On the Dirac operator with an electromagnetic potential
In 1931 papers by Moisil and Moisil-Theodorescu introduced quaternionic representation of the Dirac operator \(D=\sum_{(k=1,2,3)} {\mathbf i}_k \partial_k\), where \({\mathbf i}_k\) are the ``usual'' quaternions, so that the Dirac equation can be written in the form: \((D+f(x)I+ M^\alpha) u(x)=0\), \(f\) and \(u\) are biquaternions, i.e. complexified quaternions, and \(M^\alpha\) is the operator of right multiplication by a quaternion \(\alpha\). [See \textit{G. C. Moisil}, Bull. Sci. Math., II. Ser. 55, 168-174 (1931; Zbl 0002.19402) and \textit{G. C. Moisil} and \textit{N. Theodoresco}, Mathematica, Cluj 5, 142-159 (1931; Zbl 0002.27401)]. A similar quaternionic representation establishing a variational formulation for the Dirac equation reduced to a canonical Hamiltonian dual decomposition can be also found in 1979 article of the reviewer [\textit{V. Komkov}, J. Math. Anal. Appl. 71, 187-209 (1979; Zbl 0421.46037)]. More recent developments in this area can be found in the monograph [\textit{V. V. Kravchenko} and \textit{M. V. Shapiro}, Integral representations for spatial models of mathematical physics, Pitman Research Notes in Mathematics Series. 351, Harlow: Longman (1996; Zbl 0872.35001)]. The present article offers a new approach based on projection operators permitting a detailed analysis of the Dirac equation with electromagnetic potential. It offers decomposition of the operator: \[ D+fI=P^+ D_{\xi^+} +P^-D_{\xi-}, \] where \(P^+= 1/(2\gamma) (\gamma + f^\to)I\), \(\gamma= [\{f^\to\}^2]^{1/2}\), where any root can be taken, and \(P^-=1/(2\gamma) (\gamma-f^\to)I\). This allows the author to establish an inverse relation, basically solving certain classes of problems. A somewhat similar decomposition is carried out by the author for the eikonal equation.
scientific article; zbMATH DE number 1211539