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

Abstract

The spectrally passive, gauge-invariant, quasi-free stases ω on the C*-algebra of anticommutation relations with respect to a one-parameter quasi-free action τ are described. If the one-particle Hamiltonian H is discrete, the precise condition on the one-particle density Q of  ω is combinatorial, but if the Connes spectrum of  τ is non-zero, it implies that Q = (I+eβ (H+T))–1 for some β ≥ 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.