Does Infinity - Infinity = An Electron? (script) (Patreon)

What do you get if you take something that’s infinitely massive and combining with something else that’s negative infinitely massive? You get a single electron, at least that’s what it looks like in our most precise way of describing the quantum world.

Today’s episode is an introduction to one of the most important problems in physics. It’s the Hierarchy Problem, and its important because its solution is going to lead us beyond our current theories. The Hierarchy Problem, put simply, is that we don’t understand why the mass of the Higgs boson is so small. From our understanding of quantum field theory and the standard model of particle physics, interactions between the Higgs and the quantum vacuum should drive its mass up to enormous values. That would be catastrophic for our universe, and so we should be grateful that the small Higgs mass is somehow protected. We just don’t know how it's protected. But to fully appreciate why it’s surprising that we haven’t been able to find the mechanism for this protection, and why the capital-H hierarchy problem is such a big deal, we’re going to start with the small-h hierarchy problem involving the electron. The electron hierarchy problem existed before quantum mechanics, and was solved by quantum mechanics, once we understood the theory well enough. By giving you a sense of how the electron hierarchy problem was solved, I think you’ll really appreciate why the Higgs Hierarchy Problem is so problematic, and what its solution might mean.

We’re building on our last episode here. Pretty heavily really, so this is a rare time I’ll recommend watching that first if you haven’t already. In that episode we asked what an electron looks like if we zoom in all the way. We discovered that it was ultimately impossible to say exactly where the electron’s mass and charge lived. These properties appear distributed between the “true” or bare electron, and the frantic activity in the quantum fields around it. Well, this activity not only makes it difficult to pin an electron down, it also makes it difficult to fix the mass and charge to the sensible small values that we’ve measured. Instead, this activity tends to blow these values up to unnatural levels.

But this issue also exists in classical physics. We saw last time that the potential energy in an electron’s classical electrostatic field has a mass equivalent 20,000 times larger than the measured mass of the electron, and that’s assuming that an electron is as large as it could possibly be according to our best measurements, at around 10^-17m. And if the electron is really point-like then the mass of the field is infinite.

That can’t be right—and the most obvious thing we did wrong is to assume that classical physics applies all the way down to such small distances. The correct description of the world on such small scales is quantum mechanics—in particular, quantum field theory—QFT. In the case of the electron interacting with the electromagnetic field, the relevant version of QFT is quantum electrodynamics, QED. We’ll use QFT and QED for most of today’s discussion. But even in the classical description we can get a glimpse of the solution to the enormous field mass, and also of the even bigger problem that follows this solution.

Borrowing terms from QFT, let’s call the “true” mass of the electron the bare mass, and the energy in its surrounding field the self energy, and self-energy mass for its E=mc^2 mass equivalent.

When we measure the electron mass, we never measure the bare mass—we measure the sum of the bare and self-energy masses. We’ll call this the dressed mass, again from QFT because the bare electron is dressed in quantum fluctuations. The measured 511 eV electron mass is its dressed mass.

This equation equating the dressed to the bare plus self-energy masses can work perfectly well, even with a tiny dressed mass of 511 eV and an enormous or even infinite self-energy mass. All we need to do is to give the electron a bare mass that’s also huge, and negative and just a hair smaller in magnitude than the self-energy mass. In fact, you could even fix an infinite self-energy mass with a negative infinite bare mass—infinity minus infinity-minus-one equals one! No problem! This process of adjusting the internal contributions towards a measured quantity to eliminate infinities is called renormalization. It was first applied in cases like the electron before quantum mechanics was even invented, though has since become a key tool in quantum field theory in particular.

It sounds a little hokey because it assumes that some mass, in this case that of the bare electron, is negative. Negative masses aren’t supposed to exist. However remember that this bare mass is not something we can ever see or measure, so let’s pretend this odd assumption is OK for now. Besides, we now have a bigger problem than negative bare masses. It’s the fact that to get our tiny measured electron mass, the bare and self energy masses have to be comparatively huge numbers that are very very nearly, but just not quite the same—deviating from each other by only one part in 20,000 if we take the self-energy mass for a 10^-17 m electron.

This doesn’t feel natural somehow. It would require a type of fine tuning for these seemingly very different sources of mass. I mean, it would be one thing for them to be exactly equal, but to be only almost equal? This type of problem is generally called a naturalness or hierarchy problem. As I said, we’re going to be able to solve it for the electron, but not for the Higgs boson. That’ll be in an upcoming episode.

For now let’s focus on saving our electron, and we can do that with quantum mechanics. Last episode, when we dove in real close to the electron, we saw that the classical electric field becomes the jittery quantum electromagnetic field. In quantum field theory, those jitters are represented mathematically as the sum of activity of countless virtual particles. The EM field itself is modeled as being made of virtual photons that communicate the EM force between charged particles, and the next most important effect is the interaction between the EM and electron fields, which QFT models as the very brief production of a virtual electron-positron pair.

This extreme heft of the classical EM field is described in QFT as coming from the energy in this virtual activity. But how much self-energy mass does quantum theory predict? In QFT we use Feynman diagrams to represent the possible virtual processes that can be part of any interaction. These diagrams are really shorthand representations of components of an interaction. For any combination of ingoing and outgoing particles for any interaction, we can calculate something called the Lagrangian of the interaction by adding up all possible scenarios that could have occurred in that interaction. This allows you to calculate things like the probabilities of various interactions occurring. We can also calculate energy flowing through the various processes within the interaction, and this energy gives us the mass equivalence produced by that interaction.

