WEBVTT

00:00:00.000 --> 00:00:02.415 align:middle line:90%
[SQUEAKING]

00:00:02.415 --> 00:00:03.381 align:middle line:90%
[RUSTLING]

00:00:03.381 --> 00:00:05.313 align:middle line:90%
[CLICKING]

00:00:05.313 --> 00:00:10.388 align:middle line:90%


00:00:10.388 --> 00:00:10.930 align:middle line:90%
HONG LIU: OK.

00:00:10.930 --> 00:00:12.770 align:middle line:90%
Let us start.

00:00:12.770 --> 00:00:17.350 align:middle line:84%
So last time, we started talking
about discrete symmetries.

00:00:17.350 --> 00:00:28.880 align:middle line:90%


00:00:28.880 --> 00:00:30.655 align:middle line:84%
So if you have a
complex scalar theory--

00:00:30.655 --> 00:00:51.550 align:middle line:90%


00:00:51.550 --> 00:00:53.760 align:middle line:84%
so if you have a
complex scalar theory

00:00:53.760 --> 00:00:57.960 align:middle line:84%
and then, in addition
to U1 symmetry,

00:00:57.960 --> 00:01:00.960 align:middle line:84%
classically there are also
discrete symmetries-- firstly

00:01:00.960 --> 00:01:03.750 align:middle line:90%
is parity.

00:01:03.750 --> 00:01:07.682 align:middle line:84%
So we normally denote with
a symbol P and the act

00:01:07.682 --> 00:01:12.945 align:middle line:90%
on x mu by t taken to minus x.

00:01:12.945 --> 00:01:14.172 align:middle line:90%
OK?

00:01:14.172 --> 00:01:17.010 align:middle line:84%
So you just flip all
the spatial direction.

00:01:17.010 --> 00:01:18.540 align:middle line:90%
And you also have time reversal.

00:01:18.540 --> 00:01:23.350 align:middle line:90%


00:01:23.350 --> 00:01:23.890 align:middle line:90%
OK?

00:01:23.890 --> 00:01:24.880 align:middle line:90%
Time reversal.

00:01:24.880 --> 00:01:30.100 align:middle line:84%
And then I normally write
as a script T on x mu,

00:01:30.100 --> 00:01:32.990 align:middle line:90%
then take to the minus t and x.

00:01:32.990 --> 00:01:33.490 align:middle line:90%
OK?

00:01:33.490 --> 00:01:35.260 align:middle line:90%
It does not do anything with x.

00:01:35.260 --> 00:01:43.760 align:middle line:84%
And then we also have a
charge conjugation, which

00:01:43.760 --> 00:01:47.510 align:middle line:90%
takes phi and goes to phi star.

00:01:47.510 --> 00:01:49.640 align:middle line:90%
OK?

00:01:49.640 --> 00:01:54.650 align:middle line:84%
And so of course, our goal is
to understand the symmetries

00:01:54.650 --> 00:01:56.660 align:middle line:90%
at the quantum level.

00:01:56.660 --> 00:01:57.230 align:middle line:90%
OK?

00:01:57.230 --> 00:01:59.840 align:middle line:84%
So at the quantum level--
so you're already--

00:01:59.840 --> 00:02:11.030 align:middle line:84%
then in quantum mechanics,
symmetries are--

00:02:11.030 --> 00:02:16.980 align:middle line:90%
symmetry transformations.

00:02:16.980 --> 00:02:23.265 align:middle line:84%
So they are corresponding to
unitary, OK, transformations.

00:02:23.265 --> 00:02:23.765 align:middle line:90%
OK?

00:02:23.765 --> 00:02:26.760 align:middle line:90%


00:02:26.760 --> 00:02:27.260 align:middle line:90%
OK?

00:02:27.260 --> 00:02:29.135 align:middle line:84%
They corresponding to
unitary transformation.

00:02:29.135 --> 00:02:32.210 align:middle line:84%
We already have done that
for continuous symmetries.

00:02:32.210 --> 00:02:35.270 align:middle line:84%
And the same thing is with
the discrete symmetries.

00:02:35.270 --> 00:02:38.300 align:middle line:84%
And in quantum mechanics,
the implementation

00:02:38.300 --> 00:02:40.460 align:middle line:84%
of the discrete
symmetries are also

00:02:40.460 --> 00:02:42.140 align:middle line:90%
through some unitary operator.

00:02:42.140 --> 00:02:45.260 align:middle line:90%


00:02:45.260 --> 00:02:53.780 align:middle line:84%
And so there's a slight
subtly with the time reversal

00:02:53.780 --> 00:02:57.860 align:middle line:84%
because all the other symmetry
transformations corresponding

00:02:57.860 --> 00:03:02.735 align:middle line:84%
to the unitary transformations,
except for the time reversal.

00:03:02.735 --> 00:03:06.350 align:middle line:84%
That actually is corresponding
to the anti-unitary

00:03:06.350 --> 00:03:07.500 align:middle line:90%
transformations.

00:03:07.500 --> 00:03:11.860 align:middle line:84%
So have you seen that
in quantum mechanics?

00:03:11.860 --> 00:03:12.360 align:middle line:90%
OK.

00:03:12.360 --> 00:03:12.860 align:middle line:90%
OK.

00:03:12.860 --> 00:03:14.670 align:middle line:90%
Let me just review it here.

00:03:14.670 --> 00:03:16.980 align:middle line:90%
So yeah.

00:03:16.980 --> 00:03:19.230 align:middle line:84%
So previously, I
want to say "recall."

00:03:19.230 --> 00:03:21.930 align:middle line:90%
But we don't have to recall it.

00:03:21.930 --> 00:03:28.210 align:middle line:84%
So say-- so let's denote the
UT as the unitary operator.

00:03:28.210 --> 00:03:33.645 align:middle line:84%
So this is the unitary
operator for time reversal.

00:03:33.645 --> 00:03:37.470 align:middle line:90%


00:03:37.470 --> 00:03:39.030 align:middle line:90%
OK?

00:03:39.030 --> 00:03:44.360 align:middle line:84%
And then the claim is
that this has to be--

00:03:44.360 --> 00:03:48.520 align:middle line:90%
this should be anti-linear.

00:03:48.520 --> 00:03:52.760 align:middle line:90%


00:03:52.760 --> 00:03:55.100 align:middle line:84%
And it's not actually
ordinary unitary.

00:03:55.100 --> 00:03:56.540 align:middle line:90%
It's called the anti-unitary.

00:03:56.540 --> 00:04:00.860 align:middle line:90%


00:04:00.860 --> 00:04:03.460 align:middle line:90%
OK?

00:04:03.460 --> 00:04:08.710 align:middle line:84%
And so now let's try to explain
where those things come from.

00:04:08.710 --> 00:04:11.160 align:middle line:84%
So suppose that you
have a system which we

00:04:11.160 --> 00:04:16.430 align:middle line:90%
say is time reversal invariant.

00:04:16.430 --> 00:04:20.769 align:middle line:84%
So by definition,
the symmetry is

00:04:20.769 --> 00:04:26.770 align:middle line:84%
that the corresponding
transformation of the symmetry

00:04:26.770 --> 00:04:28.520 align:middle line:84%
should commute with
your Hamiltonian.

00:04:28.520 --> 00:04:29.020 align:middle line:90%
OK?

00:04:29.020 --> 00:04:31.130 align:middle line:84%
So this is what we
mean by the system

00:04:31.130 --> 00:04:33.560 align:middle line:90%
to be time reversal invariant.

00:04:33.560 --> 00:04:34.060 align:middle line:90%
OK?

00:04:34.060 --> 00:04:39.340 align:middle line:90%


00:04:39.340 --> 00:04:42.100 align:middle line:84%
So here, I'm just talking about
it just in quantum mechanics.

