WEBVTT
00:00:00.070 --> 00:00:01.780
The following
content is provided
00:00:01.780 --> 00:00:04.019
under a Creative
Commons license.
00:00:04.019 --> 00:00:06.870
Your support will help MIT
OpenCourseWare continue
00:00:06.870 --> 00:00:10.730
to offer high-quality
educational resources for free.
00:00:10.730 --> 00:00:13.330
To make a donation, or
view additional materials
00:00:13.330 --> 00:00:17.217
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:17.217 --> 00:00:17.842
at ocw.mit.edu.
00:00:21.177 --> 00:00:21.760
PROFESSOR: OK.
00:00:24.380 --> 00:00:28.530
Let's start with our
standard starting
00:00:28.530 --> 00:00:30.940
point of the last few lectures.
00:00:30.940 --> 00:00:34.170
That is, we are
looking at some system.
00:00:34.170 --> 00:00:36.850
We tried to describe
it by some kind
00:00:36.850 --> 00:00:43.520
of a field, statistical
field after some averaging.
00:00:43.520 --> 00:00:47.070
That is, a function of position.
00:00:47.070 --> 00:00:51.440
And we are interested
in calculating something
00:00:51.440 --> 00:00:54.710
like a partition function
by integrating over
00:00:54.710 --> 00:01:00.420
all configurations of
the statistical field.
00:01:00.420 --> 00:01:06.670
And these configurations
have some kind of a weight.
00:01:06.670 --> 00:01:09.930
This weight we choose
to write as exponential
00:01:09.930 --> 00:01:18.020
of some kind of a-- something
like an effective Hamiltonian
00:01:18.020 --> 00:01:19.900
that depends on
the configuration
00:01:19.900 --> 00:01:20.870
that you're looking at.
00:01:25.540 --> 00:01:33.080
And of course, the main
thing faced with a problem
00:01:33.080 --> 00:01:38.060
that you haven't seen before is
to decide on what the field is
00:01:38.060 --> 00:01:40.740
that you are looking
at to average.
00:01:40.740 --> 00:01:44.950
And what kind of
symmetries and constraints
00:01:44.950 --> 00:01:49.320
you want to construct in
this form of the weight.
00:01:49.320 --> 00:01:51.690
In particular, we
are sort of focusing
00:01:51.690 --> 00:01:53.800
on this Landau-Ginzburg
model that
00:01:53.800 --> 00:01:56.320
describes phase transitions.
00:01:56.320 --> 00:01:58.830
And let's say in the
absence of magnetic field,
00:01:58.830 --> 00:02:04.166
we are interested in a system
that is rotationally symmetric.
00:02:04.166 --> 00:02:07.610
But the procedure is
reasonably standard.
00:02:07.610 --> 00:02:10.190
Maybe in some
cases you can solve
00:02:10.190 --> 00:02:13.600
a particular part of this.
00:02:13.600 --> 00:02:15.080
Let's call that part beta H0.
00:02:17.690 --> 00:02:21.980
In the context that
we are working with,
00:02:21.980 --> 00:02:25.250
it's the Gaussian part that
we have looked at before.
00:02:25.250 --> 00:02:28.830
So that's the
integral over space.
00:02:28.830 --> 00:02:31.380
We used this idea of locality.
00:02:31.380 --> 00:02:34.870
We had an expansion
in all things
00:02:34.870 --> 00:02:37.610
that are consistent with this.
00:02:37.610 --> 00:02:42.270
But for the purposes of
the exactly solvable part,
00:02:42.270 --> 00:02:44.800
we focus on the Gaussian.
00:02:44.800 --> 00:02:48.460
So there is a term that is
proportional to m squared,
00:02:48.460 --> 00:02:50.833
gradient of m squared
and higher-order terms.
00:03:00.710 --> 00:03:04.010
So clearly for the
time being, I'm
00:03:04.010 --> 00:03:06.330
ignoring the magnetic field.
00:03:06.330 --> 00:03:11.180
So let's say in this
formulation the problem that we
00:03:11.180 --> 00:03:14.460
are interested is how our
partition function depends
00:03:14.460 --> 00:03:17.420
on this coefficient,
which where it goes to 0,
00:03:17.420 --> 00:03:20.740
the Gaussian weight becomes
kind of unsustainable.
00:03:23.710 --> 00:03:30.550
Now, of course, we said that the
full problem has, in addition
00:03:30.550 --> 00:03:38.190
to this beta H0, a part that
involves the interaction.
00:03:38.190 --> 00:03:41.690
So what I have done is I
have written the weight
00:03:41.690 --> 00:03:47.090
as beta H0 and a part
that is an interaction.
00:03:47.090 --> 00:03:49.080
By interaction, I
really mean something
00:03:49.080 --> 00:03:55.620
that is not solvable within
the framework of Gaussian.
00:03:55.620 --> 00:04:01.900
In our case, what was
non-solvable is essentially
00:04:01.900 --> 00:04:05.690
anything-- and there
is infinity of terms--
00:04:05.690 --> 00:04:09.950
that don't have
second-order powers of m.
00:04:09.950 --> 00:04:14.655
So we wrote terms like m to
the fourth, m to the sixth,
00:04:14.655 --> 00:04:15.976
and so forth.
00:04:22.450 --> 00:04:27.040
Now, the key to being
able to solve this problem
00:04:27.040 --> 00:04:31.130
was to make a transformation
to Fourier modes.
00:04:31.130 --> 00:04:34.100
So essentially, what
we did was to write
00:04:34.100 --> 00:04:40.140
our m of x as a sum
over Fourier modes.
00:04:40.140 --> 00:04:43.610
You could write it, let's
say, in the discrete form as e
00:04:43.610 --> 00:04:45.460
to the i q dot x.
00:04:45.460 --> 00:04:48.550
And whether I write e to
the i q dot x or minus i q
00:04:48.550 --> 00:04:51.150
dot x is not as
important as long as I'm
00:04:51.150 --> 00:04:56.260
consistent within
one session at least.
00:04:56.260 --> 00:05:01.188
And the normalization
that I used was 1 over V.
00:05:01.188 --> 00:05:04.500
And the reason I used
this normalization was
00:05:04.500 --> 00:05:08.150
that if I went to the
continuum, I could write it
00:05:08.150 --> 00:05:14.390
nicely as an integral
over q divided
00:05:14.390 --> 00:05:15.820
by the density of states.
00:05:15.820 --> 00:05:18.110
The V would disappear.
00:05:18.110 --> 00:05:22.730
e to the i q x m tilde of q.
00:05:30.510 --> 00:05:35.760
Now in particular if I
do that transformation,
00:05:35.760 --> 00:05:46.730
the Gaussian part simply
becomes 1 over V sum over q.
00:05:46.730 --> 00:05:50.800
Then the Fourier
transform of this kernel t
00:05:50.800 --> 00:06:00.091
plus k q squared and so forth
divided by 2 m of q discrete
00:06:00.091 --> 00:06:00.590
squared.
00:06:03.220 --> 00:06:06.330
Which if I go to the
continuum limit simply
00:06:06.330 --> 00:06:14.340
becomes an integral
over q t plus k q
00:06:14.340 --> 00:06:22.020
squared and so forth
over 2 m of q squared.
00:06:31.430 --> 00:06:36.290
Now, once I have the Gaussian
weight, from the Gaussian
00:06:36.290 --> 00:06:40.780
weight I can calculate
various averages.
00:06:40.780 --> 00:06:46.030
And the averages
are best described
00:06:46.030 --> 00:06:49.630
by noting that essentially
after this transformation
00:06:49.630 --> 00:06:56.300
I can also write my weight as a
product over the contributions
00:06:56.300 --> 00:07:04.000
of the different modes of
something that is of this form,
00:07:04.000 --> 00:07:08.110
e to the minus beta H0.
00:07:08.110 --> 00:07:10.450
Now written in terms
of these q modes,
00:07:10.450 --> 00:07:14.630
clearly it's a product of
independent contributions.
00:07:14.630 --> 00:07:19.050
And then of course, there will
be the u to be added later on.
00:07:19.050 --> 00:07:22.690
But when I have a product
of independent contributions
00:07:22.690 --> 00:07:26.720
for each q, I can immediately
see that if I look at,
00:07:26.720 --> 00:07:32.090
say, m evaluated for
some q, m evaluated
00:07:32.090 --> 00:07:35.820
for some different q
with the Gaussian weight.
00:07:35.820 --> 00:07:38.980
And when I calculate things
with the Gaussian weight,
00:07:38.980 --> 00:07:41.610
I put this index 0.
00:07:41.610 --> 00:07:45.320
So that's my 0 to order or
exactly solvable theory.
00:07:45.320 --> 00:07:48.580
And of course, we are
dealing here with a vector.
00:07:48.580 --> 00:07:51.350
So these things have
indices alpha and beta
00:07:51.350 --> 00:07:53.230
associated with them.
00:07:53.230 --> 00:07:58.720
And if I look at the
discrete version,
00:07:58.720 --> 00:08:04.570
I have a product over
Gaussians for each one of them.
00:08:04.570 --> 00:08:08.670
Clearly, I will
get 0 unless I am
00:08:08.670 --> 00:08:11.540
looking at the same components.
00:08:11.540 --> 00:08:15.310
And I'm looking at the same q.
00:08:15.310 --> 00:08:17.080
And in particular,
the constraint really
00:08:17.080 --> 00:08:20.385
is that q plus q prime
should add up to 0.
00:08:23.490 --> 00:08:26.340
And if those constraints
are satisfied,
00:08:26.340 --> 00:08:28.200
then I am looking at
the particular term
00:08:28.200 --> 00:08:30.260
in this Gaussian.
00:08:30.260 --> 00:08:32.299
And the expectation
value of m squared
00:08:32.299 --> 00:08:34.710
is simply the variance
that we can see
00:08:34.710 --> 00:08:40.770
is V divided by t plus k q
squared, q to the fourth,
00:08:40.770 --> 00:08:42.195
and so forth.
00:08:46.000 --> 00:08:49.780
And the thing is that most
of the time we will actually
00:08:49.780 --> 00:08:56.000
be looking at things directly
in the limit of the continuum
00:08:56.000 --> 00:09:01.230
where we replace sums q's
with integrals over q.
00:09:01.230 --> 00:09:05.810
And then we have to replace
this discrete delta function
00:09:05.810 --> 00:09:08.390
with a continuum delta function.
00:09:08.390 --> 00:09:10.750
And the procedure to
do that is that this
00:09:10.750 --> 00:09:13.060
becomes delta alpha beta.
00:09:13.060 --> 00:09:16.776
This combination
gets replaced by 2 pi
00:09:16.776 --> 00:09:21.160
to the d delta function
q plus q prime,
00:09:21.160 --> 00:09:24.952
where this is now a
direct delta function
00:09:24.952 --> 00:09:32.240
t plus k q squared plus l q
to the fourth and so forth.
00:09:32.240 --> 00:09:38.510
And the justification
for doing that is simply
00:09:38.510 --> 00:09:42.820
that the Kronecker
delta is defined such
00:09:42.820 --> 00:09:51.100
that if I sum over, let's
say, all q, the delta that
00:09:51.100 --> 00:09:58.720
is Kronecker, the
answer would be 0.
