The Burgers equations and the Born rule

Abstract

The present work demonstrates the connections between the Burgers, diffusion, Schrodinger's and Klein-Gordon's equations. The starting point is a formulation of the stochastic mechanics, which is modelled along the lines of the scale relativity theory. The resulting statistical description obeys a pair of coupled Fokker-Planck equations, which can be represented as one complexified differential equation. The paper further demonstrates the connection between the stochastic mechanics and scale relativity theory, embodied by the properties of the Burgers equation, which from this perspective appears as a stochastic geodesic equation. The main result of the article is the transparent derivation of the Born rule from the starting point of a complex stochastic process, based on a complexified Fokker-Planck formalism. (C) 2021 Elsevier Ltd. All rights reserved.