Equilibrium Statistical Ensembles and Structure of the Entropy Functional in Generalized Quantum Dynamics *

We review here the microcanonical and canonical ensembles constructed on an underlying generalized quantum dynamics and the algebraic properties of the conserved quantities. We discuss the structure imposed on the microcanonical entropy by the equilibrium conditions. 1 1. Introduction In this paper we review briefly the generalized quantum dynamics 1,2 constructed on a phase space of local noncommuting fields. We show that the equilibrium conditions on the microcanonical entropy imply that the system decomposes thermodynamically to a sequence of adiabatically independent subsystems, each with its own temperature. There is an equipartition theorem for the phase space variables of the system generated by the linear combination of conserved quantities associated with each of these independent ther-modynamic modes. We start with a review of our basic framework. 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