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Mathematical study of the Feynman path integrals and its application to quantum electro dynamic and quantum information theory

Research Project

Project/Area Number 18K03361
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionShinshu University

Principal Investigator

Ichinose Wataru  信州大学, 理学部, 特任教授 (80144690)

Project Period (FY) 2018-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords位相空間Feynman経路積分 / 制限Feynman経路積分 / Pauli方程式 / 量子連続測定 / Dirac方程式 / 相対論的Feynman経路積分 / 2重スリット実験 / Aharonov- Bohm効果 / スピン成分 / Neumannの射影公理 / Feynmanの公理 / 2重スリット実験 / Aharonov-Bohm効果 / スピンー軌道相互作用 / 非相対論的近似 / Foldy-Wouthuysen変換 / Feynman経路積分 / 量子情報理論 / 時間連続的な量子測定 / 時間連続的な量子測定理論 / 場の理論 / 量子観測理論 / 量子力学
Outline of Final Research Achievements

A mathematically rigorous study of the Feynman path integral was carried out.
(1) The restricted Feynman path integral, which represents a continuous quantum measure of the position of a particle, was formulated (2023). (2) Proved the derivation of the restricted Feynman path integral from the axioms of quantum mechanics (in submission). (3) A direct analysis of the phase space Feynman path integral was established (in preparation). (4) The Feynman path integral for the Schroedinger equation with potentials increasing on polynomial order was formulated (2018). (5) The Feynman path integral for the Dirac equation was formulated (2018). (6) Non-relativistic approximation formulations for the Dirac equation were proved (in preparation).

Academic Significance and Societal Importance of the Research Achievements

(1) 量子測定理論は、量子情報理論の重要な分野の一つである。本研究では、粒子位置に関する連続的な量子測定理論のFeynman経路積分による定式化(制限経路積分)の数学的意味付けを与えた(2023)。(2) 位相空間経路積分に関する基本的結果を導いたことにより、粒子の運動量・エネルギー等の一般の物理量に関する量子測定を定式化するFeynman経路積分の研究の準備が整った(準備中)。(3) Dirac方程式に対するFeynman経路積分を定式化することにより、量子電磁気学の経路積分による定式化の準備が整った(2018)。

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (16 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (12 results) (of which Peer Reviewed: 6 results) Presentation (4 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results)

  • [Journal Article] On the mathematical formulation of the restricted Feynman path integrals through broken line paths2023

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 60 Pages: 105-132

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the mathematical formulation of the restricted Feynman path integrals through broken line paths2022

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential2020

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Rev. Math. Phys

      Volume: 32 Pages: 1-37

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Notes on the Cauchy problem for the self-adjoint and non-self-adjoint Schroedinger equations with polynomially growing potentials2020

    • Author(s)
      Wataru Ichinose & Takayoshi Aoki
    • Journal Title

      J. Pseudo-Differ. Oper. Appl.

      Volume: 11 Issue: 2 Pages: 703-731

    • DOI

      10.1007/s11868-019-00301-6

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Feynman path integral for the magnetic Schroeinger equation with a polynomially growing electromagnetic potential2020

    • Author(s)
      Wataru Ichinose
    • Journal Title

      Rev. Math. Phys.

      Volume: 32 Issue: 01 Pages: 2050003-2050003

    • DOI

      10.1142/s0129055x20500038

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] A mathematical theory of the Feynman path integrals for continuous quantum position measurements2020

    • Author(s)
      Wataru Ichinose
    • Journal Title

      arXiv:2006.02710

      Volume: なし Pages: 1-38

    • Related Report
      2019 Research-status Report
  • [Journal Article] On the Feynman path integral for the Schroedinger equations with polynomially growing potentials in the spatial direction2019

    • Author(s)
      Wataru Ichinose
    • Journal Title

      arXiv:1901.05677

      Volume: なし Pages: 1-51

    • Related Report
      2018 Research-status Report
  • [Journal Article] Notes on the Cauchy problem for the self-adjoint and non-self-adjoint Schroedinger equations with polynomially growing potentials2019

    • Author(s)
      Wataru Ichinose、Takayoshi Aoki
    • Journal Title

      arXiv:1903.05465

      Volume: なし Pages: 1-39

    • Related Report
      2018 Research-status Report
  • [Journal Article] Feynman 経路積分の数学的結果2019

    • Author(s)
      Wataru Ichinose
    • Journal Title

      数理科学(特集「経路積分を考える - その変遷と基礎物理の発展」)

      Volume: 2 Pages: 60-68

    • Related Report
      2018 Research-status Report
  • [Journal Article] Notes on the Feynman path integral for the Dirac equation2018

    • Author(s)
      Wataru Ichinose
    • Journal Title

      J. Pseudo-Differ. Oper. Apple.

      Volume: 9 Issue: 4 Pages: 789-809

    • DOI

      10.1007/s11868-017-0227-7

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the Feynman path integral for the Schroedinger equations with polynomially growing potentials in the spatial direction2018

    • Author(s)
      Wataru Ichinose
    • Journal Title

      第57回実函数論・函数解析合同シンポジウム講演集 http://mathsoc.jp/section/fctanalysis/2018.html

      Volume: なし Pages: 108-118

    • Related Report
      2018 Research-status Report
  • [Journal Article] Feynman経路積分による量子連続測定の数学理論 - 位置とスピンの測定2018

    • Author(s)
      Wataru Ichinose
    • Journal Title

      電子情報通信学会・量子技術研究会講演集

      Volume: なし Pages: 83-85

    • Related Report
      2018 Research-status Report
  • [Presentation] A mathematical theory from the projection postulate to the restricted Feynman path integral for a continuous position measurement2021

    • Author(s)
      Wataru Ichinose
    • Organizer
      夏の作用素論シンポジウム
    • Related Report
      2021 Research-status Report
  • [Presentation] On the Feynman path integral with potentials growing polynomially in the spatial direction2018

    • Author(s)
      Wataru Ichinose
    • Organizer
      実関数論・函数解析合同シンポジウム
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Mathematical theory of the Feynman path integrals for quantum continuous measurement of positions and spin2018

    • Author(s)
      Wataru Ichinose
    • Organizer
      国際数理物理学会
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research
  • [Presentation] Feynman経路積分による量子連続測定の数学理論ー位置とスピンの測定2018

    • Author(s)
      Wataru Ichinose
    • Organizer
      電子情報通信学会第38回量子情報技術研究会
    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2024-01-30  

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