00:09:58.720 --> 00:10:01.770
Now, if I go to the
continuum limit,
00:10:01.770 --> 00:10:06.530
the sum over q I have to
replace with integral over q
00:10:06.530 --> 00:10:12.440
with a density of states, which
is V divided by 2 pi to the d.
00:10:15.660 --> 00:10:20.500
So if I want to replace
this with a continuum delta
00:10:20.500 --> 00:10:30.640
function of q, I have to get rid
of this 2 pi to the d over V.
00:10:30.640 --> 00:10:32.200
And that's what I have done.
00:10:32.200 --> 00:10:40.280
So basically, you
replace-- hopefully I
00:10:40.280 --> 00:10:42.305
didn't make a mistake.
00:10:44.901 --> 00:10:45.400
Yes.
00:10:45.400 --> 00:10:49.020
So the discrete delta I
replace with 1 over V. The V
00:10:49.020 --> 00:10:52.975
disappears and the 2
pi to the d appears.
00:10:52.975 --> 00:10:53.475
OK?
00:10:58.410 --> 00:11:06.010
Now, the thing that
makes some difficulty
00:11:06.010 --> 00:11:12.040
is that whereas the rest of
these things that we have not
00:11:12.040 --> 00:11:14.710
included as part
of the Gaussian,
00:11:14.710 --> 00:11:20.900
because of the locality I
could write reasonably simply
00:11:20.900 --> 00:11:23.660
in the space x.
00:11:23.660 --> 00:11:29.610
When I go to the space q,
it becomes complicated.
00:11:29.610 --> 00:11:33.130
Because each m of x
here I have to replace
00:11:33.130 --> 00:11:36.170
with a sum or an integral.
00:11:36.170 --> 00:11:39.460
And I have four of those m's.
00:11:39.460 --> 00:11:45.050
So here, let's say for the
first term that involves u,
00:11:45.050 --> 00:11:49.043
I have in principle to go
over an integral associated
00:11:49.043 --> 00:11:55.445
with conversion of the first
m, conversion of the second m,
00:11:55.445 --> 00:11:57.640
third m.
00:11:57.640 --> 00:12:00.790
Each one of them will carry
a factor of 2 pi to the d.
00:12:00.790 --> 00:12:04.300
So there will be three of them.
00:12:04.300 --> 00:12:07.540
And the reason I didn't
write the fourth one
00:12:07.540 --> 00:12:11.240
is because after I do all
of that transformation,
00:12:11.240 --> 00:12:17.178
I will have an integral
over x of e to the i q1
00:12:17.178 --> 00:12:22.670
dot x plus q2 dot x plus
q3 dot x plus q4 dot x.
00:12:22.670 --> 00:12:26.240
So I have an integral
of e to the i q
00:12:26.240 --> 00:12:30.860
dot x where q is the sum
of the four of them over x.
00:12:30.860 --> 00:12:32.570
And that gives me
a delta function
00:12:32.570 --> 00:12:36.630
that ensures the sum of
the four q's have to be 0.
00:12:36.630 --> 00:12:43.030
So basically, one of the
m's will carry now index q1.
00:12:43.030 --> 00:12:46.610
The other will carry index q2.
00:12:46.610 --> 00:12:49.510
The third will carry index q3.
00:12:49.510 --> 00:12:58.190
The fourth will carry index that
is minus q1 minus q2 minus q3.
00:12:58.190 --> 00:12:58.856
Yes?
00:12:58.856 --> 00:13:02.610
AUDIENCE: Does it matter
which indices for squaring it,
00:13:02.610 --> 00:13:03.846
or whatever?
00:13:03.846 --> 00:13:04.670
Sorry.
00:13:04.670 --> 00:13:06.927
Do you need a subscript
for alpha and beta?
00:13:06.927 --> 00:13:07.510
PROFESSOR: OK.
00:13:07.510 --> 00:13:08.760
That's what I was [INAUDIBLE].
00:13:08.760 --> 00:13:13.990
So this m to the fourth is
really m squared m squared,
00:13:13.990 --> 00:13:17.740
where m squared is a
vector that is squared.
00:13:17.740 --> 00:13:22.290
So I have to put the indices--
let's say alpha alpha--
00:13:22.290 --> 00:13:24.560
that are summed
over all possibility
00:13:24.560 --> 00:13:26.680
to get the dot product here.
00:13:26.680 --> 00:13:28.600
And I have to put
the indices beta
00:13:28.600 --> 00:13:33.380
beta to have the
dot products here.
00:13:33.380 --> 00:13:34.770
OK?
00:13:34.770 --> 00:13:37.660
Now when I go to the
next term, clearly I
00:13:37.660 --> 00:13:44.430
will have a whole bunch more
integrals and things like that.
00:13:44.430 --> 00:13:51.970
So the e u terms do not
look as nice and clean.
00:13:51.970 --> 00:13:55.650
They were local in real space.
00:13:55.650 --> 00:13:59.240
But when I go to this Fourier
space, they become non-local.
00:13:59.240 --> 00:14:02.190
q's that are all
over this [INAUDIBLE]
00:14:02.190 --> 00:14:07.400
are going to be coupled to each
other through this expression.
00:14:07.400 --> 00:14:10.500
And that's also why it
is called interaction.
00:14:10.500 --> 00:14:15.230
Because in some sense,
previously each q was a mode
00:14:15.230 --> 00:14:21.350
by itself and these terms give
interactions between modes that
00:14:21.350 --> 00:14:23.920
have different q's.
00:14:23.920 --> 00:14:24.420
Yes?
00:14:24.420 --> 00:14:26.003
AUDIENCE: Is there
a way to understand
00:14:26.003 --> 00:14:33.628
that physically of why you
get coupling in Fourier space?
00:14:33.628 --> 00:14:36.870
[INAUDIBLE] higher
than [INAUDIBLE].
00:14:39.680 --> 00:14:40.460
PROFESSOR: OK.
00:14:40.460 --> 00:14:46.650
So essentially, we
have a system that
00:14:46.650 --> 00:14:49.230
has translational symmetry.
00:14:49.230 --> 00:14:51.500
So when you have
translational symmetry,
00:14:51.500 --> 00:14:54.810
this Fourier vector q is
a good conserved quantity.
00:14:54.810 --> 00:14:58.080
It's like a momentum.
00:14:58.080 --> 00:15:02.270
So one thing that we have
is, in some sense, a particle
00:15:02.270 --> 00:15:05.140
or an excitation that
is going by itself
00:15:05.140 --> 00:15:08.890
with some particular momentum.
00:15:08.890 --> 00:15:13.920
But what these terms represent
is the possibility that you
00:15:13.920 --> 00:15:16.480
have, let's say,
two of these momenta
00:15:16.480 --> 00:15:19.040
coming and interacting
with each other
00:15:19.040 --> 00:15:22.370
and getting two
that are going out.
00:15:22.370 --> 00:15:24.050
Why is it [INAUDIBLE]?
00:15:24.050 --> 00:15:25.940
It's partly because
of the symmetries
00:15:25.940 --> 00:15:27.700
that we built into the problem.
00:15:27.700 --> 00:15:30.230
If I had written something
that was m cubed,
00:15:30.230 --> 00:15:33.230
I had the possibility
of 2 going to, 1 or 1
00:15:33.230 --> 00:15:34.275
going to 2, et cetera.
00:15:43.040 --> 00:15:45.580
All right.
00:15:45.580 --> 00:15:49.060
I forgot to say one
more thing, which
00:15:49.060 --> 00:15:54.890
is that for the Gaussian theory,
I can calculate essentially
00:15:54.890 --> 00:15:58.255
all expectation values
most [INAUDIBLE]
00:15:58.255 --> 00:16:01.640
in this context of the
Fourier representation.
00:16:01.640 --> 00:16:03.650
So this was an
example of something
00:16:03.650 --> 00:16:06.330
that had two factors of m.
00:16:06.330 --> 00:16:09.450
But very soon, we will see
that we would need terms
00:16:09.450 --> 00:16:13.380
that, let's say, involve
m factors of n that
00:16:13.380 --> 00:16:14.810
are multiplied each other.
00:16:14.810 --> 00:16:24.260
So I have m alpha i of
qi-- something like that.
00:16:24.260 --> 00:16:29.190
And again, 0 for the
Gaussian expectation value.
00:16:29.190 --> 00:16:31.670
And if I have written things
the way that I have-- that is,
00:16:31.670 --> 00:16:33.380
I have no magnetic field.
00:16:33.380 --> 00:16:37.260
So I have m2 minus
m symmetry, clearly
00:16:37.260 --> 00:16:41.360
the answer is going
to be 0 if l is odd.
00:16:44.060 --> 00:16:51.200
If l is even, we have this
nice property of Gaussians
00:16:51.200 --> 00:16:54.190
that we described
in 8.333, which
00:16:54.190 --> 00:17:05.890
is that this will be the sum
over all pairs of averages.
00:17:05.890 --> 00:17:10.550
So something like
m1, m2, m3, m4.
00:17:10.550 --> 00:17:15.060
You can have m1 m2
multiplying by m3 m4 average.
00:17:15.060 --> 00:17:18.500
m1 m3 average multiply
m2 m4 average.
00:17:18.500 --> 00:17:22.990
m1 m4 average multiplied
by m2 m3 average.
00:17:22.990 --> 00:17:28.770
And this is what's called
a Wick's theorem, which
00:17:28.770 --> 00:17:32.730
is an important property of
the Gaussian that we will use.
00:17:46.500 --> 00:17:53.250
So we know how to calculate
averages of things of interest,
00:17:53.250 --> 00:17:57.070
which are essentially product
of these factors of m,
00:17:57.070 --> 00:17:59.310
in the Gaussian theory.
00:17:59.310 --> 00:18:03.960
Now let's calculate
these averages
00:18:03.960 --> 00:18:05.090
in perturbation theory.
00:18:15.260 --> 00:18:23.180
So quite generally, if I want to
calculate the average of some O
00:18:23.180 --> 00:18:29.840
in a theory that involves
averaging over some function
00:18:29.840 --> 00:18:33.520
l-- so this could be some
trace, some completely
00:18:33.520 --> 00:18:38.050
unspecified things-- of a
weight that is like, say,
00:18:38.050 --> 00:18:39.690
e to the minus beta H0.
00:18:39.690 --> 00:18:43.010
A part that I can
do and a part that I
00:18:43.010 --> 00:18:48.530
want to treat as a small
change to what I can do.
00:18:48.530 --> 00:18:51.080
The procedure of
calculating the average
00:18:51.080 --> 00:18:55.210
is to multiply the
probability by the quantity
00:18:55.210 --> 00:18:56.620
that I want to average.
00:18:56.620 --> 00:19:00.446
And of course, the whole thing
has to be properly normalized.
00:19:00.446 --> 00:19:05.882
And this is the normalization,
which is the partition function
00:19:05.882 --> 00:19:06.840
that we had previously.
00:19:10.780 --> 00:19:14.000
Now, the whole idea
of perturbation
00:19:14.000 --> 00:19:17.870
is to assume that this
quantity u is small.
00:19:17.870 --> 00:19:21.456
So we start to expand
e to the minus u.
00:19:21.456 --> 00:19:23.825
I can certainly do
that very easily,
00:19:23.825 --> 00:19:26.560
let's say, in the denominator.
