So, one of the most important uses of renormalization theory,
at least in theoretical physics,
is associated with the theory of quantum electrodynamics.
In fact, the person, or the two people,
who first told us how to use renormalization theory
to understand the quantum theory of electromagnetism,
how photons and electrons could interact with each other
in cases where, for example,
electrons could interact, not just with other photons,
but with electron-positron pairs that sort of sprung out of a vacuum –
the people who first helped us understand that
were Murray Gell-Mann, our very own from Santa Fe Institute,
and Frank Low, who, in a paper in the 1950s,
talked about quantum electrodynamics at short distances.
And one of the things they figured out
was how to describe the following phenomenon.
So, you have some electron –
and I'll color the electron blue –
you have some electron in your space,
and when you ask what kind of effect
does that electron have at a distant point,
one of the things that you have to take into account,
at least in quantum electrodynamics,
is that even though this vacuum appears to be empty,
even though you sort of stipulated
that there's nothing between you and the electron,
it so turns out in fact,
that field of the electron can actually sort of tear
positrons, positively-charged antielectrons,
and electrons
out of the vacuum itself.
So, bizarrely enough – right? – you get these.
Even though you think the vacuum is empty,
quantum mechanically it's not,
and so you have this amazing phenomenon,
where these, what are called literally "virtual pairs",
so you can't actually touch them directly,
you can't sort of nip in and pull one out,
or maybe you can if you're fast enough – right? –
But sometimes you haven't put these in by hand,
and yet they exist.
And what that means in fact
is that the influence of the electron on that distant point
is modified and modulated
by all of the stuff that's happening in between.
The closer and closer you get to the electron,
the less and less those effects conspire to change the underlying theory,
but what Gell-Mann and Low noticed –
in fact, everybody was dealing with this problem –
was that the closer you got to the electron,
the more you sort of dove into this virtual cloud,
the stronger and stronger the effects of the electron itself got.
And, in fact, what happened at very small distances
was that the charge of the electron, the effect it had on you,
seemed to diverge and become infinite.
So that was a huge problem.
No one knew how to solve it – right?
And, in the end, they invented a series of mathematical tools
to help people understand how they could solve it.
Now, we don't have time to teach you quantum field theory,
let alone the renormalization problem,
but, beautifully enough, we do have a sort of toy model
for how this renormalization process works
for the kinds of physics that Gell-Mann and Low were doing,
and then later, of course, Feynman,
and then a number of other people who followed in their footsteps.
And in fact what we can do is talk, not about quantum fluctuations,
which were much harder – right? –
you had to talk about virtual worlds
and all sorts of philosophically compelling,
but mathematically difficult things.
In fact, it turns out that we can tell a good part of this story
not in terms of quantum fluctuations,
but in terms of sort of classical thermal fluctuations.
And in fact you've already seen an example of thermal fluctuations
when we introduced the Ising model.
As you remember, we had this coupling parameter, beta, in the Ising model.
The stronger the coupling was,
the more and more the particles like to align with each other.
They wanted their internal states to be the same, right?
So, now, as you decrease the coupling parameter,
or, in physics language, as you increase the temperature,
particles had a greater and greater tolerance
for, instead of both being in the same state like this,
for sometimes them to flip, right?
And you can think of this as a thermal fluctuation.
The minimal energy configuration is like this –
right? – all the particles all pointing up, or all the particles all pointing down,
but instead, if the temperature is high enough,
you sometimes get these fluctuations,
and if the temperature is high enough in the Ising model,
those fluctuations can actually propagate arbitrarily far:
you go through the critical point and then, in fact, you totally decouple,
and now they seem totally uncorrelated – okay?
So, what we're going to do now is tell the story
about the behavior of electrodynamics in the presence of thermal fluctuation,
not in the presence of these sort of strange virtual particles,
but in something that you interact with every day of your life,
although hopefully only in an intermediate fashion,
because otherwise you would die.
And that is the plasma.
So, what is a plasma?
A plasma is just a gas
where the positive nuclei have been separated
from the negatively charged electrons.
A plasma, I certainly encountered these
when I was a graduate student in astrophysics.
In fact, for example, in the case of the solar wind,
you have this gas that's coming off the Sun,
it's being bombarded by photons,
the Sun is extremely hot,
and occasionally those photons will rip the electron off an atom,
and that electron will fly off.
But the gas is so diffuse, in fact,
that it can never really find its old nuclei again – right?
It can never sort of recondense back together,
and so the solar wind is an example of a very diffuse plasma.
On earth we build plasmas all the time.
One of the most extreme cases that we do is when we try to build the tokamak,
in other words, when we try to get nuclear fusion going.
To do that, we'll take something like helium or hydrogen,
we'll heat it up as hot as we can,
separate out the positively charged nuclei
from the electrons,
and then do some magic
and try to get these guys to stick together,
and that's a whole another story,
and it's always 20 years in the future, that solution.
But plasmas exist in all sorts of places in the real world,
including, for example, in the fluorescent lights.
So in that tube you have a very rarefied gas,
and what happens is the electric field produced on either end of that tube,
enables the gas to separate out
into its constituent positively and negatively charged particles.
Okay?
So, what we're going to do,
our plasma is going to be our virtual background of particles
that occurred in quantum electrodynamics,
except now it's much easier,
we aren't having to think about
how these could be ripped apart from nothing.
Instead, what we're going to assume
is that we have some gas, let's say in a box,
and the gas is separated into its positive and negatively charged parts.