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PROFESSOR: Hey.
00:00:21.625 --> 00:00:22.125
Let's start.
00:00:24.950 --> 00:00:31.620
So a few weeks ago we started
with writing a partition
00:00:31.620 --> 00:00:34.510
function for a
statistical field that
00:00:34.510 --> 00:00:41.170
was going to capture behavior of
a variety of systems undergoing
00:00:41.170 --> 00:00:44.380
critical phase transitions.
00:00:44.380 --> 00:00:49.930
And this was obtained by
integrating over configurations
00:00:49.930 --> 00:01:00.580
of this statistical field
a rate that we wrote
00:01:00.580 --> 00:01:03.760
on the basis of a
form of locality.
00:01:06.520 --> 00:01:10.330
And terms that were
consistent with that
00:01:10.330 --> 00:01:15.596
were of the form m
squared, m to the fourth.
00:01:15.596 --> 00:01:17.192
Let's say m to the sixth.
00:01:19.790 --> 00:01:22.280
Various types of
gradient types of terms.
00:01:32.680 --> 00:01:36.927
And in principle, allowing for
a symmetry-breaking field that
00:01:36.927 --> 00:01:39.659
was more in the form of h dot 1.
00:01:46.281 --> 00:01:50.020
And again, we always
emphasized that in writing
00:01:50.020 --> 00:01:53.470
these statistical fields,
we have to do averaging.
00:01:53.470 --> 00:01:58.590
We have to get rid of a lot of
short wavelength fluctuations.
00:01:58.590 --> 00:02:01.880
And essentially, the future
m of x, although I write it
00:02:01.880 --> 00:02:06.870
as a continuum, has an
implicit short scale
00:02:06.870 --> 00:02:10.870
below which it
does not fluctuate.
00:02:10.870 --> 00:02:15.290
OK, so we tried to evaluate
this by certain point,
00:02:15.290 --> 00:02:17.260
and we didn't succeed.
00:02:17.260 --> 00:02:22.430
So we went phenomenologically
and tried to describe things
00:02:22.430 --> 00:02:24.490
on the basis of scaling theory.
00:02:24.490 --> 00:02:27.430
Ultimately, this
renormalization group
00:02:27.430 --> 00:02:33.770
procedure that we would like to
apply to something like this.
00:02:33.770 --> 00:02:36.520
Now, there is a
part of this that
00:02:36.520 --> 00:02:39.840
is actually pretty
easy to solve.
00:02:39.840 --> 00:02:43.710
And that's when
we ignore anything
00:02:43.710 --> 00:02:47.670
that is higher than
second order in m.
00:02:47.670 --> 00:02:50.840
Because once we ignore
them, we have essentially
00:02:50.840 --> 00:02:53.580
a generalized Gaussian integral.
00:02:53.580 --> 00:02:56.120
We can do Gaussian integrals.
00:02:56.120 --> 00:03:00.690
So what we are going to
do is, in this lecture,
00:03:00.690 --> 00:03:05.150
focusing on understanding
a lot about the behavior
00:03:05.150 --> 00:03:07.310
of the Gaussian
version of the theory.
00:03:07.310 --> 00:03:09.400
Which is certainly a
diminished version,
00:03:09.400 --> 00:03:13.740
because it doesn't have
lots of essential things.
00:03:13.740 --> 00:03:16.820
And then gradually putting
back all of those things
00:03:16.820 --> 00:03:20.390
that we have not considered
at the Gaussian level.
00:03:20.390 --> 00:03:23.320
In particular, we'll
try to do with them
00:03:23.320 --> 00:03:26.980
with a version of a
perturbation theory.
00:03:26.980 --> 00:03:30.120
We'll see that standard
perturbation theory has
00:03:30.120 --> 00:03:32.640
some limitations that
we will eventually
00:03:32.640 --> 00:03:38.280
resolve by using this
renormalization procedure.
00:03:38.280 --> 00:03:39.410
OK.
00:03:39.410 --> 00:03:41.750
So what happens if I do that?
00:03:41.750 --> 00:03:45.690
Why do I say that that
theory is now solve-able?
00:03:45.690 --> 00:03:48.380
And the key to
that is, of course,
00:03:48.380 --> 00:03:49.960
to go into Fourier
representation.
00:03:54.420 --> 00:03:57.890
Which, because the
theory that I wrote down
00:03:57.890 --> 00:04:01.450
has this inherent
translational of symmetry,
00:04:01.450 --> 00:04:06.420
Fourier representation
decouples the various m's
00:04:06.420 --> 00:04:09.530
that are currently connected
to their neighborhood
00:04:09.530 --> 00:04:12.430
by these gradients
and high orders.
00:04:12.430 --> 00:04:18.480
So let's introduce
a m of q, which
00:04:18.480 --> 00:04:21.660
is the Fourier
transform of m of x.
00:04:29.388 --> 00:04:32.770
Let's see. m of x.
00:04:32.770 --> 00:04:36.720
And these are all vectors.
00:04:36.720 --> 00:04:42.470
And I should really use a
different symbol, such as m
00:04:42.470 --> 00:04:46.660
[INAUDIBLE], to
indicate the Fourier
00:04:46.660 --> 00:04:50.120
components of this field m of x.
00:04:50.120 --> 00:04:53.480
But since in the context
of renormalization group
00:04:53.480 --> 00:04:57.930
we had defined a coarse grained
field that was in tilde,
00:04:57.930 --> 00:04:59.690
I don't want to do that.
00:04:59.690 --> 00:05:02.740
I hope that the
argument of the function
00:05:02.740 --> 00:05:05.940
is sufficient indicator of
whether we are in real space
00:05:05.940 --> 00:05:07.660
or in momentum space.
00:05:07.660 --> 00:05:10.870
Initially, I'll try
to put a tail on the m
00:05:10.870 --> 00:05:13.490
to indicate that I'm
doing Fourier space,
00:05:13.490 --> 00:05:16.450
but I suspect that very soon
I'll forget about the tail.
00:05:16.450 --> 00:05:20.320
So keep that in mind.
00:05:20.320 --> 00:05:25.440
So if I-- oops.
00:05:25.440 --> 00:05:26.970
OK.
00:05:26.970 --> 00:05:28.580
m of q.
00:05:28.580 --> 00:05:34.980
So if I go back and
write what this m of x
00:05:34.980 --> 00:05:49.230
is, it is an integral over 2,
2 pi to the, d to the minus iq
00:05:49.230 --> 00:05:54.930
dot x with m of q.
00:05:59.610 --> 00:06:03.190
Now, I also want
to at some stage,
00:06:03.190 --> 00:06:09.650
since it would be cleaner
to have this rate in terms
00:06:09.650 --> 00:06:15.860
of a product of q's, remind you
that this could have obtained,
00:06:15.860 --> 00:06:19.690
if I hadn't gone to
the continuum version--
00:06:19.690 --> 00:06:26.390
if I had a finite
system-- to a sum over q.
00:06:26.390 --> 00:06:28.720
And the sum over q
would be basically
00:06:28.720 --> 00:06:32.070
things that are separated
q values by multiples of 1
00:06:32.070 --> 00:06:33.960
over the size of the system.
00:06:33.960 --> 00:06:39.170
And e to the minus iq dot x.
00:06:39.170 --> 00:06:42.380
This m with the cues
that are now discretized.
00:06:45.400 --> 00:06:48.180
But let's remember that
the density of state
00:06:48.180 --> 00:06:52.230
has a factor of 1 over v.
So if I use this definition,
00:06:52.230 --> 00:06:55.825
I really should put
the 1 over v here when
00:06:55.825 --> 00:06:58.170
I go to the discrete version.
00:06:58.170 --> 00:07:00.890
And I emphasize this
because previously, we
00:07:00.890 --> 00:07:04.950
had done Fourier
decomposition where
00:07:04.950 --> 00:07:08.990
I had used the square root
of v as a normalization.
00:07:08.990 --> 00:07:15.585
It really doesn't matter
which normalization
00:07:15.585 --> 00:07:19.200
you use at the end as long
as you are consistent.
00:07:19.200 --> 00:07:23.330
We'll see the advantages of
this normalization shortly.
00:07:23.330 --> 00:07:26.735
AUDIENCE: Is there
any particular reason
00:07:26.735 --> 00:07:28.735
for using the different
sign in the exponential?
00:07:28.735 --> 00:07:30.410
PROFESSOR: Actually, no.
00:07:30.410 --> 00:07:35.140
I'm not sure even whether I
used iqx here or minus iqx here.
00:07:35.140 --> 00:07:37.870
It's just a matter
of which one you
00:07:37.870 --> 00:07:40.860
want to stick with consistently.
00:07:40.860 --> 00:07:45.460
At the end of the day, the phase
will not be that important.
00:07:45.460 --> 00:07:48.570
So even if we mistake
one form or the other,
00:07:48.570 --> 00:07:49.861
it doesn't make any difference.
00:07:55.160 --> 00:08:00.580
So if I do that, then again,
to sort of be more precise,
00:08:00.580 --> 00:08:06.300
I have to think about
what to do with gradients.
00:08:06.300 --> 00:08:14.300
Gradients, I can imagine, are
the limit of something like n
00:08:14.300 --> 00:08:21.110
at x plus A minus n
at x divided by A.
00:08:21.110 --> 00:08:24.660
If this is a gradient
in the x direction.
00:08:24.660 --> 00:08:28.840
And I have to take the
limit as A goes to 0.
00:08:28.840 --> 00:08:34.320
So when I'm thinking about this
kind of functional integral,
00:08:34.320 --> 00:08:39.940
keeping in mind that I have
a shortest landscape, maybe
00:08:39.940 --> 00:08:42.840
one way to do it is to
imagine that I discretize
00:08:42.840 --> 00:08:47.360
my system over here
into spacing of size A.
00:08:47.360 --> 00:08:52.162
And then I have a
variable on each size,
00:08:52.162 --> 00:08:57.200
and then I integrate
every place,
00:08:57.200 --> 00:09:01.530
subject this replacement
for the gradient.
00:09:01.530 --> 00:09:05.630
Again, what you do precisely
does not matter here.
00:09:05.630 --> 00:09:08.420
If you remember in
the first lecture
00:09:08.420 --> 00:09:12.760
when we were thinking
about the dl lattice system
00:09:12.760 --> 00:09:16.350
and then using these kinds
of coupling between springs
00:09:16.350 --> 00:09:19.510
that they're connecting nearest
neighbors, what ended up
00:09:19.510 --> 00:09:22.920
by using this was that
when I Fourier transformed,
00:09:22.920 --> 00:09:25.080
I had things like cosine.
00:09:25.080 --> 00:09:29.650
And then when I expanded the
cosine close to q, close to 0,
00:09:29.650 --> 00:09:32.080
I generated a series
that had q squared,
00:09:32.080 --> 00:09:34.210
q to the fourth, et cetera.
00:09:34.210 --> 00:09:40.720
So essentially, any
discretized version corresponds
00:09:40.720 --> 00:09:44.845
to an expansion like this with
sufficient [INAUDIBLE] powers
00:09:44.845 --> 00:09:47.410
of q in both.
00:09:47.410 --> 00:09:53.310
So at the end of the day, when
you go through this process,
00:09:53.310 --> 00:09:59.740
you find that you can write
the partition function after
00:09:59.740 --> 00:10:05.490
the change of variables to m
of x to m of q to doing a whole
00:10:05.490 --> 00:10:10.250
bunch of integrals
over different q's.
