Realization of spinors in Minkowski space (Q1909834)

This paper is concerned with an attempt to understand the mathematical structure of spinor solutions of the Dirac equation, and is a direct continuation of a previous paper [Russ. Math. 36, No. 5, 21-30 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 5(360), 25-35 (1992; Zbl 0780.58046)] by the same authors. The present discussion is based on the differential-geometric operator \(d+ \delta\) which defines a module on the Clifford algebra of exterior forms. This module is decomposed into the sum of a pair of simple modules which are identified with the spaces of spinors and dual spinors, respectively. This decomposition is then related to projectors on a hyperbolic plane, and it is shown that the space of spinors admits both a canonical complex structure and a non-canonical quaternionic structure, the latter being defined by a basis in an elliptic plane orthogonal to the hyperbolic plane. Finally, it is shown that the Clifford algebra may be realized as an algebra of \(2\times 2\) quaternionic matrices. The discussion is purely mathematical, but in a comment the authors suggest that the Dirac \(\psi\)-function may be interpreted as a section of a quaternionic bundle. However, no further details of this physical identification are given in the present paper. [Reviewer's Comment: The literature relating spinors to Clifford algebras is extensive, but unfortunately the authors make no direct reference to it. Clearly much of their approach and algebraic formalism is well-known. However, the relationship between spinors and quaternions is frequently overlooked in the modern literature, and some of their work seems to be new and interesting. Regrettably, the English translation of the paper is clumsy, but this does not materially obscure the mathematical discussion].