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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