00:10:10.250 --> 00:10:15.670
So, essentially you would
have-- actually, maybe I
00:10:15.670 --> 00:10:23.460
will explicitly put the
product over q outside
00:10:23.460 --> 00:10:27.060
to emphasize that essentially,
for each q I would
00:10:27.060 --> 00:10:30.370
have to do
independent integrals.
00:10:30.370 --> 00:10:33.990
Of course, for
each q mode I have,
00:10:33.990 --> 00:10:39.840
since I've gone to this
representation of a vector that
00:10:39.840 --> 00:10:46.160
is n-dimensional, I have to do
n integrals on n tilde of q.
00:10:48.750 --> 00:10:51.100
On-- m will be the tail of q.
00:10:53.720 --> 00:11:03.600
And if I had chosen
the square root
00:11:03.600 --> 00:11:08.840
of V type of
normalization, the Jacobian
00:11:08.840 --> 00:11:11.910
of the transformation from
here to here would have been 1.
00:11:11.910 --> 00:11:15.460
Because it's kind of a
symmetric way of writing things.
00:11:15.460 --> 00:11:19.120
Because I chose this
way of doing things,
00:11:19.120 --> 00:11:24.740
I will have a factor of V to
the n over 2 in the denominator
00:11:24.740 --> 00:11:26.440
here.
00:11:26.440 --> 00:11:28.940
But again, it's
just being pedantic,
00:11:28.940 --> 00:11:30.680
because at the
end of the day, we
00:11:30.680 --> 00:11:32.690
don't care about these factors.
00:11:32.690 --> 00:11:34.850
We are interested in
things like this singular
00:11:34.850 --> 00:11:39.120
part of the partition
function as it
00:11:39.120 --> 00:11:41.470
depends on these coordinates.
00:11:41.470 --> 00:11:45.230
This really just gives
you an overall constant.
00:11:45.230 --> 00:11:47.810
Of course, how many of
these constants you have
00:11:47.810 --> 00:11:55.050
would depend basically how you
have discretized the problem.
00:11:55.050 --> 00:11:59.800
But it is a constant independent
of tnh, not something
00:11:59.800 --> 00:12:01.110
that we have to worry about.
00:12:04.630 --> 00:12:10.140
Now what happens to
these Gaussian factors?
00:12:10.140 --> 00:12:13.930
Essentially, I have put
the product over q outside.
00:12:13.930 --> 00:12:17.650
So when I transform this
integral over xm squared
00:12:17.650 --> 00:12:22.790
goes over to an
integral over q, m of q
00:12:22.790 --> 00:12:25.920
squared, which then I
can write as a product
00:12:25.920 --> 00:12:28.500
over those contributions.
00:12:28.500 --> 00:12:33.770
And what you will get
is t plus, from here,
00:12:33.770 --> 00:12:38.020
you will get a Kq
squared, put in Lq
00:12:38.020 --> 00:12:41.800
to the fourth and all
kinds of order terms
00:12:41.800 --> 00:12:44.840
that I have included.
00:12:44.840 --> 00:12:52.160
Multiplying this m component
vector m of q squared.
00:12:52.160 --> 00:12:55.560
Again, reminding you
this means m of q,
00:12:55.560 --> 00:12:59.840
m of minus q, which is the
same thing as m star of q,
00:12:59.840 --> 00:13:03.700
if you go through these
procedures over here.
00:13:03.700 --> 00:13:05.670
There is 2.
00:13:05.670 --> 00:13:07.838
And this factor
of the v actually
00:13:07.838 --> 00:13:10.780
will come up over here.
00:13:10.780 --> 00:13:16.210
So previously, I had used the
normalization square root of V,
00:13:16.210 --> 00:13:19.435
and I didn't have this
factor of 1 over V.
00:13:19.435 --> 00:13:23.560
Now I have put if there,
I will have that factor.
00:13:23.560 --> 00:13:24.060
Yes?
00:13:24.060 --> 00:13:29.097
AUDIENCE: m minus q is star q
only if it is the real field,
00:13:29.097 --> 00:13:29.597
right?
00:13:29.597 --> 00:13:30.832
If m is real.
00:13:30.832 --> 00:13:31.860
PROFESSOR: Yes.
00:13:31.860 --> 00:13:34.874
And we are dealing with
the field m of q of this.
00:13:34.874 --> 00:13:36.826
AUDIENCE: And in the
case of superfluidity?
00:13:36.826 --> 00:13:38.492
PROFESSOR: In the
case of superfluidity?
00:13:41.670 --> 00:13:43.868
So let's see.
00:13:43.868 --> 00:13:51.080
So we would have a psi of q
integral d dx into the i q dot
00:13:51.080 --> 00:13:56.540
x psi of x.
00:13:56.540 --> 00:13:59.626
If I Fourier
transform this, I will
00:13:59.626 --> 00:14:04.919
get a psi star of q
integral into the x
00:14:04.919 --> 00:14:10.540
into the minus [INAUDIBLE]
x psi star of x.
00:14:10.540 --> 00:14:15.170
So what you are saying is that
in the case where psi of x
00:14:15.170 --> 00:14:25.420
is a complex number-- I
have psi1 plus ipsi2-- here
00:14:25.420 --> 00:14:26.700
I would have psi1 minus ipsi2.
00:14:29.310 --> 00:14:35.230
So here I would have
to make it a statement
00:14:35.230 --> 00:14:39.040
that the real part
and the imaginary part
00:14:39.040 --> 00:14:43.760
come when you Fourier transform
with an additional minus.
00:14:43.760 --> 00:14:45.530
But let's remember
that something
00:14:45.530 --> 00:14:47.786
like this that we
are interested is
00:14:47.786 --> 00:14:52.510
psi1 squared plus psi2 squared.
00:14:52.510 --> 00:14:55.664
So ultimately that minus sign
did not make any difference.
00:15:00.922 --> 00:15:05.660
But it's good to sort of
think of all of these issues.
00:15:05.660 --> 00:15:14.406
And in particular, we are
used to thinking of Gaussians,
00:15:14.406 --> 00:15:19.850
where I would have a scalar and
then I would have x squared.
00:15:19.850 --> 00:15:23.130
When I have this complex
number and I have psi of q,
00:15:23.130 --> 00:15:26.687
psi of minus q, then
I have a real part
00:15:26.687 --> 00:15:28.410
squared plus an
imaginary part squared.
00:15:31.380 --> 00:15:35.520
And you have to think
about whether or not
00:15:35.520 --> 00:15:39.762
you have changed the number
of degrees of freedom.
00:15:39.762 --> 00:15:46.100
If you basically integrate over
all q's, you may have problems.
00:15:46.100 --> 00:15:51.090
You may have at some point to
think about seeing psi of q
00:15:51.090 --> 00:15:54.970
and psi of minus q star
are the same thing.
00:15:54.970 --> 00:15:58.780
Maybe you have to integrate
over just the positive values.
00:15:58.780 --> 00:16:02.800
But then at each q you will
have two different variables,
00:16:02.800 --> 00:16:05.770
which is the real part
and the imaginary part.
00:16:05.770 --> 00:16:09.805
So you have to think about all
of those doublings and halvings
00:16:09.805 --> 00:16:13.100
that are involved
in this statement.
00:16:13.100 --> 00:16:16.280
And in the notes,
I have the writeup
00:16:16.280 --> 00:16:19.860
about that that you
go and precisely check
00:16:19.860 --> 00:16:22.530
where the factors of
one half and two go.
00:16:22.530 --> 00:16:25.063
But ultimately, it
looks as if you're
00:16:25.063 --> 00:16:28.170
dealing with a simple
scalar quantity.
00:16:28.170 --> 00:16:31.840
So I did not give you
that detail explicitly,
00:16:31.840 --> 00:16:35.250
but you can go and check
it in the important issue.
00:16:41.140 --> 00:16:44.410
The other term that we have.
00:16:44.410 --> 00:16:47.750
One advantage of
this normalization
00:16:47.750 --> 00:16:51.345
is that h multiplies
the integral of m
00:16:51.345 --> 00:16:57.920
of x, which is clearly this m
with a tail for q equals to 0.
00:16:57.920 --> 00:17:05.069
So that's [INAUDIBLE] mh
dotted by this m [INAUDIBLE].
00:17:10.040 --> 00:17:10.600
Yes?
00:17:10.600 --> 00:17:14.659
AUDIENCE: This is
assuming a uniform field?
00:17:14.659 --> 00:17:17.440
PROFESSOR: Yes, that's right.
00:17:17.440 --> 00:17:20.500
So we are thinking about
the physics problem,
00:17:20.500 --> 00:17:23.258
but we added the uniform field.
00:17:23.258 --> 00:17:25.862
So if you are for
some physical reason
00:17:25.862 --> 00:17:28.792
interested in a
position where you
00:17:28.792 --> 00:17:32.102
feel you can modify that,
then this would be h of q,
00:17:32.102 --> 00:17:33.131
m of minus q.
00:17:40.140 --> 00:17:47.670
Actually, one reason ultimately
to choose this normalization is
00:17:47.670 --> 00:17:54.020
that clearly what appears
here is a sum of q.
00:17:54.020 --> 00:18:00.200
If I go over to my integral over
q, then the factor of 1 over V
00:18:00.200 --> 00:18:01.110
disappears.
00:18:01.110 --> 00:18:05.560
So that's one reason-- since
mostly after this, going
00:18:05.560 --> 00:18:08.670
through the details we'll be
dealing with the continuum
00:18:08.670 --> 00:18:13.130
version-- I prefer
this normalization.
00:18:13.130 --> 00:18:20.520
And we can now do the
Gaussian integrals.
00:18:20.520 --> 00:18:24.880
Basically, there's an
overall factor of 1
00:18:24.880 --> 00:18:30.070
over V to the n over
2 for each q mode.
00:18:30.070 --> 00:18:36.980
Then each one of these
Gaussian integrals
00:18:36.980 --> 00:18:44.370
will leave me a factor of
root 2 pi times the variant.
00:18:44.370 --> 00:18:48.710
So I will get 2 pi.
00:18:48.710 --> 00:18:53.802
The variance is V
divided by t plus k
00:18:53.802 --> 00:18:57.246
q squared plus lq to the
fourth, and so forth.
00:19:00.200 --> 00:19:04.750
Square root, but there
are n components,
00:19:04.750 --> 00:19:08.400
so I will get
something like this.
00:19:08.400 --> 00:19:16.080
And then the term that
corresponds to q equals to 0
00:19:16.080 --> 00:19:19.725
does not have any of this part.
00:19:19.725 --> 00:19:21.840
So it will give a
contribution even
00:19:21.840 --> 00:19:25.370
for q equals to 0
that is like this.
00:19:25.370 --> 00:19:30.880
But you have a term that shifts
the center of integration
00:19:30.880 --> 00:19:35.890
from m equals to 0 because
of the presence of the field.
00:19:35.890 --> 00:19:40.670
So you will get a term that is
exponential of essentially--
00:19:40.670 --> 00:19:48.980
completing the square-- will
give you V divided by 2t times
00:19:48.980 --> 00:19:49.580
h squared.
00:19:57.665 --> 00:20:01.120
Now, clearly the thing
that I'm interested
00:20:01.120 --> 00:20:09.050
is log of Z as a
function of t and h.
00:20:09.050 --> 00:20:12.700
I'm interested in
t and h dependents.
