Quantum Field Theory, QFT Pt. 2 (Script) (Patreon)

Richard Feynman called it the “jewel of physics”. Of all of our mathematical descriptions of the universe, this one has produced the most stunningly precise predictions. I’m talking about quantum electrodynamics; the first true quantum field theory.

Quantum mechanics is perhaps the most unintuitive theory ever devised. And yet it’s also the most successful, in terms of sheer predictive power. Simply by following the math of quantum mechanics, incredible discoveries have been made. Its wild success tells us that the mathematical description provided by quantum mechanics reflects deep truths about reality. And by far the most successful, most predictive formulation of quantum mechanics is quantum field theory. It is our best description we have of the fundamental workings of reality. And the first part of quantum field theory that was derived – quantum electrodynamics – is the most precise, most accurate of all.

Quantum field theory – QFT – describes all elementary particles as vibrational modes in fundamental fields that exist at all points in space and time through the universe. Quantum electrodynamics – QED – provides this description for one such field – the electromagnetic field. The pillars of QED are the description of the behavior of the EM field, and the description of the behavior of the electron via the Dirac equation. We covered the Dirac equation last time, and you really should watch that episode first if you haven’t already.

Now before we starting thinking about vibrating quantum fields, or even fields at all, let’s talk about vibrations. Anyone who’s ever strummed or shredded knows that a stretched string vibrates with a certain frequency when plucked. It also vibrates with an amplitude that depends on how hard you pluck it. A larger amplitude and/or a larger frequency means the vibration carries more energy. At any point in time, every point on a vibrating string is displaced some distance from its relaxed or “equilibrium” position, and that displacement changes over time as the string oscillates back and forth.

Guitar strings are one-dimensional, but we can expand the analogy to any number of dimensions. In 2-D we have a membrane, like a drum skin – everywhere on the surface of a vibrating drum skin there’s a displacement from the flat, equilibrium state in the up-down direction. The 3-D analogy is harder to imagine; every point in space has some displacement in some imaginary extra direction – analogous to but not the same as a 4th dimension. For example, in a 3-D room full of air, sound waves are oscillations in air density. That air density has an equilibrium value – which is just the average density – but at any point in the room, a sound wave can cause air density to oscillate to higher and lower values. We describe air density as a field, because it has some value everywhere in the space of the room. That’s all a field is – some property that has some value throughout a space.

OK, let’s go quantum, and go back to the string. If this were a quantum-mechanical guitar string then there’d be a minimum amplitude for the vibration that depended on its frequency. No vibrations with amplitudes smaller than that minimum could exist, and every larger vibration would have to be a whole number – an integer multiple of the smallest amplitude. This is exactly how light behaves, as was first realized by Max Planck and proved by Einstein. Light is a wave in the electromagnetic field. The electromagnetic field is similar to the density field of a room full of air. It has a value – a field strength – everywhere in the universe. That value is usually zero, but just like the string or the air density field it can oscillate. The electromagnetic field is a quantum field, and so these oscillations have a minimum amplitude. That smallest possible oscillation above zero is an indivisible little packet of energy that we call a photon.

Quantum physics may have started with Planck’s discovery of the quantum nature of light. However, the first full formulation of quantum mechanics was Schrodinger’s equation, and it couldn’t account for light at all. We touched on the reason for that in last week’s episode. Basically, the Schrodinger equation is incompatible with Einstein’s relativity. It can’t describe things moving anywhere near the speed of light, and it implicitly assumes that forces act instantaneously. We also saw how Paul Dirac managed to find an equation describing the behavior of the electron that worked with relativity – the Dirac equation. However understanding the behavior of light and its interaction with matter required a different approach. It also required Paul Dirac. Again. That guy was a genius.

