1-2 move and its inverse 2-1 are ergodic triangulation rearrangements in 1+1 dimensions (these Pachner moves are 2-2 and 1-3 in 2d).They are special cases of the Alexander moves of combinatorial topology. The move M  splits a vertex into two at the same time slice and adds two triangles, while its inverse M_1 fuses two vertices at the same time slice and deletes two triangles. Both preserve the foliation of the spacetime. Monte Carlo sampling of the triangulations leads to a detailed balance in which moves equilibrate inverse moves. At equilibrium, the probability to be in a specific labelled triangulation of Nv+1 vertices times satisfies

2dcdtequ33

After choosing to make a move, randomly select a vertex from Nv vertices, and then randomly split one of np  past time-like edges and one of nf future time-like edges. Hence the move transition probability

2dcdtequ34

After choosing to make an inverse move, fusing the split vertices is the only way to return to the original triangulation. Hence the inverse move transition probability

2dcdtequ35

By the effective partition function, the probability of a labelled triangulation with Nv vertices is the weighted Boltzmann factor

2dcdtequ36

The probability of a move is a non-unique choice. In my simulations, I choose

2dcdtequ37

To satisfy the detailed balance, the probability of an inverse move is therefore

2dcdtequ39

Other choices may improve or worsen computational efficiency]. In practice, a Metropolis algorithm  accepts the rearrangements if these probabilities are greater than a uniformly distributed pseudo-random number between 0 and 1. For example, if P[M]=0.01, then the move is unlikely to be accepted.

In the initial spacetime configuration we have the universe before any move or antimove as shown below

2dcdtfig2-initial set up of universe

In this simulation, Ionly looked at the 1+1 dimensional universe. In this simulation, the 1+1 dimensional universe is decomposed into 2 dimensional simplices, one with two time-like edges and a space-like edge. In the simulation, the data structure stores this kind of information which turns out

to be very useful to implement the simulation. The up pointing triangles have their spatial edges on a past time slice and the down pointing triangles have their spatial edges on a future time slice. Those time-like edges are always future pointing. In this simulation, periodic boundary conditions were chosen. What this means is that the final time-slice and the first time-slice are connected so that every vertex of the beginning time-slice are identifed with every vertex on the finnal time-slice. The toroidal topology S1 x S1 was used to achieve the periodic boundary conditions.

In my simulation, I used a data structure that stores certain information about each triangle in the triangulation. This information is stored in an array. In this array, the following information

about the triangulation is stored: type of triangle (that is, if it is up pointing or down pointing), the time slice which it is located, the vertices as well as the neighbors of the triangles. Using a labeling scheme I was able to store information about the triangles in a triangulation. Each triangle was given a key starting from 0 to n-1. The type of the triangle (which is either type I or type II) was also stored. The neighbors for the different triangles were also stored. In general, the array structure for any triangle takes the form Tn = [type; time; p1; p2; p3; n1; n2; n3] where p1; p2; p3 are the vertices of the triangle and n1; n2; n3 are the neighbor of vertex 1, neighbor of vertex 2 and neighbor of vertex 3 respectively. This entire structure is composed strictly of integers. The integer assignments are in linewith the idea of reducing the problem to a counting problem. This data structure is then taken advantage of to do the combinatorial moves to split a vertex and add two triangles and antimoves to remove two triangles. The moves are done by randomly picking a vertex and splitting it, adding two triangles in the gap left behind where ever the vertex was split. The antimoves involve randomly picking vertices and deleting the triangles associated with the vertex. This is the same thing as doing a random walk through space-time. When the moves and antimoves are done, the arrays are repeatedly updated for every move and antimove. The universe is then made to evolve by repeatedly doing moves and antimoves rapidly and the size of the universe is measured every time over every 100 such combinatorial moves.

2dcdtfig5

Initial runs of the simulation give results such as those below:

2dcdtfig1-quantum fluctuations in universe size

model universe gif v1

2dcdt

 

 

 

In a  further post I will be exploring the critical value of the reduced cosmological constant.

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Quantum geometry from phase space reduction by Conrady and Freidel

In this paper the authors give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin–Sternberg and Hall that describe the commutation of quantization and reduction.This resulted to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

Introduction
Recently following the proposal of new spin foam models for gravity a number of important developments have occurred. These models allow the inclusion of a nontrivial Immirzi parameter and have been shown to satisfy two very important consistency requirements:
firstly, in the semiclassical limit they are asymptotically equivalent to the usual Regge discretisation of gravity, independently of the complexity of the underlying cell complex. Secondly, they have been shown to possess SU(2) spin network states as boundary states.

