Path integrals as analysis on path space by time slicing approximation

• Project/Area Number: 21540196
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kogakuin University
• Principal Investigator: KUMANOGO Naoto 工学院大学, 基礎・教養教育部門, 教授 (40296778)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• 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 / フーリエ積分作用素 / 確率論

Research Abstract

Using the time slicing approximation via piecewise constant paths, we gave two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. Each class is closed under addition, multiplication, translation, linear transformation and functional differentiation. Furthermore, though we need to pay attention for use, we proved the interchange of the order with some integrals and some limits, natural properties under translation and orthogonal transformation, the integration by parts with respect to functional differentiation, and the semiclassical approximation of Hamiltonian type in the phase space path integrals.

Research Products

• [Journal Article] 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 (2012)
  • Author(s): 熊ノ郷直人
  • Journal Title: 数理科学 Volume: No.586 Pages: 28-34
• [Journal Article] Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation (2011)
  • Author(s): Naoto Kumano-go
  • Journal Title: Bulletin des Sciences Mathematiques Volume: Vol.135、No.8 Pages: 936-987
  • URL: http://www.sciencedirect.com/science/article/pii/S0007449711000972
  • Peer Reviewed
• [Journal Article] ガウス過程に対する経路積分-時間分割近似法による経路空間上の解析として- (2011)
  • Author(s): 熊ノ郷直人
  • Journal Title: 数理解析研究所講究録 Volume: 1723巻 Pages: 55-78
• [Journal Article] ガウス過程に対する経路積分-時間分割近似法による経路空間上の解析として (2011)
  • Author(s): 熊ノ郷直人
  • Journal Title: 数理解析研究所講究録「経路積分と超局所解析の入門」 Volume: 1723 Pages: 55-78
• [Journal Article] Path integrals for Gaussian processes as analysis on path space by time slicing approximation (2010)
  • Author(s): Naoto Kumano-go
  • Journal Title: Integration Mathematical Theory and Applications Volume: Vol.1、No.3 Pages: 253-278
  • URL: https://www.novapublishers.com/catalog/product_info.php?products_id=20199
  • Peer Reviewed
• [Journal Article] Path integrals for Gaussian processes as analysis on path space by time slicing approximation (2010)
  • Author(s): Naoto Kumano-go
  • Journal Title: Integration : Mathematical Theory and Applications Volume: 1 Pages: 253-278
  • Peer Reviewed
• [Journal Article] Smooth functional derivatives in Feynman path integrals by time slicing approximation (2009)
  • Author(s): Naoto Kumano-go、Daisuke Fujiwara
  • Journal Title: World Scientific Pub Co Inc Pages: 429-438
• [Journal Article] Smooth functional derivatives in Feynman path integrals by time slicing approximation (2009)
  • Author(s): Naoto Kumano-go、Daisuke Fujiwara
  • Journal Title: 数理解析研究所講究録 Volume: 1648巻 Pages: 32-52
• [Journal Article] Smooth functional derivatives in Feynman path integrals by time slicing approximation (2009)
  • Author(s): Naoto Kumano-go, Daisuke Fujiwara
  • Journal Title: More Progresses in Analysis : Proceedings of the 5th International ISAAC Congress, World Scientific Pages: 429-438
• [Presentation] Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation (2012)
  • Author(s): Naoto Kumano-go
  • Organizer: The workshop "Geometric and Singular Analysis"
  • Place of Presentation: The University of Potsdam、Germany
  • Year and Date: 2012-03-16
• [Presentation] Phase space Feynman path integrals as analysis on path space (2012)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「超局所解析とその展望」
  • Place of Presentation: 日本大学
  • Year and Date: 2012-03-08
