Revising the applicability of the Stone-von Neumann theorem to scattering a quantum particle on a one-dimensional potential barrier

TL;DR

This paper demonstrates that the Stone-von Neumann theorem does not apply to quantum scattering on a one-dimensional potential barrier due to the unbounded position operator, leading to a reducible Schrödinger representation and a superselection rule separating transmission and reflection processes.

Contribution

It reveals the inapplicability of the Stone-von Neumann theorem in this context and introduces a superselection rule based on the dichotomous sectors of scattering states.

Findings

The Schrödinger representation is reducible in this scattering scenario.

A superselection rule separates transmission and reflection states.

Superpositions of different sectors are mixed states, not pure.

Abstract

It is shown that the Stone-von Neumann theorem is inapplicable to scattering a quantum nonrelativistic particle on a one-dimensional "short-range" potential barrier, since the unboundedness of the position operator plays here a crucial role. The Shcr\"odinger representation associated with this process is reducible: long before and long after the scattering event the space of its asymptotes represents the direct sum of the subspaces of left and right asymptotes. There is a dichotomous-context-induced superselection rule (SSR), in which the role of a superselection operator is played by the Pauli matrix $\sigma_3$ and the role of superselection (coherent) sectors is played by the above subspaces. By the SSR any superposition of states from different coherent sectors is a mixed state, and splitting the incident wave packet into the transmitted and reflected parts is nothing but a conversion of a pure state into a mixed one. The average values of any observable can be defined only for the transmission and reflection subprocesses. The former evolves within a single coherent sector of the Hilbert space, in the momentum representation; while the latter evolves within a single coherent sector of this space, in the coordinate representation.