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PROFESSOR: OK, let's start.
00:00:26.470 --> 00:00:30.856
So last lecture we started
with the strategy of using
00:00:30.856 --> 00:00:40.040
perturbation theory to study
our statistical themes.
00:00:40.040 --> 00:00:45.030
For example, we need to
evaluate a partition function
00:00:45.030 --> 00:00:50.220
by integrating over all
configurations of a field.
00:00:50.220 --> 00:00:54.120
Let's say n components
in d dimensions
00:00:54.120 --> 00:01:00.100
with some kind of a weight that
we can write as e to the minus
00:01:00.100 --> 00:01:05.740
beta H. And the
strategy of perturbation
00:01:05.740 --> 00:01:10.270
was to find part
of this Hamiltonian
00:01:10.270 --> 00:01:13.480
that we can calculate exactly.
00:01:13.480 --> 00:01:18.080
And the rest of it, hopefully
treating as a small quantity
00:01:18.080 --> 00:01:21.980
and doing perturbative
calculations.
00:01:21.980 --> 00:01:25.540
Now, in the context of
the Landau-Ginzburg theory
00:01:25.540 --> 00:01:32.050
that we wrote down,
this beta H0 was
00:01:32.050 --> 00:01:34.890
bets described in
terms of Fourier modes.
00:01:34.890 --> 00:01:38.398
So basically, we could
make a change of variables
00:01:38.398 --> 00:01:41.672
to integrate over all
configurations of Fourier
00:01:41.672 --> 00:01:43.380
modes.
00:01:43.380 --> 00:01:48.701
And the same breakdown
of the weight
00:01:48.701 --> 00:01:52.090
in the language
of Fourier modes.
00:01:52.090 --> 00:01:55.360
Since the underlying
theories that we were writing
00:01:55.360 --> 00:01:59.740
had translational symmetry,
every point in space
00:01:59.740 --> 00:02:04.240
was the same as any other,
the composition in to modes
00:02:04.240 --> 00:02:06.262
was immediately
accomplished by going
00:02:06.262 --> 00:02:09.199
to Fourier representation.
00:02:09.199 --> 00:02:12.030
And each component
of each q-value
00:02:12.030 --> 00:02:15.550
would correspond to essentially
an independent weight
00:02:15.550 --> 00:02:19.450
that we could expand
in some power series
00:02:19.450 --> 00:02:23.590
in this parameter q, which is
an inverse in the wave length.
00:02:23.590 --> 00:02:26.260
And the lowest
order terms are what
00:02:26.260 --> 00:02:30.440
determines the longer
and longer wavelengths.
00:02:30.440 --> 00:02:38.230
So there is some m of q
squared characterizing
00:02:38.230 --> 00:02:41.180
this part of the Hamiltonian.
00:02:41.180 --> 00:02:45.800
And since is real
space we had emphasized
00:02:45.800 --> 00:02:53.020
some form of locality, the
interaction part in real space
00:02:53.020 --> 00:02:58.660
could be written simply in terms
of a power series, let's say,
00:02:58.660 --> 00:02:59.650
in m.
00:03:03.120 --> 00:03:08.580
Which means that if we were
to then go to Fourier space,
00:03:08.580 --> 00:03:11.160
things that are
local in real space
00:03:11.160 --> 00:03:13.580
become non-local
in Fourier space.
00:03:13.580 --> 00:03:17.346
And the first of those
terms that we treated
00:03:17.346 --> 00:03:19.980
as a perturbation would
involve an integral
00:03:19.980 --> 00:03:25.985
over four factors of m tilde.
00:03:28.800 --> 00:03:33.800
Again, translational
invariance forces the four q's
00:03:33.800 --> 00:03:40.700
that appear in the
multiplication to add up to 0.
00:03:40.700 --> 00:03:48.410
So I would have m of q1
dot for dot with m of q2,
00:03:48.410 --> 00:03:54.460
m of q3 dot for
dot with m of q4,
00:03:54.460 --> 00:03:59.210
which is minus q1
minus q2 minus q3.
00:03:59.210 --> 00:04:03.280
And I can go on and
do higher order.
00:04:07.620 --> 00:04:10.260
So once we did this,
we could calculate
00:04:10.260 --> 00:04:13.670
various-- let's say, two-point
correlation functions,
00:04:13.670 --> 00:04:16.440
et cetera, in
perturbation theory.
00:04:16.440 --> 00:04:20.000
And in particular, the
two-point correlation function
00:04:20.000 --> 00:04:23.260
was related to the
susceptibility.
00:04:23.260 --> 00:04:27.020
And setting q to 0,
we found an expression
00:04:27.020 --> 00:04:33.010
for the inverse susceptibility
where the 0 order just
00:04:33.010 --> 00:04:36.270
comes from the t that
we have over here.
00:04:36.270 --> 00:04:42.580
And because of this perturbation
calculated to order of u,
00:04:42.580 --> 00:04:51.240
we had 4u n plus 2 and
integral over modes
00:04:51.240 --> 00:04:54.770
of just the variance
of the modes if y like.
00:05:03.100 --> 00:05:06.000
Now, first thing
that we noted was
00:05:06.000 --> 00:05:09.992
that the location of the point
at which the susceptibility
00:05:09.992 --> 00:05:13.110
vanishes, or
susceptibility diverges
00:05:13.110 --> 00:05:18.810
or its inverse vanishes is
no longer at t equals to 0.
00:05:18.810 --> 00:05:22.675
But we can see just setting this
expression to 0 that we have
00:05:22.675 --> 00:05:35.250
a tc, which is minus 4u n plus
2 integral d dk 2 pi to the d 1
00:05:35.250 --> 00:05:40.210
over-- let's put the k
here, k squared, potentially
00:05:40.210 --> 00:05:41.366
higher-order terms.
00:05:45.350 --> 00:05:51.760
Now, this is an integral
that in dimensions above 2--
00:05:51.760 --> 00:05:56.210
let's for time being focus
on dimensions above 2-- there
00:05:56.210 --> 00:06:00.090
is no singularity
as k goes to 0.
00:06:00.090 --> 00:06:03.850
k goes to 0, which is long
wavelength, is well-behaved.
00:06:03.850 --> 00:06:06.830
The integral could
potentially be singular
00:06:06.830 --> 00:06:10.060
if I were allowed to go
all the way to infinity,
00:06:10.060 --> 00:06:12.910
but I don't go all the infinity.
00:06:12.910 --> 00:06:18.520
All of my theories have an
underlying short wavelength.
00:06:18.520 --> 00:06:22.240
And hence, there is a
maximum in the Fourier modes
00:06:22.240 --> 00:06:26.504
which would render this
completely well-behaved
00:06:26.504 --> 00:06:28.746
integral.
00:06:28.746 --> 00:06:31.870
In fact, if I forget
higher-order term,
00:06:31.870 --> 00:06:33.150
I could put them.
00:06:33.150 --> 00:06:35.375
But if I forget
them, I can evaluate
00:06:35.375 --> 00:06:38.080
what this correction tc is.
00:06:38.080 --> 00:06:43.190
It is minus 4u n plus 2 over k.
00:06:43.190 --> 00:06:49.800
This-- I've been writing--
symmetry at surface area
00:06:49.800 --> 00:06:57.660
of a d dimensional unit sphere
divided by 2 pi to the d.
00:06:57.660 --> 00:07:03.240
And then I have the integral
of k to the d minus 3,
00:07:03.240 --> 00:07:07.660
which integrates to
lambda to the d minus 2
00:07:07.660 --> 00:07:11.780
divided by d minus 2.
00:07:11.780 --> 00:07:15.270
I wrote that
explicitly because we
00:07:15.270 --> 00:07:20.880
are going to encounter this
combination a lot of times.
00:07:20.880 --> 00:07:24.470
And so we will give
it a name k sub d.
00:07:24.470 --> 00:07:29.710
So it's just the solid angle
in d dimensions divided by 2 pi
00:07:29.710 --> 00:07:31.030
to the d.
00:07:31.030 --> 00:07:36.030
OK, so essentially, in
dimensions greater than 2,
00:07:36.030 --> 00:07:37.410
nothing much happens.
00:07:37.410 --> 00:07:42.120
There is a shift in the location
of the singularity compared
00:07:42.120 --> 00:07:43.080
to the Gaussian.
00:07:43.080 --> 00:07:45.800
Because you are no
longer at the Gaussian,
00:07:45.800 --> 00:07:49.530
you are at a theory that
has additional stabilizing
00:07:49.530 --> 00:07:51.810
terms such as m to
the fourth, et cetera.
00:07:51.810 --> 00:07:57.190
So there is no problem now for
p going to negative values.
00:07:57.190 --> 00:08:01.170
The thing that was
more interesting
00:08:01.170 --> 00:08:04.700
was that when we
looked at what happens
00:08:04.700 --> 00:08:11.170
in the vicinity of this
new tc, and to lowest order
00:08:11.170 --> 00:08:14.910
we got this form
of a divergence.
00:08:14.910 --> 00:08:21.550
And then at the next
order, I had a correction.
00:08:21.550 --> 00:08:27.470
Again, coming from this form,
4u n plus 2, an integral.
00:08:30.560 --> 00:08:34.650
And actually, this was obtained
by taking the difference of two
00:08:34.650 --> 00:08:38.080
of these factors
evaluated at p and tc.
00:08:38.080 --> 00:08:41.410
That's what gave me
the t minus tc outside.
00:08:41.410 --> 00:08:51.450
And then I had an integral that
involved two of these factors.
00:08:51.450 --> 00:08:54.690
Presumably to be
consistent to lowest order,
00:08:54.690 --> 00:08:59.220
I have to evaluate
them as small as I can.
00:08:59.220 --> 00:09:02.660
And so I would have two factors
of k squared or k squared
00:09:02.660 --> 00:09:03.570
plus something.
00:09:08.500 --> 00:09:09.918
Presumably, higher-order terms.
00:09:14.310 --> 00:09:19.350
The thing about these integrals
as opposed to the previous one
00:09:19.350 --> 00:09:21.800
is that, again,
I can try to look
00:09:21.800 --> 00:09:25.330
at the behavior at
large k and small k.
00:09:25.330 --> 00:09:28.330
At large k, no
matter how many terms
00:09:28.330 --> 00:09:31.070
I add to the series,
ultimately, I
00:09:31.070 --> 00:09:36.180
will be concerned by
cutting it off by lambda.
00:09:36.180 --> 00:09:38.390
Whereas, if I have
something that I
00:09:38.390 --> 00:09:43.030
have set t equals to 0 in both
of these denominator factors,
00:09:43.030 --> 00:09:45.920
I now have a
singularity at k goes
00:09:45.920 --> 00:09:49.200
to 0 in dimensions less than 4.
00:09:49.200 --> 00:09:52.600
The integral would blow up
in dimensions less than 4
00:09:52.600 --> 00:09:56.830
if I am allowed to go all the
way to 0, which is arbitrarily
00:09:56.830 --> 00:09:59.480
long wavelengths.
