An absolute mathematical marvel – 01/30/2024 – Marcelo Viana

In 1923, the young English mathematician and physicist Paul Dirac (1902 – 1984) entered the University of Cambridge to pursue his doctorate. He left behind the suffocating environment of his paternal home, dominated by his authoritarian father. His younger brother, Reginald, was unable to escape: he committed suicide the following year.

A great admirer of Albert Einstein’s theory of relativity, during his time at Cambridge Dirac also became interested in quantum mechanics and, in particular, the mystery of spin. The German physicist Arnold Sommerfeld proposed, in 1920, that each electron has an “inner rotation”, which was called spin, with only two possible values: positive or negative. In 1925, the Austrian Wolfgang Pauli used the concept to explain, in a very satisfactory way, the experimental behavior of electrons rotating around the nucleus of an atom.

But what is spin? The word means “rotation” in English, but it soon became clear that it was something new, with no analogue in classical physics. And why are there exactly two values ​​for the electron spin? In that same year, 1925, the Austrian Erwin Schrödinger formulated his famous equation that describes the evolution of quantum systems. But it doesn’t say anything about the spin…

At the end of 1926, Dirac bet with his German colleague Werner Heisenberg that the problem would be solved in three months (less optimistic, Heisenberg bet on at least three years). Three months later, the matter was still open, but at that point the Brit was looking for something more ambitious.

For Dirac, Schrödinger’s equation had another defect: time plays a very different role in it, while in the theory of relativity time and space are equivalent concepts, mere facets of a single fundamental entity, space-time. It was necessary to reformulate Schrödinger’s equation in order to make it compatible with the theory of relativity!

Mathematician above all, Dirac attacked the question from a mathematical point of view, leaving to worry later about the possible physical meaning of the result. Thus, at the end of 1927, he arrived at a new equation, of remarkable elegance, which describes the evolution of the electron in a way compatible with the theory of special relativity.

The news spread quickly. “It soon became clear that it was the solution. It was considered an absolute marvel,” wrote Belgian physicist Léon Rosenfeld.

But the best is yet to come.

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