Schr\"odinger's equation as a consequence of the central limit theorem without prior assumption of physical laws

Abstract

The central limit theorem has been found to apply to random vectors in complex Hilbert space. This amounts to sufficient reason to study the complex valued Gaussian, looking for relevance to quantum mechanics. Here we show that the Gaussian, with all terms fully complex, acting as a propagator, leads to Schrodinger nonrelativistic equation including scalar and vector potentials, assuming only that the norm is conserved. No prior physical laws need to be postulated. It thereby presents as a process of irregular motion analogous to the real random walk but executed under the rules of the complex number system. Inferences are 1. There is a standard view that Schrodinger equation is deterministic, while wavefunction collapse is probabilistic by Born's rule. This is opposed by the now demonstrated linkage to the central limit theorem, indicating a stochastic picture for the foundation of Schrodinger equation itself. 2. This picture is also consistent with the dynamic origin of probabilities suggested for the Born rule in the de Broglie Bohm pilot wave theory. Reasons for the primary role of C are open to discussion. The present derivation is compared with recent reconstructions of the quantum formalism, which have the aim of rationalizing its obscurities.