Semifinite von Neumann algebras in gauge theory and gravity

TL;DR

This paper investigates the conditions under which the crossed product of Type III von Neumann algebras with symmetry groups becomes semifinite, providing a framework to regulate divergences in gauge theories and gravity.

Contribution

It establishes a sufficient condition for semifiniteness of crossed products involving Type III algebras and symmetry groups, linking modular flow centrality to physical trace construction.

Findings

Identifies a sufficient condition for semifiniteness involving modular flow centrality.

Constructs a trace to compute physical expectation values in gauge theories.

Highlights implications for subregion physics 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 letter, we find a sufficient condition for the semifiniteness of the crossed product of a type III algebra with any 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 which computes physical expectation values. We comment on the importance of this result and and its implications for subregion physics in gauge theory and gravity.