Subject: renormalization
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Date: Wed, 14 Jan 2004 14:14:24 -0600

Renormalization Made Easy

John Baez

August 14, 1999
I want to explain to you the basic idea of
renormalization in quantum field theory, and I want to say a bit about how it's
related to statistical mechanics. You won't get the mathematical details here -
for that, try something like Peskin and Schroeder's book, "An Introduction to
Quantum Field Theory". Instead, I just want to give you a gut-level
understanding of this stuff.

I'm gonna assume you vaguely know what a Lagrangian for a quantum field
theory looks like, and that you know how different terms in the Lagrangian
correspond to different sorts of particle interactions, which can be drawn as
"vertices" of Feynman diagrams. But you won't need to understand this stuff in
any detail - if you've read Feynman's cute little popular book called "QED:
Strange Theory of Matter and Light", you're probably ready for what I'm about to
say.

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!

So, if someone hands you a Lagrangian, you may have to stick in some extra
terms with some extra bare coupling constants before playing the renormalization
game. If you can succeed with only finitely many extra terms, you say your
theory is "renormalizable". If you need infinitely many terms, you throw up your
hands in despair and say the theory is "nonrenormalizable". A nonrenormalizable
Lagrangian is like a hydra-headed monster that keeps needing more extra terms to
be added the more you add.

Note: when we try to take the continuum limit, we don't care if the bare
coupling constants do something screwy like go to infinity. All we care about is
whether the experimental predictions of our theory converge. If the bare
coupling constants converge we say our theory is "finite". But truly realistic
theories usually aren't this nice.

If you want to see how to tell whether a given term in your Lagrangian is
renormalizable or not, click here and I'll
work you through an example called the "4-fermion interaction". This stuff is a
wee bit more technical... if you don't want things to get technical, just keep
reading here!

2: The "Renormalization Group" Game

Okay, now I want to talk about the renormalization group. I'm deliberately
going to simplify things to the point where I'm verging on inaccuracy, but
hopefully I won't actually say anything actually *false*.

So, let's recall what we've got. We have a quantum field theory described by
a Lagrangian with a bunch of coupling constants in it - let's call them "bare"
coupling constants. We can write all these bare coupling constants in a list and
think of it as a vector: call it C. But to do calculations with this theory we
need one more number, too: we need to ignore effects going on at length scales
smaller than some distance D, called the "cutoff".

Now, starting from these numbers, we can compute the "physical" coupling
constants at any momentum scale. For example, the measured charge of the
electron depends on the momentum with which we collide two electrons. Another
way to put it is that the physical coupling constants depend on a distance
scale: for example, the measured charge of the electron depends on the distance
at which you measure its charge. These two ways of thinking about it are
equivalent, since using hbar and c we can freely convert between momentum and
inverse distance.

Let's work with distance instead of momentum, and call the distance at which
we measure the physical coupling constants D'.

So: if we know the "bare" coupling constants C and the cutoff D, we can
compute the "physical" coupling constants C' at any distance scale D'. In short:

C' = f(C,D,D')

Now let's play the "renormalization group" game. In this game, we fix the
bare coupling constants and the cutoff, and see how the physical coupling
constants C' change as we vary the distance scale D' at which we measure them.
It's fun to imagine turning a dial to adjust the distance scale D' and watching
the physical coupling constants C' move around like a little dot in
n-dimensional space, where n is the number of coupling constants. People draw
pictures of this and speak of "running coupling constants" or the
"renormalization group flow".

Note that we can play this game whether or not our field theory is
renormalizable! In the last section I talked about a different game, called
"renormalization". That game was all about letting the cutoff D go to zero. For
"renormalizable" theories there's a nice way to do it, while for
"nonrenormalizable" ones it's a real mess. But here we aren't letting D go to
zero.

So what happens if we start with a nonrenormalizable theory and play this
"renormalization group" game? Our Lagrangian will typically have a bunch of
terms in it: some nasty ones that are making the theory nonrenormalizable, and
some nice ones that would give a renormalizable theory if we just threw out the
nasty ones. Each of these terms is multiplied by a coupling constant. Now let's
look at the corresponding *physical* coupling constants as we crank up the
distance scale D'.

As we do this, the physical coupling constants in front of the nasty
nonrenormalizable terms get smaller and smaller, approaching zero! At large
distances, nonrenormalizable interactions become irrelevant!

This is an incredibly important fact, because it may explain why the quantum
field theory that seems to describe our world - the Standard Model - is
renormalizable. There may be all sorts of strange quantum gravity stuff going on
at very short distance scales - perhaps spacetime is not even a continuum! But
if at larger scales we assume that ordinary quantum field theory on flat
spacetime is a reasonably accurate approximation to what's going on, then this
renormalization group stuff assures us that at still *larger* scales,
nonrenormalizable interactions are going to look very weak.

