Zitterbewegung

The time-dependent Dirac equation is written as

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

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

:$H\; =\; \backslash beta\; mc^2\; +\; \backslash sum\_\{j\; =\; 1\}^3\; \backslash alpha\_j\; p\_j\; c$,

where $m$ is the mass of the particle, $c$ is the speed of light, $p\_j$ is the momentum operator, and $\backslash beta$ and $\backslash alpha\_j$ are matrices related to the Gamma matrices $\backslash gamma\_\backslash mu$, as $\backslash beta=\backslash gamma\_0$ and $\backslash alpha\_j=\backslash gamma\_0\backslash gamma\_j$.

In the Heisenberg picture, the time dependence of an arbitrary observable {{mvar|Q}} obeys the equation

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

In particular, the time-dependence of the position operator is given by

:$\backslash frac\{d\; x\_k(t)\}\{d\; t\}\; =\; \backslash frac\{i\}\{\backslash hbar\}\backslash left[\; H\; ,\; x\_k\; \backslash right]\; =\; c\backslash alpha\_k$.

where {{math|x_k(t)}} is the position operator at time {{mvar|t}}.

The above equation shows that the operator {{mvar|α_k}} can be interpreted as the {{mvar|k}}-th component of a "velocity operator".

Note that this implies that

:$\backslash left\backslash langle\; \backslash left(\backslash frac\{d\; x\_k(t)\}\{d\; t\}\backslash right)^2\; \backslash right\backslash rangle=c^2$,

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

To add time-dependence to {{mvar|α_k}}, one implements the Heisenberg picture, which says

:$\backslash alpha\_k\; (t)\; =\; e^\backslash frac\{i\; H\; t\}\{\backslash hbar\}\backslash alpha\_k\; e^\{-\backslash frac\{i\; H\; t\}\{\backslash hbar\}\}$.

The time-dependence of the velocity operator is given by

:$\backslash hbar\; \backslash frac\{d\; \backslash alpha\_k(t)\}\{d\; t\}\; =\; i\backslash left[\; H\; ,\; \backslash alpha\_k\; \backslash right]\; =\; 2\backslash left(i\; \backslash gamma\_k\; m\; -\; \backslash sigma\_\{kl\}p^l\backslash right)\; =\; 2i\backslash left(cp\_k-\backslash alpha\_k(t)H\backslash right)$,

where

:$\backslash sigma\_\{kl\}\; \backslash equiv\; \backslash frac\{i\}\{2\}\backslash left[\backslash gamma\_k,\backslash gamma\_l\backslash right]\; .$

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

First:

:$\backslash alpha\_k\; (t)\; =\; \backslash left(\backslash alpha\_k\; (0)\; -\; c\; p\_k\; H^\{-1\}\backslash right)\; e^\{-\backslash frac\{2\; i\; H\; t\}\{\backslash hbar\}\}\; +\; c\; p\_k\; H^\{-1\}$,

and finally

:$x\_k(t)\; =\; x\_k(0)\; +\; c^2\; p\_k\; H^\{-1\}\; t\; +\; \backslash tfrac12\; i\; \backslash hbar\; c\; H^\{-1\}\; \backslash left(\; \backslash alpha\_k\; (0)\; -\; c\; p\_k\; H^\{-1\}\; \backslash right)\; \backslash left(\; e^\{-\backslash frac\{2\; i\; H\; t\}\{\backslash hbar\}\}\; -\; 1\; \backslash right)$.

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.

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.{{Cite book |last=Thaller |first=Bernd |url=https://www.google.fr/books/edition/Advanced_Visual_Quantum_Mechanics/iq1Gi6hmTRAC?hl=en&gbpv=1&dq=%5B9%5D+Thaller+B+2005,+Advanced+Visual+Quantum+Mechanics+(New+York:+Springer).&pg=PA491&printsec=frontcover |title=Advanced Visual Quantum Mechanics |date=2005-12-06 |publisher=Springer Science & Business Media |isbn=978-0-387-27127-9 |language=en}}

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.{{Cite book|last=Greiner|first=Walter|date=1995|title=Relativistic Quantum Mechanics|url=https://link.springer.com/book/10.1007/978-3-642-88082-7|language=en-gb|doi=10.1007/978-3-642-88082-7|isbn=978-3-540-99535-7|s2cid=124404090 }}

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.Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.

