On the Connection between Nelson’s Stochastic Quantum Mechanics and Nottale’s Theory of Scale Relativity

In this paper, we review and compare the stochastic quantum mechanics of Nelson and the scale relativity theory of Nottale. We consider both nonrelativistic and relativistic frameworks and include the electromagnetic field. These theories propose a derivation of the Schrödinger and Klein–Gordon equations from microscopic processes. We show their formal equivalence. Specifically, we show that the real and imaginary parts of the complex Lorentz equation in Nottale’s theory are equivalent to the Nelson equations, which are themselves equivalent to the Madelung and de Broglie hydrodynamical representations of the Schrödinger and Klein–Gordon equations, respectively. We discuss the different physical interpretations of the Nelson and Nottale theories and stress their strengths and weaknesses. We mention potential applications of these theories to dark matter.