Fractal space-time: a geometric analogue of relativistic quantum mechanics

Abstract

Considers a 'thought experiment' in which particles are confined to move on fractal trajectories in both space and time, treating the case of a Peano-Moore trajectory in detail. Generalising these results, the author uses the classical principle of relativity and a correspondence principle to show that fractal trajectories in space with Hausdorff dimension D=2 exhibit both an uncertainty principle and a de Broglie relation. The incorporation of fractal time with D=2 places an upper bound on the macroscopic velocities of 'fractalons', which in turn requires that the macroscopic physics be Lorentz covariant. On a microscopic scale, the presence of fractal time is interpreted in terms of the appearance of particle-antiparticle pairs when observation energies become of the order of mc2. The author proposes two field equation descriptions of fractalons based on random walk space-time trajectories and subsequently relates these equations to the free particle Klein-Gordon and Dirac equations respectively.