Paper detail

Generating Functionals for Spin Foam Amplitudes

We construct a generating functional for the exact evalutation of a coherent representation of spin network amplitudes. 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 newly recognized basis of SU(2) intertwiners consisting of the monomials of the holomorphic spinor invariants. This basis is labelled by the degrees of the monomials and is thus discrete. It is also overcomplete, but contains the precise amount of data to specify points in the classical space of closed polyhedra, and is in this sense coherent. We call this new basis the discrete-coherent basis. We focus our study on the 4-valent basis, which is the first non-trivial dimension, and is also the case of interest for Quantum Gravity. We find simple relations between the new basis, the orthonormal basis, and the coherent basis. Finally we discuss the process of coarse graining moves at the level of the generating functionals and give a general prescription for arbitrary graphs. A direct relation between the polynomial of cycles in the spin network generating functional and the high temperature loop expansion of the 2d Ising model is found.