00:20:12.700 --> 00:20:14.672
So there is a bunch
of things that
00:20:14.672 --> 00:20:18.608
are constants that
I don't really care.
00:20:18.608 --> 00:20:24.890
And then there is a, from
here, minus 1/2, actually
00:20:24.890 --> 00:20:33.190
minus n 1/2 sum over q log of t
plus k q squared and so forth.
00:20:33.190 --> 00:20:37.717
And plus here, I have
V a squared over 2t.
00:20:42.690 --> 00:20:45.100
So I can define
something that's like
00:20:45.100 --> 00:20:51.425
if the energy from log of
Z divided by the volume.
00:20:54.360 --> 00:20:56.870
And you can see
that once I replace
00:20:56.870 --> 00:21:00.445
this sum of a q
with an integral,
00:21:00.445 --> 00:21:04.370
I will get a factor of volume
that I can disregard then.
00:21:04.370 --> 00:21:06.360
So there's some other constant.
00:21:06.360 --> 00:21:15.150
And then I have plus n over 2
integral over q divided by q pi
00:21:15.150 --> 00:21:22.190
to the d log of q plus k
q squared, and so forth.
00:21:22.190 --> 00:21:25.212
Minus V k squared divided by 2t.
00:21:33.970 --> 00:21:36.770
Now, again, the
question is what's
00:21:36.770 --> 00:21:41.740
the range of q's that
I have to integrate,
00:21:41.740 --> 00:21:48.178
given that I'm making things
that are coarse grained.
00:21:48.178 --> 00:21:51.890
Now, if I were to really
discretize my system
00:21:51.890 --> 00:21:55.020
and, say, put it on
q and you plot this,
00:21:55.020 --> 00:21:59.380
then the allowed values
of q would leave on
00:21:59.380 --> 00:22:01.660
the [INAUDIBLE] zone.
00:22:01.660 --> 00:22:05.370
[INAUDIBLE] zone, say, in
the different directions in q
00:22:05.370 --> 00:22:08.830
would be something
like the q that
00:22:08.830 --> 00:22:14.050
would be centered
around pi over a.
00:22:14.050 --> 00:22:17.590
But it would be centered
at 0, but then you
00:22:17.590 --> 00:22:19.808
would have pi plus pi over a.
00:22:19.808 --> 00:22:21.272
Yes?
00:22:21.272 --> 00:22:23.712
AUDIENCE: The d would
disappear, right?
00:22:23.712 --> 00:22:26.923
PROFESSOR: The d would disappear
because I divided by it.
00:22:31.640 --> 00:22:37.970
So in principle, if I had
done the discretization
00:22:37.970 --> 00:22:45.030
to a cube and plot this, I would
have been integrating over q
00:22:45.030 --> 00:22:49.840
that this would find
to a cube like this.
00:22:49.840 --> 00:22:51.670
But maybe I chose
some other lattice
00:22:51.670 --> 00:22:55.900
like a diamond
lattice, et cetera.
00:22:55.900 --> 00:22:58.862
Then the shape of this
thing would change.
00:22:58.862 --> 00:23:01.495
But what's the meaning of doing
the whole thing on a lattice
00:23:01.495 --> 00:23:03.590
anyway?
00:23:03.590 --> 00:23:06.200
The thing that I want
to do is to make sure
00:23:06.200 --> 00:23:10.400
that I have done some
averaging in order to remove
00:23:10.400 --> 00:23:13.260
short wavelength fluctuations.
00:23:13.260 --> 00:23:19.460
So a much more natural way to
do that averaging and removing
00:23:19.460 --> 00:23:22.550
short wavelength
operations is to say
00:23:22.550 --> 00:23:31.770
that my field has only
Fourier components that
00:23:31.770 --> 00:23:36.706
are from 0 to some maximum
value of lambda, which
00:23:36.706 --> 00:23:40.080
is the inverse of some radiant.
00:23:40.080 --> 00:23:43.400
And if you are worried about
the difference in integration
00:23:43.400 --> 00:23:46.470
between doing things on
this nice mirror that
00:23:46.470 --> 00:23:50.930
has nice symmetry and
maybe doing it on a cube,
00:23:50.930 --> 00:23:53.292
then the difference
is essentially
00:23:53.292 --> 00:23:59.370
the bit of integration that
you would have to do over here.
00:23:59.370 --> 00:24:02.620
But the function that
you are integrating
00:24:02.620 --> 00:24:07.930
his no singularities
for large values of q.
00:24:07.930 --> 00:24:10.020
You are interested
in the singularities
00:24:10.020 --> 00:24:13.480
of the function
when t goes to 0.
00:24:13.480 --> 00:24:15.520
And then the log
has singularities
00:24:15.520 --> 00:24:18.470
when its argument goes to 0.
00:24:18.470 --> 00:24:20.740
So I should be interested,
as far as singularities
00:24:20.740 --> 00:24:27.120
are concerned, only in the
vicinity of this point anyway.
00:24:27.120 --> 00:24:30.870
What I do out there,
whether I replace the sphere
00:24:30.870 --> 00:24:33.942
with the cube or
et cetera, will add
00:24:33.942 --> 00:24:36.616
some other non-singular
term over here,
00:24:36.616 --> 00:24:37.699
which I don't really care.
00:24:41.590 --> 00:24:43.950
Actually, if I do that,
this non-singular term
00:24:43.950 --> 00:24:46.840
here could be actually
functions of t.
00:24:46.840 --> 00:24:50.110
But they would be very perfect
and regular functions of t.
00:24:50.110 --> 00:24:52.070
Like constant plus
alpha t, plus pheta q
00:24:52.070 --> 00:24:57.190
squared, et cetera, that
have no singularities.
00:24:57.190 --> 00:25:01.730
So if I'm interested
in singularities,
00:25:01.730 --> 00:25:03.342
I am going to be
focused on that.
00:25:06.000 --> 00:25:11.900
Now actually, we
encountered this integral
00:25:11.900 --> 00:25:16.745
before when we were
looking at corrections
00:25:16.745 --> 00:25:19.756
to the saddle-point
approximation.
00:25:19.756 --> 00:25:23.830
And if you remember what
we did then was to take,
00:25:23.830 --> 00:25:27.100
let's say, C of d
of h across 0 while
00:25:27.100 --> 00:25:33.720
taking two derivatives of this
free energy with respect to t.
00:25:33.720 --> 00:25:38.056
And then we ended
up with an integral.
00:25:38.056 --> 00:25:40.900
There's a minus
sign here over d.
00:25:40.900 --> 00:25:45.980
n over 2 integral
dt 2 pi squared.
00:25:45.980 --> 00:25:48.380
2 pi to the d.
00:25:48.380 --> 00:25:51.040
Taking two derivatives
of the log.
00:25:51.040 --> 00:25:53.610
The first derivative will
give me 1 over the argument.
00:25:53.610 --> 00:25:56.910
The second derivative will give
me 1 over the argument squared.
00:25:56.910 --> 00:25:58.878
One side take care
of the minus sign.
00:26:18.558 --> 00:26:24.020
Now, I think this is
a kind of integral,
00:26:24.020 --> 00:26:30.335
after I have focused
on the singular part,
00:26:30.335 --> 00:26:34.736
that I can replace when
integrating over a sphere.
00:26:39.640 --> 00:26:43.230
Now, when I integrate
over a sphere,
00:26:43.230 --> 00:26:47.740
I may be concerned about what's
going on at small values.
00:26:47.740 --> 00:26:51.350
At q, at small values of
q, as long as t is around,
00:26:51.350 --> 00:26:53.120
I have no problem.
00:26:53.120 --> 00:26:55.380
When t goes to 0, I
will have to worry
00:26:55.380 --> 00:26:59.080
about the singularity
that comes from 1 over k,
00:26:59.080 --> 00:27:00.690
2 squared, et cetera.
00:27:00.690 --> 00:27:04.510
So that's really the singularity
that I'm interested in.
00:27:04.510 --> 00:27:07.150
Exactly what happens
at large q, I'm
00:27:07.150 --> 00:27:11.360
not really all
that interested in.
00:27:11.360 --> 00:27:20.050
And in particular, what I can
do is I can rescale things.
00:27:20.050 --> 00:27:27.730
I can call q squared
over t to the x squared.
00:27:27.730 --> 00:27:35.060
So I can essentially make
that change over there.
00:27:35.060 --> 00:27:40.890
So that whenever I
see a factor of q,
00:27:40.890 --> 00:27:44.330
I replace it with t
over k to the 1/2 x.
00:27:49.940 --> 00:27:51.417
What happens here?
00:27:51.417 --> 00:27:54.970
I have, first of all, n over 2.
00:27:54.970 --> 00:27:57.768
I have 1 over 2 pi to the d.
00:28:01.680 --> 00:28:09.340
Writing this in terms
of spherical symmetry,
00:28:09.340 --> 00:28:13.670
I will have the solid
angle d dimensions.
00:28:13.670 --> 00:28:18.510
And then I would have
q to d minus 1 q.
00:28:18.510 --> 00:28:23.530
Every time I put a factor of
q, I can replace it with this.
00:28:23.530 --> 00:28:30.130
So I would have a t over
k with a power of q/2.
00:28:30.130 --> 00:28:34.960
And then I have my integral
that becomes the x,
00:28:34.960 --> 00:28:41.230
x to the d minus 1, 1 plus
x squared plus potentially
00:28:41.230 --> 00:28:43.130
higher order things like this.
00:28:49.790 --> 00:28:56.680
Now, the upper cut-off
for x is in fact
00:28:56.680 --> 00:29:03.430
square root k over
t times lambda.
00:29:03.430 --> 00:29:10.910
And we are interested in the
limit of when t goes to 0.
00:29:10.910 --> 00:29:17.100
So that upper limit is
essentially going to infinity.
00:29:17.100 --> 00:29:20.630
Now, whether or
not this integral,
00:29:20.630 --> 00:29:24.140
if I learn to ignore
higher order terms
00:29:24.140 --> 00:29:28.890
and focus on the first
term, exists really
00:29:28.890 --> 00:29:36.950
depends on whether d is larger,
d minus 1 plus 1 d minus 4
00:29:36.950 --> 00:29:40.050
is positive or negative.
00:29:40.050 --> 00:29:44.810
And in particular,
if I learn to get rid
00:29:44.810 --> 00:29:47.830
of all those higher order terms.
00:29:47.830 --> 00:29:51.170
And basically, the
argument for that
00:29:51.170 --> 00:29:54.370
is the things that would go
with x to the fourth, et cetera,
00:29:54.370 --> 00:29:57.090
if we carry additional
factors of t--
00:29:57.090 --> 00:30:01.598
and hopefully getting rid
of them as to go to 0--
00:30:01.598 --> 00:30:03.970
will give me an
integral like this.
00:30:03.970 --> 00:30:08.562
This will exist only
if I am in dimensions
00:30:08.562 --> 00:30:11.540
d that is less than 4.
00:30:11.540 --> 00:30:12.175
Yes?
00:30:12.175 --> 00:30:13.925
AUDIENCE: Are you
missing the factors of t
00:30:13.925 --> 00:30:15.575
over t that comes
with the denominator?
00:30:18.796 --> 00:30:20.834
PROFESSOR: Yes.
00:30:20.834 --> 00:30:23.738
There is a factor
of 1 over t here.
00:30:29.550 --> 00:30:33.500
So I have to put
out the factor of t.
