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

The superanalysis stands for doing elementary and real analysis on function spaces over the superspace R^m|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.