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

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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 oo-dimensional Frechet-Grassmann 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 alge-bra on the Grassmann algebra, we define the symbol of the Pauli equation as a super Hamiltonian function on the superspace. Solving the Hamilton-Jacobi 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 prop-erties 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 equa-tion 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), 27-107, 2003-06-01

    The Mathematical Society of Japan

References:  45

Codes

  • NII Article ID (NAID)
    10015754253
  • NII NACSIS-CAT ID (NCID)
    AA00690979
  • Text Lang
    ENG
  • Article Type
    ART
  • ISSN
    02892316
  • NDL Article ID
    6589024
  • NDL Source Classification
    ZM31(科学技術--数学)
  • NDL Call No.
    Z54-F36
  • Data Source
    CJP  NDL  J-STAGE 
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