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In the first quarter of the 20th century, it was an intense time to be a physicist. It seemed like every day someone was coming out with a new theory that completely revolutionized our understanding of the universe. In 1905 Einstein published his Theory of Special Relativity, which changed the way physicists thought about space and time. Ten years later, Einstein published his Theory of General Relativity, which was even more revolutionary. OK so it was a lot of Einstein, but still.

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At about the same time, a group of physicists were beginning to study very tiny particles like electrons, as well as the weird things that can happen with light. They began to develop a theory called quantum mechanics, which contains the idea that at the smallest level, measurements of position, momentum, energy, and other quantities are uncertain.

Many physicists spent their entire careers trying to unify these two titans of 20th century physics. In 1928, one man finally succeeded, and in the process, managed to predict the existence of antimatter. As PBS Space Time explains:

Physicists trying to unite relativity and quantum mechanics had a bit of a problem. One of the key ideas of relativity is that time and space are relative, and everything depends on where you are and how fast you're moving. But that idea doesn't show up anywhere in quantum mechanics.

British Physicist Paul Dirac decided to fix this problem by combining Einstein's famous E=mc2 equation with Schroedinger's equation from quantum mechanics. What he got could only be described as an ugly mess. But Dirac saw a way to fix it.

Paul Dirac in 1928.

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However, his solution was a bit strange. In order for the math to work, he needed to add in an extra type of electron, with negative energy. Nobody knew what this was or even what it meant, but it made the end result so simple and elegant that Dirac just knew it was true.

Only a few years later, observations of cosmic rays in the upper atmosphere discovered the first antimatter particles, confirming Dirac's hypothesis. He showed that relativity and quantum mechanics could be combined after all, creating a completely new branch of physics: quantum field theory.

Source: PBS Space Time

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In 1928, One Physicist Accidentally Predicted Antimatter - Popular Mechanics