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. {{Cite web|url=https://physics.stackexchange.com/questions/28672/is-zitterbewegung-an-artefact-of-single-particle-theory|title = Dirac equation - is Zitterbewegung an artefact of single-particle theory?}}

{{See also|Feynman checkerboard}}

An alternative perspective of the physical meaning of Zitterbewegung was provided by Roger Penrose,{{cite book |last1=Penrose |first1=Roger |title=The Road to Reality |date=2004 |publisher=Alfred A. Knopf |isbn=0-224-04447-8 |edition=Sixth Printing |pages=628–632}} by observing that the Dirac equation can be reformulated by splitting the four-component Dirac spinor $\backslash psi$ into a pair of massless left-handed and right-handed two-component spinors $\backslash psi\; =\; (\backslash psi\_\{\backslash rm\; L\},\; \backslash psi\_\{\backslash rm\; R\})$ (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.

Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence.{{Cite journal |last1=Catillon |first1=P. |last2=Cue |first2=N. |last3=Gaillard |first3=M. J. |last4=Genre |first4=R. |last5=Gouanère |first5=M. |last6=Kirsch |first6=R. G. |last7=Poizat |first7=J.-C. |last8=Remillieux |first8=J. |last9=Roussel |first9=L. |last10=Spighel |first10=M. |display-authors=3 |date=2008-07-01 |title=A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling |journal=Foundations of Physics |volume=38 |issue=7 |pages=659–664 |doi=10.1007/s10701-008-9225-1 |bibcode=2008FoPh...38..659C |s2cid=121875694 |issn=1572-9516}} 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.{{cite journal|title=Quantum physics: Trapped ion set to quiver|url=http://www.nature.com/nature/journal/v463/n7277/full/463037a.html|newspaper=Nature News and Views | volume=463|issue=7277|pages=37–39|doi=10.1038/463037a|pmid=20054385|year=2010|last1=Wunderlich|first1=Christof|bibcode=2010Natur.463...37W |doi-access=free}}{{cite journal|last1=Gerritsma |first1=R. |last2=Kirchmair |first2=G. |last3=Zähringer |first3=F. |last4=Solano |first4= E. |last5=Blatt |first5=R. |last6=Roos |first6=C. F. |title=Quantum simulation of the Dirac equation|journal=Nature|year=2010|volume=463|issue=7277|doi=10.1038/nature08688|pmid=20054392|arxiv = 0909.0674 |bibcode = 2010Natur.463...68G|pages=68–71|s2cid=4322378}} Zitterbewegung-like oscillations of ultracold atoms in optical lattices were predicted in 2008.

{{cite journal

|last1= Vaishnav |first1=J. Y.

|last2= Clark |first2= C. W.

|title= Observing Zitterbewegung with Ultracold Atoms

|journal=Physical Review Letters

|year=2008

|volume=100

|issue=15

|doi= 10.1103/PhysRevLett.100.153002

|pages=153002|pmid=18518102

|arxiv=0711.3270

|bibcode=2008PhRvL.100o3002V

}}

In 2013, Zitterbewegung was simulated in a Bose–Einstein condensate of 50,000 atoms of ⁸⁷Rb confined in an optical trap.

{{cite journal

|last1=Leblanc |first1=L. J.

|last2=Beeler |first2=M. C.

|last3=Jimenez-Garcia |first3=K.

|last4=Perry |first4=A. R.

|last5=Sugawa |first5=S.

|last6=Williams |first6=R. A.

|last7=Spielman |first7=I.B.

|title=Direct observation of zitterbewegung in a Bose–Einstein condensate

|journal=New Journal of Physics|year=2013

|volume=15

|issue=7

|doi=10.1088/1367-2630/15/7/073011

|page=073011

|arxiv=1303.0914

|s2cid=119190847}}

An optical analogue of Zitterbewegung has been demonstrated in a quantum cellular automaton implemented with orbital angular momentum states of light,

{{cite journal |title=Photonic cellular automaton simulation of relativistic quantum fields: observation of Zitterbewegung

|first1=Alessia |last1=Suprano

|first2=Danilo |last2=Zia

|first3=Emanuele |last3=Polino

|first4=Davide |last4=Poderini

|first5=Gonzalo |last5=Carvacho

|first6=Fabio |last6=Sciarrino

|first7=Matteo |last7=Lugli

|first8=Alessandro |last8=Bisio

|first9=Paolo |last9=Perinotti

|journal=Phys. Rev. Research

|year=2024

|volume=6

|issue=3 |page=033136

|doi=10.1103/PhysRevResearch.6.033136

|arxiv=2402.07672|bibcode=2024PhRvR...6c3136S }}

and in photonic synthetic frequency dimensions.

