Quantum Invariance & the standard Model (Script) (Patreon)

The standard model of particle physics is the most successful, the most accurate physical theory ever developed, describing with stunning accuracy the fundamental building blocks of our universe. But even more stunning is how it was discovered: by peering deep into the symmetries of reality.

INTRO

As far as we can tell, mathematics is the language in which the universe is written. Our laws of physics are equations of motion, tuned by the fundamental constants. Previously we’ve talked a bit about the symmetries of these equations and how they lead us to conserved quantities like energy and momentum. But that’s just the tip of the theoretical iceberg - these symmetries can be portals into entirely new aspects of reality. The most amazing example of this is the standard model of particle physics. Today I’m going to open the first portal of the standard model and show you the origin of the electromagnetic field.

To appreciate the theoretical whirlwind that is the standard model, we need to introduce the idea of a gauge theory. In simple terms a gauge theory is one that has mathematical parameters, or “degrees of freedom” that can be changed without affecting the predictions of the theory. An example would be a ball rolling down a hill under constant gravitational acceleration. The speed of the ball at the bottom of the hill depends on its change in altitude. But it doesn’t matter what we defined to be altitude zero - the bottom of the hill,  sea-level, even the center of the Earth. For the equations of motion for the ball, the altitude zero-point is irrelevant. It’s what we call a gauge freedom or a gauge symmetry, and we say that the equations of motion are invariant to that parameter.

That’s a pretty basic example, but it turns out that these gauge symmetries are an important feature of most of our physical theories describing the universe - Newton’s laws of motion and gravity, Maxwell’s equations for electromagnetism, Einstein’s general relativity, and, of course, the standard model. We’re not quite sure why this is the case, but it seems to be a trend. A theory that has these gauge symmetries is called a gauge theory.-

Today we’re going to look at the simplest of the symmetries of the standard model. The standard model is ultimately based on quantum field theory, but we’re just going to use the Schrodinger equation. This is the most basic equation of motion of quantum mechanics. It describes the evolution of the wavefunction, which is the mathematical object that contains all the information about a particular physical system. 

We can’t ever see the underlying wavefunction of, say, a particle. The best we can do is make a measurement of a physical observable like position or momentum. The wavefunction can be represent different observables, and it determines the distribution of the possible results of measurement of those observables. In this episode we’ll be talking about the position wavefunction. OK, pay attention to this bit of math, it’ll be important: The square of the magnitude of this wavefunction tells us the probability distribution of a particle’s position. The position that we observe when we look at a particle is picked randomly from that distribution. This step of squaring the wavefunction is called the Born rule, and this innocuous-seeming step introduces a simple symmetry that has profound implications.

Let’s see what happens when we square the wavefunction. The wavefunction is an oscillation in quantum possibility moving through time and space. It’s no simple wave – it’s literally complex, in the mathematical sense. It has two components, one “real” and one “imaginary” and these components oscillate in sync with each other, but they’re offset - shifted in phase by a constant amount. Phase is just the wave’s current state in its up-down oscillation. When we apply the Born rule what we’re doing is squaring these two waves and adding them together. But it turns out that this value doesn’t depend on phase. The magnitude-squared of the real and imaginary components stays the same, even as those components move up and down.

It’s that magnitude squared that we can observe - it determines the particle’s position. The phase itself is fundamentally unobservable. You can shift phase by any amount and you wouldn’t change the resulting position of the particle, as long as you do the same shift to both the real and imaginary components. In fact as long as you make the same shift across the entire wavefunction, all observables are unchanged. We call this type of transformation a global phase shift, and its analogous to transforming our altitude zero point up or down by the same amount everywhere. The equations of quantum mechanics have what we call global phase invariance. Global phase is a gauge symmetry of the system.

