On a construction of a good parametrix for the Pauli equation

by Hamiltonian path-integral method-An application of superanalysis

Abstract

The superanalysis stands for doing elementary and real analysis on function spaces over the superspace _??_.^m/n 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 R³ 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 L²-scheme. In other words, we propose a quantization procedure of Feynman type for “classical mechanics with spin” using superanalysis.