Free states of the canonical anticommutation relations

Abstract

Each gauge invariant generalized free state ω A of the anticommutation relation algebra over a complex Hubert space K is characterized by an operator A on K. It is shown that the cyclic representations induced by two gauge invariant generalized free states ω A and ω B are quasi-equivalent if and only if the operators A^ -B^ and (/ -A)* -(I -B)^ are of Hubert-Schmidt class. The combination of this result with results from the theory of isomorphisms of von Neumann algebras yield necessary and sufficient conditions for the unitary equivalence of the cyclic representations induced by gauge invariant generalized free states.

Fredholm Determinants and the Statistics of Charge Transport

“…We note that (17). The expectations (23) agree because of (e iλQ g, Ne iλQ f ) = (g, Nf ) by [Q, N] = 0.…”

Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule

“…If R is semifinite, φ if a σ-finite faithful normal trace on R, H is the Hubert space of Hilbert-Schmidt operator affiliated with R, Hilbert-Schmidt relative to φ, and R is left multiplication, then an example of V Ψ is the set of vector corresponding to positive Hilbert-Schmidt operators. The inequality (5.10) correspond to the inequality \\σ -p\\ tx ^ \\σ^ -^2||Ls [7]. Hence by uniform boundedness || (Δ\ ι * + 1)…”

“…It should be remarked that (1.3), (1.4), (1.5), and (1.6) without (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) are not sufficient to characterize J r . If (1.5) is dropped, then there exists a unitary u in the center such that J = J uψ .…”

On Quasifree States of the Canonical Commutation Relations (I)

“…§ 1. Introduction A necessary and sufficient condition for the quasi-equivalence of two quasifree representations of the canonical anticommutation relations (CAR) has been derived in [11] for the gauge invariant case and in [3] for the general case. We shall derive an analogous result for the canonical commutation relations (CCR) In section 2, we review the formulation in [2].…”