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Journal of Physics Conference Series

Zitterbewegung in quantum mechanics of Proca particles

Alexander J. Silenko

Published Jan 1, 2020• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/1435/1/012057

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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.
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.
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.
The behaviour of wavepackets of the `Dirac oscillator' in (1 + 1) dimensions is examined. The Dirac oscillator is a relativistic extension of the harmonic oscillator. We construct a wavepacket in the usual Dirac representation and another one in the Foldy - Wouthuysen (FW) representation. In the nonrelativistic limit, these wavepackets are both reduced to Schrödinger's coherent wavepacket for the harmonic oscillator. The wavepacket in the Dirac representation exhibits Zitterbewegung whereas the one in the FW representation does not. We find that the wavepacket in the FW representation behaves much more like a classical particle than that of the Dirac representation. This gives insights into Ehrenfest's theorem for the Dirac equation.