In 1928, a British theoretical physicist called Paul Dirac proposed an equation of motion for the electron that encompassed both quantum mechanics and special relativity. Not only did it contain all the physics described by non-relativistic quantum mechanics (invented a few years earlier), it also contained some new physics. Of particular interest to us is the prediction of antimatter.
An important equation from special relativity relates the energy E, the momentum p and the mass m of a particle:
E 2 = p 2 c 2 + m 2 c 4
where c is the speed of light. The solutions of Dirac's equation are also solutions of this equation - he deliberately set up his equation so that this would be the case. Whenever one has a quadratic equation, there are in general two solutions. (For example, x2 = 1 has the solutions x = 1 and x = -1.) This means that the above equation for E has two solutions for a given p and m - one is the positive square root of the right-hand side, and one is the negative square root.
Rather than just dismissing the negative energy solution as being unphysical, Dirac postulated that it corresponds to a particle with the same mass as an electron but with the opposite charge. This prediction was verified a few years later when Carl Anderson, working at the California Institute of Technology, discovered the so-called positron. The positron is called the antiparticle of the electron. Its mass is equal to that of the electron, but it has the charge +e instead of -e. Whenever an electron and a positron meet they annihilate, producing electromagnetic radiation.
Dirac's prediction of antimatter was a great achievement. There is a problem associated with it, though. As far as physicists could tell, antimatter and matter should act identically, apart from having opposite charges. This means that matter and antimatter should have been created in equal amounts in the Big Bang, and so there should be just as much antimatter as matter in the Universe today. However, the Universe is overwhelmingly made up of matter, and this is one of the biggest puzzles facing physicists today.