Study on deformation quantization and noncommutative star exponentials, and its application

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03286
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Tokyo University of Science
• Principal Investigator: YOSHIOKA Akira 東京理科大学, 理学部第二部数学科, 客員教授 (40200935)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 変形量子化 / 変形量子化代数の指数関数 / convergent star product / 変形量子化の応用 / star Mittag-Leffler / star Kummer function / star gamma function / star zeta function

Outline of Final Research Achievements

Deformation quantization is to deform the usual product of functions into an associative product. The deformed product is called the star product. Using star products, we have constructed deformation of several functions and obtained the functional equations which the deformed functions satisfy. In particular, we study the deformation of exponential functions and has found that the exponential function has singular points and clarified some of its properties. The strength of the calculation of deformation quantization is that it is elementary and direct, and then, in terms of applications, it has been able to demonstrate its power in calculations of physical models. Applying it to a concrete cosmological model, we presented a formula that calculates the quantum mechanical uncertainty relation to the order of the square of Planck's constant. We also calculated eigenvalue problems in quantum mechanics.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

学術的な意義として、（１）量子計算の様々な関係式を、作用素ではなく関数を用いた初等的・直接的な式に表すこと、（２）様々な概念の非可換化を行う解析的・幾何学的な方法を与えること、の２点を挙げることができる。社会的意義として、積が具体的な表示式により与えられることから、物理学・化学および工学における量子的な計算を、作用素ではなく関数のべき級数展開を用いて直接的かつ初等的に行うことが可能であり、将来、具体的な問題の解決に役に立つことが期待される点を挙げることができる。