00:09:59.480 --> 00:10:03.430
Now in principle, if
I am not exactly at tc
00:10:03.430 --> 00:10:07.390
and I'm looking at singularity
as being away from tc,
00:10:07.390 --> 00:10:11.476
I expect on physical
grounds that fluctuations
00:10:11.476 --> 00:10:16.530
will persist up to some
correlation length.
00:10:16.530 --> 00:10:21.320
So the shortest value of k
that I should really physically
00:10:21.320 --> 00:10:27.570
be able to go to, irrespective
of how careful or careless
00:10:27.570 --> 00:10:31.640
I am with the factors of t and
t minus tc that I put here,
00:10:31.640 --> 00:10:34.830
is of the order of the
physical correlation length.
00:10:34.830 --> 00:10:37.870
And as we saw, this
means that there
00:10:37.870 --> 00:10:42.080
is a correction that
is of the form of u k
00:10:42.080 --> 00:10:47.560
to the 4-- k squared psi
to the power of 4 minus d.
00:10:47.560 --> 00:10:53.620
And I emphasized that the
dimensionless combination
00:10:53.620 --> 00:10:56.320
of the parameter
u that potentially
00:10:56.320 --> 00:10:59.880
can be added as a correction
to a number of order of 1
00:10:59.880 --> 00:11:04.650
is u k squared divided by some--
multiplied by some length scale
00:11:04.650 --> 00:11:06.775
to the power of 4 minus d.
00:11:06.775 --> 00:11:09.210
Above four dimensions,
the integral
00:11:09.210 --> 00:11:12.720
is convergent at small values.
00:11:12.720 --> 00:11:14.630
And the integral
will be dominated
00:11:14.630 --> 00:11:17.440
and the length scale
that would appear here
00:11:17.440 --> 00:11:21.440
would be some kind of a
short distance cutoff,
00:11:21.440 --> 00:11:23.170
like the averaging length.
00:11:23.170 --> 00:11:29.140
Whereas in four dimensions with
divergence of the correlation
00:11:29.140 --> 00:11:33.085
length is the thing that
will lead this perturbation
00:11:33.085 --> 00:11:35.771
theory to be kind of
difficult and [INAUDIBLE].
00:11:40.490 --> 00:11:48.690
So this is an example of a
divergent perturbation theory.
00:11:48.690 --> 00:11:50.930
So what we are
going to do in order
00:11:50.930 --> 00:11:53.930
to be able to make
sense out of it,
00:11:53.930 --> 00:11:58.600
and see how this divergence
here can be translated
00:11:58.600 --> 00:12:01.940
to a change in exponent, which
is what we are physically
00:12:01.940 --> 00:12:06.330
expecting to occur, we
reorganize this perturbation
00:12:06.330 --> 00:12:10.190
theory in a
conceptual way that is
00:12:10.190 --> 00:12:17.258
helped by this perturbative
renormalization group approach.
00:12:17.258 --> 00:12:21.490
So we keep the
perturbation theory,
00:12:21.490 --> 00:12:25.200
but change the way that we
look at perturbation theory
00:12:25.200 --> 00:12:26.910
by appealing to renormalization.
00:12:31.900 --> 00:12:36.626
So you can see that throughout
doing this perturbation theory,
00:12:36.626 --> 00:12:44.890
I end up having to do integrals
over modes that are defined
00:12:44.890 --> 00:12:49.016
in the space of the
Fourier parameter q.
00:12:49.016 --> 00:12:54.280
And a nice way to implement
this coarse graining that
00:12:54.280 --> 00:12:58.310
has led to this field
theory is to imagine
00:12:58.310 --> 00:13:03.700
that this integration is
over some sphere where
00:13:03.700 --> 00:13:09.410
the maximum inverse
wavelength that is allowed,
00:13:09.410 --> 00:13:14.090
or q number that is
allowed is some lambda.
00:13:14.090 --> 00:13:18.050
And so the task that I
have on the first line
00:13:18.050 --> 00:13:24.430
is to integrate over all
modes that live in this.
00:13:24.430 --> 00:13:26.950
And just on physical
grounds, we don't
00:13:26.950 --> 00:13:31.235
expect to get any singularities
from the modes that
00:13:31.235 --> 00:13:33.410
are at the edge.
00:13:33.410 --> 00:13:37.890
We expect to get singularities
by considering what's
00:13:37.890 --> 00:13:43.200
going on at long
wavelengths or 0q.
00:13:43.200 --> 00:13:47.980
So the idea of
renormalization group
00:13:47.980 --> 00:13:50.650
was to follow three steps.
00:13:50.650 --> 00:13:58.960
The first step was to do
coarse graining, which
00:13:58.960 --> 00:14:05.140
was to take whatever your
shortest wavelength was,
00:14:05.140 --> 00:14:10.350
make it b times
larger and average.
00:14:10.350 --> 00:14:11.700
That's in real space.
00:14:11.700 --> 00:14:13.570
In Fourier space,
what that amounts
00:14:13.570 --> 00:14:23.640
to is to get rid of all of the
variations and frequencies that
00:14:23.640 --> 00:14:26.753
are up to lambda over b.
00:14:29.980 --> 00:14:37.680
So what I can do is to say that
I had a whole bunch of modes
00:14:37.680 --> 00:14:38.962
m of q.
00:14:41.620 --> 00:14:47.120
I am going to subdivide
them into two classes.
00:14:47.120 --> 00:14:51.500
I will have the
modes sigma of q.
00:14:51.500 --> 00:14:54.950
Maybe I will write it in
different color, sigma of q.
00:14:58.180 --> 00:15:03.610
That are the ones
that sitting here.
00:15:03.610 --> 00:15:06.390
So these are the sigmas.
00:15:06.390 --> 00:15:09.520
And these correspond
to wave numbers
00:15:09.520 --> 00:15:15.240
that are between lambda
over b and lambda.
00:15:15.240 --> 00:15:20.340
And I will have a bunch
of other variables
00:15:20.340 --> 00:15:22.387
that I will call m tilde.
00:15:27.750 --> 00:15:35.080
And this wave close
to the singularity,
00:15:35.080 --> 00:15:40.930
but now removed by an
amount lambda over b.
00:15:40.930 --> 00:15:45.260
So essentially, getting
rid of the picture
00:15:45.260 --> 00:15:49.150
where it had fluctuations
at short length scale
00:15:49.150 --> 00:15:54.370
amounts to integrating
over Fourier modes that
00:15:54.370 --> 00:15:58.352
represent your field that
lie in this integral.
00:16:02.220 --> 00:16:06.585
So I want to do that as an
operation that is performed,
00:16:06.585 --> 00:16:10.580
let's say, at the level
of the partition function.
00:16:10.580 --> 00:16:16.190
So I can say that my original
integration can be broken up
00:16:16.190 --> 00:16:22.840
into integration
over this m tilde
00:16:22.840 --> 00:16:25.822
and the integration over sigma.
00:16:25.822 --> 00:16:33.860
So that's just rewriting that
rightmost integral up there.
00:16:33.860 --> 00:16:40.760
And then I have a
weight exponential.
00:16:40.760 --> 00:16:44.420
OK, let's write
it out explicitly.
00:16:44.420 --> 00:16:52.220
So the weight is composed
of beta H0 and the u.
00:16:52.220 --> 00:16:58.160
Now, we note that the beta H0
part, just as we did already
00:16:58.160 --> 00:17:00.670
for the case of the
Gaussian, does not
00:17:00.670 --> 00:17:04.869
mix up these two
classes of modes.
00:17:04.869 --> 00:17:08.460
So I can write that part
as an integral from 0
00:17:08.460 --> 00:17:16.269
to lambda over b dd 2 pi over d.
00:17:16.269 --> 00:17:19.800
And these are things
that are really inside,
00:17:19.800 --> 00:17:23.289
so I could also label
them by q lesser.
00:17:23.289 --> 00:17:27.955
So I have m tilde
of q lesser squared.
00:17:30.530 --> 00:17:35.940
And this multiplies
t plus k q lesser
00:17:35.940 --> 00:17:38.869
squared and so forth over 2.
00:17:43.560 --> 00:17:49.650
I have a similar term,
which is the modes that
00:17:49.650 --> 00:17:54.330
are between lambda
over b and lambda.
00:17:54.330 --> 00:18:01.652
So I just simply changed or
broke my overall integration
00:18:01.652 --> 00:18:06.380
in beta H0 into two parts.
00:18:06.380 --> 00:18:11.340
Now I have the higher q numbers.
00:18:11.340 --> 00:18:18.570
And these are sigma
of q larger squared.
00:18:18.570 --> 00:18:23.820
Again, same weight,
t plus k q larger
00:18:23.820 --> 00:18:26.447
squared and so forth over 2.
00:18:30.279 --> 00:18:34.460
Make sure this minus
is in line with this.
00:18:34.460 --> 00:18:36.700
And then I have,
of course, the u.
00:18:36.700 --> 00:18:42.060
So then I have a minus u.
00:18:42.060 --> 00:18:43.870
Now, I won't write
this explicitly.
00:18:43.870 --> 00:18:47.510
I will write it explicitly
on the next board.
00:18:47.510 --> 00:18:57.510
But clearly, implicitly it
involves both m tilde and sigma
00:18:57.510 --> 00:18:58.780
mixed up into each other.
00:19:07.930 --> 00:19:11.580
So I have just rewritten
my partition function
00:19:11.580 --> 00:19:16.170
after subdividing it into
these two classes of modes
00:19:16.170 --> 00:19:19.150
and just hiding all
of the complexity
00:19:19.150 --> 00:19:21.454
in this function
that mixes the two.
00:19:25.570 --> 00:19:29.690
So let's rewrite this.
00:19:29.690 --> 00:19:34.830
I have integral over the modes
that I would like to keep.
00:19:34.830 --> 00:19:37.320
The m tilde I
would like to keep.
00:19:40.010 --> 00:19:43.370
And there is a weight
associated with them
00:19:43.370 --> 00:19:46.095
that will, therefore,
not be integrated.
00:19:46.095 --> 00:19:55.870
This is the integral from 0
to lambda over b dd k lesser 2
00:19:55.870 --> 00:20:02.720
pi to the d t plus k, k lesser
q lesser squared, et cetera.
00:20:02.720 --> 00:20:10.642
And then I have tilde
of q lesser squared.
00:20:15.720 --> 00:20:22.260
Now, if I didn't have
this there, the u,
00:20:22.260 --> 00:20:27.680
I could immediately perform the
Gaussian integrals over sigmas.
00:20:27.680 --> 00:20:30.030
Indeed, we already did this.
00:20:30.030 --> 00:20:32.840
And the answer would
be e to the minus--
00:20:32.840 --> 00:20:35.860
there are n-components
to this vector.
00:20:35.860 --> 00:20:40.520
So the answer is going
to be multiplied by n.
00:20:40.520 --> 00:20:44.440
1/2 is because the square root
that I get from each mode.
00:20:44.440 --> 00:20:47.916
I get a factor of
volume integral dd
00:20:47.916 --> 00:20:53.560
q larger 2 pi to the d
integrated from lambda over v
00:20:53.560 --> 00:21:01.090
to lambda log of t plus k q
greater squared and so forth.
