CLIFFORD ALGEBRAS AND HESTENES SPINORS

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within the real Clifford algebra Cl1,3 congruent-to M2(H). Hestenes invented first in 1966 his ideal spinors phi is-an-element-of Cl1,3(2)1(1 - gamma03) and later 1967/75 he recognized the importance of his operator spinors PSI is-an-element-of Cl1,3+ congruent-to M2(C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decomposition C x Cl1,3 and not in C x M4(R) = M4(C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case psi(double dagger)gamma0psi = 0, psi(double dagger)gamma0gamma0123psi = 0. The null case is thoroughly scrutinized for the first time with a new concept called boomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P+/-, SIGMA+/- are resettled among their new relatives in End(Cl1,3). Finally, a new mapping, called tilt, is introduced to enable a transition from Cl1,3 to the (graded) opposite algebra Cl3,1 without resorting to complex numbers, that is, not using a replacement gamma(mu) --> igamma(mu).