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Semifinite von Neumann algebras in gauge theory and gravity

Shadi Ali Ahmad, Marc S. Klinger, Simon Lin

TL;DR

The paper addresses divergences in quantum field theory arising from Type III von Neumann algebras by establishing a sufficient condition for the semifiniteness of crossed products with locally compact groups that contain the modular automorphism group. Using dual weights and Haagerup’s framework, it shows that when there exists a quasi-$\alpha$-invariant, KMS-weight on the original algebra and a central modular flow within the symmetry group, the crossed product admits a semifinite trace. A key outcome is that the modular flow must be central in the symmetry group under these conditions, with a constructive trace formula $\tau(X)=\tilde{\omega}(A^{-1}X)$. These results provide a firmer mathematical basis for gauge-invariant algebras in gauge theory and gravity, aiding the regulation of entropic observables and informing subregion physics.

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.

Semifinite von Neumann algebras in gauge theory and gravity

TL;DR

The paper addresses divergences in quantum field theory arising from Type III von Neumann algebras by establishing a sufficient condition for the semifiniteness of crossed products with locally compact groups that contain the modular automorphism group. Using dual weights and Haagerup’s framework, it shows that when there exists a quasi--invariant, KMS-weight on the original algebra and a central modular flow within the symmetry group, the crossed product admits a semifinite trace. A key outcome is that the modular flow must be central in the symmetry group under these conditions, with a constructive trace formula . These results provide a firmer mathematical basis for gauge-invariant algebras in gauge theory and gravity, aiding the regulation of entropic observables and informing subregion physics.

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.
Paper Structure (11 sections, 4 theorems, 46 equations)

This paper contains 11 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Modular Theory
  3. Crossed products with locally compact groups
  4. Semifiniteness of crossed products
  5. Constructing traces
  6. Discussion
  7. Haagerup form of the crossed product
  8. Modular automorphism of the Plancherel Weight
  9. A trace on the crossed product
  10. Dual modular action on generators of the crossed product
  11. Inner implementer of the dual automorphisms

Key Result

Theorem 1

bsphack @writeauxout \newlabelcommutationcurrentlabel esphack Let $\tilde{M} = M \otimes B(L^{2}(G))$ where $M$ is part of a dynamical system $(M,G,\alpha)$ and define by $\tilde{\alpha} := \alpha \otimes \textup{Ad} \lambda_{r}$ an extension of the automorphism $\alpha$ on $M$ to $\tilde{M}$ with $

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3: Crossed product
  • Theorem 1: Commutation theorem
  • Definition 4: Haagerup's algebra
  • Definition 5: Dual weight
  • Definition 6: Quasi-invariance
  • Theorem 2: Semifiniteness
  • proof
  • Lemma 1
  • ...and 5 more