The example of the micro-
to macroeconomic transition

is a really good one to get
a sense of how coarse graining works.

So let's take an example here

to give you a sense of what we mean

when we say that we're going
to coarse grain a system.

We're going to simplify it,
we're going to renormalize it,

in the language of physics.

So, let's take just a partial description
of the entire economy that we have today.

So, here's just a sample example.

James got a job; Mary was promoted;
Sally was laid off;

Harold to parental leave; Xavier retired;
Sunshine went back to graduate school.

Well, if you're a macroeconomist,

really all you care about
is that three people lost their jobs,

or rather four people lost their jobs,
and one person got a job,

and so the net change
in employment was -3.

And so what the macroeconomist
has done there,

in the construction of her variables,

is taken a complicated story
and made it a lot simpler.

And in doing that, almost always
the following thing happens,

which is that they confuse or mix up
two entirely different descriptions.

So, if we imagine instead of
James getting a job, it was Sunshine;

instead of Mary
getting promoted, it was Sally;

and Mary got laid off.

Harold, in this case,
didn't take parental leave.

Xavier kept working instead of retiring,

and James got sick and lost his job,

and Nolan went to graduate school.

These are very different descriptions
at the micro-level;

but from the point of view

of the construction
of the macro-level economic variables,

the number of people
who were unemployed or employed

at a certain point in time,

those two descriptions are identical.

And so what coarse graining
does in other words,

is merge states of the world.

We use a lot of different terms

over the course
of this series of lectures.

One term is "equivalence classes".

So you can think these two descriptions

are in the same chronical
equivalence class,

and that equivalence class
contains all the descriptions of the world

in which the net change
in employment was -3.

You can also think of these
as if you're a computer scientist,

as irreversible mappings,

mappings that are are onto,
but not necessarily, one-to-one.

If you're a physicist, you can even
think of them, if you like –

and we'll talk about this a little bit

in the final part of this set of modules
on renormalization –

if you're a physicist,

you can also think of these
as irreversible transformations:

the entropy of the description,
the uncertainty of the description,

has now gone down.

These two different
micro-level descriptions

both map to the same final storing.

The macroeconomist –
here's another way to put it –

the macroeconomist is much more certain
about the world than the microeconomist.

Let's try to get a more intuitive sense
of coarse graining,

and we'll give you a sense
of how coarse graining plays out

in something that we understand very well,

which is the representation
of images for the human mind.

This is a woodcut here from John Tenniel

of one of the episodes
in "Alice in Wonderland".

Actually, it's from the second
of the two books,

"Alice Through the Looking Glass".

You can see Alice here
playing with a ball of yarn

and her kitten Dinah.

What I'd like you to imagine
us doing here

is simplifying this image
in different ways.

What we're going to do in particular
is coarse grain the image

in the following fashion.

So, imagine dividing
this image up into grids,

and then, at each grid point here –

if we were to zoom in to this digitized
version of Tenniel's image,

if you're going to zoom in –

I'm going to zoom in
just a little on Dinah's ear here –

if we were to zoom in,
we'll see a little patch of that image.

And what we're going to say

is for each of these patches,
we're going to represent them now.

Instead of all of the individual details –
the square is white, the square is black –

instead of all the individual details,

we're going to have that patch
just take a vote.

In fact, it's going to be a majority vote,

it's going to be like the electoral
college of images here.

So that image, there it turns out
as majority black pixels,

and so it's going to coarse grain
to a single chunky black pixel.

And we're going to go
grid square by grid square here,

taking all of the complexity
of the image within that ten by ten chunk

and compressing it down
to a single answer:

black or white, 1 or 0.

And if we go right next door
on that image,

here's a different part of Dinah,
different part of Alice's cat,

And what we can see here

is that even though this image patch
at the fine grain level is different,

the two descriptions in the different
parts of the image are different,

our course grading majority vote
will represent them identically.

Another way to put it is:

if Vermont goes Democratic,

it's basically because
everyone in Vermont is democratic.

If Pennsylvania goes Democratic –

there's a lot more non-Democratic
votes in that state,

but because of how we're going
to coarse grain the system,

we're going to say
that Pennsylvania and Vermont

both like the Democratic
candidate equally.

So, what happens when we do that?

Well, here's an example.

This is majority vote coarse graining.

What we can see here

is that on the left-hand side,
we have the original image,

and on the right-hand side
we have something

that kind of looks pretty much
like what Alice sort of looks like,

but we're missing quite a lot.

Here's another case,
this is a different rule.

In this case, here, instead of
taking the majority vote,

I did what's called [decimation],
at least in the literature,

and this is a term that was invented
by the physicist Leo Kadanoff

when he came to do
this kind of coarse graining,

not on pictures
from "Alice in Wonderland",

but on images of atoms
being spin up or spin down,

and we'll talk about that
in a later module

when we come to the Ising model.

What Leo said to do is:

Don't even bother
to average over that grid square,

just take the point
in the top left corner,

and have that define it.

You can see in this case here,
if we use that decimation rule,

that same patch of die now gets mapped,
not to black, but to white.

So what I've introduced you to here
is two ways to simplify an image,

two ways to coarse grain the image,

and, of course, different images
will now coarse grain to the same thing.

I can make many microscopic changes
in Tenniel's original illustration,

and still get back
the same final image of Alice.

Of course, the microscopic changes
that my image is insensitive to,

or that, rather, my coarse graining
of the image is insensitive to,

the microscopic changes will depend upon

the coarse graining algorithm
that I choose.

So if I do the majority vote,

then I'll be able to make some changes
that won't show up,

and if I instead take
the decimation coarse graining rule,

there's a different set of changes
that I can make at the microscopic level

that will leave the system unchanged.