Microcanonical Ensemble and Algebra of Conserved Generators for Generalized Quantum Dynamics

It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space. 1 1. Introduction Generalized quantum dynamics 1,2 is an analytic mechanics on a symplectic set of operator valued variables, forming an operator valued phase space S. These variables are defined as the set of linear transformations † on an underlying real, complex, or quaternionic Hilbert space (Hilbert module), for which the postulates of a real, complex, or quaternionic quantum mechanics are satisfied 2−6. The dynamical (generalized Heisenberg) evolution, or flow, of this phase space is generated by the total trace Hamiltonian H = TrH, where for any operator O we have