00:19:26.560 --> 00:19:30.840
I have e to the minus
beta H0, and then I
00:19:30.840 --> 00:19:38.560
have 1 minus u plus u squared
over 2 minus u cubed over 6.
00:19:38.560 --> 00:19:43.700
Basically, the usual
expansion of the exponential.
00:19:43.700 --> 00:19:49.270
In the numerator, I
have the same thing,
00:19:49.270 --> 00:19:52.610
except that there is an
I that is multiplying
00:19:52.610 --> 00:19:58.010
this expansion for the
operator, or the object that I
00:19:58.010 --> 00:19:59.542
want to calculate the average.
00:20:03.190 --> 00:20:08.680
Now, the first term that I
have in the denominator here,
00:20:08.680 --> 00:20:13.550
1 multiplied by all integrals
of e to the minus H0
00:20:13.550 --> 00:20:18.680
is clearly what I would
call the partition function,
00:20:18.680 --> 00:20:20.600
or the normalization
that I would
00:20:20.600 --> 00:20:23.920
have for the Gaussian weight.
00:20:23.920 --> 00:20:25.840
So that's the first term.
00:20:25.840 --> 00:20:30.490
If I factor that out,
then the next term
00:20:30.490 --> 00:20:33.950
is u integrated against
the Gaussian weight
00:20:33.950 --> 00:20:36.780
and then properly normalized.
00:20:36.780 --> 00:20:41.610
So the next term will be the
average of u with the Gaussian
00:20:41.610 --> 00:20:44.150
or whatever other
0 to order weight
00:20:44.150 --> 00:20:49.560
is that I can calculate things
and I have indicated that by 0.
00:20:49.560 --> 00:20:53.180
And then I would
have 1/2 average
00:20:53.180 --> 00:20:56.400
of u squared 0 and so forth.
00:20:56.400 --> 00:21:01.720
And the series in the numerator
once I factor out the Z0
00:21:01.720 --> 00:21:05.265
is pretty much the same except
that every term will have an o.
00:21:20.930 --> 00:21:26.420
The Z 0's naturally
I can cancel out.
00:21:26.420 --> 00:21:30.640
So what I have is
from the numerator o
00:21:30.640 --> 00:21:38.565
minus ou plus 1/2 ou squared.
00:21:41.550 --> 00:21:45.160
What I will do with
the denominator
00:21:45.160 --> 00:21:51.090
is to bring it in the
numerator regarding
00:21:51.090 --> 00:21:55.110
all of these as
a small quantity.
00:21:55.110 --> 00:21:59.270
So if I were to essentially
write this expression raised
00:21:59.270 --> 00:22:03.190
to the minus 1 power, I
can make a series expansion
00:22:03.190 --> 00:22:04.820
of all of these terms.
00:22:04.820 --> 00:22:09.820
So the first thing, if I
just had one over 1 minus u
00:22:09.820 --> 00:22:12.240
in the denominator,
it would come
00:22:12.240 --> 00:22:21.570
from 1 plus u plus u squared
and u cubed, et cetera.
00:22:21.570 --> 00:22:24.590
But I've only kept thing
to order of u squared.
00:22:24.590 --> 00:22:26.900
So when I then
correct it because
00:22:26.900 --> 00:22:28.820
of this thing in
the denominator,
00:22:28.820 --> 00:22:35.570
the 1/2 becomes minus
1/2 u squared 0.
00:22:35.570 --> 00:22:40.120
And then, there will be
order of u cubed terms.
00:22:40.120 --> 00:22:44.500
So the answer is the
product of two brackets.
00:22:44.500 --> 00:22:50.022
And I can reorganize that
product, again, in powers of u.
00:22:50.022 --> 00:22:54.460
The lowest-order term
is the 0 to order,
00:22:54.460 --> 00:22:57.340
or unperturbed average.
00:22:57.340 --> 00:23:03.095
And the first correction
comes from ou average.
00:23:06.250 --> 00:23:08.800
And then I have the
average of u average of u.
00:23:12.710 --> 00:23:19.020
You can see that something
like a variance or connected
00:23:19.020 --> 00:23:21.760
correlation or cumulant
appears because I
00:23:21.760 --> 00:23:25.160
have to subtract
out the averages.
00:23:25.160 --> 00:23:28.185
And then the next order
term, what will I have?
00:23:28.185 --> 00:23:31.210
I will write it as 1/2.
00:23:31.210 --> 00:23:35.490
I start with ou squared 0.
00:23:35.490 --> 00:23:38.240
Then I can multiply
this with this,
00:23:38.240 --> 00:23:44.700
so I will get minus 2 o u0 u0.
00:23:44.700 --> 00:23:49.570
And then I can multiply
o0 with those two terms.
00:23:49.570 --> 00:24:03.058
So I will have minus o0
u squared 0 plus 2 o0--
00:24:03.058 --> 00:24:05.530
this is over here.
00:24:05.530 --> 00:24:08.851
u0 squared and
higher-order terms.
00:24:16.860 --> 00:24:22.370
So basically, we can see
that the coefficients,
00:24:22.370 --> 00:24:25.530
as I have written, are going to
be essentially the coefficients
00:24:25.530 --> 00:24:32.530
that I would have if I were
to expand the exponential.
00:24:32.530 --> 00:24:36.540
So things like minus 1 to
the n over n factorial.
00:24:39.380 --> 00:24:42.680
And the leading
term in all cases
00:24:42.680 --> 00:24:48.578
is o u raised to
the n-th power 0,
00:24:48.578 --> 00:24:53.390
out of which are
subtracted various things.
00:24:53.390 --> 00:24:56.640
And the effect of
those subtractions,
00:24:56.640 --> 00:24:59.740
let's say we define
a quantity which
00:24:59.740 --> 00:25:06.060
is likely cumulants that we were
using in 8.333, which describe
00:25:06.060 --> 00:25:09.778
the subtractions that you would
have to define such an average.
00:25:16.620 --> 00:25:19.410
So that's the general structure.
00:25:19.410 --> 00:25:23.370
What this really
means-- and I sometimes
00:25:23.370 --> 00:25:27.600
call it cumulant or connected--
will become apparent
00:25:27.600 --> 00:25:28.300
very shortly.
00:25:38.880 --> 00:25:40.740
This is the general result.
00:25:40.740 --> 00:25:43.980
We have a particular
case above, which
00:25:43.980 --> 00:25:48.550
is this Landau-Ginzburg theory
perturbed around the Gaussian.
00:25:48.550 --> 00:25:53.840
So let's calculate the simplest
one of our averages, this m
00:25:53.840 --> 00:26:03.724
alpha of q m beta of q prime,
not at the Gaussian level,
00:26:03.724 --> 00:26:04.640
but as a perturbation.
00:26:07.630 --> 00:26:13.390
And actually, for
practical reasons,
00:26:13.390 --> 00:26:16.290
I will just calculate the
effect of the first term, which
00:26:16.290 --> 00:26:18.000
is u m to the fourth.
00:26:18.000 --> 00:26:21.280
So I will expand in powers of u.
00:26:21.280 --> 00:26:23.070
But once you see
that, you would know
00:26:23.070 --> 00:26:27.780
how to do it for m to the sixth
and all the higher powers.
00:26:27.780 --> 00:26:30.870
So according to
what we have here,
00:26:30.870 --> 00:26:35.860
the first term is
m alpha of q m beta
00:26:35.860 --> 00:26:42.550
of q prime evaluated
with the Gaussian theory.
00:26:42.550 --> 00:26:48.920
The next term, this one,
involves the average
00:26:48.920 --> 00:26:54.540
of this entity and the u.
00:26:54.540 --> 00:26:58.780
So our u I have
written up there.
00:26:58.780 --> 00:27:04.340
So I have minus
from the first term.
00:27:04.340 --> 00:27:07.720
The terms that are
proportional to u,
00:27:07.720 --> 00:27:13.840
I will group together
coming from here.
00:27:13.840 --> 00:27:21.180
u itself involved this
integration over q1 q2 q3.
00:27:28.500 --> 00:27:36.670
And u involves
this m i of q1 m i
00:27:36.670 --> 00:27:46.210
of q2 mj of q3 mj of minus
q1 minus q2 minus q3.
00:27:50.990 --> 00:27:53.420
And I have to multiply it.
00:27:53.420 --> 00:27:54.240
So this is u.
00:27:54.240 --> 00:27:56.720
I have to multiply it by o.
00:27:56.720 --> 00:28:03.030
So I have m alpha of
q m beta of q prime.
00:28:03.030 --> 00:28:04.240
So this is my o.
00:28:04.240 --> 00:28:05.160
This is my u.
00:28:07.670 --> 00:28:11.140
And I have to take the
average of this term.
00:28:11.140 --> 00:28:14.760
But really, the average
operates on the m's,
00:28:14.760 --> 00:28:16.030
so it will go over here.
00:28:19.850 --> 00:28:23.610
So that's the average of ou.
00:28:23.610 --> 00:28:28.620
I have to subtract from that
the average of o average of u.
00:28:28.620 --> 00:28:32.060
So let me, again,
write the next term.
00:28:32.060 --> 00:28:36.420
u will be the same
bunch of integrations.
00:28:36.420 --> 00:28:47.720
I have to do average of
o and then average of u.
00:29:06.900 --> 00:29:14.080
This completes my first
correction coming from u,
00:29:14.080 --> 00:29:17.460
and then there will be
corrections to first order
00:29:17.460 --> 00:29:20.940
coming from V. There will be
second-order corrections coming
00:29:20.940 --> 00:29:22.809
from u squared,
all kinds of things
00:29:22.809 --> 00:29:23.850
that will come into play.
00:29:26.460 --> 00:29:29.870
But the important thing, again,
to realize is the structure.
00:29:29.870 --> 00:29:34.400
u is this thing that
involves four factors of m.
00:29:34.400 --> 00:29:36.590
The averages are
over the m, so I
00:29:36.590 --> 00:29:39.560
can take them
within the integral.
00:29:39.560 --> 00:29:44.390
And so I have one case which
is an expectation value of six
00:29:44.390 --> 00:29:45.350
m's.
00:29:45.350 --> 00:29:48.020
Another case, a product of
two and a product of four.
00:29:50.680 --> 00:29:53.580
So that's why I
said I would need
00:29:53.580 --> 00:29:58.190
to know how to calculate
in the Gaussian theory
00:29:58.190 --> 00:30:03.600
product of various factors
of m because my interaction
00:30:03.600 --> 00:30:07.360
term involves various powers
of m that will be added
00:30:07.360 --> 00:30:10.010
to whatever expectation value
I'm calculating perturbation
00:30:10.010 --> 00:30:10.510
theory.
00:30:13.840 --> 00:30:17.740
So how do I calculate
an expectation
00:30:17.740 --> 00:30:22.540
that involves six factors--
certainly, it's even-- of m?
00:30:22.540 --> 00:30:28.410
I have to group-- make
all possible pairings.
00:30:28.410 --> 00:30:31.620
So this, for example,
can be paired to this.
00:30:31.620 --> 00:30:32.962
This can be paired to this.
00:30:32.962 --> 00:30:34.140
This can be paired to this.
00:30:34.140 --> 00:30:38.090
That's a perfectly
well-defined average.
