Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

TL;DR

This paper explores the projective representations of the inhomogeneous Hamilton group, revealing its role as a noninertial symmetry in quantum mechanics and extending the understanding of fundamental symmetries beyond inertial frames.

Contribution

It provides a detailed calculation of the projective representations of the inhomogeneous Hamilton group using Mackey theorems, highlighting its significance as a noninertial symmetry in quantum mechanics.

Findings

Identifies the inhomogeneous Hamilton group as a key noninertial symmetry in quantum mechanics.

Calculates the projective representations of this group using Mackey theorems.

Shows the connection between these representations and the invariance of Hamilton's equations.

Abstract

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.