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
Basically, infinity minus infinity equals some fixed finite
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.
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?