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.