00:30:38.090 --> 00:30:39.850
But you can see
that if I do this,
00:30:39.850 --> 00:30:43.370
if I pair this one,
this one, this one,
00:30:43.370 --> 00:30:46.430
I will get something that will
cancel out against this one.
00:30:51.420 --> 00:30:56.060
So basically, you can
see that any way that I
00:30:56.060 --> 00:31:03.150
do averaging that involves only
things that are coming from o
00:31:03.150 --> 00:31:07.310
and separately the
things that come from u
00:31:07.310 --> 00:31:10.840
will cancel out with the
corresponding-- oops.
00:31:14.636 --> 00:31:19.710
With the corresponding
averages that I do over here.
00:31:22.470 --> 00:31:25.630
That c stands for connected.
00:31:25.630 --> 00:31:31.650
So the only things that survive
are pairings or contractions
00:31:31.650 --> 00:31:33.950
that pick something
that is from o
00:31:33.950 --> 00:31:39.250
and connect it to
something that is from u.
00:31:39.250 --> 00:31:44.170
And the purpose of all of
these other terms at all higher
00:31:44.170 --> 00:31:48.480
orders is precisely
to remove pieces
00:31:48.480 --> 00:31:51.700
where you don't have full
connections among all
00:31:51.700 --> 00:31:53.764
of the o's and the u's
that you are dealing with.
00:31:57.640 --> 00:32:02.810
So let's see what this is.
00:32:02.810 --> 00:32:12.975
So I will show you that using
connections that involve both o
00:32:12.975 --> 00:32:27.890
and u, I will have two types of
contractions joining o and u.
00:32:34.700 --> 00:32:36.550
The first type is
something like this.
00:32:36.550 --> 00:32:40.870
I will, again, draw all-- or
write down all of the fours.
00:32:40.870 --> 00:32:53.380
So I have m alpha of q m beta of
q prime m i of q1 m i of q2 mj
00:32:53.380 --> 00:33:01.080
of q3 mj of minus q1
minus q2 minus q3.
00:33:01.080 --> 00:33:03.480
I have to take that average.
00:33:03.480 --> 00:33:05.120
And I do that the
average according
00:33:05.120 --> 00:33:09.680
to Wick's theorem as a
product of contractions.
00:33:09.680 --> 00:33:11.440
So let's pick this m alpha.
00:33:11.440 --> 00:33:15.010
It has to ultimately be
paired with somebody.
00:33:15.010 --> 00:33:18.630
I can't pair it with
m beta because that's
00:33:18.630 --> 00:33:22.500
a self-contraction and
will get subtracted out.
00:33:22.500 --> 00:33:25.900
So I can pick any
one of these fours.
00:33:25.900 --> 00:33:29.930
As far as I'm concerned,
all four are the same,
00:33:29.930 --> 00:33:36.310
so I have a choice of four
as to which one of these four
00:33:36.310 --> 00:33:38.300
operators from u I connect to.
00:33:38.300 --> 00:33:40.810
So that 4 is one of
the numerical factors
00:33:40.810 --> 00:33:44.440
that ultimately we
have to take care of.
00:33:44.440 --> 00:33:48.340
Then, the two types
comes because the next m
00:33:48.340 --> 00:33:53.980
that I pick from o I
have two possibilities.
00:33:53.980 --> 00:33:57.840
I can connect it either with the
partner of the first one that
00:33:57.840 --> 00:34:01.950
also carries index
i or I can connect
00:34:01.950 --> 00:34:06.310
to one of the things that
carries the opposite index j.
00:34:06.310 --> 00:34:10.469
So let's call type 1 where
I make the choice that I
00:34:10.469 --> 00:34:14.370
connect to the partner.
00:34:14.370 --> 00:34:17.190
And once I do that,
then I am forced
00:34:17.190 --> 00:34:19.157
to connect these two together.
00:34:22.190 --> 00:34:28.800
Now, each one of these pairings
connects one of these averages.
00:34:28.800 --> 00:34:31.790
So I can write
down what that is.
00:34:31.790 --> 00:34:37.500
So the first one connected an
alpha to an i as far as indices
00:34:37.500 --> 00:34:38.685
were concerned.
00:34:38.685 --> 00:34:46.510
It connected q to q1, so I have
2 pi to the d a delta function
00:34:46.510 --> 00:34:47.870
q plus q1.
00:34:50.630 --> 00:34:53.550
And the variance
associated with that,
00:34:53.550 --> 00:34:57.809
which is t plus k q
squared, et cetera.
00:35:04.850 --> 00:35:09.520
The second pairing
connects a beta to an i.
00:35:09.520 --> 00:35:12.890
So that's a delta beta i.
00:35:12.890 --> 00:35:20.224
And it connects q prime to q2.
00:35:20.224 --> 00:35:23.220
And so the variance
associated with that
00:35:23.220 --> 00:35:29.740
is t plus k q prime
squared and so forth.
00:35:29.740 --> 00:35:34.690
And finally, the third pairing
connects j to itself j.
00:35:34.690 --> 00:35:38.780
So I will get a delta jj.
00:35:38.780 --> 00:35:41.076
And then I have 2 pi to the d.
00:35:45.740 --> 00:35:50.240
q3 to minus q1
minus q2 minus q3.
00:35:50.240 --> 00:35:54.100
So I will get minus
q1 minus q2, and then
00:35:54.100 --> 00:35:58.628
I have t plus, say, k
q3 squared and so forth.
00:36:07.040 --> 00:36:15.700
Now, what I am supposed to
do is at the next stage,
00:36:15.700 --> 00:36:33.050
I have to sum over indices i and
j and integrate over q1 q2 q3.
00:36:37.500 --> 00:36:40.760
So when I do that,
what do I get?
00:36:40.760 --> 00:36:43.410
There is an overall
factor of minus u.
00:36:46.500 --> 00:36:48.900
Let's do the indices.
00:36:48.900 --> 00:36:54.830
When I sum over i, delta
alpha i delta beta i
00:36:54.830 --> 00:36:57.370
becomes-- actually, let
me put the factor of 4
00:36:57.370 --> 00:36:58.510
before I forget it.
00:36:58.510 --> 00:37:01.970
There is a factor
of 4 numerically.
00:37:01.970 --> 00:37:06.330
Delta alpha i delta beta i will
give me a delta alpha beta.
00:37:11.160 --> 00:37:17.200
When I integrate over
q1, q1 is set to minus q.
00:37:17.200 --> 00:37:22.050
So this after the
integration becomes q.
00:37:22.050 --> 00:37:25.120
When I integrate
over q2, the delta
00:37:25.120 --> 00:37:31.710
function q2 forces
minus q2 to be q prime.
00:37:31.710 --> 00:37:38.780
And through the process, two of
these factors of 2 pi to the d
00:37:38.780 --> 00:37:40.390
disappear.
00:37:40.390 --> 00:37:43.480
So what I'm left with
is 2 pi to the d.
00:37:43.480 --> 00:37:49.240
This delta function now
involves q plus q prime.
00:37:49.240 --> 00:37:53.310
And then in the denominator,
I have this factor of t
00:37:53.310 --> 00:37:56.030
plus k q squared.
00:37:56.030 --> 00:38:00.890
I have t plus k q prime squared.
00:38:00.890 --> 00:38:03.540
Although, q prime squared
and q squared are the same.
00:38:03.540 --> 00:38:07.920
I could have collapsed
these things together.
00:38:07.920 --> 00:38:10.855
I have one integration
left over q3.
00:38:15.700 --> 00:38:18.660
And these two
factors went outside
00:38:18.660 --> 00:38:21.120
the integral [INAUDIBLE]
independent q3.
00:38:21.120 --> 00:38:23.115
The only thing
that depends on q3
00:38:23.115 --> 00:38:26.852
is t plus k q3
squared and so forth.
00:38:31.770 --> 00:38:33.894
So that was easy.
00:38:33.894 --> 00:38:35.060
AUDIENCE: I have a question.
00:38:35.060 --> 00:38:36.710
PROFESSOR: Yes.
00:38:36.710 --> 00:38:40.780
AUDIENCE: If you're summing
over j, won't you get an n?
00:38:40.780 --> 00:38:42.620
PROFESSOR: Thank you very much.
00:38:42.620 --> 00:38:44.790
I forgot the delta jj.
00:38:44.790 --> 00:38:48.960
Summing over j, I will
get a factor of n.
00:38:48.960 --> 00:38:52.230
So what I had written
here as 4 should be 4n.
00:38:55.770 --> 00:38:56.688
Yes.
00:38:56.688 --> 00:38:59.540
AUDIENCE: This may be a
question too far back.
00:38:59.540 --> 00:39:05.090
But when you write a correlation
between two different m's,
00:39:05.090 --> 00:39:07.410
why do you write
delta function of q
00:39:07.410 --> 00:39:13.010
plus q prime instead
of q minus q prime?
00:39:13.010 --> 00:39:14.470
PROFESSOR: OK.
00:39:14.470 --> 00:39:17.950
Again, go back all
the way to here
00:39:17.950 --> 00:39:21.330
when we were doing
the Gaussian integral.
00:39:21.330 --> 00:39:27.040
I will have for
the first one, q1.
00:39:27.040 --> 00:39:31.230
For the second m,
I will write q2.
00:39:31.230 --> 00:39:33.940
So when I Fourier
transform this term,
00:39:33.940 --> 00:39:40.030
I will have e to the
i q1 plus q2 dot x.
00:39:40.030 --> 00:39:43.600
And then when I
integrate over x,
00:39:43.600 --> 00:39:48.040
I will get a delta
function q1 plus q2.
00:39:48.040 --> 00:39:52.560
So that's why I write all
of these as absolute value
00:39:52.560 --> 00:39:55.960
squared because I
could have written this
00:39:55.960 --> 00:40:02.290
as m of q m of minus q, but
I realized that m of minus q
00:40:02.290 --> 00:40:05.040
is the complex
conjugate of m of q.
00:40:05.040 --> 00:40:07.010
So all of these are
absolute values squared.
00:40:21.370 --> 00:40:27.480
Now, the second class of
contraction is-- again,
00:40:27.480 --> 00:40:37.250
write the same thing, m alpha of
q m beta of q prime m i of q1 m
00:40:37.250 --> 00:40:47.560
i of q2, mj of q3 mj of
minus q1 minus q2 minus q3.
00:40:50.260 --> 00:40:52.750
The first step is the same.
00:40:52.750 --> 00:40:57.370
I pick m alpha of q and I
have no choice but to pick
00:40:57.370 --> 00:41:01.380
one of the four
possibilities that I have
00:41:01.380 --> 00:41:06.400
for the operators
that appear in u.
00:41:06.400 --> 00:41:09.210
But for the second
one, previously I
00:41:09.210 --> 00:41:12.100
chose to connect it
to something that
00:41:12.100 --> 00:41:14.240
was carrying the same index.
00:41:14.240 --> 00:41:16.410
Now I choose to
carry it to something
00:41:16.410 --> 00:41:20.430
that carries the other
index, j in this case.
00:41:20.430 --> 00:41:23.290
And there are two things
that carry index j,
00:41:23.290 --> 00:41:26.090
so I have two choices there.
00:41:26.090 --> 00:41:30.150
And then I have the remaining
two have to be connected.
00:41:30.150 --> 00:41:31.771
Yes?
00:41:31.771 --> 00:41:34.910
AUDIENCE: Just going
back a little bit.
