2002/12/23 by Atsushi Inoue, Inoue, Atsushi
Mathematics · Physics and Astronomy · #35F10 #35L45 #36Q40 #70H99 #81S40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #math-ph #math.AP #math.MP #msc:35F10 #msc:35L45 #msc:36Q40 #msc:70H99 #msc:81S40
paper · pdf · doi:10.48550/arxiv.math-ph/0212065
49pages
arxiv created 2002/12/23 · openalex publication_date 2002/12/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of that super Weyl equation. After taking the Lie-Trotter-Kato limit with respect to the time slicing, we get the desired evolutional operator of the super Weyl equation. Bringing back this result to the matrix formulation, we have the final result. Therefore, we get a quantum mechanics with spin from a classical mechanics on the superspace which answers partly the problem of Feynman.