31/01/2024

Folha: 'An absolute mathematical marvel'

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In 1923, the young English mathematician and physicist Paul Dirac (1902-1984) entered Cambridge University to pursue his doctorate. He left behind the suffocating environment of his parents' home, dominated by his authoritarian father. His younger brother, Reginald, could not 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 had proposed, in 1920, that each electron possesses 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 orbiting 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, without any analogue in classical physics . And why are there exactly two values for the electron's spin? In that same year, 1925, the Austrian Erwin Schrödinger formulated his famous equation that describes the evolution of quantum systems. But it says nothing about spin…

At the end of 1926, Dirac made a 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 unresolved, but by then the Briton was looking for something more ambitious.

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

A mathematician above all, Dirac tackled the question from a mathematical standpoint, leaving the concern for the possible physical meaning of the result for later. 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.

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