2016 Fiscal Year Final Research Report
The theory of osicllatory integral operartors and its application to the Feynman path integral of quntum field theory
| Project/Area Number |
26400161
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Shinshu University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | Feynman 経路積分 / Dirac 方程式 / 量子電磁気学 / Schroedinger 方程式 / 量子力学 |
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| Outline of Final Research Achievements |
(1) We have constructed the Feynman path integral for a relativistic electron in the form of the sum-over-histories, over all paths of one electron in space-time that goes in any direction at any speed, forward and backward in time, especially with a countably infinite number of turns. By this result we have succeeded in introducing a countably infinite number of electrons and positrons at the same time in the theory of the Feynman path integrals as the 2nd quantization of one electron. (2) We have showed that the Feynman path integral defined in (1) is relativistically invariant, i.e. has the property of spinor under the Lorentz transformations. (3) We have showed directly from the Feynman path integral that the probability amplitude for a relativistic electron has the properties of both unitarity and causality. (4) We have constructed mathematically the Feynman path integral for the Schroedinger equations with potentials growing polynomially in the spatial direction.
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| Free Research Field |
偏微分方程式論
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