Anti-Matter and Quantum Relativity, QFT Pt 1 (Script) (Patreon)

It’s 1928. Over the past quarter century, the greatest geniuses of the modern era discovered the two keys to the fundamental nature of reality. Einstein’s theories of special and general relativity had changed forever the way we think of motion, space, and time. And the emerging field of quantum mechanics had radically altered our understanding of the fundamental building blocks of the universe. Yet this year one brilliant insight would bring these theories together and unveil the quantum fabric of reality. It would also predict the existence of antimatter.

By the late 1920s, Einstein and Planck had already shown that light is a particle as well as a wave, and Luis de Broglie had showed that all matter has a dual wave-particle nature. Bohr, Heisenberg, Born, Pauli and others pieced together a mathematical description for the weird nature of subatomic particles. Then, in 1926, Erwin Schrodinger wrote down his famous equation – the Schrodinger equation – which breathed life into the emerging model. It describes how these matter-waves – represented as wavefunctions - change over time and allowed physicists to predict the evolution of quantum systems, such as the strange interference pattern in the famous double slit experiment. 

Yet everyone knew it there was a problem with it. First, and most obvious: Schrodinger’s equation is totally incompatible with Einstein’s relativity. In relativity, the dimensions of space and time are intrinsically connected, and flow into each other as frames of reference change. But Schrodinger’s equation tracks the evolution of a particle’s wavefunction according to one and only one clock – typically the clock in the reference frame of the observer. Relativity tells us that the passage of time depends on velocity, so Schrodinger’s equation only works for slow-moving objects. That’s a problem: subatomic particles are often moving at close to the speed of light.

The other problem with the Schrodinger equation was that it described particles as simple wavefunctions - distributions of possible positions and momenta that have no internal properties. Yet we now know that many elementary particles have an internal property called spin. This doesn’t mean that they are actually rotating, but spin does result in a sort of quantum angular momentum. For example, electrons’ spin causes them to align themselves with magnetic fields, just like a rotating electric charge would. The axis of spin can point in different directions, for example up or down. 

The discovery of electron spin starts with an Austrian physicist named Wolfgang Pauli. Pauli realized that to explain electron energy levels, electrons must obey a rule that we call the Pauli exclusion principle. It states that no electron can occupy the same quantum state as another electron. In fact it applies to all particles called Fermions. In the case of electrons in atoms, it suggests we should only find one electron per atomic orbital if we count each orbital as a “quantum state”. However we actually observe two electrons per orbital and so Pauli realized there must exist a hidden quantum state. 

He introduced what we call a new “degree of freedom” internal to electrons; one that could take on one of two values. Let’s call those values “up” and “down”. That would allow two separate electrons – one “up” and one “down” – to occupy the same atomic energy level without occupying the same quantum state. Other physicists soon figured out that this new quantum state represented spin, and the “up” and “down” degrees of freedom are the direction of pointing of the angular momentum axis. We now call these 2-component wavefunctions “spinors”. 

Now it’s okay to ignore spin in the old Schrodinger equation and get approximate answers. But when a magnetic field is present, spin direction becomes very important. So for fast moving electrons AND for electrons in electromagnetic fields, Schrodinger equation gives the wrong answers. This problem consumed a brilliant British physicist, Paul Dirac. He wanted to find a fully relativistic version of the Schrodinger equation that worked for electrons. In a way he started with relativity – he wrote down Einstein’s famous equation E=mc^2, but in its full form including momentum. He then used quantum mechanical expressions for energy and momentum. The result was a huge mess. But Dirac stumbled upon a single, simple idea that caused the resulting horrendous math to collapse into an incredibly simple, beautiful equation.

That simplification required Dirac to expand the internal workings of the electron even further. Instead of having a two component spinor – up and down – as in Pauli’s theory – he needed four components. Now he had no idea what those two additional mysterious components might mean. But the resulting equation was so simple and elegant that somehow Dirac knew he was onto something. The resulting Dirac Equation describes the spacetime-evolution of this weird 4-component particle wavefunction represented by the psi (ψ). It contains the marks of both quantum mechanics in the Planck constant (h-bar), and relativity, in the speed of light, c.