00:30:33.500 --> 00:30:37.756
Write this as 1 plus k
over t plus the element
00:30:37.756 --> 00:30:39.492
of t, et cetera.
00:30:39.492 --> 00:30:41.972
So there is a
factor of 1 over t.
00:30:41.972 --> 00:30:43.460
AUDIENCE: t squared.
00:30:43.460 --> 00:30:45.196
PROFESSOR: And
that's a factor of t
00:30:45.196 --> 00:30:46.932
squared, because
that's two powers.
00:30:56.870 --> 00:31:02.385
So if I'm in dimensions d
less than 4, what I can write
00:31:02.385 --> 00:31:09.530
is that this c singular,
this as t goes to 0.
00:31:09.530 --> 00:31:13.550
The leading behavior,
this goes to the constant.
00:31:13.550 --> 00:31:17.347
So as we discussed, after all
of the mistakes that I made,
00:31:17.347 --> 00:31:22.204
there will be some overall
coefficient A. The power of t
00:31:22.204 --> 00:31:25.400
will be d over 2 minus 2.
00:31:25.400 --> 00:31:27.700
d over 2 came from
the integrations.
00:31:27.700 --> 00:31:32.310
1 over t squared came
from the denominator.
00:31:32.310 --> 00:31:36.405
And then if I were to expand
all of these other terms
00:31:36.405 --> 00:31:39.590
that we've ignored,
higher powers of-- here
00:31:39.590 --> 00:31:43.660
I will get various series
that will correct this.
00:31:43.660 --> 00:31:48.840
But the leading key dependents
in dimensions less than 4
00:31:48.840 --> 00:31:51.032
is this thing that we
had seen previously.
00:31:54.476 --> 00:31:58.340
Now I can take this, and
you see that in dimensions
00:31:58.340 --> 00:32:02.294
d less than 4,
this is a singular
00:32:02.294 --> 00:32:04.130
term that is diversion.
00:32:04.130 --> 00:32:08.100
If I were to say
what kind of thing
00:32:08.100 --> 00:32:12.330
was of the energy that
gave result to this?
00:32:12.330 --> 00:32:16.480
Then it would say that
if the energy must
00:32:16.480 --> 00:32:19.737
have had some other constant
that was proportionate of the t
00:32:19.737 --> 00:32:25.500
to the d over 2, that when
I put two derivatives,
00:32:25.500 --> 00:32:27.920
I got something like this.
00:32:27.920 --> 00:32:29.790
Of course, if the
energy could also
00:32:29.790 --> 00:32:33.940
have had a term that was linear
in t, I wouldn't have seen it.
00:32:36.770 --> 00:32:39.170
So there is a singular part.
00:32:39.170 --> 00:32:43.095
Essentially, if I were to do
that integral in dimensions
00:32:43.095 --> 00:32:47.690
less than fourth, I will
get a leading singularity
00:32:47.690 --> 00:32:49.779
that is applied.
00:32:49.779 --> 00:32:51.570
I will get a singularity
that is like this.
00:32:51.570 --> 00:32:54.520
I will get additional terms
per constant-- t, t squared,
00:32:54.520 --> 00:32:58.482
et cetera-- and
singular terms that
00:32:58.482 --> 00:32:59.880
are subbing in to this one.
00:33:03.394 --> 00:33:04.810
And then, of course,
I have a term
00:33:04.810 --> 00:33:12.856
that is minus h squared over t
if I were to include this here.
00:33:12.856 --> 00:33:23.783
So why don't I write the answer
as B minus h divided by t
00:33:23.783 --> 00:33:29.039
to the 1/2 plus d/4,
the whole thing squared.
00:33:33.830 --> 00:33:37.470
So what I did was essentially
I divided and multiplied
00:33:37.470 --> 00:33:42.600
by inputting d and put the whole
thing in the form of h divided
00:33:42.600 --> 00:33:46.410
by t to something squared?
00:33:46.410 --> 00:33:48.030
Why did I do that?
00:33:48.030 --> 00:33:53.425
It's because we had first
related a singular form
00:33:53.425 --> 00:33:59.120
for the energies in the scaling
picture that had the E to the 2
00:33:59.120 --> 00:34:03.818
minus alpha in front of
them and the function of h t
00:34:03.818 --> 00:34:05.590
to the delta.
00:34:05.590 --> 00:34:11.960
And all I wanted to emphasize
is that this picture, 2 minus
00:34:11.960 --> 00:34:16.860
alpha is d over 2.
00:34:16.860 --> 00:34:20.000
And the thing that we
call the gap exponent
00:34:20.000 --> 00:34:22.034
is 1/2 plus d over 4.
00:34:30.630 --> 00:34:32.840
Of course, I can't
use this theory
00:34:32.840 --> 00:34:36.060
as a description of the case.
00:34:36.060 --> 00:34:41.659
And the reason for that is
that the Gaussian theory
00:34:41.659 --> 00:34:45.109
exists and is well-defined
only as long as t is positive.
00:34:49.989 --> 00:34:57.310
Because once t becomes
negative, then the rate
00:34:57.310 --> 00:35:01.250
essentially becomes ill-defined.
00:35:01.250 --> 00:35:05.080
Because if I look at the
various rates that I have here,
00:35:05.080 --> 00:35:08.550
we certainly-- the
rate for q equals to 0.
00:35:08.550 --> 00:35:13.700
It is proportional
to minus t over 2v.
00:35:13.700 --> 00:35:16.670
If the t changes sign, rather
than having a Gaussian,
00:35:16.670 --> 00:35:21.900
I have essentially a rate that
is maximized as [INAUDIBLE].
00:35:21.900 --> 00:35:26.420
So clearly, again, by
issue of stability,
00:35:26.420 --> 00:35:30.940
the theory for t negative does
not describe a stable theory.
00:35:30.940 --> 00:35:33.740
And that's why n to the
fourth and all of those terms
00:35:33.740 --> 00:35:36.735
will be necessary to describe
that side of the phase
00:35:36.735 --> 00:35:37.780
transition.
00:35:37.780 --> 00:35:42.920
So if you like, this is
a kind of a description
00:35:42.920 --> 00:35:51.700
of a singularity that exists
only in this half of the space.
00:35:51.700 --> 00:35:55.250
Kind of reminiscent of coming
from the disordered side,
00:35:55.250 --> 00:36:00.730
but I don't want to give
it more reality than that.
00:36:00.730 --> 00:36:02.010
It's a mathematical construct.
00:36:02.010 --> 00:36:05.260
If we want to venture
to make the connection
00:36:05.260 --> 00:36:07.646
to the actual phase
transition, we
00:36:07.646 --> 00:36:09.329
have to prove the
n to the fourth.
00:36:19.600 --> 00:36:24.410
Now, the only reason to go
and recap this Gaussian theory
00:36:24.410 --> 00:36:30.100
is because since
it is solve-able,
00:36:30.100 --> 00:36:32.710
we can try to use
it as a toy model
00:36:32.710 --> 00:36:37.350
to apply the various steps
of renormalization group
00:36:37.350 --> 00:36:39.980
that we had outlined
last lecture.
00:36:39.980 --> 00:36:45.580
And once we understand the
steps of renormalization group
00:36:45.580 --> 00:36:50.590
for this theory, then it
gives us an anchoring point
00:36:50.590 --> 00:36:52.650
when we describe
the full theory that
00:36:52.650 --> 00:36:55.930
has n to the fourth,
et cetera-- how
00:36:55.930 --> 00:36:58.687
to sort of start with the
renormalization approach
00:36:58.687 --> 00:37:04.316
to the theory as we understand
and do the more complicated.
00:37:04.316 --> 00:37:08.060
So essentially, as I said, it's
not really a phase transition
00:37:08.060 --> 00:37:10.980
that can be described
by this theory.
00:37:10.980 --> 00:37:14.310
It's a singularity.
00:37:14.310 --> 00:37:18.910
But its value is that it is
this fully-modelled anchoring
00:37:18.910 --> 00:37:22.407
point for the full theory
that we are describing.
00:37:27.397 --> 00:37:32.886
So what we want to do is to do
an RG for the Gaussian model.
00:37:48.890 --> 00:37:51.260
So what is the procedure.
00:37:55.145 --> 00:38:00.940
We have a theory best
described in the space
00:38:00.940 --> 00:38:04.330
of variables q, the
Fourier variables.
00:38:04.330 --> 00:38:11.967
Where I have modes that
exist between 0-- very long
00:38:11.967 --> 00:38:16.075
wavelength-- to lambda, which
is the inverse of some shortest
00:38:16.075 --> 00:38:19.456
wavelength that I'm allowing.
00:38:19.456 --> 00:38:27.400
And so basically, I have
a bunch of modes m of q
00:38:27.400 --> 00:38:29.910
that are defined in
this range of qx.
00:38:32.790 --> 00:38:39.005
The first step of RG
was to coarse grain.
00:38:43.616 --> 00:38:50.225
The idea of coarse graining was
to change the scale over which
00:38:50.225 --> 00:38:56.330
you were doing the
averaging from some a to ba.
00:38:56.330 --> 00:39:04.510
So average from a to
ba of fluctuations.
00:39:07.090 --> 00:39:11.940
So once I do that,
at the end of the day
00:39:11.940 --> 00:39:16.370
I have fluctuations whose
minimum wavelength has
00:39:16.370 --> 00:39:18.175
gone from a to ba.
00:39:21.320 --> 00:39:29.750
So that means that q max, after
I go and do this procedure,
00:39:29.750 --> 00:39:37.956
is the previous q max that I
had divided by a factor of b.
00:39:37.956 --> 00:39:45.480
So basically, at the end
of the day I want to have,
00:39:45.480 --> 00:39:51.530
after coarse graining,
variables that only exist up
00:39:51.530 --> 00:39:55.468
to lambda over b.
00:39:55.468 --> 00:39:59.896
Whereas previously,
they existed after that.
00:40:02.850 --> 00:40:07.330
So this is very
easy at this level.
00:40:07.330 --> 00:40:12.800
All I can do is to
replace this m tilde of q
00:40:12.800 --> 00:40:15.520
in terms of two sets.
00:40:15.520 --> 00:40:22.400
I will call it to
be sigma if q is
00:40:22.400 --> 00:40:27.840
greater than this lambda over b.
00:40:27.840 --> 00:40:33.350
That is, everybody that
is out here, their q--
00:40:33.350 --> 00:40:36.100
I will call it q larger.
00:40:36.100 --> 00:40:41.400
Everybody that is here,
their q I will call q lesser.
00:40:41.400 --> 00:40:44.620
And all the modes
that were here,
00:40:44.620 --> 00:40:46.810
I will give them
a different name.
00:40:46.810 --> 00:40:50.320
The ones here I will call sigma.
00:40:50.320 --> 00:41:03.145
The ones here, if q
less than lambda over b,
00:41:03.145 --> 00:41:06.620
will get called m tilde.
00:41:06.620 --> 00:41:09.620
So I just renamed my variables.
00:41:09.620 --> 00:41:17.740
So essentially, right
here I had integration
00:41:17.740 --> 00:41:22.120
over all of the modes.
00:41:22.120 --> 00:41:26.403
I just renamed some of
the modes that were inside
00:41:26.403 --> 00:41:32.040
q lesser and sigma--
and tilde, the ones that
00:41:32.040 --> 00:41:35.600
are outside q greater.
00:41:35.600 --> 00:41:40.570
So what I have to do
for my Gaussian theory.
