The theory of the pseudo-differential operators and its applications to the theory of the Feynman path integral

我们项目的目的是研究物理中常用的Feynman路径积分，它是通过分段自由运动定义的，即折线路径。具体来说，我的研究计划如下。(1)渐近展开理论。(2)量子连续测量理论。(3)量子电动力学理论。虽然由于某些原因我们不得不改变了计划的一部分，但我们可以得到以下三年的研究结果：(1)我们可以给出从配分函数推导关联函数的公式的数学证明，这在物理上是众所周知的。也就是说，对于n维实连续函数J，配分函数Z(J)f可以用Feynman路径积分来定义，并且在Frechet意义下是可微的，它们的导数给出了相关函数。(2)证明了泛函z_j(q(T_J)，p(T_J))乘积的相空间Feynman路径积分的存在性，并给出了…更多的是通过运算符的方式来表示。根据这一结果，我们可以启发式地给出Feynman(1948)和Feynman-Hibbs(1965)中所给出的公式的数学定义和数学证明。(3)给出具有自旋的粒子的Feynman路径积分的定义的问题长期以来一直没有解决。(见Feynman-Hibbs，Schulman(1981)，第355页)。在我的研究中，我们可以给出一些具有自旋的粒子的Feynman路径积分的定义，证明它的存在性，并证明在一个粒子的情况下Feynman路径积分满足Pauli方程。(4)利用Feynman路径积分研究量子电动力学的形式化。在物理中通常假定的约束条件下，我们成功地实现了对高频光子的截断。利用相空间Feynman路径积分，成功地实现了不受约束条件的量子电动力学的形式化。此外，我们还成功地利用微分算子具体地给出了照片的产生和湮灭算子。较少

The aim of our project was to study the Feynman path integrals, usually used in physics, which is defined by means of piecewise free motions, i.e. broken line paths. In detail, my research plan was as follows. (1) The theory of the asymptotic expansion. (2) The theory of quantum continuous measurements. (3) The theory of the quantum electrodynamics. Though we had to change a part of my plan from some reasons, we could get the research results below for these three years.(1) We could give the mathematical proof of the formula deriving the correlation functions from the partition function, which is well known in physics. That is, for n dimensional real valued continuous function J the partition function Z(J)f can be defined by means of the Feynman path integral and is differentiable in the Frechet sense, and their derivatives give the correlation functions.(2) We could prove the existence of the phase space Feynman path integral for the product of functionals z_j(q(t_j),p(t_j)) and gave … More its representation by means of operators. From this result we could give the mathematical definition and the mathematical proof for the formulas given in Feynman (1948) and Feynman-Hibbs (1965) heuristically.(3) The problem to give the definition of the Feynman path integral for a particle with spin has been not solved for a long time (cf. p.355 in Feynman-Hibbs, Schulman(1981)). In my research we could give the definition of the Feynman path integral for some particles with spin, prove its existence and prove that the Feynman path integral satisfies the Pauli equation in the case of one particle.(4) We studied the formalization of the quantum electrodynamics by means of the Feynman path integral. We succeeded in it under the assumption cutting off the part of photons with high frequency by means of the constraint condition, which is assumed usually in physics. We also succeeded in formalizing the quantum electrodynamics without the constraint condition by means of the phase space Feynman path integral. In addition, we succeed in giving the creation and annihilation operators of photos by means of differential operators concretely. Less