00:04:42.100 --> 00:04:42.600 align:middle line:90%
OK?

00:04:42.600 --> 00:04:47.450 align:middle line:84%
Let's not even worry
about the field theory.

00:04:47.450 --> 00:04:50.720 align:middle line:84%
So now-- so suppose we
have a time reversal

00:04:50.720 --> 00:04:55.550 align:middle line:84%
invariant system, which
the time reversal operator

00:04:55.550 --> 00:04:58.380 align:middle line:90%
commutes with your Hamiltonian.

00:04:58.380 --> 00:05:04.540 align:middle line:84%
And then let's act this
UT on the wave function.

00:05:04.540 --> 00:05:05.040 align:middle line:90%
OK?

00:05:05.040 --> 00:05:09.280 align:middle line:90%
Then you get some psi prime.

00:05:09.280 --> 00:05:19.470 align:middle line:84%
So by time reversal, we
expect physically, OK,

00:05:19.470 --> 00:05:25.640 align:middle line:84%
if psi satisfy the standard
Schrodinger equation,

00:05:25.640 --> 00:05:35.540 align:middle line:84%
and then psi prime should
satisfy the Schrodinger

00:05:35.540 --> 00:05:37.430 align:middle line:90%
equation with t goes to minus t.

00:05:37.430 --> 00:05:39.690 align:middle line:90%
OK?

00:05:39.690 --> 00:05:49.410 align:middle line:84%
It's the same
Schrodinger equation,

00:05:49.410 --> 00:05:51.370 align:middle line:90%
but with t goes to minus t.

00:05:51.370 --> 00:05:51.870 align:middle line:90%
OK?

00:05:51.870 --> 00:05:53.537 align:middle line:84%
So that's what we
mean by time reversal.

00:05:53.537 --> 00:05:55.590 align:middle line:90%
OK?

00:05:55.590 --> 00:05:57.670 align:middle line:90%
So now let's try to do that.

00:05:57.670 --> 00:06:00.360 align:middle line:84%
So let's start
with this equation.

00:06:00.360 --> 00:06:08.950 align:middle line:84%
And now let's just act by UH
from the left on this equation.

00:06:08.950 --> 00:06:09.490 align:middle line:90%
OK?

00:06:09.490 --> 00:06:14.090 align:middle line:90%
So the left-hand side has UH.

00:06:14.090 --> 00:06:14.590 align:middle line:90%
Yeah.

00:06:14.590 --> 00:06:15.910 align:middle line:84%
Yeah, it means
that-- yeah, yeah.

00:06:15.910 --> 00:06:17.827 align:middle line:84%
Before too [? I leave it ?]
do one more step--

00:06:17.827 --> 00:06:20.390 align:middle line:90%


00:06:20.390 --> 00:06:24.598 align:middle line:84%
which means that
the psi prime should

00:06:24.598 --> 00:06:25.890 align:middle line:90%
satisfy the following equation.

00:06:25.890 --> 00:06:31.470 align:middle line:90%


00:06:31.470 --> 00:06:32.250 align:middle line:90%
OK?

00:06:32.250 --> 00:06:33.998 align:middle line:84%
So you just take t
equal to minus t.

00:06:33.998 --> 00:06:35.790 align:middle line:84%
And then you get the
additional minus sign.

00:06:35.790 --> 00:06:36.300 align:middle line:90%
OK?

00:06:36.300 --> 00:06:42.640 align:middle line:84%
So psi prime should
satisfy this equation.

00:06:42.640 --> 00:06:48.210 align:middle line:84%
So now let's see how you
get this equation from here

00:06:48.210 --> 00:06:52.870 align:middle line:84%
by using this relation,
OK, by using this relation.

00:06:52.870 --> 00:06:53.920 align:middle line:90%
So this is simple.

00:06:53.920 --> 00:06:59.760 align:middle line:84%
Let's just add a U on the
left of this equation.

00:06:59.760 --> 00:07:05.840 align:middle line:84%
So let me call this equation
star and this one star-star.

00:07:05.840 --> 00:07:21.870 align:middle line:84%
So if I add UT on star, then on
the right-hand side, UT commute

00:07:21.870 --> 00:07:24.495 align:middle line:90%
with H. OK?

00:07:24.495 --> 00:07:31.830 align:middle line:84%
So on the right-hand side, we
just directly get UT H psi.

00:07:31.830 --> 00:07:34.240 align:middle line:90%
And we can just commute them.

00:07:34.240 --> 00:07:37.530 align:middle line:84%
So this is the
same as H psi UT--

00:07:37.530 --> 00:07:41.340 align:middle line:90%


00:07:41.340 --> 00:07:43.350 align:middle line:84%
UT psi and this is
just psi, psi prime.

00:07:43.350 --> 00:07:45.480 align:middle line:84%
So the right-hand side
just gets psi prime.

00:07:45.480 --> 00:07:48.030 align:middle line:90%
OK?

00:07:48.030 --> 00:07:54.690 align:middle line:84%
And then from the side,
you get UT i partial t psi.

00:07:54.690 --> 00:07:58.430 align:middle line:90%


00:07:58.430 --> 00:08:00.930 align:middle line:90%
OK?

00:08:00.930 --> 00:08:04.780 align:middle line:84%
So in the ordinary situation,
we would have trouble.

00:08:04.780 --> 00:08:10.200 align:middle line:84%
So if you-- because this is
the operator. i is a c-number.

00:08:10.200 --> 00:08:12.370 align:middle line:84%
And the partial
t is some number.

00:08:12.370 --> 00:08:15.450 align:middle line:84%
So you can just directly
pass through this i

00:08:15.450 --> 00:08:16.470 align:middle line:90%
and the partial t.

00:08:16.470 --> 00:08:21.620 align:middle line:84%
But when UT act on psi, then
you get the same equation

00:08:21.620 --> 00:08:24.540 align:middle line:84%
rather than the equation
with a minus sign.

00:08:24.540 --> 00:08:25.040 align:middle line:90%
OK?

00:08:25.040 --> 00:08:26.710 align:middle line:90%
Then you have trouble.

00:08:26.710 --> 00:08:31.765 align:middle line:84%
And the only way to get that
equation is to require--

00:08:31.765 --> 00:08:38.820 align:middle line:90%


00:08:38.820 --> 00:08:47.270 align:middle line:84%
we require-- when UT pass
through i, take it to minus i.

00:08:47.270 --> 00:08:52.900 align:middle line:90%


00:08:52.900 --> 00:08:53.400 align:middle line:90%
OK?

00:08:53.400 --> 00:08:56.620 align:middle line:90%
We impose this condition.

00:08:56.620 --> 00:08:59.360 align:middle line:84%
When UT pass through
i, you get a minus i.

00:08:59.360 --> 00:08:59.860 align:middle line:90%
OK?

00:08:59.860 --> 00:09:07.390 align:middle line:90%


00:09:07.390 --> 00:09:16.890 align:middle line:84%
So this means-- so we can
define anti-linear operator.

00:09:16.890 --> 00:09:21.930 align:middle line:84%
Anti-linear operator is
defined to be the following.

00:09:21.930 --> 00:09:26.800 align:middle line:84%
It means that if you have some
operator a act on some number

00:09:26.800 --> 00:09:32.620 align:middle line:84%
c, when you pass through that
number c, you get c star a.

00:09:32.620 --> 00:09:33.390 align:middle line:90%
OK?

00:09:33.390 --> 00:09:37.950 align:middle line:84%
So anti-linear operator a is
defined to have this property.

00:09:37.950 --> 00:09:40.960 align:middle line:84%
And c is just an
ordinary c number,

00:09:40.960 --> 00:09:42.540 align:middle line:90%
OK, an ordinary c number.

00:09:42.540 --> 00:09:45.420 align:middle line:90%
So you see here UT.

