12.12 The Relativistic Dirac Equation

Relativity threw up some road blocks when quantum mechanics was first formulated, especially for the particles physicist wanted to look at most, electrons. This section explains some of the ideas.

You will need a good understanding of linear algebra to really follow the reasoning. A summary of the Dirac equation that is less heavy on the linear algebra can be found in {A.43}.

For zero spin particles, including relativity appears to be simple. The classical kinetic energy Hamiltonian for a particle in free space,

\begin{displaymath}
H = \frac 1{2m}\sum_{i=1}^3 {\widehat p}_i^2
\qquad
{\...
...hat p}_i = \frac{\hbar}{{\rm i}}\frac{\partial}{\partial r_i}
\end{displaymath}

can be replaced by Einstein's relativistic expression

\begin{displaymath}
H = \sqrt{\left(m c^2\right)^2 + \sum_{i=1}^3 \left({\widehat p}_ic\right)^2}
\end{displaymath}

where $m$ is the rest mass of the particle and $mc^2$ is the energy this mass is equivalent to. You can again write $H\psi$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E\psi$, or squaring the operators in both sides to get rid of the square root:

\begin{displaymath}
\left[
\left(m c^2\right)^2 + \sum_{i=1}^3 \left({\widehat p}_i c\right)^2
\right]\psi
=E^2\psi
\end{displaymath}

This is the “Klein-Gordon” relativistic version of the Hamiltonian eigenvalue problem. With a bit of knowledge of partial differential equations, you can check that the unsteady version, chapter 7.1, obeys the speed of light as the maximum propagation speed, as you would expect, chapter 8.6.

Unfortunately, throwing a dash of spin into this recipe simply does not seem to work in a convincing way. Apparently, that very problem led Schrö­din­ger to limit himself to the nonrelativistic case. It is hard to formulate simple equations with an ugly square root in your way, and surely, you will agree, the relativistic equation for something so very fundamental as an electron in free space should be simple and beautiful like other fundamental equations in physics. (Can you be more concise than $\vec{F}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m\vec{a}$ or $E$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mc^2$?).

So P.A.M. Dirac boldly proposed that for a particle like an electron, (and other spin $\leavevmode\kern.03em
\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ elementary particles like quarks, it turned out,) the square root produces a simple linear combination of the individual square root terms:

\begin{displaymath}
\sqrt{\left(m c^2\right)^2 + \sum_{i=1}^3 \left({\widehat ...
...
= \alpha_0 mc^2 + \sum_{i=1}^3 \alpha_i {\widehat p}_i c %
\end{displaymath} (12.18)

for suitable coefficients $\alpha_0$, $\alpha_1$, $\alpha_2$ and $\alpha_3$. Now, if you know a little bit of algebra, you will quickly recognize that there is absolutely no way this can be true. The teacher will have told you that, say, a function like $\sqrt{x^2+y^2}$ is definitely not the same as the function $\sqrt{x^2}+\sqrt{y^2}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x+y$, otherwise the Pythagorean theorem would look a lot different, and adding coefficients as in $\alpha_1x+\alpha_2y$ does not do any good at all.

But here is the key: while this does not work for plain numbers, Dirac showed it is possible if you are dealing with matrices, tables of numbers. In particular, it works if the coefficients are given by

\begin{displaymath}
\alpha_0=\left(\begin{array}{cc} 1 & 0 \ 0 & -1 \end{arra...
...ft(\begin{array}{cc}0&\sigma_z\ \sigma_z&0\end{array}\right)
\end{displaymath}

This looks like 2 $\times$ 2 size matrices, but actually they are 4 $\times$ 4 matrices since all elements are 2 $\times$ 2 matrices themselves: the ones stand for 2 $\times$ 2 unit matrices, the zeros for 2 $\times$ 2 zero matrices, and the $\sigma_x$, $\sigma_y$ and $\sigma_z$ are the so-called 2 $\times$ 2 Pauli spin matrices that also pop up in the theory of spin angular momentum, section 12.10. The square root cannot be eliminated with matrices smaller than 4 $\times$ 4 in actual size. (A derivation is in {D.71}. See also {A.35} for alternate forms of the equation.)

Now if the Hamiltonian is a 4 $\times$ 4 matrix, the wave function at any point must have four components. As you might guess from the appearance of the spin matrices, half of the explanation of the wave function splitting into four is the two spin states of the electron. How about the other half? It turns out that the Dirac equation brings with it states of negative total energy, in particular negative rest mass energy.

That was of course a curious thing. Consider an electron in what otherwise is an empty vacuum. What prevents the electron from spontaneously transitioning to the negative rest mass state, releasing twice its rest mass in energy? Dirac concluded that what is called empty vacuum should in the mathematics of quantum mechanics be taken to be a state in which all negative energy states are already filled with electrons. Clearly, that requires the Pauli exclusion principle to be valid for electrons, otherwise the electron could still transition into such a state. According to this idea, nature really does not have a free choice in whether to apply the exclusion principle to electrons if it wants to create a universe as we know it.

But now consider the vacuum without the electron. What prevents you from adding a big chunk of energy and lifting an electron out of a negative rest-mass state into a positive one? Nothing, really. It will produce a normal electron and a place in the vacuum where an electron is missing, a hole. And here finally Dirac's boldness appears to have deserted him; he shrank from proposing that this hole would physically show up as the exact antithesis of the electron, its anti-particle, the positively charged positron. Instead Dirac weakly pointed the finger at the proton as a possibility. Pure cowardice, he called it later. The positron that his theory really predicted was subsequently discovered anyway. (It had already been observed earlier, but was not recognized.)

The reverse of the production of an electron/positron pair is pair annihilation, in which a positron and an electron eliminate each other, creating two gamma-ray photons. There must be two, because viewed from the combined center of mass, the net momentum of the pair is zero, and momentum conservation says it must still be zero after the collision. A single photon would have nonzero momentum, you need two photons coming out in opposite directions. However, pairs can be created from a single photon with enough energy if it happens in the vicinity of, say, a heavy nucleus: a heavy nucleus can absorb the momentum of the photon without picking up much velocity, so without absorbing too much of the photon's energy.

The Dirac equation also gives a very accurate prediction of the magnetic moment of the electron, section 13.4, though the quantum electromagnetic field affects the electron and introduces a correction of about a tenth of a percent. But the importance of the Dirac equation was much more than that: it was the clue to our understanding how quantum mechanics can be reconciled with relativity, where particles are no longer absolute, but can be created out of nothing or destroyed according to the mass-energy relation $E$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mc^2$, chapter 1.1.2.

Dirac was a theoretical physicist at Cambridge University, but he moved to Florida in his later life to be closer to his elder daughter, and was a professor of physics at the Florida State University when I got there. So it gives me some pleasure to include the Dirac equation in my text as the corner stone of relativistic quantum mechanics.