On Quasi-equivalence of Quasifree States of the Canonical Commutation Relations

Abstract

A necessary and sufficient condition for two quasifree states of CCR's (the canonical commutation relations) to yield quasi-equivalent representations is obtained, in the most general setting where states are non gauge-invariant in general and the symplectic form defining CCR might be degenerate (either from the start or after completion relative to the topology induced by states). The criterion consists of the following two conditions: (1) The induced topologies on the test function space are equivalent (2) operators $\tilde{S}$ and ${\tilde{S}}^{\prime }$ on the completed test function space (completion relative to the induced topology, after taking quotient by the kernel of the representation) giving the two point functions $S$ and $S^{\prime }$ (relative to any common inner product giving rise to the induced topology) are such that ${\tilde{S}}^{1/2}-{\tilde{S}}^{\prime 1/2}$ is in the Hilbert Schmidt class.