Real spinors and real Dirac equation (Q2170969)

The author deals with what is known more generally as the geometric algebra approach. With its roots in the classical works of Hermann Grassmann, William Kington Clifford and Marcel Riesz, the algebra was developed by David Orlin Hestenes and later investigated by Cris Doran, Anthony Lasenby and Stephen Gull, thus offering an approach which avoids the usage of complex numbers. The geometric product combines the inner and the outer products, is real but noncommutative by construction. The main tool in the application to the Dirac equation is the same graduation which is found in the usual matrix-valued Dirac matrices. Still, in contrast to the former, this approach is representation independent. The approach of the author is more explicit than the ones by his predecessors, since he gives explicit expressions for the components of the spinors before or after the geometric analogues of the Dirac gamma matrices are applied. This allows him to prove the equivalence of the representation independent form with the canonical matrix representation by \textit{P. A. M. Dirac} [Proc. R. Soc. Lond., Ser. A 117, 610--624 (1928; JFM 54.0973.01)]. The central tool for this translation is the trace which reduces a geometric product to an element of grade zero. Via this pathway the author reaches up to scalars and vector currents and finally to the same Dirac equation found by \textit{D. Hestenes} [Space-time algebra. Foreword by Anthony Lasenby. 2nd ed. Cham: Birkhäuser/Springer (2015; Zbl 1316.83005)]. Finally, as the geometric algebra is by default closely related to gravitation, the formalism is generalised to curved spacetimes in the last part of the paper. The article is pleasant to read, as it provides the possibility to check the results by direct calculation. It can be recommended also to readers who are not expert in this field.