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PROFESSOR: Start
by reminding you

00:00:26.230 --> 00:00:29.710
of what we did in last lecture.

00:00:29.710 --> 00:00:37.260
And, in particular, there was
a little bit of confusion.

00:00:37.260 --> 00:00:41.330
It seems like, maybe some
part, I went too fast.

00:00:41.330 --> 00:00:48.125
So we'll try to-- yeah.

00:00:48.125 --> 00:00:56.540
[AUDIO OUT]

00:00:56.540 --> 00:01:03.452
So again, we start, say, by
choosing the worldsheet metric

00:01:03.452 --> 00:01:04.285
to be the Minkowski.

00:01:09.700 --> 00:01:13.040
Then the worldsheet
action just reduce

00:01:13.040 --> 00:01:16.490
to that of a bunch of
free scalar fields.

00:01:22.740 --> 00:01:26.780
Just reduced to a bunch
of free scalar fields.

00:01:26.780 --> 00:01:29.250
And those free scalar
field is a bit unusual,

00:01:29.250 --> 00:01:33.310
because, in particular, there's
a 0 component, which we have

00:01:33.310 --> 00:01:37.090
a long sine kinetic term, OK?

00:01:37.090 --> 00:01:40.210
But, of course, this
is not our full story.

00:01:40.210 --> 00:01:46.892
You still have to impose the
so-called Virosoro constraint,

00:01:46.892 --> 00:01:49.100
because [INAUDIBLE], the
stress tensor of this series

00:01:49.100 --> 00:01:53.645
should be 0, which is the
consequence of the equation

00:01:53.645 --> 00:01:55.096
of motion for gamma.

00:01:58.597 --> 00:02:01.180
But nevertheless, you can write
down the most general solution

00:02:01.180 --> 00:02:05.560
to this problem, so
let me write it here.

00:02:05.560 --> 00:02:10.115
So the most general
of a closed string,

00:02:10.115 --> 00:02:12.080
or the most general
conclusion can be

00:02:12.080 --> 00:02:16.925
written as x mu plus p mu tau.

00:02:45.340 --> 00:02:49.830
So I've now written,
what I called previous

00:02:49.830 --> 00:02:52.610
by Xl and the Xr, I have
written them explicitly

00:02:52.610 --> 00:02:55.130
in terms of the
Fourier modes, OK?

00:02:55.130 --> 00:02:56.630
And I have written
them, explicitly,

00:02:56.630 --> 00:02:58.480
in terms of Fourier modes.

00:02:58.480 --> 00:03:03.120
And in this form, so you can
compare with what we discovered

00:03:03.120 --> 00:03:05.970
before, previous,
here, because v mu,

00:03:05.970 --> 00:03:09.470
then we discussed last
time, that this v mu

00:03:09.470 --> 00:03:13.120
should be considered as
related to the center of mass,

00:03:13.120 --> 00:03:15.294
momentum, over the whole string.

00:03:15.294 --> 00:03:17.835
And, for example, for the closed
string, that's the relation.

00:03:22.760 --> 00:03:32.050
OK, so similarly,
for the open string,

00:03:32.050 --> 00:03:36.070
if you use the Neumann
boundary condition, here,

00:03:36.070 --> 00:03:37.940
use the Neumann
boundary condition,

00:03:37.940 --> 00:03:42.575
then you find, OK,
now we'll substitute

00:03:42.575 --> 00:03:49.980
the explicit expression
for the Xl and the Xr.

00:03:49.980 --> 00:03:53.130
So now, it become 2
alpha prime p mu tau.

00:03:53.130 --> 00:03:56.200
So, again, this is
our previous v mu.

00:03:56.200 --> 00:03:59.930
So this 2 is related
to the open string.

00:03:59.930 --> 00:04:04.640
Sigma only go from 0 to
pi, rather than 2 pi.

00:04:04.640 --> 00:04:09.400
And then, you can
write the oscillator,

00:04:09.400 --> 00:04:16.399
this Xl Xr in terms of the
explicit Fourier transform.

00:04:16.399 --> 00:04:18.150
So here, you only
have one set of modes.

00:04:25.090 --> 00:04:26.800
And of course, n sigma
just come from you

00:04:26.800 --> 00:04:31.120
have Xl plus Xr, and Xl,
which is equal to Xr.

00:04:31.120 --> 00:04:34.170
And yeah, when these
two become the same,

00:04:34.170 --> 00:04:37.650
when these two become the
same, then these two combine.

00:04:37.650 --> 00:04:39.300
Because the sigma
have opposite sign

00:04:39.300 --> 00:04:40.810
that combine to cause n sigma.

00:04:40.810 --> 00:04:41.789
OK?

00:04:41.789 --> 00:04:42.830
Yeah, it cause a m sigma.

00:04:50.050 --> 00:04:53.755
So this is the most
general classical solution,

00:04:53.755 --> 00:05:09.610
and in the light-cone gauge, we
say x plus can be setting to 0.

00:05:09.610 --> 00:05:16.240
And also everything related to
alpha plus, and for tilde plus,

00:05:16.240 --> 00:05:18.965
all the oscillation
modes also said to be 0.

00:05:18.965 --> 00:05:21.149
We raise it to the
plus, set it to 0,

00:05:21.149 --> 00:05:22.690
so there's only
[? tau ?] [INAUDIBLE]

00:05:22.690 --> 00:05:24.370
left in the light-cone gauge.

00:05:27.080 --> 00:05:36.980
And Virasoro constraint, they
become-- I have tau x minus.

00:05:43.650 --> 00:05:47.460
So this v plus is
the same as that,

00:05:47.460 --> 00:05:49.000
related to the p plus that way.

00:06:05.250 --> 00:06:06.820
only single v plus.

00:06:12.170 --> 00:06:15.120
OK, so you know the
equation, so the i

00:06:15.120 --> 00:06:16.960
should be considered
as a sum, OK?

00:06:16.960 --> 00:06:19.760
A sum of all directions,
or transpose directions.

00:06:22.556 --> 00:06:23.930
So, from the
Virasoro constraint,

00:06:23.930 --> 00:06:25.400
you can deduce X minus.

00:06:29.080 --> 00:06:31.600
You can deduce X minus.

00:06:31.600 --> 00:06:34.470
Yeah, again, let me just
write here, our convention's

00:06:34.470 --> 00:06:44.530
always X mu is equal to
X plus, X minus, then Xi.

00:06:44.530 --> 00:06:51.310
And i goes from 2 to D
minus 1, and X plus minus

00:06:51.310 --> 00:06:58.540
is, say, 1 over square root
2, X0 plus minus X1, OK?

00:07:05.190 --> 00:07:09.390
So, from here, you can
deduce the X minus.

00:07:09.390 --> 00:07:15.250
And also, it means that
independent variables--

00:07:15.250 --> 00:07:18.770
so this, we have determined
the X minus up to a constant,

00:07:18.770 --> 00:07:21.540
because this determines the
tau and the sigma derivative

00:07:21.540 --> 00:07:23.210
up to a constant.

00:07:23.210 --> 00:07:24.460
So, the independent variables.

00:07:29.200 --> 00:07:35.790
Xi, and then, also,
this p plus, or v

00:07:35.790 --> 00:07:41.710
plus which appears in the
plus, and then the X minus.

00:07:41.710 --> 00:07:44.510
The 0 modes for the
X minus, which is not

00:07:44.510 --> 00:07:46.370
determined by those things.

00:07:46.370 --> 00:07:49.972
And these two, this
is just a constant.

00:07:49.972 --> 00:07:51.430
And, again, these
two are constant.

00:07:51.430 --> 00:07:54.520
These are two-dimensional
fields, OK?

00:08:07.030 --> 00:08:08.140
Any questions so far?

00:08:12.070 --> 00:08:12.570
Good.

00:08:15.390 --> 00:08:18.530
So actually, [INAUDIBLE] that
the zero mode part of this

00:08:18.530 --> 00:08:19.925
equation particularly important.

00:08:22.520 --> 00:08:28.090
Yeah, let me call this
equation 1, this equation 2.

00:08:30.890 --> 00:08:33.394
So the 0 part of those equations
are particularly important.

00:08:36.700 --> 00:08:42.419
And, for example,
say, from equation 1,

00:08:42.419 --> 00:08:46.000
so the zero modes part,
for the first equation--

00:08:46.000 --> 00:08:53.440
so let's do it for closed
string-- then zero mode part,

00:08:53.440 --> 00:08:57.990
you just-- alpha
prime p minus, OK,

00:08:57.990 --> 00:08:59.620
you should take the
root of alpha tau.

00:08:59.620 --> 00:09:02.390
So you just get the
alpha prime p minus,

00:09:02.390 --> 00:09:05.720
and the right-hand side.

00:09:05.720 --> 00:09:08.460
So, right-hand side, let
me also rewrite the v

00:09:08.460 --> 00:09:14.610
plus in terms of a prime, so
this is 2 alpha prime p plus.

00:09:14.610 --> 00:09:22.940
And then, the zero mode means
that you integrate over,

00:09:22.940 --> 00:09:25.075
you integrate over the string.

00:09:34.690 --> 00:09:35.190
OK.

00:09:38.650 --> 00:09:42.230
So this is the first equation.

00:09:42.230 --> 00:09:44.190
So the zero mode equation
become like that.

00:09:47.850 --> 00:09:50.720
So, now, let me just make
a brief comment, which

00:09:50.720 --> 00:09:57.230
I, at the beginning,
I forgot to mention,

00:09:57.230 --> 00:10:01.515
last time, which apparently
causes some confusion later.

00:10:01.515 --> 00:10:08.530
It's that, if you look
at this expression,

00:10:08.530 --> 00:10:11.650
this is actually
precisely-- so if you

00:10:11.650 --> 00:10:14.040
look at that two-dimensional
field theory,

00:10:14.040 --> 00:10:17.380
this is actually
precisely the Hamiltonian,

00:10:17.380 --> 00:10:20.990
the classical Hamiltonian
for that field theory, OK?

00:10:20.990 --> 00:10:23.180
For the Xi part.

00:10:23.180 --> 00:10:26.770
This is just a free
scalar field theory.

00:10:26.770 --> 00:10:29.960
Yeah, so, in particular-- so
let me write that more precise.

00:10:29.960 --> 00:10:33.200
Let's take out this
p plus, and then this

00:10:33.200 --> 00:10:35.890
become 4 pi alpha
prime, and then

00:10:35.890 --> 00:10:39.030
this just become exactly the
Hamiltonian of that theory,

00:10:39.030 --> 00:10:42.980
because 1 and 2 for the Xi, OK?

00:10:42.980 --> 00:11:03.630
So H Xi is the Hamiltonian for
two-dimensional quantum fields.

00:11:03.630 --> 00:11:12.530
Quantum field theory of Xi, for
the transverse directions, OK?

00:11:16.630 --> 00:11:18.890
So you can also write
these explicitly

00:11:18.890 --> 00:11:21.420
in terms of those modes.

00:11:21.420 --> 00:11:28.305
In terms of those modes, then,
for example, you write p minus.

00:11:28.305 --> 00:11:32.610
Yeah, if you can also write
modes, then become p minus.

00:11:32.610 --> 00:11:40.320
You could, too, 2p
plus, pi square,

00:11:40.320 --> 00:11:43.420
plus 1 over alpha prime.

00:11:43.420 --> 00:11:47.530
So if you just substitute
those expansion into here,

00:11:47.530 --> 00:11:49.910
then you can also write it
explicitly in terms of modes.

00:12:05.820 --> 00:12:06.320
OK.

00:12:13.810 --> 00:12:17.885
And then you can combine--
so this is p minus,

00:12:17.885 --> 00:12:19.760
then you can multiply
this to the other side,

00:12:19.760 --> 00:12:22.580
combine all the p together.

00:12:22.580 --> 00:12:28.910
You can write it as M squared,
which is defined to be p mu.

00:12:28.910 --> 00:12:33.070
P mu minus P mu P
mu, which is then 2P

00:12:33.070 --> 00:12:36.000
plus P minus minus Pi squared.

00:12:38.670 --> 00:12:49.570
And then, this then become equal
to 1 over alpha prime sum m

00:12:49.570 --> 00:12:56.720
mu equal to 0 alpha minus
m i alpha m i plus alpha

00:12:56.720 --> 00:13:02.086
tilde minus m i
alpha tilde m i, OK?

00:13:07.510 --> 00:13:11.140
Then, you see that
this constraint

00:13:11.140 --> 00:13:15.820
for p prime, for p minus,
now can be written,

00:13:15.820 --> 00:13:21.350
can be rewritten in
terms of the relation

00:13:21.350 --> 00:13:25.160
of the mass of the whole string
in terms of its oscillation

00:13:25.160 --> 00:13:27.640
modes, OK?

00:13:27.640 --> 00:13:30.830
In terms of its
oscillation modes.

00:13:30.830 --> 00:13:33.200
And, similarity, for
the open string-- so,

00:13:33.200 --> 00:13:36.790
this is for the closed
string-- for the open,

00:13:36.790 --> 00:13:39.250
you could act the same thing
applies, just you only have,

00:13:39.250 --> 00:13:41.285
now, one side of modes.

00:13:51.070 --> 00:13:55.876
So remember, in those
expressions, sum over i

00:13:55.876 --> 00:13:56.665
is always assumed.

00:14:03.190 --> 00:14:06.160
And whenever I wrote
m not equal to 0,

00:14:06.160 --> 00:14:09.070
it means you always sum for
minus infinity plus infinity,

00:14:09.070 --> 00:14:11.120
and except where
m equal to 0, OK?

00:14:18.170 --> 00:14:19.427
Yes?

00:14:19.427 --> 00:14:20.010
Any questions?

00:14:22.990 --> 00:14:27.320
So these are the consequence
of the zero modes for 1.

00:14:27.320 --> 00:14:31.800
And the consequence for
the zero modes for 2,

00:14:31.800 --> 00:14:34.830
again, you can
just integrate over

00:14:34.830 --> 00:14:38.070
all direction of the string.

00:14:38.070 --> 00:14:46.030
Then the left-hand
side just get 0,

00:14:46.030 --> 00:14:49.740
because the x minus is
a periodic function,

00:14:49.740 --> 00:14:51.650
so this is a total
derivative, and so

00:14:51.650 --> 00:14:55.500
this is give you exactly 0.

00:14:55.500 --> 00:15:09.865
And then, for the constraint,
on the right-hand side,

00:15:09.865 --> 00:15:14.230
on this expression, which
can be written explicitly.

00:15:14.230 --> 00:15:17.730
So you can also, then, plus
into the explicit mode,

00:15:17.730 --> 00:15:24.876
and then they become sum m
not equal to 0, alpha minus mi

00:15:24.876 --> 00:15:32.845
alpha mi equal to [INAUDIBLE].

00:15:35.820 --> 00:15:41.440
OK, so this is sometimes called
a level matching condition.

00:15:41.440 --> 00:15:43.070
So this tells you
that the oscillation

00:15:43.070 --> 00:15:47.020
from the left-moving part--
this is for the closed string.

00:15:47.020 --> 00:15:48.870
For the open string,
this equation just

00:15:48.870 --> 00:15:50.010
does not give you anything.

00:15:50.010 --> 00:15:51.390
For the closed
string, this gives you

00:15:51.390 --> 00:15:52.431
a non-trivial constraint.

00:15:54.902 --> 00:15:55.610
Oh, I got a ring.

00:16:00.880 --> 00:16:02.575
So, for a closed
string, this just

00:16:02.575 --> 00:16:04.950
tells you that the left-moving
part and right-moving part

00:16:04.950 --> 00:16:06.680
have to be balanced.

00:16:06.680 --> 00:16:10.800
And this is related
to that string.

00:16:10.800 --> 00:16:12.680
It's periodic.

00:16:12.680 --> 00:16:14.590
Along the string, it's
periodic, so there's

00:16:14.590 --> 00:16:16.410
no special point on the string.

