On Dirac and Pauli Operators

Abstract

A unified description of the Dirac and Pauli operators based on the De Broglie principle is given. It includes descriptions of spinor modules as submodules in differential forms and the h-adic filtrations of these modules. A one-to-one correspondence between decomposable differential 2-forms of unit length and spinor modules is obtained. The Dirac operator is described as an operator acting on the h-adic filtrations of the spinor module. The Dirac-type operator is defined. It slightly differs from the Dirac operator at the one point only: its domain is the h-adic filtrations of the module of differential forms itself. It is shown that the restrictions of the Dirac-type operator to the h-adic filtrations of some spinor module is the Dirac operator if and only if the 2-form is a special solution of the Maxwell equations. The transfer operator for the Dirac operator is interpreted as an analog of the Pauli operator. A similar description of the Dirac and Pauli operators acting on vectorvalued fields is considered.