Mathematics of Quantum-Classical correspondence and the Arcsine law

• Project/Area Number: 26870696
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical informatics; Mathematical physics/Fundamental condensed matter physics
• Research Institution: Nagahama Institute of Bio-Science and Technology
• Principal Investigator: Saigo Hayato 長浜バイオ大学, バイオサイエンス学部, 准教授 (80615154)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 量子古典対応 / 逆正弦法則 / 直交多項式 / 量子ウォーク / ヘッケ環 / 量子確率論 / ヘッケ作用素 / 物理量代数 / 状態 / 代数的量子場理論 / 代数的確率論 / 相互作用フォック空間

Outline of Final Research Achievements

The idea of "quantum classical correspondence", which unifies the quantum physics and the world of classical physics, is reformulated in the framework of a branch of modern mathematics called "quantum probability theory". The universal roles played by "the Arcsine law" in the realm of mathematical sciences are clarified. For instance, the behavior and structures of important mathematical objects such as "orthogonal polynomials", "quantum walks" and "Hecke rings" are analysed in terms of the Arcsine law from the viewpoint of quantum probability.

Research Products

• [Journal Article] Dressed photons from the viewpoint of photon localization: the entrance to the off-shell science (2017)
  • Author(s): H.Saigo, I. Ojima, and M. Ohtsu
  • Journal Title: Applied Physics A Volume: 123 Issue: 12 Pages: 724-724
  • DOI: 10.1007/s00339-017-1345-z
  • Peer Reviewed
• [Journal Article] On a property of the simple random walk on Z (2017)
  • Author(s): Norio Konno, Hayato Saigo and Hiroki Sako
  • Journal Title: Yokohama Mathematical Journal Volume: 63 Pages: 1-7
  • NAID: 120006459863
  • Peer Reviewed / Open Access
• [Journal Article] Using category theory to assess the relationship between consciousness and integrated information theory (2016)
  • Author(s): Naotsugu Tsuchiya, Shigeru Taguchi, Hayato Saigo
  • Journal Title: Neuroscience Research Volume: in press Pages: 1-7
  • DOI: 10.1016/j.neures.2015.12.007
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Local state and sector theory in local quantum physics (2016)
  • Author(s): Izumi Ojima, Kazuya Okamura, Hayato Saigo
  • Journal Title: Letters in Mathematical Physics Volume: 106 Issue: 6 Pages: 741-763
  • DOI: 10.1007/s11005-016-0841-y
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The Arcsine law and an asymptotic behavior of orthogonal polynomials (2016)
  • Author(s): H. Saigo and H. Sako
  • Journal Title: Annales de l'Institut Henri Poincare D Volume: 3 Issue: 4 Pages: 405-427
  • DOI: 10.4171/aihpd/34
  • Peer Reviewed
• [Journal Article] Photon Localization Revisited (2015)
  • Author(s): Izumi Ojima and Hayato Saigo
  • Journal Title: Mathematics Volume: 3(3) Issue: 3 Pages: 897-912
  • DOI: 10.3390/math3030897
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] “Who Has Seen A Free Photon? ---From Mathematical Physics to Light-Matter Fusion Technologies---“ (2017)
  • Author(s): Motoichi Ohtsu, Izumi Ojima and Hayato Saigo
  • Organizer: The 11th Asia-Pacific Conference on Near-Field Optics,
  • Int'l Joint Research / Invited
• [Presentation] Why the Arcsine Law? (2017)
  • Author(s): Hayato Saigo
  • Organizer: Mathematical Aspects of Quantum Fields and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] 量子ウォークと同位体分離の基礎数理―量子確率論の観点から (2017)
  • Author(s): 西郷 甲矢人
  • Organizer: 九州大学マス・フォア・インダストリ研究所短期共同研究：レーザー同位体分離における量子ウォークの数理
  • Invited
• [Presentation] なぜ逆正弦法則なのか？ (2017)
  • Author(s): 西郷甲矢人
  • Organizer: 量子論にまつわる数学と数論の連携探索
  • Place of Presentation: 早稲田大学 早稲田キャンパス１９－２号館 Waseda共創館（東京都新宿区）
  • Invited
• [Presentation] 逆正弦法則、量子古典対応、量子ウォーク (2017)
  • Author(s): 西郷甲矢人
  • Organizer: 日本物理学会
  • Place of Presentation: 大阪大学 豊中キャンパス（大阪府豊中市）
  • Invited
• [Presentation] The arcsine law and an asymptotic behavior of orthogonal polynomials (2016)
  • Author(s): Hayato Saigo
  • Organizer: 4th-Yokohama Workshop on Quantum Walks
  • Place of Presentation: Yokohama Land Mark Tower: 18F room 1809 (satellite campus of Yokohama National University)（神奈川県横浜市）
  • Int'l Joint Research / Invited
• [Presentation] 直交多項式の観点から (2016)
  • Author(s): 西郷甲矢人
  • Organizer: 量子ウォークを架け橋にした学際交流
  • Place of Presentation: 横浜国立大学 常盤台キャンパス 教育文化ホール中集会室（神奈川県横浜市）
  • Invited
• [Presentation] The arcsine law and an asymptotic property of orthogonal polynomials (2015)
  • Author(s): Hayato Saigo
  • Organizer: Quantum Field and Related Topics
  • Place of Presentation: RIMS, Kyoto university, Japan
  • Year and Date: 2015-10-05
  • Int'l Joint Research / Invited
• [Presentation] Algebraic Probability and Quantum-Classical Correspondence (2014)
  • Author(s): Hayato Saigo
  • Organizer: Quantum Field and Related Topics
  • Place of Presentation: RIMS, Kyoto university
  • Year and Date: 2014-10-08
• [Presentation] The Arcsine law and an asymptotic behavior of orthogonal polynomials (2014)
  • Author(s): Hayato Saigo
  • Organizer: 16th WORKSHOP: NON-COMMUTATIVE HARMONIC ANALYSIS
  • Place of Presentation: Bedlewo, Poland
  • Year and Date: 2014-07-10
  • Invited
• [Book] 指数関数ものがたり (2018)
  • Author(s): 西郷甲矢人、能美十三
  • Total Pages: 192
  • Publisher: 日本評論社
  • ISBN: 9784535788480