Paul Dirac and Antimatter: The Mathematical Prediction Too Bold to Be Believed

Paul Adrien Maurice Dirac (1902–1984) was a British theoretical physicist born in Bristol, England. He graduated from the University of Bristol with a degree in electrical engineering (1921) and then a degree in mathematics (1923), before completing his PhD at Cambridge under Ralph Fowler in 1926 — with the first doctoral thesis on quantum mechanics ever submitted anywhere in the world.

From 1932 to 1969 he held the Lucasian Professorship of Mathematics at Cambridge — the same chair previously held by Newton and later by Stephen Hawking. Einstein wrote that “to Dirac we owe the most logically perfect presentation of quantum mechanics,” while Abdus Salam declared in 1987 that "no man except Einstein has had such a decisive influence on the course of physics in this century." Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger.

The Dirac equation, published on January 2, 1928 in the paper “The Quantum Theory of the Electron,” was the first equation that correctly described the behaviour of the electron in accordance with both quantum mechanics and Einstein's special relativity.

No one before Dirac had achieved this combination. The pre-existing Klein-Gordon equation was not linear in time and could not yield a probabilistic interpretation of the wave function. Dirac demanded linearity in time, which led him to 4×4 matrices — four components of the wave function instead of two. This four-component structure (bispinor) was a mathematical necessity without any physical justification at the time.

What was astonishing is that the equation automatically predicted the spin-½ of the electron — without Dirac having introduced it by hand — and reproduced with perfect accuracy the fine structure of hydrogen, something no previous theory had achieved from first principles.

The equation had a "problem": besides positive-energy solutions, it also admitted negative-energy solutions. In classical physics, such solutions are simply discarded as unphysical. In quantum mechanics, however, this was impossible — in 1929, Oskar Klein showed that there is inevitable mixing between positive and negative energy states.

Dirac initially proposed that these negative-energy states are already filled — a “sea” (Dirac sea) of negative-energy electrons filling the entire universe. According to Pauli's exclusion principle, no electron could “fall” into these states because they were already occupied.

Initially, Dirac hypothesized that a “hole” in this sea corresponded to the proton. Robert Oppenheimer demonstrated that if this were true, hydrogen atoms would instantly self-destruct, while Hermann Weyl showed that the “hole” must have exactly the same mass as the electron. Convinced, Dirac published a landmark prediction in 1931: there must exist an “anti-electron” with the same mass but opposite charge, which annihilates upon contact with an electron. He further proposed that every particle has a corresponding antiparticle.

On August 2, 1932, Carl David Anderson at Caltech discovered the positron — the antiparticle of the electron — using a Wilson cloud chamber exposed to cosmic radiation. A magnet surrounding the apparatus forced charged particles to curve, and Anderson observed a track with the same curvature as an electron but in the opposite direction — meaning a positive charge.

It is worth noting that Dmitri Skobeltsyn had observed similar tracks as early as 1928, while Chinese physicist Chung-Yao Chao at Caltech had anomalous results in 1929, but nobody interpreted them correctly. Patrick Blackett and Giuseppe Occhialini at the Cavendish Laboratory also discovered the positron simultaneously but delayed publication to gather more evidence.

Anderson won the 1936 Nobel Prize in Physics for the discovery. He did not coin the term “positron” — it was suggested by the Physical Review editor to whom he submitted his paper. It was the first experimental confirmation of antimatter and a triumph of theoretical physics: mathematics had predicted something nobody had ever imagined.

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The best-known application is Positron Emission Tomography (PET scan), a nuclear medicine technique used daily in hospitals worldwide. Radioactive isotopes emit positrons, which annihilate with electrons producing pairs of 511 keV photons — these photons are detected to create three-dimensional images of metabolic activity inside the human body.

At CERN, the ALPHA experiment combines positrons with antiprotons to study properties of antihydrogen — the simplest anti-atom. In 2023, a CERN-Oxford collaboration created the first electron-positron plasma beam in the laboratory with over 10 trillion pairs — enough to study the collective behaviour of antimatter plasma. A striking fact: in the human body, approximately 4,000 positrons are produced daily from the natural radioactive decay of potassium-40.

This is one of the greatest open questions in modern physics, known as “baryon asymmetry.” During baryogenesis, in the first fractions of a second after the Big Bang, matter and antimatter were produced and annihilated continuously. Theoretically, they should have been created in equal amounts — and should have mutually annihilated completely, leaving a universe filled only with radiation.

This obviously did not happen: we exist. The explanation is attributed to a small violation of CP symmetry — an asymmetry between matter and antimatter in the weak interactions. However, the CP violation measured experimentally is far too small to explain the enormous asymmetry we observe. The exact mechanism remains a mystery.

Dirac may not have given the final answer, but his prediction of antimatter opened an entire field of research. As Richard Dalitz noted, "the influence and importance of Dirac's work have increased with the decades, and physicists use daily the concepts and equations that he developed."