00:21:01.090 --> 00:21:04.190
So if I didn't have
the u, this would
00:21:04.190 --> 00:21:08.840
be the answer for doing
the Gaussian integration.
00:21:08.840 --> 00:21:12.030
But I have the u,
so what should I do?
00:21:12.030 --> 00:21:14.150
The answer is very simple.
00:21:14.150 --> 00:21:22.900
I write it as e to the minus
u m tilde m sigma average.
00:21:27.840 --> 00:21:37.270
So what I have done is to say
that with this weight, that
00:21:37.270 --> 00:21:41.490
is a Gaussian
weight for sigma, I
00:21:41.490 --> 00:21:45.760
average the function
e to the minus u.
00:21:45.760 --> 00:21:53.020
If you like, this
is a Gaussian sigma.
00:21:53.020 --> 00:21:57.960
So to sort of write it
explicitly, what I have stated
00:21:57.960 --> 00:22:03.280
is an average where
I integrate out
00:22:03.280 --> 00:22:07.920
the high-frequency
short wavelength modes
00:22:07.920 --> 00:22:14.740
is by definition integrate over
all configurations of sigma
00:22:14.740 --> 00:22:24.800
with the Gaussian weight
whatever object you have,
00:22:24.800 --> 00:22:29.490
and then normalize
by the Gaussian.
00:22:35.310 --> 00:22:41.284
Of course, in our case, our
O depends both on sigma and m
00:22:41.284 --> 00:22:44.370
tilde, so the result
of these averaging
00:22:44.370 --> 00:22:48.810
will be a function of in tilde.
00:22:48.810 --> 00:23:00.590
And indeed, I can write this as
an integral over m tilde of q
00:23:00.590 --> 00:23:05.530
with a new weight, e to the
minus beta H tilde, which
00:23:05.530 --> 00:23:08.760
only depends on m tilde
because I got rid of
00:23:08.760 --> 00:23:10.860
and I integrated
over the sigmas.
00:23:13.520 --> 00:23:22.740
And by definition,
my beta H tilde
00:23:22.740 --> 00:23:28.810
that depends only on
m tilde has a part
00:23:28.810 --> 00:23:40.380
that is the integral from 0
to lambda over b dd q lesser
00:23:40.380 --> 00:23:56.340
2 pi to the d the Gaussian
weight over the range of modes
00:23:56.340 --> 00:23:59.530
that are allowed.
00:23:59.530 --> 00:24:03.250
There is a part that is
just this constant term when
00:24:03.250 --> 00:24:06.580
I take care-- if I write
it in this fashion,
00:24:06.580 --> 00:24:09.040
there is an overall constant.
00:24:09.040 --> 00:24:15.360
Clearly, what this constant is,
is the free energy of the modes
00:24:15.360 --> 00:24:19.380
that I have integrated,
assuming that they are Gaussian,
00:24:19.380 --> 00:24:21.700
in this interval.
00:24:21.700 --> 00:24:24.670
The answer is
proportional to volume.
00:24:24.670 --> 00:24:26.610
But as usual, when
we are thinking
00:24:26.610 --> 00:24:30.270
about weights and probabilities,
overall constants don't matter.
00:24:30.270 --> 00:24:34.160
But I can certainly continue
to write that over here.
00:24:34.160 --> 00:24:39.750
So that part went to here,
this part went to here.
00:24:39.750 --> 00:24:43.130
And so the only
part that is left
00:24:43.130 --> 00:24:51.510
is minus log of d to the
minus u of m tilde and sigma
00:24:51.510 --> 00:24:53.982
after I get rid of the sigmas.
00:25:02.340 --> 00:25:06.955
So far, I have done things
that are extremely general.
00:25:09.490 --> 00:25:16.180
But now I note that I
am interested in doing
00:25:16.180 --> 00:25:19.100
perturbation.
00:25:19.100 --> 00:25:23.900
So the only place that I
haven't really evaluated things
00:25:23.900 --> 00:25:27.860
is where this u is appearing
inside the exponential log
00:25:27.860 --> 00:25:30.500
average, et cetera.
00:25:30.500 --> 00:25:34.670
So what I can do is
I can perturbatively
00:25:34.670 --> 00:25:38.376
expand this
exponential over here.
00:25:38.376 --> 00:25:43.830
So I will get log of 1 minus u,
which is approximately minus u.
00:25:43.830 --> 00:25:50.890
So the first term here would
be u averaged Gaussian.
00:25:50.890 --> 00:25:57.440
The next term will be
minus 1/2 u squared
00:25:57.440 --> 00:26:06.440
average minus u average
squared and so forth.
00:26:06.440 --> 00:26:11.850
So you can see that the
variance appeared in this stage.
00:26:11.850 --> 00:26:17.120
And generally, the l-th term
in the series would be minus 1
00:26:17.120 --> 00:26:22.470
to the l divided by l factorial.
00:26:22.470 --> 00:26:26.660
And we saw this already.
00:26:26.660 --> 00:26:32.490
The log of e to the something
is the generator of cumulants.
00:26:32.490 --> 00:26:37.890
So this would be
the l-th power of u,
00:26:37.890 --> 00:26:41.500
the cumulant here would appear.
00:26:41.500 --> 00:26:45.205
And again, the cumulant
would serve the function
00:26:45.205 --> 00:26:48.770
of cutting off connected
pieces as we shall see shortly.
00:26:55.580 --> 00:26:59.270
So that's what we
are going to do.
00:26:59.270 --> 00:27:05.290
We are going to insert
the u's, all things that
00:27:05.290 --> 00:27:07.920
go beyond the Gaussian--
but initially,
00:27:07.920 --> 00:27:13.005
just the m to the fourth
part-- inside this series
00:27:13.005 --> 00:27:18.060
and term by term calculate
the corrections to this weight
00:27:18.060 --> 00:27:24.950
that we get after we
integrate out the long lambda.
00:27:24.950 --> 00:27:30.291
Or sorry, the short wavelength
modes or the long q modes.
00:27:30.291 --> 00:27:30.790
OK?
00:27:33.870 --> 00:27:38.430
So let's focus on
this first term.
00:27:38.430 --> 00:27:45.050
So what is this u that depends
on both m tilde and sigma?
00:27:48.830 --> 00:27:52.577
And I have the expression
for u up there.
00:27:55.560 --> 00:28:07.030
So I can write it as u integral
dd q1 dd q2 dd q3 to be
00:28:07.030 --> 00:28:10.686
symmetric in all
of the four q's.
00:28:10.686 --> 00:28:15.740
I write an integration
over the fourth q,
00:28:15.740 --> 00:28:19.060
but then enforce it
by a delta function
00:28:19.060 --> 00:28:23.620
that the sum of the
q's should be 0.
00:28:29.260 --> 00:28:34.460
And then I have
four factors of m,
00:28:34.460 --> 00:28:40.540
but an m depending on
which part of the q space
00:28:40.540 --> 00:28:45.990
I am encountering is either
a sigma or an m tilde.
00:28:45.990 --> 00:28:48.320
So without doing
anything wrong, I
00:28:48.320 --> 00:28:55.070
can replace each m with
an m tilde plus sigma.
00:29:02.790 --> 00:29:07.200
So depending on where my q1
is in the integrations from 0
00:29:07.200 --> 00:29:11.630
to lambda, I will be
encountering either this
00:29:11.630 --> 00:29:13.660
or this.
00:29:13.660 --> 00:29:20.230
And then I have the dot
product of that with m tilde q2
00:29:20.230 --> 00:29:23.015
plus sigma of q2.
00:29:23.015 --> 00:29:25.230
And then I have the
dot product that
00:29:25.230 --> 00:29:33.914
would correspond to m tilde
of q3 plus sigma of q3
00:29:33.914 --> 00:29:40.096
with m tilde of q4
plus sigma of q4.
00:29:43.670 --> 00:29:45.550
So that's the structure
of my [INAUDIBLE].
00:29:49.850 --> 00:29:52.900
And again, what I have
to do in principle
00:29:52.900 --> 00:29:59.540
is to integrate out the sigmas
keeping the m tildes when
00:29:59.540 --> 00:30:02.910
I perform this
averaging over here.
00:30:09.470 --> 00:30:12.530
So let's write down,
if I were to expand
00:30:12.530 --> 00:30:16.670
this thing before
the integration, what
00:30:16.670 --> 00:30:20.580
are the types of terms
that I would get?
00:30:20.580 --> 00:30:23.264
And I'll give them names.
00:30:26.050 --> 00:30:29.800
One type of term
that is very easy
00:30:29.800 --> 00:30:36.260
is when I have m tilde of q1
dotted with m tilde of Q2,
00:30:36.260 --> 00:30:41.880
m tilde of q3 dotted
with m tilde of q4.
00:30:47.460 --> 00:30:50.670
If I expand this so
there's 2 terms per bracket
00:30:50.670 --> 00:30:54.540
and there are 4 brackets, so
there are 16 terms, only 1
00:30:54.540 --> 00:31:01.210
of these terms is of this
variety out of the 16.
00:31:01.210 --> 00:31:04.520
What I will do is
also now introduce
00:31:04.520 --> 00:31:08.030
a diagrammatic representation.
00:31:08.030 --> 00:31:15.740
Whenever I see an m tilde, I
will include a straight line.
00:31:15.740 --> 00:31:22.140
Whenever I see a sigma, I
will include a wavy line.
00:31:22.140 --> 00:31:24.790
So this entity that
I have over here
00:31:24.790 --> 00:31:28.980
is composed of four of
these straight lines.
00:31:28.980 --> 00:31:40.000
And I will indicate that by
this diagram, q1, q2, q3, q4.
00:31:40.000 --> 00:31:43.010
And the reason is, of
course-- first of all,
00:31:43.010 --> 00:31:45.136
there are four of these.
00:31:45.136 --> 00:31:49.780
So this is a so-called vertex
in a diagrammatic representation
00:31:49.780 --> 00:31:51.940
that has four lines.
00:31:51.940 --> 00:31:57.830
And secondly, the lines are not
all totally equivalent because
00:31:57.830 --> 00:32:01.030
of the way that the dot
products are arranged.
00:32:01.030 --> 00:32:06.250
Say, q1 and q2 that are dot
product together are distinct,
00:32:06.250 --> 00:32:11.020
let's say, from q1 and q3 that
are not dot product together.
00:32:11.020 --> 00:32:14.240
And to indicate that,
I make sure that there
00:32:14.240 --> 00:32:19.040
is this dotted line in
the vertex that separates
00:32:19.040 --> 00:32:22.445
and indicates which two are
dot product to each other.
00:32:27.470 --> 00:32:32.200
Now, the second class
of diagram comes
00:32:32.200 --> 00:32:36.410
when I replace one of the
m tildes with a sigma.
00:32:36.410 --> 00:32:41.240
So I have sigma of q1
dotted with m tilde
00:32:41.240 --> 00:32:51.700
of q2, m tilde of q3
dotted with m tilde of q4.
00:32:51.700 --> 00:32:55.100
Now clearly, in this
case, I had a choice
00:32:55.100 --> 00:33:00.420
of four factors of m tilde
to replace with this.
