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Communications in Mathematical Physics

, Volume 1, Issue 1, pp 14–48 | Cite as

The C*-algebras of a free Boson field

I. Discussion of the basic facts
  • D. Kastler
Article

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) Schrödinger representation, SchrödingerA*-norm and SchrödingerC*-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 Schrödinger representation. The paper ends with a discussion of several possible choices for the “free BosonC*-algebra”.

Keywords

Neural Network Statistical Physic Complex System Convolution Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1965

Authors and Affiliations

  • D. Kastler
    • 1
  1. 1.University of Aix-MarseilleFrance

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