A hole in Dirac’s theory

Dirac equation is one of the big achievements of 20th century physics, taking quantum mechanics into the real of special relativity, serving as a foundation of quantum field theory.

Paul Dirac himself describes his motivation for writing his Relativistic Electron Equation, as he calls it.

There was thus a real difficulty in making the quantum mechanics agree with relativity. That difficulty bothered me very much at the time, but it did not seem to bother other physicists, for some reason which I am not very clear about.

By asking a question that nobody cared about, he was able to write an equation that not only connected special relativity and quantum mechanics, but gave much more than that. In his own words,

Now, I found out that this equation gives the electron a spin of a half a quantum and also gives it a magnetic moment, and this spin and magnetic moment are in agreement with obsevartion. […] The new theory still allows negative energies […].

He was very troubled by the negative energy solutions. After all, energy is bounded from below, everybody knew that there must be a lowest state of energy that could be called “zero” with no energies under it. This energy state was associated with the vacuum state. Dirac being himself, in order to fix the negative energy issue, decided to redefine vacuum.

Previously, people thought of the vacuum as a region of space that is completely empty, a region of space that does not contain anything at all. Now we must adopt a new picture. […] we must set up a new picture of the vacuum in which all the negative energy states are occupied and all the positive energy states are unoccupied.

Redefining nothingness takes cojones.

We can get a departure from the vacuum state in two ways: one way is to bring attention on the holes. Well, one can look into the question of how a “hole” will move if there is an electromagnetic field present. And, it moves in roughly the same way as the elctron that fills up that “hole” would move. […] these “holes” move as though they had positive energies and positive charges instead of the usual negative charge of the electron; the “holes” appear as a new kind of particle having a positive charge.

In his new theory, the negative-energy particles could be treated as new particles with positive energies that behave like electron’s evil twins with positive charge.

Let’s recap. Dirac asked a question nobody cared about, then to answer it he made up an equation that gave negative energies that didn’t make sense. Instead of discarding his solution or his question, he reinvents the concept of vacuum and to make it work in a consistent manner, he required to invent a new unobserved physical particle. Dirac’s equation requires the existence of the anti-matter particle, the positron. That is a lot of “if”s. He was unable to fully articulate the prediction of the particle, his explanation follows.

[…] I did not dare to put forward that idea, because it seemed to me that if this new kind of particle (having the same mass as the electron and an opposite charge) existed, it would certainly have been discovered by experimenters. […] That, of course, was really quite wrong of me; it was just lack of boldness.

Still Paul was able to stick to his guns because his equation was, according to him, beautiful. And of course, no only positrons were experimentally discovered not long after, the concept of “holes” as mathematical solutions to the “lack” of electrons is a central idea in solid state physics. Without Dirac Holes there wouldn’t be transistors, for example.

Now, in a recent paper, an experimental group announced they can manipulate the quantum properties of holes just like electrons in materials.

A Coherent Single-Hole Spin in a Semiconductor

Semiconductors have uniquely attractive properties for electronics and photonics. However, it has been difficult to find a highly coherent quantum state in a semiconductor for applications in quantum sensing and quantum information processing. We report coherent population trapping, an optical quantum interference effect, on a single hole. The results demonstrate that a hole spin in a quantum dot is highly coherent.

The abstraction of “Dirac’s Hole”, only introduced for pure mathematical aesthetical reasons is now an accepted, essential, physical particle that is used and manipulated at will. Only a theoretical giant like Dirac could have pulled that off.


The only thing in this world that gives orders… is balls.
-Tony Montana

Author: minustwofish

I am a quantum physicist.