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Path integrals for Dirac equations

Research Project

Project/Area Number 26610020
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionOchanomizu University

Principal Investigator

FURUYA Kiyoko  お茶の水女子大学, 基幹研究院, 講師 (80189208)

Project Period (FY) 2014-04-01 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords発展方程式 / 経路積分 / ディラック方程式 / ベクトル値測度 / 半群
Outline of Final Research Achievements

The idea of Feynman's integral is a topic of great interest in mathematics and physics. But rigorous mathematical treatment of this integral is not enough. We defined a kind of operator-valued integration and define the path integrals.
Dirac equation is the basic equation of relativistic quantum mechanics to the fermions. For the case of space-dimension = 1, Ichinose proved the path integral for Dirac equations are represented by a scalar measure. For the case of radial Dirac equation Ichinose constructed a countably additive path space measure. But in general, for the case of space-dimension > 1, Feynman path integrals for Dirac equations are not represented by (scalar-valued) measures. We showed that the path integral for Dirac equations are represented by our vector-valued measure.

Report

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

    (13 results)

All 2015 2014

All Journal Article (6 results) (of which Peer Reviewed: 4 results,  Open Access: 2 results,  Acknowledgement Compliant: 3 results) Presentation (7 results) (of which Int'l Joint Research: 1 results,  Invited: 4 results)

  • [Journal Article] About formally self-adjoin Scher¨odinger operator with potential.2015

    • Author(s)
      KIYOKO FURUYA
    • Journal Title

      Proceedings of the 9th International Conference on Nonlinear Analysis and Convex Analysis

      Volume: 1 Pages: 99-109

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] On Formally Self-Adjoint Scher¨odingerr Operators with measurable potential.2015

    • Author(s)
      KIYOKO FURUYA
    • Journal Title

      Mathematics for Nonlinear Phenomena: Analysis and Computation ‐International Conference in honor of Professor Yoshikazu Giga on his 60th birthday‐Hokkaido University

      Volume: 163 Pages: 41-41

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On ”well posed function spaces” for hyperbolic equations of the simplest type2015

    • Author(s)
      KIYOKO FURUYA
    • Journal Title

      第41 回発展方程式研究会報告集

      Volume: 1 Pages: 47-50

    • Related Report
      2015 Annual Research Report
  • [Journal Article] On Formally Self-Adjoint Schr¨odinger operators with real potential2015

    • Author(s)
      KIYOKO FURUYA
    • Journal Title

      2015 CMS Summer Meeting Program

      Volume: 1 Pages: 70-71

    • Related Report
      2015 Annual Research Report
    • Acknowledgement Compliant
  • [Journal Article] Formally self-adjoint Scherödinger operators with real measurable potential.2014

    • Author(s)
      Furuya, Kiyoko
    • Journal Title

      J. Nonlinear Convex Anal

      Volume: 15 Pages: 1003-1017

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] A contraction semigroup for formally self-adjoin Scherödinger operator.2014

    • Author(s)
      KIYOKO FURUYA
    • Journal Title

      Proceedings of The 8th International Conference on Nonlinear Analysis and Convex Analysis-I-(Hirosaki, Japan, 2013

      Volume: 1 Pages: 41-53

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] On “well posed function spaces” for hyperbolic equations of the simplest Type2015

    • Author(s)
      KIYOKO FURUYA
    • Organizer
      第41回発展方程式研究会
    • Place of Presentation
      日本女子大学
    • Year and Date
      2015-12-25
    • Related Report
      2015 Annual Research Report
  • [Presentation] 「カナダ数学会の夏のセミナー」に参加して2015

    • Author(s)
      KIYOKO FURUYA
    • Organizer
      2015 秋の偏微分方程式セミナー
    • Place of Presentation
      大阪大学
    • Year and Date
      2015-09-10
    • Related Report
      2015 Annual Research Report
  • [Presentation] On Formally Self-Adjoint Scherodinger Operators with measurable potential.2015

    • Author(s)
      KIYOKO FURUYA
    • Organizer
      Mathematics for Nonlinear Phenomena: Analysis and Computation ‐ International Conference in honor of Professor Yoshikazu Giga on his 60th birthday
    • Place of Presentation
      札幌コンベンションセンター
    • Year and Date
      2015-08-16
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Formally Self-Adjoint Scherodinger Operators with potential.2015

    • Author(s)
      KIYOKO FURUYA
    • Organizer
      The 2015 CMS Summer Meeting
    • Place of Presentation
      Charlottetown, P.E.I.
    • Year and Date
      2015-06-05
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] About formally self-adjoin Scherödinger operator with potential2015

    • Author(s)
      Kiyoko FURUYA
    • Organizer
      The 9th International Conference on Nonlinear Analysis and Convex Anal- ysis,
    • Place of Presentation
      Rimkok Resort Hotel, Chiang Rai, THAILAND
    • Year and Date
      2015-01-21 – 2015-01-25
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 形式的に自己共役なシュレディンガー作用素について2014

    • Author(s)
      古谷 希世子
    • Organizer
      第40回発展方程式研究会
    • Place of Presentation
      日本女子大, 東京都文京区
    • Year and Date
      2014-12-25 – 2014-12-27
    • Related Report
      2014 Research-status Report
  • [Presentation] Vector Valued Measure for Path Integral Representations of a Solution for Dirac Equation2014

    • Author(s)
      Kiyoko FURUYA
    • Organizer
      ICM Satellite Conference 2014: The Fourth Asian Conference on Nonlin- ear Analysis and Optimization
    • Place of Presentation
      National Taiwan Normal Univers,Taipei, Taiwangity
    • Year and Date
      2014-08-05 – 2014-08-09
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2017-05-10  

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