Zitterbewegung

Theory

Free spin-1/2 fermion

The time-dependent Dirac equation is written as

: H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) ,

where \hbar is the reduced Planck constant, \psi(\mathbf{x},t) is the wave function (bispinor) of a fermionic particle spin-1/2, and H is the Dirac Hamiltonian of a free particle:

: H = \beta mc^2 + \sum_{j = 1}^3 \alpha_j p_j c ,

where m is the mass of the particle, c is the speed of light, p_j is the momentum operator, and \beta and \alpha_j are matrices related to the Gamma matrices \gamma_\mu , as \beta=\gamma_0 and \alpha_j=\gamma_0\gamma_j .

In the Heisenberg picture, the time dependence of an arbitrary observable Q obeys the equation

: -i \hbar \frac{d Q}{d t} = \left[ H , Q \right] .

In particular, the time-dependence of the position operator is given by : \frac{d x_k(t)}{d t} = \frac{i}{\hbar}\left[ H , x_k \right] = c\alpha_k .

where xk(t) is the position operator at time t.

The above equation shows that the operator αk can be interpreted as the k-th component of a "velocity operator".

Note that this implies that

: \left\langle \left(\frac{d x_k(t)}{d t}\right)^2 \right\rangle=c^2 ,

as if the "root mean square speed" in every direction of space is the speed of light.

To add time-dependence to αk, one implements the Heisenberg picture, which says

: \alpha_k (t) = e^\frac{i H t}{\hbar}\alpha_k e^{-\frac{i H t}{\hbar}}.

The time-dependence of the velocity operator is given by : \hbar \frac{d \alpha_k(t)}{d t} = i\left[ H , \alpha_k \right] = 2\left(i \gamma_k m - \sigma_{kl}p^l\right) = 2i\left(cp_k-\alpha_k(t)H\right) ,

where :\sigma_{kl} \equiv \frac{i}{2}\left[\gamma_k,\gamma_l\right] .

Now, because both pk and H are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.

First: :\alpha_k (t) = \left(\alpha_k (0) - c p_k H^{-1}\right) e^{-\frac{2 i H t}{\hbar}} + c p_k H^{-1} ,

and finally

: x_k(t) = x_k(0) + c^2 p_k H^{-1} t + \tfrac12 i \hbar c H^{-1} \left( \alpha_k (0) - c p_k H^{-1} \right) \left( e^{-\frac{2 i H t}{\hbar}} - 1 \right) .

The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the reduced Compton wavelength. That oscillation term is the so-called Zitterbewegung.

Gaussian wavepacket

Another way of observing the Zitterbewegung is to study the evolution of a Gaussian wavepacket. In the non-relativistic case, using Schrödinger equation a Gaussian wavepacket disperses uniformly, increasing in width and decreasing in height. Using Dirac equation, the wave packet disperses but displays an interference pattern (with features of the order of the Compton length) as it travels due to the Zitterbewegung.

Interpretation

In quantum mechanics, the Zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. The standard relativistic velocity can be recovered by taking a Foldy–Wouthuysen transformation, when the positive and negative components are decoupled. Thus, we arrive at the interpretation of the Zitterbewegung as being caused by interference between positive- and negative-energy wave components.

In quantum electrodynamics (QED) the negative-energy states are replaced by positron states, and the Zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron pairs.

More recently, it has been noted that in the case of free particles it could just be an artifact of the simplified theory. Zitterbewegung appears as due to the "small components" of the Dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct second quantized theory, or rather, it is resolved by using Feynman propagators and doing QED. Nevertheless, it is an interesting way to understand certain QED effects heuristically from the single particle picture.

Zigzag picture of fermions

An alternative perspective of the physical meaning of Zitterbewegung was provided by Roger Penrose, by observing that the Dirac equation can be reformulated by splitting the four-component Dirac spinor \psi into a pair of massless left-handed and right-handed two-component spinors \psi = (\psi_{\rm L}, \psi_{\rm R}) (or zig and zag components), where each is the source term in the other's equation of motion, with a coupling constant proportional to the original particle's rest mass m, as

: \left{\begin{matrix}\sigma^\mu \partial_\mu \psi_{\rm R} = m \psi_{\rm L}\ \overline{\sigma}^\mu \partial_\mu \psi_{\rm L} = m \psi_{\rm R} \end{matrix}\right. .

The original massive Dirac particle can then be viewed as being composed of two massless components, each of which continually converts itself to the other. Since the components are massless they move at the speed of light, and their spin is constrained to be about the direction of motion, but each has opposite helicity: and since the spin remains constant, the direction of the velocity reverses, leading to the characteristic zigzag or Zitterbewegung motion.

Experimental simulation

Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence. It has also been simulated in atomic systems that provide analogues of a free Dirac particle. The first such example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation. Zitterbewegung-like oscillations of ultracold atoms in optical lattices were predicted in 2008. |article-number=153002|pmid=18518102 In 2013, Zitterbewegung was simulated in a Bose–Einstein condensate of 50,000 atoms of 87Rb confined in an optical trap. |article-number=073011

Optical analogues of Zitterbewegung have been demonstrated in a quantum cellular automaton implemented with orbital angular momentum states of light, in photonic synthetic frequency dimensions, , and in superconducting qubits.

Zitterbewegung also occurs in the description of quasiparticles of the Bogoliubov Hamiltonian, which are described by a Dirac-like Hamiltonian with momentum-dependent mass. Other proposals for condensed-matter analogues include semiconductor nanostructures, graphene and topological insulators.

References

References

1. Exner, K.. (1891). "Über die Scintillation". F. Tempsky.
2. Breit, Gregory. (1928). "An Interpretation of Dirac's Theory of the Electron". Proceedings of the National Academy of Sciences.
3. Greiner, Walter. (1995). "Relativistic Quantum Mechanics".
4. Schrödinger, E.. (1930). "Über die kräftefreie Bewegung in der relativistischen Quantenmechanik".
5. Schrödinger, E.. (1931). "Zur Quantendynamik des Elektrons".
6. Tong, David. (2017). "Applications of Quantum Mechanics". University of Cambridge.
7. Thaller, Bernd. (2005-12-06). "Advanced Visual Quantum Mechanics". Springer Science & Business Media.
8. Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.
9. "Dirac equation - is Zitterbewegung an artefact of single-particle theory?".
10. (2004). "The Road to Reality". Alfred A. Knopf.
11. (2008-07-01). "A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling". Foundations of Physics.
12. (2010). "Quantum physics: Trapped ion set to quiver". [[Nature (journal).
13. (2010). "Quantum simulation of the Dirac equation". [[Nature (journal).
14. Schliemann, John. (2005). "Zitterbewegung of Electronic Wave Packets in III-V Zinc-Blende Semiconductor Quantum Wells". [[Physical Review Letters]].
15. Katsnelson, M. I.. (2006). "Zitterbewegung, chirality, and minimal conductivity in graphene". [[The European Physical Journal B]].
16. (2012). "Optically engineering the topological properties of a spin Hall insulator". [[Physical Review Letters]].
17. (2013). "Anomalous Electron Trajectory in Topological Insulators". [[Physical Review B]].