Nonlocality and local causality in the Schr\"odinger Equation with time-dependent boundary conditions

TL;DR

This paper demonstrates that the Schrödinger equation with moving boundary conditions can violate local causality by causing instantaneous probability current changes, and formalizes this with a thought experiment involving weak measurements.

Contribution

It reveals nonlocal effects in quantum dynamics with time-dependent boundaries and formalizes a thought experiment to illustrate causality violations in such systems.

Findings

Schrödinger equation causes instantaneous probability current changes

A gedanken device for faster-than-light communication is proposed

The system avoids typical nonphysical violations like high energies or velocities

Abstract

We investigate the nonlocal dynamics of a single particle placed in an infinite well with moving walls. It is shown that in this situation, the Schr\"odinger equation (SE) violates local causality by causing instantaneous changes in the probability current everywhere inside the well. This violation is formalized by designing a gedanken faster-than-light communication device which uses an ensemble of long narrow cavities and weak measurements to resolve the weak value of the momentum far away from the movable wall. Our system is free from the usual features causing nonphysical violations of local causality when using the (nonrelativistic) SE, such as instantaneous changes in potentials or states involving arbitrarily high energies or velocities. We explore in detail several possible artifacts that could account for the failure of the SE to respect local causality for systems involving time-dependent boundary conditions.