00:33:00.420 --> 00:33:04.780
So of the 16 terms in
this expansion, 4 of them
00:33:04.780 --> 00:33:06.880
belong to this class.
00:33:06.880 --> 00:33:10.040
Which if I were to
represent diagrammatically,
00:33:10.040 --> 00:33:12.480
I would have one of
the legs replaced
00:33:12.480 --> 00:33:16.430
with a wavy line and
all the other legs
00:33:16.430 --> 00:33:18.820
staying as solid lines.
00:33:25.530 --> 00:33:30.200
The third class of
terms correspond
00:33:30.200 --> 00:33:34.630
to replacing two of the
m tildes with sigmas.
00:33:34.630 --> 00:33:38.250
Now here again, I have a
choice whether the second one
00:33:38.250 --> 00:33:40.170
is a partner of
the first one that
00:33:40.170 --> 00:33:44.260
became sigma, such as
this one, sigma of q1
00:33:44.260 --> 00:33:51.240
dotted with sigma of
q2, m tilde of q3 dotted
00:33:51.240 --> 00:33:54.140
with m tilde of q4.
00:33:54.140 --> 00:33:56.360
And then clearly,
I could have chosen
00:33:56.360 --> 00:34:01.040
one pair or the other pair
to change into sigmas.
00:34:01.040 --> 00:34:04.830
So there are two terms
that are like this.
00:34:04.830 --> 00:34:09.050
And diagrammatically,
the wavy lines
00:34:09.050 --> 00:34:13.400
belong to the same
branch of this object.
00:34:20.409 --> 00:34:21.239
OK, next.
00:34:21.239 --> 00:34:23.270
Keep going.
00:34:23.270 --> 00:34:25.429
Actually, I have
another thing when
00:34:25.429 --> 00:34:28.840
I replace two of the
m tildes with sigma,
00:34:28.840 --> 00:34:32.389
but now belonging to two
different elements of this dot
00:34:32.389 --> 00:34:33.173
product.
00:34:33.173 --> 00:34:39.659
So I could have sigma of q1
dotted with m tilde of q2.
00:34:39.659 --> 00:34:44.699
And then I have sigma of q3
dotted with m tilde of q4.
00:34:48.830 --> 00:34:54.179
In which case, in each of the
pairs I had a choice of two
00:34:54.179 --> 00:34:56.590
for replacing m
tilde with sigma.
00:34:56.590 --> 00:34:58.390
So that's 2 times 2.
00:34:58.390 --> 00:35:01.990
There are four terms
that have this character.
00:35:01.990 --> 00:35:05.560
And if I were to represent
them diagrammatically,
00:35:05.560 --> 00:35:08.070
I would need to
put two wavy lines
00:35:08.070 --> 00:35:09.189
on two different branches.
00:35:14.460 --> 00:35:20.410
And then I have the possibility
of three things replaced.
00:35:20.410 --> 00:35:32.450
So I have sigma of q1 sigma of
q2 sigma of q3 m tilde of q4.
00:35:32.450 --> 00:35:35.800
And again, now it's
the other way around.
00:35:35.800 --> 00:35:39.120
One term is left out
of 4 to be m tilde.
00:35:39.120 --> 00:35:43.990
So this is, again,
a degeneracy of 4.
00:35:43.990 --> 00:35:50.790
And diagrammatically, I have
three lines that are wavy
00:35:50.790 --> 00:35:54.060
and one line that is solid.
00:35:54.060 --> 00:35:58.040
And at the end of
story, 6, I will
00:35:58.040 --> 00:36:10.840
have one diagram which
is all sigmas, which
00:36:10.840 --> 00:36:14.113
can be represented
essentially by all wavy lines.
00:36:21.160 --> 00:36:25.190
And to check that I didn't make
any mistake in my calculation,
00:36:25.190 --> 00:36:28.330
the sum of these
numbers better be 16.
00:36:28.330 --> 00:36:31.728
So that's 5, 7, 11, 15, 16.
00:36:34.440 --> 00:36:34.940
All right?
00:36:38.220 --> 00:36:44.320
Now, the next step of the
story is to do these averages.
00:36:44.320 --> 00:36:46.144
So I have to do the average.
00:37:02.060 --> 00:37:07.230
Now, the first term
doesn't involve any sigmas.
00:37:07.230 --> 00:37:09.610
All of my averages
here are obtained
00:37:09.610 --> 00:37:12.250
by integrating over sigmas.
00:37:12.250 --> 00:37:14.985
If there is no
sigmas to integrate,
00:37:14.985 --> 00:37:18.120
after I do the averaging
here I essentially
00:37:18.120 --> 00:37:19.680
get the same thing back.
00:37:19.680 --> 00:37:27.670
So I will get this
same expression.
00:37:27.670 --> 00:37:30.640
And clearly, that
would be a term
00:37:30.640 --> 00:37:34.710
that would contribute to
my beta H tilde, which
00:37:34.710 --> 00:37:37.460
is identical to what
I had originally.
00:37:37.460 --> 00:37:40.300
It is, again, m to the fourth.
00:37:40.300 --> 00:37:43.630
So that we understand.
00:37:43.630 --> 00:37:48.920
Now, the second
term here, what is
00:37:48.920 --> 00:37:50.620
the average that
I have to do here?
00:37:54.140 --> 00:37:58.270
I have one factor of sigma
with which I can average.
00:37:58.270 --> 00:38:03.290
But the weight that I
have is even in sigma.
00:38:03.290 --> 00:38:08.640
So the average of sigma, which
is Gaussian-distributed, is 0.
00:38:08.640 --> 00:38:10.413
So this will give me 0.
00:38:13.180 --> 00:38:17.420
And clearly, here
also there is a term
00:38:17.420 --> 00:38:19.885
that involves three
factors of sigma.
00:38:19.885 --> 00:38:22.022
Again, by symmetry this
will average out to 0.
00:38:31.850 --> 00:38:40.060
Now, there is a way of
indicating what happens here.
00:38:40.060 --> 00:38:43.370
See, what happens
here is that I will
00:38:43.370 --> 00:38:47.145
have to do an average
of this thing.
00:38:47.145 --> 00:38:50.080
The m tildes are not
part of the averaging.
00:38:50.080 --> 00:38:51.980
They just go out.
00:38:51.980 --> 00:38:54.550
The average moves all
the way over here.
00:38:57.540 --> 00:39:02.460
And the average of sigma of q1,
sigma of q2, I know what it is.
00:39:02.460 --> 00:39:06.620
It is going to be-- I could
have just written it over here.
00:39:06.620 --> 00:39:13.740
It's 2 pi to the d delta
function q1 plus q2
00:39:13.740 --> 00:39:16.560
divided by k q squared.
00:39:16.560 --> 00:39:18.590
Maybe I'll explicitly
write it over here.
00:39:18.590 --> 00:39:21.500
So what we have here is
that the average of sigma
00:39:21.500 --> 00:39:31.086
of q1 with some index sigma
of q2 with some other index
00:39:31.086 --> 00:39:33.377
is-- first of all, the two
indices have to be the same.
00:39:36.000 --> 00:39:39.332
I have a delta
function q1 plus q2.
00:39:39.332 --> 00:39:46.084
And then I have t plus k
q1 squared and so forth.
00:39:46.084 --> 00:39:48.850
So it's my usual Gaussian.
00:39:48.850 --> 00:39:53.370
So essentially, you can see
that one immediate consequence
00:39:53.370 --> 00:39:57.520
of this averaging is that
previously these things had
00:39:57.520 --> 00:39:59.770
two different momenta
and potentially
00:39:59.770 --> 00:40:01.750
two different indices.
00:40:01.750 --> 00:40:05.670
They get to be the same thing.
00:40:05.670 --> 00:40:10.690
And the fact that
the labels that
00:40:10.690 --> 00:40:14.690
were assigned to this,
the q and the index alpha,
00:40:14.690 --> 00:40:21.890
are forced to be the same,
we can diagrammatically
00:40:21.890 --> 00:40:25.570
indicate by making
this a closed line.
00:40:25.570 --> 00:40:30.230
So we are going to represent
the result of that averaging
00:40:30.230 --> 00:40:36.250
with essentially taking--
these two lines are unchanged.
00:40:36.250 --> 00:40:37.880
They can be whatever they were.
00:40:37.880 --> 00:40:42.640
These two lines really
are joined together
00:40:42.640 --> 00:40:44.840
through this process.
00:40:44.840 --> 00:40:48.850
So we indicate them that way.
00:40:48.850 --> 00:40:53.510
And similarly, when I do
the same thing over here,
00:40:53.510 --> 00:40:59.120
I do the averaging of
this and the answer
00:40:59.120 --> 00:41:05.470
I can indicate by leaving
these two lines by themselves
00:41:05.470 --> 00:41:08.845
and joining these two wavy
lines together in this fashion.
00:41:16.770 --> 00:41:19.680
Now, when you do--
this one we said is 0.
00:41:19.680 --> 00:41:23.480
So there's essentially one that
is left, which is number 6.
00:41:23.480 --> 00:41:25.395
For number 6, we
do our averaging.
00:41:27.920 --> 00:41:33.090
And for that we have to use-
for average or a product of four
00:41:33.090 --> 00:41:36.050
sigmas that are
Gaussian-distributed Wick's
00:41:36.050 --> 00:41:37.550
theorem.
00:41:37.550 --> 00:41:41.840
So one possibility is that
sigma 1 and sigma 2 are joined,
00:41:41.840 --> 00:41:45.520
and then sigma 4 and
sigma 3 have to be joined.
00:41:45.520 --> 00:41:48.760
So basically, I took
sigma 1 and sigma 2
00:41:48.760 --> 00:41:53.670
and joined them, sigma 3 and
sigma 4 that I joined them.
00:41:53.670 --> 00:41:56.210
But another possibility
is I can take
00:41:56.210 --> 00:41:58.980
sigma 1 with sigma 3 or sigma 4.
00:41:58.980 --> 00:42:01.650
So there are really two choices.
00:42:01.650 --> 00:42:05.098
And then I will have a
diagram that is like this.
00:42:19.050 --> 00:42:24.010
Now, each one of these
operations and diagrams
00:42:24.010 --> 00:42:28.430
really stands for some
integration and result.
00:42:28.430 --> 00:42:32.730
And let's for example,
pick our number 3.
00:42:38.670 --> 00:42:42.890
For number 3, what
we are supposed to do
00:42:42.890 --> 00:42:46.090
is to do the integration.
00:42:46.090 --> 00:42:47.030
Sorry.
00:42:47.030 --> 00:42:52.750
First of all, number 3 has
a numerical factor of 2.
00:42:52.750 --> 00:42:55.180
This is something that
is proportional to u
00:42:55.180 --> 00:42:56.730
when we take the average.
00:42:59.590 --> 00:43:11.710
I have in principle to do
integration over q1 q2 q3 q4.
00:43:22.895 --> 00:43:23.395
OK.
00:43:26.260 --> 00:43:34.470
The m tilde of q3 and m
tilde of q4 in this diagram
00:43:34.470 --> 00:43:37.850
were not averaged over.
00:43:37.850 --> 00:43:39.240
So that term remains.
00:43:42.140 --> 00:43:48.840
I did the averaging
over q1 and q2.
