The existence of antiparticles was predicted by Dirac a few years before the first one, the antielectron or positron, was found. The idea stemmed from the existence of negative energy states, which in a relativistic universe can not be discarded a priori. Since electrons normally seek the lowest possible energy state, Dirac posited that these extra states must all be filled with what are called virtual particles. In that case, a virtual particle could be promoted to a positive energy state, creating a real particle and leaving a hole that would behave exactly the same, but with opposite charge.
Experimentation with antimatter goes back quite some time to the cloud chambers, in which moving electrons (or positrons) leave behind trails as they move through the gas. Originally, positrons, because of the electromagnetic forces acting on them, were mistaken for electrons travelling in the opposite direction.
It is tempting to sometimes think of antimatter as consisting of negative energy, or possibly even having negative mass. However, this cannot be the case. When matter and antimatter collide, the energy released is the sum of mc^{2} of the two particles (or more accurately, the sum of of the √(p^{2}c^{2} + m^{2}c^{4}) of the particles). If antimatter had negative energy, the energy released from the two colliding particles would equal 0, since the positive and negative energies would cancel each other out.
Originally, the idea of antimatter came to Dirac when, among other things he looked at the true form of the equation E = mc^{2}, which is actually E^{2} = p^{2}c^{2} + m^{2}c^{4}, and realized that the "^{2}" sign means that the equation for energy can have two solutions, a negative energy solution and a positive energy solution. That being the case, since electrons always seek to fill the lowest possible energy state, there seemed to be nothing to stop every electron in the universe from emitting enough enery to fall into a negative energy state. To correct this he propposed a "sea of negative electrons" that would fill the universe, occupying all of the lower energy states. However, given sufficient energy, from for example, a photon, one of these particles could be lifted out of the sea of negative energy to become a positive energy particle. But when lifted out, it would leave behind a hole in the sea of negative energy  a space of zero energy, which would itself, according to the mathematics, act exactly like an electron, but with a positive charge. If an electron were to hit this area of zero energy a lower negative energy state would become available and the electron would emit enough energy to send it into that lower state, dissappearing into the sea of negative electrons.
In the lab, this would appear as a photon travelling along and suddenly splitting into an electron and positron. The positron then would hit another electron (or possibly, the same one), and energy would be released as the two particles annihilate each other.
This theory of antimatter is completely consistent with what has been observed in the laboratory, and theoretically, the antiparticle should exhibit a "normal" gravitational force.
However, nobody (including Dirac) was very satisfied with the idea that the universe was completely filled with a sea of negative electrons. Richard Feynman, however, shortly afterwards, showed that negative energy forward in time and positive energy backwards in time solutions are not allowed to the energy equation. A negative energy running backwards in time would appear to be exactly the same as a positive energy particle running forward in time except for its polarization, which would cause two particles of the same charge travelling in different directions through time to attract electromagnetically.
So, how does this fit into the idea of antimatter?
Say you have an electron, travelling forward through time, and it emits a photon with enough energy and in the right direction to send it hurling back in time. It continues along for a while, then emits another photon, which sends it hurling forward through time once again.
t5 */ t4 /\/ t3 /\/ t2 /\/ t1 /*
The Y axis is time and the X axis is position. The "*" are places where photons are emitted, the "/" and "\" trace out the path of the particle, from left to right, and the "" designate a specific point in time, labeled as t1, t2, t3, t4, and t5.
To us, observing this reaction travelling only forward in time, at T1 we see a photon split up into two particles, a positron and an electron. The electron travelling off to the right while the positron moves to the left, colliding with a regular electron at T5 and releasing energy.
In "reality" the electron starting at the left end, moves forward in time until T5 when it emits enough energy to slip into a negative energy state. It can only do so if the photon is emmitted in a way that will send it back in time, however, which is a very low probability. It continues backwards in time with negative energy but since it is travelling backwards in time the negative energy, from the viewpoint of something moving forwards in time, appears to be acting like a positive energy particle.
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