Dual formulation of spin network evolution

TL;DR

This work develops a dual formulation that links spin networks to labelled spatial triangulations in $2+1$ and $3+1$ dimensions to study causal quantum-geometric evolution. It shows that spacetime evolution can be encoded by local amplitudes of spacetime simplices (triangles and tetrahedra in $2+1$, 4-simplices in $3+1$) and clarifies how spacelike and timelike slices and foliations arise, with amplitude structures tied to moves like $1-3$, $2-2$, and $3-1$ (and Pachner moves in higher dimensions). The dual picture helps address exponential growth by reintroducing the missing $3-1$ move or by adopting a non-maximal, multifingered-time evolution akin to directed percolation, providing a controlled, graphical framework for the dynamics. It also lays groundwork for connecting spin networks with simplicial gravity approaches (Regge calculus and spin foams), offering a concrete, combinatorial foundation for the evolution of quantum geometry.

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.

Figures (16)

• Figure 1: The 2-simplex triangulating 2-dimensional space and its dual trivalent spin network. The edges of the triangle are labelled by spins and its face by an intertwiner (omitted in an SU(2) spin network).
• Figure 2: A 2-dimensional spatial manifold can be described by a (labelled) triangulation (left), or its dual trivalent spin network (right).
• Figure 3: An example of a triangulated closed surface. It gives rise to a closed spin network.
• Figure 4: Part of a four-valent spin network, five nodes, corresponds to five teterahedra of a triangulation of 3-dimensional space.
• Figure 5: The labelling of a tetrahedron used in a triangulation of 3-dimensional space. Its faces are labelled by spins and the tetrahedron itself by an intertwiner.