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PROFESSOR: OK, let's start.
00:00:25.010 --> 00:00:29.350
So today hopefully
we will finally
00:00:29.350 --> 00:00:31.810
calculate some exponents.
00:00:31.810 --> 00:00:34.600
We've been writing,
again and again,
00:00:34.600 --> 00:00:38.220
how to calculate partition
functions for systems,
00:00:38.220 --> 00:00:42.560
such as a magnet, by
integrating over configurations
00:00:42.560 --> 00:00:50.080
of all shapes of a
statistical field.
00:00:50.080 --> 00:00:58.090
And we have given weights
to these configurations that
00:00:58.090 --> 00:01:03.720
are constructed as some
kind of a function [? l ?]
00:01:03.720 --> 00:01:05.563
of these configurations.
00:01:08.550 --> 00:01:16.170
And the idea is that
presumably, if I could do this,
00:01:16.170 --> 00:01:18.680
then I could figure out
the singularities that
00:01:18.680 --> 00:01:22.140
are possible at a place
where, for example, I
00:01:22.140 --> 00:01:26.700
go from an unmagnetized
to a magnetized case.
00:01:26.700 --> 00:01:29.370
Now, one of the first
things that we noted
00:01:29.370 --> 00:01:35.980
was that in general, I can't
solve the types of Hamiltonians
00:01:35.980 --> 00:01:37.860
that I would like.
00:01:37.860 --> 00:01:41.290
And maybe what I should
do is to break it into two
00:01:41.290 --> 00:01:45.210
parts, a part that I will
treat perturbatively,
00:01:45.210 --> 00:01:47.690
and a part-- sorry,
a [INAUDIBLE]
00:01:47.690 --> 00:01:50.620
part that I can
calculate exactly,
00:01:50.620 --> 00:01:54.655
and a contribution that I can
then treat as a perturbation.
00:02:00.660 --> 00:02:03.840
Now, we saw that there
were difficulties
00:02:03.840 --> 00:02:06.890
if I attempted
straightforward perturbation
00:02:06.890 --> 00:02:09.039
type of calculations.
00:02:09.039 --> 00:02:13.870
And what we did
was to replace this
00:02:13.870 --> 00:02:18.420
with some kind of a
renormalization group approach.
00:02:18.420 --> 00:02:20.645
The idea was
something like this,
00:02:20.645 --> 00:02:24.450
that these statistical field
theories that we write down
00:02:24.450 --> 00:02:27.950
have been obtained by averaging
true microscopic degrees
00:02:27.950 --> 00:02:31.780
of freedom over some
characteristic landscape.
00:02:31.780 --> 00:02:33.940
So this field m
certainly does not
00:02:33.940 --> 00:02:37.790
have fluctuations that
are very short wavelength.
00:02:37.790 --> 00:02:41.420
And, for example, if we
were to describe things
00:02:41.420 --> 00:02:46.980
in the perspective of
Fourier components,
00:02:46.980 --> 00:02:50.510
presumably the variables
that I would have
00:02:50.510 --> 00:02:56.470
would have some maximum,
q, that is related
00:02:56.470 --> 00:02:58.650
to the inverse of
the wavelength.
00:02:58.650 --> 00:03:03.210
So there is some lambda.
00:03:03.210 --> 00:03:08.310
And if I were to in fact Fourier
transform my modes in terms
00:03:08.310 --> 00:03:15.440
of q, then these modes will be
defined [INAUDIBLE] this space.
00:03:15.440 --> 00:03:20.520
And for example,
my beta is zero.
00:03:20.520 --> 00:03:23.940
In the language of Fourier
modes would be the part
00:03:23.940 --> 00:03:29.610
that I can do exactly, which
is the part that is quadratic
00:03:29.610 --> 00:03:31.320
and Gaussian.
00:03:31.320 --> 00:03:37.250
And the q vectors would
be between the interval 0
00:03:37.250 --> 00:03:40.060
to whatever this lambda is.
00:03:40.060 --> 00:03:42.650
And the kind of thing
that I can do exactly
00:03:42.650 --> 00:03:44.270
are things that are quadratic.
00:03:44.270 --> 00:03:47.490
So I would have m of q squared.
00:03:47.490 --> 00:03:51.290
And then some
expansion, [INAUDIBLE]
00:03:51.290 --> 00:03:55.680
of q, that has a
constant plus tq squared
00:03:55.680 --> 00:03:59.920
and potentially higher
order [INAUDIBLE].
00:03:59.920 --> 00:04:04.730
So this is the Gaussian
theory that I can calculate.
00:04:04.730 --> 00:04:06.420
Problem with this
Gaussian theory
00:04:06.420 --> 00:04:10.870
is that it only is
meaningful for t positive.
00:04:10.870 --> 00:04:14.260
And in order to go to the
space where t is negative,
00:04:14.260 --> 00:04:18.390
I have to include higher order
terms in the magnetization,
00:04:18.390 --> 00:04:21.089
and those are non-perturbative.
00:04:21.089 --> 00:04:25.250
And for example, if I go back to
the description in real space,
00:04:25.250 --> 00:04:29.370
I was writing something
like um to the fourth
00:04:29.370 --> 00:04:32.780
plus higher order terms for
the expansion of this u.
00:04:35.680 --> 00:04:41.480
When we attempted to do
straightforward perturbative
00:04:41.480 --> 00:04:45.060
calculations, we encountered
some singularities.
00:04:45.060 --> 00:04:48.110
And the perturbation
didn't quite make sense.
00:04:48.110 --> 00:04:51.020
So we decided to combine
that with the idea
00:04:51.020 --> 00:04:53.360
of renormalization group.
00:04:53.360 --> 00:04:58.980
The idea there was to basically,
rather than integrate over
00:04:58.980 --> 00:05:02.590
all modes, to subdivide
the modes into two
00:05:02.590 --> 00:05:08.090
classes, the modes that
are long wavelength
00:05:08.090 --> 00:05:09.876
and I would like
to keep, and I'll
00:05:09.876 --> 00:05:15.026
call that m tilde, and the
modes that are sitting out here
00:05:15.026 --> 00:05:19.150
that I'm not interested
in because they give rise
00:05:19.150 --> 00:05:23.800
to no singularities that I
would like to get rid of.
00:05:23.800 --> 00:05:29.820
So my integration over
all set of configurations
00:05:29.820 --> 00:05:33.840
is really an integration
over both this m tilde
00:05:33.840 --> 00:05:36.670
and the sigma.
00:05:36.670 --> 00:05:42.590
And if I regard m tilde as
a span over wave numbers
00:05:42.590 --> 00:05:47.030
to either be m tilde or
sigma, I can basically
00:05:47.030 --> 00:05:49.960
write this as m
tilde plus sigma,
00:05:49.960 --> 00:05:52.180
and this is m tilde
plus sigma also.
00:05:56.280 --> 00:06:01.500
So this is just a rewriting
of the partition function
00:06:01.500 --> 00:06:06.990
where I have just changed
the names of the modes.
00:06:06.990 --> 00:06:10.860
Now, the first step in
the renormalization group
00:06:10.860 --> 00:06:15.740
is the coarse graining, which
is to average out fluctuations
00:06:15.740 --> 00:06:22.310
that have scale between a, and
let's say this in Fourier space
00:06:22.310 --> 00:06:26.030
is lambda over b, in real
space would be b times whatever
00:06:26.030 --> 00:06:32.440
your original base
scale was for average.
00:06:32.440 --> 00:06:35.100
So getting rid of
those modes would
00:06:35.100 --> 00:06:38.395
amount to basically changing
the scale over which
00:06:38.395 --> 00:06:41.630
you're averaging
by a factor of b.
00:06:41.630 --> 00:06:45.920
Once I do that, if I
can do the integration,
00:06:45.920 --> 00:06:48.880
what I will be left if
I integrate over sigma
00:06:48.880 --> 00:06:53.260
is just an integral
over m tilde.
00:06:53.260 --> 00:06:55.570
OK?
00:06:55.570 --> 00:06:59.780
Now, what would be the
form of this integration?
00:06:59.780 --> 00:07:01.980
The result.
00:07:01.980 --> 00:07:05.640
Well, first of all,
if I take the Gaussian
00:07:05.640 --> 00:07:10.760
and separate it out between
zero to lambda over b and lambda
00:07:10.760 --> 00:07:15.330
over b2 lambda and integrate
over the modes between lambda
00:07:15.330 --> 00:07:19.380
over b and lambda, just as
if I had the Gaussian, then
00:07:19.380 --> 00:07:21.600
I would get essentially
the contribution
00:07:21.600 --> 00:07:24.700
of the logarithm of
the determinants of all
00:07:24.700 --> 00:07:28.420
of these Gaussian
types of variances.
00:07:28.420 --> 00:07:32.120
So there will be a contribution
to the free energy that
00:07:32.120 --> 00:07:37.500
is effectively independent
of m tilde will depend
00:07:37.500 --> 00:07:41.627
on the rescaling factor
that you are looking at.
00:07:41.627 --> 00:07:42.460
But it's a constant.
00:07:42.460 --> 00:07:45.920
It doesn't depend on the
different configurations
00:07:45.920 --> 00:07:48.720
of the field m tilde.
00:07:48.720 --> 00:07:51.060
The other part of the
Gaussian-- so essentially,
00:07:51.060 --> 00:07:53.270
I wrote the Gaussian
as 0 to lambda
00:07:53.270 --> 00:07:55.970
over b and lambda
over b2 lambda.
00:07:55.970 --> 00:07:59.800
The part that is 0 over
lambda b will simply remain,
00:07:59.800 --> 00:08:02.110
so I will have [? beta 8 ?] 0.
00:08:02.110 --> 00:08:05.760
That now depends only
on these m tildes.
00:08:08.720 --> 00:08:14.130
Well, what do I have
to do with this term?
00:08:14.130 --> 00:08:19.350
So it's an integration over
sigma that has to be performed.
00:08:19.350 --> 00:08:22.340
I did the integration
by taking out this them
00:08:22.340 --> 00:08:25.080
as if it was a Gaussian.
00:08:25.080 --> 00:08:28.820
So effectively, the result
of the remaining integration
00:08:28.820 --> 00:08:33.110
is the average of
e to the minus u.
00:08:33.110 --> 00:08:38.390
And when I take it to the
log, I will get plus log of e
00:08:38.390 --> 00:08:44.690
to the minus u, which is a
function of m tilde and sigma,
00:08:44.690 --> 00:08:50.890
where I have integrated
out the modes that
00:08:50.890 --> 00:08:53.960
are out here, the sigmas.
00:08:53.960 --> 00:08:56.860
So it's only a
function of m tilde.
00:08:56.860 --> 00:08:59.980
They have been integrated
out using a Gaussian weight,
00:08:59.980 --> 00:09:01.580
such as the one that
I have over here.
00:09:06.700 --> 00:09:10.180
So that's formally exact.
00:09:10.180 --> 00:09:12.270
But it hasn't given
me any insights
00:09:12.270 --> 00:09:15.590
because I don't know
what that entity is.
00:09:15.590 --> 00:09:18.160
What I can do with
that entity is
00:09:18.160 --> 00:09:22.040
to make an expansion
powers of u.
00:09:22.040 --> 00:09:26.160
So I will have a minus
the average of u.
00:09:28.760 --> 00:09:32.450
And then the next term
would be the variance of u.
00:09:32.450 --> 00:09:39.870
So I will have the average of u
squared, average of u squared,
00:09:39.870 --> 00:09:42.530
and then higher order terms.
00:09:42.530 --> 00:09:49.090
So basically, this term can
be expanded as a power series
00:09:49.090 --> 00:09:51.250
as I have indicated.
00:09:51.250 --> 00:09:56.640
And again, just to make
sure, these averages
00:09:56.640 --> 00:09:59.750
are performed with
this Gaussian weight.
00:09:59.750 --> 00:10:03.090
And in particular, we've seen
that when we have a Gaussian
00:10:03.090 --> 00:10:11.300
weight, the different components
and the different q values
00:10:11.300 --> 00:10:13.690
are independent of each other.
00:10:13.690 --> 00:10:19.130
So I get here a delta alpha
beta, I get a delta of q plus q
00:10:19.130 --> 00:10:25.730
prime, and I will get
t plus k q squared
00:10:25.730 --> 00:10:28.190
and potentially higher
order powers of q
00:10:28.190 --> 00:10:30.570
that will appear in this series.
00:10:34.480 --> 00:10:37.360
Now, we kind of
started developing
00:10:37.360 --> 00:10:40.760
a diagrammatic perspective
on all of this.
00:10:40.760 --> 00:10:45.140
Something is m to
the fourth, since it
00:10:45.140 --> 00:10:49.980
was the dot product of
two factors of m squared,
00:10:49.980 --> 00:10:53.380
we demonstrate it as
a graph such as this.
