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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