00:41:34.910 --> 00:41:38.060
Are you assuming that
your integral over q3
00:41:38.060 --> 00:41:40.110
converges because you're
only integrating over
00:41:40.110 --> 00:41:42.780
the [INAUDIBLE] zone?
00:41:42.780 --> 00:41:44.066
PROFESSOR: Yes.
00:41:44.066 --> 00:41:45.470
AUDIENCE: OK.
00:41:45.470 --> 00:41:47.820
PROFESSOR: That's right.
00:41:47.820 --> 00:41:51.480
Any time I see a
divergent integral,
00:41:51.480 --> 00:41:54.290
I have a reason to
go back to my physics
00:41:54.290 --> 00:41:59.030
and see why physics
will avoid infinities.
00:41:59.030 --> 00:42:04.360
And in this case, because
all of my theories
00:42:04.360 --> 00:42:08.600
have an underlying length
scale associated with them
00:42:08.600 --> 00:42:12.900
and there is an
associated maximum value
00:42:12.900 --> 00:42:14.535
that I can go in Fourier space.
00:42:19.450 --> 00:42:23.850
The only possible
singularities that I
00:42:23.850 --> 00:42:27.500
want to get are coming
from q goes to 0.
00:42:27.500 --> 00:42:31.540
And again, if I really want
to physically cut that off,
00:42:31.540 --> 00:42:33.550
I would put the
size of the system.
00:42:33.550 --> 00:42:37.056
But I'm interested in systems
that become infinite in size.
00:42:42.040 --> 00:42:53.080
So the first term for this
way of contracting things
00:42:53.080 --> 00:42:54.380
is as follows.
00:42:54.380 --> 00:42:56.420
There are eight such terms.
00:42:56.420 --> 00:42:58.130
I should have really
put the four here.
00:42:58.130 --> 00:43:03.390
There are eight such
types of contractions.
00:43:03.390 --> 00:43:08.940
Then I have a delta alpha
i 2 pi to the d delta
00:43:08.940 --> 00:43:16.440
function q plus q1 divided by t
plus k q squared and so forth.
00:43:16.440 --> 00:43:19.740
The first contraction is
exactly the same as before.
00:43:19.740 --> 00:43:27.080
The next contraction I connect
i to j and q prime to q3.
00:43:27.080 --> 00:43:33.700
So I have a delta beta j 2
pi to the d delta function q
00:43:33.700 --> 00:43:39.240
prime going to q3 divided
by t plus k q prime squared
00:43:39.240 --> 00:43:39.800
and so forth.
00:43:42.640 --> 00:43:47.680
And the last contraction
connects an i to a j.
00:43:47.680 --> 00:43:49.710
Delta ij.
00:43:49.710 --> 00:43:54.270
I have 2 pi to the d.
00:43:54.270 --> 00:43:59.140
Connecting q2 to minus
q1 minus q2 minus q3
00:43:59.140 --> 00:44:04.200
will give me a delta function
which is minus q1 minus q3.
00:44:04.200 --> 00:44:08.950
And then I have t plus k-- I
guess in this case-- q2 squared
00:44:08.950 --> 00:44:10.342
and so forth.
00:44:17.850 --> 00:44:23.340
So once more sum ij.
00:44:23.340 --> 00:44:30.190
Integrate q1 q2 q3 and
let's see what happens.
00:44:30.190 --> 00:44:35.270
So again, it's a term that
is proportional to minus u.
00:44:35.270 --> 00:44:40.250
The numerical coefficient
that it carries is 8.
00:44:40.250 --> 00:44:47.090
And there is no n here
because when I sum over i,
00:44:47.090 --> 00:44:51.270
you can see that j is set
to be the same as alpha.
00:44:51.270 --> 00:44:56.390
Then when I sum over j, I set
alpha to be the same as beta.
00:44:56.390 --> 00:44:58.250
So there is just a
delta alpha beta.
00:45:02.400 --> 00:45:07.650
When I integrate over
q1, q1 is set to minus q.
00:45:07.650 --> 00:45:12.120
q3 is set to minus q prime.
00:45:12.120 --> 00:45:16.450
So this factor becomes the
same as q plus q prime.
00:45:20.790 --> 00:45:26.250
And the two variances,
which are in fact the same,
00:45:26.250 --> 00:45:30.610
I can continue to write
as separate entities
00:45:30.610 --> 00:45:34.200
but they're really
the same thing.
00:45:34.200 --> 00:45:36.310
And then the one
integral that is
00:45:36.310 --> 00:45:42.270
left-- I did q1 and Q3--
it's the integral over q2,
00:45:42.270 --> 00:45:51.390
2 pi to the d 1 over t plus
K q2 squared and so forth.
00:45:51.390 --> 00:45:53.790
It is, in fact, exactly
the same integral
00:45:53.790 --> 00:45:57.090
as before, except that the
name of the dummy integration
00:45:57.090 --> 00:46:01.837
variable has changed from
q2 to q3, or q3 to q2.
00:46:06.070 --> 00:46:18.890
So we have calculated m
alpha of q m beta of q prime
00:46:18.890 --> 00:46:21.730
to the lowest order in
perturbation theory.
00:46:24.560 --> 00:46:31.040
To the first order, what I
had was a delta alpha beta 2
00:46:31.040 --> 00:46:38.830
pi to the d delta function
q plus q prime divided by t
00:46:38.830 --> 00:46:40.460
plus k q squared.
00:46:44.970 --> 00:46:48.400
Now, note that all
of these factors
00:46:48.400 --> 00:46:51.030
are present in the
two terms that I
00:46:51.030 --> 00:46:52.720
had calculated as corrections.
00:46:55.400 --> 00:47:00.920
So I can factor this
out and write it
00:47:00.920 --> 00:47:06.090
as the correction as 1
plus or minus something.
00:47:06.090 --> 00:47:08.020
It is proportional to u.
00:47:10.910 --> 00:47:15.180
The coefficient is
4n plus 8, which
00:47:15.180 --> 00:47:18.560
I will write as 4 n plus 2.
00:47:21.530 --> 00:47:25.435
I took out one factor
of t plus k q squared.
00:47:25.435 --> 00:47:28.120
There is one factor
that's will be remaining.
00:47:28.120 --> 00:47:33.750
Therefore, t plus k q squared.
00:47:33.750 --> 00:47:39.280
And then I have one
integration over some variable.
00:47:39.280 --> 00:47:40.520
Let's call it k.
00:47:40.520 --> 00:47:44.180
It doesn't matter what I call
the integration variable.
00:47:44.180 --> 00:47:47.882
1 over t plus k, k
squared, and so forth.
00:47:51.990 --> 00:47:56.177
And presumably, there will
be higher-order terms.
00:48:09.100 --> 00:48:12.790
Now again, I did the
calculation specifically
00:48:12.790 --> 00:48:15.970
for the Landau-Ginzburg,
but the procedure you
00:48:15.970 --> 00:48:18.960
would have been able to
do for any field theory.
00:48:18.960 --> 00:48:20.790
You could have
started with a part
00:48:20.790 --> 00:48:22.620
that you can solve
exactly and then
00:48:22.620 --> 00:48:25.112
look at perturbations
and corrections.
00:48:28.420 --> 00:48:33.960
Now, there is, in
fact, a reason why
00:48:33.960 --> 00:48:36.570
this correction
that we calculated
00:48:36.570 --> 00:48:40.320
had exactly the same
structure of delta functions
00:48:40.320 --> 00:48:42.560
as the original one.
00:48:42.560 --> 00:48:47.070
And why I anticipate that
higher-order terms, if I were
00:48:47.070 --> 00:48:49.680
to calculate, will
preserve that structure.
00:48:52.500 --> 00:48:57.070
And the reason has to
do with symmetries.
00:48:57.070 --> 00:49:02.580
Because quite generally, I
can write for anything-- m
00:49:02.580 --> 00:49:08.930
alpha m beta of q prime without
doing perturbation theory.
00:49:08.930 --> 00:49:13.300
Again, let's remember
m alphas of q
00:49:13.300 --> 00:49:17.790
are going to be related to
m of x by inverse Fourier
00:49:17.790 --> 00:49:19.380
transformation.
00:49:19.380 --> 00:49:26.340
So m alpha of q I can write
an integral d dx e to the-- I
00:49:26.340 --> 00:49:32.810
guess by that convention, it
has to be minus i q dot x.
00:49:32.810 --> 00:49:35.670
m alpha of x.
00:49:35.670 --> 00:49:38.920
And again, m beta
of q prime I can
00:49:38.920 --> 00:49:43.700
write as minus i q
prime dot x prime.
00:49:43.700 --> 00:49:52.060
And I integrate
also over an x prime
00:49:52.060 --> 00:49:56.420
of m alpha of x m
beta of x prime.
00:49:56.420 --> 00:50:00.930
Now, these are evaluated in
real space as opposed to Fourier
00:50:00.930 --> 00:50:01.980
space.
00:50:01.980 --> 00:50:04.910
And the average goes over here.
00:50:08.172 --> 00:50:10.230
At this stage, I
don't say anything
00:50:10.230 --> 00:50:14.460
about perturbation theory,
Gaussian, et cetera.
00:50:14.460 --> 00:50:22.280
What I expect is that
this is a function,
00:50:22.280 --> 00:50:26.600
that in a system that has
translational symmetry, only
00:50:26.600 --> 00:50:28.810
depends on x minus x prime.
00:50:39.040 --> 00:50:39.540
m beta.
00:50:42.130 --> 00:50:46.180
Furthermore, in a system that
has rotational symmetry that
00:50:46.180 --> 00:50:48.020
is not spontaneously
broken, that
00:50:48.020 --> 00:50:51.580
is approaching from the
high temperature side,
00:50:51.580 --> 00:50:55.800
then just rotational
symmetry forces you
00:50:55.800 --> 00:50:58.340
that-- the only
tensor that you have
00:50:58.340 --> 00:51:01.940
has to be proportional
to delta alpha beta.
00:51:01.940 --> 00:51:06.150
So I can pick some particular
component-- let's say m1--
00:51:06.150 --> 00:51:07.700
and I can write it
in this fashion.
00:51:11.240 --> 00:51:17.020
So the rotational symmetry
explains the delta alpha beta.
00:51:17.020 --> 00:51:22.170
Now, knowing that the function
that I'm integrating over two
00:51:22.170 --> 00:51:27.620
variables actually only depends
on the relative position
00:51:27.620 --> 00:51:34.880
means that I can re-express
this in terms of the relative
00:51:34.880 --> 00:51:36.550
and center of mass coordinates.
00:51:36.550 --> 00:51:38.900
So I can express
that expression as e
00:51:38.900 --> 00:51:47.670
to the minus i q minus q
prime x minus x prime over 2.
00:51:47.670 --> 00:51:50.750
And then I will
write it as minus i
00:51:50.750 --> 00:51:56.260
q plus q prime x
plus x prime over 2.
00:51:56.260 --> 00:52:00.020
If you expand those, you will
see that all of the cross terms
00:52:00.020 --> 00:52:04.160
will vanish and I will get q
dot x and q prime dot x prime.
00:52:04.160 --> 00:52:04.990
Yes.
00:52:04.990 --> 00:52:05.906
AUDIENCE: [INAUDIBLE]?
