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

### Charles J.K. Batty

University of Oxford, UK

## 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.

## Cite this article

Charles J.K. Batty, Passive Quasi-free States of the CAR Algebra with Discrete Hamiltonians. Publ. Res. Inst. Math. Sci. 22 (1986), no. 3, pp. 487–506

DOI 10.2977/PRIMS/1195177846