00:16:16.410 --> 00:16:19.252
And then, there's no special,
and then you can actually not

00:16:19.252 --> 00:16:20.710
distinguish between
the left-moving

00:16:20.710 --> 00:16:22.160
and the right-moving part, OK?

00:16:27.320 --> 00:16:28.080
Good.

00:16:28.080 --> 00:16:31.205
So this is a the
whole classical story.

00:16:33.810 --> 00:16:39.500
This is, in some sense, the
complete classical story, OK?

00:16:39.500 --> 00:16:41.580
And then, quantum
mechanically, we just

00:16:41.580 --> 00:16:44.745
need to quantize those guys.

00:16:44.745 --> 00:16:46.119
Quantize those guys.

00:16:46.119 --> 00:16:48.285
And those are just ordinary
quantum mechanic degrees

00:16:48.285 --> 00:16:50.380
of freedom we don't
need to worry about.

00:16:50.380 --> 00:16:54.340
And this just become
a quantization

00:16:54.340 --> 00:16:57.120
of a free scalar field theory.

00:16:57.120 --> 00:16:57.620
OK.

00:17:00.310 --> 00:17:03.930
STUDENT: [INAUDIBLE] all
these modes are massive?

00:17:03.930 --> 00:17:06.602
Or is there any conditions
it can be massless?

00:17:06.602 --> 00:17:08.560
PROFESSOR: No, this is
a massless field theory.

00:17:11.284 --> 00:17:13.830
We are quantizing
this theory, right?

00:17:13.830 --> 00:17:18.035
Yeah, so this is massless scalar
field theory in two-dimension.

00:17:18.035 --> 00:17:19.780
STUDENT: But so that mass is--

00:17:19.780 --> 00:17:22.710
PROFESSOR: No, this mass
is the mass of the string.

00:17:22.710 --> 00:17:24.450
No, this is the
mass of the center

00:17:24.450 --> 00:17:27.660
of-- this is the total
mass of the string.

00:17:31.240 --> 00:17:34.490
When I say the massless, this is
a massless scalar field theory

00:17:34.490 --> 00:17:36.060
in the worldsheet.

00:17:36.060 --> 00:17:38.290
This is a spacetime description.

00:17:38.290 --> 00:17:42.480
So it's important to
separate two things.

00:17:42.480 --> 00:17:45.420
Things happening on the
worldsheet and things

00:17:45.420 --> 00:17:47.820
happening in spacetime.

00:17:47.820 --> 00:17:51.830
So this is the mass of the
string viewed as object

00:17:51.830 --> 00:17:53.850
moving in the spacetime.

00:17:53.850 --> 00:17:57.700
And when I say quantizing a
massless scalar field theory,

00:17:57.700 --> 00:18:00.640
it's to think of
those, the mode.

00:18:00.640 --> 00:18:03.440
So Xi is saying here, describe
the motion of the string.

00:18:03.440 --> 00:18:07.290
Think of them as a field
theory on the worldsheet.

00:18:07.290 --> 00:18:09.121
And that's a massless
field theory.

00:18:09.121 --> 00:18:09.620
STUDENT: OK.

00:18:09.620 --> 00:18:12.380
PROFESSOR: Right.

00:18:12.380 --> 00:18:13.030
Yes?

00:18:13.030 --> 00:18:15.874
STUDENT: [INAUDIBLE]
definition of [INAUDIBLE]?

00:18:21.530 --> 00:18:24.440
PROFESSOR: It's the conserve the
loss of charge for the string.

00:18:24.440 --> 00:18:26.781
Cause 1 into the translation.

00:18:26.781 --> 00:18:28.280
It's to conserve
the loss of charge,

00:18:28.280 --> 00:18:30.980
cause 1 into the translation.

00:18:30.980 --> 00:18:33.470
Yeah, so that's what
we derived last time.

00:18:33.470 --> 00:18:36.630
So be sure that this v--
but to be original, write v,

00:18:36.630 --> 00:18:40.180
here-- are related to the
center of mass momentum

00:18:40.180 --> 00:18:43.500
in this particular way.

00:18:43.500 --> 00:18:44.374
Yes.

00:18:44.374 --> 00:18:48.246
STUDENT: Could you use the
fact that the right [INAUDIBLE]

00:18:48.246 --> 00:18:49.214
looks like [INAUDIBLE].

00:18:52.610 --> 00:18:53.814
PROFESSOR: Sorry, which one?

00:18:53.814 --> 00:18:55.688
STUDENT: The fact that
this alpha [INAUDIBLE]

00:18:55.688 --> 00:18:57.446
P minus is actually
the Hamiltonian.

00:18:57.446 --> 00:18:58.762
Can you use it anyway?

00:18:58.762 --> 00:19:00.450
Or it's just the--

00:19:00.450 --> 00:19:02.470
PROFESSOR: No, this is a remark.

00:19:02.470 --> 00:19:04.100
And this will make
them more lateral.

00:19:08.310 --> 00:19:11.450
For us, this is not
the essential remark,

00:19:11.450 --> 00:19:18.730
but it will be more
lateral, when we work out

00:19:18.730 --> 00:19:20.990
the zero-point energy.

00:19:20.990 --> 00:19:23.320
And when we work out
zero-point energy,

00:19:23.320 --> 00:19:26.410
then that's a more
lateral thing to consider,

00:19:26.410 --> 00:19:30.280
because that will actually
give you a way of the computing

00:19:30.280 --> 00:19:31.662
zero-point energy.

00:19:31.662 --> 00:19:33.110
Yeah.

00:19:33.110 --> 00:19:35.282
Yes.

00:19:35.282 --> 00:19:38.060
STUDENT: So how do
you-- [INAUDIBLE].

00:19:38.060 --> 00:19:41.186
If you plug in this
extension, does it really

00:19:41.186 --> 00:19:43.430
direct me [INAUDIBLE]?

00:19:43.430 --> 00:19:44.178
PROFESSOR: Yeah.

00:19:44.178 --> 00:19:44.678
[LAUGHTER]

00:19:44.678 --> 00:19:46.262
STUDENT: [INAUDIBLE] n times--

00:19:46.262 --> 00:19:46.970
PROFESSOR: Sorry?

00:19:46.970 --> 00:19:50.050
STUDENT: Are you supposed
to have n time [INAUDIBLE]

00:19:50.050 --> 00:19:53.740
times the other number of alpha?

00:19:53.740 --> 00:19:55.520
PROFESSOR: Sorry,
what are you saying?

00:19:55.520 --> 00:19:57.520
STUDENT: Here, if you
plug in this [INAUDIBLE].

00:19:57.520 --> 00:20:00.333
And in the end, you've
got m times alpha m

00:20:00.333 --> 00:20:00.999
PROFESSOR: Yeah.

00:20:00.999 --> 00:20:03.231
STUDENT: But you directly
use this equation.

00:20:03.231 --> 00:20:03.897
PROFESSOR: Yeah.

00:20:03.897 --> 00:20:05.840
STUDENT: And you--
well, [INAUDIBLE]--

00:20:05.840 --> 00:20:07.950
PROFESSOR: Oh.

00:20:07.950 --> 00:20:08.940
This is trivial to see.

00:20:08.940 --> 00:20:12.090
We can see it immediately here.

00:20:12.090 --> 00:20:14.920
First, if you take the
derivative, we just get Pi,

00:20:14.920 --> 00:20:17.750
and then you just get that term.

00:20:17.750 --> 00:20:21.540
And when you take derivative
here, then you cancel the m.

00:20:21.540 --> 00:20:23.630
And then, in order to
get the zero modes,

00:20:23.630 --> 00:20:25.890
then the m and the
minus m have to cancel.

00:20:25.890 --> 00:20:28.933
So the only structure that
can happen is this one.

00:20:28.933 --> 00:20:30.660
Then, the only
thing left remaining

00:20:30.660 --> 00:20:33.900
is to check the
quotient, here, is 1.

00:20:33.900 --> 00:20:36.070
And that, we have to
do the calculation.

00:20:36.070 --> 00:20:38.997
And the rest, you don't
need to do the calculation.

00:20:38.997 --> 00:20:41.455
Yeah, the only thing to do the
calculation is for this one.

00:20:45.770 --> 00:20:46.680
Yeah.

00:20:46.680 --> 00:20:49.390
STUDENT: So just
kind of to bore you,

00:20:49.390 --> 00:20:52.120
so all we're doing
is free fields

00:20:52.120 --> 00:20:55.542
in a two-dimensional worldsheet
with the diffeomorphism

00:20:55.542 --> 00:20:57.495
and variance as
a gauge symmetry,

00:20:57.495 --> 00:21:00.280
and that gauge symmetry gives
us some complicated constraints,

00:21:00.280 --> 00:21:02.360
which we then get rid of.

00:21:02.360 --> 00:21:04.240
PROFESSOR: Solve the
constraints, yeah.

00:21:04.240 --> 00:21:05.420
STUDENT: That's
all there is to it.

00:21:05.420 --> 00:21:06.670
PROFESSOR: Yeah, that's right.

00:21:06.670 --> 00:21:08.690
So, as I said before,
solving the constraint

00:21:08.690 --> 00:21:10.800
help you two things.

00:21:10.800 --> 00:21:13.540
First, you solve
this constraint.

00:21:13.540 --> 00:21:16.210
And second, you get rid of
this X plus and X minus.

00:21:18.822 --> 00:21:21.030
Because those are the
dangerous field theory, and now

00:21:21.030 --> 00:21:22.130
we get rid of.

00:21:22.130 --> 00:21:26.107
And what's remaining,
Xi, are good.

00:21:26.107 --> 00:21:27.440
Will be, hey, they're good boys.

00:21:27.440 --> 00:21:30.555
They're good, well-behaved
field theories.

00:21:34.540 --> 00:21:36.336
Any other questions?

00:21:36.336 --> 00:21:38.510
STUDENT: Yeah, one question.

00:21:38.510 --> 00:21:41.430
So isn't knowing how to do,
basically, string theory--

00:21:41.430 --> 00:21:45.290
well, is it knowing how to
deal with the Polyakov action,

00:21:45.290 --> 00:21:46.660
and not in the light-cone gauge?

00:21:46.660 --> 00:21:47.840
PROFESSOR: Oh, sure.

00:21:47.840 --> 00:21:49.170
But that take much longer time.

00:21:51.960 --> 00:21:54.830
Even in the
light-cone gauge, I'm

00:21:54.830 --> 00:22:01.120
still trying to telling
a long story short.

00:22:01.120 --> 00:22:01.900
STUDENT: OK.

00:22:01.900 --> 00:22:05.590
And so I'm trying to
give you the essence.

00:22:05.590 --> 00:22:08.730
But make sure-- I'm trying
to tell you the essence,

00:22:08.730 --> 00:22:11.500
but at the same time, I
want you to understand.

00:22:11.500 --> 00:22:17.130
So make sure you ask,
whatever, it's not clear.

00:22:17.130 --> 00:22:21.430
But, for example, in the next
semesters of undergraduate,

00:22:21.430 --> 00:22:24.700
of string theory
for undergraduate,

00:22:24.700 --> 00:22:28.750
you will reach this
point, essentially,

00:22:28.750 --> 00:22:32.690
after maybe 15 lectures.

00:22:32.690 --> 00:22:35.740
And we've reached here maybe
only two or three lectures.

00:22:35.740 --> 00:22:40.100
And so, I'm trying to
just give you the essence,

00:22:40.100 --> 00:22:44.930
and to try not to give you
too many technical details.

00:22:44.930 --> 00:22:46.600
Say, cross checks, et cetera.

00:22:46.600 --> 00:22:48.855
There are many cross checks
you can do, et cetera.

00:22:48.855 --> 00:22:49.980
I'm not going those things.

00:22:49.980 --> 00:22:53.242
Just give you the essence.

00:22:53.242 --> 00:22:54.075
Any other questions?

00:22:57.030 --> 00:22:59.000
Good.

00:22:59.000 --> 00:23:01.100
OK.

00:23:01.100 --> 00:23:04.300
So this is the review
of the classical story.

00:23:04.300 --> 00:23:05.820
So now, let's do
the quantum story.

00:23:08.900 --> 00:23:13.280
Again, this is a quick review
of what we did last time.

00:23:17.530 --> 00:23:26.060
So, quantum mechanically,
all those become operators.

00:23:26.060 --> 00:23:28.220
All those become operators.

00:23:28.220 --> 00:23:29.780
And the alpha also
become operators.

00:23:51.760 --> 00:23:58.030
OK, so this commutator,
as we said last time,

00:23:58.030 --> 00:24:01.370
implies that 1 over
square root of alpha m i

00:24:01.370 --> 00:24:06.090
should be considered as
standard [INAUDIBLE] operators,

00:24:06.090 --> 00:24:08.840
for m greater than zero.

00:24:08.840 --> 00:24:13.420
And the mode will
be [INAUDIBLE],

00:24:13.420 --> 00:24:19.220
should be considered as
the creation operators, OK?

00:24:19.220 --> 00:24:20.880
Creation operators.

00:24:20.880 --> 00:24:26.750
And then, this is the standard
of quantization, or field

00:24:26.750 --> 00:24:27.300
theory.

00:24:27.300 --> 00:24:31.640
You reduce to an infinite
number of harmonic oscillators.

00:24:31.640 --> 00:24:37.590
And in particular,
this product just

00:24:37.590 --> 00:24:49.830
reduced to m times this Ami
dagger, Ami just become m Nmi.

00:24:49.830 --> 00:24:50.670
OK?

00:24:50.670 --> 00:24:55.732
So Nmi is the oscillator
number, occupation number, say,

00:24:55.732 --> 00:24:56.940
for each harmonic oscillator.

00:25:04.550 --> 00:25:11.550
Then, the typical
state of the system

00:25:11.550 --> 00:25:16.270
just obtained by acting
those things on the vacuum.

00:25:16.270 --> 00:25:17.920
For example, for
the open string,

00:25:17.920 --> 00:25:20.320
we only have one set of modes.

00:25:20.320 --> 00:25:23.202
So then, we would
have this form.

00:25:23.202 --> 00:25:32.610
m1 m1 alpha minus mq,
iq, mq on the vacuum.

00:25:32.610 --> 00:25:38.200
And the vacuum also have
a quantum number, p mu,

00:25:38.200 --> 00:25:43.095
because those p, those
p plus and the pi,

00:25:43.095 --> 00:25:46.140
here, are quantum operators.

00:25:46.140 --> 00:25:49.540
And so, we take the vacuum
to be eigenstate of them.

00:25:49.540 --> 00:25:51.610
And they're independent
of those alphas.

00:25:51.610 --> 00:25:56.120
So this vacuum is the only
vacuum for the oscillators,

00:25:56.120 --> 00:26:00.210
but they are still labeled
by a spacetime moment, OK?

00:26:00.210 --> 00:26:02.860
So labeled by the spacetime
momentum of the string.

00:26:05.610 --> 00:26:06.740
So for the open string.

00:26:12.370 --> 00:26:22.570
And for the closed string,
similarly, you just

00:26:22.570 --> 00:26:24.420
have two sets of modes.

00:26:24.420 --> 00:26:31.760
Minus i1 n2 alpha
minus m2 i2 and 2.

00:26:34.950 --> 00:26:37.290
And then, also,
you have the other.

00:26:37.290 --> 00:26:48.210
let me call it k1 j1 l1 alpha
minus k2 j2 l2, et cetera.

00:26:48.210 --> 00:26:50.352
In the vacuum.

00:26:50.352 --> 00:26:52.435
So this is a typical state,
for the closed string.

00:26:58.600 --> 00:27:04.310
But the allowed state should
still satisfy this constraint.

00:27:04.310 --> 00:27:06.050
OK, should still
satisfy this constraint.

00:27:08.560 --> 00:27:12.240
So this constrain
can be written as,

00:27:12.240 --> 00:27:15.300
reach the following constraint.