00:41:40.570 --> 00:41:44.350
Let's write it rather
than in this form
00:41:44.350 --> 00:41:48.574
that was discrete in
terms of the continuum.
00:41:51.360 --> 00:41:55.370
I have to iterate over all
configurations of these Fourier
00:41:55.370 --> 00:41:56.720
modes.
00:41:56.720 --> 00:41:58.793
So I have these m tilde of q's.
00:42:02.950 --> 00:42:05.921
And the wave that I have
to assign to them when
00:42:05.921 --> 00:42:11.580
I look at the continuum is
exponential, integral in dq
00:42:11.580 --> 00:42:12.760
q to pi to the d.
00:42:12.760 --> 00:42:16.736
T plus kq squared, and so forth.
00:42:19.718 --> 00:42:22.203
And tilde of q squared.
00:42:26.180 --> 00:42:33.554
And then I had the one
term that was hm of 0.
00:42:38.790 --> 00:42:43.030
What I have done is
to simply rewrite
00:42:43.030 --> 00:42:50.704
this as two sets of
integrations over the-- whoops.
00:42:50.704 --> 00:42:52.135
This was m.
00:42:55.951 --> 00:43:01.450
m, let's call is sigma
first-- sigma of q
00:43:01.450 --> 00:43:06.692
larger integrate over
m tilde of q lesser.
00:43:13.664 --> 00:43:18.670
And actually, you can
see that the modes here
00:43:18.670 --> 00:43:21.950
and the modes here don't
talk to each other.
00:43:21.950 --> 00:43:27.000
And that's really the advantage
of doing the Gaussian theory.
00:43:27.000 --> 00:43:29.930
And the thing that allowed
me to solve the problem here
00:43:29.930 --> 00:43:32.860
and also to do the
coarse graining there.
00:43:32.860 --> 00:43:36.560
Once we do things
like n to the fourth,
00:43:36.560 --> 00:43:38.856
then I will have
couplings between modes
00:43:38.856 --> 00:43:42.410
that go across between
the three sets.
00:43:42.410 --> 00:43:44.846
And then the problem
becomes difficult.
00:43:44.846 --> 00:43:48.100
But now that I
don't have that, I
00:43:48.100 --> 00:43:52.085
can actually separately write
the integral as two parts.
00:44:03.700 --> 00:44:07.000
And this is for q lesser.
00:44:07.000 --> 00:44:11.741
And for each one of them, I
essentially have the same rate.
00:44:11.741 --> 00:44:19.240
The integral over q
greater goes between lambda
00:44:19.240 --> 00:44:20.818
over d and lambda.
00:44:36.626 --> 00:44:40.578
The integral over
m tilde of q lesser
00:44:40.578 --> 00:44:43.048
is essentially the same thing.
00:44:51.446 --> 00:44:58.160
Exponential minus integral
0 to lambda over d. dv
00:44:58.160 --> 00:45:05.650
q lesser to five to the d,
t plus kq lesser squared,
00:45:05.650 --> 00:45:08.720
and so forth.
00:45:08.720 --> 00:45:14.406
And q lesser squared.
00:45:14.406 --> 00:45:22.448
And then I have the additional
term which sits at 0.
00:45:22.448 --> 00:45:31.760
It is part of the modes that
are assigned with q lesser.
00:45:36.093 --> 00:45:36.593
OK?
00:45:36.593 --> 00:45:37.620
Fine.
00:45:37.620 --> 00:45:40.590
Nothing particularly
profound here.
00:45:40.590 --> 00:45:42.090
In fact, it's very simple.
00:45:42.090 --> 00:45:46.040
It's just renaming
two sets of modes.
00:45:46.040 --> 00:45:51.420
And the averaging
that I have to do,
00:45:51.420 --> 00:45:53.764
and getting rid of
the fluctuations
00:45:53.764 --> 00:45:59.250
at short wavelength,
here is very trickier.
00:45:59.250 --> 00:46:03.938
Because this is just a
bunch of integrations
00:46:03.938 --> 00:46:09.380
that I had to do over here,
but it is only over things
00:46:09.380 --> 00:46:12.826
that are sitting close to
the edge of this [INAUDIBLE].
00:46:16.810 --> 00:46:21.292
So essentially, the
integrations over these modes
00:46:21.292 --> 00:46:24.280
is doing this
integral over here,
00:46:24.280 --> 00:46:30.260
from lambda over d to lambda,
and none of the singularities
00:46:30.260 --> 00:46:34.020
has anything to do with
the range of integration
00:46:34.020 --> 00:46:37.020
from lambda over d to lambda.
00:46:37.020 --> 00:46:40.020
So the result of
doing all of that
00:46:40.020 --> 00:46:45.206
is simply just a constant--
but not a constant.
00:46:45.206 --> 00:46:51.220
It's a function of t that
is completely non-singular
00:46:51.220 --> 00:46:55.690
and have a nice state of
expansion powers of t.
00:46:55.690 --> 00:47:01.080
A kind of [INAUDIBLE] I
call non-singular functions
00:47:01.080 --> 00:47:01.660
sometimes.
00:47:01.660 --> 00:47:03.539
Constant thing is
that eventually
00:47:03.539 --> 00:47:05.288
if you take sufficiently
high derivatives,
00:47:05.288 --> 00:47:10.900
I guess, of this value, the
t dependents [INAUDIBLE].
00:47:16.250 --> 00:47:24.550
So all of the interesting thing
is really in this m tilde of k
00:47:24.550 --> 00:47:25.800
lesser.
00:47:25.800 --> 00:47:31.220
And really, the eventual
process of renormalization
00:47:31.220 --> 00:47:35.400
in this picture is
something like this.
00:47:35.400 --> 00:47:37.330
That all of the
singularities are
00:47:37.330 --> 00:47:43.310
sitting at the center of this
kind of orange-shaped entity.
00:47:43.310 --> 00:47:46.115
And rather than biting
the whole thing,
00:47:46.115 --> 00:47:51.390
you kind of cut it slowly
and slowly from the edge,
00:47:51.390 --> 00:47:54.610
approaching to where all
of the exciting things
00:47:54.610 --> 00:47:55.680
are at the center.
00:47:55.680 --> 00:47:57.550
For this problem
of the Gaussian,
00:47:57.550 --> 00:48:00.440
it turns out to be
trivial to do so.
00:48:00.440 --> 00:48:02.980
But for the more
general problem,
00:48:02.980 --> 00:48:06.570
it can be interesting because
procedure is the same.
00:48:06.570 --> 00:48:09.030
We are interested in
what's happening here,
00:48:09.030 --> 00:48:13.450
but we gradually peel
of things that we
00:48:13.450 --> 00:48:17.540
know don't cause anything
difficult for the problems.
00:48:21.540 --> 00:48:24.600
So then I have to
multiply with this,
00:48:24.600 --> 00:48:31.440
and I have found in some sense
a probability for configurations
00:48:31.440 --> 00:48:35.490
of the coarse grain system,
which is simply given by this.
00:48:38.100 --> 00:48:42.842
But then renormalization
group has two other steps.
00:48:45.690 --> 00:48:50.520
The second step was to
say, well, in real space,
00:48:50.520 --> 00:48:53.870
as we said, the picture
that is represented
00:48:53.870 --> 00:48:57.730
by these coarse grain
variables is grainy.
00:48:57.730 --> 00:49:01.410
If my pixels were previously
one by one by one,
00:49:01.410 --> 00:49:04.746
now my pixels are d by d by d.
00:49:04.746 --> 00:49:10.380
So I can make my picture look
to have the same resolution
00:49:10.380 --> 00:49:14.630
as my initial
picture if I rescale
00:49:14.630 --> 00:49:16.590
all of the events
for a factor of t.
00:49:19.180 --> 00:49:23.310
In momentum representation,
or intuitive presentation,
00:49:23.310 --> 00:49:33.160
it corresponds to rescaling all
of the q's by a factor of B.
00:49:33.160 --> 00:49:36.640
And clearly, what
that serves to achieve
00:49:36.640 --> 00:49:44.020
is that if I replace q
lesser with B times q prime,
00:49:44.020 --> 00:49:50.820
then the maximum value will
go back to 0 to lambda.
00:49:50.820 --> 00:49:53.350
So by doing this
one in formation,
00:49:53.350 --> 00:50:00.514
I can ensure that the upper
cut-off is, in fact, lambda
00:50:00.514 --> 00:50:03.240
again.
00:50:03.240 --> 00:50:07.160
Now, there was another
thing, which in real space
00:50:07.160 --> 00:50:15.330
we said that we defined m
prime to be m tilde rescaled
00:50:15.330 --> 00:50:16.819
by some factor zeta.
00:50:21.130 --> 00:50:24.990
I had to do a change
of the contrast.
00:50:24.990 --> 00:50:27.590
I did have to do the
same change of contrast
00:50:27.590 --> 00:50:32.200
here, except that the variables
that I am dealing with here,
00:50:32.200 --> 00:50:36.470
it was in x coordinates.
00:50:36.470 --> 00:50:40.570
What I want to do it
is in the q coordinate.
00:50:40.570 --> 00:50:49.884
So I will call m with
a tail prime of q prime
00:50:49.884 --> 00:50:59.510
to be m tilde of q
prime by a factor of z.
00:50:59.510 --> 00:51:03.200
The difference between
the z and the zeta, which
00:51:03.200 --> 00:51:06.280
is real space and Fourier
space is just the fact
00:51:06.280 --> 00:51:11.210
that in going from
one to the other,
00:51:11.210 --> 00:51:14.280
you have to do
integrations over space.
00:51:14.280 --> 00:51:17.070
So dimensionally,
there is a factor of b
00:51:17.070 --> 00:51:20.670
to the d difference between
the rescaling of this quantity
00:51:20.670 --> 00:51:26.665
and that quantity,
and if you want to use
00:51:26.665 --> 00:51:32.180
or the other zeta against
b to the minus d and z.
00:51:32.180 --> 00:51:36.250
But since we would be doing
everything in Fourier space,
00:51:36.250 --> 00:51:38.500
we would just use this
factor traditionally.
00:51:42.950 --> 00:51:46.440
So if I do that, what do I find?
00:51:46.440 --> 00:51:57.430
I find that Z of t of h is
exponential of some singular,
00:51:57.430 --> 00:52:00.700
non-singular dependents.
00:52:00.700 --> 00:52:08.550
And then I have to integrate
over these new variables,
00:52:08.550 --> 00:52:11.940
m prime of q prime.
00:52:11.940 --> 00:52:13.740
Yes?
00:52:13.740 --> 00:52:17.270
AUDIENCE: In your real
space renormalization
00:52:17.270 --> 00:52:21.769
your m tilde is a
function of an x.
00:52:21.769 --> 00:52:23.435
But in your Fourier
space representation
00:52:23.435 --> 00:52:28.885
your m tilde is a
function of q prime?
00:52:28.885 --> 00:52:32.053
PROFESSOR: I guess I could have
written here x prime, also.
00:52:32.053 --> 00:52:33.094
It doesn't really matter.
00:52:42.860 --> 00:52:44.150
So do you have here?
00:52:44.150 --> 00:52:49.880
You have exponential
minus the integral.
00:52:49.880 --> 00:52:56.820
The integration for q prime now
is going back to 0 to lambda.
00:52:56.820 --> 00:53:03.390
I have db of q prime
divided by 2 pi to the d.
