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How did Paul Dirac predict the existence of the positron?

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A diagram of a cylindrical vessel containing gas, with a movable piston held down by a weight. Countless small dots beneath the piston represent gas particles in rapid, random motion, illustrating Bernoulli's model of gas pressure. — physics As conceived by Daniel Bernoulli in *Hydrodynamica* (1738), gases consist of numerous particles in rapid random motion. He assumed that the pressure of a gas is produced by the direct impact of the particles on the walls of the container.

In 1928, P.A.M. Dirac formulated a relativistic wave equation for the electron that successfully incorporated special relativity into quantum mechanics. While the equation correctly accounted for electron spin, it also yielded mathematically valid but physically puzzling solutions representing negative energy states.1234

To interpret these states, Dirac proposed that the vacuum is filled with an infinite "sea" of negative-energy electrons that are normally imperceptible. He suggested that if a negative-energy electron were excited into a positive-energy state, it would leave behind a "hole" in this sea. Dirac initially hoped these holes might be protons, but he later realized they must have the same mass as an electron and an opposite charge.42

In 1931, he concluded that these holes represented a new kind of particle—the positron—which was experimentally confirmed by Carl David Anderson in 1932. This discovery validated Dirac's mathematical formalism and established the existence of antiparticles.123