00:27:15.300 --> 00:27:18.200
It said, the oscillator
number on the left-hand side,

00:27:18.200 --> 00:27:22.650
for the left-moving
mode, or tilde.

00:27:22.650 --> 00:27:25.840
And the oscillator number
for the right-moving mode,

00:27:25.840 --> 00:27:29.090
they have to be the same.

00:27:29.090 --> 00:27:32.560
So let me rewrite this equation,
in terms of oscillator numbers.

00:27:32.560 --> 00:27:35.665
So this means i sum.

00:27:35.665 --> 00:27:38.220
So let me, now, write
them from m to infinity.

00:27:38.220 --> 00:27:40.176
So m Nm i.

00:27:55.840 --> 00:27:59.320
OK, so this just form
that equation, OK?

00:28:01.362 --> 00:28:03.570
Yeah, so this is called the
level matching condition.

00:28:11.570 --> 00:28:16.390
So, for closed string, you
cannot just act arbitrarily

00:28:16.390 --> 00:28:17.595
alpha and alpha tilde.

00:28:20.660 --> 00:28:24.740
The number of modes,
the total thing

00:28:24.740 --> 00:28:27.730
you act from the left-hand
side, from the alpha

00:28:27.730 --> 00:28:29.650
and those from the
tilde, they have

00:28:29.650 --> 00:28:33.000
to be balanced by
this equation, OK?

00:28:33.000 --> 00:28:34.900
Which is a consequence
that there's

00:28:34.900 --> 00:28:36.460
no spatial point on the string.

00:29:00.270 --> 00:29:04.750
So now, we can also
rewrite those equations

00:29:04.750 --> 00:29:07.130
at the quantum level.

00:29:07.130 --> 00:29:09.230
OK, so those equations,
the quantum level,

00:29:09.230 --> 00:29:16.970
those equation just tells
you that, for those states,

00:29:16.970 --> 00:29:19.780
the P mu are not arbitrary.

00:29:19.780 --> 00:29:21.150
P mu are not arbitrary.

00:29:21.150 --> 00:29:24.680
P mu must satisfy this
kind of constraint.

00:29:24.680 --> 00:29:27.340
P mu must satisfy this
kind of constraint.

00:29:27.340 --> 00:29:29.810
And that's can, in
turn, be integrated

00:29:29.810 --> 00:29:34.210
as the determine the
mass of the string, OK?

00:29:34.210 --> 00:29:35.640
Determine the mass
of the string.

00:29:35.640 --> 00:29:41.320
So the mass of the string,
so then for the open,

00:29:41.320 --> 00:29:46.205
mass of string can be written
1 over alpha prime sum

00:29:46.205 --> 00:29:53.440
over i sum over m
infinity m Nm i.

00:29:53.440 --> 00:29:58.140
Then plus some
zero-point energy.

00:29:58.140 --> 00:30:00.020
So this is just the
frequency of each mode.

00:30:07.620 --> 00:30:11.935
So this is just the
frequency of each mode,

00:30:11.935 --> 00:30:14.290
a frequency of each mode.

00:30:14.290 --> 00:30:16.890
And then, this is the
occupation number.

00:30:16.890 --> 00:30:22.590
So this is a standard
harmonic oscillator result.

00:30:22.590 --> 00:30:26.480
And then, at the quantum
level, of course,

00:30:26.480 --> 00:30:29.780
there's ordinary issue,
because the alpha m

00:30:29.780 --> 00:30:31.800
and m don't commute.

00:30:31.800 --> 00:30:34.200
Because [INAUDIBLE]
creation, [INAUDIBLE].

00:30:34.200 --> 00:30:36.130
So the ordering here matters.

00:30:36.130 --> 00:30:37.650
So the ordering here matters.

00:30:40.330 --> 00:30:47.340
So, in principle, yeah.

00:30:47.340 --> 00:30:47.840
It matters.

00:30:47.840 --> 00:30:50.810
Immediately, you can write it.

00:30:50.810 --> 00:30:56.430
So that will give rise to
this zero-point energy,

00:30:56.430 --> 00:30:58.510
or ordering number.

00:30:58.510 --> 00:31:01.950
So, similarity, for
the closed string,

00:31:01.950 --> 00:31:07.186
then you just have
two sets of modes.

00:31:07.186 --> 00:31:07.685
Similarly.

00:31:16.515 --> 00:31:17.015
that's a0.

00:31:21.580 --> 00:31:33.420
So a0 can be-- so this is the
place this remark is useful,

00:31:33.420 --> 00:31:38.970
because this just come
from-- essentially,

00:31:38.970 --> 00:31:41.250
this part, just come
from, essentially, it's

00:31:41.250 --> 00:31:44.960
just the Hamiltonian of the Xi.

00:31:44.960 --> 00:31:48.880
Just the Xi viewed as a field
theory on the worldsheet.

00:31:48.880 --> 00:31:51.130
And the field theory
of Xi, essentially,

00:31:51.130 --> 00:31:53.620
is a bunch of
harmonic oscillators.

00:31:53.620 --> 00:31:55.710
And for each
harmonic oscillator,

00:31:55.710 --> 00:31:58.510
we do know what is
the ordinary number.

00:31:58.510 --> 00:31:59.420
It's just 1/2.

00:31:59.420 --> 00:32:01.370
Zero-point energy, just 1/2.

00:32:01.370 --> 00:32:03.970
And then you just add
all of them together, OK?

00:32:03.970 --> 00:32:05.880
Just add all of them together.

00:32:05.880 --> 00:32:08.810
So, for example,
for the open string,

00:32:08.810 --> 00:32:11.520
this just become
alpha D minus 2.

00:32:11.520 --> 00:32:14.790
D minus 2 because that D
minus 2 derive directions.

00:32:14.790 --> 00:32:23.040
And then, your sum m equal
to infinity 1/2 omega.

00:32:23.040 --> 00:32:27.210
OK, 1/2 omega, and omega is m.

00:32:27.210 --> 00:32:30.060
Then, I told you this
beautiful trick last time,

00:32:30.060 --> 00:32:36.160
that this should be equal
to D minus 2 divided by 24 1

00:32:36.160 --> 00:32:38.320
over alpha prime,
because this guy,

00:32:38.320 --> 00:32:44.800
the sum over m 1 to infinity
m, give you minus 1/12.

00:32:44.800 --> 00:32:45.990
OK, so this is for the open.

00:32:48.750 --> 00:32:50.890
So, similarly, for
the closed string,

00:32:50.890 --> 00:32:52.860
you can do the similar thing.

00:32:52.860 --> 00:32:55.560
Just differ by sum 2, et cetera.

00:32:55.560 --> 00:33:02.670
So this give you D minus 2
24 4 divided by alpha prime.

00:33:02.670 --> 00:33:04.581
Closed.

00:33:04.581 --> 00:33:05.080
OK?

00:33:09.910 --> 00:33:16.230
So this vacuum energy--
So this can also

00:33:16.230 --> 00:33:19.180
be integrated as a vacuum
energy on the circle.

00:33:19.180 --> 00:33:21.774
Vacuum energy of this quantum
field theory on the circle.

00:33:21.774 --> 00:33:23.190
And, in quantum
field theory, this

00:33:23.190 --> 00:33:25.890
is sometimes called
the Casmir energy.

00:33:25.890 --> 00:33:30.070
And you can check yourself,
that those answers agree

00:33:30.070 --> 00:33:34.140
with the standard expression
for the Casmir energy

00:33:34.140 --> 00:33:35.572
on the circle.

00:33:35.572 --> 00:33:37.530
Yeah, if you choose here,
[INAUDIBLE] property,

00:33:37.530 --> 00:33:40.750
because we have chosen
the sides to be 2 pi.

00:33:46.540 --> 00:33:47.300
Good.

00:33:47.300 --> 00:33:55.640
So this summarized what we did,
summarized what we did so far.

00:33:58.480 --> 00:33:59.560
Any questions on this?

00:34:09.699 --> 00:34:12.150
Good, no more questions?

00:34:12.150 --> 00:34:15.670
Everything is crystal clear?

00:34:15.670 --> 00:34:17.160
I should immediately
have a quiz.

00:34:19.699 --> 00:34:20.489
Yes?

00:34:20.489 --> 00:34:26.210
STUDENT: So the 26 dimensions
comes from making this 0?

00:34:26.210 --> 00:34:27.850
PROFESSOR: No.

00:34:27.850 --> 00:34:31.480
No, if you put the 26
here, this is not 0.

00:34:31.480 --> 00:34:32.505
STUDENT: I'm sorry.

00:34:32.505 --> 00:34:35.516
I'm thinking 1
over-- I apologize.

00:34:35.516 --> 00:34:36.389
Not 0.

00:34:36.389 --> 00:34:39.630
So, you said, the 26
dimensions comes from the fact

00:34:39.630 --> 00:34:42.260
that, somehow, 26
minus 2 over 24 is 1.

00:34:42.260 --> 00:34:44.010
It's like it's numerology,
but it's like--

00:34:44.010 --> 00:34:45.520
PROFESSOR: Yeah.

00:34:45.520 --> 00:34:48.870
Yeah, that's a very
good observation.

00:34:48.870 --> 00:34:52.520
That's exactly the reason
I write in this way.

00:34:52.520 --> 00:34:53.159
Right.

00:34:53.159 --> 00:34:55.699
So let me just make a comment.

00:34:55.699 --> 00:34:56.969
Maybe I make a comment later.

00:35:00.120 --> 00:35:08.060
Anyway, so from here,
from these two expression,

00:35:08.060 --> 00:35:12.570
from this expression
and this expression,

00:35:12.570 --> 00:35:14.760
you see this picture, which
I said at the beginning.

00:35:18.190 --> 00:35:25.660
So each of these describe
a state of a string.

00:35:25.660 --> 00:35:29.930
And the state of a string, the
state of such a string, they

00:35:29.930 --> 00:35:31.900
oscillate in this
particular way.

00:35:35.500 --> 00:35:39.000
Say they have those
oscillation modes,

00:35:39.000 --> 00:35:42.250
and then it moves in
spacetime, besides your center

00:35:42.250 --> 00:35:46.390
of mass momentum, besides
center of mass momentum.

00:35:46.390 --> 00:35:52.600
So such object, you
look at it from afar,

00:35:52.600 --> 00:35:55.070
it's just like a particle, OK?

00:35:55.070 --> 00:35:57.380
It's just like a particle.

00:35:57.380 --> 00:35:59.110
So we have established it.

00:35:59.110 --> 00:36:01.540
So now, let's look
at the spectrum.

00:36:09.960 --> 00:36:20.250
So each state of a string
can be considered a map

00:36:20.250 --> 00:36:21.570
to a spacetime particle.

00:36:30.520 --> 00:36:33.580
So let's now work out.

00:36:33.580 --> 00:36:37.370
And the mass of the
particle can be worked out

00:36:37.370 --> 00:36:39.400
by those formulas.

00:36:39.400 --> 00:36:42.020
OK, so let's now
just work out what

00:36:42.020 --> 00:36:44.840
are the mightiest
particles, because we

00:36:44.840 --> 00:36:50.130
are interested in the mightiest
particles, typically, OK?

00:36:50.130 --> 00:36:53.760
So now, let's start
from the beginning.

00:36:53.760 --> 00:36:55.830
So now, let's start
with open string.

00:36:59.250 --> 00:37:02.610
So the lowest mode, of
course, is just the vacuum.

00:37:05.410 --> 00:37:07.410
P mu.

00:37:07.410 --> 00:37:08.565
OK, there's no oscillators.

00:37:11.120 --> 00:37:19.285
So, for such a mode, Nm i just
equal to 0 for all m and i, OK?

00:37:24.050 --> 00:37:26.439
So this should be just
a spacetime scalar.

00:37:26.439 --> 00:37:28.230
So this should describe
a spacetime scalar,

00:37:28.230 --> 00:37:32.680
because there's no other
quantum number, other

00:37:32.680 --> 00:37:33.980
than the momentum.

00:37:33.980 --> 00:37:36.650
So it should be a
spacetime scalar.

00:37:36.650 --> 00:37:40.230
It should describe
a scalar particle.

00:37:44.490 --> 00:37:46.925
And the mass of the particle,
we can just read from here.

00:37:50.800 --> 00:37:55.680
So the M square equal to minus--
so this is for the open string,

00:37:55.680 --> 00:37:57.584
so we use this formula.

00:37:57.584 --> 00:37:58.500
That's the only thing.

00:37:58.500 --> 00:38:00.390
Because this here is 0.

00:38:00.390 --> 00:38:03.510
So the only thing
come from a 0 term.

00:38:03.510 --> 00:38:11.650
So you're just given by
D minus 2 divided by 24 1

00:38:11.650 --> 00:38:12.688
over alpha prime.

00:38:17.680 --> 00:38:20.350
So why you need
anything in lower case

00:38:20.350 --> 00:38:26.030
is that this guy is smaller
than 0 for D greater than 2.

00:38:26.030 --> 00:38:29.390
Say, for any spacetime
dimension greater than 2,

00:38:29.390 --> 00:38:31.210
you actually find
the mass square,

00:38:31.210 --> 00:38:32.210
[INAUDIBLE] mass square.

00:38:34.890 --> 00:38:38.660
So people actually gave a fancy
name for such kind of particle.

00:38:38.660 --> 00:38:40.830
They call it tachyon.

00:38:40.830 --> 00:38:43.310
And, in the '60s,
actually, people

00:38:43.310 --> 00:38:46.940
designed an experiment to look
for such particles, particles

00:38:46.940 --> 00:38:51.150
of negative mass,
negative mass square.

00:38:51.150 --> 00:39:00.515
Anyway, we are
not going to here.

00:39:05.210 --> 00:39:10.810
Let me just say,
for the following,

00:39:10.810 --> 00:39:15.800
in the theory, if
you see excitations,

00:39:15.800 --> 00:39:18.665
if it's a negative mass
square, typically, it

00:39:18.665 --> 00:39:22.310
tells you that the system have
instability, that you're not

00:39:22.310 --> 00:39:24.580
in the lowest energy state.

00:39:24.580 --> 00:39:27.180
That you are not in
a low-energy state.

00:39:27.180 --> 00:39:32.570
So what this tells you is that
this open string propagate

00:39:32.570 --> 00:39:36.240
in the flat Minkowski
spacetime may not

00:39:36.240 --> 00:39:40.720
be the lowest configuration of
the string, but that's is OK.

00:39:40.720 --> 00:39:44.120
If you're not in the
lowest configuration,

00:39:44.120 --> 00:39:46.899
it's not a big deal, and
it just means you have not

00:39:46.899 --> 00:39:48.190
found the correct ground state.

00:39:48.190 --> 00:39:52.240
It does not mean the
theory is inconsistent, OK?

00:39:52.240 --> 00:39:56.130
And so, even though
this is unpleasant,

00:39:56.130 --> 00:39:58.361
this thing is tolerable, OK?

00:39:58.361 --> 00:39:59.360
This thing is tolerable.

00:40:02.610 --> 00:40:05.430
Any problem with this?

00:40:05.430 --> 00:40:06.350
STUDENT: [INAUDIBLE].

00:40:06.350 --> 00:40:09.470
You already set
[INAUDIBLE] m equal to 0,

00:40:09.470 --> 00:40:11.790
so that is the ground state.

00:40:11.790 --> 00:40:14.145
PROFESSOR: No, this is a
ground state on the worldsheet.

00:40:16.830 --> 00:40:20.630
But in the spacetime, this
goes one into a particle.

00:40:20.630 --> 00:40:24.450
This goes one into
excitation in the spacetime.

00:40:24.450 --> 00:40:26.795
And so this goes one into
excitation in the spacetime,

00:40:26.795 --> 00:40:28.987
with an actual mass square.

