Convergence of the Feynman path integral in the weighted Sobolev spaces and the representation of correlation functions

• ICHINOSE Wataru

  Department of Mathematical Sciences Shinshu University

抄録

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 L² 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.

収録刊行物

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

  Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ 55 (4), 957-983, 2003

  一般社団法人 日本数学会

キーワード

• Feynman path integral
• Uncertainty principle
• Correlation functions

詳細情報

• CRID: 1390282680091552256
• NII論文ID: 10013702228
• NII論文ID: 30030583242
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/1191418759
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 2003755
• NDL書誌ID: 6723231
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I6723231
• 本文言語コード: en
• データソース種別:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
• 抄録ライセンスフラグ: 使用不可