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arXiv:2006.07615 · Physical Review Research

Comment on “Nondispersive analytical solutions to the Dirac equation”

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Jul 6, 2020• 1 citation

Abstract

In our Comment we question the validity of the claim made by Campos and Cabrera [Phys. Rev. Res. 2, 013051 (2020)] that their solutions of the Dirac equation in an external time-dependent electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has an infinite time duration, there is continuous electron-positron pair creation and annihilation. Without the proper accounting for these processes, the mathematical solutions of the Dirac equation are not directly applicable to realistic physical situations. In particular, the time evolution of the average values $\ensuremath{\langle}x\ensuremath{\rangle}$ and $\ensuremath{\langle}y\ensuremath{\rangle}$ does not describe the electron trajectory but the motion of some combination of the electron and positron charge distributions with pathological properties (zitterbewegung).

DOI: 10.1103/physrevresearch.2.038001

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In our Comment we question the validity of the claim made by Campos and Cabrera [Phys. Rev. Res. 2, 013051 (2020)] that their solutions of the Dirac equation in an external time-dependent electromagnetic field describe beams of electrons. In every time-dependent field, no matter how weak, which has an infinite time duration, there is continuous electron-positron pair creation and annihilation. Without the proper accounting for these processes, the mathematical solutions of the Dirac equation are not directly applicable to realistic physical situations. In particular, the time evolution of the average values

\ensuremath{\langle}x\ensuremath{\rangle}

and

\ensuremath{\langle}y\ensuremath{\rangle}

does not describe the electron trajectory but the motion of some combination of the electron and positron charge distributions with pathological properties (zitterbewegung).

A new method was developed to solve the Dirac equation and used to find analytical solutions for time-dependent electromagnetic fields that translate a given Dirac spinor along any desired trajectory in the $x\text{\ensuremath{-}}y$ plane without distortion. As is well known, negative energy solutions describing antiparticles play a prominent role in the Dirac theory of the electron, and a separation of the part of the wave function describing only electrons from the full solutions often needs to be done. Here, we clarify how the aforementioned separation is done, which is not clear in our recent publication [Campos and Cabrera, Phys. Rev. Research 2, 013051 (2020)].

Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.

In this paper, it is presented a historical account of the formulation of the quantum relativistic wave equation of an electron – the Dirac equation, issues regarding its interpretation that arose from the very beginning, and the later formulation of this equation in relation to a quantized electron-positron field, which implies a new interpretation. The way in which solutions obtained under each interpretation of the equation relate to one another is also considered for the simple case of hydrogen-like problems.

One-electron

3+1

and

2+1

Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion, Zitterbewegung (ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron ZB: it introduces intraband frequency components into the motion, influences all the frequencies, and makes the motion stationary (not decaying in time) in case of the

2+1

Dirac equation. Finally, simulations of the

2+1

Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers, we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.