00:40:28.987 --> 00:40:31.320
And, typically, if you have
something with a actual mass

00:40:31.320 --> 00:40:32.986
square-- let me just
say one more words,

00:40:32.986 --> 00:40:37.050
here-- then that means you are
sitting on the top of a hill,

00:40:37.050 --> 00:40:40.150
and that's where you have
a actual mass square.

00:40:40.150 --> 00:40:44.420
And so, it means you are in
some kind of unstable state.

00:40:44.420 --> 00:40:48.030
But, of course, you are allowed
to sit on the top of a hill.

00:40:48.030 --> 00:40:49.080
It's not a big deal.

00:40:49.080 --> 00:40:51.802
STUDENT: Why is this
unstable, like a [INAUDIBLE].

00:40:54.530 --> 00:40:56.590
PROFESSOR: On the top of
the hill, is it stable?

00:40:56.590 --> 00:41:00.730
STUDENT: No, why [INAUDIBLE]
means [? uncomplicated. ?]

00:41:00.730 --> 00:41:04.770
PROFESSOR: Oh, if you write
a scalar field theory.

00:41:04.770 --> 00:41:08.360
Yeah, so this goes one into
a scalar field in spacetime,

00:41:08.360 --> 00:41:09.580
now, OK?

00:41:09.580 --> 00:41:14.680
So now, if I write a scalar
field theory in spacetime,

00:41:14.680 --> 00:41:17.510
say, let me call phi, with
a actual mass squared.

00:41:20.200 --> 00:41:25.140
So that means, the potential for
this mass square is like this.

00:41:25.140 --> 00:41:27.670
Then, that means that the
phi wants to increase.

00:41:30.978 --> 00:41:33.220
STUDENT: Is that the
same thing as example 4,

00:41:33.220 --> 00:41:33.500
with spontaneous--

00:41:33.500 --> 00:41:34.750
PROFESSOR: Yeah, that's right.

00:41:34.750 --> 00:41:36.540
Similar to 4, [INAUDIBLE].

00:41:36.540 --> 00:41:39.600
Except, here, we don't
know what is the bottom.

00:41:39.600 --> 00:41:40.930
We are just sitting on the top.

00:41:44.640 --> 00:41:47.480
Anyway, so later, we will
be able to find the way

00:41:47.480 --> 00:41:48.730
to get rid of this.

00:41:48.730 --> 00:41:49.370
So it's OK.

00:41:49.370 --> 00:41:52.870
So you don't need to
worry about this, here.

00:41:52.870 --> 00:41:56.060
So the second mode,
the second lowest mode,

00:41:56.060 --> 00:42:05.390
you just add alpha minus 1 on
this worldsheet ground state,

00:42:05.390 --> 00:42:06.430
OK?

00:42:06.430 --> 00:42:09.750
So now, this thing
is interesting.

00:42:09.750 --> 00:42:19.350
First, this index i, this
index i is a spacetime index.

00:42:19.350 --> 00:42:21.670
It's a spacetime index.

00:42:21.670 --> 00:42:27.750
And so, this actually
means, this transform

00:42:27.750 --> 00:42:33.720
means this state
transform as a vector

00:42:33.720 --> 00:42:41.060
under So D minus 2 the
rotation of the Xi directions.

00:42:41.060 --> 00:42:44.540
And, remember, in
the light-cone gauge,

00:42:44.540 --> 00:42:46.480
the [INAUDIBLE]
symmetry's broken,

00:42:46.480 --> 00:42:49.340
and this is the only
symmetry which is manifest.

00:42:49.340 --> 00:42:53.550
And so, that suggests this state
should be a spacetime vector,

00:42:53.550 --> 00:42:54.050
OK?

00:42:54.050 --> 00:42:55.280
Should be a spacetime vector.

00:43:01.812 --> 00:43:03.145
So now, let's work out its mass.

00:43:06.220 --> 00:43:09.730
So now, m equal to 1.

00:43:09.730 --> 00:43:15.640
So now, m equal to 1, and
so, you just have 1, here.

00:43:15.640 --> 00:43:21.560
And so, then you
just put the 1 here.

00:43:21.560 --> 00:43:32.910
So this is the alpha prime 1
minus D minus 2 divided by 24.

00:43:32.910 --> 00:43:38.000
So this is 26 minus D
divided by alpha prime.

00:43:38.000 --> 00:43:45.330
OK, so now you see the 26
divided by 24 alpha prime.

00:43:48.600 --> 00:43:50.330
So now, you see this
magic number, 26.

00:44:05.470 --> 00:44:15.040
So now, we emphasized before,
in the light-cone gauge,

00:44:15.040 --> 00:44:18.040
even though only
this So D minus 2

00:44:18.040 --> 00:44:20.070
is manifest, because you
break Lorentz symmetry.

00:44:20.070 --> 00:44:21.445
The gauge break
Lorentz symmetry.

00:44:24.120 --> 00:44:28.305
But your theory is still,
secretly, Lorentz symmetric.

00:44:28.305 --> 00:44:30.710
It should still be
Lorentz [INAUDIBLE],

00:44:30.710 --> 00:44:35.410
because the string is
propagate in the flat Minkowski

00:44:35.410 --> 00:44:37.100
spacetime.

00:44:37.100 --> 00:44:40.220
That means, all your
particle spectrum,

00:44:40.220 --> 00:44:43.840
they must fall into
[? representations ?]

00:44:43.840 --> 00:44:47.150
of the four Lorentz group, OK?

00:44:47.150 --> 00:44:49.200
They must fall into the
[? representations ?]

00:44:49.200 --> 00:44:52.080
of the four Lorentz group, even
though the Lorentz symmetry

00:44:52.080 --> 00:44:53.200
is not manifest, here.

00:44:56.060 --> 00:45:00.013
And now, let us recall
an important fact.

00:45:09.190 --> 00:45:14.550
A Lorentz vector,
a vector field.

00:45:17.090 --> 00:45:18.340
Yeah, just a vector particle.

00:45:22.560 --> 00:45:28.610
In D Minkowski
spacetime, D dimension

00:45:28.610 --> 00:45:35.650
of Minkowski spacetime, if
this particle is massive,

00:45:35.650 --> 00:45:43.420
then have D minus 1
independent components,

00:45:43.420 --> 00:45:46.490
so independent modes.

00:45:46.490 --> 00:45:52.640
And if it's massless,
then I have D

00:45:52.640 --> 00:45:56.030
minus 2 independent modes, OK?

00:45:56.030 --> 00:45:58.020
So the situation we
have for many of these

00:45:58.020 --> 00:46:00.600
is the D equal for four,
four-dimensional spacetime.

00:46:00.600 --> 00:46:04.090
So in four-dimensional
spacetime,

00:46:04.090 --> 00:46:07.210
a massless vector is a photon.

00:46:07.210 --> 00:46:10.350
Photons have two polarizations,
have two independent modes.

00:46:13.780 --> 00:46:15.200
But if you have
a massive vector,

00:46:15.200 --> 00:46:19.710
then you actually have three
polarizations, rather than two,

00:46:19.710 --> 00:46:21.450
OK?

00:46:21.450 --> 00:46:24.690
But now, we see a problem.

00:46:24.690 --> 00:46:29.050
Here, we see a vector,
but this factor only

00:46:29.050 --> 00:46:31.190
have D minus 2 components.

00:46:34.750 --> 00:46:39.560
Because i-- because these are
the only independent modes.

00:46:39.560 --> 00:46:55.765
Here, we have a vector which
has only D minus 2 components.

00:46:59.600 --> 00:47:00.780
Independent components.

00:47:00.780 --> 00:47:08.200
OK, so if you compare
with this list--

00:47:08.200 --> 00:47:09.540
because there's nothing else.

00:47:09.540 --> 00:47:13.640
Because these are the only
independent modes, here.

00:47:13.640 --> 00:47:17.050
In the last [INAUDIBLE],
there's nothing else.

00:47:17.050 --> 00:47:22.010
So by compare with our knowledge
of the Lorentz symmetry,

00:47:22.010 --> 00:47:25.820
we conclude the only way this
particle can be mathematically

00:47:25.820 --> 00:47:30.190
consistent, we said, it has
to be a massless particle.

00:47:30.190 --> 00:47:34.750
So that means M square
have to be 0, OK?

00:47:34.750 --> 00:47:40.520
So M square has to be 0, means
that D must be equal to 26.

00:47:45.770 --> 00:47:50.020
And we actually find
massless particles.

00:47:50.020 --> 00:47:52.295
We actually find the photon.

00:47:52.295 --> 00:47:55.740
So we actually find the photon
in the string excitations.

00:47:59.300 --> 00:48:00.090
Yeah, one second.

00:48:00.090 --> 00:48:01.590
Let me just finish this.

00:48:01.590 --> 00:48:08.225
For D not equal to 26,
Lorentz symmetry is lost.

00:48:17.280 --> 00:48:20.030
Lorentz symmetry is lost.

00:48:20.030 --> 00:48:23.410
It's because that
means this particle,

00:48:23.410 --> 00:48:27.590
where M square is not
0, no matter what,

00:48:27.590 --> 00:48:32.150
these states cannot fall
into a [? representation ?]

00:48:32.150 --> 00:48:34.900
of a Lorentz group.

00:48:34.900 --> 00:48:37.200
And so, being said,
Lorentz symmetry

00:48:37.200 --> 00:48:41.610
is not maintained, even
the Lorentz symmetry is

00:48:41.610 --> 00:48:44.230
a symmetry of the
classical action,

00:48:44.230 --> 00:48:47.490
but it's not maintained
at quantum level.

00:48:47.490 --> 00:48:51.880
Somehow, in the
quantization procedure,

00:48:51.880 --> 00:48:56.860
a symmetry which is in
your classical theory,

00:48:56.860 --> 00:48:58.645
it's lost, OK?

00:49:02.040 --> 00:49:06.270
And this tells you that the
quantization is not consistent.

00:49:17.720 --> 00:49:19.940
It's inconsistent.

00:49:19.940 --> 00:49:23.820
Because it means,
whatever it is,

00:49:23.820 --> 00:49:28.900
if you have something propagate
in Minkowski spacetime,

00:49:28.900 --> 00:49:31.000
is has two [INAUDIBLE]
[? representations ?]

00:49:31.000 --> 00:49:33.910
of the Lorentz group.

00:49:33.910 --> 00:49:37.900
That means that, yeah, this
just cannot be the right--

00:49:37.900 --> 00:49:40.146
you have to go back,
to redo your thing.

00:49:40.146 --> 00:49:42.270
This is not propagating in
the Minkowski spacetime.

00:49:45.990 --> 00:49:51.620
So, alternatively-- so
this is a conclusion

00:49:51.620 --> 00:49:55.430
that the D must be
equal to 26, OK?

00:49:55.430 --> 00:49:58.560
So you can reach the same
conclusion the following way.

00:50:01.080 --> 00:50:08.190
So, right now, so the way we did
this, we said we fudge this 0.

00:50:08.190 --> 00:50:11.240
Yeah, we did not
fudge it, but we

00:50:11.240 --> 00:50:16.360
did something to an infinite
sum, and find a valid answer.

00:50:16.360 --> 00:50:19.500
Yeah, we have to do an infinite
sum of positive numbers,

00:50:19.500 --> 00:50:22.810
and then find the active number.

00:50:22.810 --> 00:50:26.140
And then we find, somehow,
there's something.

00:50:26.140 --> 00:50:29.505
D minus 2 and D minus
2 somehow missing 1.

00:50:29.505 --> 00:50:30.005
Anyway.

00:50:33.880 --> 00:50:36.690
Anyway, but this is
actually a deep story.

00:50:36.690 --> 00:50:40.710
It's not, say, just missing
1, or something like that.

00:50:40.710 --> 00:50:43.680
So you can reach
the same conclusion

00:50:43.680 --> 00:50:45.540
by doing the following.

00:50:45.540 --> 00:50:52.480
Say, you put here a 0 as
undetermined quotient.

00:50:52.480 --> 00:50:57.830
And it turns out, the same
0-- yeah, so you can check.

00:50:57.830 --> 00:51:01.400
So here, it tells you that
Lorentz symmetry is lost.

00:51:01.400 --> 00:51:05.120
So you can double check
this conclusion as follows.

00:51:05.120 --> 00:51:08.340
So, remember, I said that
that classical action

00:51:08.340 --> 00:51:10.650
is [INAUDIBLE] on
the Lorentz symmetry,

00:51:10.650 --> 00:51:12.430
and then this conserve
the loss of charge,

00:51:12.430 --> 00:51:15.660
because one to the
Lorentz transformation.

00:51:15.660 --> 00:51:19.035
And those charge,
they become generators

00:51:19.035 --> 00:51:21.010
of Lorentz symmetry
at the quantum level,

00:51:21.010 --> 00:51:23.660
just in quantum field theory.

00:51:23.660 --> 00:51:29.320
And then, by consistency,
those Lorentz--

00:51:29.320 --> 00:51:33.320
conserve the Lorentz charges,
as a quantum operator,

00:51:33.320 --> 00:51:37.930
they must satisfy
Lorentz algebra, OK?

00:51:37.930 --> 00:51:41.820
Then, you can check
with a general D,

00:51:41.820 --> 00:51:43.970
and with a general a0.

00:51:46.850 --> 00:51:51.350
And that Lorentz algebra
is only satisfied

00:51:51.350 --> 00:51:54.880
in the D equal to 26
dimension, and this a0 given

00:51:54.880 --> 00:51:56.330
by these formulas.

00:51:56.330 --> 00:52:00.360
OK, so that will be a
rigorous way to derive it.

00:52:00.360 --> 00:52:02.790
Rigorous way to derive it,
but we are not doing it here,

00:52:02.790 --> 00:52:05.040
because that will take
a little bit of time.

00:52:07.902 --> 00:52:11.718
STUDENT: Does that also involve
the [INAUDIBLE] of the small--

00:52:11.718 --> 00:52:13.625
PROFESSOR: Hm?

00:52:13.625 --> 00:52:19.440
STUDENT: That [INAUDIBLE] of
the [INAUDIBLE], alpha mode?

00:52:19.440 --> 00:52:20.106
PROFESSOR: Yeah.

00:52:20.106 --> 00:52:20.606
Yes.

00:52:20.606 --> 00:52:22.137
Yeah.

00:52:22.137 --> 00:52:23.012
STUDENT: [INAUDIBLE].

00:52:27.154 --> 00:52:28.100
PROFESSOR: Sorry?

00:52:28.100 --> 00:52:28.975
STUDENT: [INAUDIBLE].

00:52:31.294 --> 00:52:32.210
PROFESSOR: No, no, no.

00:52:32.210 --> 00:52:35.835
You're just assuming
some general a0, here.

00:52:35.835 --> 00:52:39.970
You determine this by
requiring that the Lorentz

00:52:39.970 --> 00:52:41.100
algebra is satisfied.

00:52:41.100 --> 00:52:46.672
STUDENT: [INAUDIBLE], in
terms of this [INAUDIBLE].

00:52:46.672 --> 00:52:48.130
PROFESSOR: Yeah,
in terms of alpha.

00:52:48.130 --> 00:52:49.332
That's right.

00:52:49.332 --> 00:52:51.490
STUDENT: Then, [INAUDIBLE].

00:52:51.490 --> 00:52:54.850
PROFESSOR: Oh, those
commutators are fine.

00:52:54.850 --> 00:52:57.637
Those commutators are just
from standard quantization.

00:52:57.637 --> 00:53:02.210
STUDENT: Oh,
there's [INAUDIBLE].

00:53:02.210 --> 00:53:05.660
PROFESSOR: Sorry,
which one over 24?

00:53:05.660 --> 00:53:07.580
No, forget about 1/24.

00:53:07.580 --> 00:53:09.430
There's no 1/24.

00:53:09.430 --> 00:53:12.020
There's no 1/24.

00:53:12.020 --> 00:53:15.850
You just write a 0, here.

00:53:15.850 --> 00:53:17.620
It's undetermined constant.

00:53:17.620 --> 00:53:19.180
And check it by consistency.

