Stopping the CCR flow and its isometric cocycles

TL;DR

This paper introduces a method using non-commutative stopping times to halt the CCR flow and its isometric cocycles, establishing a connection between stopping times and endomorphisms of operator algebras.

Contribution

It demonstrates how to stop CCR flows and cocycles using non-commutative stopping times, creating a new link between stopping time semigroups and operator algebra endomorphisms.

Findings

Stopped CCR flows produce homomorphisms into endomorphisms.

Operators from stopped cocycles satisfy cocycle relations.

Method applies to arbitrary index CCR flows.

Abstract

It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e., left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation.