We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral
∫Ωx,yF(γ)eiνS(γ)D(γ) of a functional
F(γ), cf. [13]. Here
Ωx,y is the set of paths
γ(t) in R
d starting from a point
y∈ R
d at time
0 and arriving at
x∈ R
d at time
T,
S(γ) is the action of
γ and
ν=2πh−1, with Planck's constant
h. Assuming that
p(γ) is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals: $ \int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma) $ $ = -\int_{\Omega_{x,y}} F(\gamma) {\rm Div}\, p(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma) -i\nu \int_{\Omega_{x,y}} F(\gamma)DS(\gamma)[p(\gamma)]e^{i\nu S(\gamma)}{\cal D}(\gamma). $ (1) Here
DF(γ)[p(γ)] and
DS(γ)[p(γ)] are differentials of
F(γ) and
S(γ) evaluated in the direction of
p(γ), respectively, and
Divp(γ) is divergence of vector field
p(γ). This formula is an analogy to Elworthy's integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case
F(γ∗)=0. Here
γ∗ is the stationary point of the phase
S(γ).
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.