Path integrals for Dirac equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 26610020
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Ochanomizu University
• Principal Investigator: FURUYA Kiyoko お茶の水女子大学, 基幹研究院, 講師 (80189208)
• Project Period (FY): 2014-04-01 – 2016-03-31
• Keywords: 発展方程式 / 経路積分 / ディラック方程式 / ベクトル値測度 / 半群

Outline of Final Research Achievements

The idea of Feynman's integral is a topic of great interest in mathematics and physics. But rigorous mathematical treatment of this integral is not enough. We defined a kind of operator-valued integration and define the path integrals.
Dirac equation is the basic equation of relativistic quantum mechanics to the fermions. For the case of space-dimension = 1, Ichinose proved the path integral for Dirac equations are represented by a scalar measure. For the case of radial Dirac equation Ichinose constructed a countably additive path space measure. But in general, for the case of space-dimension > 1, Feynman path integrals for Dirac equations are not represented by (scalar-valued) measures. We showed that the path integral for Dirac equations are represented by our vector-valued measure.

Free Research Field

数物系科学