Generating Functionals for Spin Foam Amplitudes by Hnybida

This week I have been reading the PhD thesis ‘Generating Functionals for Spin Foam Amplitudes’ by Jeff Hnybida. This is a few useful topic because the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

In the various approaches to Quantum Gravity such as Loop Quantum Gravity, Spin Foam Models and Tensor-Group Field theories use invariant tensors on a group, called intertwiners, as the basic building block of transition amplitudes. For the group SU(2) the contraction of these intertwiners in the pattern of a graph produces spin network amplitudes.

In this paper a generating functional for the exact evaluation of a coherent representation of these spin network amplitudes is constructed. This generating functional is defined for arbitrary graphs and depends only on a pair of spinors for each edge. The generating functional is a meromorphic polynomial in the spinor invariants which is determined by the cycle structure of the graph.
The expansion of the spin network generating function is given in terms of a basis of SU(2) intertwiners consisting of the monomials of the holomorphic spinor invariants. This basis, the discrete-coherent basis, is labelled by the degrees of the monomials and is  discrete. It also contains the precise amount of data needed to specify points in the classical space of closed polyhedra.

The focus the paper is on the 4-valent basis, which is the case of interest for Quantum Gravity. Simple relations between the discrete-coherent basis, the orthonormal basis, and the coherent basis are found.

The 4-simplex amplitude in this basis depends on 20 spins and is referred to as the 20j symbol. The 20j symbol is the exact evaluation of the coherent 4-simplex amplitude. 


The asymptotic limit of the 20j symbol is found to give a generalization of the Regge action to Twisted Geometry.

3d quantum gravity

A triple of edge vectors meeting at a node must be invariant under the local rotational gauge transformations


There is only one invariant rank three tensor on SU(2) up to normalization: The Wigner 3j symbol or Clebsch-Gordan coefficient. The 3j symbol has the interpretation as a quantum triangle and its three spins correspond to the lengths of its three edges, which close to form a triangle due to the SU(2) invariance. Contracting four 3j symbols in the pattern of a tetrahedron gives the  well-known 6j symbol which is the amplitude for each tetrahedron.

Coherent BF Theory
The coherent intertwiners are  a coherent state representation of the space of invariant tensors on SU(2). The exact evaluations computed later are a result of a special exponentiating property of
coherent states. Each SU(2) coherent state is labelled by a spinor |z 〉,  |z] denotes its contragradient version. Using a bra-ket notation for the spinors


such that given two spinors z and w the two invariants which can be formed by contracting with either epsilon or delta are denoted


The exponentiating property of the coherent states corresponds to the fact that the spin j representation is simply the tensor product of 2j copies of the spinor |z〉⊗2j   . A coherent rank n tensor on SU(2) is therefore the tensor product of n exponentiated spinors.

To make the coherent tensor invariant we group average using the Haar measure


which is the denition of the Livine-Speziale intertwiner.

The coherent 6j symbol is constructed by contracting 4 coherent intertwiners in the pattern of a tetrahedron. Labeling each vertex by i = 1,..,4 and edges by pairs (ij) this amplitude depends on 6 spins jij = jji and 12 spinors |zij 〉≠ |zji〉 where the upper index denotes the vertex and the lower index the connected vertex. The coherent
amplitude in 3d is given by


The asymptotics of the coherent amplitude have been studied extensively, however the actual evaluation of these amplitudes was not known. While the asymptotic analysis is important to check the semi-classical limit, the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

To obtain the exact evaluation we use a special property of the Haar measure on SU(2) to express the group integrals above as Gaussian integrals. The generating functional is defined as


we are able to compute the Gaussian integrals in above, not just for the tetrahedral graph but for any arbitrary graph. Performing the Gaussian integrals produces a determinant depending purely on the spinors. The determinant can be evaluated in general and can be expressed in terms of loops of the spin network graph.

For example, after integration and evaluating the determinant, the generating functional of the 3-simplex takes the form:


4d quantum gravity

General Relativity in four dimensions is not topological, but it can  be formulated by a constrained four dimensional BF theory. That is if B is constrained to be of the form


for a real tetrad 1-form e then the BF action becomes the Hilbert Palatini action for General Relativity. The aim of the spin foam program is to formulate a discretized version of these constraints that can break the topological invariance of BF theory and give rise
to the local degrees of freedom of gravity.

The advantage of formulating GR as a constrained BF theory is that, instead of quantizing Plebanski’s action, we can instead use the topological nature of BF theory to quantize  the discretized BF action and impose the  discretized constraints at the quantum level.
The first model of this type was proposed by Barret and Crane.
While this is not a quantization of a constrained system in the sense of Dirac it is a quantization of the Gupta-Bleuler type which was realised by Livine and Speziale  and led to corrected versions of the Barret-Crane model by Engle, Livine, Pereira, Rovelli  and by Freidel, Krasnov.

The behaviour of our spin network generating functional under
general coarse graining moves is a simple transformation of the coarse grained action in terms of lattice paths. For a square lattice, the generating functional expressed as sums over loops similar to gives precisely the partition function for the 2d Ising model.


Since the Ising model and its renormalization are very well understood this example could provide a toy model for which one could base a study of the more complicated spin foam renormalization.

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