On the Stochastic Origin of Quantum Mechanics

The quantum Liouville equation, which describes the phase space dynamics of a quantum system of fermions, is analyzed from stochastic point of view as a particular example of the Kramers-Moyal expansion. Quantum mechanics is extended to relativistic domain by generalizing the Wigner-Moyal equation. Thus, an expression is derived for the relativistic mass in the Wigner quantum phase space presentation. The diffusion with an imaginary diffusion coefficient is discussed. An imaginary stochastic process is proposed as the origin of quantum mechanics.

One of many existing interpretations of quantum mechanics is the stochastic one, which is summarized in the seminal Nelson paper [1]. The Schrödinger equation is derived there from a real Wiener process, but the Nelson approach says nothing about the evolution of the quantum probability in the momentum space, which is compulsory for the complete mechanical treatment. Moreover, it is well known that the instant velocity of a Wiener process is infinite and thus the Nelson description is not equivalent to quantum mechanics. Obviously, the correct stochastic analysis requires consideration in the phase space. In the phase space formulation of quantum mechanics, the Schrödinger equation transforms to the quantum Liouville (Wigner-Moyal) equation [2][3][4] which is governing the evolution of the Wigner quasi-probability density ( , , ) W p q t . The traditional Liouville equation, being a milestone of classical statistical mechanics [5], follows from Eq. (1) in the classical limit 0  . The structure of the Wigner-Moyal equation hints already the stochastic origin of quantum mechanics. In stochastic theory [6], Eq. (1) is a particular example of the well-known Kramers-Moyal equation [4] The functions 1 Re ( / 2) nn nq iU      represent jump moments in the momentum subspace, which are also known as the Kramers-Moyal coefficients [6]. It is evident from their definition in Eq. (2) that the higher jump moments describe stochastic processes non-differentiable in the common sense. Since quantum mechanics is time reversible, there are no diffusion terms in Eq. (2), because 0 n  for any even n . While  [7]. The Pawula theorem [6] states that either 0 n  3 n  or all jump moments are meaningful. It follows from positivity of the probability density, which is, however, not the case of the Wigner quasi-probability density W in quantum mechanics. Nevertheless, the Pawula theorem imposes some restrictions on the external potential: () Uq could be constant, linear, harmonic or a general function with infinite number of q -derivatives. For the particular potentials above the corresponding Eq. (2) is purely classical, since In classical mechanics, the Hamilton function defines a system in mechanical sense and it governs the whole evolution of the particles momenta and coordinates. Since H is a sum of the particles kinetic and potential energies, one can generalize further Eq. (2) in a dual Kramers-Moyal form, symmetric on the particles momenta and coordinates, The jump moments here are given by

HW EW
 is the stationary non-relativistic Wigner function, while its eigenvalue E is the non-relativistic particle energy. Thus, the relativistic correction in Eq. (5) seems to be the correct one.
Introducing the partial Fourier transformation in the coordinate subspace, where the Fourier image of ( , , ) W p q t along the particle coordinate is that the non-relativistic energy of a quantum particle is Hence, it is straightforward to recognize that Eq. (7) represents the relativistic mass of a quantum particle in the Wigner phase space representation. The Einstein non-quantum relativistic mass 0 k M  follows in the classical limit 0  . In the case of a massless particle ( 0 m  ) at its lower energy level as a particle ( 0 p  ), for instance, the corresponding quantum wave energy is the zero-level vacuum ones, Thus, Eq. (7) properly accounts for the quantum wave-particle dualism.
The Kramers-Moyal equation (3) , which reduces to c    for a massless particle.
It is important to find out what stochastic process is driving the quantum motion. It is well known that the Schrödinger equation for a free particle is, in fact, the diffusion equation with imaginary diffusion constant /2 im . Thus, the quantum motion is a Brownian movement with imaginary stochastic force [7]. The classical diffusion equation for the evolution of the probability density ( , ) ( , , ) q t W p q t dp   of a Brownian particle reads where D is a real diffusion coefficient. The solution of Eq. (9)