Klein-Gordon Equation for Spinors

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Abstract
Spinors obeying the Dirac equation also obey the Klein-Gordon equation, but the converse is not true. In this paper we make a systematic study of four-component spinors obeying the Klein-Gordon equation, with special regard for the additional solutions. The starting point is the Lagrangian density L=ψ̄,μγμγνψ,ν−m2ψ̄ψ, and we first develop from it the theory of a classical spinor field. We then proceed to the canonical quantization of this field and are confronted by some anticommutators of creation and annihilation operators equal to −1, and the subsequent need for an indefinite metric in Hilbert space. Quantum electrodynamics can be reformulated, and in spite of a modified fermion propagator, gradient coupling, and vertices with two photon lines, the amplitude for Compton scattering to order e2 is the usual one. Special problems arising for massless fermions are indicated, and we note that the four-fermion point interaction is now renormalizable. Some interesting variations of strong interactions also become possible.