  The Hilbert space of the unitary irreducible representations of a Lie group
that is a quantum dynamical group are identified with the quantum state space.
Hermitian representation of the algebra are observables. The eigenvalue
equations for the representation of the set of Casimir invariant operators
define the field equations of the system.
  A general class of dynamical groups are semidirect products K *s N for which
the representations are given by Mackey's theory. The homogeneous group K must
be a subgroup of the automorphisms of the normal group N.
  The archetype dynamical group is the Poincare group. The field equations
defined by the representations of the Casimir operators define the basic
equations of physics; Klein-Gordon, Dirac, Maxwell and so forth.
  This paper explores a more general dynamical group candidate that is also a
semi-direct product but where the 'translation' normal subgroup N is now the
Heisenberg group.
  The relevant automorphisms of the Heisenberg group are the symplectic group.
This together with the requirement for an orthogonal metric leads to the
pseudo-unitary group for the homogeneous group K.
  The physical meaning and motivation of this group, called the quaplectic
group, is presented and the Hermitian irreducible representations of the
algebra are determined. As with the Poincare group, choice of the group defines
the Hilbert space of representations that are identified with quantum particle
states. The field equations that are the eigenvalue equations for the
representation of the Casimir operators, are obtained and investigated. The
theory embodies the Born reciprocity principle and a new relativity principle.
