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

我们的项目的目的是研究Feynman路径积分，通常用于物理学，这是通过分段自由运动，即折线路径定义。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，我们不得不改变我的计划的一部分，我们可以得到下面的研究结果，这三年。(1)对于物理学中已知的由配分函数导出关联函数的公式，我们可以给出数学证明。也就是说，对于n维真实的值连续函数J，配分函数Z（J）f可以通过Feynman路径积分定义，并且在Frechet意义下是可微的，它们的导数给出相关函数。(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 ...更多信息 通过操作符表示。从这个结果出发，我们可以直观地给出Feynman（1948）和Feynman-Hibbs（1965）公式的数学定义和数学证明。(3)给出自旋粒子的费曼路径积分的定义的问题已经很长时间没有解决了（参见。第355页，Feynman-Hibbs，Schulman（1981））。在我的研究中，我们给出了某些自旋粒子的Feynman路径积分的定义，证明了它的存在性，并证明了在单粒子情况下Feynman路径积分满足Pauli方程。(4)利用费曼路径积分研究了量子电动力学的形式化。我们在物理学中通常假设的约束条件下，截断了高频光子的部分，从而成功地解决了这一问题。我们还成功地利用相空间的费曼路径积分形式化了无约束条件的量子电动力学。此外，我们还利用微分算符的方法，成功地具体给出了照片的产生算符和湮灭算符。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