On the exact evaluation of spin networks
We introduce a fully coherent spin network amplitude whose expansion generates all SU spin networks associated with a given graph. We then give an explicit evaluation of this amplitude for an arbitrary graph. We show how this coherent amplitude can be obtained from the specialization of a generating functional obtained by the contraction of parametrized intertwiners à la Schwinger. We finally give the explicit evaluation of this generating functional for arbitrary graphs.
Spin networks consist of graphs labeled by representations of SU. They play a fundamental role in quantum gravity in two respects. First, they arise as a basis of states for 4 dimensional quantum gravity formulated in terms of Ashtekar-Barbero variables AL (). Second, in spin foam models the evaluations of spin networks appear in the expression of the quantum transition amplitude between boundary spin network states. In 3 dimensions the amplitude associated with a tetrahedron is the 6j symbol or the spin network evaluation of a tetrahedral graph PR (). Similarly, in 4 dimensions the amplitude associated with a 4-simplex is given by a product of two 15j-symbols with labels being related via the Immirzi parameter EPRL (); FK (). These results have triggered an extensive study of the properties of these symbols especially concerning their asymptotic properties. It has been known for a while that the 6j symbol is asymptotically related to the cosine of the Regge action, and moreover it has been shown more recently that these results extend to the 15j symbol BarrettAs1 (); BarrettAs2 (); FC1 (); FC2 (). In order to understand the asymptotic property of the spin network evaluation, a key insight was to express these amplitudes in terms of coherent states and express the amplitudes as a functional of spinor variables. Furthermore, the coherent state representation has been found to admit a corresponding geometrical interpretation in terms of polyhedral poly () and twisted geometries twistedgeo () where the spin labels represent the area of a polygonal face. It was also recognized that further simplifications of the amplitudes and other structures involving the coherent intertwiners could be achieved by summing the amplitudes over the spins with certain weight while keeping the total spin associated with each of the vertices fixed. In this case the amplitudes were found to exhibit an extra U symmetry that renders certain computations extremely efficient UN1 (); UN2 (); UN3 (). What we propose here is to go one step further and consider fully coherent spin network amplitudes obtained by summing over all spins with a specific weight. Such a proposal has also been developed recently by Livine and Dupuis LD () in order to write spin foam models more efficiently.
What we show is that by carefully choosing the weight of the spin network amplitudes we can compute them exactly for an arbitrary graph. This is our main result. These fully coherent amplitudes contain as their expansion coefficients the spin network evaluations for arbitrary spins and therefore can be understood as a generating functional for all spin networks evaluations. Similar generating functionals have been studied before, first by Schwinger Schwinger () and then further developed by Bargmann Bargmann () (see also Labarthe ()). Much later, an explicit evaluation of the generating functional for the chromatic (or Penrose Penrose ()) evaluation of a spin network on a planar trivalent graph was given by Westbury Westbury (). More recently, Garoufalidis et al. Garoufalidis () extended the evaluation of the chromatic generating functional to non planar graphs and Costantino and Marche Costantino () to the case where holonomies along the edges are present. In our case we focus on a slightly different generating functional that does not generate the chromatic evaluation but rather the usual spin network evaluation (i.e. the one obtained by the contraction of intertwiners). This evaluation differs from the chromatic one by an overall sign BarrettPR () and is much simpler in the non planar case. We present new techniques that allow this generating functional to be represented as a Gaussian integral and finally as the reciprocal of a polynomial. Furthermore, these results are valid for graphs of full generality such as those which are non-planar or of higher valency.
Ii Coherent evaluation of the vertex and general graphs
One of the key recent developments concerning spin foam amplitudes has been the ability to express them in terms of SU coherent states. In the following we denote the coherent states and their contragradient version by
The bracket between these two spinors is purely holomorphic and anti-symmetric with respect to the exchange of with . We will also denote the conjugate states by and .
The vertex amplitude for SU BF theory, expressed in terms of the SU coherent states, depends on spins and depends holomorphically on spinors ; it is based on a 4-simplex graph and is given by:
One of the key advantages of expressing the vertex amplitude in terms of coherent states is the ease with which to compute its asymptotic properties BarrettAs1 (); BarrettAs2 (); FC2 (). However, even if the asymptotic property of this amplitude is known, we do not know how to to compute it explicitly. What we are going to show is that by resumming these amplitudes in terms of a fully coherent amplitudes where is summed over, we can get an exact expression for the vertex amplitude.
