Quantum Mechanics and the Special Theory of Relativity from a Random Walk.

Gerhard Grössing

Frederick T. Wall

Random walks with step lengths equal to the shortest possible physically meaningful distances are considered from the point of view of special relativity involving two observers moving uniformly with respect to each other. A requirement of statistical equivalence of the probability distributions seen by those observers leads to the Lorentz transformations, provided a randomly moving particle shifts from one submicroscopic cell of uncertainty to a neighbor with a speed equivalent to that of light. Ordinary smooth motion would appear to involve a tremendous amount of submicroscopic back and forth randomness subject to a statistical bias favoring a particular direction. The diffusive nature of the motion naturally leads to a spreading of the probability distribution.

M. Faber

We describe particles in a potential by a special diffusion process, the maximal entropy random walk (MERW) on a lattice. Since MERW originates in a variational problem, it shares the linear algebra of Hilbert spaces with quantum mechanics. The Born rule appears from measurements between equilibrium states in the past and the same equilibrium states in the future. Introducing potentials by the observation that time, in a gravitational field running in different heights with a different speed, MERW respects the rule that all trajectories of the same duration are counted with equal probability. In this way, MERW allows us to derive the Schrödinger equation for a particle in a potential and the Darwin term of the nonrelativistic expansion of the Dirac equation. Finally, we discuss why quantum mechanics cannot be simply a result of MERW, but, due to the many analogies, MERW may pave the way for further understanding.

Joginder Singh Galsin

Mark Denny

The exact harmonic oscillator energy distribution, including zero-point energy, is derived from a random walk calculation. The Planck distribution for blackbody radiation is found as a special case, in which the number of oscillators is not constrained. Connection with the correspondence principle sheds light upon the special historical role of harmonic oscillators in the development of quantum mechanics.

Erkki Brändas