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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 015 270 737  
Author(s):  
Inoue, Atsushi:  
Shorttitle:  
A partial solution for Feynman's problem  
Source:  
Electron. J. Differ. Equ. 2000, Conf. 04, 121  145, electronic only (2000): http://www.emis.de/journal/EJDE/confproc/04/i1/abstr.html  
Classification:  
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
superanalysis; Weyl equation; external electromagnetic field depending on time; parametrix; Fujiwara method for fermions  
Review:  
Classical mechanics (i.e., formally speaking, a system of partial differential equations, PDE) played the role of a core of physics until the discovery of its limitations, roughly speaking, at the large and short distances. Relativity and quantum theory offered its adequate complements in these respective domains. Feynman integrals provide a fairly universal language to be used in the latter case. Unfortunately, the related mathematics is still incomplete: One of open problems addressed in the paper concerns the quantum description of particles with spin. More specifically, the author contemplates and proposes new methodology of solving systems of PDE in the Feynman spirit (i.e., in the author's interpretation, by the method of characteristics), provided that the standard Heisenberg noncommutativity (of multiplication ``q" and differentiation ``p") becomes complemented by a certain ``spinrelated" (i.e., matrix) noncommutativity. The proposal is based on the Fujiwara's ``good parametrix" reformulation of Feynma integral. It enables a ``regularization" of the Feynman's measure (which, strictly speaking, does not exist). As long as the infinitesimal parametrix gives the Weyl quantization, the Weyl equation is considered (with timedependent external electromagnetic field for definiteness). The core of the idea lies in using the grassmanian variables for fermions. In such a setting, the author's main result is the definition of the generalized ``good parametrix". PS: The reader may benefit very much form the very well written appendix which offers a compact review of the supersymmetric analogue of standard analysis. In this context, I would recommend also another well written fresh basic reference, viz., the book ``Superanaliz" by A. Yu. Chrennikov which is available, I am afraid, only in Russian (Nauka, Moscow, 1997).  
Remarks to the editors:  
I am not entirely sure if the reference to this (purely electronically published) text is complete, indeed.  