• [Presentation] Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation (2012)
  • Author(s): Naoto Kumano-go
  • Organizer: The workshop "Recent Developments in Stochastic Analysis"
  • Place of Presentation: EPFL、Switzerland
  • Year and Date: 2012-01-31
• [Presentation] 区分的陪特性経路による相空間経路積分-計算例を中心に (2011)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「確率解析とその周辺」
  • Place of Presentation: 佐賀大学
  • Year and Date: 2011-11-13
• [Presentation] 区分的陪特性経路による相空間の経路積分-計算例を中心に- (2011)
  • Author(s): 熊ノ郷直人
  • Organizer: 数理科学セミナー
  • Place of Presentation: 松本市浅間温泉みやま荘
  • Year and Date: 2011-09-25
• [Presentation] Phase space Feynman path integrals by time slicing approximation (2011)
  • Author(s): Naoto Kumano-go
  • Organizer: RIMS共同研究「Introductory Workshop on Feynman Path Integral and Microlocal Analysis」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Path integrals for Gaussian processes as analysis on path space by time slicing approximation (2010)
  • Author(s): Naoto Kumano-go
  • Organizer: Global COE International Mini-Workshop "Microlocal analysis and partial differential equations"
  • Place of Presentation: 東京大学
  • Year and Date: 2010-11-20
• [Presentation] Path integrals for Gaussian processes as analysis on path space by time slicing approximation (2010)
  • Author(s): 熊ノ郷直人
  • Organizer: 研究集会「確率解析とその周辺」
  • Place of Presentation: 岡山大学
  • Year and Date: 2010-11-12
• [Presentation] Path integrals for Gaussian processes as analysis on path space by time slicing approximation (2010)
  • Author(s): Naoto Kumano-go
  • Organizer: The 10th International Conference "Path integrals 2010"
  • Place of Presentation: The Howard University、USA
  • Year and Date: 2010-07-13
• [Presentation] ガウス過程に対する経路積分-時間分割近似法による経路空間上の解析として (2010)
  • Author(s): 熊ノ郷直人
  • Organizer: RIMS共同研究「経路積分と超局所解析の入門」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-05-26
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclasscial approximations (2010)
  • Author(s): Naoto Kumano-go
  • Organizer: The workshop "Operator on singular spaces"
  • Place of Presentation: The University of Potsdam、Germany
  • Year and Date: 2010-03-10
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2010)
  • Author(s): Naoto Kumano-go
  • Organizer: The meeting "Feynman Path Integrals and Their Applications"
  • Place of Presentation: Swansea University、UK
  • Year and Date: 2010-01-18
• [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 (2010)
  • Author(s): 熊ノ郷直人
  • Organizer: ENCOUNTER with MATHEMATICS
  • Place of Presentation: 中央大学
  • Year and Date: 2010-01-09
• [Presentation] Phase space Feynman path integrals and their semi-classical approximations (2009)
  • Author(s): Naoto Kumano-go
  • Organizer: RIMS Conference "Microlocal Analysis and Related Topics"
  • Place of Presentation: 関西学院大学大阪梅田キャンパス
  • Year and Date: 2009-10-21
• [Presentation] Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations (2009)
  • Author(s): Naoto Kumano-go
  • Organizer: NIMS hot topics workshop on Infinite Dimensional Analysis and Related Topics
  • Place of Presentation: Hannam University、Korea
  • Year and Date: 2009-06-24
• [Presentation] Feynman integrals (2009)
  • Author(s): Naoto Kumano-go
  • Organizer: Homi Bhabna Birth Centenary Symposium on Hyperbolic Partial Differential Equations and Related Topics
  • Place of Presentation: TIFR-CAM、India
• [Book] 数理解析研究所講究録1723「経路積分と超局所解析の入門」 (2011)
  • Author(s): 熊ノ郷直人、山崎晋、岡田靖則
  • Publisher: 京都大学数理解析研究所
• [Book] 数理解析研究所講究録1723「経路積分と超局所解析の入門」 (2011)
  • Author(s): 編集:熊ノ郷直人、山崎晋、岡田靖則
  • Total Pages: 190
  • Publisher: 京都大学数理解析研究所