Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group

TL;DR

This paper identifies the inhomogeneous symplectic group as the maximal quantum symmetry group that preserves Heisenberg relations, using projective and unitary representations of its central extension.

Contribution

It establishes that the maximal quantum symmetry group is the inhomogeneous symplectic group and explicitly computes its unitary irreducible representations.

Findings

Maximal quantum symmetry group is the inhomogeneous symplectic group.

Projective representations correspond to unitary representations of the group's central extension.

Explicit computation of irreducible representations using Mackey theorems.

Abstract

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations, that are fundamental to quantum mechanics, must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.