Fun With Physics

Renormalization

1. The Game Called "Renormalization"

Okay, let's see.... let's consider a quantum field theory whose Lagrangian has a few free parameters — masses and charges and so. Just to sound cool, let's call all of these numbers "coupling constants". Now to get finite answers from this theory, we need to impose a "frequency cutoff". We do this by simply ignoring all waves in our fields that have a a frequency higher than some fixed value. This works best after we replace "t" by "it" everywhere in our equations, so let's do that — this is called a "Wick rotation" by the experts. Now we're working with a theory on Euclidean spacetime, and the frequency cutoff can also be thought of as a distance cutoff. In other words, it amounts to ignoring effects that involve fields varying on distance scales shorter than some distance D.

In what follows, you have to keep your eye on the parameters in the theory: I'm gonna keep shuffling them around, so to check that I'm not conning you, you have to make sure there's always the same number of 'em around — sort of like watching a magician playing a shell game. So make sure you see what we're starting with! Our Lagrangian has some numbers in it called "coupling constants", but our theory really has one more parameter: the cutoff scale D.

Now our Lagrangian has some coupling constants in it, but it's hard to measure these directly. Even though they have names like "mass", "charge" and so on, these parameters aren't what you directly measure by colliding particles in an accelerator. In fact, if you try to measure the charge of the electron (say) by smashing two electrons into each other in an accelerator, seeing how much they repel each other, and naively using the obvious formula to determine their charge, the answer you get will depend on their momenta in the center-of-mass frame — or in other words, how hard you smashed them into each other. The same is true for the electron mass and any other coupling constants there are in the Lagrangian of our theory. They have a "bare" value — the value that appears in the Lagrangian — and a "physical" value — the value you measure by doing an experiment and an obvious naive sort of calculation. The "physical" values depend on the "bare" values, the cutoff D, and a momentum scale p.

(Of course, we could cleverly try to use a less naive formula to determine the bare values of the coupling constants from experiment, but let's not do that — let's just use the stupid obvious formula that neglects the funky quantum effects that are making the physical values differ from the bare values! By being deliberately "naive" here, we're actually being very smart here — as you'll eventually see.)

There are all sorts of games we can play now. The simplest, oldest game is this. We can measure the physical coupling constants at some momentum scale p, and then figure out which bare coupling constants would give these physical values — assuming some cutoff D. Then we can try to take a limit as D → 0, adjusting the bare coupling constants as we take the limit, in order to keep the predicted physical coupling constants at their experimentally determined values. This "continuum limit", if it exists, will be a theory without any shortest distance scale in it. That's very important if you think spacetime is a continuum!

This game is called "renormalization".

Sometimes you win this game — and sometimes you lose. The main thing to worry about is this: even if certain bare coupling constants are zero, the corresponding physical coupling constants may be nonzero. For example, if you start with a Lagrangian in which the mass of some particle is zero, you might not have bothered to include that mass among your bare coupling constants. But its physical mass (measured at some momentum scale) can still be nonzero. In this case, we say the particle "acquires a mass through its interactions with other particles". This sort of thing happens all the time.

What this means is that to succeed in adjusting the bare coupling constants to fit the experimentally observed physical coupling constants, we need to start with a Lagrangian that has enough bare coupling constants to begin with. You can't expect to fit N numbers with fewer than N numbers!