What do the matrices $\alpha_k$ represent in the Dirac equation?

Question

I have been scouring the internet for an answer. All I have managed to find are the matrices for $k=1,2,3,4,5$. However, I still have no idea they represent, within the equation. Am I correct in saying the $\alpha_k$ have no effect when the electron is at rest?

Answer 1

You have to be careful about the interpretation of the $\alpha$ matrices. However, if you must assign an interpretation to $\alpha_{k}$, it is the velocity of the Dirac fermion (in units of $c$). There are several facts that indicate this. If you use Hamilton's equation $\dot{x}_{k}=\partial H/\partial p_{k}$ (or the equivalent Heisenberg equation of motion), you find that $\dot{x}_{k}=c\alpha_{k}$. Similarly, if you calculate the Lorentz force, you find $q(c\vec{\alpha}\times\vec{B})$.

However, the fact that $\alpha_{k}$ is a matrix means that the velocity it represents has some counterintuitive properties. For one thing, it is not possible to specify more than one component of the velocity at a time, since $[\alpha_{j},\alpha_{k}]\neq0$. Nor is the velocity a constant of the motion for a free particle, because $[\alpha_{j},H]\neq0$ Moreover, the only eigenvalues of $\alpha_{k}$ are $\pm1$, which suggests that if it were possible to measure the instantaneous velocity, we would always find it moving with speed $c$. These properties are quite different from what we expect to find for the speed; we would normally expect the speed to be the conserved quantity $\vec{p}c^{2}/E$, which magnitude strictly less than $c$.

The resolution of these paradoxes is that a massive spin-$\frac{1}{2}$ particle is always undergoing rapid oscillations, with frequency $\omega\gtrsim2mc^{2}/\hbar.$ These oscillations are known as zitterbewegung ("quivering motion" in German). The instantaneous velocity in any given direction is always $c$, but the rapid oscillations ensure that the particle does not make it very far in any direction before turning around the other way. The average speed, when averaged over time scales greater than $\omega^{-1}$ is then give by the usual $\vec{p}c^{2}/E$. The zitterbewegung arises because to have a well localized (to less than about a Compton wavelength) wave function for a Dirac particle, it is necessary to include both positive-energy and negative-energy Fourier modes. Interference between these modes (which can be interpreted as exchange interactions with virtual fermion-antifermion pairs) leads to the rapid oscillations.

This may seem to contradict special relativity. However, the rules for what relativity "allows" are normally derived under the assumption that the velocity components commute with everything else in the theory. When the components of $\dot{\vec{x}}$ do not commute with each other, and even a single component $\dot{x}_{k}(t)$ does not commute with itself at different times $t$, the "rules" are different. Since the average velocity over longer times has all the usual features that we expect from the relativistic velocity, it may be tempting to dismiss the zitterbewegung as an unphysical artifact. However, the zitterbewegung actually does have observable consequences; the high-frequency oscillations are responsible for the Darwin term in the nonrelativistic expansion of the Dirac Hamiltonian.

Sakurai's Advanced Quantum Mechanics has an excellent discussion of all these issues.