A FINITE AND DISCRETE MODEL FOR SINGLE FERMION MASS RENORMALIZATION: Derivation of the free particle Dirac equation*

We assume that a single particle of mass m cannot be localized to better than +h/2mc. Using our understanding of finite and discrete measurement accuracy, the single particle transition in 1+1 space-time from (O, O) to (z, t) can then be characterized by a scale factor N and two integers r, 1defined by z = N(r – l)h/mc and ct = N(r + l)h/mc 2. The average velocity over this finite interval is v = fic. The square of the average momentum is p2 = #mc2. We show that the solution of the free particle Dirac equation with these boundary conditions can be derived by assuming that the (unobserved) trajectories connecting the two endpoints are all constructed from Nr steps to the right and NZ steps to the left, with velocity +C or —c respectively; each single step has length h/me. We attribute this Zitterbewegung to the emission and absorption of transverse photons to and from the background radiation, each of which necessarily flips the spin. We assert that the symmetry . ...-. condition on the background radiation that this radiation be undetectable in free particle motion, plus the assumption that the starting and ending spin state must be the same, constitutes the essential requirement for successful single particle mass renormalization in our simple model. We then show that these requirements suffice to determine finite series which uniquely correspond to the (truncated) series . . solution of the corresponding free particle Dirac equation with the same boundary conditions. We sketch how to extend the model to 3+1 dimensions. The connection of oti model to the derivation of Maxwell’s equations from finite and discrete spacetime measurement accuracy is briefly discussed.