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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 015 270 737
Inoue, Atsushi:
A partial solution for Feynman's problem
Electron. J. Differ. Equ. 2000, Conf. 04, 121 - 145, electronic only (2000): http://www.emis.de/journal/EJDE/conf-proc/04/i1/abstr.html
Primary Classification:
81S40Path integrals
Secondary Classification:
81S30Phase space methods including Wigner distributions, etc.
35Q40Equations from quantum mechanics
81T60Supersymmetric field theories
58D30Applications in quantum mechanics Feynman path integrals, relativity, fluid dynamics, etc.
superanalysis; Weyl equation; external electromagnetic field depending on time; parametrix; Fujiwara method for fermions

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 non-commutativity (of
multiplication ``q" and differentiation ``p") becomes complemented
by a certain ``spin-related" (i.e., matrix) non-commutativity.

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 time-dependent
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.

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