The algebraic foundations of Dirac notation and path integrals are examined. It is shown that a formulation of quantum mechanics on the basis of the theory of C* alge- bras permits these questions to be analyzed from a common standpoint.

Abstract

In connection with the existence of irreducible representations of commutation relations which are not equivalent from the unitary standpoint, with a number of degrees of freedom 3N = ~, it is pertinent to study the algebraic foundations of quantum mechanics for the case 3N < ~, i.e., under the applicability conditions of von Neumann's theorem of the unitary equivalence of any representations of commutation relations to the direct sum of Schr~dinger relations. One reason for this situation is that the discussion often involves the limit N--~ V-+~ N/V==-!,so that Von Neumann's theorem does not apply. It is thus interesting to examine the process by which a Hilbert space arises from purely algebraic postulates, without direct appeal to the GNS construction. As we will show below, this question is intimately related to the basis for Dirac notation; we will also see that the well-known representation of the transition amplitude <xt x'l'L~ by a path integral and thus the Feynman construction of quantum mechanics can be given an algebraic footing. These aspects of our results are closely related to a study by Berezin [i], who does not work from the C* algebra of R generated by commutation relations, as we will do, but from representations of the algebras of functions defined in a phase space. We will not attempt to treat the problem in its most general form in this paper. The proofs will be given for some simple examples, but the necessary generalizations are easily made. i. We make a transition from the Weyl basis in the C* algebra of R, X(q p) to the coordiA hate basis D(~ y) and we write the latter out explicitly (h = i)