00:09:45.420 --> 00:09:49.560 align:middle line:84%
Is order for UT to be ready to
be a time reversal operator.

00:09:49.560 --> 00:09:53.190 align:middle line:84%
And UT had to be
anti-linear, OK, so that,

00:09:53.190 --> 00:09:56.370 align:middle line:84%
when you pass through i, it
takes i equal to minus i.

00:09:56.370 --> 00:09:59.220 align:middle line:84%
Because the minus i is the
complex conjugate of i.

00:09:59.220 --> 00:09:59.750 align:middle line:90%
OK?

00:09:59.750 --> 00:10:00.250 align:middle line:90%
Good.

00:10:00.250 --> 00:10:01.260 align:middle line:90%
Any questions on this?

00:10:01.260 --> 00:10:04.755 align:middle line:90%


00:10:04.755 --> 00:10:05.255 align:middle line:90%
Good?

00:10:05.255 --> 00:10:09.350 align:middle line:90%


00:10:09.350 --> 00:10:10.850 align:middle line:90%
So yeah.

00:10:10.850 --> 00:10:16.941 align:middle line:90%
So UT should be anti-linear.

00:10:16.941 --> 00:10:21.090 align:middle line:90%


00:10:21.090 --> 00:10:22.680 align:middle line:90%
OK.

00:10:22.680 --> 00:10:26.110 align:middle line:84%
So also, for
anti-linear operator,

00:10:26.110 --> 00:10:28.470 align:middle line:84%
the adjoints should be
defined differently.

00:10:28.470 --> 00:10:29.160 align:middle line:90%
Yes?

00:10:29.160 --> 00:10:31.710 align:middle line:84%
AUDIENCE: [INAUDIBLE]
you started off

00:10:31.710 --> 00:10:35.070 align:middle line:84%
with trying to show
star-star using

00:10:35.070 --> 00:10:37.380 align:middle line:90%
this-- how UT acts on star.

00:10:37.380 --> 00:10:41.545 align:middle line:84%
But then you kind of inserted
this requirement ad hoc

00:10:41.545 --> 00:10:42.420 align:middle line:90%
to satisfy star star.

00:10:42.420 --> 00:10:43.020 align:middle line:90%
HONG LIU: That's right.

00:10:43.020 --> 00:10:43.812 align:middle line:90%
Yeah, that's right.

00:10:43.812 --> 00:10:45.600 align:middle line:90%
AUDIENCE: So I'm a bit confused.

00:10:45.600 --> 00:10:49.700 align:middle line:84%
Then how do you come up-- why
the star-star [INAUDIBLE]??

00:10:49.700 --> 00:10:51.640 align:middle line:84%
HONG LIU: No, the
star-star is the definition

00:10:51.640 --> 00:10:52.450 align:middle line:90%
of time reversal.

00:10:52.450 --> 00:10:55.090 align:middle line:90%
Right?

00:10:55.090 --> 00:10:58.510 align:middle line:84%
Star-star is what physically
we mean by time reversal.

00:10:58.510 --> 00:11:01.120 align:middle line:84%
If you want something to
implement the time reversal,

00:11:01.120 --> 00:11:03.530 align:middle line:90%
it has to achieve star-star.

00:11:03.530 --> 00:11:04.030 align:middle line:90%
Yeah.

00:11:04.030 --> 00:11:07.080 align:middle line:90%


00:11:07.080 --> 00:11:09.720 align:middle line:90%
Other questions?

00:11:09.720 --> 00:11:10.220 align:middle line:90%
OK.

00:11:10.220 --> 00:11:10.720 align:middle line:90%
Good.

00:11:10.720 --> 00:11:13.990 align:middle line:84%
And then we derived
that in order

00:11:13.990 --> 00:11:19.030 align:middle line:84%
to achieve the time reversal,
then UT has to be anti-linear.

00:11:19.030 --> 00:11:22.390 align:middle line:84%
And then because
ant-linear operator

00:11:22.390 --> 00:11:26.020 align:middle line:84%
has this weird
property, the adjoint

00:11:26.020 --> 00:11:28.150 align:middle line:84%
of anti-linear
operator-- the definition

00:11:28.150 --> 00:11:29.450 align:middle line:90%
should also be changed.

00:11:29.450 --> 00:11:29.950 align:middle line:90%
OK?

00:11:29.950 --> 00:11:45.620 align:middle line:84%
So the adjoint of
anti-linear operator A

00:11:45.620 --> 00:11:52.590 align:middle line:84%
is defined to be, say, if
you have psi, A dagger chi.

00:11:52.590 --> 00:11:53.580 align:middle line:90%
OK?

00:11:53.580 --> 00:11:58.140 align:middle line:84%
If A is an ordinary
operator, and then

00:11:58.140 --> 00:12:00.060 align:middle line:84%
the definition of the
adjoint is that this

00:12:00.060 --> 00:12:02.320 align:middle line:90%
is equal to A psi chi.

00:12:02.320 --> 00:12:02.820 align:middle line:90%
OK?

00:12:02.820 --> 00:12:04.900 align:middle line:84%
So this is a
standard definition.

00:12:04.900 --> 00:12:05.531 align:middle line:90%
Yes?

00:12:05.531 --> 00:12:06.156 align:middle line:90%
AUDIENCE: Yeah.

00:12:06.156 --> 00:12:08.343 align:middle line:84%
So I was just
confused about how you

00:12:08.343 --> 00:12:13.390 align:middle line:84%
deal with making multiplication,
scalar multiplication

00:12:13.390 --> 00:12:18.462 align:middle line:84%
noncommutative and I was
wondering why you couldn't do--

00:12:18.462 --> 00:12:23.090 align:middle line:84%
wouldn't it make sense to just
say that UT and H anti commute?

00:12:23.090 --> 00:12:27.260 align:middle line:90%
HONG LIU: UT and H anti-commute?

00:12:27.260 --> 00:12:30.700 align:middle line:90%


00:12:30.700 --> 00:12:33.370 align:middle line:90%
So by taking the--

00:12:33.370 --> 00:12:36.300 align:middle line:90%
yeah, you can also do that.

00:12:36.300 --> 00:12:38.050 align:middle line:90%
It just does not work as well.

00:12:38.050 --> 00:12:38.980 align:middle line:90%
Yeah.

00:12:38.980 --> 00:12:40.300 align:middle line:90%
You see whatever it works.

00:12:40.300 --> 00:12:41.440 align:middle line:90%
Yeah, yeah.

00:12:41.440 --> 00:12:45.160 align:middle line:84%
It's much easier just to make
the UT to be anti-linear.

00:12:45.160 --> 00:12:45.790 align:middle line:90%
Yeah.

00:12:45.790 --> 00:12:46.390 align:middle line:90%
Yeah.

00:12:46.390 --> 00:12:47.200 align:middle line:90%
Yeah, yeah.

00:12:47.200 --> 00:12:49.390 align:middle line:84%
For this particular
equation, it can work.

00:12:49.390 --> 00:12:52.120 align:middle line:84%
But then you have to be against
the principle that somehow

00:12:52.120 --> 00:12:54.250 align:middle line:90%
H commute with the symmetry.

00:12:54.250 --> 00:12:55.030 align:middle line:90%
Yeah, yeah.

00:12:55.030 --> 00:12:57.598 align:middle line:84%
That is considered to be
more sacred a principle.

00:12:57.598 --> 00:12:58.098 align:middle line:90%
Yeah.

00:12:58.098 --> 00:13:01.770 align:middle line:90%


00:13:01.770 --> 00:13:03.660 align:middle line:90%
Other questions?

00:13:03.660 --> 00:13:07.940 align:middle line:90%
And indeed-- yeah, yeah.

00:13:07.940 --> 00:13:10.540 align:middle line:90%
Right.