In fact, this may explain why gravity is so weak! If we treat quantum gravity
perturbatively as a quantum field theory on flat spacetime, it's
nonrenormalizable. If we assume the gravitational constant is reasonably large
near the Planck scale, and we follow the renormalization group flow, we find
that it's very small at macroscopic distance scales. In fact, we even get the
right order of magnitude. But this isn't surprising: it's really just the magic
of dimensional analysis.

This sort of idea goes back to Kenneth Wilson who won the Nobel prize in
physics in 1982, for work he did around 1972 on the renormalization group and
critical points in statistical mechanics. His ideas are now important not only
in statistical mechanics but also in quantum field theory. For a nice short
summary of the "Wilsonian philosophy of renormalization", let me paraphrase
Peskin and Schroeder:

In Chapter 10 we took the philosophy that the distance cutoff D
should be disposed of by taking the limit D -> 0 as quickly as possible. We
found that this limit gives well-defined predictions only if the Lagrangian
contains no coupling constants with dimensions of length^d with d > 0. From
this viewpoint, it seemed exceedingly fortunate that quantum electrodynamics,
for example, contained no such coupling constants since otherwise this theory
would not yield well-defined predictions.

Wilson's analysis takes just the opposite point of view, that any quantum
field theory is defined fundamentally with a distance cutoff D that has some
physical significance. In statistical mechanical applications, this distance
scale is the atomic spacing. In quantum electrodynamics and other quantum
field theories appropriate to elementary particle physics, the cutoff would
have to be associated with some fundamental graininess of spacetime, perhaps
the result of quantum fluctuations in gravity. We discuss some speculations on
the nature of this cutoff in the Epilogue. But whatever this scale is, it lies
far beyond the reach of present-day experiments. Wilson's arguments show that
this this circumstance *explains* the renormalizability of quantum
electrodynamics and other quantum field theories of particle interactions.
Whatever the Lagrangian of quantum electrodynamics was at the fundamental
scale, as long as its couplings are sufficiently weak, it must be described at
the energies of our experiments by a renormalizable effective Lagrangian.

3: Ultraviolet and Infrared Fixed Points

In the last section I described the "renormalization group" game. Now I want
to explain "ultraviolet and infrared fixed points" of the renormalization group,
but first let me summarize what I already said. We have a quantum field theory
described by a Lagrangian with a bunch of terms multipled by numbers called
"bare" coupling constants - we call the list of all of them C. We ignore effects
going on at length scales smaller than some distance D called the "cutoff". And
now we can compute stuff....

In particular, we can compute the so-called "physical" coupling constants C'
as measured at any given length scale D'. And we can watch how C' changes as we
slowly crank D' up. This is called the "renormalization group flow".

Various things can happen. I already said a bit about this: I said that for
nonrenormalizable terms in the Lagrangian, the physical coupling constants
shrink as we increase D'.

In fact we can say more: they scale roughly like D' to some negative power.
If you're smart, you can even guess what this power is by staring at the term in
question and doing some dimensional analysis! Using Planck's constant and the
speed of light you can express all units in terms of length. If a particular
bare coupling constant c in front of some term in the Lagrangian has dimensions
of length to the power d, then the corresponding physical constant c' will scale
roughly like D' to the power -d. More precisely:

c'/c ~ (D'/D)^{-d}

In particular, this term will be nonrenormalizable if d is greater than zero.
(For a more thorough explanation of this criterion, and also some loopholes,
click here.)

Of course, another way to put this is that for nonrenormalizable theories,
the physical coupling constants *grow* as we *decrease* D'. This is another way
to see why nonrenormalizable theories are "bad" - they involve interactions that
get ridiculously strong at short distance scales. Why is this bad? Well, it's
certainly bad if you're trying to do perturbation theory and think of the
interaction as a small perturbation. It may not always be bad in any more
profound sense, because there *are* nonrenormalizable theories that are
perfectly consistent, mathematically speaking.

On the other hand, if d is less than zero we say our term in the Lagrangian
is "superrenormalizable". In this case the physical coupling constant scales
roughly like D' to some *positive* power. In the same sense that
nonrenormalizable theories are not nice, superrenormalizable theories are
super-nice.

Finally, for "renormalizable" theories, the physical coupling constants scale
roughly like D to the zeroth power - i.e., they're roughly constant. They are
right on the brink between nasty and nice. We actually have to do a more careful
analysis to see if they are nasty or nice. For example, quantum electrodynamics
is renormalizable, but it turns out to be nasty: at first the charge of the
electron looks almost constant as we decrease D', but it actually grows -
logarithmically at first, but then faster and faster. On the other hand, lots of
nonabelian gauge theories are nice: the coupling constant slowly *shrinks to
zero* as we decrease D'. We say they are "asymptotically free".