{{cite journal|title=Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions

|last1=Cheng |first1=Dali

|last2=Wang |first2=Kai

|last3=Roques-Carmes |first3=Charles

|last4=Lustig |first4=Eran

|last5=Long |first5=Olivia Y.

|last6=Wang |first6=Heming

|last7=Fan |first7=Shanhui

|journal=Nature

|year=2025

|volume=637

|issue=8044 |pages=52–56 |doi=10.1038/s41586-024-08259-2

|pmid=39743600 |arxiv=2406.00321|bibcode=2025Natur.637...52C }}

Zitterbewegung also occurs in the description of quasiparticles of the Bogoliubov Hamiltonian, which are described by a Dirac-like Hamiltonian with momentum-dependent mass.

{{cite journal |last1=Yan |first1=Li |last2=Hong-Yi |first2=Su |last3=Fu-Lin |first3=Zhang |last4=Jing-Ling |first4=Chen |last5=Chun-Feng |first5=Wu |last6=Kwek |first6=Leong-Chuan |year=2015 |title=Zitterbewegung in Bogoliubov's System |journal=Communications in Theoretical Physics |volume=63 |issue=2 |pages=145–148 |doi=10.1088/0253-6102/63/2/05|arxiv=1307.6912 |bibcode=2015CoTPh..63..145L }}

Other proposals for condensed-matter analogues include semiconductor nanostructures, graphene and topological insulators.{{cite journal|title=Zitterbewegung of Electronic Wave Packets in III-V Zinc-Blende Semiconductor Quantum Wells |last=Schliemann|first=John |journal=Physical Review Letters |volume=94 |number=20 |year=2005|pages=206801 |arxiv=cond-mat/0410321|doi=10.1103/PhysRevLett.94.206801 |pmid=16090266|bibcode=2005PhRvL..94t6801S |s2cid=118979437}}{{cite journal|title=Zitterbewegung, chirality, and minimal conductivity in graphene |last=Katsnelson |first=M. I. |journal=The European Physical Journal B |volume=51 |number=2 |year=2006 |pages=157–160 |arxiv=cond-mat/0512337|doi=10.1140/epjb/e2006-00203-1 |bibcode=2006EPJB...51..157K |s2cid=119353065 }}{{cite journal|title=Optically engineering the topological properties of a spin Hall insulator |last1=Dóra |first1=Balász |last2=Cayssol |first2=Jérôme |last3=Simon |first3=Ference |last4=Moessner |first4=Roderich |arxiv=1105.5963 |journal=Physical Review Letters |volume=108 |number=5 |year=2012 |pages=056602 |doi=10.1103/PhysRevLett.108.056602|pmid=22400947 |bibcode=2012PhRvL.108e6602D |s2cid=15507388 }}{{cite journal|title=Anomalous Electron Trajectory in Topological Insulators |last1=Shi |first1=Likun |last2=Zhang |first2=Shoucheng |last3=Cheng |first3=Kai |journal=Physical Review B |volume=87 |number=16 |year=2013 |page=161115 |doi=10.1103/PhysRevB.87.161115 |arxiv=1109.4771|bibcode=2013PhRvB..87p1115S |s2cid=118446413 }}

• Casimir effect
• Lamb shift

{{reflist}}

• {{cite book |first= A. |last= Messiah |title= Quantum Mechanics |volume= II |chapter= XX, Section 37 |pages= 950–952 |date= 1962 |chapter-url= https://archive.org/details/QuantumMechanicsVolumeIi |chapter-format= pdf |isbn= 9780471597681 }}

• [https://www.newscientist.com/channel/fundamentals/mg19526112.300-the-word-zitterbewebung.html Zitterbewegung in New Scientist]

Category:Quantum field theory