Path space picture for Feynman--Kac averages

In a Wiener process the paths x(t) identical with Z(t) execute the diffusive behavior of a free particle, and that is reflected by the (Wiener) measure μ/sub z/[x(.)] on path space (concentrated on continuous path x(t), 0 equal to or less than t equal to or less than T, say). The general Feynman-Kac average of a functional f[x(.)] on path space in the presence of a potential lambdaV(x,t) is given by the exponential representing the distortion of the distribution of the free-particle paths introduced by the potential. Alternatively, as is shown, for a given lambdaV, it is possible to construct a new Markov process with continuous sample paths x(t) identical with Y(t), the diffusive behavior of which already includes the effect of the potential in such a way that the Feynman-Kac average may be expressed in the form, F = ∫ f[Rx(.)]dμ/sub z/[x(.)]. The variable R is a map on path space under which Z(t) → Y(t). The new process Y(t) satisfies an imaginary time Newtonian equation of motion appropriate for the potential and in the presence of additional ''quantum forces'' as generated by the Wiener process Z(t). In understanding these two equivalent descriptions of the average F, it may be helpful to observe that the first (Z-) description is like the interaction picture of quantum mechanics while the second (Y-) description corresponds to the Heisenberg picture. A number of interesting properties of the Y-process and their significance for quantum mechanics are exhibited. (U.S.)