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arXiv:2008.05954 · Physical Review A

Zitterbewegung of massless particles

Alexander J. Silenko

Published Jun 21, 2022• 11 citations

Abstract

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.

DOI: 10.1103/physreva.105.062211

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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.

Abstract Zitterbewegung of massive and massless scalar bosons and a massive Proca (spin-1) boson is analyzed. The equations describing the evolution of the velocity and position of the scalar boson in the generalized Feshbach–Villars representation and the corresponding equations for the massive Proca particle in the Sakata–Taketani representation are equivalent to each other and to the well-known equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy–Wouthuysen representation. Since the position and velocity operators in the Foldy–Wouthuysen representation and their transforms to other representations are the quantum-mechanical counterparts of the corresponding classical variables, Zitterbewegung is not observable.

Abstract Zitterbewegung of a massive Proca (spin-1) boson is analyzed. The equations of motion of a massive Proca particle in the Sakata-Taketani representation are equivalent to the corresponding equations for the Dirac particle. However, Zitterbewegung does not appear in the Foldy-Wouthuysen representation. Zitterbewegung is not observable because the quantum-mechanical counterparts of the classical position and velocity are the position and velocity operators in the Foldy-Wouthuysen representation and their transforms to other representations.

The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.

The expressions for the dynamical operators corresponding to a Bargmann-Wigner particle of mass κ and arbitrary spin s are given, making use of the Foldy-Wouthuysen and Chakrabarti transformations for the Bargmann-Wigner equations given by Pursey, and Azcárraga and Boya, respectively. The operators selecting the positive and negative energy states are also given in the two representations. The operators for arbitrary spin are studied and a condition for an operator be constant of motion is given. Finally, the Newton-Wigner position operator and those obtained in the Foldy-Wouthuysen and Chakrabarti generalized representations are compared.

The Hamiltonian and other relevant physical operators for a free particle of arbitrary spin and mass, based upon the generalized Foldy-Wouthuysen transformation, are developed in terms of the spin-matrix polynomials. The general theory of these polynomials, given in an earlier paper, is reviewed briefly. It is a general feature of these expansions that the number of terms is determined by the spin of the particle, but the coefficients are spin-independent. The massless limit is examined in some detail and the invariant metric operator is discussed.