00:13:10.540 --> 00:13:11.040 align:middle line:90%
OK.

00:13:11.040 --> 00:13:12.110 align:middle line:90%
Good.

00:13:12.110 --> 00:13:14.640 align:middle line:84%
So the way we define the adjoint
should also be different.

00:13:14.640 --> 00:13:16.620 align:middle line:90%
So this is standard definition.

00:13:16.620 --> 00:13:20.860 align:middle line:84%
So for anti-linear, you
have this definition.

00:13:20.860 --> 00:13:23.080 align:middle line:84%
And then you put
the additional star.

00:13:23.080 --> 00:13:24.890 align:middle line:90%
OK?

00:13:24.890 --> 00:13:26.380 align:middle line:90%
You put an additional star.

00:13:26.380 --> 00:13:28.130 align:middle line:84%
And the reason you put
the additional star

00:13:28.130 --> 00:13:32.420 align:middle line:84%
is precisely due
to this property.

00:13:32.420 --> 00:13:33.620 align:middle line:90%
OK?

00:13:33.620 --> 00:13:38.330 align:middle line:84%
Because remember, if you
put a complex conjugate,

00:13:38.330 --> 00:13:40.022 align:middle line:84%
put the c number
here, and then you

00:13:40.022 --> 00:13:41.480 align:middle line:84%
should be able to
take the c number

00:13:41.480 --> 00:13:45.320 align:middle line:90%
outside this kind of overlap.

00:13:45.320 --> 00:13:47.570 align:middle line:84%
But in order to be
compatible with the property

00:13:47.570 --> 00:13:50.090 align:middle line:84%
of the anti-linear
operator, then you

00:13:50.090 --> 00:13:54.560 align:middle line:84%
have to define the adjoint
to be this way so that you

00:13:54.560 --> 00:13:55.700 align:middle line:90%
have a consistent story.

00:13:55.700 --> 00:13:57.580 align:middle line:90%
OK?

00:13:57.580 --> 00:14:00.730 align:middle line:84%
Just I will leave it to
you yourself as an exercise

00:14:00.730 --> 00:14:03.520 align:middle line:84%
to show this is actually
the right definition.

00:14:03.520 --> 00:14:04.540 align:middle line:90%
OK?

00:14:04.540 --> 00:14:13.510 align:middle line:84%
And then we can define
the anti-unitary

00:14:13.510 --> 00:14:16.510 align:middle line:84%
as that, if you have an
operator with anti-unitary,

00:14:16.510 --> 00:14:24.910 align:middle line:90%
it means U phi, U chi.

00:14:24.910 --> 00:14:27.430 align:middle line:84%
So standard unitary
condition is just

00:14:27.430 --> 00:14:30.640 align:middle line:84%
that this is equal to chi--
and the psi and the chi.

00:14:30.640 --> 00:14:32.000 align:middle line:90%
OK?

00:14:32.000 --> 00:14:36.910 align:middle line:84%
And now for the anti-unitary,
again, you just put the star.

00:14:36.910 --> 00:14:37.410 align:middle line:90%
OK?

00:14:37.410 --> 00:14:38.327 align:middle line:90%
You just put the star.

00:14:38.327 --> 00:14:43.780 align:middle line:90%


00:14:43.780 --> 00:14:44.280 align:middle line:90%
OK?

00:14:44.280 --> 00:14:51.030 align:middle line:84%
So the claim is that the UT,
the time reversal operator,

00:14:51.030 --> 00:14:55.920 align:middle line:84%
should be an anti-unitary
operator, uh, in order

00:14:55.920 --> 00:14:59.080 align:middle line:84%
to be a symmetry, in
order to be a symmetry.

00:14:59.080 --> 00:14:59.580 align:middle line:90%
OK?

00:14:59.580 --> 00:15:05.520 align:middle line:84%
So this is just a brief
review of time reversal

00:15:05.520 --> 00:15:06.750 align:middle line:90%
in quantum mechanics.

00:15:06.750 --> 00:15:12.760 align:middle line:84%
So now let's look at the quantum
version of those symmetries

00:15:12.760 --> 00:15:13.760 align:middle line:90%
in quantum field theory.

00:15:13.760 --> 00:15:14.260 align:middle line:90%
OK?

00:15:14.260 --> 00:15:17.870 align:middle line:90%
So first, let's do the warm-up.

00:15:17.870 --> 00:15:22.220 align:middle line:84%
Before let's do it
for the Dirac theory,

00:15:22.220 --> 00:15:31.080 align:middle line:84%
let's warm up for complex scalar
fields, which is much simpler.

00:15:31.080 --> 00:15:31.580 align:middle line:90%
OK?

00:15:31.580 --> 00:15:34.140 align:middle line:90%


00:15:34.140 --> 00:15:38.490 align:middle line:84%
So in this theory, the
action is invariant, we said,

00:15:38.490 --> 00:15:39.490 align:middle line:90%
under those symmetries.

00:15:39.490 --> 00:15:39.990 align:middle line:90%
OK?

00:15:39.990 --> 00:15:43.880 align:middle line:84%
If we just flip the
spatial direction,

00:15:43.880 --> 00:15:47.218 align:middle line:84%
clearly that Lagrangian
is invariant.

00:15:47.218 --> 00:15:49.760 align:middle line:84%
If we leave the time direction
because the time derivative is

00:15:49.760 --> 00:15:51.260 align:middle line:90%
quadratic, it's also invariant.

00:15:51.260 --> 00:15:55.520 align:middle line:84%
And again, if you
change phi to phi star,

00:15:55.520 --> 00:15:56.810 align:middle line:90%
the Lagrangian is invariant.

00:15:56.810 --> 00:15:58.250 align:middle line:90%
OK.

00:15:58.250 --> 00:16:01.910 align:middle line:84%
So quantum mechanically,
those symmetries

00:16:01.910 --> 00:16:10.190 align:middle line:84%
should be implemented by
some unitary operator.

00:16:10.190 --> 00:16:10.760 align:middle line:90%
OK.

00:16:10.760 --> 00:16:17.630 align:middle line:84%
So now the parity
transformation on the field

00:16:17.630 --> 00:16:20.840 align:middle line:84%
to the corresponding to
you start with phi x.

00:16:20.840 --> 00:16:25.060 align:middle line:90%
And then you go to phi prime x.

00:16:25.060 --> 00:16:30.720 align:middle line:84%
So phi prime x should
be equal to phi px.

00:16:30.720 --> 00:16:31.320 align:middle line:90%
OK?

00:16:31.320 --> 00:16:33.900 align:middle line:90%
This means that the--

00:16:33.900 --> 00:16:38.030 align:middle line:84%
and then just put-- yeah, you
can also put the phase here.

00:16:38.030 --> 00:16:39.170 align:middle line:90%
OK?

00:16:39.170 --> 00:16:41.450 align:middle line:84%
So you can show that this
transformation certainly is

00:16:41.450 --> 00:16:43.520 align:middle line:90%
a symmetry of this Lagrangian.

00:16:43.520 --> 00:16:44.900 align:middle line:90%
OK?

00:16:44.900 --> 00:16:47.600 align:middle line:84%
Yeah, because as I said,
if you just flip it,

00:16:47.600 --> 00:16:49.127 align:middle line:90%
it does not change anything.

00:16:49.127 --> 00:16:50.960 align:middle line:84%
But you can also put
an arbitrary phase here

00:16:50.960 --> 00:16:52.080 align:middle line:90%
in principle.

00:16:52.080 --> 00:16:52.580 align:middle line:90%
OK?

00:16:52.580 --> 00:16:54.622 align:middle line:84%
You can in principle put
an arbitrary phase here.

00:16:54.622 --> 00:16:58.850 align:middle line:84%
Because they all only depend
on the phi and phi dagger.