00:53:09.150 --> 00:53:19.490
Now, you see that every time
I have a q-- V or q, q lesser,
00:53:19.490 --> 00:53:24.250
in fact-- I have
to go to q prime
00:53:24.250 --> 00:53:28.592
by introducing a
factor of the inverse.
00:53:28.592 --> 00:53:33.440
So there will be a total
factor of V to the minus V
00:53:33.440 --> 00:53:37.246
that comes from
this integration.
00:53:37.246 --> 00:53:40.000
And that will multiply t.
00:53:40.000 --> 00:53:45.590
That will multiply
kb to the minus d.
00:53:45.590 --> 00:53:51.650
But then here I have to q's
because of the q squared there.
00:53:51.650 --> 00:53:58.940
Again, doing the same thing, I
will get V to the d minus two.
00:53:58.940 --> 00:54:02.790
I had q plus 2, if you like.
00:54:02.790 --> 00:54:09.195
And then the next l would
be lb to the d minus 4.
00:54:09.195 --> 00:54:12.130
And you can see that as I have
higher and higher derivatives
00:54:12.130 --> 00:54:15.850
of q, I get higher and
higher powers with negative
00:54:15.850 --> 00:54:16.350
[INAUDIBLE].
00:54:21.060 --> 00:54:28.760
But then I have m tilde that I
want to replace with m prime.
00:54:28.760 --> 00:54:34.090
And that process will give
me a factor of z squared.
00:54:34.090 --> 00:54:39.340
And then I have m prime
of q prime squared.
00:54:43.225 --> 00:54:46.410
There is no integration
for this terms.
00:54:46.410 --> 00:54:48.260
It's just one mode.
00:54:48.260 --> 00:54:53.460
But each mode I have
rescaled by a factor of z.
00:54:53.460 --> 00:55:03.262
So I will have a term that
is z h dot m prime of 0.
00:55:13.690 --> 00:55:17.470
So what we see is that
what we have managed to do
00:55:17.470 --> 00:55:21.590
is to make the Gaussian
integration over
00:55:21.590 --> 00:55:26.410
here precisely the same thing
as the Gaussian integration
00:55:26.410 --> 00:55:28.650
that I started with.
00:55:28.650 --> 00:55:33.486
So I can conclude
that this function tnh
00:55:33.486 --> 00:55:39.090
that I am interested in has
a path that is non-singular.
00:55:42.780 --> 00:55:45.670
But its singular
part is the same
00:55:45.670 --> 00:55:52.427
as the same z calculated for
a bunch of new parameters.
00:55:55.840 --> 00:56:02.250
And in particular, the
new t is v to the minus
00:56:02.250 --> 00:56:07.755
d z squared the old t.
00:56:07.755 --> 00:56:15.320
The new k is b to the minus
d minus 2 z squared q.
00:56:15.320 --> 00:56:21.515
The new L would be to the
minus d minus 4 z squared L,
00:56:21.515 --> 00:56:23.500
and so forth.
00:56:23.500 --> 00:56:27.858
And the new h is zh.
00:56:27.858 --> 00:56:28.792
Yes?
00:56:28.792 --> 00:56:33.465
AUDIENCE: There should be q
prime squared and q prime 4?
00:56:33.465 --> 00:56:34.090
PROFESSOR: Yes.
00:56:49.342 --> 00:56:50.326
Yes.
00:56:50.326 --> 00:56:53.770
This is my day to do a
lot of algebraic errors.
00:57:00.440 --> 00:57:00.940
OK.
00:57:05.180 --> 00:57:07.630
So what is the
change in parameters?
00:57:07.630 --> 00:57:10.970
So I wrote it over there.
00:57:10.970 --> 00:57:17.610
So this kind of captures
the very simplest type
00:57:17.610 --> 00:57:20.160
of renormalization.
00:57:20.160 --> 00:57:24.510
Actually, all I did
was a scaling analysis.
00:57:24.510 --> 00:57:29.260
If I were to change
positions by a factor of b
00:57:29.260 --> 00:57:34.460
and change the magnitude of my
field m by a factor z or zeta,
00:57:34.460 --> 00:57:37.302
this is the kind of
results that I will get.
00:57:40.470 --> 00:57:46.500
Now, how can we make this
capture the kind of picture
00:57:46.500 --> 00:57:52.190
that we have over here in the
language of renormalization?
00:57:52.190 --> 00:57:57.830
Want to be able to
change two parameters
00:57:57.830 --> 00:57:59.918
and reach a fixed point.
00:58:03.590 --> 00:58:09.940
So we know that kind of
[INAUDIBLE] that t and h
00:58:09.940 --> 00:58:11.020
have to go to 0.
00:58:11.020 --> 00:58:15.910
They are the variables
that determine essentially
00:58:15.910 --> 00:58:19.260
whether you are at this
said similar point.
00:58:19.260 --> 00:58:24.460
So if t and h I forget, the
next most important term that
00:58:24.460 --> 00:58:29.850
comes into play is k prime,
which is some function of k.
00:58:29.850 --> 00:58:32.680
And if I want to be
at the fixed point,
00:58:32.680 --> 00:58:36.540
I may want to choose
the factor z such
00:58:36.540 --> 00:58:40.960
that k prime is the same as k.
00:58:40.960 --> 00:58:50.390
So choose z such
that k prime is k.
00:58:50.390 --> 00:58:54.346
And that tells me
immediately that z
00:58:54.346 --> 00:58:57.552
would be b to the power
of 1 plus d over 2.
00:59:02.760 --> 00:59:09.756
If I choose that particular
form of z, then what do I get?
00:59:09.756 --> 00:59:15.132
I get t prime is z
squared b to the minus b.
00:59:15.132 --> 00:59:20.600
So when I do that, I
will get b squared t.
00:59:20.600 --> 00:59:27.780
I get that h prime
is just z times h.
00:59:27.780 --> 00:59:33.280
So it is b to the 1
plus b over 2 times h.
00:59:33.280 --> 00:59:41.291
These are both directions that
as b becomes larger than 1,
00:59:41.291 --> 00:59:45.970
b prime becomes larger than th
prime, becomes larger than h.
00:59:45.970 --> 00:59:48.040
These are relevant directions.
00:59:48.040 --> 00:59:53.230
I would associate with them
eigenvalues y dt minus 2.
00:59:53.230 --> 00:59:54.316
Divide h.
00:59:54.316 --> 00:59:56.696
That is 1 plus d over 2.
01:00:00.510 --> 01:00:07.300
So if I go according to the
scaling construction that we
01:00:07.300 --> 01:00:14.310
had before, f singular
of tnh is t to the power
01:00:14.310 --> 01:00:19.470
d over y dt, some
scaling function of h,
01:00:19.470 --> 01:00:24.640
g to the power of
divide h over y dt.
01:00:24.640 --> 01:00:27.480
This is what we have
established before.
01:00:27.480 --> 01:00:32.430
With these values I will
get t to the d over 2,
01:00:32.430 --> 01:00:41.888
some scaling function of h, t to
the power of 1/2 plus d over 4.
01:00:44.660 --> 01:00:51.700
We can immediately compare this
expression and this expression
01:00:51.700 --> 01:00:52.670
that we have over here.
01:00:52.670 --> 01:00:52.915
Yes?
01:00:52.915 --> 01:00:53.540
AUDIENCE: Wait.
01:00:53.540 --> 01:00:54.970
What's the reason
to choose scale
01:00:54.970 --> 01:00:57.011
as the parameter that maps
onto itself and not L?
01:00:57.011 --> 01:00:57.670
PROFESSOR: OK.
01:00:57.670 --> 01:00:59.310
I'll come to that.
01:00:59.310 --> 01:01:05.140
So having gone this far,
let's see what l is doing.
01:01:05.140 --> 01:01:07.730
So if I put here-- you
can see that clearly L
01:01:07.730 --> 01:01:12.540
has v to the minus
2 compared to k.
01:01:12.540 --> 01:01:15.620
So currently, the way
that we established,
01:01:15.620 --> 01:01:19.860
L prime is b to the minus 2m.
01:01:19.860 --> 01:01:22.310
If I had a higher
derivative, it would
01:01:22.310 --> 01:01:26.530
be b to a minus larger
number, et cetera.
01:01:26.530 --> 01:01:32.390
So L, out of these other things,
are irrelevant variables.
01:01:32.390 --> 01:01:36.950
So they are essentially
under rescaling,
01:01:36.950 --> 01:01:40.832
under looking at the system
in larger and larger scale,
01:01:40.832 --> 01:01:42.125
they will go to 0.
01:01:42.125 --> 01:01:48.410
And I did get a system that has
the same topological structure
01:01:48.410 --> 01:01:51.133
as what I had established here.
01:01:51.133 --> 01:01:53.625
Because I have to
tune two parameters
01:01:53.625 --> 01:01:56.880
in order to reach
the critical point.
01:01:56.880 --> 01:02:00.510
Let's say I had
chosen something else.
01:02:00.510 --> 01:02:11.530
If I had chosen z such
that L prime equals to L.
01:02:11.530 --> 01:02:13.790
I could do that.
01:02:13.790 --> 01:02:19.780
Then all of the derivatives that
are higher factors of q in this
01:02:19.780 --> 01:02:22.810
[INAUDIBLE], they would
be all irrelevant.
01:02:22.810 --> 01:02:28.140
But then I would
have k, t, and h
01:02:28.140 --> 01:02:30.219
all with irrelevant variables.
01:02:34.390 --> 01:02:39.000
So yeah, it could be that
there is some physics.
01:02:39.000 --> 01:02:42.040
I mean, certainly
mathematically I
01:02:42.040 --> 01:02:48.410
can ask the system what
happens if k goes to 0.
01:02:48.410 --> 01:02:50.670
I kind of ignore
the k dependencies
01:02:50.670 --> 01:02:54.340
that I have in all
of these expressions,
01:02:54.340 --> 01:02:59.660
but there are going to be
singular dependencies on k.
01:02:59.660 --> 01:03:04.070
So if there is indeed
some experimental system
01:03:04.070 --> 01:03:10.170
in which you have to tune,
in addition temperature,
01:03:10.170 --> 01:03:12.160
something that has
to do with the way
01:03:12.160 --> 01:03:14.740
that the spins or
degrees of freedom
01:03:14.740 --> 01:03:17.885
are coupled to each other,
and that coupling changes sign
01:03:17.885 --> 01:03:20.056
from being positive
to being negative,
01:03:20.056 --> 01:03:23.995
you go from one type of behavior
to another type of behavior,
01:03:23.995 --> 01:03:26.350
maybe this would be
a good thing for it.
01:03:26.350 --> 01:03:29.040
But you can see the
kind of structure
01:03:29.040 --> 01:03:34.190
you would get if k has to go
to 0, you go from a structure
01:03:34.190 --> 01:03:36.860
where things want to be
in the same direction
01:03:36.860 --> 01:03:39.410
to things that want
to be anti-parallel.
01:03:39.410 --> 01:03:43.370
And then clearly you need higher
order terms to stabilize things
01:03:43.370 --> 01:03:45.340
so that your
singularity does not
01:03:45.340 --> 01:03:48.880
go all the way to 0
wavelength, et cetera.
01:03:48.880 --> 01:03:54.090
So one can actually come
up with physical systems
01:03:54.090 --> 01:03:56.226
that kind of resemble
that, were there
01:03:56.226 --> 01:03:58.942
is some landscape
that is also chosen.