These new developments link with the evolution of the understanding of the simplicity constraints. These simplicity constraints state in any dimension that the bivector used to
contract the curvature tensor in the Einstein action comes from a frame field. In a simplicial context this constraint splits into three classes: there are the face simplicity constraints (among bivectors associated to the same face), the cross–simplicity constraints (among
bivectors associated to faces of the same tetrahedron) and the volume constraints (among opposite bivectors). All these constraints are quadratic, but the volume constraint, because it depends on different tetrahedra, also involves the connection and is therefore extremely hard to quantise. The first major advance came from the work of Baez, Barrett and Crane who showed that it is possible to linearise the volume constraint and replace it by a constraint living only at one tetrahedron. This new constraint is the closure
constraint and the discrete analogue of the Gauss law generating gauge transformations. The second key insight in this direction came from the work of Engle, Pereira and Rovelli who showed that one can linearise the simplicity and cross simplicity constraints, opening
the path toward a new way of constructing spin foam models and allowing the incorporation of the Immirzi parameter.
The purpose of the paper is to show that the new spin foam model can be written explicitly in terms of a sum of amplitudes, where all simplicity constraints are imposed strongly in the path integral, and where all the boundary spin networks have a geometrical interpretation even before taking the semiclassical limit.

Along with the construction of new models there is also a merging of two lines of thought on spin foam models that developed in parallel for a long time and are now  intersecting.

One line of thought, can be traced back to the seminal work of Barbieri – see post: ‘Quantum tetrahedra and simplicial spin networks, who realized that spin network states of loop quantum gravity can be understood by applying the quantization procedure to a collection of geometric tetrahedra in 3 spatial dimensions. This important work suggested that spin network states and quantum gravity may be about quantizing geometric structures. This idea was then quickly applied to the problem of quantization of a geometric simplex in R4, with the result being the construction of the Barrett–Crane model. This line of thought was also developed by Engle, Livine, Pereira, Rovelli. The main problem in this approach is that the geometry associated with the quantum tetrahedra is fuzzy and cannot be resolved sharply due to the noncommutativity of the geometric ‘flux’ operators. This problem prevents a priori a sharp coupling between neighboring 4d building blocks, and that was the main issue with the Barrett-Crane model.

The second line of thought can be traced back to the work of Reisenberger who proposed to think about quantum gravity directly in terms of a path integral approach in which we integrate over classical configurations living on a 2–dimensional spine and where
spin foam models arise from a type of bulk discretization. This line of reasoning is related to the Plebanski reformulation of gravity as a constraint SO(4) BF theory. In this approach it is quite easy to get a consistent gluing condition, however, it is much less trivial to find models which have a natural and simple algebraic expression in terms
of recoupling coefficients and it is even more of a challenge to get SU(2) boundary spin networks as boundary states.

The merging between these two approaches starts with a work of Livine and Speziale who proposed to label spin network states not with the usual intertwiner basis, but with ‘coherent intertwiners’ that are labelled by four vectors whose norm is fixed to be the area
of the faces of the tetrahedron. This has led to a very efficient and geometrical way of deriving the new spin foam models
There is however a caveat – the coherent intertwiner resolves the fuzzyness of the quantum space of a tetrahedron by construction, but the classical configuration no longer satisfies the closure constraint unless one takes the semiclassical limit and it is therefore not geometrical.

The resolution of these problems proposed by the authors in ths paper here is to use coherent states that are associated to the geometrical configuration – see post:’ A semiclassical tetrahedron‘ and relate these ‘geometrical’ states to the usual spin network
states or the coherent intertwiner states.

This paper gives a construction of such states and shows that they form an over-complete basis of the space of four–valent intertwiners and gives the explicit isomorphism between this new basis of states, where the closure constraint is imposed strongly, and the coherent intertwiner basis. The proof of this isomorphism amounts essentially to showing  that ‘quantisation commutes with reduction’ and utilises heavily the work of Guillemin and  Sternberg on geometrical quantisation.

The main formula of the paper expresses the decomposition of the identity in the space of four–valent intertwiners Hj= (Vj1 .· · · .Vj4)as an integral over coherent states satisfying the closure constraint, and so as an integral over the space of classical tetrahedra:

quantum geometry

 

This identity can be applied it to the spin foam models: it allows the formulation the path integrals in terms of variables on which the closure constraints are imposed strongly. These variables are well–suited to analyze the asymptotic large spin behaviour of amplitudes, and are used in this paper to derive the asymptotics of the FK vertex amplitude.

The paper describes in detail the phase space of the classical tetrahedron and shows that it can be obtained as a symplectic quotient. It states the classical part of the Guillemin–Sternberg isomorphism, which is a isomorphism between a constrained phase space divided by the action of the gauge group and the unconstrained phase space divided by the complexification of the group. It also gives a detailed discussion of coherent states and their link with geometrical quantisation and a construction of the isomorphism between the new geometrical basis and the coherent intertwiner basis.This enables the FK. spin foam model to be written in terms of the new tetrahedral states and use this to derive the asymptotics of the vertex amplitude.

 

 

 

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