L. de la Peña

Stochastic theory for classical and quantum mechanical systems

We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newtons second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrödinger equation, which is derived here with no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section.

THE QUANTUM HARMONIC OSCILLATOR REVISITED: A NEW LOOK FROM STOCHASTIC ELECTRODYNAMICS

We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first‐order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.

Does quantum mechanics accept a stochastic support

Arguments are given in favor of a stochastic theory of quantum mechanics, clearly distinguishable from Brownian motion theory. A brief exposition of the phenomenological theory of stochastic quantum mechanics is presented, followed by a list of its main results and perspectives. A possible answer to the question about the origin of stochasticity is given in stochastic electrodynamics by assigning a real character to the vacuum radiation field. This theory is shown to reproduce important quantum mechanical results, some of which are presented explicitly to illustrate its potentialities. Finally the main problems and some perspectives of research within stochastic electrodynamics are discussed.

Quantum Theory and Linear Stochastic Electrodynamics

We discuss the main results of Linear Stochastic Electrodynamics, starting from a reformulation of its basic assumptions. This theory shares with Stochastic Electrodynamics the core assumption that quantization comes about from the permanent interaction between matter and the vacuum radiation field, but it departs from it when it comes to considering the effect that this interaction has on the statistical properties of the nearby field. In the transition to the quantum regime, correlations between field modes of well-defined characteristic frequencies arise, which coincide with the transition frequencies of quantum mechanics and are therefore directly related with the energy quantization. The Heisenberg equations of motion of (non-relativistic) quantum electrodynamics are thus obtained. After a detailed consideration of the significance of the approximations made, we present a discussion on some of the most delicate or controversial features of quantum mechanics from the perspective provided by the present theory.

The classical motion of an extended charged particle revisited

SummaryWe analyse the rotational dynamics of a nonrelativistic extended self-interacting particle. The equation of motion is a linear integrodifferential one, that gives place to a strictly causal behaviour, under a certain condition, as in the translational case. The motion is endowed with memory. Considering the interaction with an external magnetic field, we compute the classical gyromagnetic ratio and state the existence of stable modes, characterized by a precession frequency (that in a certain limit reduces to the Larmor one). There are no polarization phenomena.RiassuntoSi analizza la dinamica rotazionale di una particella non relativistica estesa autointeragente. L’equazione di moto è un’equazione lineare integrodifferenziale che dà luogo a un comportamento strettamente causale, in certe condizioni, come nel caso di translazione. Il moto è dotato di memoria. Considerando l’interazione con un campo magnetico esterno, si calcola il classico rapporto giromagnetico e si stabilisce l’esistenza di modi stabili, caratterizzati da una frequenza di precessione (che in un certo limite si riduce a quella di Larmor). Non c’è alcun fenomeno di polarizzazione.РезюмеМы анализируем ротационную динамику нерелятивистской протяженной само-взаимодействующей частицы. Уравнение движния явлется линейным интегро-дифференциальным уравнением, которое при определенном условии приводит к строго причинному поведению, как в трансляционном случае. Движение обладает памятью. Рассматривая взаимодействие с внешним магнитным полем, мы вычисляем классическое гиромагнитное отношение и показываем существование устойчивых мод, характеризуемых частотой прецессии (кпторая в пределе сводится к ларморовской частоте). Явления поляризации не существуют.

Quantum phenomena and the zeropoint radiation field. II

A previous paper was devoted to the discussion of a new version of stochastic electrodynamics (SED) and to the study of the conditions under which quantum mechanics can be derived from it, in the radiationless approximation. In this paper further effects on matter due to the zeropoint field are studied, such as atomic stability, radiative transitions, the Lamb shift, etc., and are shown to be correctly described by the proposed version of SED. Also, a detailed energy-balance condition and a fluctuation-dissipation relation are established; it is shown in particular that equilibrium is attained only with a field spectrum ∼Ω3.The proposed approach is shown to suggest an understanding of quantum mechanics as a kind of limitcycle theory. Finally, a brief discussion is included about the nonchaotic behavior of the (bounded) SED system in the quantum regime, as measured by Lyapunov exponents.

A Bell inequality involving position, momentum and energy

Abstract A Bell inequality involving only variables with a classical analog is derived, and it is shown to be violated by quantum mechanics.

The spin of the electron according to stochastic electrodynamics

By making use of the method of moments we study some aspects of the statistical behavior of the nonrelativistic harmonic oscillator according to stochastic electrodynamics. We show that the random rotations induced on the particle by the zero-point field account for the magnitude of the spin of the electron, the result differing from the correct one(3/4)h2 by a factor of2. Assuming that the measurement of a spin projection may be effectively taken into account by considering the action of only the subensemble of the field with the corresponding circular polarization, the calculated value of the spin projection comes out to be the correct one within a factor of order unity. The radiative corrections give rise to both the Lamb shift and the anomalous magnetic moment of the electron, the latter being evaluated to within a factor of2. The magnetic and gyromagnetic properties of the electron come out to be in agreement with quantum mechanics. Interference effects are shown to occur when evaluating the average value of the square of the angular momentum.

Quantum behavior derived as an essentially stochastic phenomenon

We study the behavior of the atomic matter immersed in a zero-point radiation field, which is taken as the source of the endless stochastic motion of electrons. The starting point is a statistical phase-space equation for a particle in interaction with the field. Under certain natural approximations, the system reaches a stationary condition; the corresponding reduced description obtained in configuration space is just the Schrodinger quantum mechanics. Quantum nonlocality is revisited from this point of view and is shown to appear as an artifact of the reduced description. The theory presented also serves to explain why usual quantum mechanics is not amenable to a phase-space formulation.

Electromagnetic vacuum-particle interaction as the source of quantum phenomena

A nonperturbative treatment has been introduced to study the steady-state dynamics of the (classical) electron in interaction with the random zero-point electromagnetic field. Here we use a statistical procedure to transfer this description to the particles configuration or momentum space, obtaining in the radiationless approximation the equations of quantum mechanics. Inclusion of the radiative terms under a perturbative scheme leads to nonrelativistic quantum electrodynamics.