00:43:48.840 --> 00:43:51.760
When I did that averaging,
I, first of all,
00:43:51.760 --> 00:43:56.720
got a delta alpha alpha because
those were two things that
00:43:56.720 --> 00:43:58.780
were dot product to
each other, so they
00:43:58.780 --> 00:44:01.690
were carrying the same
index to start with.
00:44:01.690 --> 00:44:06.220
I have a 2 pi to the d, a
delta function q1 plus q2.
00:44:06.220 --> 00:44:11.485
And I have t plus k q1 squared.
00:44:17.780 --> 00:44:19.790
Now, delta alpha alpha.
00:44:19.790 --> 00:44:25.060
Summing over alpha
gives a factor of n.
00:44:25.060 --> 00:44:28.420
And when you look
at these diagrams,
00:44:28.420 --> 00:44:32.836
quite generally
whenever you see a loop,
00:44:32.836 --> 00:44:37.080
with a loop you would associate
the factor of n because
00:44:37.080 --> 00:44:42.080
of the index that runs
and gets summed over.
00:44:42.080 --> 00:44:47.840
So this answer is going to
be proportional to 2 u n.
00:44:51.660 --> 00:44:57.960
Now, q1 and q2 are
said to be 0, the sum.
00:44:57.960 --> 00:44:59.810
So this is 0.
00:44:59.810 --> 00:45:05.140
So q3 and q4 have
to add up to 0.
00:45:05.140 --> 00:45:09.290
So the part that involves
q3 and q4, the m tilde,
00:45:09.290 --> 00:45:13.720
essentially I will
get an integral dd--
00:45:13.720 --> 00:45:15.980
let's say whatever q3.
00:45:15.980 --> 00:45:19.810
It doesn't matter because
it's an index of integration.
00:45:19.810 --> 00:45:26.732
I have m tilde of q3 squared.
00:45:26.732 --> 00:45:31.180
And again, q3, it
is something that
00:45:31.180 --> 00:45:33.590
goes with one of these m tildes.
00:45:33.590 --> 00:45:38.952
So this is an integration that I
have to do between 0 and lambda
00:45:38.952 --> 00:45:39.944
over b.
00:45:42.920 --> 00:45:45.130
So there is essentially
one integration
00:45:45.130 --> 00:45:48.702
left because q1 and q2
are left to be the same.
00:45:48.702 --> 00:45:49.660
So this is an integral.
00:45:49.660 --> 00:45:52.450
Let's call the
integration variable
00:45:52.450 --> 00:45:56.240
that was q1-- I could k,
it doesn't matter-- 2 pi
00:45:56.240 --> 00:46:02.140
to the d the same integral
that we've seen before.
00:46:02.140 --> 00:46:07.640
Except that since this
originated from the sigmas,
00:46:07.640 --> 00:46:11.870
the integration here is
from lambda over b lambda.
00:46:19.160 --> 00:46:24.340
So this is basically a number
that I can take, say, out here
00:46:24.340 --> 00:46:27.640
and regard as a
coefficient that multiplies
00:46:27.640 --> 00:46:32.890
a term that is m tilde squared.
00:46:32.890 --> 00:46:37.990
And similarly, 4.
00:46:37.990 --> 00:46:38.490
4.
00:46:38.490 --> 00:46:41.410
We said we have four
diagrams of his variety,
00:46:41.410 --> 00:46:46.820
so this would be a
contribution that is 4u.
00:46:46.820 --> 00:46:49.620
I can read out the whole--
write out the whole thing.
00:46:49.620 --> 00:46:52.130
Certainly, I have all of this.
00:46:52.130 --> 00:46:54.290
I have all of this.
00:46:54.290 --> 00:47:03.980
In that case, I have m tilde
q3 m tilde-- well, let's see.
00:47:03.980 --> 00:47:08.860
I have m tilde of
q2 m tilde of q4.
00:47:08.860 --> 00:47:13.730
And they carry different
indices because they
00:47:13.730 --> 00:47:18.320
came from two
different dot products.
00:47:18.320 --> 00:47:23.725
And then I have to do
an average over sigma 1
00:47:23.725 --> 00:47:27.580
and sigma 3 which carry
different indices that
00:47:27.580 --> 00:47:29.410
are for beta.
00:47:29.410 --> 00:47:38.654
2 pi to the d delta function
q1 plus q3 e plus k q1 squared.
00:47:44.170 --> 00:47:49.490
Again, since q1 plus q3 is 0 and
the sum of the four q's is 0,
00:47:49.490 --> 00:47:52.675
these two have to add up to 0.
00:47:52.675 --> 00:47:57.270
So the answer, again, will
be written as 4u integral 0
00:47:57.270 --> 00:48:02.955
to lambda over b dd of
some q divided by 2 pi
00:48:02.955 --> 00:48:07.890
to the b m tilde of q squared.
00:48:07.890 --> 00:48:09.620
And then actually,
the same integration,
00:48:09.620 --> 00:48:18.812
lambda over d lambda dd
k 2 pi to the d 1 over 2
00:48:18.812 --> 00:48:20.255
plus k squared.
00:48:28.440 --> 00:48:31.990
So out of the six
terms, two are 0.
00:48:31.990 --> 00:48:35.590
Two are explicitly
calculated over here.
00:48:35.590 --> 00:48:39.200
One is trivially
just m to the fourth.
00:48:39.200 --> 00:48:43.940
The last one is basically
summing up all of these things.
00:48:46.750 --> 00:48:52.805
But these explicitly do
not depend on m tilde.
00:48:52.805 --> 00:48:58.290
So I'll just call the result of
doing all of this sum V delta f
00:48:58.290 --> 00:49:01.260
v at level 1.
00:49:01.260 --> 00:49:08.230
In the same way that integrating
the modes sigma that I'm not
00:49:08.230 --> 00:49:10.870
interested and
averaging over them
00:49:10.870 --> 00:49:15.380
gave a constant of integration,
that constant of integration
00:49:15.380 --> 00:49:20.280
gets corrected to
order of u over here.
00:49:20.280 --> 00:49:23.373
I don't need to explicitly
take care of it.
00:49:28.650 --> 00:49:32.010
So given all of
this information,
00:49:32.010 --> 00:49:36.760
let's write down what our last
line from the previous board
00:49:36.760 --> 00:49:37.670
is.
00:49:37.670 --> 00:49:43.000
So our intent was to
calculate a weight that
00:49:43.000 --> 00:49:48.050
governed these
coarse-grained modes.
00:49:48.050 --> 00:49:50.730
And our answer is
that, first of all,
00:49:50.730 --> 00:49:58.977
we will get a bunch of constants
delta f v 0 plus delta f v 1
00:49:58.977 --> 00:50:00.060
that we don't really care.
00:50:00.060 --> 00:50:04.040
They're just an
overall change that
00:50:04.040 --> 00:50:05.940
doesn't matter for
the probabilities.
00:50:05.940 --> 00:50:10.650
It's just contribution
to the free energy.
00:50:10.650 --> 00:50:14.180
And then we start to get things.
00:50:14.180 --> 00:50:17.980
And to the lowest
order what we had
00:50:17.980 --> 00:50:24.740
was replacing the
Gaussian weight,
00:50:24.740 --> 00:50:31.070
but only over this permitted
set of wavelengths.
00:50:31.070 --> 00:50:35.460
So I have dd q
lesser, let's say,
00:50:35.460 --> 00:50:43.550
2 pi to the d t plus k q
lesser squared and so forth.
00:50:43.550 --> 00:50:50.150
Divided by 2 m
tilde of q squared.
00:50:54.370 --> 00:50:59.150
Then, term number 1 in
the series gave me what?
00:50:59.150 --> 00:51:06.290
It gave me something that
was equivalent to my u
00:51:06.290 --> 00:51:09.160
if I were to Fourier
transform back to real space,
00:51:09.160 --> 00:51:10.920
m to the fourth.
00:51:10.920 --> 00:51:19.050
Except that my cutoff has
been shifted by lambda over b.
00:51:19.050 --> 00:51:22.210
So I don't want to
bother to write down
00:51:22.210 --> 00:51:25.660
that full form in
terms of Fourier modes.
00:51:25.660 --> 00:51:28.796
Essentially, if I want
to write this explicitly,
00:51:28.796 --> 00:51:31.390
it is just like that
line that I have,
00:51:31.390 --> 00:51:33.575
except that for the
integrations I'll
00:51:33.575 --> 00:51:37.190
have to explicitly indicate
0 to lambda over b.
00:51:39.950 --> 00:51:46.900
So the only terms that
we haven't included
00:51:46.900 --> 00:51:48.400
are the ones that are over here.
00:51:51.120 --> 00:51:54.740
Now, you look at
those terms and you
00:51:54.740 --> 00:51:59.290
find that the structure
of these terms
00:51:59.290 --> 00:52:01.720
is precisely what
we have over here.
00:52:07.080 --> 00:52:10.870
Except that there
is a modification.
00:52:10.870 --> 00:52:15.500
There is a constant term
that is added from this one
00:52:15.500 --> 00:52:18.666
and there's a constant term
that is added from that one.
00:52:18.666 --> 00:52:23.000
So the effect of those
things I can capture
00:52:23.000 --> 00:52:31.090
by changing the parameters
t to something else t tilde.
00:52:31.090 --> 00:52:36.230
So you can see that to order
of u squared that I haven't
00:52:36.230 --> 00:52:39.720
calculated, to order
of u, the only effect
00:52:39.720 --> 00:52:45.110
of this coarse graining is to
modify this one parameter so
00:52:45.110 --> 00:52:48.140
that t goes to t tilde.
00:52:48.140 --> 00:52:52.590
It certainly depends on how
much I coarse grain things.
00:52:52.590 --> 00:52:59.330
And this is the original t
plus the sum of these things.
00:52:59.330 --> 00:53:12.670
So I will have 2-- n
plus 2 u integral dd k 2
00:53:12.670 --> 00:53:17.364
pi to the d 1 over
t plus k k squared.
00:53:17.364 --> 00:53:18.780
And presumably,
higher-order terms
00:53:18.780 --> 00:53:24.980
are allowed, going from
0 to lambda over here.
00:53:24.980 --> 00:53:26.390
AUDIENCE: Question.
00:53:26.390 --> 00:53:29.080
PROFESSOR: Yes.
00:53:29.080 --> 00:53:29.886
Question?
00:53:29.886 --> 00:53:30.850
AUDIENCE: Yeah.
00:53:30.850 --> 00:53:35.090
When you have an integration
over x, if you have previously
00:53:35.090 --> 00:53:41.480
defined lambda to be
a cutoff in k space,
00:53:41.480 --> 00:53:45.180
might it be-- is it
1 over b lambda then?
00:53:45.180 --> 00:53:48.135
Or, is it b over lambda?
00:53:48.135 --> 00:53:48.637
Because--
00:53:48.637 --> 00:53:49.220
PROFESSOR: OK.
00:53:52.870 --> 00:53:53.460
You're right.
00:53:53.460 --> 00:53:57.460
So previously, maybe the
best way to write this
00:53:57.460 --> 00:54:05.120
would have been that there
is a shortest length scale a.
