The classical limit of a state on the Weyl algebra

TL;DR

This paper characterizes when states on the Weyl algebra have a classical limit as a probability measure, linking regularity to countable additivity, and proposes a way to modify the observable collection to ensure all states have physical classical limits.

Contribution

It establishes a precise criterion for the classical limit of states on the Weyl algebra and suggests a method to modify the observable set for physical classical limits.

Findings

A state is regular iff its classical limit is a countably additive Borel probability measure.

The classical limit of a state can be characterized as a probability measure on phase space.

Modifying the collection of observables can ensure all states have physical classical limits.

Abstract

This paper considers states on the Weyl algebra of the canonical commutation relations over the phase space R^{2n}. We show that a state is regular iff its classical limit is a countably additive Borel probability measure on R^{2n}. It follows that one can "reduce" the state space of the Weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical.