Theory of path integrals by time slicing approximation and its applications

• Project/Area Number: 18740077
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Kogakuin University
• Principal Investigator: KUMANOGO Naoto Kogakuin University, 工学部, 准教授 (40296778)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,830,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000); Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
• Keywords: 経路積分 / 関数方程式 / 関数解析学 / 数理物理 / 振動積分 / 擬微分作用素 / 準古典近似 / 確率解析 / 確率論 / フーリエ積分作用素 / Fourier積分作用素

Research Abstract

区分的に陪特性な経路を用いた時間分割近似法により、相空間(ハミルトン型)経路積分の理論を構成した。厳密に言えば、時間分割近似法が、運動量の始点と位置の終点に関して広義一様収束する、かなり一般的な汎関数のクラスを与えた。このクラスは不確定性原理にかかわらないように一部の汎関数を除外しているため、和や積という演算が自由にできる。応用として、ハミルトン型の摂動展開や準古典近似を証明した。

Research Products

• [Journal Article] Phase space path integrals and their semiclassical approximations (2008)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: Proceedings of the 9th International Conference 'Path Integrals, New Trend and Perspectives', World Scientific Pages: 102-107
• [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2008)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: Bulletin des Sciences mathematiques 132巻 Pages: 313-357
  • Peer Reviewed
• [Journal Article] Feynman path integrals and semiclassical approximation (2008)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: RIMS Kokyuroku Bessatsu B5 : Algebraic Analysis and Exact WKB Analysis for Systems of Differential Equations(eds. Takashi Aoki, Naoto Kumano-go and Susumu Yamazaki) Pages: 241-263
  • Peer Reviewed
• [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2008)
  • Author(s): Naoto Kumanogo and Daisuke Fujiwara
  • Journal Title: Bulletin des Sciences Mathematiques 132 Pages: 313-357
  • Peer Reviewed
• [Journal Article] Feynman path integrals and Semiclassical approximation (2008)
  • Author(s): N., Kumano-go,D., Fujiwara
  • Journal Title: RIMS Kokyuroku Bessatsu B5 Pages: 241-263
  • Peer Reviewed
• [Journal Article] Path integrals as analysis on path space by time slicing approximation (2007)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: PAMM. Proc. Appl. Math. Mech. 7巻 Pages: 1130101-1130102
  • NAID: 130007835721
• [Journal Article] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth (2006)
  • Author(s): Daisuke Fujiwara, Naoto Kumano-go
  • Journal Title: Journal of the Mathematical Society of Japan 58巻 Pages: 837-867
  • NAID: 10018381149
  • Peer Reviewed
• [Journal Article] An improved remainder estimate of stationary phase method for some oscillatory integrals over a space of large dimension (2006)
  • Author(s): Daisuke Fujiwara, Naoto Kumano-go
  • Journal Title: Funkcialaj Ekvacioj 49巻 Pages: 59-86
  • NAID: 130000140660
  • Peer Reviewed
• [Journal Article] An improved remainder estimate of stationary phase method for some oscillatory integrals over a space of large dimension (2006)
  • Author(s): D.Fujiwara, N.Kumano-go
  • Journal Title: Funkcialaj Ekvacioj 49 Pages: 59-86
  • NAID: 130000140660
• [Journal Article] The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth (2006)
  • Author(s): D.Fujiwara, N.Kumano-go
  • Journal Title: Journal of the Mathematical Society of Japan Vol. 58 No. 3 Pages: 837-867
  • NAID: 10018381149
• [Journal Article] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations
  • Author(s): N., Kumano-go・D., Fujiwara
  • Journal Title: Bulletin des Sciences Mathematiques (In Press)
  • Peer Reviewed
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2009)
  • Author(s): Naoto Kumano-go
  • Organizer: AMS Joint Mathematics Meeting
  • Place of Presentation: Washington D. C., USA
  • Year and Date: 2009-01-06
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2009)
  • Author(s): Naoto Kumanogo
  • Organizer: American Mathematical Society
  • Place of Presentation: Washington D.C
  • Year and Date: 2009-01-06
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2008)
  • Author(s): Naoto Kumano-go
  • Organizer: The workshop 'Decay and regularity for solutions of differential equations and dynamical systems'
  • Place of Presentation: The University of Cagliari, Italy
  • Year and Date: 2008-09-26
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2008)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「完全WKB解析と超局所解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-05-26
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2008)
  • Author(s): Naoto Kumano-go
  • Organizer: International Symposium "Function Spaces and Partial Differential Equations"
  • Place of Presentation: 大阪大学
  • Year and Date: 2008-02-19
• [Presentation] Feynman経路積分,時間分割近似法による経路空間上の解析として (2007)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「微分方程式の総合的研究」日本数学会函数方程式論分科会
  • Place of Presentation: 東京大学
  • Year and Date: 2007-12-14
• [Presentation] Feynman経路積分、時間分割近似法による経路空間上の解析として (2007)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「確率論とPDE」
  • Place of Presentation: 広島大学
  • Year and Date: 2007-10-15
• [Presentation] Phase space path integrals and their semi-classical approximations (2007)
  • Author(s): Naoto Kumano-go
  • Organizer: The 9th International Conference 'Path Integrals-New Trends and Perspectives' Max-Planck-Institut fur Physik komplexer Systeme
  • Place of Presentation: Dresden, Germany
  • Year and Date: 2007-09-25
• [Presentation] Path integrals as analysis on path space by time slicing approximation (2007)
  • Author(s): Naoto Kumano-go
  • Organizer: The 6th International Congress on Industrial and Applied Mathematics
  • Place of Presentation: Zurich, Switzerland
  • Year and Date: 2007-07-18
• [Presentation] Feynman path integrals as analysis on path space by time slicing approximation (2007)
  • Author(s): Naoto Kumano-go
  • Organizer: The workshop 'Microlocal Analysis and Harmonic Analysis in Inverse Problems'
  • Place of Presentation: CIRM, Marseille, France
  • Year and Date: 2007-03-29
• [Presentation] 経路積分と準古典近似 (2006)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「微分方程式系の代数解析と完全WKB解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2006-12-14
• [Presentation] Smooth functional derivatives in Feynman path integrals by time slicing approximation (2006)
  • Author(s): Naoto Kumano-go
  • Organizer: The conference 'Partial Differential Equations on Noncompact and Singular Manifolds'
  • Place of Presentation: The University of Potsdam, Germany
  • Year and Date: 2006-08-11
• [Presentation] Feynman path integrals as analysis on path space by time slicing approximation (2006)
  • Author(s): Naoto Kumano-go
  • Organizer: 研究集会「逆問題ワークショップ」
  • Place of Presentation: 筑波大学
  • Year and Date: 2006-07-10
• [Presentation] Smooth functional derivatives in Feynman path integrals by time slicing approximation (2006)
  • Author(s): Naoto Kumano-go
  • Organizer: The conference 'The Feynman Integral and Related Topics in Mathematics and Physics'
  • Place of Presentation: The University of Nebraska Lincoln, USA
  • Year and Date: 2006-05-13