Dynamics in Quantum Statistical Mechanics

Abstract

In these lectures we describe and illustrate various general properties of dynamics in quantum statistical mechanics. Our principal interest is the construction of a dynamical description for a thermodynamic system. This construction corresponds essentially to integration of the equations of motion. In ordinary quantum mechanics the dynamics can be described by the Schrödinger equation determined by a symmetric operator H. The equation has a unique solution if, and only if, H is self-adjoint. Alternatively, one may work in the Heisenberg picture and the self-adjointness of H is necessary and sufficient to give a dynamical group acting on the algebra of observables. In statistical mechanics the Heisenberg picture appears to be more useful and one can often formulate the equations of motion in terms of a symmetric derivation δ acting on an algebra of observables. Integration of the equations of motion is then dependent on certain ‘self-adjointness’ properties of δ. Various possibilities occur depending on the system under consideration.