Hamilton relativity group for noninertial states in quantum mechanics

TL;DR

This paper develops a Hamilton relativity group framework for noninertial states in quantum mechanics, linking group automorphisms with quantum observables and deriving invariants including a generalized spin.

Contribution

It introduces the inhomogeneous Hamilton group as the relativity group for noninertial frames and computes its central extension and Casimir invariants.

Findings

Derived the inhomogeneous Hamilton group for noninertial frames

Computed the group's central extension and Casimir invariants

Identified a generalized spin invariant applicable to noninertial states

Abstract

Physical states in quantum mechanics are rays in a Hilbert space. Projective representations of a relativity group transform between the quantum physical states that are in the admissible class. The physical observables of position, time, energy and momentum are the Hermitian representation of the Weyl-Heisenberg algebra. We show that there is a consistency condition that requires the relativity group to be a subgroup of the group of automorphisms of the Weyl-Heisenberg algebra. This, together with the requirement of the invariance of classical time, results in the inhomogeneous Hamilton group that is the relativity group for noninertial frames in classical Hamilton's mechanics. The projective representation of a group is equivalent to unitary representations of its central extension. The central extension of the inhomogeneous Hamilton group and its corresponding Casimir invariants are computed. One of the Casimir invariants is a generalized spin that is invariant for noninertial states. It is the familiar inertial Galilean spin with additional terms that may be compared to noninertial experimental results.