Relativity implications of the quantum phase

TL;DR

This paper explores how the quantum phase influences the symmetry groups in quantum mechanics, revealing a maximal quantum symmetry related to the conformally scaled inhomogeneous symplectic group and its implications for relativistic quantum theories.

Contribution

It identifies the maximal quantum symmetry preserving the Heisenberg relations as projective representations of a conformally scaled inhomogeneous symplectic group, extending the understanding of quantum symmetries beyond relativistic groups.

Findings

Quantum phase leads to projective representations of symmetry groups.

The maximal quantum symmetry is the projective representations of a conformally scaled inhomogeneous symplectic group.

Implications for noninertial quantum theory and quantum gravity are discussed.

Abstract

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations. We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.