Application of the pseudo-differential operotous to the Feynmon path

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The aim of our project was to give the rigorous meaning to the Feynman path integrals used in physics. In detail, we study : (1) We give the rigorous meaning to the Feynman path integrals in phase space.(2) We give the rigorous meaning to the path integral representations of correlation functions and generating functions Z (J) in quantum mechanics and also quantum free field. Doc.N.Kumano-go was made responsible for a part of the study of the path integral by means of Fourier integral operators.We can say about our reserch result that our aim was completed except for the study of quntum free field as will be stated. In the paper [1] we proved convergence of the Feynman path integral in confuguration space independently of the choice of electromagnetic potentials, which implies the gauge invariance of the Feynman path integrals and generalizes the results proved in the author's paper in C.M.P (1997). From this result we can get the path integral representation of Z (J) in quantumm mechanics. In [2] we modified the definition of the path integral in phase space used in physics and showed its gauge invariance, convergence and the fact that this path integral in phase space is equal to the path integral in confuguration space familiar in physics. In [3] we gave the rigorous meaning to the path integral representation of correlation functions including not only position operators but also momentum operators. Here the path integrals in phase space in [2] was essentially used. In [4,5] Kumano-go gave a simple proof of convergence of the kernel of the path integral defined through piecewise classical paths by using the Fourier integral operators. In [6], usnig the same method, Kumano-go gave a simple proof of the path integral defined through broken line paths.