00:53:19.180 --> 00:53:21.420
Determined by consistency.

00:53:21.420 --> 00:53:23.240
STUDENT: [INAUDIBLE].

00:53:23.240 --> 00:53:24.380
PROFESSOR: Yeah, no.

00:53:24.380 --> 00:53:25.654
Yes?

00:53:25.654 --> 00:53:31.920
STUDENT: [INAUDIBLE] we'll have
to work [INAUDIBLE] anyway.

00:53:31.920 --> 00:53:36.652
So why do we worry so
much about [INAUDIBLE]?

00:53:36.652 --> 00:53:37.360
PROFESSOR: Sorry?

00:53:37.360 --> 00:53:40.236
STUDENT: I mean, we'll have
to work D minus 4 dimensions

00:53:40.236 --> 00:53:40.820
anyway.

00:53:40.820 --> 00:53:45.401
So then, should we
just begin with, like,

00:53:45.401 --> 00:53:47.750
Minkowski space D
is just not correct?

00:53:47.750 --> 00:53:49.870
PROFESSOR: Right, yeah.

00:53:49.870 --> 00:53:53.810
So will find, actually,
this conclusion

00:53:53.810 --> 00:53:56.730
does not depend on the details.

00:53:56.730 --> 00:54:00.890
It does not depend on details.

00:54:00.890 --> 00:54:06.580
Yeah, so you can generalize
this to more general case.

00:54:06.580 --> 00:54:08.920
Say, curve spacetime, et cetera.

00:54:08.920 --> 00:54:13.460
And the [INAUDIBLE]
spacetime [INAUDIBLE]-- then,

00:54:13.460 --> 00:54:16.780
you'll find the same
conclusion will happen.

00:54:16.780 --> 00:54:18.320
The same conclusion will happen.

00:54:18.320 --> 00:54:20.656
And then, you reduce
to four dimensions,

00:54:20.656 --> 00:54:22.280
then you will find
the fourth dimension

00:54:22.280 --> 00:54:24.710
a massive vector which only
have two polarizations.

00:54:27.230 --> 00:54:28.055
Yes?

00:54:28.055 --> 00:54:29.955
STUDENT: Maybe related
to this question.

00:54:29.955 --> 00:54:32.674
We have no evidence that
Lorentz symmetry holds

00:54:32.674 --> 00:54:33.965
in the compactified dimensions.

00:54:33.965 --> 00:54:36.140
So why do we want
to keep it there?

00:54:36.140 --> 00:54:38.500
PROFESSOR: We just
say, doesn't matter.

00:54:38.500 --> 00:54:41.030
It's only a question
as far as you

00:54:41.030 --> 00:54:43.880
have some uncompact directions.

00:54:43.880 --> 00:54:46.600
As far as you have some
Lorentz directions, then

00:54:46.600 --> 00:54:49.400
this will apply.

00:54:49.400 --> 00:54:50.230
Yes.

00:54:50.230 --> 00:54:52.206
STUDENT: Kind of going back a
little bit to the [INAUDIBLE]

00:54:52.206 --> 00:54:52.706
thing.

00:54:52.706 --> 00:54:56.010
How do we know for a fact that
the string tension causes--

00:54:56.010 --> 00:54:58.453
because, for example, I
think, in QCD strings,

00:54:58.453 --> 00:55:01.374
isn't string tension negative?

00:55:01.374 --> 00:55:02.290
PROFESSOR: Not really.

00:55:02.290 --> 00:55:05.000
How do you define a
negative string tension?

00:55:05.000 --> 00:55:06.170
STUDENT: A negative tension?

00:55:06.170 --> 00:55:07.544
I don't know.

00:55:07.544 --> 00:55:09.492
I just read that,
the QCD string,

00:55:09.492 --> 00:55:11.930
they have negative
string tension.

00:55:11.930 --> 00:55:17.559
PROFESSOR: No, I think, here--
so alpha prime is a scale.

00:55:17.559 --> 00:55:18.475
It's a physical scale.

00:55:21.830 --> 00:55:25.400
Tension is-- it's
defined to be positive.

00:55:25.400 --> 00:55:27.050
Just by definition,
it's positive.

00:55:27.050 --> 00:55:27.550
Yeah.

00:55:31.540 --> 00:55:33.600
Other questions?

00:55:33.600 --> 00:55:40.850
OK, so let me just
say a little bit more

00:55:40.850 --> 00:55:52.600
regarding-- so now we
have found a tachyon

00:55:52.600 --> 00:55:54.530
and a massless
vector, and, also, we

00:55:54.530 --> 00:55:57.960
have fixed the spacetime
dimension to be 26.

00:55:57.960 --> 00:56:00.140
We have fixed spacetime
dimension to 26.

00:56:02.650 --> 00:56:08.657
So now, if you now
fix 0 equal to 26,

00:56:08.657 --> 00:56:10.490
then the higher excitations
are all massive.

00:56:23.530 --> 00:56:28.420
OK, so for the photon,
essentially, it's this guy.

00:56:28.420 --> 00:56:30.300
So this guy's inactive.

00:56:30.300 --> 00:56:32.190
This guy cancel this guy.

00:56:32.190 --> 00:56:35.570
So, when you go to
higher excitation,

00:56:35.570 --> 00:56:38.910
then this guy will
dominate, and the m square

00:56:38.910 --> 00:56:40.470
will be all positive.

00:56:40.470 --> 00:56:44.530
And the scale is controlled by
this 1 over alpha prime, OK?

00:56:44.530 --> 00:56:57.110
So will be all massive with
spacing, given by, say, 1 over

00:56:57.110 --> 00:56:58.280
alpha prime.

00:56:58.280 --> 00:57:02.930
So, we said, spacing m squared
given by 1 over alpha prime.

00:57:02.930 --> 00:57:08.120
For example, the next
level would be-- so alpha

00:57:08.120 --> 00:57:15.090
minus 1 alpha minus 1 some
i some j, or alpha minus 2 i

00:57:15.090 --> 00:57:18.370
acting on 0, P, OK?

00:57:18.370 --> 00:57:21.370
So those things acting
on-- to avoid confusion,

00:57:21.370 --> 00:57:23.470
let me write it more clearly.

00:57:23.470 --> 00:57:29.290
So this acting on 0, P, and
alpha minus 2 [INAUDIBLE]

00:57:29.290 --> 00:57:33.250
on 0, P. OK?

00:57:33.250 --> 00:57:37.390
So this would be like a tensor,
because these have two index.

00:57:37.390 --> 00:57:39.860
And this, again, like a vector.

00:57:39.860 --> 00:57:41.769
Again, like a vector.

00:57:41.769 --> 00:57:44.060
So those would be, obviously,
the mass square of 1 over

00:57:44.060 --> 00:57:44.940
alpha prime.

00:57:48.730 --> 00:57:52.010
And you can check that,
actually, they actually fall

00:57:52.010 --> 00:57:54.340
into the four [INAUDIBLE]
[? representations ?]

00:57:54.340 --> 00:57:57.570
of the Lorentz group, OK?

00:57:57.570 --> 00:57:59.490
For [INAUDIBLE]
[? representation ?]

00:57:59.490 --> 00:58:02.230
of the Lorentz group.

00:58:02.230 --> 00:58:03.040
Yes?

00:58:03.040 --> 00:58:08.679
STUDENT: Where is the one with
only one index [INAUDIBLE]?

00:58:08.679 --> 00:58:09.970
PROFESSOR: Sorry. say it again?

00:58:09.970 --> 00:58:12.650
STUDENT: So the one with one
index, since it's massive,

00:58:12.650 --> 00:58:15.420
it's supposed to have P minus
1 degrees of freedom, right?

00:58:15.420 --> 00:58:21.330
PROFESSOR: They have-- oh.

00:58:21.330 --> 00:58:26.590
What's happening-- that's a
very good question-- so, what's

00:58:26.590 --> 00:58:30.370
happening is that
this should give

00:58:30.370 --> 00:58:33.190
a tensor [? representation, ?]
but this not enough.

00:58:33.190 --> 00:58:37.620
And this acts together to form
a tensor [? representation. ?]

00:58:37.620 --> 00:58:42.266
Yeah, because i only going
from to 2 to D minus 1,

00:58:42.266 --> 00:58:43.641
so you need to
add them together.

00:58:47.460 --> 00:58:47.960
Good.

00:58:50.950 --> 00:58:53.310
So just to summarize
story for the open string,

00:58:53.310 --> 00:58:54.170
we find the tachyon.

00:58:54.170 --> 00:58:57.990
We find the massless vector,
which can be integrated, maybe,

00:58:57.990 --> 00:58:58.866
as a photon.

00:58:58.866 --> 00:59:00.740
And then, you find lots
of massive particles,

00:59:00.740 --> 00:59:07.020
infinite number of
massive particles, OK?

00:59:07.020 --> 00:59:10.698
So any other questions, or
do you want to have a break?

00:59:13.770 --> 00:59:16.031
We are a little bit out of time.

00:59:16.031 --> 00:59:18.030
Yeah, maybe let me give
you three minutes break.

00:59:21.920 --> 00:59:24.790
Yeah, let's have a break, now.

00:59:24.790 --> 00:59:29.730
So, again, the lowest
state is just the 0, P.

00:59:29.730 --> 00:59:31.570
And then, again, all the N is 0.

00:59:37.390 --> 00:59:39.210
All the N are 0.

00:59:39.210 --> 00:59:41.445
So we just read the
answer from here.

00:59:44.015 --> 00:59:47.090
Then M square, for
the closed string,

00:59:47.090 --> 00:59:50.520
you just equal to a0
for the closed string.

00:59:50.520 --> 01:00:00.420
So, here, is now m squared
minus 4 divided by alpha prime.

01:00:00.420 --> 01:00:04.510
D minus 2 divide 24.

01:00:04.510 --> 01:00:10.630
So, again, this is tachyonic
for D greater than 2, OK?

01:00:10.630 --> 01:00:15.570
Tachyonic for D greater than
2, and this is a scalar.

01:00:15.570 --> 01:00:18.785
And this is a scalar, because
there's no other quantum

01:00:18.785 --> 01:00:19.285
number.

01:00:26.289 --> 01:00:28.330
Yeah, so now we are familiar
with these tachyons,

01:00:28.330 --> 01:00:31.732
so we don't need
to worry about it.

01:00:31.732 --> 01:00:32.815
So let's look at the next.

01:00:35.880 --> 01:00:39.750
So, next, naively,
you may say, let's do

01:00:39.750 --> 01:00:48.790
this one as open string case,
but this is not allowed.

01:00:52.380 --> 01:00:54.191
This is not allowed.

01:00:54.191 --> 01:00:54.690
Why?

01:00:54.690 --> 01:00:55.731
STUDENT: [INAUDIBLE].

01:00:55.731 --> 01:00:56.730
PROFESSOR: That's right.

01:00:56.730 --> 01:00:59.510
It does not satisfy
this condition.

01:00:59.510 --> 01:01:02.280
Because, this one, you only have
the left-moving excitations,

01:01:02.280 --> 01:01:04.960
but does not have the
right-moving excitations.

01:01:04.960 --> 01:01:07.270
You are not balanced.

01:01:07.270 --> 01:01:16.460
So you also need to add-- so the
next one will be this guy, OK?

01:01:19.970 --> 01:01:22.370
So, now-- oh, here, have a j.

01:01:26.370 --> 01:01:29.510
Now have a j.

01:01:29.510 --> 01:01:37.500
So this, we'll have m
squared 26 minus D divided

01:01:37.500 --> 01:01:40.750
by 20 alpha 4 alpha prime.

01:01:40.750 --> 01:01:41.940
6 alpha prime.

01:01:50.970 --> 01:01:58.430
Again, now, look for what
representations of the Lorentz

01:01:58.430 --> 01:02:00.585
group will give you this.

01:02:03.230 --> 01:02:08.910
OK, you'll find none, unless
you're in the D equal to 26,

01:02:08.910 --> 01:02:09.410
OK?

01:02:14.060 --> 01:02:21.680
The same story happens,
again, only for D equal to 26,

01:02:21.680 --> 01:02:29.070
fall into the representations
of Lorentz group.

01:02:35.180 --> 01:02:43.710
And reach the m squared,
again, it's massless.

01:02:43.710 --> 01:02:44.820
m square is massless.

01:02:53.550 --> 01:02:56.970
So, it turns out,
actually, this does not

01:02:56.970 --> 01:02:59.759
transform under any reducible
[? representation ?]

01:02:59.759 --> 01:03:00.550
of a Lorentz group.

01:03:03.930 --> 01:03:06.040
It's actually a
reducible, so it can be

01:03:06.040 --> 01:03:10.060
separated into several subsets.

01:03:10.060 --> 01:03:13.190
So this can be further
decomposed to--

01:03:30.760 --> 01:03:37.000
So you can take all
the i and j together,

01:03:37.000 --> 01:03:47.820
take the same i, some i.

01:03:47.820 --> 01:03:52.060
So take this guy,
take these two index

01:03:52.060 --> 01:03:56.460
to be the same, and the
sum of all the directions.

01:03:56.460 --> 01:04:00.285
So this does not transform
under the rotation

01:04:00.285 --> 01:04:02.290
of i's, so this is a scalar.

01:04:05.850 --> 01:04:07.280
But it's a massless scalar.

01:04:11.160 --> 01:04:17.660
And you can also
have the situation--

01:04:17.660 --> 01:04:20.265
so all the state are in
this [? representation ?]

01:04:20.265 --> 01:04:24.100
of both states, so
they are D minus 2.

01:04:24.100 --> 01:04:28.840
So there are D minus 2
times D minus 2 of them.

01:04:28.840 --> 01:04:31.450
D minus 2 times D
minus 2 of them.

01:04:31.450 --> 01:04:34.250
So this, D minus
2, say, one of them

01:04:34.250 --> 01:04:37.960
can be decomposing to a scalar.

01:04:37.960 --> 01:04:43.680
And I can also take the linear
slope [INAUDIBLE] of them

01:04:43.680 --> 01:04:45.145
with a symmetric traceless.

01:04:51.320 --> 01:04:54.840
So the trace part is,
essentially, this scalar.

01:04:54.840 --> 01:04:58.460
And I can also take
a traceless part.

01:04:58.460 --> 01:05:02.849
Traceless e i j, OK,
because the trace part

01:05:02.849 --> 01:05:04.140
is already covered by this one.

01:05:04.140 --> 01:05:07.110
I don't want to repeat.

01:05:07.110 --> 01:05:09.810
And this is precisely
the generalization,

01:05:09.810 --> 01:05:14.390
what we normally call the
spin-2 representation,

01:05:14.390 --> 01:05:17.100
to general dimension.

01:05:17.100 --> 01:05:23.230
OK, so we have found a
massless spin-2 particle.

01:05:23.230 --> 01:05:25.330
So this is a massless
spin-2 particle.

01:05:31.644 --> 01:05:33.060
And then, you can
also, of course,

01:05:33.060 --> 01:05:36.220
take it to be antisymmetric.

01:05:39.100 --> 01:05:40.675
So that's the only
possibility, now.

01:05:46.390 --> 01:05:47.550
bij, antisymmetric bij.

01:05:57.450 --> 01:05:59.200
So these are called
antisymmetric.

01:05:59.200 --> 01:06:03.169
So these will give rise
to an antisymmetric tensor

01:06:03.169 --> 01:06:03.710
in spacetime.

01:06:08.430 --> 01:06:11.600
So this is an object
with a 2 index,

01:06:11.600 --> 01:06:15.014
and the 2 index are
antisymmetric, OK?

01:06:26.340 --> 01:06:35.120
So, similarly, the higher
modes are all massive.

01:06:41.730 --> 01:06:48.090
For example, at the next
level, next mass level,

01:06:48.090 --> 01:06:51.500
m square is equal to 4
divided by alpha prime.

01:07:04.610 --> 01:07:07.237
Any questions on this?

