How a dubious math trick became a law of physics, a stock trading analogy
You can use renormalization in any statistical model as long as it matches what is actually happening under the hood.
You can use renormalization in any statistical model as long as it matches what is actually happening under the hood.
Math professors tell their students never to divide by zero because it is “undefined”.
But what about subtracting infinity?
While 0 is actually a number on the real line, infinity isn’t even that. The real numbers continue towards infinity, but all that means is that for any given real number, x, there is a real number larger than x. To get infinity, you need the “extended” real number line, as if infinity is part of the Director’s Cut and you have to pay $20 more for it.
But what if I really want to?
Say I have a cool function, G, that calculates what the value of Google’s stocks are going to be tomorrow. The only problem is that G is infinite. I know, however, that I can fix G by subtracting another function H from it. H is also infinite. That gives me a new function G’ that is finite and it works perfectly! I’m rich!
I’m a little worried though. G is the function that I figured should predict the stocks and H is just some math trick I came up with. Why does that even work?
On the other hand, what’s the big deal? As long as I get my money, I shouldn’t need to worry about it. Maybe G isn’t predicting anything real and G’ is the real function after all.
And that’s more or less the story of renormalization, one of the least understood but most powerful success stories in high energy physics, where what started as a math trick that made mathematicians cringe went on to become a law of physics.
Physicists had to figure it all out the hard way with a lot of kicking and screaming too. As quantum mechanics became quantum field theory in the 1940s and 50s and they wanted to understand Quantum Electrodynamics (QED), they kept running into infinities.
Richard Feynman, genius son of the Atom Bomb, safe cracker, and bongo player, had devised a method of calculating predictions of how particles would behave when they interact in a particle accelerator as they are smashed to pieces.
These “Feynman diagrams” were a tool for predicting the outcomes of experiments based on all the things that can happen to particles that collide together and are later detected. Unlike all the heavy operator math and calculus of previous generations, his diagrams were a turn-the-crank style algorithm that beautifully associated pictures of particle interactions with the mathematics.
If you look above you can see a diagram I made, you have two electrons coming in from the top and exiting at the bottom. These electrons repel one another. In order to interact, electrons use photons as their force carriers, so the photons with the little Greek gamma letters are propagating between the two electrons, causing them to repel. Something funny is happening there in the middle. The photons, instead of just propagating between the electrons, create an electron-positron pair in the middle. The pair appears from one photon annihilates and becomes the other photon.
The loop in the middle is a closed loop, meaning that it is hidden from detection. Nobody sees it and we can only infer its existence based on what we measure when the electrons collide with a detector.
It was those little loops that caused all the trouble. It is possible for anything to happen in there including massive exchanges of energy that simply disappear before the electrons hit the detector. In fact, there is no limit to the energy of the interactions as long as energy and momentum are the same at the end.
Let’s go back to my Google stock predictor analogy. Suppose my top secret mathematics has some way to model the volume of stock trading based on interacting stock traders. In my analogy traders are the electrons and trades are the photons. The stocks they move are the energy and momentum. If I model traders and trades as continuous fields in a statistical model then it is very similar to a quantum system.
Weird things can happen in field theories like this. For example, I can have two traders interact by a trade that suddenly spawns two more traders who make massive buys and sells and vanish without a trace as in my diagram. You could call them “middlemen”.
My model is a set of differential equations so it models stocks continuously and doesn’t have any maximum size trade. Suppose the total number of stocks in my model has to be the same from one day to the next, but traders can buy and sell as much as they want provided it all works out. My equations are set up so that they can theoretically model an infinite number of stock trades in an infinitesimal period of time. It may seem odd but I argue it works. It doesn’t matter that that doesn’t really happen.
Even with the infinite volume of trading, my formula could still be finite. One thing I have to do in my stock model is sum up over all the stock trades. That sum, even over infinities may or may not be infinite itself.
Gabriel’s Horn is an example where you can sum over something infinite to get something finite:
Gabriel’s Horn is an infinitely long horn that, nevertheless, has a finite volume. So you could pour a finite amount of water into it and fill it up but you could not measure it with a tape measurer.
