Poincare and Heisenberg quantum dynamical symmetry: Casimir invariant field equations of the quaplectic group

TL;DR

This paper explores the role of the quaplectic group, combining Poincare and Heisenberg symmetries, in deriving quantum field equations through Casimir invariants, extending the understanding of symmetries in quantum physics.

Contribution

It introduces the quaplectic group as a unified symmetry framework and derives general field equations using Mackey representation theory.

Findings

Derived field equations from the Casimir invariants of the quaplectic group.

Showed the quaplectic group contains multiple Poincare subgroups and the Heisenberg group.

Extended the symmetry-based derivation of quantum field equations beyond the Poincare group.

Abstract

The unitary irreducible representations of a Lie group defines the Hilbert space on which the representations act. If this Lie group is a physical quantum dynamical symmetry group, this Hilbert space is identified with the physical quantum state space. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. The Poincare group is the archetypical example with the unitary representations defining the Hilbert space of relativistic particle states and the Klein-Gordon, Dirac, Maxwell equations are obtained from the representations of the Casimir invariant operators eigenvalue equations. The representation of the Heisenberg group does not appear in this derivation. The unitary representations of the Heisenberg group, however, play a fundamental role in nonrelativistic quantum mechanics, defining the Hilbert space and the basic momentum and position commutation relations. Viewing the Heisenberg group as a generalized non-abelian "translation" group, we look for a semidirect product group with it as the normal subgroup that also contains the Poincare group. The quaplectic group, that is derived from a simple argument using Born's orthogonal metric hypothesis, contains four Poincare subgroups as well as the normal Heisenberg subgroup. The general set of field equations are derived using the Mackey representation theory for general semidirect product groups.