00:10:53.380 --> 00:10:56.430
And we also introduced
a convention
00:10:56.430 --> 00:11:03.280
where there solid lines
would correspond to m tilde.
00:11:03.280 --> 00:11:09.230
Let's say wavy lines
would correspond to sigma.
00:11:09.230 --> 00:11:12.560
And essentially,
what I have to do
00:11:12.560 --> 00:11:17.630
is to write this
object according
00:11:17.630 --> 00:11:22.780
to this, where each
factor of m is replaced
00:11:22.780 --> 00:11:27.410
by two factors, which is the sum
of this entity and that entity
00:11:27.410 --> 00:11:29.690
diagrammatically.
00:11:29.690 --> 00:11:34.660
So that's two to the four,
or 16 different possibilities
00:11:34.660 --> 00:11:38.740
that I could have
once I expand this.
00:11:38.740 --> 00:11:42.430
And what was the answer that
we got for the first term
00:11:42.430 --> 00:11:44.170
in the series?
00:11:44.170 --> 00:11:51.450
So if I take u0 average, the
kind of diagrams that I can get
00:11:51.450 --> 00:11:55.180
is essentially keeping
this entity as it is.
00:11:55.180 --> 00:12:02.010
So essentially, I will get the
original potential that I have.
00:12:02.010 --> 00:12:04.380
Rather than m to the
fourth, I will simply
00:12:04.380 --> 00:12:07.010
have the equivalent m
tilde to the fourth.
00:12:07.010 --> 00:12:09.780
So basically,
diagrammatically this
00:12:09.780 --> 00:12:13.050
would correspond to this entity.
00:12:13.050 --> 00:12:15.890
There was a whole bunch
of things that cancel out
00:12:15.890 --> 00:12:21.260
to zero in the diagram that
I had with only one leg
00:12:21.260 --> 00:12:25.620
when I took the average
because I had a leg by itself,
00:12:25.620 --> 00:12:29.370
which would make it an odd
average, would give me zero.
00:12:29.370 --> 00:12:32.850
So I didn't have to
put any of these.
00:12:32.850 --> 00:12:38.050
And then I had
diagrams where I had
00:12:38.050 --> 00:12:42.030
two of the lines
replaced by wavy lines.
00:12:42.030 --> 00:12:47.610
And so then I would get
a contribution to u.
00:12:47.610 --> 00:12:53.545
There was a factor of 4n
plus-- sorry, 2n plus 4.
00:12:57.690 --> 00:13:01.624
That came from
diagrams in which I
00:13:01.624 --> 00:13:06.130
took two of the legs
that were together,
00:13:06.130 --> 00:13:11.560
and the other two I made wavy,
and I joined them together.
00:13:11.560 --> 00:13:13.810
And essentially,
I had the choice
00:13:13.810 --> 00:13:17.080
of picking this pair of
legs or that pair of legs,
00:13:17.080 --> 00:13:20.330
so that gave me a factor of two.
00:13:20.330 --> 00:13:24.020
And something that we will
see again and again, whenever
00:13:24.020 --> 00:13:27.430
we have a loop that
goes around by itself,
00:13:27.430 --> 00:13:30.470
it corresponds to something
like a delta alpha alpha,
00:13:30.470 --> 00:13:35.280
which, when you sum over alpha,
will give you a factor of n.
00:13:35.280 --> 00:13:38.320
The other
contribution, the four,
00:13:38.320 --> 00:13:47.260
came from diagrams
in which I had
00:13:47.260 --> 00:13:50.680
two wavy lines on
different branches.
00:13:50.680 --> 00:13:54.060
And since they came originally
from different branches,
00:13:54.060 --> 00:13:58.490
there wasn't a repeated helix to
sum and give me a factor of n.
00:13:58.490 --> 00:14:00.560
So I just have as
a factor of two
00:14:00.560 --> 00:14:04.380
from choice of one branch
or the other branch.
00:14:04.380 --> 00:14:06.780
So that was a factor of four.
00:14:06.780 --> 00:14:10.820
And then associated with
each one of these diagrams,
00:14:10.820 --> 00:14:16.680
there was then an integration
over the index, k,
00:14:16.680 --> 00:14:20.050
that characterized
these m tildes,
00:14:20.050 --> 00:14:23.460
in fact, the sigmas that
had been integrated over.
00:14:23.460 --> 00:14:27.780
So I would have an integral
from lambda over b2 lambda.
00:14:27.780 --> 00:14:32.210
Let's call that
dbk 2 pi to the d 1
00:14:32.210 --> 00:14:35.233
over the variance, which
is what I have here,
00:14:35.233 --> 00:14:37.275
t plus k, k squared,
[INAUDIBLE].
00:14:42.970 --> 00:14:46.940
There are diagrams then with
three wavy lines, which again
00:14:46.940 --> 00:14:49.770
gave me zero because
the average of three--
00:14:49.770 --> 00:14:53.799
average of an odd number with
a Gaussian weight is zero.
00:14:53.799 --> 00:14:55.340
And then there were
a bunch of things
00:14:55.340 --> 00:15:00.210
that would correspond
to all legs being wavy.
00:15:00.210 --> 00:15:02.590
There was something
like this, and there
00:15:02.590 --> 00:15:05.090
was something like this.
00:15:05.090 --> 00:15:09.980
And basically, I didn't
really have to calculate them.
00:15:09.980 --> 00:15:12.900
So I just wrote the
answer to those things
00:15:12.900 --> 00:15:17.530
as being a contribution
to the free energy
00:15:17.530 --> 00:15:20.380
and overall constant,
such as the constant
00:15:20.380 --> 00:15:24.720
that I have over here, but
not at the next order in u,
00:15:24.720 --> 00:15:26.343
but independent of
the configurations.
00:15:29.420 --> 00:15:33.830
So this was straightforward
perturbation.
00:15:33.830 --> 00:15:35.730
I forgot something
very important
00:15:35.730 --> 00:15:49.070
here, which is that this
entire coefficient was also
00:15:49.070 --> 00:15:53.770
coupled to these solid lines,
whose meaning is that it
00:15:53.770 --> 00:16:01.540
is an integral over q 2 pi to
the d m tilde of q squared,
00:16:01.540 --> 00:16:04.880
where the waves and
numbers that are sitting
00:16:04.880 --> 00:16:09.133
on these solid lines naturally
run from 0 to lambda.
00:16:14.340 --> 00:16:26.370
So we can see that if I were to
add this to what I have above,
00:16:26.370 --> 00:16:30.080
I see that my z has
now been written
00:16:30.080 --> 00:16:32.940
as an integral over
these modes that I'm
00:16:32.940 --> 00:16:41.080
keeping of a new weight that
I will call beta h tilde,
00:16:41.080 --> 00:16:42.290
depending on these m tildes.
00:16:46.350 --> 00:16:53.240
Where this beta h tilde is,
first of all, these terms
00:16:53.240 --> 00:16:55.180
that are proportional.
00:16:55.180 --> 00:16:57.940
That's with a v here also.
00:16:57.940 --> 00:17:01.670
To contributions of
the free energy coming
00:17:01.670 --> 00:17:04.420
from the modes that I
have integrated out,
00:17:04.420 --> 00:17:07.869
either at the zero order or
at the first order so far.
00:17:11.300 --> 00:17:22.329
I have the u, exactly the
same u as I had before,
00:17:22.329 --> 00:17:24.089
but now acting on m tilde.
00:17:24.089 --> 00:17:27.230
So four factors of m tilde.
00:17:27.230 --> 00:17:34.330
The only thing that happened is
that the Gaussian contribution
00:17:34.330 --> 00:17:42.230
now running from 0
to lambda over b,
00:17:42.230 --> 00:17:49.650
that is proportional to
m tilde of q squared,
00:17:49.650 --> 00:17:54.490
is now still a series, such
as the one that I had before,
00:17:54.490 --> 00:17:57.310
where the coefficient that
was a constant has changed.
00:18:00.080 --> 00:18:02.020
All the other terms
in the series,
00:18:02.020 --> 00:18:04.600
the term that is
proportional to q squared,
00:18:04.600 --> 00:18:09.060
q to the fourth, et cetera,
are left exactly as before.
00:18:11.670 --> 00:18:19.600
So what happened is that this
beta h tilde pretty much looks
00:18:19.600 --> 00:18:23.010
like the beta h
that I started with,
00:18:23.010 --> 00:18:26.030
with the only difference
being that t tilde is
00:18:26.030 --> 00:18:30.110
t plus essentially
what I have over there,
00:18:30.110 --> 00:18:38.190
4u n plus 2 integral
lambda over b2 lambda
00:18:38.190 --> 00:18:46.120
ddk 2 pi to the d, 1 over t
plus k, k squared, and so forth.
00:18:49.530 --> 00:18:53.000
But quite importantly,
the parameter
00:18:53.000 --> 00:18:57.490
that I would associate with
coefficient of q squared
00:18:57.490 --> 00:18:59.690
is left unchanged.
00:18:59.690 --> 00:19:02.580
If I had a coefficient
of q to the fourth,
00:19:02.580 --> 00:19:05.060
its coefficient
would be unchanged.
00:19:05.060 --> 00:19:07.790
And I have a coefficient for u.
00:19:07.790 --> 00:19:11.810
Its coefficient
is unchanged also.
00:19:11.810 --> 00:19:14.370
So the only thing
that happened is
00:19:14.370 --> 00:19:20.800
that the parameter that
corresponded to t got modified.
00:19:20.800 --> 00:19:24.160
And you actually
should recognize this
00:19:24.160 --> 00:19:28.250
as the inverse
susceptibility, if I
00:19:28.250 --> 00:19:33.290
were to integrate all
the way from 0 to lambda.
00:19:33.290 --> 00:19:37.630
And when we did that, this
contribution was singular.
00:19:37.630 --> 00:19:40.265
And that's why straightforward
perturbation theory
00:19:40.265 --> 00:19:41.770
didn't make sense.
00:19:41.770 --> 00:19:44.040
But now we are not
integrating to 0,
00:19:44.040 --> 00:19:46.160
which would have
given the singularity.
00:19:46.160 --> 00:19:48.560
We are just integrating
over the shell
00:19:48.560 --> 00:19:52.680
that I have indicated outside.
00:19:52.680 --> 00:20:02.920
So this step was the first
step of renormalization group
00:20:02.920 --> 00:20:04.450
that we call coarse graining.
00:20:11.500 --> 00:20:16.080
But rg had two other steps.
00:20:16.080 --> 00:20:19.270
That was rescale.
00:20:19.270 --> 00:20:23.180
Basically, the
theory that I have
00:20:23.180 --> 00:20:27.090
has a cut-off that
is lambda over b.
00:20:27.090 --> 00:20:31.290
So it looks grainier
in real space.
00:20:31.290 --> 00:20:35.250
So what I can do in real
space is to shrink it.
00:20:35.250 --> 00:20:40.170
In Fourier space, I have
to blow up my momenta.
00:20:40.170 --> 00:20:43.770
So essentially,
whenever I see q,
00:20:43.770 --> 00:20:47.730
I replace it with the
inverse q prime so
00:20:47.730 --> 00:20:53.640
that q prime, that is bq, runs
from zero to lambda, restoring
00:20:53.640 --> 00:20:57.180
the cut-off that
I had originally.
00:20:57.180 --> 00:21:05.740
And the next step
was to renormalize,
00:21:05.740 --> 00:21:15.520
which amounted to replacing the
field m tilde with a new field
00:21:15.520 --> 00:21:20.362
m prime after multiplying or
rescaling by a factor of z
00:21:20.362 --> 00:21:21.070
to be determined.
00:21:24.680 --> 00:21:28.720
Now, this amounts to simple
dimensional analysis.
00:21:28.720 --> 00:21:33.360
So I go back into my equation,
and whenever I see q,
00:21:33.360 --> 00:21:36.510
I replace it with
b inverse q prime.
00:21:36.510 --> 00:21:40.740
So from the integration, I get
a factor of b to the minus d,
00:21:40.740 --> 00:21:45.760
multiplying t tilde, replace
m tilde by z times m prime.
00:21:45.760 --> 00:21:48.140
So that's two factors of z.
00:21:48.140 --> 00:21:54.150
So what I get is that t prime
is z squared b to the minus d,
00:21:54.150 --> 00:21:56.830
this t tilde that I
have indicated above.
00:22:00.290 --> 00:22:05.960
Now, k prime is
also something that
00:22:05.960 --> 00:22:07.570
appears in the Gaussian term.
00:22:07.570 --> 00:22:08.970
So it has a z squared.
00:22:08.970 --> 00:22:11.310
It came from two factors of m.
