Geometrization of statistical mechanics

Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert-space formulation of classical statistical thermodynamics, we show that the specification of a canonical polarity on the real projective space RPn induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, equilibrium thermal states are determined by a Hamiltonian gradient how with respect to this metric. This flow is characterized by the property that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is extended to establish a new framework for the quantum-mechanical dynamics of equilibrium thermal states. In this case, the relevant phase space is the complex projective space CPn, here regarded as a real manifold Gamma endowed with the Fubini-Study metric and a compatible symplectic structure. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the non-uniqueness of the infinite-temperature state. We are then led to formulate a geometric characterization of the standard KMS relation often considered in the context of C* algebras. Finally, we develop a theory of the quantum microcanonical and canonical ensembles, based on the geometry of the quantum phase space Gamma. The example of a quantum spin-1/2 particle in a heat bath is studied in detail.