00:52:09.065 --> 00:52:09.690
PROFESSOR: Yes.
00:52:09.690 --> 00:52:10.260
Thank you.
00:52:18.420 --> 00:52:24.156
So now I can change
integration variables
00:52:24.156 --> 00:52:33.110
to the relative coordinate and
the center of mass coordinate
00:52:33.110 --> 00:52:37.330
rather than integrating
over x and x prime.
00:52:37.330 --> 00:52:41.410
The integration over the
center of mass, the x plus x
00:52:41.410 --> 00:52:47.060
prime variable, couples
to q plus q prime.
00:52:47.060 --> 00:52:49.800
So it will immediately
tell me that the answer has
00:52:49.800 --> 00:52:56.350
to be proportional
to q plus q prime.
00:52:56.350 --> 00:53:00.820
I had already established that
there is a delta alpha beta.
00:53:00.820 --> 00:53:05.500
So the only thing that is
left is the integration
00:53:05.500 --> 00:53:10.980
over the relative coordinate
of e to the minus i
00:53:10.980 --> 00:53:17.900
some q-- either one of them.
00:53:17.900 --> 00:53:19.030
q dot r.
00:53:19.030 --> 00:53:22.685
Since q prime is minus q,
I can replace it with e
00:53:22.685 --> 00:53:27.060
to the i q dot the
relative coordinate.
00:53:27.060 --> 00:53:29.800
m1 of r m1 of 0.
00:53:36.590 --> 00:53:42.310
So that's why for a system
that has translational symmetry
00:53:42.310 --> 00:53:45.280
and rotational
symmetry, this structure
00:53:45.280 --> 00:53:48.880
of the delta functions
is really imposed
00:53:48.880 --> 00:53:51.460
for this type of
expectation value.
00:53:51.460 --> 00:53:57.450
Naturally, perturbation
theory has to obey that.
00:53:57.450 --> 00:54:04.160
But then, this is a quantity
that we had encountered before.
00:54:06.810 --> 00:54:09.060
If you recall when
we were scattering
00:54:09.060 --> 00:54:13.600
light out of the
system, the amplitude
00:54:13.600 --> 00:54:18.460
of something that was
scattered was proportional
00:54:18.460 --> 00:54:23.940
to the Fourier transform of
the correlation function.
00:54:23.940 --> 00:54:27.860
And furthermore,
in the limit where
00:54:27.860 --> 00:54:34.540
S is evaluated for q equal
to 0, what we are doing
00:54:34.540 --> 00:54:38.440
is we're essentially integrating
the correlation function.
00:54:38.440 --> 00:54:41.155
We've seen that the integrals
of correlation function
00:54:41.155 --> 00:54:42.613
correspond to the
susceptibilities.
00:54:47.770 --> 00:54:52.200
So you may have thought
that what I was calculating
00:54:52.200 --> 00:54:55.780
was a two-point correlation
function in perturbation
00:54:55.780 --> 00:54:56.340
theory.
00:54:56.340 --> 00:54:58.992
But what I was
actually leading up to
00:54:58.992 --> 00:55:01.690
is to know what the
result is for scattering
00:55:01.690 --> 00:55:02.980
from this theory.
00:55:02.980 --> 00:55:04.800
And in some limit
of it, I've also
00:55:04.800 --> 00:55:07.510
calculated what the
susceptibility is,
00:55:07.510 --> 00:55:09.065
how the susceptibility
is corrected.
00:55:11.930 --> 00:55:16.440
And again, if you recall
the typical structure
00:55:16.440 --> 00:55:20.750
that people see for
S of q is that S of q
00:55:20.750 --> 00:55:25.996
is something like 1 over
something like this.
00:55:25.996 --> 00:55:28.770
This is the
Lorentzian line shapes
00:55:28.770 --> 00:55:31.000
that we had in scattering.
00:55:31.000 --> 00:55:34.680
And clearly, the
Lorentzian line shape
00:55:34.680 --> 00:55:40.600
is obtained by Fourier
transformation and expectation
00:55:40.600 --> 00:55:44.900
values of these
expansions that we make.
00:55:44.900 --> 00:55:49.780
So it kind of makes
sense that rather
00:55:49.780 --> 00:55:55.940
than looking at this quantity,
I should look at its inverse.
00:55:55.940 --> 00:55:59.770
So I have calculated
s of q, which
00:55:59.770 --> 00:56:03.290
is the formula that
I have up there.
00:56:03.290 --> 00:56:15.520
So this whole thing
here is S of q.
00:56:18.490 --> 00:56:23.310
If I calculate its
inverse, what do I get?
00:56:23.310 --> 00:56:25.370
First of all, I
have to invert this.
00:56:25.370 --> 00:56:33.600
I have t plus k q squared, which
is what would have given me
00:56:33.600 --> 00:56:36.680
the Lorentzian if I
were to invert it.
00:56:36.680 --> 00:56:39.740
And now we have found the
correction to the Lorentzian
00:56:39.740 --> 00:56:44.610
if you like, which
is this object raised
00:56:44.610 --> 00:56:47.210
to the power of minus 1.
00:56:47.210 --> 00:56:49.750
But recall that I've
only calculated things
00:56:49.750 --> 00:56:52.520
to lowest order in u.
00:56:52.520 --> 00:56:57.110
So whenever I see something
and I'm inverting it
00:56:57.110 --> 00:57:00.020
just like I did
over here, I better
00:57:00.020 --> 00:57:02.900
be consistent to order of u.
00:57:02.900 --> 00:57:07.850
So to order of u, I can take
this thing from the numerator,
00:57:07.850 --> 00:57:10.780
bring it to the num--
from denominator
00:57:10.780 --> 00:57:13.490
to numerator at the expense
of just changing the sign.
00:57:32.580 --> 00:57:34.740
Order of u squared
that we haven't really
00:57:34.740 --> 00:57:35.656
bothered to calculate.
00:57:41.110 --> 00:57:44.420
So now it's nice because you
can see that when I expand this,
00:57:44.420 --> 00:57:48.890
this factor will
cancel that factor.
00:57:48.890 --> 00:57:52.900
So the inverse has the
structure that we would like.
00:57:52.900 --> 00:57:58.750
It is t plus something
that is a constant,
00:57:58.750 --> 00:58:04.260
doesn't depend on
q, 4 n plus 2 u.
00:58:04.260 --> 00:58:05.600
Well, actually, no.
00:58:05.600 --> 00:58:06.210
Yeah.
00:58:06.210 --> 00:58:12.250
Because this denominator
gets canceled.
00:58:12.250 --> 00:58:20.150
I will get 4 n plus 2
u integral over k 2 pi
00:58:20.150 --> 00:58:26.690
to the d 1 over t plus k
k squared and so forth.
00:58:26.690 --> 00:58:31.100
And then I have my k q squared.
00:58:31.100 --> 00:58:33.580
And presumably, I will
have higher-order terms
00:58:33.580 --> 00:58:36.042
both in u and higher
powers of q, et cetera.
00:58:43.920 --> 00:58:51.960
And in particular, the
inverse of the susceptibility
00:58:51.960 --> 00:58:53.790
is simply the first part.
00:58:53.790 --> 00:58:56.150
Forget about the k q squared.
00:58:56.150 --> 00:58:58.420
So the inverse of
susceptibility is
00:58:58.420 --> 00:59:11.190
t plus 4 n plus 2 u integral
d dk 2 pi to the d 1 over t
00:59:11.190 --> 00:59:14.690
plus k k squared and so forth.
00:59:14.690 --> 00:59:17.045
Plus order of things
that we haven't computed.
00:59:35.900 --> 00:59:40.000
So why is it interesting
to look at susceptibility?
00:59:40.000 --> 00:59:46.590
Because susceptibility is
one of the quantities--
00:59:46.590 --> 00:59:52.270
it always has to be positive--
that we were associating
00:59:52.270 --> 00:59:56.700
previously with
singular behavior.
00:59:56.700 --> 01:00:01.340
And in the absence of the
perturbative correction
01:00:01.340 --> 01:00:03.980
from the Gaussian,
the susceptibility
01:00:03.980 --> 01:00:05.460
we calculated many times.
01:00:05.460 --> 01:00:08.040
It was simply 1 over t.
01:00:08.040 --> 01:00:11.290
If I had added a
field, the field h
01:00:11.290 --> 01:00:12.770
would have changed
the free energy
01:00:12.770 --> 01:00:16.370
by an amount that would be
h squared over 2t as we saw.
01:00:16.370 --> 01:00:18.270
Take two derivatives,
I will get 1
01:00:18.270 --> 01:00:20.440
over t for the susceptibility.
01:00:20.440 --> 01:00:25.760
So the 0 order sustainability
that I will indicate by chi sub
01:00:25.760 --> 01:00:31.340
0 was something that was
diverging at t equals to 0.
01:00:31.340 --> 01:00:34.670
And we were identifying
the critical exponent
01:00:34.670 --> 01:00:38.830
of the divergence as
gamma equals to 1.
01:00:38.830 --> 01:00:44.330
So here, I would have
added gamma 0 equals to 1.
01:00:44.330 --> 01:00:46.830
Because of the
linear divergence--
01:00:46.830 --> 01:00:49.410
and the linear divergence
can be traced back
01:00:49.410 --> 01:00:52.930
to the linearity
of the vanishing
01:00:52.930 --> 01:00:55.000
of the inverse susceptibility
at temperature.
01:00:59.020 --> 01:01:01.540
Now, let's see whether we
have calculated a correction
01:01:01.540 --> 01:01:04.139
to gamma.
01:01:04.139 --> 01:01:05.680
Well, the first
thing that you notice
01:01:05.680 --> 01:01:15.370
that if I evaluated the new chi
inverse at 0, all I need to do
01:01:15.370 --> 01:01:18.970
is to put 0 in this formula.
01:01:18.970 --> 01:01:23.890
I will get 4 n plus 2 u.
01:01:23.890 --> 01:01:31.740
This integral d dk 2 pi to
the d 1 over k k squared.
01:01:31.740 --> 01:01:33.722
I set t equals to 0 here.
01:01:36.814 --> 01:01:42.580
Now this is, indeed, an integral
that if I integrate all the ay
01:01:42.580 --> 01:01:45.150
to infinity would diverge on me.
01:01:45.150 --> 01:01:48.690
I have to put an upper cutoff.
01:01:48.690 --> 01:01:50.070
It's a simple integral.
01:01:50.070 --> 01:01:57.106
I can write it as
integral 0 to lambda dk
01:01:57.106 --> 01:02:04.650
k to the d minus 1 with some
surface of a d dimensional
01:02:04.650 --> 01:02:06.230
sphere.
01:02:06.230 --> 01:02:09.650
I have this 2 pi
to the d out front
01:02:09.650 --> 01:02:13.000
and I have a k squared here.
01:02:13.000 --> 01:02:14.960
I can put the k out here.
01:02:14.960 --> 01:02:18.730
So you can see that this is an
integral that's just a power.
01:02:18.730 --> 01:02:20.710
I can simply do that.
01:02:20.710 --> 01:02:32.180
The answer ultimately will be
4 n plus 2 u Sd 2 pi to the d.
01:02:32.180 --> 01:02:35.830
There is a factor of 1 over
k that comes into play.