00:22:11.310 --> 00:22:15.870
But because it had an
additional factor of q squared
00:22:15.870 --> 00:22:23.720
rather than b to the minus d,
it is b to the minus d minus 2.
00:22:23.720 --> 00:22:28.740
And I can do the same analysis
for higher order terms going
00:22:28.740 --> 00:22:32.150
with higher powers of
q in the expansion that
00:22:32.150 --> 00:22:33.240
appears in the Gaussian.
00:22:35.830 --> 00:22:38.020
But then we get to
the non-linear terms,
00:22:38.020 --> 00:22:42.550
and the first linearity that
we have kept is this u prime.
00:22:42.550 --> 00:22:47.170
And what we see is it goes
with four factors of m.
00:22:47.170 --> 00:22:49.960
So there will be
z to the fourth.
00:22:49.960 --> 00:22:53.400
If I write things
in Fourier space,
00:22:53.400 --> 00:22:56.210
m to the fourth in real
space in Fourier space
00:22:56.210 --> 00:23:00.410
would involve m of
g1, m of q2, m of q3.
00:23:00.410 --> 00:23:04.660
And the fourth m, that is
minus q1, minus q2, minus q3.
00:23:04.660 --> 00:23:06.740
But there will be
three integrations
00:23:06.740 --> 00:23:10.730
over q, which gives me three
factors of b to the minus 3.
00:23:14.280 --> 00:23:17.500
So these are pretty much
exactly what we had already
00:23:17.500 --> 00:23:23.850
seen for the Gaussian model--
forgot the k-- except that we
00:23:23.850 --> 00:23:27.840
replaced this t that was
appearing for the Gaussian
00:23:27.840 --> 00:23:30.480
model with t tilde which
is what I have up here.
00:23:34.440 --> 00:23:38.860
Now, you have to choose
z such that the theory
00:23:38.860 --> 00:23:45.300
looks as much as possible as
the original way that I had.
00:23:45.300 --> 00:23:48.590
And as I mentioned,
our anchoring point
00:23:48.590 --> 00:23:50.730
would be the Gaussian.
00:23:50.730 --> 00:23:52.680
So for the Gaussian
model, we saw
00:23:52.680 --> 00:23:56.240
that the appropriate choice,
so that ultimately we
00:23:56.240 --> 00:24:01.240
were left with the right
number of relevant directions
00:24:01.240 --> 00:24:05.820
was to set this
combination to 1, which
00:24:05.820 --> 00:24:10.250
means that I have
to choose z to be
00:24:10.250 --> 00:24:12.270
v to the power of
1 plus d over 2.
00:24:17.522 --> 00:24:22.300
Now, once I choose
that factor for z,
00:24:22.300 --> 00:24:24.430
everything else
becomes determined.
00:24:24.430 --> 00:24:30.930
This clearly has two factors of
b with respect to the original.
00:24:30.930 --> 00:24:32.575
So this becomes b squared.
00:24:35.890 --> 00:24:39.570
This you have to do
a little bit of work.
00:24:39.570 --> 00:24:42.230
Z to the fourth
would be b to the 4
00:24:42.230 --> 00:24:47.150
plus 2d, then minus 3d
gives me b to the 4 minus d.
00:24:50.480 --> 00:24:55.460
And I can similarly determine
what the dimensions would
00:24:55.460 --> 00:25:01.220
be for additional terms
that appear in the Gaussian,
00:25:01.220 --> 00:25:03.855
as well as additional
nonlinearities that
00:25:03.855 --> 00:25:04.910
could appear here.
00:25:04.910 --> 00:25:09.600
All of them, by this analysis,
I can assign some power of b.
00:25:14.220 --> 00:25:19.680
So this completes
the rg in the sense
00:25:19.680 --> 00:25:24.340
that at least at this order
in perturbation theory,
00:25:24.340 --> 00:25:27.420
I started with my
original theory,
00:25:27.420 --> 00:25:30.660
and I see how the
parameters of the new theory
00:25:30.660 --> 00:25:35.490
are obtained if I were to
rescale and renormalize
00:25:35.490 --> 00:25:36.691
by this factor of b.
00:25:39.280 --> 00:25:43.600
Now, we did one thing else,
which is quite common,
00:25:43.600 --> 00:25:46.410
which is rather than
choosing factors
00:25:46.410 --> 00:25:53.780
like b equals to 2 or 3, making
b to be infinitesimally small,
00:25:53.780 --> 00:25:58.740
at least on the picture
that I have over there.
00:25:58.740 --> 00:26:05.300
What I'm doing is I'm
making this b very close
00:26:05.300 --> 00:26:08.800
to 1, which means
that effectively I'm
00:26:08.800 --> 00:26:10.470
putting the modes
that I'm getting
00:26:10.470 --> 00:26:15.030
rid of in a tiny
shell around lambda.
00:26:17.710 --> 00:26:24.350
So I have chosen b to be
slightly larger than 1
00:26:24.350 --> 00:26:26.720
by an amount delta l.
00:26:26.720 --> 00:26:29.460
And I expect that
all of the parameters
00:26:29.460 --> 00:26:33.180
will also change
very slightly, such
00:26:33.180 --> 00:26:38.430
that this t prime
evaluated at scale v
00:26:38.430 --> 00:26:42.660
would be what I had
originally, plus something
00:26:42.660 --> 00:26:45.640
that vanishes as
delta l goes to zero
00:26:45.640 --> 00:26:50.510
and presumably is linear
in delta l dt by dl.
00:26:50.510 --> 00:26:56.741
And similarly, I can
do the same thing for u
00:26:56.741 --> 00:27:01.540
and all the other
parameters of the theory.
00:27:01.540 --> 00:27:08.300
Once I do that, these
jumps from one parameter
00:27:08.300 --> 00:27:14.230
to another parameter can
be translated into flows.
00:27:14.230 --> 00:27:21.270
And, for example, dt by
dl gets a contribution
00:27:21.270 --> 00:27:25.350
from writing b squared
as 1 plus 2 delta l.
00:27:25.350 --> 00:27:30.310
That is proportional
to 2 times t.
00:27:30.310 --> 00:27:32.590
And then there's
another contribution
00:27:32.590 --> 00:27:35.020
that is order of delta l.
00:27:35.020 --> 00:27:39.750
Clearly, if b equals to 1,
this integral would vanish.
00:27:39.750 --> 00:27:43.580
So if b is very close
to 1, this integral
00:27:43.580 --> 00:27:45.920
is off the order of delta l.
00:27:45.920 --> 00:27:50.560
And what it is is just
evaluating the integrand when
00:27:50.560 --> 00:27:55.910
k equals to lambda at the
shell, and then multiplying
00:27:55.910 --> 00:27:58.760
by the volume of
that shell, which
00:27:58.760 --> 00:28:01.450
is the surface area
times the thickness.
00:28:01.450 --> 00:28:05.430
So I will get from here a
contribution order of delta l.
00:28:05.430 --> 00:28:12.270
I have divided through by
delta l, which is 4u m plus 2 1
00:28:12.270 --> 00:28:17.650
over t plus k lambda squared
[INAUDIBLE] integrand.
00:28:17.650 --> 00:28:22.460
And then I have the surface
area divided by 2 pi to the d
00:28:22.460 --> 00:28:25.390
that we have always called kd.
00:28:25.390 --> 00:28:28.970
And then lambda to
the d is the product
00:28:28.970 --> 00:28:31.710
of lambda to the d
minus 1 and lambda
00:28:31.710 --> 00:28:34.390
delta l, which comes
from the thickness.
00:28:34.390 --> 00:28:36.860
The delta l I have taken out.
00:28:36.860 --> 00:28:40.960
And this whole thing is the
order of u contribution.
00:28:40.960 --> 00:28:43.100
And then they had
a term that is du
00:28:43.100 --> 00:28:48.920
by dl, which is 4
minus d times u.
00:28:48.920 --> 00:28:57.070
So this is the result of
doing this perturbative rg
00:28:57.070 --> 00:28:59.660
to the lowest order
in this parameter u.
00:29:02.970 --> 00:29:08.790
Now, these things are really
the important parameters.
00:29:08.790 --> 00:29:11.970
There will be other parameters
that I have not specifically
00:29:11.970 --> 00:29:13.120
written down.
00:29:13.120 --> 00:29:16.390
And next lecture, we will
deal with all of them.
00:29:16.390 --> 00:29:18.900
But let's focus on these two.
00:29:18.900 --> 00:29:21.180
So I have one
parameter, which is
00:29:21.180 --> 00:29:26.270
t, the other
parameter, which is u.
00:29:26.270 --> 00:29:34.570
But u can only be positive
for the theory to make sense.
00:29:34.570 --> 00:29:37.930
I said that originally the
Gaussian theory only makes
00:29:37.930 --> 00:29:42.860
sense if t is positive because
once t becomes negative,
00:29:42.860 --> 00:29:47.470
then the weight gets shifted
to large values of m.
00:29:47.470 --> 00:29:48.870
It is unphysical.
00:29:48.870 --> 00:29:52.220
So for physicalness of
the Gaussian theory,
00:29:52.220 --> 00:29:57.960
I need to confine myself
to the t positive plane.
00:29:57.960 --> 00:30:01.920
Now that I have u, I can
have t that is negative
00:30:01.920 --> 00:30:04.635
and um to the fourth,
as long as u positive,
00:30:04.635 --> 00:30:08.910
will make the
weight well behaved.
00:30:08.910 --> 00:30:12.900
So this entire plane
is now accessible.
00:30:12.900 --> 00:30:15.570
Within this plane,
there is a point
00:30:15.570 --> 00:30:18.480
which corresponds to
a fixed point, a point
00:30:18.480 --> 00:30:22.550
that if I'm at that location,
then the parameters no longer
00:30:22.550 --> 00:30:23.410
change.
00:30:23.410 --> 00:30:28.230
Clearly, if u does not change, u
at the fixed point should be 0.
00:30:28.230 --> 00:30:31.130
If u at the fixed point is
0 and t does not change,
00:30:31.130 --> 00:30:33.000
t at the fixed point is 0.
00:30:33.000 --> 00:30:36.760
So this is the fixed point.
00:30:36.760 --> 00:30:40.620
Since I'm looking at a
two-dimensional projection,
00:30:40.620 --> 00:30:44.830
there will be two
eigendirections associated
00:30:44.830 --> 00:30:47.920
with moving away from
this fixed point.
00:30:47.920 --> 00:30:52.180
If I stick with the
axis where u is 0,
00:30:52.180 --> 00:30:54.650
you can see that u will stay 0.
00:30:54.650 --> 00:30:57.980
But then dt by dl is 2t.
00:30:57.980 --> 00:31:01.430
So if I'm on the axis
that u equals to 0,
00:31:01.430 --> 00:31:03.590
I will stay on this axis.
00:31:03.590 --> 00:31:06.540
So that's one of
my eigendirections.
00:31:06.540 --> 00:31:09.850
And along this
eigendirection, I will
00:31:09.850 --> 00:31:15.080
be flowing out with
an eigenvalue of 2.
00:31:18.110 --> 00:31:24.380
Now, in general however, let's
say if I go to t equals to 0,
00:31:24.380 --> 00:31:28.860
you can see that if t is
0, but u positive dt by dl
00:31:28.860 --> 00:31:30.330
is positive.
00:31:30.330 --> 00:31:33.640
So basically, the
u direction you
00:31:33.640 --> 00:31:39.660
will be going if you start
on the t equals to 0 axis,
00:31:39.660 --> 00:31:42.000
you will generate a positive t.
00:31:42.000 --> 00:31:44.850
And the typical
flows that you would
00:31:44.850 --> 00:31:47.480
have would be in this direction.
00:31:47.480 --> 00:31:49.960
Actually, I should draw
it with a different color.
00:31:49.960 --> 00:31:54.070
So quite generically,
the flows are like this.
00:31:57.740 --> 00:32:08.960
But there is a direction along
which the flow is preserved.
00:32:08.960 --> 00:32:11.330
So there is a straight line.
00:32:11.330 --> 00:32:14.530
This straight line you can
calculate by setting dt
00:32:14.530 --> 00:32:20.136
by dl divided by du by dl
to be the ratio of t over u.
00:32:20.136 --> 00:32:23.650
You can very easily
find that it corresponds
00:32:23.650 --> 00:32:26.790
to a line of t
being proportional
00:32:26.790 --> 00:32:28.940
to u with a negative slope.
00:32:28.940 --> 00:32:31.350
And the eigenvalue
along that direction
00:32:31.350 --> 00:32:34.540
is determined by 4 minus d.
00:32:34.540 --> 00:32:37.640
So that the picture that I
have actually drawn for you
00:32:37.640 --> 00:32:42.140
here corresponds to
dimensions greater than four.
