Little-used Mathematical Structures in Quantum Mechanics II. Representations of the CCR and Superseparability

TL;DR

This paper explores the mathematical structures of inequivalent representations of the canonical commutation relations in quantum mechanics, proposing experiments to observe superseparability and discussing implications for many-worlds interpretations.

Contribution

It introduces the concept of superseparability in quantum states and proposes experimental tests to observe inequivalent representations of CCR in single-particle systems.

Findings

Proposes two experiments to detect superseparability.

Links inequivalent representations to topological and geometrical quantities.

Suggests implications for many-worlds interpretations.

Abstract

It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that are both simple and well-known. The representations are unitarily inequivalent because the spectra of the position and momentum operators are different, and spectra are invariant under unitary transformations. The existence of these representations can have consequences that run from the merely unexpected to the barely conceivable. To start with, states of a single particle that belong to inequivalent representations will always be mutually orthogonal; they will never interfere with each other. This property, called superseparability elsewhere, is well-defined mathematically, but has not yet been observed. This article suggests two single-particle interference experiments that may reveal its existence. The existence of inequivalent irreducibile representations may be traced to the existence of different self-adjoint extensions of symmetric operators on infinite-dimensional Hilbert spaces. Analysis of the underlying mathematics reveals that some of these extensions can be interpreted in terms of topological, geometrical and physical quantities that can be controlled in the laboratory. The tests suggested are based on these interpretations. In conclusion, it is pointed out that mathematically rigorous many-worlds interpretations of quantum mechanics may be possible in a framework that admits superseparability.