Zitterbewegung

The time-dependent Dirac equation is written as

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

where ${\displaystyle \hbar }$ is the (reduced) Planck constant, ${\displaystyle \psi (\mathbf {x} ,t)}$ is the wave function (bispinor) of a fermionic particle spin-½, and H is the Dirac Hamiltonian of a free particle:

${\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}$,

where ${\textstyle m}$ is the mass of the particle, ${\textstyle c}$ is the speed of light, ${\textstyle p_{j}}$ is the momentum operator, and ${\displaystyle \beta }$ and ${\displaystyle \alpha _{j}}$ are matrices related to the Gamma matrices ${\textstyle \gamma _{\mu }}$, as ${\textstyle \beta =\gamma _{0}}$ and ${\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}}$.

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

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

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

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

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

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

Note that this implies that

${\displaystyle \left\langle \left({\frac {\partial x_{k}(t)}{\partial t}}\right)^{2}\right\rangle =c^{2}}$,

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

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

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

The time-dependence of the velocity operator is given by

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

where

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

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:

${\displaystyle \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}$,

and finally

${\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}$.

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

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 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.[3]