  We address the issue of when generalized quantum dynamics, which is a
classical symplectic dynamics for noncommuting operator phase space variables
based on a graded total trace Hamiltonian ${\bf H}$, reduces to Heisenberg
picture complex quantum mechanics. We begin by showing that when ${\bf H}={\bf
Tr} H$, with $H$ a Weyl ordered operator Hamiltonian, then the generalized
quantum dynamics operator equations of motion agree with those obtained from
$H$ in the Heisenberg picture by using canonical commutation relations. The
remainder of the paper is devoted to a study of how an effective canonical
algebra can arise, without this condition simply being imposed by fiat on the
operator initial values. We first show that for any total trace Hamiltonian
which involves no noncommutative constants, there is a conserved
anti--self--adjoint operator $\tilde C$ with a structure which is closely
related to the canonical commutator algebra. We study the canonical
transformations of generalized quantum dynamics, and show that $\tilde C$ is a
canonical invariant, as is the operator phase space volume element. The latter
result is a generalization of Liouville's theorem, and permits the application
of statistical mechanical methods to determine the canonical ensemble governing
the equilibrium distribution of operator initial values. We give arguments
based on a Ward identity analogous to the equipartition theorem of classical
statistical mechanics, suggesting that statistical ensemble averages of Weyl
ordered polynomials in the operator phase space variables correspond to the
Wightman functions of a unitary complex quantum mechanics, with a conserved
operator Hamiltonian and with the standard canonical commutation relations
obeyed by Weyl ordered operator strings. Thus there is a well--defined sense in
