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

伪微分算子理论及其在费曼路径积分理论中的应用

基本信息

批准号：
16540145
负责人：
ICHINOSE Wataru
金额：
$ 2.37万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540145/
关键词：
Feynman path integral partition function correlation function path integral of the functional spin Pauli equation quantum electrodynamics creation and annihilation operators 量子測定理論 spin 汎函数微分 Pauli方程重み付きFeynman経路積分

项目摘要

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

我们项目的目的是研究通常用于物理学的Feynman路径积分，该积分是通过分段无动作（即破裂的线路路径）定义的。详细说明，我的研究计划如下。（1）不对称扩展理论。（2）量子连续测量的理论。（3）量子电子的理论。尽管我们不得不从计划的某些原因中将一部分计划更改，但在这三年中，我们可以在下面获得研究结果。（1）我们可以给出从分区功能中得出相关函数的公式的数学证明，该功能在物理学中是众所周知的。也就是说，对于n维真实的连续函数，分区函数z（j）f可以通过feynman路径的整数来定义，并且在特定意义上是可以区分的，并且它们的导数赋予了相关函数。（2）我们可以证明，我们可以证明了相位空间feynman path of Functionals z_j JJ（q _j）的阶段空间途径的存在（qjj）（q jj），p _j JJ（t _jj）操作员。从这个结果，我们可以给出数学定义和数学定义和数学证明，用于Feynman（1948）和Feynman-Hibbs（1965）启发性的公式。（3）给出了Feynman Path与Spin的粒子积分的定义的问题很长时间没有解决（请参见Feynman-Hibbs，Schulman（1981）。在我的研究中，我们可以给出一些带有旋转的颗粒的Feynman路径积分的定义，证明其存在并证明Feynman Path的积分在一个粒子的情况下满足了Pauli方程。（4）我们通过Feynman Path Integral的手段研究了量子电子的格式化。我们通过约束条件以高频的方式削减了照片的一部分，我们成功地将其成功地缩小了。我们还成功地通过相位空间Feynman路径积分来制定量子电动力学，而没有约束条件。此外，我们成功地通过差分运算符赋予了照片的创建和歼灭操作员。较少的