Reviews in Mathematical Physics

We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions.

Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results.

All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra with all or a part of the following assumptions:

(I) The interaction is even, namely, the dynamics α_t commutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.)

(II) The domain of the generator δ_α of α_t contains the set of all strictly local elements of .

(III) The set is the core of δ_α.

(IV) The dynamics α_t commutes with lattice translation automorphism group τ of .

A major technical tool is the conditional expectation from onto its C*-subalgebras for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems.

In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain , commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption.

If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation covariant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result.

Everything goes in parallel for spin lattice systems without the evenness assumption (I).

The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.