March 16, 2020

*Answered by: Jonathan Gorard*

## How do your models relate to spin networks, spin foams and loop quantum gravity?

One can think of the Wolfram model as being a significant generalization of the concept of a spin network or spin foam in loop quantum gravity. In standard loop quantum gravity, a spin network is a combinatorial structure for representing the quantum state of a gravitational field on a three-dimensional spacelike hypersurface as a directed graph, whilst a spin foam is a higher-dimensional version of the same idea, with the directed graph now replaced with a topological 2-complex, representing the overall quantum state of the entire four-dimensional spacetime. In this way, spin networks and spin foams may be thought of as being directly analogous to spatial hypergraphs and (multiway) causal graphs for Wolfram model systems. However, the crucial distinction is that, in the case of spin networks, edges correspond explicitly to irreducible representations of some predefined compact Lie group (with vertices corresponding to the intertwiners of the adjacent representations), whereas the spatial hyperedges in the case of Wolfram model systems correspond to abstract relations between elementary elements, with no group structure explicitly defined (in other words, all salient algebraic and geometrical features of Wolfram model systems are purely emergent, as opposed to being “burned in” to the underlying combinatorial structure, as is the case in spin networks and spin foams).