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Path integrals - Expansion of analysis on path space by time slicing approximation

Research Project

Project/Area Number 24540193
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionKogakuin University

Principal Investigator

KUMANOGO NAOTO  工学院大学, 基礎・教養教育部門, 教授 (40296778)

Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords経路積分 / 関数方程式論 / 関数解析学 / 数理物理 / 擬微分作用素 / フーリエ積分作用素 / 振動積分 / 確率解析 / 関数方程式 / 確率論
Outline of Final Research Achievements

We gave a general class of functionals for which the phase space path integrals of higher order parabolic type have a mathematically rigorous meaning. For any functional belonging to this class, the time slicing approximation of the phase space path integral, converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Because this class of functionals is closed under addition and multiplication, we can produce many functionals which are path integrable. Furthermore, though we need to pay attention for use, the interchange of the order of the phase space path integrals and some integrals, and the interchange of the order of the phase space path integrals and some limit operations, are valid.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (12 results)

All 2015 2014 2013 2012 Other

All Journal Article (6 results) (of which Peer Reviewed: 2 results,  Acknowledgement Compliant: 2 results) Presentation (6 results) (of which Invited: 1 results)

  • [Journal Article] Phase space Feynman path integrals of higher order parabolic type with general functional as integrand2015

    • Author(s)
      Naoto Kumano-go, A. S. Vasudeva Murthy
    • Journal Title

      Bulletin des Sciences Mathematiques

      Volume: In Press Issue: 5 Pages: 495-537

    • DOI

      10.1016/j.bulsci.2014.11.001

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Phase space Feynman path integrals as analysis on path space2014

    • Author(s)
      熊ノ郷 直人
    • Journal Title

      数理解析研究所講究録

      Volume: 1902 Pages: 1-20

    • Related Report
      2014 Annual Research Report
    • Acknowledgement Compliant
  • [Journal Article] Phase space path integrals as analysis on path space2013

    • Author(s)
      Naoto Kumano-go
    • Journal Title

      数理解析研究所講究録

      Volume: 1861 Pages: 83-99

    • Related Report
      2013 Research-status Report
  • [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations2013

    • Author(s)
      Naoto Kumano-go, Daisuke Fujiwara
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B37 Pages: 113-136

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Phase space Feynman path integrals - calculation examples via piecewise bicharacteristic paths2012

    • Author(s)
      Naoto Kumano-go
    • Journal Title

      数理解析研究所講究録

      Volume: 1797 Pages: 167-186

    • Related Report
      2012 Research-status Report
  • [Journal Article] Phase space Feynman path integrals - as analysis on path space via piecewise constant paths2012

    • Author(s)
      Naoto Kumano-go
    • Journal Title

      数理解析研究所講究録

      Volume: 1797 Pages: 187-203

    • Related Report
      2012 Research-status Report
  • [Presentation] Path integrals for Gaussian Processes as analysis on path space2015

    • Author(s)
      Naoto Kumano-go
    • Organizer
      Workshop "Geometric and Singular Analysis"
    • Place of Presentation
      The University of Potsdam, Germany
    • Year and Date
      2015-02-10
    • Related Report
      2014 Annual Research Report
  • [Presentation] Phase space Feynman path integrals of higher order parabolic type2014

    • Author(s)
      Naoto Kumano-go
    • Organizer
      RIMS Joint Research "Introductory 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
      2014-10-10
    • Related Report
      2014 Annual Research Report
  • [Presentation] Phase space Feynman path integrals as analysis on path space

    • Author(s)
      熊ノ郷 直人
    • Organizer
      RIMS研究集会「スペクトル・散乱理論とその周辺」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Research-status Report
  • [Presentation] Phase space Feynman path integrals of m-th order parabolic type for general functionals

    • Author(s)
      Naoto Kumano-go
    • Organizer
      Workshop 'Geometric and Singular Analysis'
    • Place of Presentation
      The University of Potsdam, Germany
    • Related Report
      2013 Research-status Report
  • [Presentation] 相空間の経路積分-経路空間上の解析として-

    • Author(s)
      熊ノ郷 直人
    • Organizer
      日本数学会秋季総合分科会、函数方程式論特別講演
    • Place of Presentation
      九州大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

    • Author(s)
      Naoto Kumano-go
    • Organizer
      Special Session on Interplays Between Feynman Operational Calculus, Wiener and Feynman Integrals, Physics, and Analysis on Wiener Space, AMS Joint Mathematics Meetings
    • Place of Presentation
      San Diego, USA
    • Related Report
      2012 Research-status Report

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Published: 2013-05-31   Modified: 2019-07-29  

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