Dichotomous-context-induced superselection rule for a one-dimensional completed scattering

TL;DR

This paper demonstrates that a superselection rule induced by the physical context divides the state space in one-dimensional scattering, leading to a complex process with distinct subprocesses of transmission and reflection.

Contribution

It introduces a novel superselection rule in quantum scattering that separates the process into coherent subprocesses and clarifies their dynamics.

Findings

Superselection rule divides the state space into two sectors.

Pure initial states become mixed final states after scattering.

Transmission and reflection are treated as separate coherent subprocesses.

Abstract

It is shown that in the state space of a quantum particle involved in a one-dimensional completed scattering (OCS) there is a superselection rule (SSR) induced by the dichotomous physical context that determines the properties of a quantum ensemble of scattering particles in the disjoint spatial regions located on the different sides of the potential barrier at asymptotically large distances from it. At the initial stage of the OCS this context is associated, in the general case, with two sources of particles located on the different sides of the barrier; and at the final stage it is associated, in the general case, with two particle detectors located on different sides of the barrier. The role of a superselection operator is played by the Pauli matrix $\sigma_3$ that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.