A two-component wave equation for particles of spin 1/2 and nonzero rest mass - II

Abstract

Spin-1/2 particles with nonzero rest mass have hitherto been described by the field equation of Dirac,i.e. by a four-component wave function. «Why must the wave function have four components?» was a question posed by Feynman and Gell-Mann which they answered by using the second-order equation [(i∇μ=Aμ)2+σ·(B+iE)]ϕ=m2ϕ with ϕ a two-component spinor. The same equation—sometimes called the relativistic Pauli equation—was studied further by Brown, Tonin and Pietschmann with specific ends in view. However, a two-component wave equation which is essentially a Schrodinger equation insomuch as it retains the formiħ(∂ϕ/∂t)=Hϕ, and is, therefore,first order in ∂/∂t has been considered impossible, which it certainly is, if the operators used are confined to the type α∂, where α's are matrices with complex elements and ∂'s are differential operators. Biedenharn and collaborators in 1971–1972 presented a two-component alternative to Dirac’s equation which was of the Schrodinger type: their approach, however, is completely different from ours. We drop all the above restrictions but one, in these papers, and discuss, in part I, the qualifications of the equation (∂0+σk∂k)ψ=−ℵTψ, where ∂μ ≡ ∂/t6xμ,σk are the Pauli matrices,T is the linear operator which changes the sign oft, η=m0c/ħ and ψ a two-component wave function. The equation of the particle interacting with an electromagnetic field is dealt with in part II. In part I we have established that both components of all the solutions of the above equation satisfy the Klein-Gordon equation and that a 1-1 correspondence can be set up between its solutions and the positive-energy solutions of the Dirac equation which preserves inner products (when suitably defined for our case). Covariance under the proper Lorenz group followed by covariance under space and time inversions and translations has then been shown. Eigenfunctions of energy-momentum and helicity have been explicitly found and it is shown that causality is preserved and a Green’s function exists. A list appears, and the end, of the points discussed in part II which further the acceptability of the theory.