Loop prolongations and three-cocycles in simulated magnetic fields from rotating reference frames

TL;DR

This paper extends the mathematical framework of non-relativistic quantum mechanics to include all accelerating frames using loop prolongations and three-cocycles, explaining observed phase shifts in neutron interferometry and optical lattices.

Contribution

It introduces non-associative extensions called loop prolongations of the Galilean line group to incorporate non-inertial reference frames in quantum mechanics.

Findings

Derived phase shifts from Earth's rotation in neutron interferometry.

Explained rotational effects in optical lattices as consequences of loop prolongations.

Established a rigorous mathematical foundation for non-inertial quantum reference frames.

Abstract

We show that the Wigner-Bargmann program of grounding non-relativistic quantum mechanics in the unitary projective representations of the Galilei group can be extended to include all non-inertial reference frames. The key concept is the \emph{Galilean line group}, the group of transformations that ties together all accelerating reference frames, and its representations. These representations are constructed under the natural constraint that they reduce to the well-known unitary, projective representations of the Galilei group when the transformations are restricted to inertial reference frames. This constraint can be accommodated only for a class of representations with a sufficiently rich cocycle structure. Unlike the projective representations of the Galilei group, these cocycle representations of the Galilean line group do not correspond to central extensions of the group. Rather, they correspond to a class of non-associative extensions, known as \emph{loop prolongations}, that are determined by three-cocycles. As an application, we show that the phase shifts due to the rotation of the earth that have been observed in neutron interferometry experiments and the rotational effects that lead to simulated magnetic fields in optical lattices can be rigorously derived from the representations of the loop prolongations of the Galilean line group.