GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS H Definition 0.1 is a variation of a definition given by Cameron and Storvick [1] and earlier, in the exponential case which has traditionally been of most interest, by Nelson [40]. For X purely imaginary (the Feynman case), the requirements in Definition 0.1 for the existence of K.(F) are more stringent than the requirements in either of [1 or40]. Hence, when KX(F) exists in our sense, it will certainly exist in the sense of [1], The "integral" introduced in [1] has been studied in several later papers including, for example, [18,19]. The interested readers may wish to check some of the references in [1,18,19]. They should also note the differences in notation between this paper and earlier papers such as [1,18,19]. A reader not familiar with the difficulties involved in defining the Feynman integral may wonder why the Wiener integral in (0.13) is not used to define K.(F) for all X e C^. We do not wish to go into this in detail but remark that formula (0.13) can be rewritten with X appearing 1/2 as the scaling m ° X ' of the measure m rather than in the argument of the functions in the integrand. The problems associated with using (0.13) to define KA(F) for X e C+\R are then due to the fact that scaled Wiener "measure" is not countably additive for nonreal scalings, a result due to Cameron. (See, e.g., [9, Theorem 5.1.1, p. 217].) REMARK 0.2. In order to avoid possible misunderstandings, we mention that the present theory is different in spirit and in purpose from the approach to the Feynman integral developed by the second author in [29-32]. In particular, no attempt is made here to treat very general potentials (see G above). On several occasions, however, we shall use some of the same techniques or encounter similar difficulties. REMARK 0.3. The physical interpretations that we give throughout this paper refer to the quantum mechanical case, i.e., to X purely imaginary. The standard quantum mechanical case corresponds to e = -iV, with V real- valued, as well as X = -i (i := /^T ). By contrast, the diffusion case would correspond to e = -V and x = 1.

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