Dual formulation of spin network evolution

TL;DR

This paper presents a dual formulation of spin network evolution using labelled triangulations, clarifying causal structure and addressing exponential growth issues, thus providing a more rigorous graphical framework for spin network dynamics.

Contribution

It introduces a dual representation of spin network evolution via labelled triangulations, linking it to simplicial gravity and clarifying causal and combinatorial structures.

Findings

Causal evolution rules for spin networks are clarified.

Exponential growth in the model is mitigated.

A more rigid graphical framework for spin network evolution is established.

Abstract

We illustrate the relationship between spin networks and their dual representation by labelled triangulations of space in 2+1 and 3+1 dimensions. We apply this to the recent proposal for causal evolution of spin networks. The result is labelled spatial triangulations evolving with transition amplitudes given by labelled spacetime simplices. The formalism is very similar to simplicial gravity, however, the triangulations represent combinatorics and not an approximation to the spatial manifold. The distinction between future and past nodes which can be ordered in causal sets also exists here. Spacelike and timelike slices can be defined and the foliation is allowed to vary. We clarify the choice of the two rules in the causal spin network evolution, and the assumption of trivalent spin networks for 2+1 spacetime dimensions and four-valent for 3+1. As a direct application, the problem of the exponential growth of the causal model is remedied. The result is a clear and more rigid graphical understanding of evolution of combinatorial spin networks, on which further work can be based.