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首页> 外文期刊>Journal of the Mathematical Society of Japan >Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions
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Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions

机译：加权Sobolev空间中Feynman路径积分的收敛性和相关函数的表示。

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摘要

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 zeco. 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.
机译：有很多方法可以使费曼路径积分具有严格的含义。在本文中，特别研究了通过虚线路径确定时间分段近似的方法。证明了随着离散参数趋于zeco，Feynman路径积分在时间上的这些分时的近似积分在构型空间和相空间中都收敛于L〜2空间。在本文中表明，这些时间切片近似积分也在某些加权Sobolev空间中收敛。接下来，作为该收敛结果在加权Sobolev空间中的应用，研究了位置和动量算符的相关函数的路径积分表示。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，随着离散参数趋于零，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不确定性原理。

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