For various reasons I’ve been having fun thinking about renormalization, so I thought I’d try to put together a post about renormalization in words (rather than equations).

The standard canon for field theory students learn is that when a calculation diverges, one has to (1) regularize the divergence and then (2) renormalize the theory. This means that one has to first parameterize the divergence so that one can work with explicitly finite quantities that only diverge in some limit of the parameter. Next one has to recast the entire theory for self-consistency with respect to this regularization.

While there are some subtleties involved in picking a regularization scheme, we shall not worry about this much and will instead focus on what I think is really interesting: renormalization, i.e. the [surprising] behavior of theories as one changes scale.

The details of the general regularization-renormalization procedure can be found in every self-respecting quantum field theory textbook, but it can often be daunting to understand what’s going on physically rather than just technically. This is what I’d like to try to explore a bit.

First of all, renormalization is something that is intrinsically woven into quantum field theory rather than a trick’ to get sensible results (as was often said in the 1960s). One way of looking at this is to say that we do not renormalize because our calculations find infinities, but rather because of the nature of quantum corrections in an interacting theory.

Recall the Lehmann-Symanzik-Zimmerman (LSZ) reduction procedure. Ha ha! Just kidding, nobody remembers the LSZ reduction formalism unless they find themselves in the unenviable position of teaching it.

Here’s what’s important: we understand the properties of free fields because their Lagrangian is quadratic and the path integral can be solved explicitly. But non-interacting theories are boring, so we usually play with interacting theories as perturbations on free theories. When we do this, however, things get a little out-of-whack.

Statements about free field propagators, for example, are no longer strictly true because of the new field interactions. The two-point Greens function is no longer the simple propagator of a field from one point to another, but now takes into account self-interactions of the field along the way. This leads one to the Lehmann-Kallen form of the propagator and the spectral density function which encodes intermediate bound states.

You can go back and read about those things in your favorite QFT text, but the point is this: we like to use “nice” properties of the free theory ito work with our interacting theory. In order to maintain these “nice properties” we are required to rescale (renormalize) our fields and couplings. For example, we would like to maintain that a field’s propagator has a pole of unity residue at its physical mass, the field operator annihilates the vacuum, and that the field is properly normalized. Assuming these properties, the LSZ reduction procedure tells us that we can calculate S-matrix elements in the usual way.

Suppose we start with a model, represented by some Lagrangian. We call this the bare Lagrangian. This is just something some theorist wrote down. The bare Lagrangian has parameters (masses, couplings), but they’re “just variables” — i.e. they needn’t be directly’ related to measurable quantities. We rescale fields and shift couplings to fit the criteria of LSZ,

$\phi = Z^{-1/2}\phi_{bare}$
$g = g_{bare} + \delta g$.

We refer to these as the renormalized field and renormalized couplings. These quantities are finite and can be connected to experiments.

When we do calculations and find divergences, we can (usually) absorb these quantities into the bare field and couplings. Thus the counter terms $\delta g$ and the field strength renormalization $Z$ are also formally divergent, but in a way that cancels the divergence of the bare field and couplings.

That sets everything up for us. We haven’t really done anything, mind you, just set up all of the clockwork. In fact, the real beauty is seeing what happens when we let go and see what the machine does (the renormalization group). I’ll get to this in a future post.

Further reading: For beginning field theorists, I strongly recommend the heuristic description of renormalization in Zee’s QFT text. A good discussion of LSZ and the Lehmann-Kallen form is found in the textbooks by Srednicki and Ticciati. Finally, for one of the best discussions of renormalization, the Les Houches lectures “Methods in Field Theory” (a paperback version is available for a reasonable price) is fantastic.