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,