4 G.W. JOHNSON and M.L. LAPIDUS
ignore the diagonal of the square. We point out that Feynman's convention
(0.4) is suited for Lebesgue measure I, a continuous measure, so that the
diagonal s- , = s2 of the square is a set of measure zero. In this sense,
our theory is broader than parts of Feynman's operational calculus.
We are using the expression "Feynman!s operational calculus" as though
it has a precise meaning. However, a key problem is to give a precise de-
finition and interpretation of this calculus and to demonstrate how to use
it effectively in particular in carrying out the disentangling process and
in developing a functional calculus. The reader might be interested and
surprised to read Feynman's own comments [11, p. 108] on the difficulty of
putting his methods on a rigorous basis and on the need for further mathe-
matical development.
The class of functionals on Wiener space that we are able to treat is
quite large. In fact, under pointwise mulitplication and equipped with a
natural norm, it forms a commutative Banach algebra A consisting of cer-
tain series of products of functionals of the form (0.2). With the help
of the basic results of Section 2, we show in Section 6 that each func-
tional in A possesses operator-valued Wiener and Feynman integrals, en-
larging in the process the class of functionals for which the operator-
valued Feynman integral is known to exist. Further, each of these
operators can be disentangled in the form of a generalized Dyson series.
Related but much smaller Banach algebras of functionals were studied
by Johnson and Skoug in [18 and 19]; [19, pp. 121-123] is especially
relevant. The functionals in [19] are generated by functionals of the
form (0.2) with 8 varying but with n fixed as Lebesgue measure. The
resulting Dyson series are much simpler. The emphasis in [19] was some-
what different, and, in particular, no attempt was made to relate the
results to Feynman's operational calculus.
Feynman's paper [11],in conjunction with the present work and that
of Lapidus in [33,34], suggests additional questions which we anticipate
investigating in a subsequent paper that will further develop Feynman1s
operational calculus for noncommuting operators.
We mention the works of Nelson [41] and Maslov [36]which are also
related to Feynman's operational calculus. They have little in common,
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