In the case of our free electron, the ingoing and outgoing particles are just the electron itself, and so we just consider interactions that leave the electron unchanged. But that still includes lots of activity—for example, emitting and absorbing one or more virtual photons—and that’s what the basic EM field is in this model. But those virtual photons may engage in all sorts of shenanigans with virtual electron-positron pairs prior to reabsorption.

In order to calculate the self-energy and corresponding self-energy mass of the electron, Feynman tells us we need to add up not only every possible interaction, but also every way each of those interactions can play out. For example, every energy and momentum that the intermediate virtual particles could have had.

For simple cases where the electron emits and reabsorbs a photon that’s no big deal—the energy of that photon is strictly limited by the law of energy conservation. But for cases where there are entire loops of activity in the intermediate virtual space, the possible energies of these intermediate particles are not constrained by the energy of the electron. Quantum mechanics tells us they could be anything, and quantum field theory tells us that we must add all possibilities. The result of summing all possible energies in all possible Feynman diagrams is infinite energy,

which means contributing infinite self-energy mass. Something still seems off.

Because of the Heisenberg uncertainty principle, high energy corresponds to measuring at short distances. So the choice to sum energies up to arbitrarily large numbers is equivalent to trying to observe the electron with infinite resolution. By this logic, a truly point-like electron should have infinite self-energy—so that’s the same bad answer we got from classical electromagnetism.

But renormalization can also fix this issue. In the classical case we imagined a negative bare mass to the electron. In QFT we instead need to cancel the problematic Feynman diagrams one by one. This is done by essentially breaking these “divergent” diagrams into two parts—a regular term with its infinite contribution and a counter-term that cancels that infinity. That still sounds hokey, don’t worry—the founders of this approach, people like Richard Feynman, Julian Schwinger and Freeman Dyson, also worried about that. But the results were stunningly successful. Quantum field theory, and in particular quantum electrodynamics gave incredibly precise predictions of the quantum behavior of things like the electron. But the theory didn’t predict the actual mass of the electron. The reason QED works is that all this “hokey” renormalization is grounded by the fact that we can measure the mass of the electron, so all of this canceling that occurs under the hood is …

made concrete by calibrating it to reality.

It's still being called “wweeping infinities under the rug”, and it feels awkward, but renormalization was put into much more comfortable context by Ken Wilson, who pointed out that these infinities arose only because we’d tried to push quantum mechanics too far. Remember when we did the self-energy calculation with classical electromagnetism, the self-energy mass got out of hand when we started to apply Coulomb’s law way beyond its validity, into the quantum realm. But maybe something similar is happening here—maybe all these convenient counter-terms that cancel our infinities represent real physics beyond the stuff directly described by our quantum field theory.

Now we know for sure that quantum mechanics also has limits to its validity. Quantum mechanics doesn’t include gravity, so we know that it must be what we call an effective theory—emergent from some deeper theory—just as classical physics emerges from the quantum. We know for sure that any of our quantum field theories fails at the Planck scale—lengths smaller than around 10^-35 m, or energies larger than 10^28 eV.

So what if we just stop adding energies when they reach a certain cutoff where we know our theory is no longer valid? The process of doing that is called regularization. If we set the Planck scale as our cutoff, and don’t include energies above that we do get a finite mass, but that mass is also at the Planck scale—now 10^22 times larger than the measured electron mass. A universe with elementary particles this massive would have collapsed before it started expanding. We can still fix that crazy mass using renormalization, but now we’re asking Planck-scale physics to cancel out the positive self-energy of the electron all the way down to the electron scale. That means a near-perfect canceling to one part in 10^22. And that is a much worse hierarchy problem than we figured with classical physics..

In the case of the electron, we actually don’t have to go all the way to the Planck scale to find new physics to solve the electron’s hierarchy problem. I mentioned last time that when we zoom in real close to the electron so that the energy in the EM field approaches the rest mass of the electron itself, a new quantum process becomes important. Virtual positrons are now able to annihilate with the real electron, causing the companion virtual electron to take its place in reality. We saw how this leads to an apparent blurring of the location of the electron, which is annoying. But it also solves our hierarchy problem, which is great.

It turns out that this particular interaction has a negative contribution to the self-energy of the electron—and this perfectly cancels the leading positive component of the self-energy, leaving only second-order contributions. That remaining un-canceled self-energy of the electron is much smaller than the canceled part. In fact it’s similar to the bare mass, and scales linearly with the bare mass. We don’t need any coincidental fine-tuning to do this canceling because there’s an in-built symmetry between the negative and positive energy components of the field. You can think of this symmetry as being the matter-antimatter symmetry. It’s the fact that the electron has an antiparticle that protects it from being blown up to the Planck scale. More formally, its mass is protected by the chiral symmetry of the particle.

So, woohoo! We saved the electron’s mass from being “unnatural” by finding some new physics to make its smallness not a complete fluke. Unlike the case of the Higgs boson, whose unnaturalness is yet to be understood, and very soon we’ll get back to why this is such a big deal. We’ll also come back to some of the tools we saw—renormalization and regularization. These are actually pretty profound subjects in themselves, and teach us something about the emergence of effective theories from deeper layers, and the connection between the energetic scales of spacetime.