Now, to get ready for my explanation about what all this has to do with
2nd-order phase transitions, let's just introduce some concepts to help us tie
all these ideas together. We've seen that sometimes when we keep making D'
smaller and smaller, the physical coupling constants C' approach some particular
value. I've just talked about the case when they approach zero, but other cases
are important too! Whenever this sort of thing happens, we say the limiting
value of C' is an "ultraviolet fixed point of the renormalization group". Here
"ultraviolet" refers to the fact that we are looking at ever smaller distance
scales.

Similarly, if C' approaches some value when D' keeps getting larger, we say
that value is an "infrared fixed point".

For example, suppose we have a superrenormalizable or asymptotically free
theory with just one coupling constant. Then as we keep making D' smaller, the
physical coupling constant approaches zero, so zero is an ultraviolet fixed
point. Of course "zero" here corresponds to a *free* field theory with no
interaction at all. So free theories are ultraviolet fixed points of
superrenormalizable or asymptotically free theories. Similarly, free theories
are infrared fixed points of nonrenormalizable theories, and certain
renormalizable but nasty theories like quantum electrodynamics.

4: Second-Order Phase Transitions

Okay, now I'm going to finish by describing Wilson's ideas relating
renormalization to 2nd-order phase transitions. First of all, what's a 2nd-order
phase transition?

Actually, FIRST of all, what's a FIRST-order phase transition?

The most familiar examples are when ice melts or liquid water boils: we have
two phases of matter, and the internal energy changes discontinuously as we go
from one phase to another. But look at this phase diagram, which I borrowed from
Scott Lanning:

\
\ liquid X (critical point)
| \ /
^ | solid \ /
| | \ /
P | /
R | /
E | /
S | /
S | / gas
U |/
R |
E |_______________________________________

TEMPERATURE --->

We see something interesting: the sharp boundary between liquid and gas
phases fizzles out at a point called the "critical point". Above this point
there is no real difference between a liquid and gas! This critical point is a
"2nd-order phase transition", because while the internal energy doesn't change
discontinuously there, its first derivative becomes infinite there.

Right at the critical point, something very cool happens: the system
transforms in a simple way under scaling! What does this mean? Well, if you get
some water right at the critical point, it looks "opalescent" like a moonstone.
If you stare at it carefully, you'll see a bunch of liquid water droplets of all
different sizes floating around in steam. However, if you look closely at any of
these droplets, you see they are full of bubbles of steam, and if you look
closely at the steam, you see it's full of little droplets of liquid! It's like
a random fractal: no matter how closely you look, you see the same thing. You
can't tell if you're looking at water droplets in steam or bubbles of steam in
water, and there is no distinguished length scale... at least until you get down
to the scale of atoms, that is.

Building on insights due to Landau, Kadanoff and others, Wilson realized that
you could come up with a very precise theory of critical points by taking
advantage of this symmetry under change of scale. In particular, this theory
lets us understand so-called "critical exponents".

To explain this, let me switch to a simpler example of a critical point.
Consider a ferromagnet like a crystal of iron. At temperatures above a certain
point called the Curie temperature, the iron will not be magnetized. But as we
cool it below the Curie point the spins of certain electrons in the atoms will
line up and the iron will become magnetized. If there is an external magnetic
field around when we cool the iron below the Curie temperature, the spins will
line up with this magnetic field. Suppose the magnetic field points along the z
axis - either up or down. Then we have the following phase diagram:

^ |
| |
M |
A | magnetized up
G |
N |
E |
T | ----------TEMPERATURE-->---------X unmagnetized
I | (critical
C | point)
|
F | magnetized down
I |
E |
L |
D |

The sharp boundary between the "up" and "down" magnetized phases fizzles out
at the Curie temperature. The Curie temperature is a critical point! Right at
this critical point the magnet displays symmetry under scaling. If we look at
the atoms in the crystal lattice and see which ones are "spin-up" and which ones
are "spin-down", at the critical point we see regions of spin up and regions of
spin down, but all these regions are speckled with smaller regions of the
opposite type, and so on... on down to the length scale set by the crystal
lattice itself.

To describe this scaling symmetry a bit more mathematically, let's simplify
things a bit and imagine that for each point x in the crystal lattice we have a
variable s(x) which equals 1 if that atom is spin-up and -1 if it's spin-down.
When the crystal is in thermal equilibrium this variable keeps randomly flipping
sign, so we can think of it as a random variable. This means we can talk about
its mean, standard deviation and stuff like that.