00:54:05.120 --> 00:54:08.760
So I should really
indicate what's
00:54:08.760 --> 00:54:13.070
happening here as shortest
length scale having gone
00:54:13.070 --> 00:54:17.770
to v. And there is
always a relationship
00:54:17.770 --> 00:54:23.330
between the a and lambda,
which is inverse relation,
00:54:23.330 --> 00:54:26.835
but there are factors of 2
pi and things like that which
00:54:26.835 --> 00:54:28.676
I don't really want to bother.
00:54:28.676 --> 00:54:29.610
It doesn't matter.
00:54:33.980 --> 00:54:34.480
Yes.
00:54:34.480 --> 00:54:36.915
AUDIENCE: Shouldn't
it be t tilde is
00:54:36.915 --> 00:54:40.319
equal to 2 plus 4 multiplied by?
00:54:40.319 --> 00:54:41.110
PROFESSOR: Exactly.
00:54:41.110 --> 00:54:41.680
Good.
00:54:41.680 --> 00:54:46.750
Because the coefficients
here are divided by 2.
00:54:46.750 --> 00:54:47.975
So that 2 I forgot.
00:54:47.975 --> 00:54:49.175
And I should restore it.
00:54:52.520 --> 00:54:57.800
And if I had gone a
little bit further,
00:54:57.800 --> 00:55:01.734
I would have then started
comparing this formula
00:55:01.734 --> 00:55:05.320
with this formula.
00:55:05.320 --> 00:55:08.890
And I realized that I
should have had the 4.
00:55:08.890 --> 00:55:14.130
Clearly, the two formula are
telling me the same thing.
00:55:14.130 --> 00:55:16.330
You can see that they
are almost exactly
00:55:16.330 --> 00:55:20.760
the same with the exception
of how much integration.
00:55:20.760 --> 00:55:21.657
Yes?
00:55:21.657 --> 00:55:22.740
AUDIENCE: One other thing.
00:55:22.740 --> 00:55:25.710
Are bounds of those
integrals for your tb,
00:55:25.710 --> 00:55:29.540
shouldn't they be
lambda over d to lambda?
00:55:29.540 --> 00:55:30.165
PROFESSOR: Yes.
00:55:33.630 --> 00:55:36.170
Lambda over d to lambda.
00:55:36.170 --> 00:55:39.520
And it is because of
that that I don't really
00:55:39.520 --> 00:55:43.210
have to worry because we
saw that when we were doing
00:55:43.210 --> 00:55:46.830
straightforward
perturbation theory,
00:55:46.830 --> 00:55:50.000
the reason that perturbation
theory was blowing up
00:55:50.000 --> 00:55:54.000
in my face was integrating
all the way to the origin.
00:55:54.000 --> 00:56:00.040
And the trick of renormalization
group is by averaging of there,
00:56:00.040 --> 00:56:06.190
I don't really yet
reach the singularity
00:56:06.190 --> 00:56:09.690
that I would have
at k equals to 0.
00:56:09.690 --> 00:56:13.860
This integral by itself is not
problematic at k equals to 0,
00:56:13.860 --> 00:56:15.410
but future integrals would.
00:56:19.640 --> 00:56:22.378
Any other mistakes?
00:56:22.378 --> 00:56:24.760
No.
00:56:24.760 --> 00:56:27.450
All right.
00:56:27.450 --> 00:56:29.430
But the other part
of this story is
00:56:29.430 --> 00:56:34.400
that the effect of this coarse
graining at lowest order mu
00:56:34.400 --> 00:56:38.600
was to modify this parameter t.
00:56:38.600 --> 00:56:41.260
But importantly,
to do nothing else.
00:56:41.260 --> 00:56:45.432
That is, we can see that
in this Hamiltonian that we
00:56:45.432 --> 00:56:49.340
had written, the
k that is rescaled
00:56:49.340 --> 00:56:51.610
is the same as the old k.
00:56:51.610 --> 00:56:57.970
And the u is the
same as the old u.
00:56:57.970 --> 00:57:06.330
That is, to the lowest order the
only effect of coarse graining
00:57:06.330 --> 00:57:10.955
was to modify the parameter
that covers coefficient
00:57:10.955 --> 00:57:11.736
of t squared.
00:57:18.880 --> 00:57:21.940
But we have not
completed our task
00:57:21.940 --> 00:57:24.866
of constructing a
renormalization group.
00:57:27.715 --> 00:57:31.080
Renormalization group
had three steps.
00:57:31.080 --> 00:57:34.820
The most difficult step,
which was the coarse graining
00:57:34.820 --> 00:57:40.220
we have completed, but now we
have generated a grainy picture
00:57:40.220 --> 00:57:44.660
in which the shortest
wavelengths are a factor of b
00:57:44.660 --> 00:57:47.160
larger than what we had before.
00:57:47.160 --> 00:57:50.850
In order to make our
pictures look the same,
00:57:50.850 --> 00:57:59.326
we had to do steps
2 and 3 of rg.
00:57:59.326 --> 00:58:03.060
Step 2 was to shrink
all of the lengths
00:58:03.060 --> 00:58:10.800
in real space, which amounts
to q prime being b times q.
00:58:10.800 --> 00:58:15.650
And that will restore the
upper part of the q integration
00:58:15.650 --> 00:58:17.510
to be lambda.
00:58:17.510 --> 00:58:22.120
And there was a
rescaling that we
00:58:22.120 --> 00:58:26.540
had to perform for the magnitude
of the fluctuations which
00:58:26.540 --> 00:58:34.690
amounted to replacing the m
tilde of q with z N prime.
00:58:34.690 --> 00:58:38.290
So this would be, if
you like, q lesser
00:58:38.290 --> 00:58:41.200
and this would be m prime.
00:58:51.410 --> 00:58:53.220
Now, the reason these
steps are trivial
00:58:53.220 --> 00:58:56.140
is because just
whenever I see a q,
00:58:56.140 --> 00:58:59.700
I replace it with
b inverse q prime.
00:58:59.700 --> 00:59:05.190
Whenever I see an m tilde,
I replace it with z m prime.
00:59:05.190 --> 00:59:14.140
So then I will find the
Hamiltonian that characterizes
00:59:14.140 --> 00:59:22.030
the m prime variables, the
rg implemented variables,
00:59:22.030 --> 00:59:26.764
which is-- OK, there is a
bunch of constants out front.
00:59:26.764 --> 00:59:29.942
There is the fv.
00:59:29.942 --> 00:59:34.860
Actually, it's 0 delta fv1.
00:59:34.860 --> 00:59:40.030
In the sign, I had
to put as plus.
00:59:40.030 --> 00:59:42.140
But really, it doesn't matter.
00:59:46.740 --> 00:59:51.660
Then, I go and write
down what I have.
00:59:51.660 --> 00:59:58.310
The integration over q
prime after the rescaling is
00:59:58.310 --> 01:00:04.225
performed goes back to the same
cutoff or the same shortest
01:00:04.225 --> 01:00:05.500
wavelength as before.
01:00:08.060 --> 01:00:12.160
Except that when I do
this replacement of q
01:00:12.160 --> 01:00:18.040
lesser with q prime, I will get
a factor of b to the minus d
01:00:18.040 --> 01:00:19.716
down here.
01:00:19.716 --> 01:00:21.860
And there are b integrations.
01:00:21.860 --> 01:00:24.650
And then I have t tilde.
01:00:24.650 --> 01:00:32.720
The next term is k tilde, but
k tilde is the same thing as k.
01:00:32.720 --> 01:00:34.230
Goes with q squared.
01:00:34.230 --> 01:00:37.440
So this becomes q prime
squared, and then I
01:00:37.440 --> 01:00:40.040
will get a b to the minus 2.
01:00:40.040 --> 01:00:43.766
Higher orders will get more
factors of b to the minus 2
01:00:43.766 --> 01:00:46.040
over 2.
01:00:46.040 --> 01:00:51.590
And then I have m tilde,
which is replaced by z.
01:00:51.590 --> 01:00:53.578
And there are two of them.
01:00:53.578 --> 01:00:57.805
I will get m prime
of q prime squared.
01:01:03.395 --> 01:01:05.770
If I were to
explicitly now write
01:01:05.770 --> 01:01:12.340
the factors that go
in construction of u,
01:01:12.340 --> 01:01:17.410
since u had three
integrations over q--
01:01:17.410 --> 01:01:21.980
left board over there-- I
will get three integrations
01:01:21.980 --> 01:01:33.160
over q prime giving me a
factor of b to the minus 3d.
01:01:36.780 --> 01:01:44.087
And then I have four factors
of m tilde that become m prime.
01:01:56.820 --> 01:02:01.450
Along the way, I will
pick up four factors of z.
01:02:01.450 --> 01:02:04.658
And then, of course,
order of u squared.
01:02:07.860 --> 01:02:14.050
So under this three
steps of rg, what
01:02:14.050 --> 01:02:21.430
happened was that I generated
t prime, which was z
01:02:21.430 --> 01:02:28.650
squared b to the
minus d t tilde.
01:02:28.650 --> 01:02:34.390
I generated u prime,
which was z to the fourth
01:02:34.390 --> 01:02:40.702
b to the minus 3d
u, the original u.
01:02:40.702 --> 01:02:44.510
And I generated
the k prime, which
01:02:44.510 --> 01:02:50.570
was z squared b to
the minus d minus 2 k.
01:02:50.570 --> 01:02:53.667
And I could do the same for
various other parameters.
01:02:58.440 --> 01:03:01.655
Now again, we come
up with this issue
01:03:01.655 --> 01:03:06.760
of what to choose for zeta.
01:03:06.760 --> 01:03:08.630
Sorry, for z.
01:03:08.630 --> 01:03:11.770
And what I had
said previously was
01:03:11.770 --> 01:03:15.070
that we went through all
of these exercise of doing
01:03:15.070 --> 01:03:18.530
the Gaussian model
via rg in order
01:03:18.530 --> 01:03:21.150
to have an anchoring point.
01:03:21.150 --> 01:03:24.640
And there we saw that the
thing that we were interested
01:03:24.640 --> 01:03:28.840
was to look at the point where
k prime was the same as k.
01:03:28.840 --> 01:03:32.770
So let's stick with
that choice and see
01:03:32.770 --> 01:03:34.720
what the consequences are.
01:03:34.720 --> 01:03:38.885
So choose z such that
k prime is the same
01:03:38.885 --> 01:03:43.720
a k, which means the I
choose my z to be b to the 1
01:03:43.720 --> 01:03:47.100
plus d over 2 exactly
as I had done previously
01:03:47.100 --> 01:03:49.105
for the Gaussian model.
01:03:49.105 --> 01:03:52.900
So now I can substitute
these values.
01:03:52.900 --> 01:03:55.660
And therefore,
see that following
01:03:55.660 --> 01:04:02.080
a rescaling by a factor of
b, the value of my t prime.
01:04:02.080 --> 01:04:03.970
z squared b to the minus d.
01:04:03.970 --> 01:04:08.830
I will get b to the power
of 2 plus d minus d.
01:04:08.830 --> 01:04:11.960
So that would give me b squared.
