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

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03361
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Shinshu 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
• 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

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).

Free Research Field

偏微分方程式論

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)。