Let us denote and define the following vertex amplitude which now depends only on the spinors:
Since it can be easily seen that such a series admits a non-zero radius of convergence. This amplitude can be thought of as a generating functional for the vertex amplitude , where the magnitudes of the determine which spin the amplitude is peaked on. Let us finally note that the particular set of coefficients we use in order to sum the coherent state amplitudes is motivated by the U perspective UN1 (); UN2 (). That is, if we denote by the SU coherent intertwiners and by the SU coherent states we have the relation
where is the coherent intertwiner LS (). The idea to use generalized coherent states which include the sum over all spins in a way compatible with the U symmetry, has been already proposed in UN3 () in order to treat all simplicity constraints arising in the spin foam formulation of gravity on the same footing. One ambiguity concerns the choice of the measure factor used to perform the summation over the total spin . The choice to sum states as differs form the one taken in UN3 (), but is ultimately justified for us by the fact that the spin network amplitude can be exactly evaluated.
The definition of the generally coherent amplitude is not limited to the 4-simplex, it can be extended to any spin network: More generally, let be an oriented graph with edges denoted by and vertices by . We assign two spinors to each oriented edge , one for and one for the reverse oriented edge . We also assign spins to every edge and define
where (resp. ) is the starting (resp. terminal vertex) of the edge .
Given this data we define a functional depending on and holomorphically on all given by:
where we define to be the set of edges of and the set of vertices. Finally, we introduce the following amplitude depending purely on the spinors
The main motivation for this definition comes from the fact that it can be explicitly evaluated, and this follows from the fact that this amplitude can be expressed as a Gaussian integral.
The fully coherent amplitude can be evaluated as a Gaussian integral
where ; and is a 2 by 2 matrix which vanishes if there is no edge between and . If is an edge of , is given by
This Gaussian integral can be evaluated giving
Given a group element we can construct a unit spinor where . Using the the decomposition of the identity , we can express the group product as
where we used the decomposition of the identity . On the other hand, given a spinor we can construct a group element , for which . Any function of and its conjugate can be viewed as a function of this element and therefore can be viewed, when restricted to unit spinors as a function on SU. Lets now suppose that is homogeneous of degree in , i.e. for . Then we can express the group integration as a Gaussian integral over spinors
where is the normalized Haar measure (for proof see the appendix). Therefore can be written as a Gaussian integral
Using the relation we can write the integrand as where the 2 by 2 matrix is given by
and vanishes if there is no edge between and . This Gaussian integral can be easily evaluated giving the determinant formula (10). ∎
We now want to evaluate this determinant explicitly. To do this we require the following definitions.
A loop of is a set of edges such that and . A simple loop of is a loop in which for , that is each edge enters at most once. A non trivial cycle of is a simple loop of in which for , i.e. it is a simple loop in which each vertex is traversed at most once. A disjoint cycle union of is a collection of non trivial cycles of which are pairwise disjoint (i.e. do not have any common edges or vertices). Given a non trivial cycle we define the quantity
where is the number of edges of whose orientation agrees with the chosen orientation of , and . Finally, given a disjoint cycle union we define
With these definitions we present the final expression for the vertex amplitude in the following theorem.
where the sum is over all disjoint cycle unions of .
The proof of this result is detailed in the appendix, and is due to the following special property of the matrix .
The Matrix defined in Eq. (9) is what we call a scalar loop matrix. That is for any collection of indices of where is the size of the quantity
is proportional to the identity.
This property allows us to prove the following lemma from which the theorem follows:
If is a scalar loop matrix composed of 2 by 2 block matrices then
where the sum is over all collections of pairwise disjoint cycles of which cover , and is the signature of viewed as a permutation of .
Evaluating this sum leads to our main theorem.
Let us first illustrate this theorem on one of the simplest graphs: the theta graph . This graph consists of two vertices with edges running between them. The amplitude for this graph depends on spinors denoted for the spinors attached to the first vertex and for the ones attached to the second vertex. The orientation of all the edges is directed from to where labels the edges of . For this graph the only cycles which have non-zero amplitudes are of length 2. Further, since there are only two vertices, each disjoint cycle union consists of a single nontrivial cycle. The amplitude associated to such a cycle going along the edge and then is given by
Therefore, from our general formula we have
We now illustrate the theorem for cases of the 3-simplex and the 4-simplex. In a -simplex there is exactly one oriented edge for any pair of vertices and so we can label cycles by sequences of vertices. We choose the orientation of the simplex to be such that positively oriented edges are given by for . Associated to the oriented edge we assign the spinors
Given a non trivial cycle of a -simplex we define its amplitude by
For the 3-simplex we have four non-trivial cycles of length and three non-trivial cycles of length . Since each of these cycles share a vertex or edge with every other, the only disjoint cycle unions are those which contain one non-trivial cycle. Therefore, after taking into account the sign convention the 3-simplex amplitude is given by
The sign in front of is determined in the following way. First, there is one which comes from the cycle union having one non trivial cycle and two because the non trivial cycle contains the two edges and which have a positive orientation. Thus the sign is negative.
