A SIMPLE DEFINITION OF THE FEYNMAN INTEGRAL, WITH APPLICATIONS 3
the Banach algebras S1 and S" defined in [k]. (The definition of S" involves no
integration in function space).
In section h we shall present a translation theorem and show that the sequential
v
Feynman integral remains invariant under orthogonal transformations of E
In section 5 we present a Fubini theorem giving conditions for permuting sequential
Feynman integrals with Lebesgue type integrals or with infinite series.
In section 6 we give applications of the sequential Feynman integral to the
Schroedinger equation and to the quadratic potentials of Johnson and Skoug [9]-
In section 7 we shall show the relationship to other sequential definitions of the
Feynman integral. In particular the sequential Wiener integral [3] and the Truman
integral [13], [1^]. Our work and that of Truman are closely related to that of
Albeverio and H/egh - Krohn [1] and [3]. Johnson [7] shows the relationship between our
space S (given in [k]) to the space 5(H) of Fresnel integrable functionals of
Albeverio and Hpegh-Krohn. Indeed, in a recent communication, Johnson has pointed out
that our space S defined above is identical (and not merely isometrically isomorphic)
to the space 5(H) .
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