The math is over my head, but if QM can be derived by applying the heisenberg uncertainty relation to the mathematics of brownian motion (which is a combination of probability theory and classical or pre-quantum physics) it suggests, at least to me, that non-local connections are more common and pervasive than mainstream science has been willing to recognize.
Harry Quantum equations from Brownian motion http://www.nrcresearchpress.com/toc/cjp/89/2 full text can be downloaded. Abstract: random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schro¨dinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schro¨dinger equation can be transformed into the usual Schro¨dinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation. From the Discussion section: "In the light of these results, it may be concluded that the classical equations for an ensemble of averages of excess of parity in Brownian movement can he transformed into the usual Schro¨dinger quantum equation by imposing the Heisenberg uncertainty relation between posi tion and momentum of Brownian particles, and the similar classical equations can be transformed into the usual Dirac relativistic quantum equations by imposing the uncertainty relation between energy and time associated with Brownian particle, without using a formal analytic continuation and wave-particle duality. Here, the derivation of quantum equations is a sensible classical scheme to produce many-particle simulations of quantum mechanics, where the quantum equations exist as a description of classical theory (Brownian movement). It may be emphasized that from this classical theory, the nonrelativistic quantum mechanics (i.e., Schro¨- dinger equation) originates as the consequence of Heisenberg’s uncertainty relation between position and momentum, while the relativistic quantum mechanics (i.e., Dirac equation) is the consequence of Heisenberg’s uncertainty relation between energy and time associated with Brownian particles. The usual formal analytic continuation, which is necessary to relate the classical and quantum equations, is completely absent here, and hence the interpretation of quantum mechanics is direct, without the problems of measurement usually associated with the usual quantum mechanics. The underlying microscopic model used here is a simple random walk model, whose probabilistic description is completely classical. The classical Schro¨dinger equations derived here appear as a description of a second-order effect in the ensembles of diffusing particles. In this context, the real and the imaginary parts of the solutions of Schro¨dinger’s equation are observable properties of ensembles of random walks, in the same way as solutions of diffusion equations are real observable properties of such ensembles. In this sense, we have a classical microscopic model of the Schro¨- dinger equation, which is as direct as the random walk model of diffusion. There is a similar analogy between the usual telegraph equations [11, 12], describing particle densities of Brownian movements on a lattice site and the classical Dirac equations derived here, and hence the derivation of (quantum-) Schro¨dinger and Dirac equations from Brownian motion in our work is not just a coincidence. Furthermore, here the Heisenberg uncertainty relations are not the consequence of wave-particle duality but are the result of quantum limits (2.19) and (3.24), imposed on Brownian motion, which automatically transform the classical Schro¨- dinger and Dirac equations, respectively, to the usual corresponding quantum equations in the nonrelativistic and relativistic frame works, respectively. These results support the recent work [20] on the role of a generalized uncertainty principle in the development of quantum mechanics from a classical context. These results partially support the earlier work [4–8], showing that the quantum mechanical equations are the derived properties of the binomial distribution, no formal analytic continuation is required to produce them. Some results of this paper shall be helpful in framing the foundation of space-time path formalism [21] for relativistic quantum mechanics. We shall undertake the study of this problem in our forthcoming paper.
