Why Be Regular? Part II

TL;DR

This paper defends the importance of regular representations of the Weyl algebra in quantum mechanics, arguing that algebraic methods justify their focus over non-regular representations proposed by Halvorson.

Contribution

It offers a novel algebraic perspective to justify the emphasis on regular representations, countering Halvorson's advocacy for non-regular ones in quantum theory.

Findings

Regular representations are justified through algebraic methods.

Non-regular representations do not provide the same physical insights.

Algebraic approach clarifies the role of regularity in quantum representations.

Abstract

We provide a novel perspective on "regularity" as a property of representations of the Weyl algebra. In Part I, we critiqued a proposal by Halvorson [2004, "Complementarity of representations in quantum mechanics", Studies in History and Philosophy of Modern Physics 35(1), pp. 45--56], who advocates for the use of the non-regular "position" and "momentum" representations of the Weyl algebra. Halvorson argues that the existence of these non-regular representations demonstrates that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. In this sequel, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.