Regularization is a method in quantum field theory which makes it possible to renormalize a theory. This is essential in quantum field theory because of the following problem:

Let's say you've written down a theory—you've specified all your types of particles, their (bare) masses, and the ways they can interact. Then you try to calculate something, like the probability for a certain scattering process (perhaps using feynman diagrams). The probability you calculate equals infinity. This is a problem. You were hoping for an answer somewhere between zero and one. Instead you got infinity. Clearly there's something wrong with your theory.

Unfortunately, the only way that we can get rid of these infinities is to cancel them out with other infinities. When writing down your theory, you need to explicitly write down infinite values for quantities in a particular way such that all the infinities end up canceling when you calculate physical quantities.

Basically, infinity minus infinity equals some fixed finite value.

How can we be sure the above "equation" is mathematically sound? "Infinity minus infinity" could equal any real number. Indeed, it can even equal infinity, or negative infinity! Clearly, before we can try canceling out terms, we need to "wrangle" all of these infinities. This is where regularization comes into play.

Regularization is a temporary change you make to your theory which makes it drastically nonphysical, but makes all of your calculated quantities finite. You can then see which terms are going to blow up when you remove the regularization, so that you can insert counter-terms to cancel those terms, then remove the regularization. It's a hideous process, but can be somewhat interesting if you don't have to be the one doing the calculation.

Here are some examples of regulators:

1. Momentum Cutoff

The general reason you need to renormalize is that you're adding up trajectories from every possible momentum that a particle can have. Since there is no “maximum momentum” that a particle can have, this gives a sum (integral, actually) which can often add up to infinity. How can we regulate this? The simplest way is to set a maximum momentum, Λ, truncating your sum. After adding counter-terms, take the limit where Λ goes to infinity, and you’ll end up with finite quantities. One reason this regulator is horribly nonphysical is that it violates special relativity: an observer in another reference frame would need a different value for Λ.

2. Discretization of Spacetime

Because of the quantum mechanical interplay between position and momentum (see Heisenberg Uncertainty Principle), this regulator is mathematically equivalent to a momentum cutoff. However, this is probably the “least nonphysical” of all regulators. You make the assumption that space and time are not continuous, but are divided up into a discrete lattice with some characteristic length scale, “a”. The reason this is a “less nonphysical” regulator is that it could actually be true. It may indeed be the case that spacetime has some discrete lattice that is smaller than any length scale that physicists have yet been able to measure. Since any physical measurement of distance has some small degree of error, we will never be able to say with certainty that spacetime is continuous. However, this is still a problematic regulator, since it not only violates relativity, but it also implies that space is no longer rotationally symmetric. After adding counter-terms, take the limit where “a” goes to zero.

3. Pauli-Villars Regulator

This regulator adds a new particle into your theory, whose contributions cancel out infinities from other particles. This regulator is nice, because you’re not imposing any strange conditions on spacetime, just adding a particle to your theory. It’s not perfect, though, because it essentially has to be the same particle as the one you’re trying to regulate, but with a relative minus sign. This means you might have a boson which obeys fermion statistics, and for a physicist that makes absolutely no physical sense. So, after adding counter-terms, take the limit where your nonphysical particle’s mass goes to infinity, and its contributions will vanish.

4. Dimensional Regularization

Without question, this is the most nonphysical of all regulators. It seems plausible at first; change the number of dimensions of your theory (i.e. instead of four dimensions, you’re now in “d” dimensions). For mathematical reasons that I’m not going to get into, this will give finite results if d is not an integer. That’s right; we now drop the assumption that the number of dimensions in our theory is an integer. In fact, d no longer has to be a real number. We could be working in “4½ dimensions”, or “4 + i dimensions”. How can we physically understand this? Simple: we can’t. By definition, the number of dimensions of a mathematical space has to be an integer, but we just pretend that’s not true for the time being and just say d = 4 + ε, where ε is some small complex variable. Our results become finite, and we can add counter-terms, then take the limit as ε goes to zero, and we are back in four dimensions. Interestingly enough, while this method is ridiculously nonphysical, it preserves relativity and all other symmetries, which often makes it a preferred regulator for physicists.

In summary, physicists will do the darndest things to make their theory work. You might wonder, “Is all of this work really worth it?” You tell me: using regulation and renormalization, physicists have been able to construct an incredibly accurate and precise description of everything in the universe (sans gravity). Don’t the ends justify the means of doing a little fuzzy math?

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