Let’s push this a little further to see how far this symmetry goes. This time we’ll shift the phase by different amounts at different locations, while still keeping the real and imaginary shifts the same at each location. This position-dependent phase shift is called a local phase shift instead of a global one. We’ll try that because we already know that the magnitude-squared of the wavefunction should still stay the same under local phase shifts.-

Let’s see what that would look like. A global phase shift looks like this, where all points move by the same amount. However if we do a local phase shift, say only at this point here, only that location changes as if it was part of the shifted wave, making a discontinuous spike. If you allow this sort of local phase shift you can change each point in a different way and really mess up the wavefunction. That shouldn’t change our probabilities for the positions of a particle. But what about observables besides position?

Well, according to the basic Schrodinger equation - we just ruined everything. Among other things, messing with local phase really screws up our prediction for the particle’s momentum. Momentum is related to the average steepness of the wavefunction. Change the shape of that wavefunction with local phase shifts and you actually break conservation of momentum. Local phase is not a gauge symmetry of the basic Schrodinger equation. OK, that was a bust. I guess we’re done here.

Alright, wait just a second. Just for funsies, maybe can just change the Schrodinger equation to find a version that really is invariant to local phase shifts. To do that, we’d need to alter the part of the Schrodinger equation that gives us the momentum of our particle - the momentum operator. After all, momentum is what got screwed up. It turns out that we can add a mathematical term to the momentum operator that’s specially designed to undo any mess we make to the phase of the wavefunction. If we tune this term correctly, it absorbs any local changes we make to the phase. And what is that extra term? It’s something we call a vector potential. I won’t go into that right now - but the important and absolutely bizarre thing about this mathematical entity is that it looks like something very familiar. It looks exactly like the type of vector potential that you would have in the presence of an electromagnetic field. 

So we’ve discovered that the only way for particles to have local phase invariance is for us to introduce a new fundamental field that pervades all of space. And it turns out that field exists, and it’s the electromagnetic field. This is totally crazy - we just rediscovered electromagnetism be insisting on a gauge symmetry that we had no right to expect in the first place. But we didn’t just rediscover the EM field - we learned a ton about it. By discovering how it fits into the Schrodinger equation, we have unlocked its quantum behavior. We now know how it interacts with particles of matter to give them this symmetry. 

We also learned about the origin of electric charge, which we now see is a coupling term. Any particle that has this kind of charge will interact with and be affected by the electromagnetic field, and be granted local phase invariance. But the reverse is also true, in order to have this particular type of local phase invariance, particles must possess electric charge. By the way, applying Noether’s theorem tells us there is a conserved quantity associated with any symmetry – in this case the symmetry is local phase invariance, and the conserved quantity is electric charge.

At this point we only need a couple of extra steps to produce the full description of electromagnetism in the quantum world – quantum electrodynamics, or QED. First we need to upgrade the Schrodinger equation to the Dirac equation so it works with Einstein’s special relativity - we talked about that in this episode. Then we need to apply quantum principles to our field – like considering its internal or self-energy and allowing quantized oscillations in the field itself. Those oscillations in our new electromagnetic field turn out to be the photon. 

But what about all those fundamental particles without electric charge? Neutral particles like neutrinos? To understand those we’d need to go beyond the Schrodinger equation, and to explore new gauge symmetries. It turns out that local phase invariance is just the simplest of the larger suite of gauge symmetries of the standard model. Those symmetries are obtusely named U(1), SU(2) and SU(3), and they predict the fields that give rise to electromagnetism, the weak, and the strong nuclear forces respectively. 

The fields that arise from these gauge symmetries are called gauge fields, and all have their associated oscillations - their associated particles. These are the gauge bosons: the photon for electromagnetism, the W and Z bosons of the weak interaction and the gluon of the strong interaction. Together they govern the interactions of the matter particles of the standard model, which we’ll come back to in future episodes.

Perhaps the greatest mystery here is not the nature of the quantum fields, nor the connection between symmetry and the fundamental forces. Perhaps it’s the fact that, by pure exploration of mathematics, delving many layers of abstraction deeper than our capacity for intuition, we are lead to true discoveries about physical reality. And following those mathematical labyrinths reveals physical theories with stunning predictive power - like the Standard Model of Particle Physics. Mathematics truly seems to be the language in which the universe is written. We should be amazed that we can learn that language, and through it comprehend the underlying nature of Space Time.