00:32:42.140 --> 00:32:47.000
In dimensions greater than four
along this other direction,
00:32:47.000 --> 00:32:49.940
you will be flowing
towards the fixed point.
00:32:49.940 --> 00:32:52.790
And in general, the flows
look something like this.
00:33:00.980 --> 00:33:02.530
So what does that mean?
00:33:02.530 --> 00:33:05.490
Again, the whole thing
that we wrote down
00:33:05.490 --> 00:33:07.400
was supposed to
describe something
00:33:07.400 --> 00:33:10.770
like a magnet at
some temperature.
00:33:10.770 --> 00:33:13.910
So when I fix my
temperature of the magnet,
00:33:13.910 --> 00:33:18.680
I presumably reside at
some particular point
00:33:18.680 --> 00:33:20.800
on this diagram.
00:33:20.800 --> 00:33:24.330
Let's say in the
phase that is up here,
00:33:24.330 --> 00:33:28.860
eventually I can see that I
go to large t and u goes to 0.
00:33:28.860 --> 00:33:31.845
So the eventual weight is
very much like a Gaussian,
00:33:31.845 --> 00:33:34.760
e to the tm squared over 2.
00:33:34.760 --> 00:33:38.030
So this is essentially
independent patches
00:33:38.030 --> 00:33:42.360
of the system randomly pointing
to different directions.
00:33:42.360 --> 00:33:46.560
If I change my system to
have a lower temperature,
00:33:46.560 --> 00:33:49.720
I will be looking at
a point such as this.
00:33:49.720 --> 00:33:51.200
As I lower the
temperature, I will
00:33:51.200 --> 00:33:54.600
be looking at some
other point presumably.
00:33:54.600 --> 00:33:56.880
But all of these
points that correspond
00:33:56.880 --> 00:33:59.630
to lowering
temperatures, if I also
00:33:59.630 --> 00:34:05.200
now look at increasing land
scale, will flow up here.
00:34:05.200 --> 00:34:09.340
Presumably, if I
go below tc, I will
00:34:09.340 --> 00:34:13.800
be flowing in the other
direction, where t is negative,
00:34:13.800 --> 00:34:16.400
and then the u is
needed for stability,
00:34:16.400 --> 00:34:19.139
which means that I have
to spontaneously choose
00:34:19.139 --> 00:34:21.880
a direction in which
I order things.
00:34:21.880 --> 00:34:27.730
So the benefit of doing this
renormalization and this study
00:34:27.730 --> 00:34:32.530
was that in the absence
of u, I could not
00:34:32.530 --> 00:34:36.350
achieve the low temperature
part of the system.
00:34:36.350 --> 00:34:40.000
With the addition of u, I
can describe both sides,
00:34:40.000 --> 00:34:44.150
and I can see on the
rescaling which set of points
00:34:44.150 --> 00:34:46.880
go to what is the analog of
the high temperature, which
00:34:46.880 --> 00:34:50.650
set of points go to what is
the analog of low temperature.
00:34:50.650 --> 00:34:53.750
And the point that
corresponds to the transition
00:34:53.750 --> 00:34:58.390
between the two is on the basing
of attraction of the Gaussian
00:34:58.390 --> 00:35:01.360
fixed point that
is asymptotically,
00:35:01.360 --> 00:35:04.730
the theory would be described
by just gradient of m squared.
00:35:07.260 --> 00:35:11.130
But this picture does
not work if I go too
00:35:11.130 --> 00:35:12.840
d that is less than four.
00:35:15.670 --> 00:35:20.320
And d less than four,
I can again draw u.
00:35:20.320 --> 00:35:24.040
I can draw t.
00:35:24.040 --> 00:35:30.110
And I will again find
the fixed point at 0, 0.
00:35:30.110 --> 00:35:34.350
I will again find
an eigendirection,
00:35:34.350 --> 00:35:40.640
at u equals to 0, which pushes
things out along the u equals
00:35:40.640 --> 00:35:42.490
to 0 axis.
00:35:42.490 --> 00:35:46.380
Going from d of above
four to d of below four
00:35:46.380 --> 00:35:48.630
does not really
materially change
00:35:48.630 --> 00:35:52.660
the location of this other
eigendirection by much.
00:35:52.660 --> 00:35:55.480
It pretty much
stays where it was.
00:35:55.480 --> 00:36:00.910
The thing that it does
change is the eigenvalue.
00:36:00.910 --> 00:36:05.590
So basically, here I
will find that the flow
00:36:05.590 --> 00:36:07.070
is in this direction.
00:36:07.070 --> 00:36:11.550
And if I were to generalize
the picture that I have,
00:36:11.550 --> 00:36:17.740
I would get things that
would be going like this
00:36:17.740 --> 00:36:19.455
or going like this.
00:36:23.770 --> 00:36:26.820
Once again, there are
a set of trajectories
00:36:26.820 --> 00:36:29.300
that go on one side,
a set of trajectories
00:36:29.300 --> 00:36:31.150
that go on the other side.
00:36:31.150 --> 00:36:35.390
And presumably, by
changing temperature,
00:36:35.390 --> 00:36:37.540
I will cross from one
set of trajectories
00:36:37.540 --> 00:36:40.520
to the other set
of trajectories.
00:36:40.520 --> 00:36:43.250
But the thing is
that the point that
00:36:43.250 --> 00:36:47.530
corresponds to hitting the
basin that separates the two
00:36:47.530 --> 00:36:53.050
sets of trajectories, I don't
know what it corresponds to.
00:36:53.050 --> 00:36:56.100
Here, for d greater
than 4, it went
00:36:56.100 --> 00:36:57.930
to the Gaussian fixed point.
00:36:57.930 --> 00:37:00.350
Here currently, I don't
know where it is going.
00:37:02.970 --> 00:37:09.590
So I have no understanding at
this level of what the scale
00:37:09.590 --> 00:37:13.960
invariant properties are
that describe magnets
00:37:13.960 --> 00:37:17.720
in three dimensions at
their critical temperature.
00:37:23.030 --> 00:37:27.960
Now, the thing is that the
resolution and everything
00:37:27.960 --> 00:37:36.230
that we need comes from
staring more at this expansion
00:37:36.230 --> 00:37:37.860
that we had.
00:37:37.860 --> 00:37:42.060
We can see that this is an
alternating theory because I
00:37:42.060 --> 00:37:44.630
started with e to the minus u.
00:37:44.630 --> 00:37:50.140
And so the next term is likely
to have the opposite sign
00:37:50.140 --> 00:37:52.450
to the first term.
00:37:52.450 --> 00:37:58.380
So I anticipate that at the end
of doing the calculation, if I
00:37:58.380 --> 00:38:02.340
go to the next order,
there will be a term here
00:38:02.340 --> 00:38:06.270
that is minus vu squared.
00:38:06.270 --> 00:38:09.940
Actually, there will be a
contribution to dt by dl also
00:38:09.940 --> 00:38:14.590
that is minus, let's
say, au squared.
00:38:14.590 --> 00:38:18.140
So I expect that if I were to
do things at the next order,
00:38:18.140 --> 00:38:22.040
and we will do that
in about 15 minutes,
00:38:22.040 --> 00:38:25.140
I will get these kinds of terms.
00:38:25.140 --> 00:38:27.660
Once I have that
kind of term, you
00:38:27.660 --> 00:38:33.160
can see that I anticipate
then a fixed point occurring
00:38:33.160 --> 00:38:38.699
at the location u star, which
is 4 minus d divided by b.
00:38:42.830 --> 00:38:47.020
And then, by looking in the
vicinity of this fixed point,
00:38:47.020 --> 00:38:49.460
I should be able to
determine everything
00:38:49.460 --> 00:38:53.400
that I need about
the phase transition.
00:38:53.400 --> 00:39:00.140
But then you can ask, is this
a legitimate thing to do?
00:39:00.140 --> 00:39:04.810
I have to make sure I do
things self consistently.
00:39:04.810 --> 00:39:07.580
I did a perturbation
theory assuming
00:39:07.580 --> 00:39:11.990
that u is a small quantity,
so that I can organize things
00:39:11.990 --> 00:39:16.960
in power of u, u
squared, u cubed.
00:39:16.960 --> 00:39:23.740
But what does it mean that I
have control over powers of u?
00:39:23.740 --> 00:39:28.550
Once I have landed at this
fixed point, where at the fixed
00:39:28.550 --> 00:39:33.600
point, u has a value that
is fixed and determined.
00:39:33.600 --> 00:39:37.120
It is this 4 minus d over b.
00:39:37.120 --> 00:39:41.340
So in order for the series
to make sense and be
00:39:41.340 --> 00:39:45.080
under control, I
need this u star
00:39:45.080 --> 00:39:49.800
to be under control
as a small parameter.
00:39:49.800 --> 00:39:53.560
So what knob do I have
to ensure that this u
00:39:53.560 --> 00:39:56.180
star is a small parameter?
00:39:56.180 --> 00:39:59.800
Turns out that practically
the only knob that I have
00:39:59.800 --> 00:40:02.540
is that this 4 minus
d should be small.
00:40:05.070 --> 00:40:09.830
So I can only make this
into a systematic theory
00:40:09.830 --> 00:40:15.300
by making it into an expansion
in a small quantity, which
00:40:15.300 --> 00:40:18.130
is 4 minus d.
00:40:18.130 --> 00:40:19.920
Let's call that epsilon.
00:40:19.920 --> 00:40:23.490
And now we can hopefully,
at the end of the day,
00:40:23.490 --> 00:40:28.940
keep track of appropriate
powers of epsilon.
00:40:28.940 --> 00:40:31.940
So the Gaussian theory
describes properly
00:40:31.940 --> 00:40:34.900
the behavior at four dimensions.
00:40:34.900 --> 00:40:38.340
At 4 minus epsilon
dimensions, I can
00:40:38.340 --> 00:40:40.820
figure out where
this fixed point is
00:40:40.820 --> 00:40:44.630
and calculate things correctly.
00:40:44.630 --> 00:40:46.500
All right?
00:40:46.500 --> 00:40:50.230
So that means that I need
to do this calculation
00:40:50.230 --> 00:40:51.500
of the variance of u.
00:40:54.720 --> 00:41:00.600
So what I will do here
is to draw a diagram
00:41:00.600 --> 00:41:03.250
to help me do that.
00:41:03.250 --> 00:41:04.990
So let's do something like this.
00:41:29.510 --> 00:41:32.510
OK, let's do
something like this.
00:42:06.760 --> 00:42:10.830
Six, seven rows
and seven columns.
00:42:10.830 --> 00:42:15.060
The first row is to just tell
you what we are going to plot.
00:42:15.060 --> 00:42:19.420
So basically, I need
a u squared average,
00:42:19.420 --> 00:42:23.750
which means that I need
to have two factors of u.
00:42:23.750 --> 00:42:28.838
Each one of them depends
on m tilde and sigma.
00:42:28.838 --> 00:42:35.260
And so I will
indicate the two sets.
00:42:35.260 --> 00:42:38.920
Actually already,
we saw when we were
00:42:38.920 --> 00:42:43.210
doing the case of the
first order calculation,
00:42:43.210 --> 00:42:48.600
how to decompose this
object that has four lines.
00:42:48.600 --> 00:42:53.920
And we said, well, the first
thing that I can do is to just
00:42:53.920 --> 00:42:56.840
use the m's.
00:42:56.840 --> 00:42:58.840
The next thing
that I can do is I
00:42:58.840 --> 00:43:02.810
could replace one of
the m's with a sigma.
00:43:02.810 --> 00:43:07.030
And there was a choice
of four ways to do so.
00:43:07.030 --> 00:43:15.040
Or I could choose to replace
two of the m's with wavy lines.
00:43:15.040 --> 00:43:17.900
And question was, the right
branch or the left branch?
00:43:17.900 --> 00:43:20.590
So there's two of these.
00:43:20.590 --> 00:43:28.190
I could put the wavy lines
on two different branches.
00:43:28.190 --> 00:43:32.820
And there was four
ways to do this one.
00:43:32.820 --> 00:43:40.120
I could have three wavy
lines, and the one solid line
00:43:40.120 --> 00:43:44.260
could then be in one
of four positions.
00:43:44.260 --> 00:43:52.250
Or I had all wavy
lines, so there is this.
00:43:52.250 --> 00:43:57.320
So that's one of my factors of u
on the vertical for this table.
00:43:57.320 --> 00:44:00.290
On the horizontal, I
will have the same thing.
00:44:00.290 --> 00:44:03.400
I will have one of these.
00:44:03.400 --> 00:44:08.080
I will have four of these.
00:44:08.080 --> 00:44:13.260
I will have two of these.