00:16:58.850 --> 00:16:59.870 align:middle line:90%
OK?

00:16:59.870 --> 00:17:05.359 align:middle line:90%
And so this is p.

00:17:05.359 --> 00:17:07.880 align:middle line:90%
And then this is the t.

00:17:07.880 --> 00:17:10.400 align:middle line:84%
Then corresponding
to phi x, then it

00:17:10.400 --> 00:17:17.220 align:middle line:84%
goes to some phi prime x
equal to, again, some phase--

00:17:17.220 --> 00:17:20.220 align:middle line:90%
so phi tx.

00:17:20.220 --> 00:17:22.970 align:middle line:90%
OK?

00:17:22.970 --> 00:17:31.260 align:middle line:84%
And then the charge conjugation
normally is called C.

00:17:31.260 --> 00:17:33.930 align:middle line:84%
So parity is normally
called the P. OK?

00:17:33.930 --> 00:17:37.215 align:middle line:84%
So here, it's
corresponding to phi

00:17:37.215 --> 00:17:47.880 align:middle line:84%
x, goes to phi prime x
equal to eta P phi star x.

00:17:47.880 --> 00:17:48.380 align:middle line:90%
OK?

00:17:48.380 --> 00:17:53.230 align:middle line:90%


00:17:53.230 --> 00:17:56.630 align:middle line:84%
So in each case, such
kind of transformation

00:17:56.630 --> 00:17:59.013 align:middle line:84%
should be generated by
some unitary operator.

00:17:59.013 --> 00:18:00.680 align:middle line:84%
So for example, this
should be generated

00:18:00.680 --> 00:18:05.165 align:middle line:90%
by UP phi x UP dagger.

00:18:05.165 --> 00:18:07.770 align:middle line:90%


00:18:07.770 --> 00:18:08.310 align:middle line:90%
OK?

00:18:08.310 --> 00:18:11.670 align:middle line:84%
And then similarity for
this one, we call the UT--

00:18:11.670 --> 00:18:12.540 align:middle line:90%
UT dagger.

00:18:12.540 --> 00:18:15.300 align:middle line:84%
And here, you see it
in UC dagger similarly.

00:18:15.300 --> 00:18:16.710 align:middle line:90%
OK?

00:18:16.710 --> 00:18:18.210 align:middle line:84%
So at the quantum
level, this should

00:18:18.210 --> 00:18:21.300 align:middle line:84%
be generated by some action
of some unitary operator.

00:18:21.300 --> 00:18:22.020 align:middle line:90%
Yes?

00:18:22.020 --> 00:18:24.520 align:middle line:84%
AUDIENCE: Do you always have
this U1 symmetry or do you have

00:18:24.520 --> 00:18:27.550 align:middle line:84%
-- is there an interacting
theory where you have one power

00:18:27.550 --> 00:18:28.430 align:middle line:90%
of phi.

00:18:28.430 --> 00:18:29.273 align:middle line:90%
But we don't have--

00:18:29.273 --> 00:18:29.940 align:middle line:90%
HONG LIU: Sorry?

00:18:29.940 --> 00:18:32.010 align:middle line:84%
AUDIENCE: So you can
add an arbitrary phase?

00:18:32.010 --> 00:18:33.930 align:middle line:90%
HONG LIU: No, no.

00:18:33.930 --> 00:18:35.550 align:middle line:84%
I'm going to talk
about the phase.

00:18:35.550 --> 00:18:35.730 align:middle line:90%
OK?

00:18:35.730 --> 00:18:37.230 align:middle line:84%
I'm going to talk
about the phase.

00:18:37.230 --> 00:18:42.270 align:middle line:84%
So if you are even looking
at that Lagrangian,

00:18:42.270 --> 00:18:45.060 align:middle line:84%
and then you see you put the
phase, this is invariant.

00:18:45.060 --> 00:18:46.530 align:middle line:90%
OK.

00:18:46.530 --> 00:18:52.590 align:middle line:84%
But now the question is, can
those phases be arbitrary?

00:18:52.590 --> 00:18:53.090 align:middle line:90%
OK?

00:18:53.090 --> 00:18:55.520 align:middle line:84%
The question is then, can
those phases be arbitrary?

00:18:55.520 --> 00:19:02.160 align:middle line:90%


00:19:02.160 --> 00:19:05.880 align:middle line:84%
So in the case of this
complex scalar, actually

00:19:05.880 --> 00:19:07.440 align:middle line:90%
it should not be.

00:19:07.440 --> 00:19:10.870 align:middle line:84%
So let's use the
example of the parity.

00:19:10.870 --> 00:19:16.040 align:middle line:84%
So we know that when you
do parity twice, OK--

00:19:16.040 --> 00:19:18.920 align:middle line:84%
if you do parity twice,
you should go back

00:19:18.920 --> 00:19:20.270 align:middle line:90%
to your original.

00:19:20.270 --> 00:19:21.770 align:middle line:84%
Yeah, your system
does not change.

00:19:21.770 --> 00:19:22.270 align:middle line:90%
OK?

00:19:22.270 --> 00:19:25.200 align:middle line:84%
If you just do reflections while
your system does not change.

00:19:25.200 --> 00:19:30.960 align:middle line:84%
So that means that, when you
do this twice, you should--

00:19:30.960 --> 00:19:32.640 align:middle line:84%
you haven't changed
anything at all.

00:19:32.640 --> 00:19:33.330 align:middle line:90%
OK?

00:19:33.330 --> 00:19:37.202 align:middle line:84%
So when we do it twice, the
phi should come back to itself.

00:19:37.202 --> 00:19:39.800 align:middle line:84%
And indeed, this P-squared,
if you do it twice,

00:19:39.800 --> 00:19:40.980 align:middle line:90%
comes back to itself.

00:19:40.980 --> 00:19:43.450 align:middle line:84%
But then this phase should
also come back to itself.

00:19:43.450 --> 00:19:43.950 align:middle line:90%
OK?

00:19:43.950 --> 00:19:47.480 align:middle line:90%
So that means that the--

00:19:47.480 --> 00:19:51.480 align:middle line:84%
eta P-squared should
be equal to 1.

00:19:51.480 --> 00:19:51.980 align:middle line:90%
OK?

00:19:51.980 --> 00:19:54.980 align:middle line:90%


00:19:54.980 --> 00:19:58.310 align:middle line:84%
So if you wanted the symmetry
to come back to itself

00:19:58.310 --> 00:20:03.460 align:middle line:84%
and then eta a P, then it
can only be plus-minus 1.

00:20:03.460 --> 00:20:04.970 align:middle line:90%
But it actually can be minus 1.

00:20:04.970 --> 00:20:05.470 align:middle line:90%
OK?

00:20:05.470 --> 00:20:06.720 align:middle line:90%
It does not have to be plus 1.

00:20:06.720 --> 00:20:13.930 align:middle line:84%
So recall when eta P is equal
to plus 1, we call it a scalar.

00:20:13.930 --> 00:20:18.807 align:middle line:84%
When eta P is a minus 1,
we call it a pseudoscalar.

00:20:18.807 --> 00:20:24.280 align:middle line:90%


00:20:24.280 --> 00:20:26.050 align:middle line:90%
We call it pseudoscalar.

00:20:26.050 --> 00:20:28.020 align:middle line:90%
OK?

00:20:28.020 --> 00:20:40.510 align:middle line:84%
And a similar thing can be
said about the eta t and eta c.

00:20:40.510 --> 00:20:41.010 align:middle line:90%
OK?

00:20:41.010 --> 00:20:55.510 align:middle line:90%


00:20:55.510 --> 00:21:00.220 align:middle line:84%
So by definition, this
unitary operator acts on phi.

00:21:00.220 --> 00:21:03.720 align:middle line:90%
It should act as this.

