Eighty years ago, the Dirac equation suggested that high-speed electrons should jitter violently. And at last, physicists have found evidence to support this.

In 1928, Paul Dirac combined two of the most successful theories of the twentieth century: quantum theory, a set of laws describing Nature at its smallest; and special relativity, the description of objects that are moving at speeds that are close to the speed of light. But the universe can be small and fast, right?  Oh, yes.

The merger of quantum theory with special relativity is embodied in the Dirac equation which predicts the existence of antimatter (identical to normal matter but with opposite electric charge); a weird, quantum concept called spin; as well as the spectrum of the hydrogen atom. Moreover, the equation represents the transition from the small-scale, low-energy world of quantum theory to the small-scale but high-energy world of particle physics.

Yet despite these remarkable successes, peculiar predictions of the Dirac equation, such as results called the Klein Paradox and Zitterbewegung remain, even now, difficult to observe in the laboratory.

A year after Dirac’s discovery, the physicist Oskar Klein applied the equation to the problem of firing electrons at the subatomic equivalent of a mountain – a potential barrier. In order to travel on unhindered, intuitively the electron would need to be energetic enough to surmount the barrier. Traditional quantum theory says the opposite: even if the electron doesn’t have enough energy to climb over the barrier, there is still a chance of it being found on the other side. As an analogy, imagine being trapped in a hole that was apparently too deep to climb out of yet still finding yourself able to escape – that is the bizarre prediction of quantum theory.

Strange as this might seem, when Klein applied the new-fangled Dirac equation (suitable for high energies) to the problem, the result became even stranger. Klein showed that, under certain conditions, the barrier now becomes transparent. In fact, as the size of the barrier increases towards infinite, the electron is always transmitted. In other words, as the hole is dug deeper and deeper, it becomes easier and easier to escape from it!

In 2006 the Klein Paradox was observed in the laboratory. In contrast, Zitterbewegung – the tendency for high-energy electrons to quiver – has only just been seen. But what makes the electrons quiver? The answer is that quantum theory allows a particle to be in a combination of its matter and antimatter states simultaneously. The two states interfere, adding up in some places and cancelling out in others. This leads to the quivering motion but unfortunately it is at such a high frequency that it has never been observed in the laboratory. Until now. Well, sort of.

A team of Austrian and Spanish physicists recently succeeded in simulating the jittering of an electron using a calcium ion trapped in position by lasers. This system is mathematically identical to the case of the high-energy electron so if the calcium ion jitters then it is likely that electrons jitter too. There is one key difference: the calcium ion can be made to jitter at a much lower frequency, allowing the Zitterbewegung to be detected.

It is a major breakthrough. Simulations such as this are important as they are a stepping stone to modelling realistic systems occurring in chemistry and high-temperature superconductivity – the ability of certain materials to conduct electricity with zero electrical resistance. Currently, these systems are beyond the scope of modern computers.

But if we are to harness the power of superconductivity, we will need to understand it.

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