Relativistic wave equations

Abstract

The classical relativistic connexion between the energy p t of a free article and its momentum p x , p y , p z , namely, p t 2 — p x 2 — p y 2 — p z 2 — m 2 = 0, (1) leads in the quantum theory to the wave equation { p t 2 — p x 2 — p y 2 — p z 2 — m 2 } ψ = 0, (2) where the p 's are understood as the operators iħ ∂/∂ t , — iħ ∂/∂ x . . The general theory of the physical interpretation of quantum mechanics requires a wave equation of the form { p t — H} ψ = 0, (3) where H is a Hermitian operator not containing p t , and is called the Hamiltonian. The obvious equation of the form (3) which one gets from (2), namely, { p t — ( p x 2 + p y 2 + p z 2 + m 2 ) ½ } ψ = 0, is unsatisfactory on account of the square root, which makes the application of Lorentz transformations very complicated. By allowing our particle to have a spin, we can get wave equations of the form (3) which are consistent with (2) and do not involve square roots. An example, applying to the case of a spin of half a quantum, namely, the equation { p t + α x p x + α y p y + α z p z + α m m } ψ = 0, (4) where the four α 's are anti-commuting matrices whose squares are unity, is well known, and has been found to give a satisfactory description of the electron and positron. The present paper will be concerned with other examples, applying to spins greater than a half. The elementary particles known to present-day physics, the electron, positron, neutron, and proton, each have a spin of a half, and thus the work of the present paper will have no immediate physical application. All the same, it is desirable to have the equation ready for a possible future discovery of an elementary particle with a spin greater than a half, or for approximate application to composite particles. Further, the underlying theory is of considerable mathematical interest.