01:04:11.960 --> 01:04:16.850
And I have t plus 4u n plus 2.
01:04:16.850 --> 01:04:21.490
This integral from
lambda over b to lambda
01:04:21.490 --> 01:04:28.560
to dk 2 pi to the
d 1 over d plus k,
01:04:28.560 --> 01:04:30.909
k squared, and so forth.
01:04:30.909 --> 01:04:35.750
Presumably, order of u squared.
01:04:35.750 --> 01:04:44.255
And that my factor
of u rescaled by b.
01:04:44.255 --> 01:04:49.670
I have to put four factors of z.
01:04:49.670 --> 01:04:52.900
So I will get b
to the 4 plus 2d,
01:04:52.900 --> 01:04:57.740
and then 3d gets subtracted,
so I will get b to the 4 minus
01:04:57.740 --> 01:05:00.330
d and u again.
01:05:00.330 --> 01:05:02.730
But presumably,
order of u squared.
01:05:15.470 --> 01:05:20.130
So the first factors
are precisely the things
01:05:20.130 --> 01:05:25.280
that we had done and obtained
for the Gaussian model.
01:05:25.280 --> 01:05:29.440
So the only thing that we
gained by this exercise so far
01:05:29.440 --> 01:05:31.045
is this correction to t.
01:05:33.720 --> 01:05:37.670
We will see that that correction
is not that important.
01:05:37.670 --> 01:05:41.480
And in order to really
gain an understanding,
01:05:41.480 --> 01:05:44.030
we have to go to the next order.
01:05:44.030 --> 01:05:48.700
But let's use this
opportunity to also make
01:05:48.700 --> 01:05:51.120
some changes of
terminology that is useful.
01:05:53.980 --> 01:05:58.450
So clearly, the way that
we had constructed this--
01:05:58.450 --> 01:06:00.220
and in particular,
if you are thinking
01:06:00.220 --> 01:06:04.600
about averaging over spins,
et cetera, in real space--
01:06:04.600 --> 01:06:06.500
the natural thing
to think about is
01:06:06.500 --> 01:06:10.350
that maybe your b is a factor
of 2, or a factor of 3.
01:06:10.350 --> 01:06:14.010
You sort of group things by
a factor of twice as much
01:06:14.010 --> 01:06:16.440
and then do the averaging.
01:06:16.440 --> 01:06:19.230
But when you look at
things from the perspective
01:06:19.230 --> 01:06:25.620
of this momentum shell,
this b can be anything.
01:06:25.620 --> 01:06:30.730
And it turns out to be useful
just as a language tool, not as
01:06:30.730 --> 01:06:35.670
a conceptual tool, to make
this b to be very close to 1
01:06:35.670 --> 01:06:39.900
so that effectively you are
just removing a very, very
01:06:39.900 --> 01:06:45.380
tiny, thin shell
around the boundary.
01:06:45.380 --> 01:06:54.620
So essentially what I am saying
is choose b that is almost 1,
01:06:54.620 --> 01:06:58.550
but maybe a little
bit shifted from 1.
01:07:01.340 --> 01:07:06.060
Then clearly, as b
goes to 1, then t prime
01:07:06.060 --> 01:07:11.020
has to go to t--
prime has to go to u.
01:07:11.020 --> 01:07:15.580
So what I can do is I
can define t prime scaled
01:07:15.580 --> 01:07:22.850
by a factor of 1 plus delta l
to be t plus a small amount dt
01:07:22.850 --> 01:07:25.830
by dl.
01:07:25.830 --> 01:07:27.835
And in order of delta l squared.
01:07:31.930 --> 01:07:39.570
And similarly, I can write u
prime to be u plus delta l du
01:07:39.570 --> 01:07:44.680
by dl and higher order.
01:07:44.680 --> 01:07:53.230
And the reason is that if we now
look at the parameter space--
01:07:53.230 --> 01:07:58.060
t, u, whatever-- the
effect of this procedure
01:07:58.060 --> 01:08:01.610
rather than being some jump
from some point to another point
01:08:01.610 --> 01:08:04.100
because we rescaled
by a factor of b
01:08:04.100 --> 01:08:07.850
is to go to a nearby point.
01:08:07.850 --> 01:08:11.980
So these things,
dt by dl, du by dl,
01:08:11.980 --> 01:08:15.690
essentially point
to the direction
01:08:15.690 --> 01:08:19.279
in which the parameters
would change.
01:08:19.279 --> 01:08:23.260
And so they allow you
to replace these jumps
01:08:23.260 --> 01:08:28.464
that you have by things that are
called flows in this parameter
01:08:28.464 --> 01:08:29.580
space.
01:08:29.580 --> 01:08:34.819
So basically, you have
constructed flows and vectors
01:08:34.819 --> 01:08:39.390
that describe the flows
in this parameter space.
01:08:39.390 --> 01:08:48.399
Now, if I do that
over here, we can also
01:08:48.399 --> 01:08:51.310
see some other things emerging.
01:08:51.310 --> 01:08:57.279
So b squared I can write
as 1 plus 2 delta l.
01:09:00.779 --> 01:09:02.920
And then here, I have t.
01:09:06.330 --> 01:09:12.000
Now clearly here, if d was
1, the integral would be 0.
01:09:12.000 --> 01:09:14.955
I am integrating
over a tiny shell.
01:09:14.955 --> 01:09:18.430
So the answer here
when b goes to 1
01:09:18.430 --> 01:09:24.760
is really just the area
of that sphere multiplied
01:09:24.760 --> 01:09:26.720
by the thickness.
01:09:26.720 --> 01:09:28.790
And so what do I get?
01:09:28.790 --> 01:09:35.109
I get 4u n plus 2 is just
the overall coefficient.
01:09:35.109 --> 01:09:39.649
Then, what is the surface area?
01:09:39.649 --> 01:09:43.609
I have the solid angle
divided by 2 pi to the d.
01:09:43.609 --> 01:09:46.475
OK, you divide solid
angle divided by 2 pi
01:09:46.475 --> 01:09:49.960
to the d to be kd.
01:09:49.960 --> 01:09:53.350
What's the value of
the integrand on shell?
01:09:53.350 --> 01:09:56.530
On shell, I have to
replace k with the lambda.
01:09:56.530 --> 01:10:01.680
So I will get t plus
k lambda squared.
01:10:01.680 --> 01:10:09.000
Actually, the surface area is
Sd lambda to the d minus 1.
01:10:09.000 --> 01:10:14.000
But then the thickness
is lambda delta l.
01:10:21.164 --> 01:10:23.320
The second one is
actually very easy.
01:10:23.320 --> 01:10:28.755
It is 1 plus 4 minus
b delta l times u.
01:10:32.870 --> 01:10:42.460
So you can see that if I match
things to order of delta l,
01:10:42.460 --> 01:10:45.010
I get the following rg flows.
01:10:47.550 --> 01:10:53.650
So the left-hand side is
t plus delta l dt by dl.
01:10:53.650 --> 01:10:56.920
The right-hand side-- if I
expand, there is a factor of t.
01:10:56.920 --> 01:10:59.020
So that it gets rid of.
01:10:59.020 --> 01:11:03.930
And I'm left with a term that is
proportional to delta l, whose
01:11:03.930 --> 01:11:07.370
coefficient on the
left-hand side is dt by dl.
01:11:07.370 --> 01:11:11.393
And the right-hand side, I
get either a factor of 2t
01:11:11.393 --> 01:11:16.460
from multiplying the 2 delta
l with t or from multiplying
01:11:16.460 --> 01:11:20.140
the 1 with the result
of this integration.
01:11:20.140 --> 01:11:27.035
I will get a 4u n
plus 2 kd lambda
01:11:27.035 --> 01:11:32.170
to the d divided by t
plus k lambda squared.
01:11:32.170 --> 01:11:35.820
And I don't need to
evaluate any integrals.
01:11:38.410 --> 01:11:44.390
And the second flow equation
for the parameter u du by dl
01:11:44.390 --> 01:11:46.650
is 4 minus d u.
01:11:56.540 --> 01:12:00.680
Now, clearly in the
language of flows,
01:12:00.680 --> 01:12:06.282
a fixed point is when
there is no flow.
01:12:06.282 --> 01:12:12.560
So I could have dt by dl
du by dl need to be 0.
01:12:15.210 --> 01:12:18.530
And du by dl being 0
immediately tells ms
01:12:18.530 --> 01:12:23.000
that u star has to be 0.
01:12:23.000 --> 01:12:25.630
And if I set u
star equals to 0, I
01:12:25.630 --> 01:12:29.960
will see that t
star has to be 0.
01:12:29.960 --> 01:12:33.370
So these equations
have one and only
01:12:33.370 --> 01:12:36.480
one fixed point at this order.
01:12:36.480 --> 01:12:42.900
And then looking for relevance
and irrelevance of going away
01:12:42.900 --> 01:12:46.855
from the fixed point can
be captured by linearizing.
01:12:46.855 --> 01:12:51.660
That is, I write my t to
be t star plus delta l.
01:12:51.660 --> 01:12:54.765
Of course, my t star
and u star are both 0.
01:12:54.765 --> 01:12:59.560
But in general, if
they were not at 0,
01:12:59.560 --> 01:13:07.650
I would linearize my in general,
non-linear rg recursions
01:13:07.650 --> 01:13:11.620
by going slightly away
from the fixed point.
01:13:11.620 --> 01:13:14.940
And then the linearized
form of the equation
01:13:14.940 --> 01:13:21.030
would say that going away be a
small amount delta t delta u--
01:13:21.030 --> 01:13:25.350
and there could be more and
more of these operators-- an
01:13:25.350 --> 01:13:32.800
be written in terms of a matrix
multiply delta t and delta u.
01:13:32.800 --> 01:13:36.460
And clearly, the matrix
for u is very simple.
01:13:36.460 --> 01:13:42.600
It is simply proportional
to 4 minus d delta u.
01:13:42.600 --> 01:13:45.720
The matrix for t?
01:13:45.720 --> 01:13:47.170
Well, there's two terms.
01:13:47.170 --> 01:13:50.060
First of all, there is this 2.
01:13:50.060 --> 01:13:53.070
And then if I am
taking a derivative,
01:13:53.070 --> 01:13:55.340
there will be a derivative
of this expression
01:13:55.340 --> 01:13:57.700
because there is a
t-dependence here.
01:13:57.700 --> 01:13:59.930
But ultimately, since
I'm evaluating it
01:13:59.930 --> 01:14:06.230
at u star equals to 0, I don't
need to include that term here.
01:14:06.230 --> 01:14:08.800
But if I now make
a variation in u,
01:14:08.800 --> 01:14:15.870
I will get an
off-diagonal term here,
01:14:15.870 --> 01:14:19.086
which is k lambda squared.
01:14:19.086 --> 01:14:22.394
So these two can
combine with each other.
01:14:27.180 --> 01:14:32.730
Now, looking for
relevance or irrelevance
01:14:32.730 --> 01:14:36.110
is then equivalent-- previously,
we were talking about it
01:14:36.110 --> 01:14:42.520
in terms of the full equations
with b that could have been
01:14:42.520 --> 01:14:45.190
anything, but now we
have gone to this limit
01:14:45.190 --> 01:14:47.210
of infinitesimal b.
