An integration by parts formula for Feynman path integrals (Q392589)

This paper gives a proof of the integration-by-parts relation appearing in the abstract NEWLINE\[NEWLINE\begin{multlined} \int\limits_{\Omega_{x,y}} DF\left(\gamma\right)\left[p\left(\gamma\right)\right] e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right)\\ = - \int\limits_{\Omega_{x,y}} F\left(\gamma\right) \mathrm{Div}p\left(\gamma\right) e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right) - i \nu \int\limits_{\Omega_{x,y}} F\left(\gamma\right) DS\left(\gamma\right)\left[p\left(\gamma\right)\right] e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right).\end{multlined} NEWLINE\]NEWLINE The value of this paper is the careful definition of all quantities and the rigorous proof of the above formula. The authors define the path integral by a time-slicing procedure in eq.(9). The functional \(F(\gamma)\) has to be a \(m\)-smooth, where \(m\)-smoothness is defined in definition (1.1). \(p(\gamma)\) is the tangent vector field to the path \(\gamma\) and \(p(\gamma)\) has to be an \(m'\)-admissible vector field, where \(m'\)-admissibility is defined in definition (3.1). Furthermore the authors require that two out of the three quantities \(DF(\gamma)[p(\gamma)]\), \(F(\gamma) \mathrm{Div}p(\gamma)\) or \(F(\gamma) DS(\gamma)[p(\gamma)]\) are \(F\)-integrable, where the concept of \(F\)-integrability is defined by requiring that the corresponding time-slicing limit of the path-integral exists. In addition, there is the condition \(T \leq \mu\), which relates the path intervall \([0,T]\) and the potential \(V(t,x)\) through eq.(5) and eq.(6). Under these conditions the authors prove the integration-by-parts relation.NEWLINENEWLINEThe last section of the paper gives an application of the integration-by-parts relation towards the semiclassical asymptotic behaviour of Feynman path integrals.