Zitterbewegung

Zitterbewegung ("trembling motion" from German) is a hypothetical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation. The existence of such motion was first proposed by Erwin Schrödinger in 1930 as a result of his analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (at the speed of light) of the position of an electron around the median, with an angular frequency of $2mc^{2}/\hbar$, or approximately 1.6×10²¹ radians per second. A re-examination of Dirac theory, however, shows that interference between positive and negative energy states may not be a necessary criterion for observing zitterbewegung.

Zitterbewegung of a free relativistic particle has never been observed. However, it has been simulated twice. First, with a trapped ion, by putting it in an environment such that the non-relativistic Schrödinger equation for the ion has the same mathematical form as the Dirac equation (although the physical situation is different). Then, in 2013, it was simulated in a setup with Bose-Einstein condensates.

Theory

The time-dependent Dirac equation

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

where $H$ is the Dirac Hamiltonian for an electron in free space

$H=\left(\alpha _{0}mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}\,c\right)$

in the Heisenberg picture implies that any operator Q obeys the equation

$-i\hbar {\frac {\partial Q}{\partial t}}(t)=\left[H,Q\right]\,.$

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

$\hbar {\frac {\partial x_{k}}{\partial t}}(t)=i\left[H,x_{k}\right]=\hbar c\alpha _{k}$

where $\alpha _{k}\equiv \gamma _{0}\gamma _{k}$.

The above equation shows that the operator $\alpha _{k}$ can be interpreted as the kth component of a "velocity operator". To add time-dependence to $\alpha _{k}$, one implements the Heisenberg picture, which says

$\alpha _{k}(t)=e^{iHt/\hbar }\alpha _{k}e^{-iHt/\hbar }$

The time-dependence of the velocity operator is given by

$\hbar {\frac {\partial \alpha _{k}}{\partial t}}(t)=i\left[H,\alpha _{k}\right]=2(i\gamma _{k}m-\sigma _{kl}p^{l})=2i(p_{k}-\alpha _{k}H)$

where $\sigma _{kl}\equiv {\frac {i}{2}}[\gamma _{k},\gamma _{l}]$.

Now, because both $p_{k}$ and $H$ are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator. First:

$\alpha _{k}(t)=(\alpha _{k}(0)-cp_{k}H^{-1})e^{-2iHt/\hbar }+cp_{k}H^{-1}$

Then:

$x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{1 \over 2}i\hbar cH^{-1}(\alpha _{k}(0)-cp_{k}H^{-1})(e^{-2iHt/\hbar }-1)$

where $x_{k}(t)$ is the position operator at time $t$.

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

Interestingly, the "Zitterbewegung" term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. This can be achieved by taking a Foldy Wouthuysen transformation. Thus, we arrive at the interpretation of the "Zitterbewegung" as being caused by interference between positive- and negative-energy wave components.

See also

• Casimir effect
• Lamb shift
• Stochastic electrodynamics: Zitterbewegung is explained as an interaction of a classical particle with the zero-point field.
• Foldy-Wouthuysen transformation

Further reading

• E. Schrödinger, Über die kräftefreie Bewegung in der relativistischen Quantenmechanik ("On the free movement in relativistic quantum mechanics"), Berliner Ber., pp. 418–428 (1930); Zur Quantendynamik des Elektrons, Berliner Ber, pp. 63–72 (1931)
• A. Messiah, Quantum Mechanics Volume II, Chapter XX, Section 37, pp. 950–952 (1962)

External links

• The Zitterbewegung Interpretation of Quantum Mechanics, an alternative explanation in addition to positive-negative energy states interference.
• Zitterbewegung in New Scientist
• Geometric Algebra in Quantum Mechanics
• Summary of trapped ion simulation