Analytic Representation of The Dirac Equation

Abstract

In this paper we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wave functions, as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behavior of the particle-antiparticle relationship. We interpret the zitterbewegung and the result that a velocity measurement (of a Dirac particle) at any instant in time is, as reflections of the fact that the Dirac equation makes a spatially extended particle appear as a point in the present by forcing it to oscillate between the past and future at speed c. From this we infer that, although the form of the Dirac equation serves to make space and time appear on an equal footing mathematically, it is clear that they are still not on an equal footing from a physical point of view. On the other hand, the Foldy-Wouthuysen transformation, which connects the Dirac and square root operator, is unitary. Reflection on these results suggests that a more refined notion (than that of unitary equivalence) may be required for physical systems.