00:21:03.720 --> 00:21:05.980 align:middle line:90%
So now we can--

00:21:05.980 --> 00:21:11.340 align:middle line:84%
and now remember
the phi is expressed

00:21:11.340 --> 00:21:18.000 align:middle line:84%
in terms of the creation
and annihilation operators.

00:21:18.000 --> 00:21:32.800 align:middle line:90%


00:21:32.800 --> 00:21:35.410 align:middle line:90%
OK?

00:21:35.410 --> 00:21:41.248 align:middle line:90%
So by requiring-- yes?

00:21:41.248 --> 00:21:43.105 align:middle line:84%
AUDIENCE: [INAUDIBLE]
scalar or pseudoscalar

00:21:43.105 --> 00:21:45.515 align:middle line:84%
is it true that if
eta P is equal to one

00:21:45.515 --> 00:21:47.602 align:middle line:90%
then eta T and eta C also be 1?

00:21:47.602 --> 00:21:48.310 align:middle line:90%
HONG LIU: No, no.

00:21:48.310 --> 00:21:49.370 align:middle line:90%
They don't have to be.

00:21:49.370 --> 00:21:51.090 align:middle line:90%
No, no.

00:21:51.090 --> 00:21:52.330 align:middle line:90%
They don't have to be.

00:21:52.330 --> 00:21:54.930 align:middle line:84%
They can be-- say
this equal to 1.

00:21:54.930 --> 00:21:56.830 align:middle line:90%
eta T can be minus 1.

00:21:56.830 --> 00:21:57.370 align:middle line:90%
Yeah, yeah.

00:21:57.370 --> 00:21:58.180 align:middle line:90%
They don't have to be.

00:21:58.180 --> 00:21:58.680 align:middle line:90%
Yeah.

00:21:58.680 --> 00:22:02.284 align:middle line:90%


00:22:02.284 --> 00:22:03.200 align:middle line:90%
AUDIENCE: [INAUDIBLE]

00:22:03.200 --> 00:22:03.783 align:middle line:90%
HONG LIU: Yes?

00:22:03.783 --> 00:22:07.057 align:middle line:84%
AUDIENCE: If eta is one, its a
scalar in what sense, I guess?

00:22:07.057 --> 00:22:07.640 align:middle line:90%
HONG LIU: Huh?

00:22:07.640 --> 00:22:09.175 align:middle line:84%
AUDIENCE: It's
scalar in what sense?

00:22:09.175 --> 00:22:13.290 align:middle line:84%
Especially for the other eta's
don't have a [INAUDIBLE] Like,

00:22:13.290 --> 00:22:17.052 align:middle line:84%
I don't understand what is
a scalar it is equal to 1.

00:22:17.052 --> 00:22:18.760 align:middle line:84%
HONG LIU: You just
say it's a convention.

00:22:18.760 --> 00:22:20.820 align:middle line:90%
It's a name.

00:22:20.820 --> 00:22:23.260 align:middle line:84%
Yeah, we just want to
distinguish these two cases.

00:22:23.260 --> 00:22:25.560 align:middle line:84%
So this is-- so when
you say it's a scalar,

00:22:25.560 --> 00:22:27.390 align:middle line:84%
we should say this is
the ordinary scalar.

00:22:27.390 --> 00:22:29.100 align:middle line:90%
s this is called a pseudoscalar.

00:22:29.100 --> 00:22:29.640 align:middle line:90%
Yeah.

00:22:29.640 --> 00:22:32.860 align:middle line:84%
We just want to distinguish
these two cases.

00:22:32.860 --> 00:22:33.360 align:middle line:90%
Yeah.

00:22:33.360 --> 00:22:36.523 align:middle line:90%


00:22:36.523 --> 00:22:37.190 align:middle line:90%
Other questions?

00:22:37.190 --> 00:22:37.760 align:middle line:90%
Yes?

00:22:37.760 --> 00:22:42.435 align:middle line:90%
AUDIENCE: So if [INAUDIBLE]

00:22:42.435 --> 00:22:43.060 align:middle line:90%
HONG LIU: Yeah.

00:22:43.060 --> 00:22:45.475 align:middle line:90%
AUDIENCE: [INAUDIBLE]

00:22:45.475 --> 00:22:46.100 align:middle line:90%
HONG LIU: Yeah.

00:22:46.100 --> 00:22:50.660 align:middle line:90%
So it's getting complicated.

00:22:50.660 --> 00:22:54.980 align:middle line:84%
You can you can try to invent
eight names for these, where

00:22:54.980 --> 00:22:57.080 align:middle line:90%
there are eight possibilities.

00:22:57.080 --> 00:23:01.450 align:middle line:84%
And people normally don't bother
to invent eight names for it.

00:23:01.450 --> 00:23:05.630 align:middle line:84%
And since the parity
is often used,

00:23:05.630 --> 00:23:09.620 align:middle line:84%
and so they just talk
about the parity.

00:23:09.620 --> 00:23:10.520 align:middle line:90%
OK?

00:23:10.520 --> 00:23:12.080 align:middle line:90%
They just define it this way.

00:23:12.080 --> 00:23:14.690 align:middle line:84%
We normally don't give a
name, let's say, in this case.

00:23:14.690 --> 00:23:15.740 align:middle line:90%
Yeah.

00:23:15.740 --> 00:23:18.590 align:middle line:84%
You just specify whether
it's 1 or minus 1.

00:23:18.590 --> 00:23:21.770 align:middle line:84%
And so the reason we
give name for this

00:23:21.770 --> 00:23:23.300 align:middle line:90%
is because of the pion.

00:23:23.300 --> 00:23:26.640 align:middle line:90%
And pions are pseudoscalars.

00:23:26.640 --> 00:23:27.140 align:middle line:90%
OK?

00:23:27.140 --> 00:23:29.600 align:middle line:84%
They actually transform
under the parity--

00:23:29.600 --> 00:23:31.580 align:middle line:90%
by minus sign.

00:23:31.580 --> 00:23:37.550 align:middle line:84%
The Higgs would be the ordinary
scalar that transforms as 1.

00:23:37.550 --> 00:23:41.930 align:middle line:90%


00:23:41.930 --> 00:23:43.420 align:middle line:90%
Good?

00:23:43.420 --> 00:23:47.540 align:middle line:90%
So if I have this expression--

00:23:47.540 --> 00:23:52.180 align:middle line:84%
so by requiring
UP phi dagger phi

00:23:52.180 --> 00:23:58.660 align:middle line:84%
x UP dagger equal to
that thing-- yeah.

00:23:58.660 --> 00:24:05.530 align:middle line:84%
You could do this, say,
plus-minus eta P phi Px.

00:24:05.530 --> 00:24:10.618 align:middle line:84%
So by requiring this, we can
work out how UP act on a and b.

00:24:10.618 --> 00:24:11.118 align:middle line:90%
OK?

00:24:11.118 --> 00:24:14.290 align:middle line:90%


00:24:14.290 --> 00:24:17.980 align:middle line:84%
We can work out how
UP worked on a and b.

00:24:17.980 --> 00:24:21.770 align:middle line:84%
So you find that
the parity acting on

00:24:21.770 --> 00:24:29.890 align:middle line:84%
ak UP dagger equal to
the same eta P factor.

00:24:29.890 --> 00:24:33.275 align:middle line:90%
And then take it to a minus k.

00:24:33.275 --> 00:24:33.775 align:middle line:90%
OK?

00:24:33.775 --> 00:24:38.950 align:middle line:84%
So this makes complete
sense because we integrate

00:24:38.950 --> 00:24:40.680 align:middle line:90%
the k as momentum, right?

00:24:40.680 --> 00:24:44.695 align:middle line:84%
So under reflection
of a spatial direction

00:24:44.695 --> 00:24:46.490 align:middle line:84%
your momentum also
change direction.

