A path space picture for Feynman-Kac averages

Abstract

In a Wiener process the paths x( t) ≡ Z( t) execute the diffusive behavior of a free particle, and that is reflected by the (Wiener) measure μ z [ x(·)] on path space (concentrated on continuous paths x( t), 0 ⩽ t ⩽ T, say). The general Feynman-Kac average of a functional T[ x(·)] on path space in the presence of a potential λV( x, t) is given by F = N ∫ F[x(·:)]exp −λ ∫ 0 t V(x(t′),t′)dt′ dν z[x(·)] the exponential representing the distortion of the distribution of the free-particle paths introduced by the potential. Alternatively, as we show, for a given λV, it is possible to construct a new Markov process with continuous sample paths x( t) ≡ 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(·)]dν z[x(·)] Here, 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.