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arXiv:1411.1854 · AIP conference proceedings

The problem of motion: The statistical mechanics of Zitterbewegung

Kevin H. Knuth

Published Jan 1, 2015• 8 citations
arXivHTMLPDF

Abstract

Around 1930, both Gregory Breit and Erwin Schroedinger showed that the eigenvalues of the velocity of a particle described by wavepacket solutions to the Dirac equation are simply $\pm$c, the speed of light. This led Schroedinger to coin the term Zitterbewegung, which is German for "trembling motion", where all particles of matter (fermions) zig-zag back-and-forth at only the speed of light. The result is that any finite speed less than $c$, including the state of rest, only makes sense as a long-term average that can be thought of as a drift velocity. In this paper, we seriously consider this idea that the observed velocities of particles are time-averages of motion at the speed of light and demonstrate how the relativistic velocity addition rule in one spatial dimension is readily derived by considering the probabilities that a particle is observed to move either to the left or to the right at the speed of light.

DOI
10.1063/1.4906026

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Around 1930, both Gregory Breit and Erwin Schroedinger showed that the eigenvalues of the velocity of a particle described by wavepacket solutions to the Dirac equation are simply ±\pm±c, the speed of light. This led Schroedinger to coin the term Zitterbewegung, which is German for "trembling motion", where all particles of matter (fermions) zig-zag back-and-forth at only the speed of light. The result is that any finite speed less than ccc, including the state of rest, only makes sense as a long-term average that can be thought of as a drift velocity. In this paper, we seriously consider this idea that the observed velocities of particles are time-averages of motion at the speed of light and demonstrate how the relativistic velocity addition rule in one spatial dimension is readily derived by considering the probabilities that a particle is observed to move either to the left or to the right at the speed of light.
The Assessment of the interpretation of Zitterbewegung for free particle solution using various models has been carried out in this work where we firstly considered by using the free particle solution for which Dirac equation and projection operator were carried out.. ln this case, it was invariably revealed that the solution have two doubly degenerate eigenvalues representing positive and negative state that was further support by the analysis which was carried out using Heisenberg’s equation and the representation in relativistic velocity and semi-classical equation of motion for acceleration., but in this second case, the results obtained were observed to have two terms, first terms standing in for rapidly oscillatory motion and the second term representing average motion of the particle in x-direction. The first term in the expression is the one deemed to signify zitterbewegung which is one considered to be sas a result of the fluctuation resulting from the interference between negative and positive energy state while the other term is the normal state terms signifying the average motion of the particle. This therefore confirms the explicit nature of using Dirac equation in handling problems involving the behavior of free particles in field as compared to the use of semiclassical equation of motion.
This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.
Dirac's Relativistic Wave Equation implies a measured electron velocity of ±c\pm c±c in any direction, in contradiction to Special Relativity and observation. It is shown in this article that this anomalous electron velocity reveals an internal structure of the electron whereby the mass and the charge of the electron cannot be described by the same position operator. The measured velocity of electron mass is always less than ccc in any direction but charge can be displaced at the speed of light. This speed is realizable only when the electron is in a state that is a superposition of positive and negative energy states, also known as a zitterbewegung state. It is shown that in zitterbewegung it is the charge and not the mass that undergoes rapid spatial oscillation, and that there are measurable consequences of this charge zitterbewegung. Zitterbewegung of charge also occurs in an entangled electron-positron pair created by a strong electric field.
In this paper, we study the Zitterbewegung, as the most anomalous and controversial effect of the Dirac equation. For this purpose, we do not establish a study framework, but we carry out a dissection of the Dirac solution for the velocity operator from a wave optics perspective, which will provide us (such as it is) with all the information about the Zitterbewegung, about the true sense and legality of the solution, and of the Dirac equation itself. In particular, we will obtain that said solution of the Dirac equation is composed of two wave packets coupled through the phase functions, and that it is the destructive and constructive confrontation of their group velocities that will give rise to the classical speed of the particle and the quantum anomalous term, respectively. This coupling is the kinetic aspect of a more general energetic development and of a corpuscular etiology, from which we derive a geometry that will then be an active part of any related phenomenology. A geometry will allow us to uncover some primary questions about physical principles, kinetic energy, spin and velocities, as well as differentiate between wave packets, photons and particles.
Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.