An algebraic interpretation of the Wheeler-DeWitt equation

TL;DR

This paper reveals a deep algebraic link between the Wheeler-DeWitt equation in quantum gravity and tensor category associators, providing a new perspective on 3D quantum gravity models.

Contribution

It establishes a direct algebraic interpretation of the Wheeler-DeWitt equation using tensor categories and 6j-symbols, connecting quantum gravity with algebraic structures.

Findings

The Wheeler-DeWitt equation corresponds to the pentagon identity in tensor categories.

A key asymptotic formula relates 6j-symbols to rotation matrices.

This connection offers a new algebraic framework for 3D quantum gravity.

Abstract

We make a direct connection between the construction of three dimensional topological state sums from tensor categories and three dimensional quantum gravity by noting that the discrete version of the Wheeler-DeWitt equation is exactly the pentagon for the associator of the tensor category, the Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula relating 6j-symbols to rotation matrices given by Edmonds.