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The theory of osicllatory integral operartors and its application to the Feynman path integral of quntum field theory

Research Project

Project/Area Number 26400161
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionShinshu University

Principal Investigator

ICHINOSE Wataru  信州大学, 学術研究院理学系, 教授 (80144690)

Co-Investigator(Kenkyū-buntansha) 佐々木 格  信州大学, 学術研究院理学系, 准教授 (50558161)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
KeywordsFeynman 経路積分 / Dirac 方程式 / 量子電磁気学 / Schroedinger 方程式 / 量子力学 / Feynman経路積分 / Dirac方程式 / Schroedinger方程式 / Feynman propagator / 空間方向に多項式オーダーで増大するポテンシャル / 場の理論 / 相対論的な因果律 / 緩増加超関数空間
Outline of Final Research Achievements

(1) We have constructed the Feynman path integral for a relativistic electron in the form of the sum-over-histories, over all paths of one electron in space-time that goes in any direction at any speed, forward and backward in time, especially with a countably infinite number of turns. By this result we have succeeded in introducing a countably infinite number of electrons and positrons at the same time in the theory of the Feynman path integrals as the 2nd quantization of one electron. (2) We have showed that the Feynman path integral defined in (1) is relativistically invariant, i.e. has the property of spinor under the Lorentz transformations. (3) We have showed directly from the Feynman path integral that the probability amplitude for a relativistic electron has the properties of both unitarity and causality. (4) We have constructed mathematically the Feynman path integral for the Schroedinger equations with potentials growing polynomially in the spatial direction.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (16 results)

All 2017 2016 2015 2014

All Journal Article (5 results) (of which Peer Reviewed: 4 results,  Acknowledgement Compliant: 2 results) Presentation (11 results) (of which Int'l Joint Research: 2 results,  Invited: 6 results)

  • [Journal Article] Essential spectrum of the discrete Laplacian on a perturbed periodic graph2017

    • Author(s)
      I. Sasaki and A. Suzuki
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 446 Pages: 1863-1881

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Notes on the Feynman path integral for the Dirac equation2017

    • Author(s)
      W. Ichinose
    • Journal Title

      arXiv. org

      Volume: 1705.04040 Pages: 1-29

    • Related Report
      2016 Annual Research Report
    • Acknowledgement Compliant
  • [Journal Article] On the construction of the Feynman path integral for the Dirac equation2016

    • Author(s)
      W. Ichinose
    • Journal Title

      Suuriken Kokyuroku

      Volume: 印刷中

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the construction of the Feynman path integral for the Dirac equation2015

    • Author(s)
      W. Ichinose
    • Journal Title

      Suuriken Kokyuroku

      Volume: 1958 Pages: 63-80

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Enhanced Binding of an N-particle System Interacting with a Scalar Field II. Relativistic Version2015

    • Author(s)
      F. Hiroshima, I. Sasaki
    • Journal Title

      Publ. RIMS

      Volume: 51 Issue: 4 Pages: 655-690

    • DOI

      10.4171/prims/168

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Presentation] Embedded Eigenvalue and von Neumann-Wigner Potential for the Relativistic Schroedinger Operator2016

    • Author(s)
      I. Sasaki
    • Organizer
      QMath13 - Mathematical Results in Quantum Physics
    • Place of Presentation
      Georgia 工科大学(USA)
    • Year and Date
      2016-10-08
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research
  • [Presentation] On the Cauchy problem for the Schroedinger equations with polynomially growing potentials in the spatial direction2016

    • Author(s)
      W. Ichinose
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学理工学部
    • Year and Date
      2016-09-29
    • Related Report
      2016 Annual Research Report
  • [Presentation] The Feynman path integral for the Schroedinger equations with polynomially growing potentials in the spatial direction2016

    • Author(s)
      W. Ichinose
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学理工学部
    • Year and Date
      2016-09-29
    • Related Report
      2016 Annual Research Report
  • [Presentation] On the Embedded Eigenvalue of Relativistic Schrodinger Operator2016

    • Author(s)
      I. Sasaki
    • Organizer
      偏微分方程式姫路研究集会
    • Place of Presentation
      兵庫県姫路市市民センター
    • Year and Date
      2016-03-03
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Feynman path integral for the Dirac equation in L^2 space2015

    • Author(s)
      W. Ichinose
    • Organizer
      信州大学偏微分方程式研究集会
    • Place of Presentation
      信州大学理学部
    • Year and Date
      2015-06-12 – 2015-06-13
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the construction of the Feynman path integral for the Dirac equations2015

    • Author(s)
      W. Ichinose
    • Organizer
      松本偏微分方程式研究集会
    • Place of Presentation
      信州大学理学部
    • Year and Date
      2015-06-12
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Dirac方程式に対するFeynman経路積分 (1)ー 無限遠方の過去と未来を行き交う電子,2015

    • Author(s)
      W. Ichinose
    • Organizer
      日本数学会関数方程式分科会
    • Place of Presentation
      明治大学理工学部
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] Dirac方程式に対するFeynman経路積分 (2) ー 因果律の相対論的不変性(有限伝播性)2015

    • Author(s)
      W. Ichinose
    • Organizer
      日本数学会関数方程式分科会
    • Place of Presentation
      明治大学理工学部
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] On the Feynman path integral for the Dirac equations2015

    • Author(s)
      W. Ichinose
    • Organizer
      愛媛大学解析セミナー
    • Place of Presentation
      愛媛大学理学部
    • Year and Date
      2015-01-24
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the construction of the Feynman path integral for the Dirac equations2014

    • Author(s)
      W. Ichinose
    • Organizer
      RIMS Joint Reserch "Inroductory Workshop on Path Integrals and Pseudo-Differential Operators
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2014-10-08 – 2014-10-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On the construction of the Feynman path integral for the Schroedingier and the Dirac equations in the space of tempered distributions2014

    • Author(s)
      W. Ichinose
    • Organizer
      夏の作用素論セミナー
    • Place of Presentation
      長浜勤労福祉会館( 滋賀県長浜市)
    • Year and Date
      2014-09-07 – 2014-09-09
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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