Why Be Regular? Part I

TL;DR

This paper critically examines the concept of regularity in Weyl algebra representations, challenging prior interpretations and advocating for a focus on regular representations using algebraic methods.

Contribution

It offers a novel critique of non-regular representations and provides a justification for emphasizing regular representations in quantum mechanics.

Findings

Identifies obstacles to interpreting non-regular representations as definite position or momentum.

Challenges Halvorson's proposal on non-regular representations.

Proposes algebraic methods to justify focusing on regular representations.

Abstract

We provide a novel perspective on "regularity" as a property of representations of the Weyl algebra. We first critique 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 argues that the non-regular "position" and "momentum" representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.