Quantum Statistical Mechanics in Phase Space and the Classical Limit

Moorad Alexanian

A Hilbert space formulation of quantum statistical mechanics is developed by using the notion of myriotic fields. The nonseparable nature of the Hilbert space is investigated, and it is shown that in the classical limit, ℏ → 0, the Hilbert space approaches, in the absence of point eigenvalues, that Hilbert space appropriate to classical statistical mechanics.

Mario Castagnino

N. N. Bogolyubov, A. M. Kurbatov, A. S. Shumovskiǐ

Phil Attard

The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave function symmetrization is also given. The method is tested for quantum harmonic oscillators. For both the boson and fermion cases, the grand potential and the average energy obtained by numerical quadrature over classical phase space are shown to agree with the known analytic results. A mean field approximation is given which is suitable for condensed matter, and which allows the quantum statistical mechanics of interacting particles to be obtained in classical phase space.

M. J. Everitt

Here we argue that spinor structure arises naturally if relativistic statistical mechanics is formulated directly on phase spacetime. Requiring a first-order phase-spacetime description that retains both mass-shell branches leads to a Clifford factorisation of the relativistic constraint and hence to a $4\times4$ spinor-matrix distribution function. We show that deformation quantisation leads to a phase-space formulation of spin quantum mechanics. We argue that projection onto positive- and negative-energy sectors recovers the standard relativistic classical transport equations in the appropriate scalar limits, while the corresponding left- and right- stargenvalue equations reproduce the constraint structure of the Dirac-Wigner formulation. The result is a phase-space route from relativistic statistical mechanics to spinor quantum mechanics, in which spin algebra emerges as the internal structure required by any relativistic statistical theory containing both mass-shell branches and the dimensions of angular momentum from quantum non-locality.

S. Twareque Ali, Eduard Prugovečki