00:24:46.490 --> 00:24:49.810 align:middle line:84%
And so it takes ak
to get to a minus k.

00:24:49.810 --> 00:24:52.000 align:middle line:84%
And then, similarly,
you can work it

00:24:52.000 --> 00:24:59.230 align:middle line:84%
out the UT acting
on ak UT dagger.

00:24:59.230 --> 00:25:04.780 align:middle line:84%
It gives you eta
T, also a minus k.

00:25:04.780 --> 00:25:07.040 align:middle line:90%
OK?

00:25:07.040 --> 00:25:08.440 align:middle line:90%
And similarly, with the bk.

00:25:08.440 --> 00:25:08.940 align:middle line:90%
OK?

00:25:08.940 --> 00:25:16.610 align:middle line:90%


00:25:16.610 --> 00:25:17.900 align:middle line:90%
So this also makes sense.

00:25:17.900 --> 00:25:19.580 align:middle line:84%
When you reverse
your time direction,

00:25:19.580 --> 00:25:21.290 align:middle line:90%
your momentum also changes sign.

00:25:21.290 --> 00:25:22.010 align:middle line:90%
OK?

00:25:22.010 --> 00:25:25.080 align:middle line:90%
Your momentum also changes sign.

00:25:25.080 --> 00:25:29.870 align:middle line:84%
So now the charge conjugation is
a little bit more interesting.

00:25:29.870 --> 00:25:30.370 align:middle line:90%
OK?

00:25:30.370 --> 00:25:33.310 align:middle line:84%
So the UC for the
charge conjugation

00:25:33.310 --> 00:25:35.650 align:middle line:84%
is a little bit
more interesting.

00:25:35.650 --> 00:25:43.330 align:middle line:84%
So the charge conjugation
takes ak dagger

00:25:43.330 --> 00:25:46.444 align:middle line:90%
equal to eta C equal to b.

00:25:46.444 --> 00:25:47.308 align:middle line:90%
OK?

00:25:47.308 --> 00:25:51.280 align:middle line:90%
And similarly, it takes the--

00:25:51.280 --> 00:25:52.690 align:middle line:90%
yeah.

00:25:52.690 --> 00:25:57.130 align:middle line:84%
So you see that what
charge conjugation does

00:25:57.130 --> 00:26:00.610 align:middle line:84%
is to exchange particle
and antiparticle,

00:26:00.610 --> 00:26:04.660 align:middle line:84%
OK, exchange particle
and antiparticle.

00:26:04.660 --> 00:26:05.210 align:middle line:90%
Good.

00:26:05.210 --> 00:26:07.285 align:middle line:84%
Any questions on
this scalar field?

00:26:07.285 --> 00:26:13.760 align:middle line:90%


00:26:13.760 --> 00:26:15.280 align:middle line:90%
Good.

00:26:15.280 --> 00:26:17.185 align:middle line:90%
So now let's look at the--

00:26:17.185 --> 00:26:20.110 align:middle line:84%
with this scalar
field story, now we

00:26:20.110 --> 00:26:29.380 align:middle line:84%
can look at a Dirac field,
which is more intricate, OK,

00:26:29.380 --> 00:26:32.240 align:middle line:90%
which is more intricate.

00:26:32.240 --> 00:26:33.655 align:middle line:90%
So let's write down the--

00:26:33.655 --> 00:26:36.410 align:middle line:90%


00:26:36.410 --> 00:26:39.320 align:middle line:84%
so let's forget about
the previous thing.

00:26:39.320 --> 00:26:44.255 align:middle line:84%
Let's write down the action
for the Dirac theory.

00:26:44.255 --> 00:26:52.140 align:middle line:90%


00:26:52.140 --> 00:26:52.640 align:middle line:90%
OK?

00:26:52.640 --> 00:26:55.150 align:middle line:84%
And then let's also write
down this equation of motion.

00:26:55.150 --> 00:27:01.550 align:middle line:90%


00:27:01.550 --> 00:27:02.050 align:middle line:90%
OK.

00:27:02.050 --> 00:27:03.600 align:middle line:84%
Now let me call
this equation star

00:27:03.600 --> 00:27:05.080 align:middle line:90%
and erase the earlier star.

00:27:05.080 --> 00:27:08.490 align:middle line:90%


00:27:08.490 --> 00:27:10.950 align:middle line:84%
So now let's first understand
how the parity should

00:27:10.950 --> 00:27:14.390 align:middle line:90%
act on the Dirac field.

00:27:14.390 --> 00:27:16.648 align:middle line:90%
OK?

00:27:16.648 --> 00:27:18.940 align:middle line:84%
So now, actually, everything
with the Dirac field story

00:27:18.940 --> 00:27:19.720 align:middle line:90%
becomes tricky.

00:27:19.720 --> 00:27:20.620 align:middle line:90%
OK?

00:27:20.620 --> 00:27:22.210 align:middle line:90%
It becomes tricky.

00:27:22.210 --> 00:27:24.940 align:middle line:84%
So naively, if you
wanted to just try

00:27:24.940 --> 00:27:30.280 align:middle line:84%
to generalize the
scalar story, we

00:27:30.280 --> 00:27:33.070 align:middle line:84%
say, let's imagine the Dirac
field should transform.

00:27:33.070 --> 00:27:39.580 align:middle line:90%


00:27:39.580 --> 00:27:40.080 align:middle line:90%
OK?

00:27:40.080 --> 00:27:43.890 align:middle line:84%
I'm just-- to say
for the parity,

00:27:43.890 --> 00:27:51.210 align:middle line:84%
for example, psi Px,
OK, with some phase,

00:27:51.210 --> 00:27:54.040 align:middle line:90%
OK, maybe with some phase.

00:27:54.040 --> 00:27:59.130 align:middle line:84%
So we want to naively generalize
to the scalar story, that

00:27:59.130 --> 00:28:01.520 align:middle line:90%
would be what you do.

00:28:01.520 --> 00:28:02.020 align:middle line:90%
OK?

00:28:02.020 --> 00:28:04.900 align:middle line:84%
That should be how
the parity transform.

00:28:04.900 --> 00:28:11.770 align:middle line:84%
But now remember psi
now is a four vector.

00:28:11.770 --> 00:28:15.490 align:middle line:84%
Nobody tells you
that, under parity,

00:28:15.490 --> 00:28:18.370 align:middle line:84%
the different component of
this four vector cannot get

00:28:18.370 --> 00:28:20.250 align:middle line:90%
reshuffled.

00:28:20.250 --> 00:28:20.750 align:middle line:90%
OK?

00:28:20.750 --> 00:28:22.460 align:middle line:84%
Remember, under Lorentz
transformations,

00:28:22.460 --> 00:28:24.570 align:middle line:90%
they do get reshuffled.

00:28:24.570 --> 00:28:25.070 align:middle line:90%
OK?

00:28:25.070 --> 00:28:26.185 align:middle line:84%
In Lorentz
transformations, remember

00:28:26.185 --> 00:28:28.310 align:middle line:84%
that for spinor the Lorentz
transformation not only

00:28:28.310 --> 00:28:33.770 align:middle line:84%
act on x but also on the
internal space of the spinners.

00:28:33.770 --> 00:28:36.650 align:middle line:84%
So you would expect the most
general parity conservation

00:28:36.650 --> 00:28:40.560 align:middle line:84%
should also act on
the internal space.

00:28:40.560 --> 00:28:41.060 align:middle line:90%
OK?

00:28:41.060 --> 00:28:43.250 align:middle line:84%
So we would expect the
phase is not enough.

00:28:43.250 --> 00:28:47.746 align:middle line:84%
So we actually have to put
a matrix here and that's

00:28:47.746 --> 00:28:53.430 align:middle line:84%
called D. And whatever phase
will be absorbed in this D. OK?