When the external magnetic field is zero, the mean of s(x) is zero:

<s(x)> = 0

because each atom has a 50-50 chance of being spin-up or spin-down. This
isn't particularly interesting. What's interesting is the mean of the *product*
of s(x) and s(y) for two different points in the lattice, x and y:

<s(x)s(y)>

This is called a "2-point function". It measures the *correlation* of spins
at different points in the lattice, since it equals 1 if the two spins always
point the same way and 0 if they are completely uncorrelated.

The 2-point function only depends on the distance between x and y. Away from
the critical point it decays exponentially with distance (at least when the
external magnetic field is zero), and this exponental decay determines a special
length scale called the "correlation length":

<s(x)s(y)> ~ exp(-|x-y|/L)

But as we approach the critical point, the correlation length goes to
infinity, and right *at* the critical point, the 2-point function decays like
some power of distance:

<s(x)s(y)> ~ 1/|x-y|^d

The number d is an example of what we call a "critical exponent".

A similar thing is true for all the higher "n-point functions", at least if
we define them correctly, which I won't bother to do here. They all satisfy nice
power laws at the critical point. This is what people mean when they say that a
system at a critical point transforms simply under scaling.

Now, I'm oversimplifying something important here, so I'd better explain it.
These power laws like

<s(x)s(y)> ~ 1/|x-y|^d

are really only approximate! Actually this is obvious, because the left hand
side can't get bigger than 1, while the right hand side goes to infinity as
|x-y| goes to zero. In reality, the the 2-point function behaves in a very
complicated way when the distance between our two atoms is very small. It's only
when the distance gets *big* that things simplify and the power law becomes a
better and better approximation.

Does this remind you of anything?

It should: this is where the renormalization group comes in! We can imagine
"zooming out" on our crystal, looking at it from ever larger distance scales. As
we do, things simplify: we can forget about individual atoms and approximate the
situation by a field theory defined in the continuum. In fact, we can try to use
one of the field theories that we've been talking about in the previous
sections! Remember, quantum field theory in Euclidean space is just the same as
statistical mechanics. Quantum field theory needs a cutoff, but we've got one:
the distance between atoms in our crystal. So we're all set: we can write down
some Lagrangian and start playing the renormalization group game to see what
happens as we zoom out.

You may be suspicious here: how are we ever going to guess which Lagrangian
corresponds to our original problem involving a crystal of iron? After all, iron
is complicated stuff!

Luckily, it's not so bad. At short distance scales, to get a decent
approximation to our original problem, we may need to start with a really
complicated Lagrangian. However, suppose we do this. Then as we zoom out to
large distance scales, the renormalization group game says that the Lagrangian
will simplify. For example, we've already seen that nonrenormalizable terms in
the Lagrangian become "irrelevant" as we go to large distance scales: the
physical coupling constants in front of them go to zero!

More generally, we shouldn't be at all surprised if our physical coupling
constants approach an infrared fixed point as we zoom out, letting the distance
scale approach infinity. This is exactly what infrared fixed points are all
about! Even better, all sorts of theories with different bare coupling constants
can approach the same infrared fixed point. We say two different theories, or
two different physical systems, are in the same "universality class" if they
approach the same infrared fixed point as we crank up the distance scale.

For example, when we're studying what happens at the Curie temperature, lots
of different ferromagnets lie in the same universality class. Indeed, it turns
out that you can study a lot of them using slight variations of one of the
simplest quantum field theories of all: the phi^4 theory.

There is a lot more to say, and I'm too tired to say most of it, but there's
one thing I *must* tell you, just to wrap up some loose ends. Wilson's real
triumph was to calculate critical exponents like the number d in the power law
for the 2-point function:

<s(x)s(y)> ~ 1/|x-y|^d

How did he do it? Well, Landau already had one way to do this, which gives
just the results you would guess using dimensional analysis. But that method
didn't always give the right answers. To get the right answers, it helps to
realize that n-point functions are closely related to physical coupling
constants. In fact, while I never actually *defined* the physical coupling
constants, they are really just a way of extracting some information about
n-point functions. So if we calculate the "running of coupling constants" using
the renormalization group game, we can work out the critical exponents.

To actually do this, it helps to use something called the "Callan-Symanzik
equation", but I'm not going to explain this - for this, you should probably
read a book on quantum field theory. But don't worry about this too much; be
happy if you feel you get my main point here, which is that 2ND-ORDER PHASE
TRANSITIONS CORRESPOND TO INFRARED FIXED POINTS OF THE RENORMALIZATION GROUP!

For more on the Callan-Symanzik equation, and an application to quantum
gravity, try "week139"
in my series This Week's Finds.

The author feels that this technique of deliberately lying will actually
make it easier for you to learn the ideas. - Donald Knuth

mailto:[email-masked]

© 1999 John Baez

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