Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions
Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions
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加权Sobolev空间中费曼路径积分的收敛性及相关函数的表示
DOI：
10.2969/jmsj/1191418759
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发表时间：
2003
影响因子：
0.7
通讯作者：
W. Ichinose
中科院分区：
文献类型：
--
作者：
W. Ichinose
。有许多方法可以给费曼路径积分赋予严格的意义。本文重点研究了用折线路径确定时间切片近似的方法。证明了当离散化参数趋于零时，Feynman路径积分在定态fi空间和相空间的时间切片近似积分在L 2空间收敛.本文证明了这些时间分片近似积分在某些加权Soblev空间中也是收敛的。其次，作为这一收敛结果在加权Soblev空间中的应用，研究了关联函数的位置算子和动量算子的路径积分表示。我们注意到它们的路径积分表示是在相空间中给出的。结果表明，当离散化参数趋于零时，相关函数的近似积分收敛或发散。我们注意到近似积分的发散反映了量子力学中的测不准原理。
. There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L 2 space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.
维纳空间上的薛定谔算子
DOI：
--
发表时间：
2007
期刊：
Communications on Stochastic Analysis 1
影响因子：
--
作者：
Sugitani;K.ほか;H. Sunagawa;落合啓之;谷口正信;A. Kishimoto;I. Shigekawa
通讯作者：
I. Shigekawa