00:44:13.260 --> 00:44:16.490
I will have four of these.
00:44:19.150 --> 00:44:28.747
I will have four of these, and
one which is all wavy lines.
00:44:36.380 --> 00:44:40.670
Now I have to put
two of these together
00:44:40.670 --> 00:44:43.900
and then do the average.
00:44:43.900 --> 00:44:47.590
Now clearly, if I put
two of these together,
00:44:47.590 --> 00:44:49.290
there's no average to be done.
00:44:49.290 --> 00:44:52.770
I will get something that
is order of m to the fourth.
00:44:52.770 --> 00:44:56.410
But remember that I'm
calculating the variance.
00:44:56.410 --> 00:45:00.110
So that would subtract
from the average squared
00:45:00.110 --> 00:45:01.540
of the same quantity.
00:45:01.540 --> 00:45:03.790
It's a disconnected piece.
00:45:03.790 --> 00:45:07.640
And I have stated that
anything that is disconnected
00:45:07.640 --> 00:45:09.820
will not contribute.
00:45:09.820 --> 00:45:13.460
And in particular, there is no
way to join this to anything.
00:45:13.460 --> 00:45:17.660
So everything that we
log here in this row
00:45:17.660 --> 00:45:21.040
would correspond
to no contribution
00:45:21.040 --> 00:45:25.190
once I have subtracted out
the average of u squared.
00:45:25.190 --> 00:45:27.610
And there is symmetry
in this table.
00:45:27.610 --> 00:45:30.470
So the corresponding
column is also
00:45:30.470 --> 00:45:33.496
all things that are
disconnected entities.
00:45:39.260 --> 00:45:42.050
All right.
00:45:42.050 --> 00:45:45.330
Now let's see the next one.
00:45:45.330 --> 00:45:49.880
I have a wavy line here, a
sigma here, and a sigma here.
00:45:49.880 --> 00:45:53.020
I can potentially
join them together
00:45:53.020 --> 00:45:57.870
into a diagram that looks
something like this.
00:46:01.850 --> 00:46:06.820
So I will have this, this.
00:46:06.820 --> 00:46:08.490
I have a leg here.
00:46:08.490 --> 00:46:12.820
I will have this line
gets joined to that line.
00:46:12.820 --> 00:46:15.580
And then I have
this, this, this.
00:46:19.401 --> 00:46:21.280
Now, what is that beast?
00:46:21.280 --> 00:46:27.820
It is something that has
six factors of m tilde.
00:46:27.820 --> 00:46:29.600
So this is something
that is order
00:46:29.600 --> 00:46:35.750
of m tilde to the sixth power.
00:46:35.750 --> 00:46:37.900
So the point is
that we started here
00:46:37.900 --> 00:46:40.710
saying that I should
put every term that
00:46:40.710 --> 00:46:42.075
is consistent with symmetry.
00:46:44.650 --> 00:46:48.230
I just focused on the
first fourth order term,
00:46:48.230 --> 00:46:49.940
but I see this is one
of the things that
00:46:49.940 --> 00:46:53.180
happens under
renormalization group.
00:46:53.180 --> 00:46:55.530
Everything that is
consistent with symmetry,
00:46:55.530 --> 00:46:57.840
even if you didn't put it
there at the beginning,
00:46:57.840 --> 00:46:59.970
is likely to appear.
00:46:59.970 --> 00:47:02.580
So this term appeared
at this order.
00:47:02.580 --> 00:47:04.880
You have to think of
ultimately whether that's
00:47:04.880 --> 00:47:07.210
something to worry about or not.
00:47:07.210 --> 00:47:09.110
I will deal with that next time.
00:47:09.110 --> 00:47:10.640
It is not something
to worry about.
00:47:10.640 --> 00:47:13.390
But let's forget about
that for the time being.
00:47:13.390 --> 00:47:18.750
Next term, I have one wavy line
here and two wavy lines there.
00:47:18.750 --> 00:47:21.260
So it's something
that is sigma cubed.
00:47:21.260 --> 00:47:24.490
Against the Gaussian
weight, it gives me 0.
00:47:24.490 --> 00:47:29.740
So because of it being an odd
term, I will get a 0 here.
00:47:29.740 --> 00:47:32.355
What color [INAUDIBLE] a 0 here.
00:47:37.770 --> 00:47:45.670
Somehow I need this row
to be larger in connection
00:47:45.670 --> 00:47:46.825
with future needs.
00:47:54.900 --> 00:47:57.370
Next one is also
something that involves
00:47:57.370 --> 00:48:02.140
three factors of sigma,
so it is 0 by symmetry.
00:48:02.140 --> 00:48:06.510
And again, since this
is a diagram that
00:48:06.510 --> 00:48:13.010
has symmetry along the diagonal,
there will be 0's over here.
00:48:13.010 --> 00:48:15.030
Next diagram.
00:48:15.030 --> 00:48:17.730
I can somehow join
things together
00:48:17.730 --> 00:48:21.630
and create something
that has four legs.
00:48:21.630 --> 00:48:23.970
It will look
something like this.
00:48:23.970 --> 00:48:31.530
I will have this leg.
00:48:31.530 --> 00:48:41.310
This leg can be joined,
let's say, with this leg,
00:48:41.310 --> 00:48:44.940
giving me something out here.
00:48:44.940 --> 00:48:48.905
And these two wavy lines
can be joined together.
00:48:48.905 --> 00:48:49.780
That's a possibility.
00:48:52.360 --> 00:48:53.020
You say, OK.
00:48:53.020 --> 00:48:55.210
This is a diagram
that corresponds
00:48:55.210 --> 00:49:00.130
to four factors of m tilde.
00:49:00.130 --> 00:49:05.990
So that should
contribute over here.
00:49:05.990 --> 00:49:09.790
Actually, the answer
is that diagram is 0.
00:49:09.790 --> 00:49:12.840
The reason for that
is the following.
00:49:12.840 --> 00:49:16.680
Let's look at this
vortex over here.
00:49:16.680 --> 00:49:19.850
It describes four momenta
that have come together.
00:49:23.260 --> 00:49:27.400
And the sum of the
four has to be 0.
00:49:27.400 --> 00:49:28.710
Same thing holds here.
00:49:31.360 --> 00:49:33.530
The sum of these
four has to be 0.
00:49:37.440 --> 00:49:50.990
Now, if we look at
this diagram, once I
00:49:50.990 --> 00:49:53.780
have joined these
two together, I
00:49:53.780 --> 00:49:58.000
have ensured that the
sum of these two is 0.
00:49:58.000 --> 00:49:59.660
The sum of all of four is 0.
00:49:59.660 --> 00:50:01.820
The sum of these two is 0.
00:50:01.820 --> 00:50:04.380
So the sum of these
two should be 0 too.
00:50:04.380 --> 00:50:06.190
But that's not allowed.
00:50:06.190 --> 00:50:10.385
Because one of them is outside
this shell, and the other
00:50:10.385 --> 00:50:12.650
is inside the shell.
00:50:12.650 --> 00:50:17.310
So just kinematically,
there's no choice of momenta
00:50:17.310 --> 00:50:21.190
that I could make that would
give a contribution to this.
00:50:21.190 --> 00:50:24.710
So this is 0 because
of what I will
00:50:24.710 --> 00:50:26.910
write as momentum
type of conservation.
00:50:30.120 --> 00:50:34.710
Again, because of that, I
will have here as 0 momentum
00:50:34.710 --> 00:50:37.630
down here.
00:50:37.630 --> 00:50:42.640
The next diagram has one sigma
from here and four sigmas.
00:50:42.640 --> 00:50:44.970
So that's an odd
number of sigmas.
00:50:44.970 --> 00:50:47.410
So this will be 0
too, just because
00:50:47.410 --> 00:50:49.939
of up-down symmetry in m tilde.
00:50:54.530 --> 00:51:00.450
So we are gradually getting
rid of places in this table.
00:51:00.450 --> 00:51:03.260
But the next one is
actually important.
00:51:03.260 --> 00:51:07.150
I can take these two and
join them to those two
00:51:07.150 --> 00:51:10.700
and generate a diagram
that looks like this.
00:51:10.700 --> 00:51:12.700
So I have these two hands.
00:51:12.700 --> 00:51:17.730
These two hands get joined to
the corresponding two hands.
00:51:17.730 --> 00:51:21.440
And I have a diagram
such as this.
00:51:21.440 --> 00:51:22.300
Yes.
00:51:22.300 --> 00:51:27.934
AUDIENCE: [INAUDIBLE] Is
there another way for them
00:51:27.934 --> 00:51:28.961
to join also?
00:51:28.961 --> 00:51:30.460
PROFESSOR: Yes,
there is another way
00:51:30.460 --> 00:51:32.780
which suffers exactly
the same problem.
00:51:32.780 --> 00:51:38.050
Ultimately, because you
see the problem is here.
00:51:38.050 --> 00:51:40.540
I will have to join
two of them together,
00:51:40.540 --> 00:51:42.810
and the other two
will be incompatible.
00:51:49.850 --> 00:51:55.110
Now, just to sort of give you
ultimately an idea, associated
00:51:55.110 --> 00:52:00.510
with this diagram there will be
a numerical factor of 2 times 2
00:52:00.510 --> 00:52:05.400
from the horizontal times
the vertical choices.
00:52:05.400 --> 00:52:08.150
But then there's
another factor of 2
00:52:08.150 --> 00:52:10.320
because this diagram
has two hands.
00:52:10.320 --> 00:52:12.370
The other diagram has two hands.
00:52:12.370 --> 00:52:15.350
They can either join like this,
or they can join like this.
00:52:15.350 --> 00:52:19.650
So there's two possibilities
for the crossing.
00:52:19.650 --> 00:52:25.240
If you kind of look ahead to
the indices that carry around,
00:52:25.240 --> 00:52:27.770
these two are part
of the same branch.
00:52:27.770 --> 00:52:29.630
They carry the same index.
00:52:29.630 --> 00:52:33.470
These two would be carrying
the same index, let's say j.
00:52:33.470 --> 00:52:36.570
These two would be carrying
the same index, j prime.
00:52:36.570 --> 00:52:39.290
So when I do the sum, I
will have a sum over j
00:52:39.290 --> 00:52:41.790
and j prime of delta j, j prime.
00:52:41.790 --> 00:52:44.620
I will have a sum
over j delta jj, which
00:52:44.620 --> 00:52:47.130
will give me a factor of n.
00:52:47.130 --> 00:52:51.110
Any time you see a closed loop,
you generate a factor of n,
00:52:51.110 --> 00:52:53.250
just like we did over here.
00:52:53.250 --> 00:52:55.346
It generated a factor of n.
00:52:55.346 --> 00:52:57.550
OK, so there's that.
00:52:57.550 --> 00:53:01.040
The next diagram looks
similar, but does not
00:53:01.040 --> 00:53:02.870
have the factor of n.
00:53:02.870 --> 00:53:09.650
I have from over there the two
hands that I have to join here.
00:53:09.650 --> 00:53:14.385
I have to put my
hands across, and I
00:53:14.385 --> 00:53:18.160
will get something like this.
00:53:18.160 --> 00:53:24.230
So it's a slightly
different-looking diagram.
00:53:24.230 --> 00:53:31.700
The numerical factor that goes
with that is 2 times 4 times 2.
00:53:31.700 --> 00:53:33.054
There is no factor of n.
00:53:35.720 --> 00:53:38.230
Now, again, because
of symmetry, there's
00:53:38.230 --> 00:53:41.330
a corresponding entity
that we have over here.
00:53:41.330 --> 00:53:43.800
If I just rotate that,
I will essentially
00:53:43.800 --> 00:53:46.470
have the same diagram.
00:53:46.470 --> 00:53:49.995
Opposite way, I have
essentially that.
00:53:55.280 --> 00:54:01.350
The two hands reach
across to these
00:54:01.350 --> 00:54:03.870
and give me something
that is like this.
00:54:08.680 --> 00:54:10.478
To that, sorry.
00:54:13.620 --> 00:54:17.250
They join to that one.
00:54:17.250 --> 00:54:20.330
And the corresponding
thing here looks like this.
00:54:25.440 --> 00:54:30.000
Numerical factors, this
would be 2 times 4 times 2.
00:54:30.000 --> 00:54:31.790
It is exactly the same as this.
00:54:31.790 --> 00:54:34.622
This would be 4 times 4 times 2.
00:54:38.730 --> 00:54:41.090
At the end of the day,
I will convince you
00:54:41.090 --> 00:54:48.700
that this block of four diagrams
is really the only thing
00:54:48.700 --> 00:54:51.390
that we need to compute.
00:54:51.390 --> 00:54:54.580
But let's go ahead and
see what else we have.
