Holomorphic versions of the Fabrey-Glimm representations of the canonical commutation relations

Abstract

The space cutoff is h, ki ~IK 2, #(k~) = (#2 + tkil2)1/2 For v of a more general form, lower parameter j, and upper cutoff a, they show convergence of ( ~ , q~, ~¢ ~p) ex(~) for ~b, ~p in a dense subset N of Fock space, as ~ 00. T;, is a truncated version of e ~*(~) and X(a) is the renormalization. The closure of the inductive limit of ~ over the lower parameters defines a Hilbert space which carries a Weyl representation of the CCR (canonical commutation relations). The Bargmann-Segal complex wave representation for the free field has as Hilbert space H2(K'cx, d#), the completion of the tame holomorphic functionals on K'cx, the complex distributions, which are square-integrable with respect to the Gaussian cylinder set measure # on K'. The finite-dimensional case has been discussed by Bargmann [1] and the infinite dimensional case by Segal [15; 16]. Creation operators on HZ(K ', dl~) are diagonalized and annihilation operators are differentiations. We construct an analogue to the complex wave representation for the interaction case as a countable inductive limit of spaces of the following form: completion of the tame holomorphic functional s on K'cx in the space of functionals which are square integrable with respect to a countably additive measure associated with Tj. This space carries a representation of the CCR for which creation is a multiplication operator and annihilation is, formally, differentiation plus multiplication by the log derivative of Tj. The representation is unitarily equivalent to the Glimm-Fabrey representation. For a fixed lower parameter j and upper cutoff a we construct H2(K ', drlj~), where dqj~ = I Tj.~l 2 II r~ II-2 d#. In order to show that the t/j~ converge to a countably additive measure, we analyze the characteristic functions L~(h) of tb. ~ and,