The C*-algebras of a free Boson field

Abstract

We give a systematic description of severalC*-algebras associated with a free Boson field. In this first part the structure of the one-particle space enters only through its symplectic form σ and a directed absorbing set of finite-dimensional subspaces on which σ is non-degenerate. The Banach *-algebras ℒ1 (\(\mathfrak{E}\), σ) and ℳ1 (\(\mathfrak{E}\), σ) of absolutely continuous resp. bounded measures on a finite-dimensional symplectic space (\(\mathfrak{E}\), σ), with their “twisted convolution product” stemming from Weyl's commutation relations, are studied as the analogues of the ℒ1 resp. ℳ1 algebras of a locally compact group. The fundamental “vacuum idempotent” Ω determines their (unique) Schrodinger representation, SchrodingerA*-norm and SchrodingerC*-completions and . After a study of these one proceeds to a construction as an inductive limit of the algebras ℳ1(ℌ, σ) and for an infinite-dimensional symplectic space (ℌ, σ). The “Fock representations” (with the corresponding “field operators”) are presented as the infinite-dimensional generalization of the Schrodinger representation. The paper ends with a discussion of several possible choices for the “free BosonC*-algebra”.