Mathematics discovered antimatter – 02/07/2024 – Marcelo Viana

On January 2, 1928, British mathematician and physicist Paul Dirac (1902–1984) published in the journal “Proceedings of the Royal Society” one of the most important works in the history of physics, the result of more than a year of dedication. In it, Dirac reformulated Schrödinger’s wave equation, which describes the way quantum systems evolve over time, in the language of Einstein’s theory of special relativity.

A few years earlier, Arnold Sommerfeld had proposed that electrons have an “internal rotation”, given by a number that takes only two values, positive or negative. This helped explain experimental observations about the behavior of electrons revolving around atomic nuclei. But it was nothing more than an ad hoc hypothesis, which was neither supported by evidence nor explained what spin is. And Schrödinger’s equation said nothing about the matter…

With Dirac everything changed: the new equation implies that the electron’s spin must exist, with exactly two possible values. In fact, instead of one solution describing the electron, the Dirac equation has two solutions, one for each value of the spin. It was the achievement he had been aiming for since he took on this challenge.

The problem is that Dirac’s equation has more solutions: four in total! How to interpret the other two? It would be easy to dismiss them on the grounds that they have no physical meaning. But this was repugnant to Dirac’s mathematical sensibility. Instead, he suggested that the two unexpected solutions corresponded to another subatomic particle, which no one had predicted, and spent the next few years justifying this claim.

In 1931, he had become convinced that it was an “antielectron”, with the same mass as the electron, but an electrical charge of the opposite sign (positive). When electron and antielectron meet, they annihilate each other in an explosion of pure energy. American physicist Richard Feynmann would later show that the antielectron, or positron, is an electron that moves backwards in time, a very intriguing idea.

Studying cosmic radiation, in 1932, the American physicist Carl Anderson experimentally proved the existence of the positron. Dirac won the Nobel Prize in Physics the following year, and Anderson was similarly distinguished in 1936. Today, quantum mechanics predicts that every subatomic particle has an antiparticle, and this prediction has been confirmed experimentally.

Stop and think, dear reader, dear reader: all this unsuspected wealth of nature – the existence of antimatter – was discovered in a mathematical equation. Isn’t it incredible?

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