Application of the pseudo-differential operotous to the Feynmon path

伪微分运算在Feynmon路径中的应用

• 批准号: 10640176
• 负责人: ICHINOSE Wataru
• 金额: $ 1.86万
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640176/
• 关键词: pendo-diffenntial; fouri intogral oporotoo; Schrodinsyr eguotion; path integral; Feynman

The aim of our project was to give the rigorous meaning to the Feynman path integrals used in physics. In detail, we study : (1) We give the rigorous meaning to the Feynman path integrals in phase space.(2) We give the rigorous meaning to the path integral representations of correlation functions and generating functions Z (J) in quantum mechanics and also quantum free field. Doc.N.Kumano-go was made responsible for a part of the study of the path integral by means of Fourier integral operators.We can say about our reserch result that our aim was completed except for the study of quntum free field as will be stated. In the paper [1] we proved convergence of the Feynman path integral in confuguration space independently of the choice of electromagnetic potentials, which implies the gauge invariance of the Feynman path integrals and generalizes the results proved in the author's paper in C.M.P (1997). From this result we can get the path integral representation of Z (J) in quantumm mechanics. In [2] we modified the definition of the path integral in phase space used in physics and showed its gauge invariance, convergence and the fact that this path integral in phase space is equal to the path integral in confuguration space familiar in physics. In [3] we gave the rigorous meaning to the path integral representation of correlation functions including not only position operators but also momentum operators. Here the path integrals in phase space in [2] was essentially used. In [4,5] Kumano-go gave a simple proof of convergence of the kernel of the path integral defined through piecewise classical paths by using the Fourier integral operators. In [6], usnig the same method, Kumano-go gave a simple proof of the path integral defined through broken line paths.

我们的项目的目的是给严格意义的费曼路径积分在物理学中使用。具体研究内容如下：（1）给出了相空间中Feynman路径积分的严格意义。(2)给出了量子力学和量子自由场中关联函数和生成函数Z（J）的路径积分表示的严格意义。熊野吾博士负责用傅里叶积分算子研究路径积分的一部分工作。我们可以说，除了下面要说明的量子自由场的研究之外，我们的研究结果已经完成。在文[1]中，我们证明了Feynman路径积分在配置空间中的收敛性与电磁势的选择无关，这暗示了Feynman路径积分的规范不变性，推广了作者在C.M.P（1997）中所证明的结果。由这个结果可以得到量子力学中Z（J）的路径积分表示。在文[2]中，我们对物理学中相空间中的路径积分的定义作了修改，并证明了它的规范不变性、收敛性以及相空间中的路径积分与物理学中常见的组态空间中的路径积分等价。在文[3]中，我们给出了关联函数的路径积分表示的严格意义，它不仅包括位置算符，而且还包括动量算符。这里基本上使用了[2]中相空间中的路径积分。Kumano-go在文献[4，5]中利用Fourier积分算子给出了由分段经典路径定义的路径积分的核的收敛性的一个简单证明。Kumano-go在[6]中用同样的方法给出了由折线路径定义的路径积分的一个简单证明。

项目成果

N.Kumano-go: "A proot H.Kumano-go-Taniguchi theorem for malti-products of Fourier integral operators" Osaka Journal of Mathematics. 35・2. 357-380 (1998)

N.Kumano-go：“傅里叶积分算子的马尔蒂积的原始 H.Kumano-go-Taniguchi 定理” 大阪数学杂志 35・2（1998）。

W.Ichinose: "The phase space Feynman path integrol with gouge inoaniance and its conwegence"Deview in Mathematical Physical. 12-11. 1451-1463 (2000)

W.Ichinose：“相空间费曼路径积分与凿痕 inoaniance 及其聚合”数学物理中的评论。

W.Ichinose: "The Feynman path integral represan tation of Green funations o fposition and momentum operatain"Suriden Kopuroku. (to appoar). (2001)

W.Ichinose：“位置和动量操作的绿色函数的费曼路径积分表示”Suriden Kopuroku。

D.Fujiwara et al: "A simple proof of H.Kumano-go-Taniguch's Theorem"Surikon Kokuuku. 1056. 59-67 (1998)

D.Fujiwara 等人：“H.Kumano-go-Taniguch 定理的简单证明”Surikon Kokuuku。

N.Kumamo-go: "Estimates for the time slicing appoximations of the Feynmen path into grols"Kogakuin University Peport. 87. 1-8 (1999)

N.Kumamo-go：“将 Feynmen 路径的近似值切片为 grols 的时间估计”Kogakuin University Peport。