Stochastic quantum mechanics is a framework for describing the dynamics of particles that are subjected to an intrinsic random processes as well as various external forces.
[1][2] The derivation can be based on the extremization of an action in combination with a quantization prescription.
[3] 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.
[1] The first relatively coherent stochastic theory of quantum mechanics was put forward by Hungarian physicist Imre Fényes.
[4][5][6][7] Louis de Broglie[8] felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another.
[7] The theory of stochastic quantum mechanics is ascribed to Edward Nelson, who independently discovered a derivation of the Schrödinger equation within this framework.
This interpretation is well-known from the context of statistical mechanics,[9] and Brownian motion in particular.
[11] On the other hand, proving the existence of a probability measure that defines the quantum mechanical path integral faces difficulties,[12][13] and it is not guaranteed that such a probability measure can be generated by a stochastic process.
[14][15][16][17] 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.
It introduces the quantum fluctuations as the result of new stochastic law of nature called the background hypothesis.
[2] 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 derivation heavily relies on tools from Lagrangian mechanics and stochastic calculus.
The postulates of the theory can be summarized in a stochastic quantization condition that was formulated by Nelson.
[21] Nowadays, stochastic quantization more commonly refers to a framework developed by Parisi and Wu in 1981.
However, there exist velocity fields, defined using conditional expectations.
It follows that in the Stratonovich formulation the second order part of the velocity vanishes, i.e.
through the addition of a total derivative term to the original action, such that where
Thus, since the dynamics should not be affected by the addition of a total derivative to the action, the action is gauge invariant under the above redefinition of the potentials for an arbitrary differentiable function
Due to the presence of the second order derivative term, the gauge invariance is broken.
However, this can be restored by adding a derivative of the vector potential to the Lagrangian.
[23] As the divergent term is constant, it does not contribute to the equations of motion.
For this reason, this term has been discarded in early works on stochastic mechanics.
This issue is known as Wallstrom's criticism,[24][25] and can be resolved by properly taking into account the divergent term.
[22][26] It starts from the definition of canonical momenta: The Hamiltonian in the Stratonovich formulation can then be obtained by the first order Legendre transform: In the Itô formulation, on the other hand, the Hamiltonian is obtained through a second order Legendre transform:[27] The stochastic action can be extremized, which leads to a stochastic version of the Euler-Lagrange equations.
In the Stratonovich formulation, these are given by For the Lagrangian, discussed in previous section, this leads to the following second order stochastic differential equation in the sense of Stratonovich: where, the field strength is given by
In this case, one starts by defining Hamilton's principal function.
, Hamilton's principal function is defined as where it is assumed that the process
However, the Hamiltonian, in the second Hamilton-Jacobi equation is now obtained using a second order Legendre transform.
Due to the equivalence relations on Hamilton's principal function, the opposite statement is also true: for any solution of these complex diffusion equations, one can construct a stochastic process