The Dirac equation perfectly predicts the motion of electrons at any speed, even in an electromagnetic field. It was a major victory. But it opened up even more questions than it answered. To begin with, what on the Earth were those two extra degrees of freedom in the 4-component electron? The answer came from trying to calculate the energy of the electron using this equation. It predicted something totally bizarre. It allowed electrons to exist in states of negative energy. If true, that would lead to some bizarre effects. For example, a lone electron moving in an electromagnetic field could keep releasing energy as light … infinitely, and sink to lower and lower to infinite negative energy states. There was no bottom of the energy well. We know perfectly well that this doesn’t happen.

Dirac came up with an idea to explain this. We call it the Dirac sea. Imagine an infinitely deep ocean of electrons that exist everywhere in the universe. These electrons occupy all of the negative energy states, all the way from negative infinity up to zero. The only time we can actually interact with an electron is when one has positive energy, which would leave it sitting on top of this sea. This is where the Pauli Exclusion Principle comes back. If the energy states of this imaginary ocean are completely full, then that one extra electron can’t lose any more energy. It just floats on top of the sea.

The idea of the Dirac sea leads to its own weird predictions. Remove one electron from the surface and it leaves a hole. That hole should act like a particle all by itself – a bit like an eddy on the surface of a pool of water. It would move around, it would have inertia – acting like it had the mass of the missing electron. It would also act like it had opposite electric charge to an electron – a positive charge. And if a positive-energy electron found one of these holes it would “fall in”, annihilating both and releasing all of the energy bound up in their masses. Of course there IS something in our universe that acts exactly like holes in the Dirac sea. It’s called anti-matter. Dirac had just predicted its existence.

Now the Dirac sea itself doesn’t really exist. But it was one of the first attempts to describe something very real: the idea of a quantum field. We now know that every elementary particle has an associated “field” that fills all of space. These fields are more like membranes than infinitely deep oceans. They have a very definite energy – usually zero. The elementary particles that we know and love are just regions where a field has a bit more energy. That energy manifests as vibrations in the field. Quantum field theory is a VERY deep topic and will be the subject of upcoming episodes. For now, let’s get to the bottom of these holes.

Dirac’s negative energy solutions describe antimatter, not holes in the Dirac sea. Only a few years after Dirac wrote down his equation in 1928, the positron – the antimatter electron – was spotted in cosmic rays by Carl Anderson. Antimatter is very real, but what is it? Well, it’s a vibration in the same quantum field as it’s regular matter counterpart. Antimatter’s existence is fundamentally tied to these weird, 4-component electrons that Dirac invented to make his equation work. Those two extra components correspond the up and down spins of the electron’s antimatter counterpart. Two spin directions for the electron, two for the positron. A 4-component spinor.

In fact the electron and the positron cannot exist without each other. They are two sides of the same coin – positive and negative energy solutions of the same type of vibration in the electron field. It’s actually a tiny bit more complicated than that, and way more awesome, but there’ll be time for that in the future. So ALL elementary particles have a quantum field and all have an antimatter counterpart. Just as with the holes in the Dirac sea, antimatter particles have the same mass as their counterparts, but opposite charge. That mass is very real – it’s not negative mass, despite this “negative energy” description. When matter-antimatter counterparts find each other they annihilate, releasing an awful lot of very real energy. A penny of antimatter could be used to launch a good-sized rocket into orbit.

Dirac’s incredible insight in combining quantum mechanics and relativity revealed an entire flip-side of our universe with its prediction of antimatter. It was also a key step in the discovery of quantum fields and quantum field theory and the development of the standard model of particle physics, which have become our best description of the underlying workings of reality. And that’s a quantum rabbit hole that we’ll jump into very soon, right here on Space Time.