The form of representations of the canonical commutation relations for Bose fields and connection with finitely many degrees of freedom

Abstract

Given a representation of the canonical commutation relations (CCR) for Bose fields in a separable (or, under an additional assumption, nonseparable) Hilbert space ℌ it is shown that there exists a decreasing sequence of finite and quasi-invariant measures μ _n on the space

of all linear functionals on the test function space

, such that ℌ can be realized as the direct sum of the$L_{\mu _n }^2$, the space of all μ _n -square-integrable functions on

. In this realizationU(f) becomes multiplication by

. The action ofV(g) is similar as in the case of cyclicU(f) which has been treated byAraki andGelfand. But different$L_{\mu _n }^2$ 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 ofLew. All results are independent of the dimensionality of

and hold in particular for dim

. 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.