Passive Quasi-free States of the CAR Algebra with Discrete Hamiltonians

Abstract

The spectrally passive, gauge-invariant, quasi-free stases $\omega$ on the C*-algebra of anticommutation relations with respect to a one-parameter quasi-free action $\tau$ are described. If the one-particle Hamiltonian $H$ is discrete, the precise condition on the one-particle density $Q$ of  $\omega$ is combinatorial, but if the Connes spectrum of  $\tau$ is non-zero, it implies that $Q=(I+e^{\beta (H+T)}{)}^{–1}$ for some $\beta \ge 0$ and some operator $T$ of bounded trace norm, apart from some degenerate possibilities. If $H$ has both discrete and continuous parts, these results can be combined with those of de Cannière for the purely continuous case.