01:14:47.210 --> 01:14:52.190
Then, what I have to do
is to find the eigenvalues
01:14:52.190 --> 01:14:54.973
of the matrix that I
have for these flows.
01:14:54.973 --> 01:14:58.460
Now, a matrix that has this
structure where there's
01:14:58.460 --> 01:15:01.790
0 on one side of the
diagonal, I immediately
01:15:01.790 --> 01:15:07.260
know that the eigenvalues
are 2 and 4 minus d.
01:15:07.260 --> 01:15:12.130
So basically, depending
on whether I'm
01:15:12.130 --> 01:15:15.060
in dimensions greater
than 4 or less than 4,
01:15:15.060 --> 01:15:20.410
I will have either two or
one relevant direction.
01:15:20.410 --> 01:15:23.780
In particular, if I look
at what is happening
01:15:23.780 --> 01:15:28.730
for d that is greater than
4-- for d greater than 4,
01:15:28.730 --> 01:15:31.970
I will just have one
relevant direction.
01:15:31.970 --> 01:15:36.480
And if I look at the
behavior and the flows
01:15:36.480 --> 01:15:44.000
that I am allowed to have in
the two parameters t and u.
01:15:44.000 --> 01:15:47.530
Now, if my Hamiltonian
only has t and u,
01:15:47.530 --> 01:15:49.620
I'm only allowed
to look at the case
01:15:49.620 --> 01:15:52.530
where u is positive
in order not to have
01:15:52.530 --> 01:15:55.140
weights that are unbounded.
01:15:55.140 --> 01:16:00.142
My fixed point occurs at 0, 0.
01:16:00.142 --> 01:16:10.480
And then one simple thing is
that if u is 0, it will stay 0.
01:16:10.480 --> 01:16:13.950
And then dt by dl is 2t.
01:16:13.950 --> 01:16:18.320
So I know that I always
have an eigen-direction that
01:16:18.320 --> 01:16:23.810
is along the t and is flowing
away with a velocity of 2
01:16:23.810 --> 01:16:24.405
if you like.
01:16:27.660 --> 01:16:32.250
The other eigen-direction is
not the axis t equals to 0.
01:16:32.250 --> 01:16:37.060
Because if t is 0
originally but 0 is nonzero,
01:16:37.060 --> 01:16:41.940
this term will generate some
positive amount of t for me.
01:16:41.940 --> 01:16:45.085
So if I start
somewhere on this axis,
01:16:45.085 --> 01:16:49.210
t will go in the direction
of becoming positive.
01:16:49.210 --> 01:16:52.060
Above four dimensions,
you will become less.
01:16:52.060 --> 01:16:54.280
So above four
dimension, you can see
01:16:54.280 --> 01:16:57.930
that the general
trend of flows is
01:16:57.930 --> 01:17:01.092
going to be something like this.
01:17:01.092 --> 01:17:04.700
Indeed, there has to be
a second eigen-direction
01:17:04.700 --> 01:17:07.430
because I'm dealing
with a 2 by 2 matrix.
01:17:07.430 --> 01:17:09.110
And if you look at
it carefully, you'll
01:17:09.110 --> 01:17:14.360
find that the second
eigen-direction is down here
01:17:14.360 --> 01:17:17.730
and corresponds to a
negative eigenvalue.
01:17:17.730 --> 01:17:22.290
So I basically would be having
flows that go towards that.
01:17:22.290 --> 01:17:25.180
And in general, the
character of the flows,
01:17:25.180 --> 01:17:30.380
if I have parameters
somewhere here,
01:17:30.380 --> 01:17:31.919
they would be flowing there.
01:17:31.919 --> 01:17:33.915
If I have parameters
somewhere here,
01:17:33.915 --> 01:17:35.412
they would be flowing there.
01:17:38.410 --> 01:17:43.220
So physically, I have something
like iron in five dimension,
01:17:43.220 --> 01:17:43.990
for example.
01:17:43.990 --> 01:17:47.880
And it corresponds to
being somewhere here.
01:17:47.880 --> 01:17:51.213
When I change the temperature
of iron in five dimension
01:17:51.213 --> 01:17:55.900
and I execute some
trajectory such as this,
01:17:55.900 --> 01:17:59.680
all of the points that are on
this side of the trajectory
01:17:59.680 --> 01:18:01.980
on their flow will
go to a place where
01:18:01.980 --> 01:18:04.780
u becomes small and
t becomes positive.
01:18:04.780 --> 01:18:09.130
So I will essentially go to
this Gaussian-like fixed point
01:18:09.130 --> 01:18:12.520
that describes
independent spins.
01:18:12.520 --> 01:18:14.780
All the things
that are down here,
01:18:14.780 --> 01:18:16.950
which previously in
the Gaussian model
01:18:16.950 --> 01:18:19.130
we could not describe,
because the Gaussian model
01:18:19.130 --> 01:18:21.870
did not allow me to
go to t negative.
01:18:21.870 --> 01:18:23.670
Because now I have
a positive view,
01:18:23.670 --> 01:18:25.600
I have no problem with that.
01:18:25.600 --> 01:18:29.410
And I find that essentially
I go to large negative t
01:18:29.410 --> 01:18:31.270
corresponding to
some positive u.
01:18:31.270 --> 01:18:34.621
I can figure out what the
amount of magnetization is.
01:18:34.621 --> 01:18:36.900
And so then in
between them, there
01:18:36.900 --> 01:18:41.790
is a trajectory that separates
paramagnetic and ferromagnetic
01:18:41.790 --> 01:18:42.370
behavior.
01:18:42.370 --> 01:18:45.985
And clearly, that
trajectory at long scales
01:18:45.985 --> 01:18:48.420
corresponds to the
fixed point that
01:18:48.420 --> 01:18:52.690
is simply the gradient squared.
01:18:52.690 --> 01:18:55.640
Because all the other
terms went to 0, so
01:18:55.640 --> 01:18:57.950
we know all of the
correlation functions,
01:18:57.950 --> 01:19:02.400
et cetera, that we should
see in this system.
01:19:02.400 --> 01:19:12.690
Unfortunately, if I look for
d less than 4, what happens
01:19:12.690 --> 01:19:16.520
is that I still have
the same fixed point.
01:19:16.520 --> 01:19:19.410
And actually , the
eigen-directions don't change
01:19:19.410 --> 01:19:20.760
all that much.
01:19:20.760 --> 01:19:23.500
u equals to 0 is still
an eigen-direction
01:19:23.500 --> 01:19:26.390
that has relevance, too.
01:19:26.390 --> 01:19:30.334
But the other direction
changes from being
01:19:30.334 --> 01:19:32.190
irrelevant to being relevant.
01:19:32.190 --> 01:19:35.470
So I have something like this.
01:19:35.470 --> 01:19:38.600
And the natural
types of flows that I
01:19:38.600 --> 01:19:41.250
get kind of look like this.
01:19:45.820 --> 01:19:49.890
And now if I take my
iron in three dimensions
01:19:49.890 --> 01:19:53.950
and change its temperature,
I go from behavior
01:19:53.950 --> 01:19:57.120
that is kind of
paramagnetic-like
01:19:57.120 --> 01:20:00.240
to ferromagnetic-like.
01:20:00.240 --> 01:20:04.120
And there is a transition point.
01:20:04.120 --> 01:20:06.180
But that transition
point I don't
01:20:06.180 --> 01:20:07.983
know what fixed
point it goes to.
01:20:07.983 --> 01:20:09.000
I have no idea.
01:20:11.780 --> 01:20:14.550
So the only difference
by doing this analysis
01:20:14.550 --> 01:20:18.190
from what we had done just
on the basis of scaling
01:20:18.190 --> 01:20:20.420
and Gaussian theory,
et cetera, is
01:20:20.420 --> 01:20:23.080
that we have located
the shift in t
01:20:23.080 --> 01:20:27.640
with u, which is what
we had done before.
01:20:27.640 --> 01:20:30.780
So was it worth it?
01:20:30.780 --> 01:20:34.000
The answer is up to here, no.
01:20:34.000 --> 01:20:37.080
But let's see if you had
gone one step further what
01:20:37.080 --> 01:20:39.470
would have happened.
01:20:39.470 --> 01:20:42.670
The series is an
alternating series.
01:20:42.670 --> 01:20:46.860
So the next order term
that I get here I expect
01:20:46.860 --> 01:20:49.300
will come with some
negative u squared.
01:20:49.300 --> 01:20:52.880
So let's say there will
be some amount of work
01:20:52.880 --> 01:20:53.750
that I have to do.
01:20:53.750 --> 01:20:57.680
And I calculate something
that is minus a u squared.
01:20:57.680 --> 01:21:00.005
I do some amount of
work and I calculate
01:21:00.005 --> 01:21:04.410
something that is
minus b u squared.
01:21:04.410 --> 01:21:09.340
But then you can see that if
I search for the fixed point,
01:21:09.340 --> 01:21:13.570
I will find another
one at u star, which
01:21:13.570 --> 01:21:17.340
is 4 minus d over b.
01:21:17.340 --> 01:21:21.890
So I will find another point.
01:21:21.890 --> 01:21:26.310
And indeed, we'll find
that things that left here
01:21:26.310 --> 01:21:29.860
will go to that point.
01:21:29.860 --> 01:21:34.650
And so that point has
one direction here.
01:21:34.650 --> 01:21:37.310
It was relevant that
becomes irrelevant.
01:21:37.310 --> 01:21:39.530
And the other
direction is the analog
01:21:39.530 --> 01:21:43.780
of this direction, which
still remains relevant,
01:21:43.780 --> 01:21:45.740
but with modified exponents.
01:21:45.740 --> 01:21:49.870
We will figure out what that is.
01:21:49.870 --> 01:21:52.300
So the next step
is to find this b,
01:21:52.300 --> 01:21:55.490
and then everything
would be resolved.
01:21:55.490 --> 01:22:01.920
The only thing to then realize
is, what are we perturbing it?
01:22:01.920 --> 01:22:04.210
Because the whole
idea of perturbation
01:22:04.210 --> 01:22:09.190
theory is that you should
have a small parameter.
01:22:09.190 --> 01:22:13.905
And if we are
perturbing a u and then
01:22:13.905 --> 01:22:17.720
basing our results on
what is happening here,
01:22:17.720 --> 01:22:20.330
the location of this
fixed point better
01:22:20.330 --> 01:22:23.080
be small-- close
to the original one
01:22:23.080 --> 01:22:26.010
around which I am perturbing.
01:22:26.010 --> 01:22:28.590
So what do I have to make small?
01:22:28.590 --> 01:22:30.860
4 minus d.
01:22:30.860 --> 01:22:33.700
So we thought we were
making a perturbation in u,
01:22:33.700 --> 01:22:37.840
but in order to have a small
quantity the only thing that we
01:22:37.840 --> 01:22:40.760
can do is to stay very
close to four dimension
01:22:40.760 --> 01:22:43.630
and make a perturbation
expansion around four
01:22:43.630 --> 01:22:45.180
dimension.