Quantum mechanics as a statistical theory

Abstract

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

Moyal products—a new perspective on quasi-Hermitian quantum mechanics

“…The Moyal product [21] was introduced by Moyal to facilitate Wigner's phase space formulation of quantum mechanics. More recently it was revived in the context of non-commutative systems (see e.g.…”

“…An isomorphism with the operator product can be established by defining the Moyal product of functions A(α, β) and B(α, β) [21,22] …”

Функции Вигнера Для Задачи Ландау В Некоммутативной Квантовой Механике

“…Исходя из квантово-механической волновой функции Вигнер построил функцию распределения, которая является вещественной, но не всюду положительной. За-тем Мойал записал эволюционное уравнение для данного распределения, введя для этого свою знаменитую скобку [17]. Функция Вигнера распределения вероятности для одновременных значений координат x и импульсов p в фазовом пространстве размерности 2d определяется в терминах волновой функции…”

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for wigner functions

“…There is a formulation of quantum mechanics [4], [5], [6] -I will refer to it as the phase-space formulation -that has become popular as an arena in which to investigate such matters as quantum chaos [7], quantum tomography [8], [9], the relationship between quantum and classical mechanics [10], [11], and the nature of quantization [12], [13]. What I would like to draw attention to here is that the phase-space formulation is purely real -there are no complex numbers to be seen in the defining equations.…”