Unfortunately, my stock trading formula isn’t as nicely behaved as Gabriel’s Horn. Even after I integrate it, it is still infinite.
I figure out a solution that makes it finite: cut it off at a maximum stock trade. There’s just one small problem: I don’t know what that cut off is. While there might be an “official” cutoff, there is no unofficial one, just between traders. I can hazard a guess, but there has to be a better way.
Suppose, instead, that I represent the cutoff as an unknown quantity and, instead of guessing what it is, I just try to roll that into some other part of my model that I can measure. Suppose I have a number in my model that represents the strength of interaction between stock traders, how much they influence one another. Two traders come into contact with one another, see the trades each has made, and make new decisions. Now, if I’m modeling traders as making infinite size trades in infinitesimal periods of time, maybe I can just take all those individual, high speed trade interactions and roll them up into some meta-trade that is the sum total of all the trades between traders made in the interval of time that I chose. That is, I pretend that a sequence of trades is one trade which helps a lot of the huge trades cancel each other out. This idea of grouping the trades into “blocks” is sometimes called the “block spin” approach.
Now, the ratio of the sum of stock trades over the interval I’m measuring to the maximum size trade cutoff that I don’t know is a measure of the scale of the meta-trade. Since I don’t know what that ratio is, I have to assume it is included in the interaction strength and other values I can measure.
I also want to assume that trade interactions get weaker over smaller time intervals. There is less time to react and traders are more focused on buying and selling and less on each other.
So now I’ve made a few assumptions:
(a) There is a cutoff in trade size beyond which no trades occur.
(b) My interaction strength is different depending on the period of time over which I am measuring trades because of this meta-trade idea.
(c) My measured interaction strength depends on the cutoff and goes to zero at that point.
Let’s introduce a few terms:
Assumption (a) is called my “regularization”. I am making my sum or integral “regular” by cutting it off. My assumption (b) is called a “running coupling” which just means that my interaction strength “runs” as I change the time interval I’m measuring over. (In particle accelerators, time intervals and energy are related, so you’ll hear people say “energy level” in physics.) The final part is “renormalization group flow” to a “fixed point”. The fixed point is zero interaction at the cutoff and the running coupling “flows” to it.
Since my measured interaction depends on the cutoff and effectively goes to zero as I approach that mysterious cutoff, I can use a little mathematical trick now to remove the infinity from my original function G. I just sum up stock trade volume to my cutoff and then any terms that depend on the cutoff I pull into my interaction term that I’m going to measure.
And poof! That cutoff disappears. Where did it go? What happened is that I measured it indirectly by measuring the interaction strength. The strength flows so that it hits zero at the cutoff so they are interdependent. The beauty is that I don’t need to know where that point is as long as the interaction disappears there.
Of course, maybe my traders just get more and more aggressive as the time get shorter, so their interaction strength increases as my time interval decreases and trade size increases. What then? Well, good luck with that. Your theory is what we call “nonrenormalizable”. In that case, you either need a better theory or you need to know what actually happens at the cutoff. Do they get in a fist fight? Are the police called? Does the stock market crash for a billionth of a second?
It turns out that in both physics and stock trading renormalization is not a mathematical trick. It is a model for how things actually work in the real world. Coupling constants for interacting particles do change at different energy levels. (Also mass and charge.) I’ve tried to make that obvious with my stock trading analogy. You can’t trade infinite stocks in an infinitesimal length of time. Likewise, you don’t have infinite energy interactions between particles at infinitesimal distances.
A lot of physics textbooks focus on how “odd” it is that the theories of forces and matter are all “renormalizable” and that they all just barely qualify for it. It is a lucky break, but there is no physical reason why a theory should be renormalizable. Nor is there any physical reason why a theory should require it in the first place. What is important is that renormalization works because of what is physically happening in those theories. The universe appears to be built so that the smallest scales behave nicely.
Gravity is an exception. Like the violent traders, at the smallest scales and highest energies tightly curved black holes form that are anything but weakly interacting. Is that really what’s happening down there? We have no way of knowing. Gravity hasn’t been measured at scales in the quantum realm. All we know is that the theory we have isn’t renormalizable and for good reason.