Let me first mention another problem with the Schrodinger equation. It’s terrible for many-particle systems. It follows the changing position and momentum and generally the quantum state of every individual particle. But that’s extremely inefficient. See, two of the same type of elementary particle are indistinguishable from each other. If you take a pair of electrons or photons in two different quantum states and make them swap places with each other then nothing changes. Following the quantum state of every individual particle is like trying to do your finances by tagging and tracking the movement of each individual dollar. No one cares which dollar is which – we only care how many dollars end up in whose bank account.

But bean-counting this way is not just inefficient in quantum mechanics – it gives the wrong answers. A given quantum event or interaction can happen in multiple different ways, and the probability of the interaction depends on correctly counting those different possible ways. If you try to track individual particles you’re at risk of double-counting. You end up with multiple arrangements of particles that are actually the same arrangement due to the particles being identical to each other. That means you get the wrong probabilities.

Dirac’s solution was to NOT try to track the changing states of individual photons. Instead of quantizing particles’ physical properties, like position and momentum, as did Schrodinger, Dirac quantized the electromagnetic field itself. He imagined each point in space as being an oscillator – technically, a simple harmonic oscillator – just like an oscillating spring. The oscillation at each point can be complicated, but it has to be built up from a number of minimum-amplitude quantum oscillations – which is to say, photons. So Dirac described a space of quantum states – including position and momentum-slash-frequency, like an infinite array of springs. His mathematics then kept track of the number of “particles” or quantum oscillations in each of these states. This automatically avoids double-counting because the math doesn’t even try to track the movement of individual photons – only the shifting number in each quantum state.

Dirac wasn’t the first to come up with this idea, but he was the first to successfully apply it to describing electromagnetic interactions. He named the resulting theory quantum electrodynamics. He also coined the name “second quantization” for the process of counting the changing number of quantum oscillations, or “particles” per state. Schrodinger’s approach of tracking the changing ;quantum state of each particle became “first quantization”.

There’s another reason this second quantization is better at describing the interaction of light and matter. See, Schrodinger’s approach has no idea how to destroy a particle; all it can do is move particles around via their evolving wavefunction. Yet in particle interactions, particles are created and destroyed all the time. An electron can absorb or emit a photon, an electron and a positron can annihilate each other and create two photons, and that can even happen the other way around. But the second quantization is all about creating and destroying particles. This new capability became essential for the description of subatomic processes.

The resulting quantum electrodynamics describes the interaction of matter and radiation with stunning success. It is one of the most carefully tested theories in all of physics. For example, it eventually allowed scientists to predict with incredible precision the tiny difference in atomic electron energy levels due to electron spins – spins interacting with magnetic fields in so-called hyperfine splitting, or spins interacting with vacuum-energy fluctuations in the Lamb shift. It predicts the relative value of the fine-structure constant to a precision of one part in a billion. No other theory in physics has done so well.

Spurred by its success in describing electromagnetism, physicists soon extended the second quantization approach to other elementary particles. It required different rules for the fields – for example, the Pauli Exclusion Principle tells us you can only have one fermion – so electron, quark, etc. per quantum state, rather than infinite particles in the case of the photon. Nonetheless second-quantization works for all elementary particles. That tells us something extremely important. Remember, this approach began with thinking of photons as oscillations in the electromagnetic field. So does this mean that all particles are also oscillations in fields? Yes, that’s exactly what it means. In fact every base elementary particle has its own field. It IS its own field.

This is the postulate of quantum field theory. Fields are fundamental, and particles and their anti-matter counterparts are just the ways in which that field vibrates. There’s an electron field, whose oscillations are what we know as the electron and the anti-electron; there are fields for every type of quark/anti-quark pair, for every type of force-carrying particle – so-called bosons, like photons and gluons, and of course for the famous Higgs boson.

The calculations of QED and all quantum field theory is about counting the number of ways a quantum phenomenon can occur. That’s a challenge, because there are infinite ways in which everything can occur. In fact a huge part of quantum field theory is about taming the infinities that arise in any calculation. To do that efficiently, we’re going to need another genius. We’re going to need Richard Feynman.  And we’re going to need another episode of Space Time.