 Let's look more closely at the Dirac equation's problem of negative energy solutions. We'll use normal units where the speed of light appears explicitly. The positive energy solutions have a minimum for momentum P equals zero. So the lowest positive energy level is E1 equals MC squared. Higher energy levels correspond to higher momentum, the electron moving faster and faster. And this continues without limit towards E equals infinity. Suppose an electron has energy E2. It can drop to energy E1 with the emission of a photon. If this is the lowest possible energy state, the ground state, then the electron would remain there indefinitely. The negative energy solutions mirror the positive ones. The highest negative energy is minus E1, corresponding to zero momentum and electron at rest. Lower energy states correspond to higher momentum, the electron moving faster and faster. And this continues without limit towards E equals minus infinity. The existence of negative energy states means that an electron with energy E1 can transition to energy minus E1 with the emission of a photon. From there, it may continue falling to ever lower energy states and emitting photons without end. So it seems every electron in existence would fall down this negative energy ladder, emitting an infinite amount of radiation in the process. Thankfully, this doesn't happen, but it makes the Dirac equation as it stands unviable. In 1929, Dirac submitted a workaround of this problem in a paper titled A Theory of Electrons and Protons. In the section subtitled Solution of the Negative Energy Difficulty, after noting the problem, he offered a solution. The Pauli Exclusion Principle, however, will come into play and prevent more than one electron going into any one state. Let us assume there are so many electrons in the world that all the states of negative energy are occupied, except perhaps a few of small velocity. We shall have an infinite number of electrons in negative energy states, and indeed an infinite number per unit volume all over the world, but if their distribution is exactly uniform, we should expect them to be completely unobservable. Only the small departures from the exact uniformity brought about by some of the negative energy states being unoccupied, can we hope to observe. Dirac went on to suggest that these unoccupied negative energy states might actually be protons. So, Dirac says, yes, there are an infinite number of negative energy electron states, but they're all full, and the infinite number of electrons that fill them are invisible. They have no observable effect. It's hard to overstate how woo-woo sounding this idea is. As far as the negative energy states go, it's invisible electrons all the way down. So within any milliliter-sized region of space, there are an infinite number of electrons, with velocities ranging from zero to arbitrarily close to the speed of light. Yet we can't see them, or otherwise observe any effect due to their presence. This explanation became known as the Dirac C. Needless to say, it was controversial, and was the target of serious criticism. In the opinion of Werner Heisenberg, quote, the saddest chapter of modern physics is and remains the Dirac theory. I regard it as learned trash, which no one can take seriously. Here's the thing about science, though. Ultimately, people's opinions don't matter. It doesn't matter if an idea seems ridiculous and laughable. There is only one scientific authority, and that's nature. What matters is if an idea leads to testable predictions, and if those predictions can be verified. Dirac supposed that just as negative energy electrons could emit photons, they could also absorb them. Therefore, a photon of energy 2mc squared could cause an electron to transition from the highest negative energy state to the lowest positive energy state. Half of that energy, mc squared, would correspond to the mass m of the electron. The other mc squared would also correspond to a mass m. Moreover, taking a negative charge away from nothing would leave a positive charge behind. Dirac called this empty negative energy state a hole and predicted it would behave like a particle with the electron's mass and opposite charge. Here's a cartoon representation of this prediction. The electrons in the box represent the invisible Dirac C. On the left, a photon excites one of the Dirac C electrons into a positive energy state wherein it becomes visible. This leaves a hole in the Dirac C. The energy of the system has increased by 2mc squared. The electric charge has not changed because we've merely changed the state of a pre-existing electron. Dirac's claim, shown at right, is that this will appear the same as if we created two particles with positive energy, an electron and a positively charged particle with the same mass as the electron, which for now will call a positive electron. The energy of this system has also increased by 2mc squared and the charge has unchanged since we've created both negative and positive charges, which cancel out. At left, suppose the electron above the hole moves downward. In effect, the hole has moved upward. At right, this would be equivalent to the positive electron moving upward. As this continues, what we would observe is a positive electron moving through space. In 1931, Dirac submitted his prediction for publication. He claimed, A hole, if there were one, would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an anti-electron. And there you have the prediction of antimatter. There was no experimental evidence for the existence of such a bizarre thing. This was motivated solely, one might say, by Dirac wanting to fix a seeming shortcoming of his equation. He continued, We should not expect to find any of them in nature on account of their rapid rate of recombination with electrons. But if they could be produced experimentally in high vacuum, they would be quite stable and amenable to observation. So, according to Dirac, it should be possible for the energy of a photon to be transformed into an electron-anti-electron pair. The electron and anti-electron can be distinguished by applying a magnetic field, which causes moving charges of different signs to curve in opposite directions. In 1932, Carl Anderson was using a so-called cloud chamber to observe cosmic rays. A charged particle traveling through a cloud chamber ionizes molecules along its path. Those ions then cause a super-saturated vapor in the chamber to condense into a visible cloud of droplets. The result is to make the particles track through the chamber visible. Anderson observed an event of the type predicted by Dirac, confirming the existence of the anti-electron, what we now call a positron. Here's a later result that shows the process more clearly. At the point marked by the arrow, we see two particles diverge. They curve in different directions in the magnetic field, allowing us to identify the upper particle as a positron and the lower particle as an electron. At the time Anderson published his result, people were using different names for this particle, including the anti-electron, the positive electron, and simply the positive. Anderson suggested the name positron, which is stuck. He also suggested that by symmetry the electron be called the negatron, a name that didn't stick. So, in spite of the bizarre woo-woo nature of the Dirac-C concept, Dirac and his equation were vindicated. The main weakness of his theory became one of its greatest strengths. Still, there's something strange about this Dirac-C idea. In the cartoon at left, we need the Dirac-C as a place to first pull an electron out of and second, as something for the whole to exist and move in. This is what the Dirac equation predicts. To tune it right, there's no need for a Dirac-C. We simply create an electron-positron pair. This is what is experimentally observed. Isn't this a more elegant explanation? It doesn't require space to be filled with an infinite number of invisible negative energy electrons. So why can't the theory predict that directly? The problem is with quantum mechanics itself. First, we started with the assumption that psi is the wave function of a single electron. There's nowhere for other particles to fit in the theory. Second, in both the Schrodinger and Dirac equations, the probability at any time that the electron is somewhere in space is always 100%. Therefore, projecting this back in time, it can never have been created. And projecting it into the future, it can never be destroyed. It follows that if an electron appears, it must have already existed and merely transitioned from an invisible state to a visible state. And, if it disappears, it must have merely transitioned to an invisible state in which it still exists. The Dirac-C provides the required invisible state and it provides a way to account for the properties of a positron as a whole in the Dirac-C in a theory that only describes electrons. What in time became clear is that, to quote Stephen Weinberg, relativistic wave mechanics in the sense of a relativistic quantum theory of a fixed number of particles is an impossibility. Another theoretical framework was needed in which to express these ideas. Eventually, quantum electrodynamics, the first quantum field theory, was able to recast the Dirac equation into a formalism that gives a rigorous and coherent description of electrons, positrons, photons, and their interactions without negative energies and without the Dirac-C.