00:54:54.580 --> 00:54:58.000
If I take this thing
that has two hands,
00:54:58.000 --> 00:55:00.470
try to join this thing
that has three hands,
00:55:00.470 --> 00:55:05.160
I will get, of course,
0, based on symmetry.
00:55:05.160 --> 00:55:09.950
If I take this term
with two hands,
00:55:09.950 --> 00:55:11.760
join this thing
with four hands, I
00:55:11.760 --> 00:55:14.610
will generate a
bunch of diagrams,
00:55:14.610 --> 00:55:18.170
including, for
example, this one.
00:55:18.170 --> 00:55:23.520
I can do this.
00:55:23.520 --> 00:55:27.280
There are other diagrams also.
00:55:27.280 --> 00:55:33.040
So these are ultimately diagrams
with two hands left over.
00:55:33.040 --> 00:55:38.390
So they will be contributions
to m tilde squared.
00:55:38.390 --> 00:55:43.230
And they will indeed
give me modifications
00:55:43.230 --> 00:55:44.710
of this term over here.
00:55:47.920 --> 00:55:51.230
But we don't need
to calculate them.
00:55:51.230 --> 00:55:52.500
Why?
00:55:52.500 --> 00:55:55.190
Because we want to do
things consistently
00:55:55.190 --> 00:55:58.220
to order of epsilon.
00:55:58.220 --> 00:56:04.100
In the second equation, we
start already with epsilon u.
00:56:04.100 --> 00:56:07.760
So this term was order
of epsilon squared.
00:56:07.760 --> 00:56:10.140
Since u star will
be order of epsilon,
00:56:10.140 --> 00:56:12.050
this term will be
epsilon squared.
00:56:12.050 --> 00:56:14.120
The two terms I
have to evaluate,
00:56:14.120 --> 00:56:16.470
they are both of the same order.
00:56:16.470 --> 00:56:18.590
But in the first
equation, I already
00:56:18.590 --> 00:56:21.740
have a contribution that
is order of epsilon.
00:56:21.740 --> 00:56:25.970
If I'm calculating things
consistently to lowest order,
00:56:25.970 --> 00:56:29.170
I don't need to calculate
this explicitly.
00:56:29.170 --> 00:56:31.520
I would need to
calculate it explicitly
00:56:31.520 --> 00:56:33.470
if I wanted to calculate
things to order
00:56:33.470 --> 00:56:37.090
of epsilon squared, which
I'm not about to do.
00:56:37.090 --> 00:56:40.460
But to our order,
this diagram exists,
00:56:40.460 --> 00:56:42.660
but we don't need to evaluate.
00:56:42.660 --> 00:56:47.040
Again, going because of the
symmetry along the diagonal
00:56:47.040 --> 00:56:50.170
of the diagram, we
have something here
00:56:50.170 --> 00:56:54.770
that is order of m tilde
squared that we don't evaluate.
00:56:54.770 --> 00:56:57.570
OK, let's go further.
00:56:57.570 --> 00:57:00.280
Over here, we have two
hands, three hands.
00:57:00.280 --> 00:57:03.220
By symmetry, it will be zero.
00:57:03.220 --> 00:57:06.820
Over here, we have
two hands, four hands.
00:57:06.820 --> 00:57:13.710
I will get a whole
bunch of other things
00:57:13.710 --> 00:57:16.740
that are order of
m tilde squared.
00:57:16.740 --> 00:57:24.500
So there are other terms that
are of this same form that
00:57:24.500 --> 00:57:26.730
would modify the
factor of a, which
00:57:26.730 --> 00:57:30.340
I don't need to
explicitly evaluate.
00:57:30.340 --> 00:57:31.910
All right.
00:57:31.910 --> 00:57:34.770
What do we have left?
00:57:34.770 --> 00:57:37.610
There is a diagram here
that is interesting
00:57:37.610 --> 00:57:41.720
because it also gives
me a contribution that
00:57:41.720 --> 00:57:51.930
is order of m tilde squared,
which we may come back to
00:57:51.930 --> 00:57:53.120
at some point.
00:57:53.120 --> 00:57:56.840
But for the time being,
it's another thing
00:57:56.840 --> 00:58:00.400
that gives us a
contribution to a.
00:58:00.400 --> 00:58:01.700
Here, what do we have?
00:58:01.700 --> 00:58:04.400
We have three hands, four
hands, zero by symmetry,
00:58:04.400 --> 00:58:07.150
zero by symmetry.
00:58:07.150 --> 00:58:10.410
Down here, we have
no solid hands.
00:58:10.410 --> 00:58:13.850
So we will get a whole
bunch of diagrams,
00:58:13.850 --> 00:58:18.920
such as this one, for
example, other things, which
00:58:18.920 --> 00:58:21.960
collectively will
give a second order
00:58:21.960 --> 00:58:24.860
correction to the free energy.
00:58:24.860 --> 00:58:28.942
It's another constant that
we don't need to evaluate.
00:58:35.540 --> 00:58:41.345
So let's pick one of
these diagrams, this one
00:58:41.345 --> 00:58:44.400
in particular, and
explicitly see what that is.
00:58:52.750 --> 00:58:56.530
It came out of putting
two factors of u together.
00:58:56.530 --> 00:58:57.620
Let's be explicit.
00:58:57.620 --> 00:59:04.540
Let's call the momenta
here q1, q2, and k1, k2.
00:59:04.540 --> 00:59:10.090
And the other u here came
from before I joined them,
00:59:10.090 --> 00:59:13.460
there was a q3, q4.
00:59:13.460 --> 00:59:16.397
There was a k1 prime, k2 prime.
00:59:19.320 --> 00:59:22.850
So let's say the first
u-- this is a diagram that
00:59:22.850 --> 00:59:26.280
will contribute at
order of u squared.
00:59:26.280 --> 00:59:28.745
Second order terms
in the series all
00:59:28.745 --> 00:59:30.510
come with a factor of one half.
00:59:30.510 --> 00:59:34.100
It is u to the n
divided by n factorial.
00:59:34.100 --> 00:59:38.600
So this would be explicitly
u squared over 2.
00:59:38.600 --> 00:59:41.230
For the choice of
the left diagram,
00:59:41.230 --> 00:59:43.990
we said there were
two possibilities.
00:59:43.990 --> 00:59:46.120
For the choice of
right diagram, there
00:59:46.120 --> 00:59:48.900
were two branches, one of
which I could have taken.
00:59:48.900 --> 00:59:51.480
In joining the two
hands together,
00:59:51.480 --> 00:59:53.940
I had a degeneracy of two,
so I have all of that.
00:59:57.570 --> 01:00:04.500
A particular one of these
is an integral over q1, q2.
01:00:04.500 --> 01:00:08.590
And from here, I would have
integrations over q3, q4.
01:00:12.170 --> 01:00:16.250
These are all
integrations that are
01:00:16.250 --> 01:00:19.570
for variables that are
in the inner shell.
01:00:19.570 --> 01:00:22.970
So this is lambda
to lambda over b.
01:00:22.970 --> 01:00:26.830
I have integrations from
lambda over v2 lambda
01:00:26.830 --> 01:00:33.420
for the variables k1,
k2, k1 prime, k2 prime.
01:00:38.610 --> 01:00:43.720
And if I explicitly
decided to write all four
01:00:43.720 --> 01:00:46.910
momenta associated with
a particular index,
01:00:46.910 --> 01:00:50.310
I have to explicitly include
the delta functions that
01:00:50.310 --> 01:00:54.664
say the sum of the momenta
has to add up to 0.
01:01:08.840 --> 01:01:14.530
Now, what I did was to drawing
these two sigmas together.
01:01:14.530 --> 01:01:17.350
So I calculated one of
those Gaussian averages
01:01:17.350 --> 01:01:20.410
that I have over there.
01:01:20.410 --> 01:01:23.080
Actually, before
I do that, I note
01:01:23.080 --> 01:01:26.010
that these pairs
are dotted together.
01:01:26.010 --> 01:01:29.825
So I have m tilde of
q1 dotted with m tilde
01:01:29.825 --> 01:01:42.580
of q3, m tilde of q4 dotted
with m tilde of q1, q2, q3, q4.
01:01:42.580 --> 01:01:43.880
These two are dot products.
01:01:43.880 --> 01:01:46.620
These two are dot products.
01:01:46.620 --> 01:01:51.260
Here, I joined the
two sigmas together.
01:01:51.260 --> 01:01:53.560
The expectation
value gives me 2 pi
01:01:53.560 --> 01:01:57.770
to the d at delta
function k1 plus k1
01:01:57.770 --> 01:02:04.240
prime divided by t plus k,
k1 squared, and so forth.
01:02:04.240 --> 01:02:06.110
And the delta
function, if I call
01:02:06.110 --> 01:02:11.540
these indices j, j,
j prime, j prime,
01:02:11.540 --> 01:02:13.590
I will have a delta j j prime.
01:02:17.220 --> 01:02:20.440
And from the lower two that
I have connected together,
01:02:20.440 --> 01:02:27.460
I have 2 pi to the d delta
function k2 plus k2 prime.
01:02:27.460 --> 01:02:33.550
Another delta j j prime, t plus
k, k2 squared, and so forth.
01:02:38.050 --> 01:02:40.190
Now I can do the integrations.
01:02:40.190 --> 01:02:47.350
But first of all, numerical
factors, I will get 4u squared.
01:02:47.350 --> 01:02:50.560
As I told you, delta j
j prime, delta j j prime
01:02:50.560 --> 01:02:51.910
will give me delta jj.
01:02:51.910 --> 01:02:54.780
Sum over j, I will
get a factor of n.
01:02:54.780 --> 01:02:59.070
That's the n that I
anticipated and put over there.
01:02:59.070 --> 01:03:07.250
I have the integrations 0 to
lambda over v, ddq1, ddq2,
01:03:07.250 --> 01:03:13.370
ddq3, ddq4, 2 pi to the 4d.
01:03:16.980 --> 01:03:25.780
And then this m tilde
q1 m tilde q2 m tilde q3
01:03:25.780 --> 01:03:27.220
dotted with m tilde q4.
01:03:30.570 --> 01:03:32.500
Now, note the following.
01:03:32.500 --> 01:03:36.940
If I do the integration
over k1 prime,
01:03:36.940 --> 01:03:40.570
k1 prime is set to minus k1.
01:03:40.570 --> 01:03:43.220
If I do the integration
over k2 prime,
01:03:43.220 --> 01:03:47.510
k2 prime is set to minus k2.
01:03:47.510 --> 01:03:51.410
If I now do the
integration over k2,
01:03:51.410 --> 01:03:56.560
k2 is set to minus q1
minus q2 minus k1, which
01:03:56.560 --> 01:04:02.340
if I insert over here, will give
me a delta function that simply
01:04:02.340 --> 01:04:06.988
says that the four external
q's have to add up to 0.
01:04:13.320 --> 01:04:17.490
So there is one integration
that is left, which is over k1.
01:04:17.490 --> 01:04:19.750
So I have to do
the integral lambda
01:04:19.750 --> 01:04:25.650
over v2 lambda, dd
of k1 2 pi to the d.
01:04:25.650 --> 01:04:31.120
So basically, there's k1 running
across the upper line gives me
01:04:31.120 --> 01:04:37.500
a factor of 1 over t plus
k, k1 squared, and so forth.
01:04:37.500 --> 01:04:39.350
And then there is
what is running
01:04:39.350 --> 01:04:42.720
along the bottom line,
which is k2 squared.
01:04:42.720 --> 01:04:48.790
And k2 squared is the
same thing as q1 plus q2
01:04:48.790 --> 01:04:51.115
plus k1, the whole
thing squared.
01:04:59.230 --> 01:05:05.580
So the outcome of
doing the averages that
01:05:05.580 --> 01:05:09.360
appear in this
integral is to generate
01:05:09.360 --> 01:05:14.710
a term that is proportional
to m to the fourth, which
01:05:14.710 --> 01:05:20.390
is exactly what we
had, with one twist.
01:05:20.390 --> 01:05:22.380
The twist is that
the coefficient that
01:05:22.380 --> 01:05:28.710
is appearing here actually
depends on q1 and q2.
01:05:28.710 --> 01:05:32.430
Of course, q1 and q2
being inner momenta
01:05:32.430 --> 01:05:36.840
are much smaller than k1, which
is one of the shared momenta.
01:05:36.840 --> 01:05:39.010
So in principle,
I can expand this.
01:05:39.010 --> 01:05:49.560
I can expand this as
ddk 2 pi to the d lambda
01:05:49.560 --> 01:05:56.310
over b2 lambda, 1 over
t plus k, k squared.
01:05:56.310 --> 01:06:02.680
I've renamed k1 to k squared to
lowest order in q, this is 0.
