Zitterbewegung

 ( Quantum Field Theory ) 

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

where $\backslash hbar$ is the reduced Planck constant, $\backslash psi(\backslash mathbf\{x\},t)$ is the wave function (bispinor) of a particle spin-1/2, and 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 obeys the equation

$-i\; \backslash hbar\; \backslash frac\{d\; Q\}\{d\; t\}\; =\; \backslash left\; .$

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

$\backslash frac\{d\; x\_k(t)\}\{d\; t\}\; =\; \backslash frac\{i\}\{\backslash hbar\}\backslash left\; =\; c\backslash alpha\_k$.

where is the position operator at time .

The above equation shows that the operator $\backslash alpha\_k$ can be interpreted as the -th component of a "velocity operator".

Note that this implies that

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

where we squared both sides of the expression and used the property that $\backslash alpha\_k^2=I\_4$. The expectation value is now as if the "root mean square speed" in every direction of space is the speed of light.

To add time-dependence to , 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

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

where

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

Now, because both and 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.