Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. I. Representations with Cyclic Field

Abstract

We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.