01:07:07.237 --> 01:07:10.231
STUDENT: This antisymmetric
tensor, what is that?

01:07:10.231 --> 01:07:13.730
STUDENT: It's like
a [INAUDIBLE].

01:07:13.730 --> 01:07:15.760
PROFESSOR: It's a
antisymmetric tensor.

01:07:15.760 --> 01:07:17.826
STUDENT: But what's the spin?

01:07:20.627 --> 01:07:22.710
PROFESSOR: Yeah, normally,
in the fourth dimension

01:07:22.710 --> 01:07:25.490
will be something what
we normally call 1 comma

01:07:25.490 --> 01:07:27.620
1, a representation
of the Lorentz group.

01:07:30.952 --> 01:07:34.200
Yeah, it's not 0, 2.

01:07:34.200 --> 01:07:37.261
It's what we normally
call 1 comma 1, yeah.

01:07:37.261 --> 01:07:37.760
Yes.

01:07:37.760 --> 01:07:41.220
STUDENT: It's a form
of bij [INAUDIBLE]?

01:07:41.220 --> 01:07:44.280
PROFESSOR: No, this
is just arbitrary.

01:07:44.280 --> 01:07:49.740
Yeah, so this is your state
space, at this level, right?

01:07:49.740 --> 01:07:53.460
And so, the general state
would be [INAUDIBLE] of them.

01:07:53.460 --> 01:07:57.620
And those states, they
transform separately,

01:07:57.620 --> 01:07:58.720
on the Lorentz symmetry.

01:07:58.720 --> 01:08:00.390
So we separate them.

01:08:00.390 --> 01:08:02.180
And so, for example,
symmetric strings,

01:08:02.180 --> 01:08:04.950
they transform separately on
the Lorentz transformation

01:08:04.950 --> 01:08:07.310
under these guys.

01:08:07.310 --> 01:08:10.500
So this should cause 1 into
different spacetime fields.

01:08:10.500 --> 01:08:15.590
So each of those things
should correspond

01:08:15.590 --> 01:08:17.760
to a spacetime field, OK?

01:08:23.590 --> 01:08:25.955
So, now, let me just
summarize what we have found.

01:08:36.620 --> 01:08:40.810
So we have found--
so let's collect

01:08:40.810 --> 01:08:42.810
the massless
excitations we've found.

01:08:42.810 --> 01:08:44.851
Because, as we will see,
the massless excitations

01:08:44.851 --> 01:08:46.100
are the most important one.

01:08:46.100 --> 01:08:50.069
Let me also mention,
let me just emphasize.

01:08:53.260 --> 01:08:55.850
In physics, it's always
massless particle give you

01:08:55.850 --> 01:08:59.370
something interesting, OK?

01:08:59.370 --> 01:09:02.270
For example, here,
even for D not

01:09:02.270 --> 01:09:07.430
equal to 26, those massive
particles-- as I said,

01:09:07.430 --> 01:09:10.390
maybe I did not emphasize--
even for D not equal to 26,

01:09:10.390 --> 01:09:13.319
these massive particles that
do form into representations

01:09:13.319 --> 01:09:15.722
or Lorentz symmetry,
for any dimension,

01:09:15.722 --> 01:09:18.762
only for those
massless particles, OK?

01:09:18.762 --> 01:09:21.399
It almost seems funny.

01:09:21.399 --> 01:09:24.810
And so, of course, we also
know the massless [INAUDIBLE]

01:09:24.810 --> 01:09:27.200
that give rise to long-range
[INAUDIBLE], et cetera.

01:09:27.200 --> 01:09:31.070
So now, let's say, let's
collect the massless particles.

01:09:35.450 --> 01:09:36.324
Massless excitations.

01:09:41.550 --> 01:09:45.689
So we will write them in
terms of the spacetime fields.

01:09:45.689 --> 01:09:51.479
So, for the open
string, essentially, we

01:09:51.479 --> 01:09:53.640
find the massless vector field.

01:09:56.320 --> 01:09:58.280
So this is our photon.

01:09:58.280 --> 01:10:00.830
So, at the moment,
I put it as a,

01:10:00.830 --> 01:10:04.550
quote, photon, because we only
find the massless particle.

01:10:04.550 --> 01:10:11.180
We don't know whether this is
our own beloved photon, yet.

01:10:11.180 --> 01:10:16.640
And, for the closed string,
then we find a symmetric tensor.

01:10:18.704 --> 01:10:20.620
So this is what we
normally call the graviton.

01:10:24.580 --> 01:10:26.750
So again, quoted.

01:10:26.750 --> 01:10:28.980
We find the massless
spin-2 particle,

01:10:28.980 --> 01:10:30.610
which, if you write
in terms of field,

01:10:30.610 --> 01:10:35.180
would be like a
symmetric tensor,

01:10:35.180 --> 01:10:39.740
Or B mu mu, which is
antisymmetric tensor.

01:10:39.740 --> 01:10:42.220
And now, this mu mu, this
all wrong in all spacetime

01:10:42.220 --> 01:10:43.590
dimensions, OK?

01:10:43.590 --> 01:10:45.280
And then we have a phi.

01:10:45.280 --> 01:10:46.910
Then you have a scalar field.

01:10:46.910 --> 01:10:49.440
So this is just called
antisymmetric tensor,

01:10:49.440 --> 01:10:52.930
and this phi is
called a [INAUDIBLE].

01:10:52.930 --> 01:10:56.630
Phi, which is that scalar field,
is often called a [INAUDIBLE].

01:11:08.470 --> 01:11:12.380
So, so far, even we
call them photon, call

01:11:12.380 --> 01:11:16.340
this one photon,
and the h mu, they

01:11:16.340 --> 01:11:20.586
are not-- to call them photon
and the graviton is, actually,

01:11:20.586 --> 01:11:21.960
a little bit
cheating, because we

01:11:21.960 --> 01:11:24.380
don't know whether they
really behave like a photon

01:11:24.380 --> 01:11:26.860
or like a graviton, OK?

01:11:26.860 --> 01:11:30.020
We just find a massless
spin-1 particle

01:11:30.020 --> 01:11:32.735
and a massless spin-2 particle.

01:11:32.735 --> 01:11:36.430
But, actually, there is
something very general

01:11:36.430 --> 01:11:39.345
one can say, just from
general principle.

01:11:43.990 --> 01:11:46.280
Just from general
principle, one can

01:11:46.280 --> 01:11:58.050
show, based on
Lorentz covariance,

01:11:58.050 --> 01:12:02.790
and, say, an absence of
[INAUDIBLE] physical states,

01:12:02.790 --> 01:12:13.870
et cetera, say
[INAUDIBLE], et cetera,

01:12:13.870 --> 01:12:16.430
just based on those
general principle,

01:12:16.430 --> 01:12:33.620
one can argue that,
at the low energies,

01:12:33.620 --> 01:12:50.090
that the dynamics of any
massless vector field

01:12:50.090 --> 01:12:55.820
should be Maxwell.

01:12:59.270 --> 01:13:05.180
And, for the massless
spin-2 particle,

01:13:05.180 --> 01:13:06.990
must be Einstein gravity, OK?

01:13:15.910 --> 01:13:20.060
So that's why, say,
tomorrow, supposing

01:13:20.060 --> 01:13:24.150
if you mend this theory
yourself, and suppose

01:13:24.150 --> 01:13:28.780
that's a quantum theory,
and that theory just

01:13:28.780 --> 01:13:33.220
happens to have a
massive spin-2 particle,

01:13:33.220 --> 01:13:36.060
then you don't have
to do calculations.

01:13:36.060 --> 01:13:38.830
Then you say, if my
theory is consistent,

01:13:38.830 --> 01:13:43.804
this spin-2 particle must
behave like a graviton, OK?

01:13:43.804 --> 01:13:46.012
STUDENT: Are you saying that
there's no other Lorentz

01:13:46.012 --> 01:13:46.928
invariant [INAUDIBLE]?

01:13:49.079 --> 01:13:51.620
PROFESSOR: Yeah, essentially,
you can show, the low energies,

01:13:51.620 --> 01:13:54.800
it's always just based on
gauge symmetry, et cetera.

01:13:54.800 --> 01:13:57.460
The only thing you can
have is [INAUDIBLE],

01:13:57.460 --> 01:14:00.840
yeah, is Einstein gravity.

01:14:00.840 --> 01:14:03.810
Good.

01:14:03.810 --> 01:14:08.790
And this can be, actually,
checked explicitly.

01:14:08.790 --> 01:14:11.820
So now, let me erase
those things, now.

01:14:11.820 --> 01:14:13.305
Don't need them.

01:14:18.760 --> 01:14:22.410
So this can actually
be checked explicitly.

01:14:22.410 --> 01:14:28.052
So, in string theory, not only
can you find the spectrum,

01:14:28.052 --> 01:14:30.640
you can also compute
the scattering amplitude

01:14:30.640 --> 01:14:33.320
among those particles.

01:14:33.320 --> 01:14:37.290
OK, so I said [INAUDIBLE]
before, essentially, perform

01:14:37.290 --> 01:14:39.420
path integrals with
some initial string

01:14:39.420 --> 01:14:43.550
states going to some final
string states, et cetera, OK?

01:14:43.550 --> 01:14:47.530
So, of course, this
will be too far for us,

01:14:47.530 --> 01:14:48.710
so we will not go into that.

01:14:48.710 --> 01:14:50.120
Let me just tell you the answer.

01:14:53.400 --> 01:14:56.690
So you can confirm this.

01:14:56.690 --> 01:15:00.040
So this expectation,
this confirmed.

01:15:07.130 --> 01:15:26.815
This confirmed by explicit
string theory calculation

01:15:26.815 --> 01:15:32.865
of scattering of
these particles, OK?

01:15:41.040 --> 01:15:44.240
For example, let's
consider-- so we

01:15:44.240 --> 01:15:48.730
have this massless spin-2
particle, which I called h.

01:15:48.730 --> 01:15:52.600
So let's consider, you start
with initial state with two h,

01:15:52.600 --> 01:15:56.220
then scatter to get its own two
final stage, which, again, h.

01:15:56.220 --> 01:15:58.625
So this is, say, graviton
graviton scattering.

01:15:58.625 --> 01:16:03.180
Start with two
graviton, scatter them.

01:16:03.180 --> 01:16:06.520
So, as in string theory, we'll
be, say, at the lowest order,

01:16:06.520 --> 01:16:09.375
we will have a
diagram like this.

01:16:09.375 --> 01:16:11.490
So a nice [INAUDIBLE] order.

01:16:19.600 --> 01:16:20.900
I'm not drawing very well.

01:16:31.940 --> 01:16:35.080
Anyway, I hope this is clear.

01:16:51.080 --> 01:16:54.320
So you start with two
initial string states.

01:16:54.320 --> 01:16:57.040
You scatter into some
final states, OK?

01:16:57.040 --> 01:16:59.660
So this is an obvious string
theory scattering diagram.

01:16:59.660 --> 01:17:02.400
You can, in principle, compute
this using path integral, which

01:17:02.400 --> 01:17:04.590
I outlined earlier.

01:17:04.590 --> 01:17:06.950
Of course, we will not
compute this path integral.

01:17:06.950 --> 01:17:09.009
And so, you see, there
are two vertex here.

01:17:09.009 --> 01:17:10.300
One is proportionate to string.

01:17:10.300 --> 01:17:14.680
You have two string
merging to one string,

01:17:14.680 --> 01:17:18.060
and then you have a string
that's split into two.

01:17:18.060 --> 01:17:20.430
So they're two.

01:17:20.430 --> 01:17:22.605
Remember, each of
[INAUDIBLE] equals

01:17:22.605 --> 01:17:24.360
1 and 2 [INAUDIBLE] string, OK?

01:17:27.000 --> 01:17:30.050
So this will be the
process, in string theory.

01:17:30.050 --> 01:17:32.810
So this will be the
process in string theory.

01:17:32.810 --> 01:17:39.820
So now, If you go
to low energies,

01:17:39.820 --> 01:17:41.520
low energies means
that, if you can

01:17:41.520 --> 01:17:44.170
see that the energy of the
initial and the final particles

01:17:44.170 --> 01:17:46.695
to be much, much smaller
than 1 over alpha prime.

01:17:46.695 --> 01:17:49.430
So, remember, 1
over alpha prime,

01:17:49.430 --> 01:17:53.342
it's the scale which go from
massless to massive particles.

01:17:53.342 --> 01:17:55.550
So, if you can see the very
low-energy process, which

01:17:55.550 --> 01:17:59.060
E is much, much smaller than
1 over alpha prime, then

01:17:59.060 --> 01:18:01.720
the contribution of the
mass-- so, in some sense,

01:18:01.720 --> 01:18:05.210
in the string, in this
intermediate channel, when you

01:18:05.210 --> 01:18:07.140
go from 2 initial
state to 2 final state,

01:18:07.140 --> 01:18:09.760
this intermediate channel,
the infinite number

01:18:09.760 --> 01:18:16.560
of string states can participate
in this intermediate process.

01:18:16.560 --> 01:18:19.935
But, in the process, if your
energy's sufficiently low,

01:18:19.935 --> 01:18:24.790
then, from your common sense,
we can do a calculation.

01:18:24.790 --> 01:18:27.020
And then, the contribution
of the massive state

01:18:27.020 --> 01:18:29.410
becomes, actually,
not important.

01:18:29.410 --> 01:18:32.490
So, essentially,
what is important is

01:18:32.490 --> 01:18:37.030
those massless particles
propagating between them.

01:18:37.030 --> 01:18:41.510
And then, you can show that it's
actually just precisely reduced

01:18:41.510 --> 01:18:44.400
to the Einstein gravity.

01:18:44.400 --> 01:18:46.660
Precisely reduced
to Einstein gravity.

01:18:46.660 --> 01:18:52.650
So more explicitly-- so yeah,
so when you go to low energies,

01:18:52.650 --> 01:19:02.780
then only massless modes
exchange, dominate.

01:19:08.350 --> 01:19:09.410
Terminates.

01:19:09.410 --> 01:19:11.660
And then, you find that the
answer is precisely agreed

01:19:11.660 --> 01:19:15.140
in the [INAUDIBLE] limit.

01:19:15.140 --> 01:19:22.850
Agree with that from
Einstein gravity.

01:19:29.300 --> 01:19:32.025
Because you not
only have graviton,

01:19:32.025 --> 01:19:36.040
you also have this B and phi,
they are also massless modes.

01:19:36.040 --> 01:19:39.420
So this is a slight
generalization

01:19:39.420 --> 01:19:42.140
of Einstein gravity.

01:19:42.140 --> 01:19:48.230
It's Einstein gravity coupled
to such B and the phi, OK?

01:19:52.390 --> 01:19:57.210
So, in fact, you can write
down the so-called low-energy

01:19:57.210 --> 01:19:58.090
effective action.

01:20:01.180 --> 01:20:04.870
So-called low-energy effective
action, and we call LEE, here.

01:20:09.924 --> 01:20:10.840
let me see, like this.

01:20:19.850 --> 01:20:25.190
So phi is this phi, here, and
R is the standard reach scalar

01:20:25.190 --> 01:20:28.810
for the Einstein gravity.

01:20:34.000 --> 01:20:50.720
H is this reindexed
tensor formed out of B.

01:20:50.720 --> 01:20:57.620
So H square just a kinetic
term for B. So, more precisely,

01:20:57.620 --> 01:21:01.330
you can show that the
scattering amplitude

01:21:01.330 --> 01:21:04.205
you obtained from
string theory, then you

01:21:04.205 --> 01:21:11.710
take a low-energy limit, that
answer precisely the same

01:21:11.710 --> 01:21:15.910
as the scattering amplitude you
calculated from this theory,

01:21:15.910 --> 01:21:20.020
say, expanded around
Minkowski spacetime, OK?

