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物理学中通常使用的费曼路径积分，它是通过分段自由运动（即折线路径）来定义的。具体而言，我的研究计划如下：（1）理论。（2）量子连续测量理论。（3）量子电动力学理论，虽然由于某些原因不得不改变部分计划，但是这三年我们还是得到了以下的研究成果。（ 1）我们可以给出从配分函数导出相关函数的公式的数学证明，这在物理学中是众所周知的，即对于n维实值连续函数J，配分函数Z(J)f可以通过费曼方程来定义。路径积分和在 Frechet 意义上可微，它们的导数给出了相关函数。 (2) 我们可以证明泛函乘积的相空间费曼路径积分的存在性z_j(q(t_j),p(t_j)) 并通过运算符给出其表示形式。从这个结果我们可以给出 Feynman (1948) 和 Feynman-Hibbs 中给出的公式的数学定义和数学证明。 1965）启发式地。（3）给出具有自旋的粒子的费曼路径积分的定义的问题长期以来一直没有得到解决（参见第355页） Feynman-Hibbs, Schulman(1981)）在我的研究中，我们可以给出一些具有自旋的粒子的费曼路径积分的定义，证明其存在性，并证明费曼路径积分在单个粒子的情况下满足泡利方程。 (4)我们利用费曼路径积分研究了量子电动力学的形式化，我们在假设用约束条件切断了高频部分的情况下取得了成功。我们还成功地用相空间费曼路径积分形式化了无约束条件的量子电动力学，并用微分算子具体地给出了光的产生和湮灭算子。