Stochastic quantum mechanics is a framework for describing the dynamics of particles that are subjected to intrinsic random processes as well as various external forces. The framework provides a derivation of the diffusion equations associated to these stochastic particles. It is best known for its derivation of the Schrödinger equation as the Kolmogorov equation for a certain type of conservative (or unitary) diffusion. The derivation can be based on the extremization of an action in combination with a quantization prescription. This quantization prescription can be compared to canonical quantization and the path integral formulation, and is often referred to as Nelson's stochastic quantization or stochasticization. As the theory allows for a derivation of the Schrödinger equation, it has given rise to the stochastic interpretation of quantum mechanics. This interpretation has served as the main motivation for developing the theory of stochastic mechanics. In the 1930s both Erwin Schrodinger and Reinhold Furth recognised a similarity between the equations of classical diffusion and the formalism of quantum theory, but the first relatively coherent stochastic theory of quantum mechanics was put forward in 1946 by Hungarian physicist Imre Fényes. Louis de Broglie felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another. The theory of stochastic quantum mechanics is ascribed to Edward Nelson, who independently discovered a derivation of the Schrödinger equation within this framework. This theory was also developed by Davidson, Guerra, Ruggiero, Pavon and others.
Stochastic interpretation of quantum mechanics The stochastic interpretation interprets the paths in the path integral formulation of quantum mechanics as the sample paths of a stochastic process. It posits that quantum particles are localized on one of these paths, but observers cannot predict with certainty where the particle is localized. The only way to locate the particle is by performing a measurement. An observer can only predict probabilities for the outcomes of such a measurement based on their earlier measurements and their knowledge about the forces that are acting on the particle. This interpretation is well-known from the context of statistical mechanics, and Brownian motion in particular. Hence, according to the stochastic interpretation, quantum mechanics should be interpreted in a way similar to Brownian motion. However, in the case of Brownian motion, the existence of a probability measure (called the Wiener measure) that defines the statistical path integral is well established, and this measure can be generated by a stochastic process called the Wiener process. On the other hand, proving the existence of a probability measure that defines the quantum mechanical path integral faces difficulties, and it is not guaranteed that such a probability measure can be generated by a stochastic process. Stochastic mechanics is the framework concerned with the construction of such stochastic processes that generate a probability measure for quantum mechanics. For a Brownian motion, it is known that the statistical fluctuations of a Brownian particle are often induced by the interaction of the particle with a large number of microscopic particles. In this case, the description of a Brownian motion in terms of the Wiener process is only used as an approximation, which neglects the dynamics of the individual particles in the background. Instead it describes the influence of these background particles by their statistical behavior. The stochastic interpretation of quantum mechanics is agnostic about the origin of the quantum fluctuations of a quantum particle. It introduces the quantum fluctuations as the result of a new stochastic law of nature called the background hypothesis. This hypothesis can be interpreted as a strict implementation of the statement that `God plays dice', but it leaves open the possibility that this dice game is replaced by a hidden variable theory, as in the theory of Brownian motion. The remainder of this article deals with the definition of such a process and the derivation of the diffusion equations associated to this process. This is done in a general setting with Brownian motion and Quantum mechanics as special limits, where one obtains respectively the heat equation and the Schrödinger equation. The derivation heavily relies on tools from Lagrangian mechanics and stochastic calculus.
Stochastic quantization
The postulates of the theory can be summarized in a stochastic quantization condition that was formulated by Nelson. For a non-relativistic theory on R n {\displaystyle \mathbb {R} ^{n}} this condition states:
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