For the 4-simplex we have ten 3-cycles, fifteen 4 cycles, and twelve 5 cycles and again the disjoint cycle unions consist of only single cycles. We define the 3-cycle amplitude to be
the 4-cycle amplitude to be
and the 5-cycle amplitude to be
Finally, the 4-simplex amplitude is given by
Iii Intertwiners and The vertex Amplitude
The goal of this section is to understand more deeply the relationship between the coherent evaluation of 3 and 4-valent graphs like the 3 and 4-simplex and the usual evaluation of spin network.
In order to express the coherent evaluation and , in terms of the 6j and 15j symbols respectively, we need to know the relationship between the coherent intertwiner and the normalised 3j symbol. This relationship is well known for 3-valent intertwiners Bargmann (); holomorph (); UN2 (); livine-bonzom (), however we will give an independent and elegant derivation that will allow us to understand this relationship in the unknown 4-valent case ( for an exception see UN2 ()).
iii.1 The n-valent intertwiner
It is well-known Schwinger (); Bargmann () that the spin representation can be understood in terms of holomorphic functions on spinor space which are homogeneous of degree . In this formulation a holomorphic and orthonormal basis corresponding to the diagonalisation of is given by
where are the components of the spinor . This basis is orthonormal with respect to the Gaussian measure
and is the Lebesgue measure on . In fact these basis elements are the bracket between the usual states and the coherent states
. In this representation it is straightforward to construct a basis of -valent intertwiners, i.e. functions of which are invariant under and homogeneous of degree in . A complete basis of these intertwiners is labeled by integers with and given by
where the sign factor is chosen for convenience111For instance in the trivalent case we take so that the ordering correspond to the cyclic ordering with instead of .. By homogeneity the integers must satisfy the conditions
and when these conditions are satisfied we write .
We now would like to understand the relationship between this basis of intertwiners and the coherent intertwiners, and in particular the scalar product between these states. In order to investigate this, let us introduce the normalised intertwiner basis
Intuitively, the theta graph consists of two -valent intertwiners with pairs of legs identified. Indeed, expanding the theta graph amplitude (21) in a power series yields an expression in terms of these intertwiners
This shows that is a generating functional for the -valent intertwiners. Given the definition (6) of the amplitude in terms of coherent intertwiners, this implies that
This shows that the relation between the coherent intertwiner and the normalised -valent intertwiner is given by
where we have introduce the state .
We now have to understand the normalization properties of . In order to do so, it is convenient to introduce another generating functional defined by
The remarkable fact about this generating functional, which follows from (36), is that it can be written as the evaluation of the following integral
We can now compute
where and in the second line we performed the Gaussian integral.
Using (38) to write this equality in terms of the intertwiner basis we get
where we have used that is the complex conjugate222 We have of . This shows that the combination
is a projector onto the space of SU intertwiners of spin .
In the case there is only one intertwiner. Indeed, given the homogeneity restriction requires which can be easily solved by
In this case the fact that is a projector implies that form an orthonormal basis, . In other word we can write
where the coefficients are the Wigner 3 symbols.
Using the relationship (37) between the normalised and coherent intertwiners and the definition (7) of the amplitude in terms of coherent intertwiners we can evaluate the 3-simplex amplitude in terms of the 6 symbol as
Here and this signs comes from the fact that the oriented graph for the 6 symbol differs from the generic orientation we have chosen by a change of order of the edge and (see e.g. LJ () for the definition of the 6). Note that it is also interesting to consider the amplitude
although this amplitude cannot be evaluated exactly, unlike . This amplitude does however possess interesting asymptotic properties.
Iv Generating Functionals
We would like now to provide a direct evaluation of the scalar product between two intertwiners. In order to do so we introduce the following generating functional which depends holomorphically on spinors and complex numbers
This functional was first consider by Schwinger Schwinger (). We now compute the scalar product between two such intertwiners
If we denote by and the two components of the spinor , and use that together with the antisymmetry of , this integral reads
with . We can easily integrate over , since the integrand is linear in and we obtain:
where and . In the case where this determinant can be explicitly evaluated and it is given by
In the case the explicit evaluation gives
Note that the Plücker identity tells us that when .
By expanding the LHS of (49) for
we see that the generating functional contains information about the scalar products of the new intertwiners. The property of this scalar product is studied in Ljeff ().
For general we notice that
and since is antisymmetric we can express the determinant as the square of a pfaffian as
where , up to , and is the submatrix of consisting of the rows and columns indexed by . In particular we have and for
By the pfaffian expansion formula for consists of terms, all of which contain a factor for some . For instance . Therefore if then we have relations in which case the scalar product has the form
where . This shows that when , we recover the amplitude we computed initially. This is not a coincidence, this is always true for any graph as we now show.
iv.1 General evaluation
Given an oriented graph we define a generating functional that depends holomorphically on parameters associated with a pair of edges meeting at .
where the integral is over one spinor per edge of and we integrate a product of intertwiners for each vertex . If is a -valent vertex with outgoing edges and incoming edges we define
We then have the following lemma