Generalized Quantum Dynamics as Pre-Quantum Mechanics

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