TY - JOUR

T1 - On a construction of a good parametrix for the Pauli equation by Hamiltonian path-integral method — an application of superanalysis in memory of Professor Masahisa ADACHI

AU - Inoue, Atsushi

AU - Maeda, Yoshiaki

PY - 2003

Y1 - 2003

N2 - The superanalysis stands for doing elementary and real analysis on function spaces over the superspace Rm|n with value or. Here, and are ∞-dimensional Fréchet-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 Schrödinger equations with spin) on R3 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 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 properties but also has the explicit dependence on the Planck constant ħ. The Lie product formula for these parametrices yields a desired evolutional operator of the Pauli equation in the L2-scheme. In other words, we propose a quantization procedure of Feynman type for “classical mechanics with spin” using superanalysis.

AB - The superanalysis stands for doing elementary and real analysis on function spaces over the superspace Rm|n with value or. Here, and are ∞-dimensional Fréchet-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 Schrödinger equations with spin) on R3 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 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 properties but also has the explicit dependence on the Planck constant ħ. The Lie product formula for these parametrices yields a desired evolutional operator of the Pauli equation in the L2-scheme. In other words, we propose a quantization procedure of Feynman type for “classical mechanics with spin” using superanalysis.

KW - Fréchet-Grassmann algebra

KW - Spin

KW - quantization of Feynman type

KW - super Hamilton flow

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U2 - 10.4099/math1924.29.27

DO - 10.4099/math1924.29.27

M3 - Article

AN - SCOPUS:84903887248

VL - 29

SP - 27

EP - 107

JO - Japanese Journal of Mathematics

JF - Japanese Journal of Mathematics

SN - 0289-2316

IS - 1

ER -