  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.
