On Quasifree States of the Canonical Commutation Relations (I)

Abstract

A self-dual CCR algebra is defined and arbitrary quasifree state is realized in a Fock type representation of another self-dual CCR algebra of a double size as a preparation for a study of quasi-equivalence of quasifree states. § 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]. A self-dual algebra is defined when a linear space K, an antilinear involution F of K and a hermitian form j" on K satisfying f(Tf^ Fg) = -/(/i #)* are given. In section 3, we define a quasifree state in terms of a nonnegative hermitian form S on K such that S(f, g)-S(r g , r/) = r(/, g). In section 4, the structure of S relative to (K, 7% /") is analyzed. In section 5, basic properties of a Fock representation are stated and a result in [1] is quoted. A Fock type representation is defined as a generalization of a Fock representation to the case of degenerate 7* (i.e.

Polynomial Weyl Representations

“…We may consider replacing the Fock representation by another Weyl representation and performing the same type of operation to generate further Weyl representations. For convenience, a quasifree representation might be chosen as basic: quasifree representations admit a standard description in terms of Fock representations by means of a doubling procedure; see [1] and [4]. When comparing Weyl representations in such a case, it would be sensible to replace the notion of unitary equivalence by the more general notion of quasiequivalence.…”

Hadamard state in Schwarzschild–de Sitter spacetime

“…Then we say that ξ is a regular direction for ω, with respect to the continuous one-parametric group of * -automorphisms {α t } t∈R induced by the action of the Killing vector field § X, if there exists some h ∈ C ∞ 0 (R 2 ) and an open neighborhood V of ξ in R 2 {0} such that, for each s ∈ N + , there are C s , λ s > 0 so that…”

The Charge Quantum Numbers of Gauge Invariant Quasi-Free Endomorphisms

“…In fact, the latter depends on the choice of the Fock state, i. e. on the choice of P 1 . The (selfdual) CCR algebra C(K, κ) over (K, κ) is the simple *-algebra which is generated by 1 and elements f ∈ K, subject to the commutation relation [AS72,Ara72] …”