Clifford algebras and Hestenes spinors

Abstract

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl _1,3 ≃ M₂ (H). Hestenes invented first in 1966 hisideal spinors$\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )$ and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl ⁺ _1,3 ≃ 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 decompositionC ⊗ Cl _1,3 andnot inC ⊗ 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 ψ^†γ₀ψ=0, ψ^†γ₀γ₀₁₂₃ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. 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_±, Σ_± are resettled among their new relatives in End(Cl _1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl _1,3 to the (graded) opposite algebra Cl _3,1 without resorting to complex numbers, that is, not using a replacement γ_μ →iγ_μ.