Fractal space-time and the statistical mechanics of random walks

Abstract

We illustrate some of the ideas involved in fractal space-time using familiar deterministic fractals. Starting with the objective of reproducing the Heisenburg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of particles moving on fractal trajectories in space and time. With this qualitative picture in mind we then explore exactly solvable models to verify that our ideas are mathematically consistent. We find that the Schrödinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schrödinger equation has a straightforward microscopic model which is not, however, appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and this unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechanics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. Finally, we discuss an exactly solvable model which incorporates fractal time. The calculation produces the Dirac equation in 1 + 1 dimensions and because of intrinsic space-time loops, constitutes a model with the potential to exhibit the wave-particle duality found in nature.