01:03:58.942 --> 01:04:02.910
But for this very simplest
thing that we are doing,
01:04:02.910 --> 01:04:05.440
this is what is going on.
01:04:05.440 --> 01:04:09.080
But you could have also
asked the other question.
01:04:09.080 --> 01:04:12.010
So clearly we
understand what happens
01:04:12.010 --> 01:04:15.790
if you choose z so
that some term is fixed
01:04:15.790 --> 01:04:18.750
and everything above it is
relevant, everything below it
01:04:18.750 --> 01:04:20.370
is irrelevant.
01:04:20.370 --> 01:04:31.998
But why not choose z
such that t is fixed?
01:04:31.998 --> 01:04:35.225
So that's going to
be b to the d over 2,
01:04:35.225 --> 01:04:37.120
then t prime equals to t.
01:04:40.048 --> 01:04:45.060
If I choose that, then
clearly the coupling k
01:04:45.060 --> 01:04:46.295
will already be irrelevant.
01:04:49.270 --> 01:04:54.640
So this is actually a
reasonable fixed point.
01:04:54.640 --> 01:04:57.360
It's a fixed one that
corresponds to a system
01:04:57.360 --> 01:05:00.530
where k has gone
to 0, which means
01:05:00.530 --> 01:05:04.150
that the different points
don't talk to each other.
01:05:04.150 --> 01:05:06.180
Remember, when we
were discussing
01:05:06.180 --> 01:05:09.410
the behavior of correlation
lens at fixed points,
01:05:09.410 --> 01:05:11.995
there was two possibilities--
either the correlation
01:05:11.995 --> 01:05:14.570
lens was infinite or it was 0.
01:05:14.570 --> 01:05:20.730
So if I choose this, then k
prime will go eventually to 0.
01:05:20.730 --> 01:05:24.410
I go towards a system in
which the degrees of freedom
01:05:24.410 --> 01:05:28.630
are completely decoupled
from each other.
01:05:28.630 --> 01:05:30.020
Perfectly well-behaved.
01:05:30.020 --> 01:05:34.380
Fixed behavior that corresponds
to 0 correlation lens.
01:05:34.380 --> 01:05:38.200
And you can see that if
I go through this formula
01:05:38.200 --> 01:05:44.640
that I told you over
here, zeta in real space
01:05:44.640 --> 01:05:49.250
would be b to the
minus d over 2.
01:05:49.250 --> 01:05:53.960
And what that means
is that if you average
01:05:53.960 --> 01:05:59.130
independent variables
over a size b,
01:05:59.130 --> 01:06:02.290
the scale of fluctuation is
because of the central limit
01:06:02.290 --> 01:06:04.720
theeorem is the square
root of the volume.
01:06:04.720 --> 01:06:05.720
So that's how it scales.
01:06:08.580 --> 01:06:12.310
So essentially, what's
at the end of the story?
01:06:12.310 --> 01:06:16.820
That's a behavior in which there
is only one coefficient event--
01:06:16.820 --> 01:06:18.150
forget about h.
01:06:18.150 --> 01:06:23.480
The eventual rate is just t over
2m squared at different points.
01:06:23.480 --> 01:06:26.120
That's the central limit here.
01:06:26.120 --> 01:06:30.660
So through a different route, we
have rediscovered, if you like,
01:06:30.660 --> 01:06:32.300
the central limit theorem.
01:06:32.300 --> 01:06:36.330
Because if you average lots
of uncorrelated variables,
01:06:36.330 --> 01:06:39.700
you will generate
Gaussian rates.
01:06:39.700 --> 01:06:45.070
So what we are really
after in this language
01:06:45.070 --> 01:06:49.922
is how to generalize the
central limit theorem, how to--
01:06:49.922 --> 01:06:53.290
as we find the analog of
a Gaussian, the degrees
01:06:53.290 --> 01:06:55.730
of freedom that
are not correlated
01:06:55.730 --> 01:06:59.580
but talk to their neighborhood.
01:06:59.580 --> 01:07:04.110
So the kind of field
theory that we are after
01:07:04.110 --> 01:07:06.950
are these generalizations
of central limit theorem
01:07:06.950 --> 01:07:10.700
to the types of
field theories that
01:07:10.700 --> 01:07:12.025
have some locality enablement.
01:07:15.985 --> 01:07:16.980
AUDIENCE: Question.
01:07:16.980 --> 01:07:17.680
PROFESSOR: Yes.
01:07:17.680 --> 01:07:22.660
AUDIENCE: So wherever you can
define the renormalization
01:07:22.660 --> 01:07:25.720
you're finding different z's?
01:07:25.720 --> 01:07:26.380
PROFESSOR: Yes.
01:07:26.380 --> 01:07:29.660
AUDIENCE: We can tune
how many parameters we
01:07:29.660 --> 01:07:30.770
want to be able to--
01:07:30.770 --> 01:07:31.880
PROFESSOR: Exactly.
01:07:31.880 --> 01:07:32.900
Yes.
01:07:32.900 --> 01:07:36.300
And that's where the
physics comes into play.
01:07:36.300 --> 01:07:40.030
Mathematically, there's a whole
set of different fixed points
01:07:40.030 --> 01:07:43.390
that you can construct for
choosing different z's.
01:07:43.390 --> 01:07:45.540
You have to decide
which one of them
01:07:45.540 --> 01:07:47.260
corresponds to the
physical problem
01:07:47.260 --> 01:07:49.036
that you are working on.
01:07:49.036 --> 01:07:49.970
AUDIENCE: Yes.
01:07:49.970 --> 01:07:53.052
So if the fixed point
stops being just defined
01:07:53.052 --> 01:07:57.955
by the nature of the
system, but it's also
01:07:57.955 --> 01:08:00.925
depends on how we
define renormalization?
01:08:00.925 --> 01:08:04.390
On mathematical
descriptions and--
01:08:07.855 --> 01:08:13.176
PROFESSOR: If by how we
define renormalization looking
01:08:13.176 --> 01:08:15.080
to choose z, yes,
I agree with you.
01:08:15.080 --> 01:08:16.510
Yes.
01:08:16.510 --> 01:08:20.990
But again, you have
this possibility
01:08:20.990 --> 01:08:25.005
of looking at the system
at different scales.
01:08:25.005 --> 01:08:30.260
But we have been very agnostic
about what that system is.
01:08:30.260 --> 01:08:33.300
And so you how many
ways of doing things.
01:08:33.300 --> 01:08:36.474
Ultimately, you need some
reality to come and choose
01:08:36.474 --> 01:08:38.609
among these different ways.
01:08:38.609 --> 01:08:41.444
Yes?
01:08:41.444 --> 01:08:42.860
AUDIENCE: So you
do want to keep k
01:08:42.860 --> 01:08:46.930
a relevant variable in
group problems, right?
01:08:46.930 --> 01:08:49.666
PROFESSOR: No.
01:08:49.666 --> 01:08:50.999
I make k to be a fixed variable.
01:08:50.999 --> 01:08:53.590
AUDIENCE: Oh, exactly.
01:08:53.590 --> 01:08:57.939
Why don't you add a small
amount, like an absolute
01:08:57.939 --> 01:09:02.020
to the power of bf and
[INAUDIBLE] point z.
01:09:02.020 --> 01:09:04.680
Plus or minus, doesn't matter.
01:09:04.680 --> 01:09:07.880
Why the equality
assumption exactly?
01:09:07.880 --> 01:09:10.310
And the smaller one
doesn't change anything?
01:09:10.310 --> 01:09:16.420
All the other variables
like L become irrelevant?
01:09:16.420 --> 01:09:18.029
PROFESSOR: OK.
01:09:18.029 --> 01:09:25.090
So the point is that it
is b raised to some power.
01:09:25.090 --> 01:09:30.890
So here I had, I don't
know, Katie k prime was k.
01:09:30.890 --> 01:09:33.130
And you say, why not
kb to the absolute?
01:09:33.130 --> 01:09:34.834
AUDIENCE: Yeah, exactly.
01:09:34.834 --> 01:09:37.420
PROFESSOR: Now,
the thing that I'm
01:09:37.420 --> 01:09:41.649
interested in what happens
at larger and larger scale.
01:09:41.649 --> 01:09:44.920
So in principle, I
should be able to make v
01:09:44.920 --> 01:09:47.390
as large as I want.
01:09:47.390 --> 01:09:52.585
So I don't have the
freedom that you mentioned.
01:09:52.585 --> 01:09:56.516
And you are right in
the sense that, OK,
01:09:56.516 --> 01:09:58.470
what does it mean
whether this ratio is
01:09:58.470 --> 01:10:01.355
larger than or
smaller than what?
01:10:01.355 --> 01:10:04.970
But the point is that
once you have selected
01:10:04.970 --> 01:10:07.914
some parameter in your system--
L or whatever you have,
01:10:07.914 --> 01:10:12.210
some value-- you can, by
playing around with this,
01:10:12.210 --> 01:10:15.950
choose a value of V
for any epsilon such
01:10:15.950 --> 01:10:18.730
that you reach that limit.
01:10:18.730 --> 01:10:26.090
So by doing this, you in a
sense have defined a lens scale.
01:10:26.090 --> 01:10:28.760
The lens scale would
depend on epsilon,
01:10:28.760 --> 01:10:30.637
and you would have
different behaviors,
01:10:30.637 --> 01:10:33.160
whether you have shorter
than that lens scale
01:10:33.160 --> 01:10:35.556
or larger than that lens scale.
01:10:35.556 --> 01:10:39.410
So this has to be done precisely
because of this freedom
01:10:39.410 --> 01:10:41.318
of making b larger, and so on.
01:10:47.996 --> 01:10:50.080
Now, if you are dealing
with a finite system
01:10:50.080 --> 01:10:53.032
and you can't make your b
much larger than something
01:10:53.032 --> 01:10:54.740
or whatever, then
you're perfectly right.
01:10:59.110 --> 01:10:59.640
Yes?
01:10:59.640 --> 01:11:02.210
AUDIENCE: Physically,
z or zeta should
01:11:02.210 --> 01:11:08.445
be whatever type quantity
is needed to actually make
01:11:08.445 --> 01:11:10.721
it look exactly the same--
where it keeps coming out.
01:11:10.721 --> 01:11:11.720
PROFESSOR: Exactly, yes.
01:11:11.720 --> 01:11:12.261
That's right.
01:11:12.261 --> 01:11:14.300
AUDIENCE: And then we
know, because we already
01:11:14.300 --> 01:11:16.200
know that we have two
relevant variables,
01:11:16.200 --> 01:11:19.010
that z has to look this
way for a system that
01:11:19.010 --> 01:11:20.160
has two relevant variables.
01:11:20.160 --> 01:11:22.651
PROFESSOR: For the
Gaussian one, right.
01:11:22.651 --> 01:11:23.276
AUDIENCE: Yeah.
01:11:23.276 --> 01:11:25.840
But then if we had a
different kind of system,
01:11:25.840 --> 01:11:28.460
then actually, just going
from the physical perspective,
01:11:28.460 --> 01:11:30.836
we would need a different z
to make things look the same.
01:11:30.836 --> 01:11:32.543
And that would give
us a different number
01:11:32.543 --> 01:11:33.390
of variables here.
01:11:33.390 --> 01:11:34.229
PROFESSOR: Yes.
01:11:34.229 --> 01:11:34.770
That's right.