01:02:35.830 --> 01:02:37.880
The integral of
this will give me
01:02:37.880 --> 01:02:43.938
the upper cutoff to the d
minus 2 divided by d minus 2.
01:02:47.270 --> 01:02:54.480
So what we find is that the
corrected susceptibility
01:02:54.480 --> 01:03:00.650
to lowest order does not
diverge at t equals to 0.
01:03:00.650 --> 01:03:03.980
Its inverse is a finite value.
01:03:03.980 --> 01:03:07.270
So actually, you
can see that I've
01:03:07.270 --> 01:03:11.060
added something positive
to the denominator
01:03:11.060 --> 01:03:15.860
so the value of susceptibility
is always reduced.
01:03:15.860 --> 01:03:17.250
So what does that mean?
01:03:17.250 --> 01:03:20.970
Does that mean that the
susceptibility does not
01:03:20.970 --> 01:03:23.350
have a singularity anymore?
01:03:23.350 --> 01:03:24.760
The answer is no.
01:03:24.760 --> 01:03:29.340
It's just that the location of
the singularity has changed.
01:03:29.340 --> 01:03:31.520
The presence of
u m to the fourth
01:03:31.520 --> 01:03:35.900
gives some additional stiffness
that you have to overcome.
01:03:35.900 --> 01:03:38.930
t equals to 0 is not
sufficient for you.
01:03:38.930 --> 01:03:43.470
You have to go to
some other point tc.
01:03:43.470 --> 01:03:45.720
So I expect that this
thing will actually
01:03:45.720 --> 01:03:52.090
diverge at a new point
tc that is negative.
01:03:52.090 --> 01:03:58.730
And if it diverges, then
its inverse will be 0.
01:03:58.730 --> 01:04:05.120
So I have to solve the
equation tc plus 4 n
01:04:05.120 --> 01:04:17.250
plus 2 u integral d dk 2 pi
to the d of 1 over tc plus k
01:04:17.250 --> 01:04:19.406
k squared and so forth.
01:04:23.350 --> 01:04:26.620
So this seems like an implicit
[INAUDIBLE] equation in tc
01:04:26.620 --> 01:04:30.430
because I have to evaluate the
integral that depends on tc,
01:04:30.430 --> 01:04:33.930
and then have to set
that function to 0.
01:04:33.930 --> 01:04:37.140
But again, we have calculated
things only correctly
01:04:37.140 --> 01:04:38.680
to order of u.
01:04:38.680 --> 01:04:43.160
And you can see already that
tc is proportional to u.
01:04:43.160 --> 01:04:48.000
So this answer here is something
that is order of u presumably.
01:04:48.000 --> 01:04:51.410
And a u compared
to the u out front
01:04:51.410 --> 01:04:54.500
will give me a correction
that is order of u squared.
01:04:54.500 --> 01:04:57.532
So I can ignore this
thing that is over here.
01:05:00.770 --> 01:05:05.430
So to order of u, I know
that tc is actually minus
01:05:05.430 --> 01:05:07.450
what I had calculated before.
01:05:07.450 --> 01:05:14.400
So I get that tc is
minus this 4 n plus 2 u k
01:05:14.400 --> 01:05:22.380
Sd lambda to the d minus
2 2 pi to the d d minus 2.
01:05:22.380 --> 01:05:26.900
It doesn't matter what it is,
it's some non-universal value.
01:05:26.900 --> 01:05:29.470
Point is that, again,
the location of the phase
01:05:29.470 --> 01:05:33.250
transition certainly will
depend on the parameters
01:05:33.250 --> 01:05:34.790
that you put in your theory.
01:05:34.790 --> 01:05:39.210
We readjusted our theory
by putting m to the fourth.
01:05:39.210 --> 01:05:43.505
And certainly, that will change
the location of the transition.
01:05:43.505 --> 01:05:46.050
So this is what we found.
01:05:46.050 --> 01:05:50.390
The location of the
transition is not universal.
01:05:50.390 --> 01:05:53.680
However, we kind
of hope and expect
01:05:53.680 --> 01:05:58.700
that the singularity, the
divergence of susceptibility
01:05:58.700 --> 01:06:00.180
has a form that is universal.
01:06:00.180 --> 01:06:04.770
There is an exponent that
is characteristic of that.
01:06:04.770 --> 01:06:08.720
So asking the question of
how these corrected chi
01:06:08.720 --> 01:06:13.420
divergence diverges
at this tc is the same
01:06:13.420 --> 01:06:18.440
as asking the question of how
its inverse vanishes at tc.
01:06:18.440 --> 01:06:21.530
So what I am interested
is to find out
01:06:21.530 --> 01:06:27.660
what the behavior of chi inverse
is in the vicinity of the point
01:06:27.660 --> 01:06:28.810
that it goes to 0.
01:06:32.177 --> 01:06:35.650
So basically, what
this singularity is.
01:06:35.650 --> 01:06:39.080
This is, of course, 0.
01:06:39.080 --> 01:06:42.830
By definition, chi
inverse of tc is 0.
01:06:42.830 --> 01:06:51.640
So I am asking how chi vanishes
its inverse when I approach tc.
01:06:51.640 --> 01:06:57.395
So I have the formula for
chi once I substitute t,
01:06:57.395 --> 01:07:00.170
once I substitute tc
and I subtract them.
01:07:00.170 --> 01:07:03.656
To lowest order I
have t minus tc.
01:07:03.656 --> 01:07:11.770
To next order, I have this 4 u
n plus 2 integral over k 2 pi
01:07:11.770 --> 01:07:13.710
to the d.
01:07:13.710 --> 01:07:23.430
I have for chi inverse of t 1
over t plus k k squared minus 1
01:07:23.430 --> 01:07:27.490
over tc plus k k squared.
01:07:27.490 --> 01:07:30.040
And terms that I
have not calculated
01:07:30.040 --> 01:07:32.003
are certainly
order of u squared.
01:07:36.000 --> 01:07:39.430
Now, you can see that
if I combine these two
01:07:39.430 --> 01:07:42.820
into the same denominator
that is the product,
01:07:42.820 --> 01:07:46.954
in the numerator I will
get a factor of t minus tc.
01:07:46.954 --> 01:07:49.280
The k q squareds cancel.
01:07:49.280 --> 01:07:55.910
So I can factor out this t
minus tc between the two terms.
01:07:55.910 --> 01:08:00.170
Both terms vanish
at t equals to tc.
01:08:00.170 --> 01:08:06.050
And then I can look at
what the correction is
01:08:06.050 --> 01:08:09.595
to one, just like I did before.
01:08:09.595 --> 01:08:12.670
The correction is going
to-- actually, this
01:08:12.670 --> 01:08:17.153
will give me tc minus t,
so I will have a minus 4 u
01:08:17.153 --> 01:08:25.240
n plus 2 integral
d dk 2 pi to the d.
01:08:25.240 --> 01:08:34.600
The product of two of these
factors, t plus k-- tc plus k k
01:08:34.600 --> 01:08:44.585
squared t plus k k squared,
and then order of u squared.
01:08:49.535 --> 01:08:50.525
OK?
01:08:50.525 --> 01:08:51.515
Happy with that?
01:08:55.990 --> 01:09:01.410
Now again, this tc we've
calculated is order of u.
01:09:01.410 --> 01:09:05.260
And consistently, to calculate
things to order of u,
01:09:05.260 --> 01:09:06.340
I can drop that.
01:09:09.130 --> 01:09:13.960
And again, consistently to
doing things to order of u,
01:09:13.960 --> 01:09:16.729
I can add a tc here.
01:09:16.729 --> 01:09:19.790
And that's also a correction
that is order of u.
01:09:19.790 --> 01:09:22.770
And this answer
would not change.
01:09:22.770 --> 01:09:26.000
The justification of
why I choose to do that
01:09:26.000 --> 01:09:30.069
will become apparent
shortly, but it's consistent
01:09:30.069 --> 01:09:32.460
that this is left.
01:09:32.460 --> 01:09:37.220
So what I find at
this stage is that I
01:09:37.220 --> 01:09:42.109
need to evaluate an
integral of this form.
01:09:58.750 --> 01:10:01.800
And again, with all
of these integrals,
01:10:01.800 --> 01:10:06.850
we better take a look as to
what the most significant
01:10:06.850 --> 01:10:09.950
contribution to the integral is.
01:10:09.950 --> 01:10:16.730
And clearly, if I
look at k goes to 0,
01:10:16.730 --> 01:10:20.580
there are various
factors out there
01:10:20.580 --> 01:10:24.920
that as long as t
minus tc is positive,
01:10:24.920 --> 01:10:28.870
I will have no worries
because this k squared
01:10:28.870 --> 01:10:31.930
will be killed off
by factors of k
01:10:31.930 --> 01:10:36.170
to the d minus 1 in
dimensions above 2.
01:10:36.170 --> 01:10:39.620
But if I go to large
k-values, I find
01:10:39.620 --> 01:10:43.880
that large k-values,
the singularity
01:10:43.880 --> 01:10:48.790
is governed by k to
the power of d minus 4.
01:10:48.790 --> 01:10:57.670
So as long as I'm
dealing with things
01:10:57.670 --> 01:11:01.130
that have some upper
cutoff, I don't
01:11:01.130 --> 01:11:05.455
have to worry about it even
in dimensions greater than 4.
01:11:05.455 --> 01:11:12.380
In dimensions
greater than 4, what
01:11:12.380 --> 01:11:15.470
happens is that the
integral is going
01:11:15.470 --> 01:11:21.700
to be dominated by the
largest values of k.
01:11:21.700 --> 01:11:26.360
But those largest values of
k will be cutoff by lambda.
01:11:26.360 --> 01:11:29.410
The answer ultimately
will be proportional to 1
01:11:29.410 --> 01:11:35.980
over k squared, and then k
to the power of d minus 4
01:11:35.980 --> 01:11:38.680
replaced by lambda
to the d minus 4--
01:11:38.680 --> 01:11:42.210
various overall coefficient
of d minus 4 or whatever.
01:11:42.210 --> 01:11:43.190
It doesn't matter.
01:11:46.130 --> 01:11:49.430
On the other hand, if you go
to dimensions less than 4--
01:11:49.430 --> 01:11:50.850
again, larger than
2, but I won't
01:11:50.850 --> 01:11:58.500
write that for the time being--
then the behavior at large k
01:11:58.500 --> 01:12:00.820
is perfectly convergent.
01:12:00.820 --> 01:12:05.340
So you are integrating
a function that goes up,
01:12:05.340 --> 01:12:06.720
comes down.
01:12:06.720 --> 01:12:11.440
You can extend the integration
all the way to infinity,
01:12:11.440 --> 01:12:14.480
end up with a definite integral.
01:12:14.480 --> 01:12:18.290
We can rescale all
of our factors of k
01:12:18.290 --> 01:12:23.150
to find out what that definite
integral is dependent on.
01:12:23.150 --> 01:12:30.000
And essentially, what it does
is it replaces this lambda
01:12:30.000 --> 01:12:34.170
with the characteristic value
of k that corresponds roughly
01:12:34.170 --> 01:12:35.550
to the maximum.
01:12:35.550 --> 01:12:37.910
And that's going to
occur at something
01:12:37.910 --> 01:12:45.200
like t minus tc over
k to the power of 1/2.