01:21:20.020 --> 01:21:22.060
So this is Einstein
gravity coupled

01:21:22.060 --> 01:21:24.660
to [? home ?] scalar field, OK?

01:21:31.451 --> 01:21:31.950
Yes?

01:21:31.950 --> 01:21:33.838
STUDENT: So what
about higher loops

01:21:33.838 --> 01:21:37.620
or higher energies [INAUDIBLE]?

01:21:37.620 --> 01:21:40.440
PROFESSOR: Of course, then,
it will not be the same.

01:21:40.440 --> 01:21:44.510
This is low-energies, OK?

01:21:44.510 --> 01:21:46.080
So let me make one more remark.

01:21:51.220 --> 01:21:53.110
Make one more remark.

01:21:53.110 --> 01:22:01.580
In Einstein gravity,
say, like this.

01:22:01.580 --> 01:22:05.150
So Einstein gravity coupled
to the matter field.

01:22:05.150 --> 01:22:07.900
So, when I say Einstein
gravity, I always

01:22:07.900 --> 01:22:10.620
imply Einstein gravity
plus some other matter

01:22:10.620 --> 01:22:13.040
field which you can add.

01:22:13.040 --> 01:22:16.400
So, in Einstein gravity,
start your scattering process.

01:22:16.400 --> 01:22:19.930
It's that lowest
order, we all know,

01:22:19.930 --> 01:22:23.990
is proportional to G Newton .

01:22:23.990 --> 01:22:28.490
OK, so this is the same G
Newton, the G newton observed.

01:22:31.180 --> 01:22:37.150
Yeah, let me call this, say,
scattering for this is A4.

01:22:37.150 --> 01:22:41.380
And then, the scattering
for the Einstein gravity

01:22:41.380 --> 01:22:43.970
is proportionate to G Newton.

01:22:43.970 --> 01:22:46.750
So one graviton is
changed, and it's

01:22:46.750 --> 01:22:48.530
proportionate to
Newton constant.

01:22:48.530 --> 01:22:52.610
This is an attractive
force between two objects.

01:22:52.610 --> 01:22:57.330
And, if you look at
this string diagram,

01:22:57.330 --> 01:23:00.670
then this string diagram is
proportional to gs square.

01:23:00.670 --> 01:23:04.910
So we conclude that the relation
between the Newton constant

01:23:04.910 --> 01:23:08.370
and the string coupling must
be G Newton proportional to gs

01:23:08.370 --> 01:23:11.722
square, up to some,
say, dimensional numbers

01:23:11.722 --> 01:23:12.805
or some numerical factors.

01:23:16.130 --> 01:23:17.550
So this is very
important relation

01:23:17.550 --> 01:23:18.910
you should always keep in mind.

01:23:24.540 --> 01:23:29.240
So now, here's the
important point.

01:23:29.240 --> 01:23:35.096
[INAUDIBLE] just asked, what
happens at the loop levels?

01:23:35.096 --> 01:23:36.970
So you can compare the
three-level processes,

01:23:36.970 --> 01:23:38.844
because, actually, find
they agree very well.

01:23:38.844 --> 01:23:43.000
We say, what happens
at loop levels?

01:23:43.000 --> 01:23:44.660
So now, let me call
this equation 1.

01:23:48.420 --> 01:24:01.309
So, at loop level, this 1
is notoriously divergent.

01:24:01.309 --> 01:24:03.350
So if you can calculate
some scattering amplitude

01:24:03.350 --> 01:24:09.810
to the loop level, then find
the results are divergent.

01:24:09.810 --> 01:24:11.440
In particular, more
and more divergent

01:24:11.440 --> 01:24:17.280
when you go to more and
more higher loops, OK?

01:24:17.280 --> 01:24:23.092
More divergent at higher orders
in [INAUDIBLE] series, say,

01:24:23.092 --> 01:24:23.925
at the higher loops.

01:24:28.430 --> 01:24:31.670
So that's what we normally mean,
say a theory is non-realizable.

01:24:41.470 --> 01:24:46.020
So this tells you-- so, if
you take this gravity theory--

01:24:46.020 --> 01:24:49.290
so this is just, essentially,
our Einstein gravity

01:24:49.290 --> 01:24:54.200
coupled to some fields-- if
you take the Einstein gravity,

01:24:54.200 --> 01:24:58.090
expand the long
flat space, quantize

01:24:58.090 --> 01:25:03.640
that spin-2 excitations,
then that will fail.

01:25:03.640 --> 01:25:06.520
Because, at certain point, you
don't know what you are doing,

01:25:06.520 --> 01:25:10.350
because you get all divergences,
which you cannot renormalize.

01:25:10.350 --> 01:25:13.800
OK, you can normalize.

01:25:13.800 --> 01:25:18.800
So, of course,
what this tells you

01:25:18.800 --> 01:25:22.090
is that this equation
itself likely

01:25:22.090 --> 01:25:25.710
does not describe
the right UV physics.

01:25:25.710 --> 01:25:27.840
So that's why you see
all these divergences,

01:25:27.840 --> 01:25:31.602
because you maybe lost some
more important physics,

01:25:31.602 --> 01:25:32.810
which you cannot renormalize.

01:25:38.860 --> 01:25:43.150
But now, so this is supposed
to only agree with the string

01:25:43.150 --> 01:25:47.310
theory at low energies, which
the maximum modes are not

01:25:47.310 --> 01:25:48.260
important.

01:25:48.260 --> 01:25:51.820
But, in string theory, there
are this infinite number

01:25:51.820 --> 01:25:54.480
if massive modes, et cetera.

01:25:54.480 --> 01:25:59.940
So you find, in string
theory, if you do similar loop

01:25:59.940 --> 01:26:07.920
calculating string theory,
the string loop diagrams,

01:26:07.920 --> 01:26:10.020
magically, are all UV finite.

01:26:16.000 --> 01:26:19.810
Or UV finite, so there's
no such divergences.

01:26:19.810 --> 01:26:22.990
There's no such divergences.

01:26:22.990 --> 01:26:25.309
So this is the first hint.

01:26:25.309 --> 01:26:27.100
So this was the first
hint of string theory

01:26:27.100 --> 01:26:30.820
as a consistent
theory of gravity.

01:26:30.820 --> 01:26:40.660
And because, at least,
at the pertubative level,

01:26:40.660 --> 01:26:45.930
you can really quantize
massive spin-2 particles,

01:26:45.930 --> 01:26:51.460
and to calculate their physics
in the self-consistant way, OK?

01:26:54.400 --> 01:26:57.450
Any questions on this?

01:26:57.450 --> 01:26:58.269
Yes.

01:26:58.269 --> 01:26:59.727
STUDENT: In that
pertubation, you'd

01:26:59.727 --> 01:27:02.760
have to use all those upper--

01:27:02.760 --> 01:27:04.900
PROFESSOR: Yeah, that's crucial.

01:27:04.900 --> 01:27:07.677
So that's why this kind of
thing is not good enough,

01:27:07.677 --> 01:27:09.760
because that does not have
enough degrees freedom.

01:27:09.760 --> 01:27:11.850
So, in string theory, you have
all these additional degrees

01:27:11.850 --> 01:27:14.030
of freedom that make your
UV structure completely

01:27:14.030 --> 01:27:14.530
difference.

01:27:14.530 --> 01:27:15.027
Yes.

01:27:15.027 --> 01:27:16.777
STUDENT: Then if you
take that [INAUDIBLE]

01:27:16.777 --> 01:27:19.500
and you just [INAUDIBLE]
from above on [INAUDIBLE],

01:27:19.500 --> 01:27:20.991
will it make [INAUDIBLE]?

01:27:24.040 --> 01:27:28.000
PROFESSOR: Yeah,
so this will work,

01:27:28.000 --> 01:27:32.200
as what normally works as a
low-energy effective theory.

01:27:32.200 --> 01:27:34.710
A low-energy [INAUDIBLE].

01:27:34.710 --> 01:27:41.310
When you consistently integrate
out the massive modes.

01:27:41.310 --> 01:27:45.390
And yeah, so, even
at loop level,

01:27:45.390 --> 01:27:48.550
this can capture, indeed, at
low energies, in my loop level,

01:27:48.550 --> 01:27:54.270
this can capture some of
the string theory with that.

01:27:54.270 --> 01:27:59.120
But you have to normalize
properly, et cetera, yeah.

01:27:59.120 --> 01:27:59.620
Yes.

01:27:59.620 --> 01:28:02.540
STUDENT: Does this imply
that, for large objects,

01:28:02.540 --> 01:28:05.320
not things on these very
small scales, that we

01:28:05.320 --> 01:28:07.430
should reproduce the
Einstein field questions?

01:28:07.430 --> 01:28:08.412
PROFESSOR: Yeah.

01:28:08.412 --> 01:28:10.120
STUDENT: This is
sufficient to show that?

01:28:10.120 --> 01:28:13.420
PROFESSOR: Yeah, that's
what it tells you.

01:28:13.420 --> 01:28:18.140
Yeah, for example,
if you measure

01:28:18.140 --> 01:28:23.670
the gravity between you and me,
you won't see the difference.

01:28:23.670 --> 01:28:27.022
Yeah, actually, you
will see a difference.

01:28:27.022 --> 01:28:28.730
So this theory is a
little bit different,

01:28:28.730 --> 01:28:30.430
because of this
massive scalar field.

01:28:34.600 --> 01:28:38.420
So, in ordinary gravity,
the attractive force

01:28:38.420 --> 01:28:41.570
between you and me just
come from graviton.

01:28:41.570 --> 01:28:43.710
But in this theory,
because this scalar is

01:28:43.710 --> 01:28:48.210
massless and, actually, have
additional attractive force.

01:28:48.210 --> 01:28:50.220
And so, this theory is,
actually, not the same.

01:28:53.080 --> 01:28:56.740
So this theory, even though
it's very similar generalization

01:28:56.740 --> 01:28:59.070
of Einstein gravity,
but actually

01:28:59.070 --> 01:29:02.190
give you a different
gravity force.

01:29:02.190 --> 01:29:04.300
So that's why, with
the string theory,

01:29:04.300 --> 01:29:07.530
each going to describe the real
life, somehow the scalar field

01:29:07.530 --> 01:29:09.480
has to become massive.

01:29:09.480 --> 01:29:12.720
Some other mechanism has to
make the scalar field massive.

01:29:12.720 --> 01:29:14.337
STUDENT: I see.

01:29:14.337 --> 01:29:16.670
PROFESSOR: Yeah, of course,
we also don't observe B mu m

01:29:16.670 --> 01:29:18.378
and this also, obviously,
become massive.

01:29:22.670 --> 01:29:26.234
STUDENT: And will we find
out that's the mechanism

01:29:26.234 --> 01:29:27.970
to make [INAUDIBLE]?

01:29:27.970 --> 01:29:29.550
PROFESSOR: Yeah.

01:29:29.550 --> 01:29:33.924
So this is one of the
very important questions,

01:29:33.924 --> 01:29:35.340
since early days
of string theory.

01:29:35.340 --> 01:29:38.500
People have been trying to look
for all kinds of mechanisms

01:29:38.500 --> 01:29:40.794
to do it, et cetera, yeah.

01:29:40.794 --> 01:29:43.650
STUDENT: So there is an
agreed-upon way of doing this?

01:29:43.650 --> 01:29:45.140
Or is it still sort
of [INAUDIBLE]?

01:29:45.140 --> 01:29:47.960
PROFESSOR: It depend, yeah.

01:29:47.960 --> 01:29:50.620
This goes to a Lex point.

01:29:50.620 --> 01:29:55.040
Yeah, wait for my Lex point.

01:29:55.040 --> 01:29:56.029
Yes.

01:29:56.029 --> 01:29:59.736
STUDENT: Can I ask, how do we
know that this effective theory

01:29:59.736 --> 01:30:01.777
couples all the fields as
Einstein's theory does,

01:30:01.777 --> 01:30:05.610
though quandrant derivative?

01:30:05.610 --> 01:30:07.230
PROFESSOR: So what do you mean?

01:30:07.230 --> 01:30:13.370
You can add-- the
coupling between them,

01:30:13.370 --> 01:30:16.940
this and gravity is through
the-- Yeah, here, of course,

01:30:16.940 --> 01:30:19.240
you should use
quadrant derivative.

01:30:19.240 --> 01:30:20.760
Yeah.

01:30:20.760 --> 01:30:24.760
Yeah, here, I did not-- I'm not
very careful in defining this,

01:30:24.760 --> 01:30:29.010
but here, you use quadrant
derivative, et cetera.

01:30:29.010 --> 01:30:32.286
STUDENT: What about coupling
to the open strings,

01:30:32.286 --> 01:30:35.110
or to photons, or to matter?

01:30:35.110 --> 01:30:37.032
PROFESSOR: Would
be the same thing.

01:30:37.032 --> 01:30:38.431
STUDENT: As?

01:30:38.431 --> 01:30:40.680
PROFESSOR: As what you would
expect, that [INAUDIBLE].

01:30:40.680 --> 01:30:48.820
Just saying, it just governed
by general covariance.

01:30:48.820 --> 01:30:51.440
General covariance have
to arise at low energies.

01:30:51.440 --> 01:30:55.130
STUDENT: So that general
principle is effective

01:30:55.130 --> 01:30:57.080
including matter, as well?

01:30:57.080 --> 01:30:58.720
PROFESSOR: Yeah,
then you can check.

01:30:58.720 --> 01:30:59.500
They can check.

01:30:59.500 --> 01:31:01.860
It it's a string theory,
anything you can check

01:31:01.860 --> 01:31:03.571
is consistent with
that principle.

01:31:07.470 --> 01:31:08.030
Good.

01:31:08.030 --> 01:31:09.320
So now, it's another point.

01:31:09.320 --> 01:31:12.490
So let me just
make a side remark

01:31:12.490 --> 01:31:16.170
on the physical consequence
of this scales field.

01:31:16.170 --> 01:31:18.350
So we see that this
scalar field is important,

01:31:18.350 --> 01:31:25.670
because it, actually, can
mediate, say, attractive force.

01:31:25.670 --> 01:31:28.330
But, actually, there's
another very important role

01:31:28.330 --> 01:31:30.890
of this scalar field task.

01:31:36.400 --> 01:31:40.630
It's that, if you look at this
low-energy effective action--

01:31:40.630 --> 01:31:46.360
so let me now
write this G Newton

01:31:46.360 --> 01:31:47.770
as 1 over g string square.

01:31:50.480 --> 01:31:52.770
Then this have the
structure proportional to 1

01:31:52.770 --> 01:32:00.800
over g string square times
exponential minus 2 phi, OK.

01:32:00.800 --> 01:32:04.570
So now, there's a
very important thing.

01:32:04.570 --> 01:32:14.280
So now you see, if phi
behave non-trivially--

01:32:14.280 --> 01:32:18.900
this is, actually, modify this
guy-- it's actually factively--

01:32:18.900 --> 01:32:25.220
yeah, because this is multiplied
by the Einstein scalar--

01:32:25.220 --> 01:32:30.290
so this, effectively, modifies
your Newton constant, OK?

01:32:30.290 --> 01:32:32.086
In fact, they modify
the Newton constant.

01:32:35.780 --> 01:32:40.610
In fact, you can actually
integrate this gs

01:32:40.610 --> 01:32:46.710
as the expectation
value of these phi, OK?

01:32:46.710 --> 01:32:50.700
That's the expectation
value of this phi.

01:32:50.700 --> 01:32:52.580
Yeah, because if you
can change your phi,

01:32:52.580 --> 01:32:54.660
and then change
effective gs, then the gs

01:32:54.660 --> 01:32:58.224
can reintegrate as
expectation value of phi.

01:33:01.540 --> 01:33:06.710
So this is something very
important and very deep,

01:33:06.710 --> 01:33:12.150
because, remember, gs is,
essentially, The only parameter

01:33:12.150 --> 01:33:13.250
in string theory.

