Vol. 45, No. 1, 1973

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Feynman integrals of non-factorable finite-dimensional functionals

Gerald William Johnson and David Lee Skoug

Vol. 45 (1973), No. 1, 257–267
Abstract

Let C0[a,b] denote the space of continuous functions x on [a,b] such that x(a) = 0. Recently Cameron and Storvick defined an operator-valued Feynman integral Jq(F) of functionals F on C0[a,b]. Let F(x) = f(x(t1),,x(tn)) where a = t0 < t1 < < tn = b. The present authors earlier established the existence of Jq(F) for functionals F as above under the assumption that f is factorable and bounded. In the present paper it is shown that with the factorability assumption completely removed, J1(F) may fail to exist even with f required to be in Lp(Rn) for 1 p . On the other hand it is shown that Jq(F) does exist under the rather surprising condition that f L2112 where L2112 is the set of all complex-valued measurable functions f on Rn(n 2) such that f2112 ≡∥f< where

f { ∫ ∞ [∫ ∞       ∫ ∞
(n − 2)
−∞   −∞        −∞
  ∫ ∞                                 ]2   }
× (    |f(u ,⋅⋅⋅ ,u  )|2du )1∕2du  ⋅⋅⋅ du      du
− ∞    1      n    1     2      n−1    n12.
Another positive result shows that if F is an analytic function of a finite sum of factorable functions, then Jq(F) exists.

Mathematical Subject Classification
Primary: 28A40
Milestones
Received: 8 July 1970
Revised: 27 October 1972
Published: 1 March 1973
Authors
Gerald William Johnson
David Lee Skoug