Semifinite von Neumann algebras in gauge theory and gravity

Abstract

von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a useful tool for constructing gauge-invariant algebras. The crossed product of a Type III algebra with its modular automorphism group is semifinite, which means that the crossed product regulates divergences in local quantum field theories. In this article, we find a sufficient condition for the semifiniteness of the crossed product of a Type III algebra with locally compact group containing the modular automorphism group. Our condition surprisingly implies the centrality of the modular flow in the symmetry group, and we provide evidence for the necessity of this condition. Under these conditions, we construct an associated trace that computes physical expectation values. We comment on the importance of this result and its implications for subregion physics in gauge theory and gravity. Published by the American Physical Society 2025