01:12:45.200 --> 01:12:50.430
So I will get d minus 4 over 2.
01:12:50.430 --> 01:12:54.990
There is some overall definite
integral that I have to do,
01:12:54.990 --> 01:12:58.440
which will give me some
numerical coefficient.
01:12:58.440 --> 01:13:00.680
But at this time,
let's forget about
01:13:00.680 --> 01:13:02.410
the numerical coefficient.
01:13:02.410 --> 01:13:05.190
Let's see what the structure is.
01:13:05.190 --> 01:13:08.543
So the structure then
is that chi inverse
01:13:08.543 --> 01:13:16.150
of t, the singularity
that it has
01:13:16.150 --> 01:13:20.940
is t minus tc to the 0 order.
01:13:20.940 --> 01:13:24.370
Same thing as you would have
predicted for the Gaussian.
01:13:24.370 --> 01:13:27.270
And then we have a
correction, which
01:13:27.270 --> 01:13:32.820
is this minus something that
goes after all of these things
01:13:32.820 --> 01:13:33.860
with some coefficient.
01:13:33.860 --> 01:13:36.140
I don't care what
that coefficient is.
01:13:36.140 --> 01:13:39.205
u n plus 2 divided by k squared.
01:13:42.070 --> 01:13:48.090
And then multiplied by lambda
to the power of d minus 4,
01:13:48.090 --> 01:13:55.420
or t minus tc over k to the
power of d minus 4 over 2.
01:13:55.420 --> 01:13:57.535
And then presumably,
higher-order terms.
01:14:00.830 --> 01:14:05.000
And whether you have
the top or the bottom
01:14:05.000 --> 01:14:09.403
will depend on d greater
than 4 or d less than 4.
01:14:24.430 --> 01:14:26.850
So you see the problem.
01:14:26.850 --> 01:14:32.110
If I'm above four
dimensions, this term
01:14:32.110 --> 01:14:34.240
is governed by the upper cutoff.
01:14:34.240 --> 01:14:37.510
But the upper cutoff
is just some constant.
01:14:37.510 --> 01:14:41.520
So all that happens
is that the dependence
01:14:41.520 --> 01:14:44.590
remains as being
proportional to t minus tc.
01:14:44.590 --> 01:14:47.630
The overall amplitude
is corrected
01:14:47.630 --> 01:14:50.270
by something that depends on u.
01:14:50.270 --> 01:14:51.170
You are not worried.
01:14:51.170 --> 01:14:54.120
You say that the leading
singularity is the same thing
01:14:54.120 --> 01:14:55.500
as I had before.
01:14:55.500 --> 01:14:58.770
Gamma will stay to be 1.
01:14:58.770 --> 01:15:04.380
I try to do that in less
than four dimensions
01:15:04.380 --> 01:15:10.790
and I find that as I approach
tc, the correction that I had
01:15:10.790 --> 01:15:12.858
actually itself
becomes divergent.
01:15:16.210 --> 01:15:20.850
So now I have to throw out
my entire perturbation theory
01:15:20.850 --> 01:15:26.270
because I thought I was making
an expansion in quantity
01:15:26.270 --> 01:15:28.520
that I can make
sufficiently small.
01:15:28.520 --> 01:15:31.680
So in usual perturbation
theory, you say
01:15:31.680 --> 01:15:35.730
choose epsilon less than 10
to the minus 100, or whatever,
01:15:35.730 --> 01:15:38.560
and then things will
be small correction
01:15:38.560 --> 01:15:40.790
to what you had at 0 order.
01:15:40.790 --> 01:15:44.740
Here, I can choose my u
to be as small as I like.
01:15:44.740 --> 01:15:49.550
Once I approach tc, the
correction will blow up.
01:15:49.550 --> 01:15:52.680
So this is called a divergent
perturbation theory.
01:15:52.680 --> 01:15:53.180
Yes.
01:15:53.180 --> 01:15:55.596
AUDIENCE: So could we have
known a priori that we couldn't
01:15:55.596 --> 01:15:58.290
get a correction to gamma
from the perturbation theory
01:15:58.290 --> 01:15:59.995
because the only way
for gamma to change
01:15:59.995 --> 01:16:01.786
is for the correction
to have a divergence?
01:16:04.550 --> 01:16:07.910
PROFESSOR: You are presuming
that that's what happens.
01:16:07.910 --> 01:16:11.560
So indeed, if you
knew that there
01:16:11.560 --> 01:16:15.530
is a divergence with an exponent
that is larger than gamma,
01:16:15.530 --> 01:16:17.320
you probably could
have guessed that you
01:16:17.320 --> 01:16:20.410
wouldn't get it this way.
01:16:20.410 --> 01:16:24.720
Let's say that we are choosing
to proceed mathematically
01:16:24.720 --> 01:16:29.090
without prior knowledge of what
the experimentalists have told
01:16:29.090 --> 01:16:33.414
us, then we can
discover it this way.
01:16:33.414 --> 01:16:34.830
AUDIENCE: I was
thinking if you're
01:16:34.830 --> 01:16:38.760
looking for how gamma changes
due to the higher-order things,
01:16:38.760 --> 01:16:43.210
if we found that our
perturbation diverged
01:16:43.210 --> 01:16:44.940
with a lower
exponent than gamma,
01:16:44.940 --> 01:16:46.810
then the leading one
would still be there,
01:16:46.810 --> 01:16:47.870
original gamma would be gone.
01:16:47.870 --> 01:16:48.060
PROFESSOR: Yes.
01:16:48.060 --> 01:16:48.560
AUDIENCE: And then
if it's higher,
01:16:48.560 --> 01:16:50.140
then we have the same problem.
01:16:50.140 --> 01:16:51.480
PROFESSOR: That's right.
01:16:51.480 --> 01:16:53.840
So the problem that
we have is actually
01:16:53.840 --> 01:16:59.840
to somehow make sense of this
type of perturbation theory.
01:16:59.840 --> 01:17:01.530
And as you say, it's correct.
01:17:01.530 --> 01:17:02.910
We could have actually guessed.
01:17:02.910 --> 01:17:07.350
And I'll give you another
reason why the perturbation
01:17:07.350 --> 01:17:10.100
theory would not have worked.
01:17:10.100 --> 01:17:12.310
But the only thing
that we can really do
01:17:12.310 --> 01:17:14.370
is perturbation
theory, so we have
01:17:14.370 --> 01:17:18.260
to be clever and figure
out a way of making sense
01:17:18.260 --> 01:17:22.340
of this perturbation theory,
which we will do by combining
01:17:22.340 --> 01:17:25.400
it with the normalization group.
01:17:25.400 --> 01:17:28.850
But a better way or another
way to have seen maybe
01:17:28.850 --> 01:17:33.630
why this does not work is
good old-fashioned dimensional
01:17:33.630 --> 01:17:35.360
analysis.
01:17:35.360 --> 01:17:40.960
I have within the exponent of
the weight that I wrote down
01:17:40.960 --> 01:17:47.920
terms that are of this formula,
t m squared k gradient of m
01:17:47.920 --> 01:17:53.080
squared u m to the
fourth and so forth.
01:17:53.080 --> 01:17:57.170
Since whatever is in
the exponent should
01:17:57.170 --> 01:18:01.240
be dimensionless-- I usually
write beta H for example--
01:18:01.240 --> 01:18:06.210
we know that this t
has some dimension.
01:18:06.210 --> 01:18:11.060
Square the dimension of m
multiplied by length to the d.
01:18:11.060 --> 01:18:13.906
This should be dimensionless.
01:18:13.906 --> 01:18:20.930
Similarly, k m squared again.
01:18:20.930 --> 01:18:24.830
Because of the gradient
l to the d minus 2,
01:18:24.830 --> 01:18:28.870
that combination should
be dimensionless.
01:18:28.870 --> 01:18:37.470
And my u m to the fourth l to
the d should be dimensionless.
01:18:41.400 --> 01:18:46.130
So we can get rid of the
dimensions of m by dividing,
01:18:46.130 --> 01:18:52.040
let's say, u m to the fourth
with the square of k m squared.
01:18:52.040 --> 01:18:57.570
So we can immediately
see that u divided by k
01:18:57.570 --> 01:19:02.822
squared, I get rid of
the dimensions of m.
01:19:02.822 --> 01:19:08.010
l to the power of d
l to the 2d minus 4,
01:19:08.010 --> 01:19:11.550
giving me l to the 4
minus d is dimensionless.
01:19:19.030 --> 01:19:24.530
So any perturbation theory that
I write down ultimately where
01:19:24.530 --> 01:19:30.630
I have some quantity x,
which is at 0 order 1,
01:19:30.630 --> 01:19:34.770
and then I want to make a
correction where u appears,
01:19:34.770 --> 01:19:38.330
I should have
something, u over k
01:19:38.330 --> 01:19:44.090
squared, and then some power of
length to make the dimensions
01:19:44.090 --> 01:19:46.280
work out.
01:19:46.280 --> 01:19:49.650
So what lengths do I
have available to me?
01:19:49.650 --> 01:19:53.910
One length that I have is
my microscopic length a.
01:19:53.910 --> 01:19:59.250
So I could have put here a
to the power of 4 minus d.
01:19:59.250 --> 01:20:01.490
But there is also
an emergent length
01:20:01.490 --> 01:20:03.760
in the problem, which is
the correlation length.
01:20:08.600 --> 01:20:15.480
And there is no reason why
the dimensionless form that
01:20:15.480 --> 01:20:19.380
involves the correlation
length should not appear.
01:20:19.380 --> 01:20:23.720
And indeed, what we have
over here to 0 order,
01:20:23.720 --> 01:20:28.580
our correlation length had
the exponent 1/2 divergence.
01:20:28.580 --> 01:20:32.560
So this is really the 0
order correlation length
01:20:32.560 --> 01:20:37.400
that is raised to the
power of 4 minus d.
01:20:37.400 --> 01:20:43.140
So even before doing
the calculation,
01:20:43.140 --> 01:20:47.500
we could have guessed
on dimensional ground
01:20:47.500 --> 01:20:51.640
that it is quite possible
that we are expanding in u,
01:20:51.640 --> 01:20:52.910
we think.
01:20:52.910 --> 01:20:56.020
But at the end of the day,
we are expanding in u c
01:20:56.020 --> 01:20:57.850
to the power of 4 minus d.
01:20:57.850 --> 01:21:00.890
And there is no way that that's
a small quantity on approaching
01:21:00.890 --> 01:21:02.690
the phase transition.
01:21:02.690 --> 01:21:05.730
And that hit us on
the face and also is
01:21:05.730 --> 01:21:10.060
the reason why I replaced this
t over here with t minus tc
01:21:10.060 --> 01:21:14.020
because the only place where I
expect singularities to emerge
01:21:14.020 --> 01:21:16.835
in any of these
expansions is at tc.
01:21:16.835 --> 01:21:22.010
I arranged things so they would
appear at the right place.
01:21:22.010 --> 01:21:27.870
So should we throw out
perturbation theory completely
01:21:27.870 --> 01:21:29.680
since the only
thing that we can do
01:21:29.680 --> 01:21:32.080
is really perturbation theory?
01:21:32.080 --> 01:21:33.920
Well, we have to
be clever about it.
01:21:33.920 --> 01:21:37.051
And that's what we
will do next lectures.