Abstract
The evaluation of a relativistic spin network for the classical case of the Lie group SU(2) is given by an integral formula over copies of SU(2). For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometries for a 4-simplex.
Highlights
A relativistic spin network is a graph embedded in M3 with a non-negative integer, the spin, labelling each edge of the graph
An evaluation of relativistic spin networks was defined in [1], The evaluation gives an invariant of the isotopy class of the labelled embedded graph
In [1], we discussed the case of 4-valent graphs, giving a canonical formula
Summary
A relativistic spin network is a graph embedded in M3 with a non-negative integer, the spin, labelling each edge of the graph. An evaluation of relativistic spin networks was defined in [1], The evaluation gives an invariant of the isotopy class of the labelled embedded graph. An integral formula is presented for the evaluation in the classical case, which is when the parameter A is specialised to ±1. In this case, the invariant is a rational number which depends only on the graph and not on the embedding. Spin n corresponds to the representation of dimension n+1 These can be regarded as the irreducible representations of the compact Lie group SU(2), and so the classical invariant can be written in terms of this group
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