The Crossed Product, Modular (Tomita) Dynamics and its Role in the Transition of Type $III$ to Type $II_{\infty}$ v.Neumann Algebras and Connections to Quantum Gravity

TL;DR

This paper investigates how the crossed product and modular dynamics facilitate the transition from type III to type II_infinity von Neumann algebras, linking this process to quantum gravity effects and the behavior of quantum fluctuations.

Contribution

It provides a novel interpretation of the modular evolution's role in the algebraic transition, connecting it to quantum gravity phenomena and analyzing changes in algebraic structures.

Findings

Modular dynamics influence the type transition of von Neumann algebras.

Quantum fluctuations are affected by modular evolution, suggesting a link to quantum gravity.

Properties of projectors and partial isometries change during the algebraic transition.

Abstract

We analyse the role of the crossed product and the modular (Tomita) dynamics in the transition of type $III$ to type $II_{\infty}$ v.Neumann algebras which was recently observed in papers by Witten et al. In a preceding paper we argued that type $II_{\infty}$ v.Neumann algebras display certain features which we attributed to quantum gravity effects. We claim that the action of the modular evolution on the quantum fluctuations can be understood as an aspect of quantum gravity. We mention in this context the work of Sakharov on induced gravity. Furthermore we analyse the change of the properties of projectors and partial isometries in the transition from type $III$ to type $II_{\infty}$.