There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L2 space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.
Convergence of a time-slicing approximation is studied in a general way of the Feynman path integral in phase space with an electromagnetic potential. In the present paper a new approximation of the Feynman path integral in phase space, different from the familiar one in physics, is proposed so that its convergence can be shown independently of the choice of electromagnetic potentials and that gauge invariance of the Feynman path integral defined by its limit can be shown directly. It is also shown that the Feynman path integral in phase space defined above can be expressed in the form of the Feynman path integral in configuration space defined in the familiar way in physics, which Feynman himself stated heuristically.