The application of the theory of the pseudo-diffrential operators to the Feynman path integral

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The aim of our project was to give the rigorous meaning to the Feynman path integral usually used in physics, which is defined by the method of the time-slicing approximation determined through broken line paths. In detail, we study : (1)We show the convergence of the phase space Feynman path integral of the functional as the discretization parameter tends to zero and give its expression in terms of the operator notation. (2)We show the existence of the generating functional Z(J) of the source J in quantum mechanics and also quantum free field theory. Next we show the functional differentiability of Z(J) in J and that its derivative gives the correlation function. (3)We give the mathematical proof to the perturbation theory of the path integral.Our aim has been completed except for the study of Z(J) of quantum free field theory and (3). First, we studied the convergence of the phase space Feynman path integral that is usually used in physics. We proved that its path integral converges … More to the solution of the corresponding Schrodinger equation in L^2 space. Next, we proved the convergence of the phase space path integral not only in L^2 space but also in the generalized Sobolev spaces B^a. By means of this result we studied the convergence of the phase space path integral of the functional II^N_<j=1> q(t_j)p(t_j). We gave their expressions in terms of the operator notation and showed that these path integrals give correlation functions, the canonical commutator relation and etc. In addition, we generalized this result to the phase space path integral of the functional II^N_<J=1> z_j(q(t_j),p(t_j)). We gave the necessary and sufficient condition on z_j(x, p) (j=1,2,【triple bond】,N) for this phase space path integral to be convergent. We also gave the expression in terms of the operator notation when the path integral was convergent. From this result we could give the rigorous proofs of the heuristic results in Feynman's celebrated paper 1948. Next, we showed the existence of the generating functional Z(J)f (f∈B^a) for any J∈X, where X is the set of all R^n-valued continuous functions on [O, T]. We also proved that the functional Z(J)f : X→B^a is Frechet differentiable in J and its Frechet derivatives give correlation functions. The paper on this result is in preparation. In relation to our project each investigator studied the integral operator of the oscillatory type, the ground state transition of Bose field, and BMO space and Besov space and its applications to the differential equations Less