On a construction of a good parametrix for the Pauli equation by Hamiltonian pathintegral method : An application of superanalysis

 Inoue Atsushi INOUE Atsushi
 DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES TOKYO INSTITUTE OF TECHNOLOGY

 Maeda Yoshiaki MAEDA Yoshiaki
 DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES AND TECHNOLOGY KEIO UNIVERSITY
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Author(s)

 Inoue Atsushi INOUE Atsushi
 DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES TOKYO INSTITUTE OF TECHNOLOGY

 Maeda Yoshiaki MAEDA Yoshiaki
 DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCES AND TECHNOLOGY KEIO UNIVERSITY
Abstract
The superanalysis stands for doing elementary and real analysis on function spaces over the superspace _??_.<sup>m/n</sup> with value R or C. Here, R and C are oodimensional FrechetGrassmann algebras which play the role of R and C in the standard theory, respectively. Using this analysis, we construct a parametrix of the Pauli equation (=the Schrodinger equations with spin) on R<sup>3</sup> from ‘classical objects’. More precisely, by using the differential operator representations of the Clifford algebra on the Grassmann algebra, we define the symbol of the Pauli equation as a super Hamiltonian function on the superspace. Solving the HamiltonJacobi and continuity equations corresponding to that Hamiltonian function, we construct a certain Fourier Integral Operator on superspace, which gives a parametrix of the Pauli equation. This parametrix is called “good” because it has not only the ordinary approximation properties but also has the explicit dependence on the Planck constant h. The Lie product formula for these parametrices yields a desired evolutional operator of the Pauli equation in the L<sup>2</sup>scheme. In other words, we propose a quantization procedure of Feynman type for “classical mechanics with spin” using superanalysis.
Journal

 Japanese journal of mathematics. New series

Japanese journal of mathematics. New series 29(1), 27107, 20030601
The Mathematical Society of Japan
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