  This thesis is concerned with the representation theory of the Heisenberg
group and its applications to both classical and quantum mechanics. We continue
the development of $p$-mechanics which is a consistent physical theory capable
of describing both classical and quantum mechanics simultaneously.
$p$-Mechanics starts from the observation that the one dimensional
representations of the Heisenberg group play the same role in classical
mechanics which the infinite dimensional representations play in quantum
mechanics.
  In this thesis we introduce the idea of states to $p$-mechanics.
$p$-Mechanical states come in two forms: elements of a Hilbert space and
integration kernels. In developing $p$-mechanical states we show that quantum
probability amplitudes can be obtained using solely functions/distributions on
the Heisenberg group. This theory is applied to the examples of the forced,
harmonic and coupled oscillators. In doing so we show that both the quantum and
classical dynamics of these systems can be derived from the same source. Also
using $p$-mechanics we simplify some of the current quantum mechanical
calculations.
  We also analyse the role of both linear and non-linear canonical
transformations in $p$-mechanics. We enhance a method derived by Moshinsky for
studying the passage of canonical transformations from classical to quantum
mechanics. The Kepler/Coulomb problem is also examined in the $p$-mechanical
context. In analysing this problem we show some limitations of the current
$p$-mechanical approach. We then use Klauder's coherent states to generate a
Hilbert space which is particularly useful for the Kepler/Coulomb problem.
