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We derive the classical Hamilton-Jacobi equation from first principles as thenatural description for smooth stochastic processes when one neglectsstochastic velocity fluctuations. The Schr\"{o}dinger equation is shown to bethe natural exact equation for describing smooth stochastic processes. Inparticular, processes with up to quadratic stochastic fluctuations areelectromagnetically coupled quantum point particles. The stochastic derivationoffers a clear geometric picture for Quantum Mechanics as a locally realistichidden variable theory. While that sounds paradoxical, we show that Bell'sformula for local realism is incomplete. If one includes smooth stochasticfluctuations for the hidden variables, local realism is preserved and quantummechanics is obtained. Quantum mechanics should therefore be viewed as a"nondeterministic, non-Bell locally realistic hidden variable theory". Sincethe description is simply a stochastic process, it should be relativelystraightforward to create mesoscopic analogue systems that show all thehallmarks of Quantum Mechanics, including super-Bell correlations. In fact, anysystem that can be described by the linear time evolution of a density matrixis both a stochastic and a Quantum system from our point of view, since we showthat the existence of a transition probability density directly implies theexistence of a density matrix. Systems for which the stochastic degrees offreedom vary smoothly over time have quantum Hamiltonians with standard kineticterms.

A First Principles Derivation of Classical and Quantum Mechanics as the Natural Theories for Smooth Stochastic Paths