01:33:13.250 --> 01:33:15.780
The only dimension is
parameter in string theory,

01:33:15.780 --> 01:33:18.400
which characterize the
strings of the string.

01:33:21.760 --> 01:33:25.950
And now, we see this
constant is not arbitrary.

01:33:25.950 --> 01:33:30.309
It's actually determined
by some dynamic field, OK?

01:33:30.309 --> 01:33:32.100
So that actually means
that, string theory,

01:33:32.100 --> 01:33:33.710
there's no free parameter.

01:33:33.710 --> 01:33:36.310
There's no free
dimensionless parameter.

01:33:36.310 --> 01:33:39.285
Everything, in some sense,
determined by dynamics.

01:33:43.250 --> 01:33:45.420
So this is a very
remarkable feature.

01:33:45.420 --> 01:33:47.870
So this makes people think,
in the early days of string

01:33:47.870 --> 01:33:53.990
theory, that you actually
may be able to derive

01:33:53.990 --> 01:33:55.815
the mass of the electron.

01:33:55.815 --> 01:33:57.940
Because there's no free
parameter in string theory,

01:33:57.940 --> 01:33:59.690
so you should be able
to derive everything

01:33:59.690 --> 01:34:01.920
from first principle.

01:34:01.920 --> 01:34:10.585
Anyway, but this also create
a problem for the issue

01:34:10.585 --> 01:34:12.620
I just mentioned.

01:34:12.620 --> 01:34:17.110
Again, this is a side remark,
but it's a fun remark.

01:34:25.426 --> 01:34:27.050
But string theory,
we mentioned before,

01:34:27.050 --> 01:34:29.830
is a summation of a topology.

01:34:29.830 --> 01:34:32.660
And the topology's
weighted by g string.

01:34:32.660 --> 01:34:34.870
So, if g string
is small, they you

01:34:34.870 --> 01:34:36.730
only need to look at
the lowest topology,

01:34:36.730 --> 01:34:40.340
because the higher topology are
suppressed by higher power of g

01:34:40.340 --> 01:34:41.780
string, OK?

01:34:41.780 --> 01:34:44.260
And, particularly, if g
string become [INAUDIBLE] 1,

01:34:44.260 --> 01:34:48.620
then, to calculate
such a scattering,

01:34:48.620 --> 01:34:52.000
then you need to sum if all
plausible topologies, and then

01:34:52.000 --> 01:34:53.720
that would be unmanageable
problem, which

01:34:53.720 --> 01:34:55.000
we don't know how to do.

01:34:55.000 --> 01:35:00.260
And so, want g string
to be very small,

01:35:00.260 --> 01:35:02.340
so that, actually,
we can actually

01:35:02.340 --> 01:35:05.410
control this theory, OK?

01:35:05.410 --> 01:35:08.760
But now, we have a difficulty.

01:35:08.760 --> 01:35:11.310
But we also said
we want the scalar

01:35:11.310 --> 01:35:16.120
field to to develop a mass.

01:35:16.120 --> 01:35:19.930
We want this scalar
field to develop a mass.

01:35:19.930 --> 01:35:22.020
And, turns out, this
is actually not easy,

01:35:22.020 --> 01:35:24.850
to arrange this scalar
field to develop a mass,

01:35:24.850 --> 01:35:27.570
and, at the same time,
to make this expectation

01:35:27.570 --> 01:35:29.150
to be very small.

01:35:29.150 --> 01:35:34.036
K. And, actually, that turns
out to be a non-trivial problem.

01:35:34.036 --> 01:35:35.577
Yeah, so it's actually
not that easy.

01:35:40.250 --> 01:35:41.805
OK, so my last comment.

01:35:47.120 --> 01:35:49.530
So, earlier, we
said, the tachyon.

01:35:52.850 --> 01:35:54.360
So what we described
so far, these

01:35:54.360 --> 01:35:58.500
are called the bosonic string,
because we only have bosons.

01:35:58.500 --> 01:35:59.745
We only have bosons.

01:35:59.745 --> 01:36:01.570
It's called the bosonic string.

01:36:01.570 --> 01:36:05.074
So this bosonic string can
be generalized to what's

01:36:05.074 --> 01:36:06.240
called a superstring theory.

01:36:13.290 --> 01:36:15.410
Of course, superstring.

01:36:15.410 --> 01:36:17.830
So what superstring
does is the following.

01:36:17.830 --> 01:36:25.130
After you fix this gauge,
again, the superstring

01:36:25.130 --> 01:36:28.760
can be written as some
covariant worldsheet theory

01:36:28.760 --> 01:36:30.890
with some intrinsic metric.

01:36:30.890 --> 01:36:33.110
And, in a superstring,
after you fix this gauge,

01:36:33.110 --> 01:36:36.260
the worldsheet metric
to be Minkowski, then

01:36:36.260 --> 01:36:38.799
the superstring action can
be written as the following.

01:36:41.700 --> 01:36:50.520
You just, essentially, have the
previously free scalar action.

01:36:50.520 --> 01:36:53.300
But then, you add some fermions.

01:36:59.404 --> 01:37:00.320
You add some fermions.

01:37:05.302 --> 01:37:07.260
So these are just some
two-dimensional fermions

01:37:07.260 --> 01:37:09.089
living on the worldsheet.

01:37:13.819 --> 01:37:15.860
Yeah, so these are
two-dimensional spinner fields

01:37:15.860 --> 01:37:17.720
living on the worldsheet.

01:37:17.720 --> 01:37:19.360
So the reason you
can add such a thing,

01:37:19.360 --> 01:37:21.930
because those things don't
have obvious geometric

01:37:21.930 --> 01:37:23.140
interpretation.

01:37:23.140 --> 01:37:26.230
So you can consider them as
describe some internal degrees

01:37:26.230 --> 01:37:28.800
of freedom of the string.

01:37:28.800 --> 01:37:32.800
And so, these guys provide the
spacetime in the interpretation

01:37:32.800 --> 01:37:34.630
of moving the spacetime.

01:37:34.630 --> 01:37:38.190
And those are just some
additional internal degrees

01:37:38.190 --> 01:37:40.670
of freedom on the worldsheet.

01:37:40.670 --> 01:37:44.260
It turns out, by adding
these fermions, actually,

01:37:44.260 --> 01:37:46.540
things change a lot.

01:37:46.540 --> 01:37:49.612
Actually, they do
not change very much.

01:37:49.612 --> 01:37:51.570
It turns out, things does
not change very much,

01:37:51.570 --> 01:37:53.410
because this is a free
fermion series, also

01:37:53.410 --> 01:37:55.380
very easy to quantize.

01:37:55.380 --> 01:37:58.210
And everything we
did before just

01:37:58.210 --> 01:38:01.900
carry over, except you
need to add those fermions

01:38:01.900 --> 01:38:04.134
You need to also quantize
those free fermions.

01:38:09.081 --> 01:38:09.580
Turns

01:38:09.580 --> 01:38:12.360
Out, when you do that,
there are actually

01:38:12.360 --> 01:38:22.200
two different
quantization scheme,

01:38:22.200 --> 01:38:25.880
quantization procedure,
quantization process.

01:38:25.880 --> 01:38:29.520
Two different
quantizations exist.

01:38:29.520 --> 01:38:34.965
So, when you add these
fermions with no tachyon.

01:38:34.965 --> 01:38:36.590
So, you actually can
get rid of tachyon

01:38:36.590 --> 01:38:40.940
by including these fermions, OK?

01:38:40.940 --> 01:38:48.600
So then, the lowest mode is
just your massless mode, OK?

01:38:48.600 --> 01:38:53.000
And the reason that you can
have more than one quantization

01:38:53.000 --> 01:38:55.180
is that, when you
have fermions--

01:38:55.180 --> 01:38:57.980
and this is fermion
defined on the circle.

01:38:57.980 --> 01:38:59.750
So fermion on the
circle, you can define

01:38:59.750 --> 01:39:01.910
to be periodic or antiperiodic.

01:39:01.910 --> 01:39:03.329
So now, you have some choices.

01:39:03.329 --> 01:39:05.120
And, depending on
whether you chose fermion

01:39:05.120 --> 01:39:07.220
to be periodic,
antiperiodic, et cetera,

01:39:07.220 --> 01:39:10.490
and then the story
become different, OK?

01:39:10.490 --> 01:39:14.110
And, sorry, we're
not going into there.

01:39:14.110 --> 01:39:18.515
But, in principle, just waste
enough time, in principle,

01:39:18.515 --> 01:39:20.330
now, you can do it yourself.

01:39:20.330 --> 01:39:23.430
Because, just quantize
free field theory.

01:39:23.430 --> 01:39:26.225
Actually, you still
cannot do it yourself.

01:39:26.225 --> 01:39:28.350
Yeah, this there's still
a little bit more subtlety

01:39:28.350 --> 01:39:32.250
than that, but the
principle is very similar.

01:39:32.250 --> 01:39:36.640
So, in this case, you
get rid of tachyon.

01:39:36.640 --> 01:39:42.960
So these two procedures cause
type IIA and type IIB string.

01:39:42.960 --> 01:39:45.045
So they give you two
type of string theory.

01:39:45.045 --> 01:39:49.217
One is called type IIA,
and one is called type IIB.

01:39:49.217 --> 01:39:51.050
So, also, a lot of the
important difference,

01:39:51.050 --> 01:39:55.830
instead of D equal to 26, now
only requires D equal to 10,

01:39:55.830 --> 01:39:56.330
OK?

01:40:04.420 --> 01:40:08.160
So now, let me just write
down the massive spectrum.

01:40:08.160 --> 01:40:11.030
So now, because
you have fermions,

01:40:11.030 --> 01:40:15.010
now, actually, this
spacetime particle

01:40:15.010 --> 01:40:15.970
can also have fermions.

01:40:15.970 --> 01:40:18.552
Previously, it's all
bosonic particles.

01:40:18.552 --> 01:40:20.260
Now, by adding these
worldsheet fermions,

01:40:20.260 --> 01:40:21.968
it turns out that
these can also generate

01:40:21.968 --> 01:40:23.860
the spacetime fermions.

01:40:23.860 --> 01:40:27.460
It can generate
spacetime fermions.

01:40:27.460 --> 01:40:29.623
So now, the massive spectrum.

01:40:33.780 --> 01:40:35.900
So now, these are
the lowest particles.

01:40:35.900 --> 01:40:39.010
There's no tachyon.

01:40:39.010 --> 01:40:43.060
So these, now, become all
10 dimensional fields.

01:40:43.060 --> 01:40:51.370
So for type IIA, again, you have
this graviton, this B mu mu,

01:40:51.370 --> 01:40:53.580
then you have this
theta [INAUDIBLE].

01:40:53.580 --> 01:40:55.904
And then you have a
lot of active fields

01:40:55.904 --> 01:40:57.070
come from the closed string.

01:40:57.070 --> 01:40:58.665
And so, this is for
the closed string.

01:41:01.990 --> 01:41:04.700
Now, you actually
have a gauge field.

01:41:04.700 --> 01:41:07.990
Also, in the closed string, you
want gauge field and the three

01:41:07.990 --> 01:41:11.150
form tensor fields.

01:41:11.150 --> 01:41:15.026
So you have three
indexes, 40 antisymmetric.

01:41:15.026 --> 01:41:16.790
STUDENT: Is the gauge
field just coming

01:41:16.790 --> 01:41:19.424
from the fermion bilinear?

01:41:19.424 --> 01:41:20.590
PROFESSOR: Yeah, yeah, yeah.

01:41:20.590 --> 01:41:20.930
Right.

01:41:20.930 --> 01:41:21.550
That's right.

01:41:21.550 --> 01:41:26.770
They are related, actually,
to fermion bilinears, yeah.

01:41:26.770 --> 01:41:29.950
So those things are
exactly the same as before,

01:41:29.950 --> 01:41:32.096
and then you get some
additional fields.

01:41:32.096 --> 01:41:34.345
So these are normally called
the Ramond-Ramond fields.

01:41:42.220 --> 01:41:43.564
And then, plus fermions.

01:41:43.564 --> 01:41:44.980
So let me write
down the fermions.

01:41:47.550 --> 01:41:50.230
It turns out that, these
theories, actually was magic.

01:41:50.230 --> 01:41:53.410
So, each of those fields, they
have some super fermionic part

01:41:53.410 --> 01:41:56.650
So it's actually a
supersymmetric theory.

01:41:56.650 --> 01:41:59.930
So then, you also have type IIB.

01:41:59.930 --> 01:42:06.510
And, again, the bosonic string,
you have H mu mu, B mu mu, phi.

01:42:06.510 --> 01:42:11.860
And then, you have the
larger scalar field, chi,

01:42:11.860 --> 01:42:16.180
then the larger antisymmetric
tensor C mu mu 2,

01:42:16.180 --> 01:42:17.800
and it's in the four form field.

01:42:21.950 --> 01:42:25.870
OK, so this, again, is
so-called the Ramond-Ramond.

01:42:25.870 --> 01:42:26.735
Then plus fermions.

01:42:33.490 --> 01:42:36.240
And then, the [INAUDIBLE]
theory of them

01:42:36.240 --> 01:42:40.280
becomes, so-called, type IIA
and type IIB supergravity.

01:42:40.280 --> 01:42:43.190
So these are supersymmetric
theories, and then

01:42:43.190 --> 01:42:47.370
the corresponding generation of
gravity for the supergravity.

01:42:47.370 --> 01:42:50.162
So we do not need
to go in to there.

01:42:54.780 --> 01:42:55.300
Yes.

01:42:55.300 --> 01:42:56.480
STUDENT: So, if you
were going to do

01:42:56.480 --> 01:42:58.646
string theory, for example,
for a the generalization

01:42:58.646 --> 01:43:00.700
on membranes, as you
had mentioned before,

01:43:00.700 --> 01:43:02.470
if you were doing
this on two manifolds,

01:43:02.470 --> 01:43:05.630
would you also have to
include anion contributions?

01:43:05.630 --> 01:43:08.460
PROFESSOR: You may.

01:43:08.460 --> 01:43:10.180
I don't know.

01:43:10.180 --> 01:43:13.630
Nobody have succeeded
in doing this.

01:43:13.630 --> 01:43:14.590
Yeah, you may.

01:43:18.430 --> 01:43:27.350
OK, actually, today I have
to be particularly slow,

01:43:27.350 --> 01:43:31.180
because I have a much
more grand plan for today.

01:43:33.990 --> 01:43:36.680
That also means that
there's definitely--

01:43:36.680 --> 01:43:41.740
you can only do the first
two problems in your pset.

01:43:41.740 --> 01:43:43.620
There's only four
problems in your pset,

01:43:43.620 --> 01:43:46.310
but you can only
do the first two.

01:43:46.310 --> 01:43:49.650
So you want to start,
either, the first two,

01:43:49.650 --> 01:43:52.335
or you want to wait until next
week to do it all together.

01:43:56.040 --> 01:43:59.170
STUDENT: If we do it
all together next week,

01:43:59.170 --> 01:44:02.950
so how about the next homework?

01:44:02.950 --> 01:44:04.364
[INAUDIBLE] one week later?

01:44:04.364 --> 01:44:05.030
PROFESSOR: Yeah.

01:44:09.400 --> 01:44:14.981
OK, yeah, then let's
just defer to one week.

01:44:14.981 --> 01:44:15.730
Yeah, next Friday.

01:44:21.590 --> 01:44:26.270
Yeah, then the next
time-- so these, in sum,

01:44:26.270 --> 01:44:31.250
conclude our basic discussion
of the string theory.

01:44:31.250 --> 01:44:35.520
You have seen most of the
magics of string theory,

01:44:35.520 --> 01:44:36.785
even though it's very fast.

01:44:40.810 --> 01:44:45.070
So, next time, we will
talk about D-branes,

01:44:45.070 --> 01:44:50.560
which is another piece of
magic from string theory.

01:44:50.560 --> 01:44:52.110
Yeah.