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2006 Fiscal Year Final Research Report Summary

The theory of the pseudo-differential operators and its applications to the theory of the Feynman path integral

Research Project

Project/Area Number 16540145
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionShinshu University

Principal Investigator

ICHINOSE Wataru  Shinshu University, Math. Sci., Professor, 理学部, 教授 (80144690)

Co-Investigator(Kenkyū-buntansha) MORIMOTO Yoshinori  Kyoto University, Inte. Human Stud., Professor, 総合人間学部, 教授 (30115646)
HIROSHIMA Fumio  Kyusyu University, Fac. Math., Associate Professor, 数理学研究院, 助教授 (00330358)
KUMANO-GO Naoto  Kogakuin University, Math., Associate Professor, 工学部, 助教授 (40296778)
TANIUCHI Yasushi  Shinshu University, Math. Sci., Associate Professor, 理学部, 助教授 (80332675)
OTOBE Yoshiki  Shinshu University, Math. Sci., Lecturer, 理学部, 講師 (30334882)
Project Period (FY) 2004 – 2006
KeywordsFeynman path integral / partition function / correlation function / path integral of the functional / spin / Pauli equation / quantum electrodynamics / creation and annihilation operators
Research Abstract

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

  • Research Products

    (10 results)

All 2008 2007 2006 2005

All Journal Article (10 results)

  • [Journal Article] A mathematical note on the Feynman path integral for the quantum electrodynamics2008

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Surikaisekiken Kokyuroku (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A mathematical theory of the Feynman path integral for the generalized Pauli equations2007

    • Author(s)
      Wataru Ichinose
    • Journal Title

      J.Math.Soc.Japan 59・2

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] L^2 stability and boundedness of the Fourier integral operators applied to the theory of the Feynman path integral2007

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Progress in Analysis, the proceeding of the 5^<th> ISSAC Congress, World Scientific Vol.2

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A mathematical theory of the Feynman path integral for the generalized Pauli equations2007

    • Author(s)
      Wataru Ichinose
    • Journal Title

      J. Math. Soc. Japan 59(2) (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] L^2 stability and boundedness of the Fourier integral operators applied to the theory of the Feynman path integral2007

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Progress in Analysis, the proceeding of the 5^<th> ISSAC Congress, World Scientific Vol.2 (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A mathematical theory of the phase space Feynman path integral of the functional2006

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Commun.Math.Phys. 265・3

      Pages: 739-779

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the functional derivatives of the generating functional for correlation functions2006

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Surikaisekiken Kokyuroku 1479

      Pages: 130-141

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A mathematical theory of the phase space Feynman path integral of the functional2006

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Commun. Math. Phys. 265(3)

      Pages: 739-779

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On the Feynman path integral for particles with spin2005

    • Author(s)
      Wataru Ichinose
    • Journal Title

      日本数学会2005年度秋期総合分科会,函数解析分科会特別講演(招待講演)アブストラクト

      Pages: 50-55

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the Feynman path integral for particles with spin2005

    • Author(s)
      Wataru Ichinose
    • Journal Title

      The abstract of the invited talk in the section of the functional Analysis in the Congress of the Mathematical Society of Japan

      Pages: 50-55

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2008-05-27  

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