01:11:37.530 --> 01:11:42.900
Now, in terms of that
practically in all cases
01:11:42.900 --> 01:11:44.920
we either are dealing
with a phase that
01:11:44.920 --> 01:11:48.005
has 0 correlation
on that, and then
01:11:48.005 --> 01:11:51.190
this Gaussian behavior
and central limit theorem
01:11:51.190 --> 01:11:55.345
is what we are dealing-- and the
averaging is by 1 over volume.
01:11:55.345 --> 01:11:57.700
Or we have something
that is very pretty
01:11:57.700 --> 01:12:01.420
close to this big [INAUDIBLE]
that we have now discovered,
01:12:01.420 --> 01:12:04.460
which is just the
gradient squared.
01:12:04.460 --> 01:12:07.490
And that has its own
scaling according
01:12:07.490 --> 01:12:10.750
to these powers that
I have found here,
01:12:10.750 --> 01:12:14.326
and I will explain
that more deeply.
01:12:14.326 --> 01:12:16.910
It turns out that at
the end of the day,
01:12:16.910 --> 01:12:20.910
that when we look at
real phase transitions,
01:12:20.910 --> 01:12:26.680
all of these exponents will
change, but not too much.
01:12:26.680 --> 01:12:30.590
So this Gaussian fixed point
is actually in some sense
01:12:30.590 --> 01:12:34.890
rather close to where
we want to end up.
01:12:34.890 --> 01:12:38.560
So that's why it's also an
important anchoring point,
01:12:38.560 --> 01:12:40.440
as I just mentioned.
01:12:45.140 --> 01:12:48.510
Again, I said that
essentially what we did
01:12:48.510 --> 01:12:52.620
was take the rate that
we had originally,
01:12:52.620 --> 01:12:54.400
and we did a rescaling.
01:12:54.400 --> 01:13:02.125
So basically, we
replace x by-- let
01:13:02.125 --> 01:13:06.682
me get the directions there.
01:13:06.682 --> 01:13:12.730
So we replace x by bx prime.
01:13:12.730 --> 01:13:16.323
If I had started
being in real space,
01:13:16.323 --> 01:13:22.470
I would have replaced
m with zeta m prime.
01:13:22.470 --> 01:13:28.765
m after getting rid of
some degrees of freedom.
01:13:28.765 --> 01:13:32.160
Again, zeta m prime.
01:13:32.160 --> 01:13:36.946
Before I just do that to the
rate that I had written before,
01:13:36.946 --> 01:13:39.550
there was a beta h.
01:13:39.550 --> 01:13:46.870
Which was we could derive
d d x t over 2m squared, um
01:13:46.870 --> 01:13:50.314
to the fourth and
higher order terms,
01:13:50.314 --> 01:13:59.170
k over 2 gradient m squared, L
over 2 Laplacian of m squared
01:13:59.170 --> 01:14:01.870
and so forth.
01:14:01.870 --> 01:14:06.350
Just do this
replacement of things.
01:14:06.350 --> 01:14:08.720
What do I get?
01:14:08.720 --> 01:14:14.230
I get that t prime
is b to the d.
01:14:14.230 --> 01:14:18.430
Whenever I see x, I
replace it with dx prime.
01:14:18.430 --> 01:14:22.470
Whenever I see m, I replace
it with zeta m prime.
01:14:22.470 --> 01:14:23.840
So I get here the zeta squared.
01:14:26.850 --> 01:14:34.331
u prime would be b to
the d zeta to the fourth.
01:14:38.259 --> 01:14:44.642
k prime would be b to the
b minus 2 zeta squared.
01:14:44.642 --> 01:14:48.240
L prime would be
b to the d minus 4
01:14:48.240 --> 01:14:49.629
zeta squared, and so forth.
01:14:52.410 --> 01:14:52.920
Essentially.
01:14:52.920 --> 01:15:01.700
All I did was replace x with b
times x prime and m with zeta m
01:15:01.700 --> 01:15:02.990
prime.
01:15:02.990 --> 01:15:05.190
If I do that
throughout, you can see
01:15:05.190 --> 01:15:08.210
how the various
factors will change.
01:15:08.210 --> 01:15:11.020
So I didn't do all of
these integrations,
01:15:11.020 --> 01:15:13.580
et cetera that I did over here.
01:15:13.580 --> 01:15:17.230
I just did the dimensional
analysis, if you like.
01:15:17.230 --> 01:15:21.560
And within that dimensional
analysis now in real space,
01:15:21.560 --> 01:15:29.550
if I set k prime to be k, you
can see that zeta is d to the 2
01:15:29.550 --> 01:15:31.015
minus d over 2.
01:15:37.460 --> 01:15:42.670
And again, you can see
that once I have fixed k,
01:15:42.670 --> 01:15:46.150
all of the things that have the
same power of m but two higher
01:15:46.150 --> 01:15:49.670
derivatives would get a
factor of b to the minus 2,
01:15:49.670 --> 01:15:53.030
just as we had over here.
01:15:53.030 --> 01:15:55.410
Again, with this
choice, you can check
01:15:55.410 --> 01:16:00.420
that if I put it back here,
I would get b squared.
01:16:00.420 --> 01:16:07.320
But let's imagine that I have
a generalization of m to the n.
01:16:07.320 --> 01:16:12.950
If I have a term that
multiplies m to some power p--
01:16:12.950 --> 01:16:18.420
with the coefficient up-- then
under this kind of rescaling
01:16:18.420 --> 01:16:28.190
I will get up prime is b to the
d zeta to the power of p, up.
01:16:32.890 --> 01:16:36.030
And with this choice
of zeta, what do I get?
01:16:36.030 --> 01:16:38.840
I will get b to the d.
01:16:38.840 --> 01:16:48.464
And then I will get plus p
1 minus d over 2 times up.
01:16:51.750 --> 01:16:56.914
Which I can define to be b
to some power yp times up.
01:16:56.914 --> 01:16:57.872
Look here to make sure.
01:17:01.250 --> 01:17:07.230
So my yp, the
dimension of something
01:17:07.230 --> 01:17:11.790
that multiplies
m to some power p
01:17:11.790 --> 01:17:18.982
is simply p plus
d 1 minus p or 2.
01:17:24.910 --> 01:17:28.280
And let's check some things.
01:17:28.280 --> 01:17:29.205
I have y1.
01:17:32.190 --> 01:17:35.290
y1 would correspond to a
magnetic field, something
01:17:35.290 --> 01:17:38.760
that is proportional
to the m itself.
01:17:38.760 --> 01:17:43.860
And if I push p close to 1,
I will get 1 plus d over 2.
01:17:43.860 --> 01:17:49.374
And that is, indeed, the
yh that we had over here.
01:17:49.374 --> 01:17:52.552
1 plus d over 2.
01:17:52.552 --> 01:17:55.400
So this is yh.
01:17:55.400 --> 01:18:00.095
If I ask what is
multiplying m squared,
01:18:00.095 --> 01:18:02.060
I put p equals to 2 here.
01:18:02.060 --> 01:18:07.220
I will get 2, and then here
I would get 1 minus 2 over 2.
01:18:07.220 --> 01:18:09.589
So that's the same thing.
01:18:09.589 --> 01:18:11.630
This is the thing that we
were calling before yt.
01:18:14.640 --> 01:18:17.710
We didn't include any
nq term in the theory,
01:18:17.710 --> 01:18:19.110
didn't make sense to us.
01:18:19.110 --> 01:18:22.050
But we certainly
included the u that
01:18:22.050 --> 01:18:24.670
was multiplied in
m to the fourth.
01:18:24.670 --> 01:18:27.442
AUDIENCE: So is the p
[INAUDIBLE] in the yp?
01:18:32.876 --> 01:18:40.576
PROFESSOR: There is p times
1 plus d 1 minus p over 2.
01:18:40.576 --> 01:18:42.548
p over 2.
01:18:42.548 --> 01:18:44.027
Just rewrote it.
01:18:46.990 --> 01:18:51.900
If I look at 4, here would be 4.
01:18:51.900 --> 01:18:56.210
And then I would put 1 minus
4 over 2, which is 1 minus 2,
01:18:56.210 --> 01:18:57.750
which is minus 1.
01:18:57.750 --> 01:18:59.810
So I would get 4 minus z.
01:19:03.670 --> 01:19:09.302
If I look at y6, I
would get 6 minus 2d.
01:19:09.302 --> 01:19:09.843
And so forth.
01:19:15.140 --> 01:19:18.780
So if I just do
dimensional analysis
01:19:18.780 --> 01:19:22.080
and I say that I start with a
fixed point that corresponds
01:19:22.080 --> 01:19:27.740
to gradient of m squared,
and everybody else 0,
01:19:27.740 --> 01:19:30.880
and I ask, if in the
vicinity of the fixed point
01:19:30.880 --> 01:19:33.660
where k is fixed
and everybody else
01:19:33.660 --> 01:19:39.310
is 0 I put on a little bit
up any of these other terms,
01:19:39.310 --> 01:19:41.200
what happens?
01:19:41.200 --> 01:19:45.970
And I find that what happens
is that certainly the h
01:19:45.970 --> 01:19:48.285
term, the term that is
linear, will be relevant.
01:19:48.285 --> 01:19:53.250
The term that is m
squared is relevant.
01:19:53.250 --> 01:19:57.720
Whether or not all the
other terms in the series--
01:19:57.720 --> 01:20:00.320
like m to the fourth, m
to the sixth, et cetera--
01:20:00.320 --> 01:20:04.670
will be relevant
depends on dimension.
01:20:04.670 --> 01:20:10.240
So once more we've hit
this dimensional fork.
01:20:10.240 --> 01:20:13.720
So the term m to
the fourth that we
01:20:13.720 --> 01:20:19.090
said is crucial to getting this
theory to have some meaning--
01:20:19.090 --> 01:20:23.080
and there's no reason for it
to be absent-- is, in fact,
01:20:23.080 --> 01:20:24.580
relevant.
01:20:24.580 --> 01:20:27.030
In fact, close to
three dimensions
01:20:27.030 --> 01:20:29.700
you would say that that's
really the only other term that
01:20:29.700 --> 01:20:31.210
is relevant.
01:20:31.210 --> 01:20:34.890
And you'd say, well,
it's almost good enough.
01:20:34.890 --> 01:20:37.950
But almost good enough
is not sufficient.
01:20:37.950 --> 01:20:42.390
If we want to describe a
physical theory that has only
01:20:42.390 --> 01:20:47.170
two relevant directions, we
cannot use this fixed point,
01:20:47.170 --> 01:20:51.110
because this fixed point has
three relevant directions
01:20:51.110 --> 01:20:54.370
in three dimensions.
01:20:54.370 --> 01:21:01.070
We have to deal
with this somehow.
01:21:01.070 --> 01:21:03.080
So what will we do?
01:21:03.080 --> 01:21:08.110
Next is to explicitly
include this m to the fourth.
01:21:08.110 --> 01:21:11.100
In fact, we will include
all the other terms, also.
01:21:11.100 --> 01:21:13.810
But we will see that all the
other terms, all the higher
01:21:13.810 --> 01:21:16.670
powers, are irrelevant
in the same sense
01:21:16.670 --> 01:21:18.530
that all of these
higher derivative terms
01:21:18.530 --> 01:21:19.740
are irrelevant.
01:21:19.740 --> 01:21:22.950
But that m to the
fourth term is something
01:21:22.950 --> 01:21:25.760
that we really have
to take care of.
01:21:25.760 --> 01:21:27.310
And we will do that.