Quantum Mechanics in non-inertial reference frames: time-dependent rotations and loop prolongations

TL;DR

This paper extends a quantum mechanics framework to include rotational accelerations in non-inertial frames, using loop prolongations of the Galilean line group to account for fictitious forces like centrifugal and Coriolis effects.

Contribution

It introduces loop prolongations of the Galilean line group for quantum mechanics in rotating non-inertial frames, expanding beyond traditional group representations.

Findings

Recovered centrifugal and Coriolis forces from loop representations

Extended previous linear acceleration models to include rotations

Showed that non-associative loops are necessary for general non-inertial quantum transformations

Abstract

This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelarations. We show that the incorporation of rotational accelerations requires a class of \emph{loop prolongations} of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-stablished framework for implementing symmetry transformations in quantum mechanics.