FORM OF REPRESENTATIONS OF CANONICAL COMMUTATION RELATIONS FOR BOSE FIELDS AND CONNECTION WITH FINITELY MANY DEGREES OF FREEDOM

Given a representation of the canonical commutation relations (CCR) for Bose fields in a separable (or, under an additional assumption, nonseparable) Hilbert space {Hilbert space} it is shown that there exists a decreasing sequence of finite and quasi-invariant measures μn on the space[Figure not available: see fulltext.] of all linear functionals on the test function space[Figure not available: see fulltext.], such that {Hilbert space} can be realized as the direct sum of the {Mathematical expression}, the space of all μn-square-integrable functions on[Figure not available: see fulltext.]. In this realization U(f) becomes multiplication by[Figure not available: see fulltext.]. The action of V(g) is similar as in the case of cyclic U(f) which has been treated by Araki and Gelfand. But different {Mathematical expression} can be mixed now. Simply transcribing the results in terms of direct integrals one obtains a form of the representations which turns out to be essentially the direct integral form of Lew. All results are independent of the dimensionality of[Figure not available: see fulltext.] and hold in particular for dim[Figure not available: see fulltext.]. Thus one has obtained a form of the CCR which is the same for a finite and an infinite number of degrees of freedom. From this form it is in no way obvious why there is such a great distinction between the finite and infinite case. In order to explore this question we derive von Neumanns theorem about the uniqueness of the Schrödinger operators in a constructive way from this dimensionally independent form and show explicitly at which point the same procedure fails for the infinite case. © 1969 Springer-Verlag.