Continuum calculus. II. The heterogeneous continuous functional differentiation applied to the Feynman path integral

Abstract

The continuum calculus proposed previously [L. L. Lee, J. Math. Phys. 17, 1988 (1976)] is here extended to the study of continuous differentiation of a functional. The result, called HCFD, is shown to be the inverse operation of the functional integration for Feynman path integrals, in analogy to the case in ordinary differential calculus. The class of ’’separable’’ functionals is defined, which are useful in the derivation of the theory, playing a role similar to that of the characteristic functions in the Lebesgue theory of integration. A Radon–Nikodym type derivative is introduced in the definition of the continuous derivative for a general Banach algebra. This development constitutes a functional calculus of the continuum type. Comparisons with other types of functional derivatives are also made.