01:06:02.680 --> 01:06:13.140
And then I can expand this thing
as 1 plus k q1 plus q2 squared
01:06:13.140 --> 01:06:17.300
and so forth,
divided by t plus k k
01:06:17.300 --> 01:06:23.210
squared raised to
the minus 1 power.
01:06:23.210 --> 01:06:27.320
Point is that if I
set the q's to 0,
01:06:27.320 --> 01:06:31.190
I have obtained a
constant addition
01:06:31.190 --> 01:06:35.150
to the coefficient of
my m to the fourth.
01:06:35.150 --> 01:06:39.650
But I see that further
down, I have generated also
01:06:39.650 --> 01:06:42.620
terms that depend on q.
01:06:42.620 --> 01:06:45.630
What kind of terms
could these be?
01:06:45.630 --> 01:06:47.930
If I go back to
real space, these
01:06:47.930 --> 01:06:50.420
are terms that are
after order of m
01:06:50.420 --> 01:06:55.900
to the fourth, which was, if you
remember, m squared m squared.
01:06:55.900 --> 01:06:59.300
But carry additional
gradients with them.
01:06:59.300 --> 01:07:04.100
So, for example, it could
be something like this.
01:07:04.100 --> 01:07:07.760
It has two factors of
q, various factors of n.
01:07:07.760 --> 01:07:12.870
Or it could be something like m
squared gradient of m squared.
01:07:12.870 --> 01:07:18.440
The point is that we have
again the possibility, when
01:07:18.440 --> 01:07:22.030
we write our most general
term, to introduce
01:07:22.030 --> 01:07:25.650
lots and lots of non-linearities
that I didn't explicitly
01:07:25.650 --> 01:07:27.130
include.
01:07:27.130 --> 01:07:31.040
But again, I see, if I
forget them at the beginning,
01:07:31.040 --> 01:07:34.730
the process will
generate them for you.
01:07:34.730 --> 01:07:37.720
So I should have really
included these types of terms
01:07:37.720 --> 01:07:41.870
in the beginning because they
will be generated under the RG,
01:07:41.870 --> 01:07:44.700
and then I can track
their evolution of all
01:07:44.700 --> 01:07:46.190
of the parameters.
01:07:46.190 --> 01:07:49.180
I started with 0 for
this type of parameter.
01:07:49.180 --> 01:07:50.830
I generated it out of nothing.
01:07:50.830 --> 01:07:53.640
So I should really go
back and put it there.
01:07:53.640 --> 01:07:56.780
But for the time being,
let's again ignore that.
01:07:56.780 --> 01:07:59.970
And next time, I'll
see what happens.
01:07:59.970 --> 01:08:05.500
So what we find at
the end of evaluating
01:08:05.500 --> 01:08:15.800
all of these diagrams is that
this beta h tilde evaluated
01:08:15.800 --> 01:08:19.100
at the second
order, first of all,
01:08:19.100 --> 01:08:26.109
has a bunch of constants
which in principle, now
01:08:26.109 --> 01:08:28.220
we can calculate
to the next order.
01:08:33.600 --> 01:08:42.029
Then we find that
we get terms that
01:08:42.029 --> 01:08:47.979
are proportional to m tilde
squared, the Gaussian.
01:08:51.240 --> 01:08:58.740
And I can get the terms that
I got out of second order
01:08:58.740 --> 01:09:00.170
and put it here.
01:09:00.170 --> 01:09:04.220
So I have my
original t tilde now
01:09:04.220 --> 01:09:11.520
evaluated at order of u squared.
01:09:11.520 --> 01:09:16.590
Because of all those diagrams
that I said I have to do.
01:09:16.590 --> 01:09:22.410
I will get a k tilde q
squared and so forth,
01:09:22.410 --> 01:09:27.680
all of them multiplying
m tilde squared.
01:09:27.680 --> 01:09:33.560
And I see that I
generated terms that
01:09:33.560 --> 01:09:37.270
are of the order of m
tilde to the fourth.
01:09:37.270 --> 01:09:44.569
So I have ddq1,
ddq4, 2 pi to the 4d,
01:09:44.569 --> 01:09:58.880
2 pi to the d delta function,
q1, q4, m tilde q1, m tilde q2,
01:09:58.880 --> 01:10:05.410
m tilde q3, m tilde q4.
01:10:05.410 --> 01:10:11.590
And what I have to
lowest order is u.
01:10:11.590 --> 01:10:13.780
And then I have a
bunch of terms that
01:10:13.780 --> 01:10:17.600
are of the form
something like this.
01:10:17.600 --> 01:10:22.300
So they are corrections
that are proportional
01:10:22.300 --> 01:10:28.150
to the integral lambda
over b lambda ddk 2 pi
01:10:28.150 --> 01:10:35.110
to the d, 1 over t plus
k, k squared squared.
01:10:37.740 --> 01:10:44.300
So essentially, I took
this part of that diagram.
01:10:44.300 --> 01:10:47.590
That diagram has a
contribution at order
01:10:47.590 --> 01:10:51.120
of u squared, which is 4n.
01:10:51.120 --> 01:10:54.950
So if I had written it
as u squared over 2,
01:10:54.950 --> 01:10:59.160
I would have put
8n, the 8 coming
01:10:59.160 --> 01:11:02.470
from just the multiplication
that I have there, 2 times
01:11:02.470 --> 01:11:05.578
2 times 2 times n.
01:11:05.578 --> 01:11:09.140
Now, if you calculate
the other three diagrams
01:11:09.140 --> 01:11:13.600
that I have boxed,
you'll find that they
01:11:13.600 --> 01:11:17.690
give exactly the same
form of the contribution,
01:11:17.690 --> 01:11:21.580
except that the numerical
factor for them is different.
01:11:21.580 --> 01:11:32.890
I will get 16, 1632, adding
up together to a factor of 64
01:11:32.890 --> 01:11:35.120
here.
01:11:35.120 --> 01:11:40.230
And then the point is that I
will generate additional terms
01:11:40.230 --> 01:11:43.380
that are, let's say,
order of q squared
01:11:43.380 --> 01:11:46.360
and so forth,
Which are the kinds
01:11:46.360 --> 01:11:48.192
of terms that I
had not included.
01:11:52.350 --> 01:11:58.430
So what we find is
that-- question?
01:12:02.390 --> 01:12:04.370
AUDIENCE: Where did
you add the t total?
01:12:07.340 --> 01:12:08.930
PROFESSOR: OK.
01:12:08.930 --> 01:12:12.170
So let's maybe write
this explicitly.
01:12:12.170 --> 01:12:15.560
So what would be the coefficient
that I have to put over here?
01:12:19.890 --> 01:12:24.610
I have t at the 0-th order.
01:12:24.610 --> 01:12:33.640
At order of u, I calculated
4u n plus 2 integral.
01:12:42.700 --> 01:12:47.540
The point is that when I
add up all of those diagrams
01:12:47.540 --> 01:12:51.170
that I haven't
explicitly calculated,
01:12:51.170 --> 01:12:54.550
I will get a correction
here that is order
01:12:54.550 --> 01:12:59.105
of u squared whose
coefficient I will call a.
01:13:03.180 --> 01:13:07.140
But then this is the 0-th
order in the momenta,
01:13:07.140 --> 01:13:11.340
and then I have to
go and add terms
01:13:11.340 --> 01:13:14.940
that are at the order of q
squared and higher order terms.
01:13:23.910 --> 01:13:26.370
So this was, again, the
course graining step
01:13:26.370 --> 01:13:28.020
of RG, which is the hard part.
01:13:28.020 --> 01:13:33.110
The rescaling and
renormalization are simple.
01:13:33.110 --> 01:13:35.600
And what they give me
at the end of the day
01:13:35.600 --> 01:13:40.130
are the modifications
to dt by dl and du
01:13:40.130 --> 01:13:42.810
by dl that we expected.
01:13:42.810 --> 01:13:46.210
dt by dl we already wrote.
01:13:46.210 --> 01:13:47.860
0-th order is 2t.
01:13:47.860 --> 01:13:50.040
First order is a correction.
01:13:50.040 --> 01:13:55.260
4u n plus 2 integral, which,
when evaluated on the shell,
01:13:55.260 --> 01:14:01.010
gives me kt lambda to
the d t plus k lambda
01:14:01.010 --> 01:14:02.170
squared and so forth.
01:14:05.120 --> 01:14:10.790
Now, this a here would
involve an integration.
01:14:10.790 --> 01:14:14.830
Again, this integration
I evaluated on the shell.
01:14:14.830 --> 01:14:18.130
So the answer will
be some a that
01:14:18.130 --> 01:14:23.330
will depend on t,
k, and other things,
01:14:23.330 --> 01:14:26.180
would be a contribution
that is order of u squared.
01:14:29.020 --> 01:14:32.350
I haven't explicitly
calculated what this a is.
01:14:32.350 --> 01:14:36.100
It will depend on all of
these other parameters.
01:14:36.100 --> 01:14:40.010
Now, when I calculate the u
by dl, I will get this 4 minus
01:14:40.010 --> 01:14:44.120
d times u to the lowest order.
01:14:44.120 --> 01:14:49.540
To the next order, I
essentially get this integral.
01:14:49.540 --> 01:15:06.830
So I have minus 4n minus
4n plus 8 u squared.
01:15:06.830 --> 01:15:09.760
Evaluate that
integral on the shell.
01:15:09.760 --> 01:15:14.850
kd lambda to the
d t plus k lambda
01:15:14.850 --> 01:15:16.640
squared and so forth squared.
01:15:19.440 --> 01:15:21.330
And presumably,
both of these will
01:15:21.330 --> 01:15:24.260
have corrections at
higher orders, order
01:15:24.260 --> 01:15:25.627
of u squared, et cetera.
01:15:29.740 --> 01:15:34.700
So this generalizes the
picture that we had over here.
01:15:37.260 --> 01:15:40.300
Now we can ask, what
is the fixed point?
01:15:46.180 --> 01:15:49.400
In fact, there will
be two of them.
01:15:49.400 --> 01:15:56.700
There is the old Gaussian
fixed point at t star
01:15:56.700 --> 01:15:59.620
u star equals to 0.
01:15:59.620 --> 01:16:03.460
Clearly, if I said t and u
equals to 0, I will stay at 0.
01:16:03.460 --> 01:16:06.190
So the old fixed
point is still there.
01:16:06.190 --> 01:16:09.010
But now I have a
new fixed point,
01:16:09.010 --> 01:16:12.860
which is called the ON fixed
point because it explicitly
01:16:12.860 --> 01:16:16.040
depends on the
symmetry of the order
01:16:16.040 --> 01:16:19.750
parameter, the number
of components, n,
01:16:19.750 --> 01:16:21.240
as well as dimensionality.
01:16:21.240 --> 01:16:23.520
It's called the ON fixed point.
01:16:23.520 --> 01:16:31.430
So setting this to 0, I will
find that u star is essentially
01:16:31.430 --> 01:16:35.780
epsilon divided by
whatever I have here.
01:16:35.780 --> 01:16:46.530
I have 4n plus 8
kd lambda to the d.
01:16:46.530 --> 01:16:51.810
In the numerator,
I would have t star
01:16:51.810 --> 01:16:55.850
plus k lambda squared squared.
01:16:59.500 --> 01:17:03.920
And then I can
substitute that over here
01:17:03.920 --> 01:17:06.820
to find what t star is.
01:17:06.820 --> 01:17:16.250
So t star would be minus 2
n plus 2 kd lambda to the d
01:17:16.250 --> 01:17:21.230
divided by t star
plus k lambda squared,
01:17:21.230 --> 01:17:26.870
et cetera, times u star, which
is what I have the line above,
01:17:26.870 --> 01:17:30.790
which is t star plus
k lambda squared,
01:17:30.790 --> 01:17:37.960
et cetera, squared, divided by
4n plus 8 kd lambda to the t.
01:17:46.290 --> 01:17:49.610
Now, over here,
this is in principle
01:17:49.610 --> 01:17:54.130
an implicit equation for t star.
01:17:54.130 --> 01:17:57.700
But I forgot the epsilon
that I have here.
01:17:57.700 --> 01:18:02.150
But it is epsilon multiplying
some function of t star.
01:18:02.150 --> 01:18:05.430
So clearly, t star
is order of epsilon.
01:18:05.430 --> 01:18:10.700
And I can set t star equal to
0 in all of the calculation,
01:18:10.700 --> 01:18:14.340
if I'm calculating things
consistently to epsilon.
01:18:14.340 --> 01:18:19.060
You can see that this kd
lambda to the d cancels that.
01:18:19.060 --> 01:18:25.350
One of these factors cancels
what I have over here.
01:18:25.350 --> 01:18:32.290
At the end of the day, I will
get minus n plus 2 divided
01:18:32.290 --> 01:18:37.740
by n plus 8 k lambda
squared epsilon.
01:18:42.110 --> 01:18:46.190
And similarly, over here
I can get rid of t star
01:18:46.190 --> 01:18:48.070
because it's already
order of epsilon,
01:18:48.070 --> 01:18:50.740
and I have epsilon out here.
01:18:50.740 --> 01:18:55.670
So the answer is going
to be k squared lambda
01:18:55.670 --> 01:19:02.420
to the power of 4 minus
d divided by 4n plus
01:19:02.420 --> 01:19:06.850
8 kd lambda to the d.
01:19:06.850 --> 01:19:10.112
Presumably, both of these
plus order of epsilon squared.
01:19:14.540 --> 01:19:17.040
So you can see that,
as anticipated,
01:19:17.040 --> 01:19:19.750
there's a fixed
point at negative 2 t
01:19:19.750 --> 01:19:21.870
star and some particular u star.
01:19:21.870 --> 01:19:22.990
There was a question.
01:19:22.990 --> 01:19:23.864
[INAUDIBLE]
01:19:23.864 --> 01:19:24.739
AUDIENCE: [INAUDIBLE]
01:19:28.690 --> 01:19:29.999
PROFESSOR: What is unnecessary?
01:19:29.999 --> 01:19:30.874
AUDIENCE: [INAUDIBLE]
01:19:33.800 --> 01:19:36.250
AUDIENCE: You already
did 4 minus [INAUDIBLE].
01:19:36.250 --> 01:19:38.202
AUDIENCE: The u started.
01:19:38.202 --> 01:19:39.949
Yeah, that one.
01:19:39.949 --> 01:19:40.615
PROFESSOR: Here.
01:19:40.615 --> 01:19:41.980
AUDIENCE: Erase it.
01:19:41.980 --> 01:19:44.061
PROFESSOR: Oh,
lambda to the d is 0.
01:19:44.061 --> 01:19:44.560
Right.
01:19:44.560 --> 01:19:45.200
Thank you.
01:19:45.200 --> 01:19:47.590
AUDIENCE: For t star,
does factor of 2.
01:19:47.590 --> 01:19:50.450
PROFESSOR: t star, does
it have a factor of 2?
01:19:50.450 --> 01:19:52.310
Yes, 2 divided by 4.
01:19:52.310 --> 01:19:53.640
There is a factor of 2 here.
01:19:57.690 --> 01:19:58.190
Thank you.
01:20:02.960 --> 01:20:03.680
Look at this.
01:20:03.680 --> 01:20:08.790
You don't really see
much to recommend it.
01:20:08.790 --> 01:20:13.830
The interesting
thing is to find what
01:20:13.830 --> 01:20:17.800
happens if you are not
exactly at the fixed point,
01:20:17.800 --> 01:20:20.130
but slightly shifted.
01:20:20.130 --> 01:20:26.240
So we want to see what happens
if t is t star plus delta t,
01:20:26.240 --> 01:20:32.860
u is u star plus delta u,
if I shift a little bit.
01:20:32.860 --> 01:20:37.610
If I shift a little bit,
linearizing the equation
01:20:37.610 --> 01:20:42.540
means I want to know
how the new shifts are
01:20:42.540 --> 01:20:45.030
related to the old shift.
01:20:45.030 --> 01:20:48.880
And essentially doing
things at the linear level
01:20:48.880 --> 01:20:53.600
means I want to construct a
two-by-two matrix that relates
01:20:53.600 --> 01:20:58.470
the changes delta t delta
u to the shifts originally
01:20:58.470 --> 01:21:00.690
of delta t and delta u.
01:21:00.690 --> 01:21:02.970
What do I have to
do to get this?
01:21:02.970 --> 01:21:07.240
What I have to do is to take
derivatives of the terms for dt
01:21:07.240 --> 01:21:10.560
by dl with respect to
t, with respect to u.
01:21:10.560 --> 01:21:12.700
Take the derivative
with respect to t.
01:21:12.700 --> 01:21:13.380
What do I get?
01:21:13.380 --> 01:21:15.120
I will get two.
01:21:15.120 --> 01:21:24.900
I will get minus 4u n plus 2
kd lambda to the father of d
01:21:24.900 --> 01:21:30.640
divided by t plus k
lambda squared squared.
01:21:30.640 --> 01:21:35.160
So the derivative of 1 over t
became minus 1 over t squared.
01:21:35.160 --> 01:21:36.700
There is a second order term.
01:21:36.700 --> 01:21:39.810
So there will be a derivative
of that with respect
01:21:39.810 --> 01:21:43.730
to t multiplying u squared.
01:21:43.730 --> 01:21:46.480
I want you to calculate it.
01:21:46.480 --> 01:21:49.655
Delta u, if I make
a change in u,
01:21:49.655 --> 01:21:55.300
there will be a shift
here, which is 4n
01:21:55.300 --> 01:22:01.140
plus 2 kd lambda
to the d divided
01:22:01.140 --> 01:22:05.390
by t plus k lambda squared.
01:22:05.390 --> 01:22:08.370
From the second order
term, I will get minus 2au.
01:22:11.950 --> 01:22:14.750
For the second equation,
if I take the derivative
01:22:14.750 --> 01:22:23.700
of this variation in t, I will
get a plus 4n plus 8u squared
01:22:23.700 --> 01:22:29.180
kd lambda to the d t
plus k lambda squared
01:22:29.180 --> 01:22:32.690
and so forth cubed.
01:22:32.690 --> 01:22:43.150
And the fourth place, I will
get epsilon minus 8 n plus 8 kd
01:22:43.150 --> 01:22:50.190
lambda to the d u divided
by t plus k lambda
01:22:50.190 --> 01:22:51.820
squared and so forth squared.
01:22:55.888 --> 01:23:00.950
Now, I want to evaluate this
matrix at the fixed point.
01:23:00.950 --> 01:23:04.460
So I have to linearize in
the vicinity of fixed point.
01:23:04.460 --> 01:23:10.300
Which means that I put the
values of t star and u star
01:23:10.300 --> 01:23:11.142
everywhere here.
01:23:20.900 --> 01:23:26.064
And then I have to calculate
the eigenvalues of this matrix.
01:23:26.064 --> 01:23:29.870
Now, note that this
element of the matrix
01:23:29.870 --> 01:23:33.410
is proportional
to u star squared.
01:23:33.410 --> 01:23:36.453
So this is certainly
evaluated at the fixed point
01:23:36.453 --> 01:23:40.310
order of epsilon squared.
01:23:40.310 --> 01:23:42.760
Order of epsilon
squared to me is zero.
01:23:42.760 --> 01:23:45.670
I don't see order
of epsilon squared.
01:23:45.670 --> 01:23:47.050
So I can get rid of this.
01:23:47.050 --> 01:23:49.890
Think of a zero
here at this order.
01:23:49.890 --> 01:23:52.570
Which means that
the matrix now has
01:23:52.570 --> 01:23:55.480
zeroes on one side of
the diagonal, which
01:23:55.480 --> 01:23:57.645
means that what
is appearing here
01:23:57.645 --> 01:24:01.070
are exactly the eigenvalues.
01:24:01.070 --> 01:24:04.220
Let's calculate
the eigenvalue that
01:24:04.220 --> 01:24:06.760
corresponds to this element.
01:24:06.760 --> 01:24:09.230
I will call it yu.
01:24:09.230 --> 01:24:18.700
It is epsilon minus 8 n plus
8 kd lambda to the d t star.
01:24:18.700 --> 01:24:22.470
Well, since I'm calculating
things to order of epsilon,
01:24:22.470 --> 01:24:25.350
I can ignore that
t star down there.
01:24:25.350 --> 01:24:30.820
I have k squared lambda
to the four or k squared,
01:24:30.820 --> 01:24:33.220
lambda squared, and
so forth squared.
01:24:33.220 --> 01:24:35.730
Multiplied by u star.
01:24:35.730 --> 01:24:38.830
Where is my u star?
u star is here.
01:24:38.830 --> 01:24:43.975
So it is multiplied by n plus 2.
01:24:47.860 --> 01:24:51.110
Sorry, my u star is up here.
01:24:51.110 --> 01:25:02.840
k squared lambda to the 4
minus d 4 n plus 8 kd epsilon.
01:25:02.840 --> 01:25:04.170
Right.
01:25:04.170 --> 01:25:06.500
Now the miracle happens.
01:25:06.500 --> 01:25:10.960
So k squared cancels
the k squared.
01:25:10.960 --> 01:25:14.050
Lambda to the four
and lambda to the d
01:25:14.050 --> 01:25:16.940
cancel this lambda
to the four minus d.
01:25:16.940 --> 01:25:18.560
The kd cancels the kd.
01:25:18.560 --> 01:25:22.340
The n plus 8 cancels
the n plus a.
01:25:22.340 --> 01:25:24.140
8 cancels the 2.
01:25:24.140 --> 01:25:27.050
The answer is epsilon
minus 2 epsilon,
01:25:27.050 --> 01:25:28.080
which is minus epsilon.
01:25:32.260 --> 01:25:34.138
OK?
01:25:34.138 --> 01:25:39.010
[LAUGHTER]
01:25:39.010 --> 01:25:43.235
So this direction has
become irrelevant.
01:25:43.235 --> 01:25:47.960
The epsilon here turn
to a minus epsilon.
01:25:47.960 --> 01:25:51.200
This irrelevant
direction disappeared.
01:25:51.200 --> 01:25:54.530
There is this relevant
direction that is left,
01:25:54.530 --> 01:25:57.350
which is a slightly
shifted version of what
01:25:57.350 --> 01:26:02.170
my original [INAUDIBLE]
direction was.
01:26:02.170 --> 01:26:04.730
And you can calculate yt.
01:26:04.730 --> 01:26:08.180
So you go to that
expression, do the same thing
01:26:08.180 --> 01:26:09.970
that I did over here.
01:26:09.970 --> 01:26:11.800
You'll find that at
the end of the day,
01:26:11.800 --> 01:26:19.630
you will find 2 minus n plus
2 over n plus 8 epsilon.
01:26:19.630 --> 01:26:24.140
All these unwanted things, like
kd's, these lambdas, et cetera,
01:26:24.140 --> 01:26:25.610
disappear.
01:26:25.610 --> 01:26:30.580
You expected at the end of
the day to get pure numbers.
01:26:30.580 --> 01:26:32.450
The exponents are pure numbers.
01:26:32.450 --> 01:26:34.820
They don't depend on anything.
01:26:34.820 --> 01:26:38.350
So we had to carry
all of this baggage.
01:26:38.350 --> 01:26:41.590
And at the end of the day, all
of the baggage miraculously
01:26:41.590 --> 01:26:43.580
disappears.
01:26:43.580 --> 01:26:48.760
We get a fixed point that has
only one relevant direction,
01:26:48.760 --> 01:26:51.510
which is what we always wanted.
01:26:51.510 --> 01:26:53.770
And once we have
the exponent, we
01:26:53.770 --> 01:26:55.960
can calculate
everything that we want,
01:26:55.960 --> 01:26:58.820
like the exponent for
divergence of correlation length
01:26:58.820 --> 01:27:00.950
is the inverse of that.
01:27:00.950 --> 01:27:04.825
You can calculate how it
has shifted from one half.
01:27:04.825 --> 01:27:10.510
It is something like n plus
2 over n plus 8 epsilon.
01:27:10.510 --> 01:27:14.820
And we see that the exponents
now explicitly depend
01:27:14.820 --> 01:27:18.320
on dimensionality of space
because of this epsilon.
01:27:18.320 --> 01:27:20.060
They explicitly
depend on the number
01:27:20.060 --> 01:27:23.370
of components of your
order parameter n.
01:27:23.370 --> 01:27:28.200
So we have managed, at least
in some perturbative sense,
01:27:28.200 --> 01:27:33.490
to demonstrate that there exists
a kind of scale invariance
01:27:33.490 --> 01:27:37.900
that characterizes this
ON universality class.
01:27:37.900 --> 01:27:40.950
And we can calculate
exponents for that, at least
01:27:40.950 --> 01:27:42.580
perturbatively.
01:27:42.580 --> 01:27:44.720
In the process of
getting that number,
01:27:44.720 --> 01:27:47.070
I did things rapidly
at the end, but I also
01:27:47.070 --> 01:27:49.890
swept a lot of
things under the rug.
01:27:49.890 --> 01:27:54.050
So the task of next lecture is
to go and look under the rug
01:27:54.050 --> 01:27:57.240
and make sure that we haven't
put anything that is important
01:27:57.240 --> 01:27:58.742
away.