00:28:53.430 --> 00:28:57.440 align:middle line:84%
Of course, you'll also
include the phase.

00:28:57.440 --> 00:28:58.970 align:middle line:90%
And yeah.

00:28:58.970 --> 00:29:01.160 align:middle line:84%
So for notation
convenience, I will also

00:29:01.160 --> 00:29:10.020 align:middle line:84%
write it as psi prime x
prime equal to D psi x.

00:29:10.020 --> 00:29:10.520 align:middle line:90%
OK?

00:29:10.520 --> 00:29:13.650 align:middle line:90%
So the x prime equal to Px.

00:29:13.650 --> 00:29:15.530 align:middle line:90%
OK?

00:29:15.530 --> 00:29:17.450 align:middle line:84%
So these two are
the same equation.

00:29:17.450 --> 00:29:18.750 align:middle line:90%
OK?

00:29:18.750 --> 00:29:31.605 align:middle line:84%
So you just write, D is a
matrix acting on spinor space.

00:29:31.605 --> 00:29:36.450 align:middle line:90%


00:29:36.450 --> 00:29:38.580 align:middle line:90%
OK?

00:29:38.580 --> 00:29:44.650 align:middle line:84%
So in what sense
we say the Dirac

00:29:44.650 --> 00:29:47.840 align:middle line:90%
theory is parity invariant?

00:29:47.840 --> 00:29:48.340 align:middle line:90%
OK?

00:29:48.340 --> 00:29:49.840 align:middle line:84%
Let me just
formulate it in terms

00:29:49.840 --> 00:29:51.223 align:middle line:90%
of the equation of motion.

00:29:51.223 --> 00:29:52.890 align:middle line:84%
You can also equivalently
reformulate it

00:29:52.890 --> 00:29:53.950 align:middle line:90%
in terms of the action.

00:29:53.950 --> 00:29:54.790 align:middle line:90%
OK?

00:29:54.790 --> 00:29:56.058 align:middle line:90%
Yes?

00:29:56.058 --> 00:29:58.220 align:middle line:90%
AUDIENCE: [INAUDIBLE]

00:29:58.220 --> 00:30:01.220 align:middle line:84%
HONG LIU: Oh, we will
talk about the behavior.

00:30:01.220 --> 00:30:03.050 align:middle line:90%
This is a good question.

00:30:03.050 --> 00:30:04.300 align:middle line:90%
It does not have to.

00:30:04.300 --> 00:30:06.760 align:middle line:84%
And we will talk about
the property of T. Yeah.

00:30:06.760 --> 00:30:11.580 align:middle line:90%


00:30:11.580 --> 00:30:13.020 align:middle line:90%
Do you have other questions?

00:30:13.020 --> 00:30:15.650 align:middle line:90%


00:30:15.650 --> 00:30:16.790 align:middle line:90%
Good.

00:30:16.790 --> 00:30:21.870 align:middle line:84%
So what do we mean by the
theory is parity invariant?

00:30:21.870 --> 00:30:22.800 align:middle line:90%
OK?

00:30:22.800 --> 00:30:27.540 align:middle line:84%
By the theory-- we say Dirac
theory is parity invariant.

00:30:27.540 --> 00:30:31.110 align:middle line:90%


00:30:31.110 --> 00:30:35.560 align:middle line:84%
We say if Dirac theory
have a parity symmetry,

00:30:35.560 --> 00:30:37.650 align:middle line:84%
you can always define
a parity operation.

00:30:37.650 --> 00:30:41.920 align:middle line:84%
So when we say Dirac equation
have a parity symmetry,

00:30:41.920 --> 00:30:46.570 align:middle line:84%
it means that if we
start with a psi which

00:30:46.570 --> 00:30:51.480 align:middle line:84%
satisfies the Dirac
equation, and then there

00:30:51.480 --> 00:31:03.810 align:middle line:84%
should exist such a D, such a
matrix, such a transformation

00:31:03.810 --> 00:31:18.680 align:middle line:84%
so that at this
transformed psi still

00:31:18.680 --> 00:31:24.000 align:middle line:84%
satisfies the Dirac equation
but in the reflected space.

00:31:24.000 --> 00:31:24.500 align:middle line:90%
OK?

00:31:24.500 --> 00:31:27.320 align:middle line:90%
So you want to reflect.

00:31:27.320 --> 00:31:32.550 align:middle line:84%
So x prime goes right
into the reflected space.

00:31:32.550 --> 00:31:37.685 align:middle line:84%
And as we say the theory
have a parity symmetry-- it

00:31:37.685 --> 00:31:41.460 align:middle line:84%
means in the reflected space you
should have the same equation.

00:31:41.460 --> 00:31:41.960 align:middle line:90%
OK?

00:31:41.960 --> 00:31:44.040 align:middle line:84%
You should have the same
form of the equation.

00:31:44.040 --> 00:31:48.200 align:middle line:84%
So that means that this psi
prime, when you're writing it

00:31:48.200 --> 00:31:50.540 align:middle line:84%
in the reflected
space, should have

00:31:50.540 --> 00:31:54.360 align:middle line:84%
the same form of the equation
as the original Dirac equation.

00:31:54.360 --> 00:31:58.370 align:middle line:84%
So this is the same concept
as this Lorentz covariance

00:31:58.370 --> 00:31:59.480 align:middle line:90%
we described earlier.

00:31:59.480 --> 00:32:00.920 align:middle line:90%
OK?

00:32:00.920 --> 00:32:01.990 align:middle line:90%
Yes?

00:32:01.990 --> 00:32:04.532 align:middle line:84%
AUDIENCE: Why doesn't the parity
transform the gamma matrices

00:32:04.532 --> 00:32:06.530 align:middle line:90%
matrices?

00:32:06.530 --> 00:32:10.095 align:middle line:84%
Because we changed d
mu but not the gamma?

00:32:10.095 --> 00:32:10.970 align:middle line:90%
HONG LIU: Yeah, yeah.

00:32:10.970 --> 00:32:12.750 align:middle line:84%
Gamma matrix is
just some numbers.

00:32:12.750 --> 00:32:13.250 align:middle line:90%
Right?

00:32:13.250 --> 00:32:15.730 align:middle line:84%
Because the gamma matrix is
always just some numbers.

00:32:15.730 --> 00:32:17.390 align:middle line:90%
They're not spacetime variables.

00:32:17.390 --> 00:32:20.875 align:middle line:84%
And so they're not-- so they
don't change under the parity.

00:32:20.875 --> 00:32:23.710 align:middle line:90%


00:32:23.710 --> 00:32:25.350 align:middle line:90%
Other questions?

00:32:25.350 --> 00:32:26.450 align:middle line:90%
Yes?

00:32:26.450 --> 00:32:29.910 align:middle line:84%
AUDIENCE: Is there any reason
why D has to be a matrix?

00:32:29.910 --> 00:32:31.282 align:middle line:90%
Why does it have to be linear?

00:32:31.282 --> 00:32:31.990 align:middle line:90%
HONG LIU: No, no.

00:32:31.990 --> 00:32:33.300 align:middle line:90%
D does not have to be a matrix.

00:32:33.300 --> 00:32:36.690 align:middle line:84%
It's just that D
can be a matrix.

00:32:36.690 --> 00:32:40.520 align:middle line:84%
But we want to find the
D so that we have this.

00:32:40.520 --> 00:32:41.576 align:middle line:90%
But D can be identity.

00:32:41.576 --> 00:32:43.576 align:middle line:84%
AUDIENCE: Could it be
some complicated function,

00:32:43.576 --> 00:32:44.340 align:middle line:90%
like non-linear?

00:32:44.340 --> 00:32:47.800 align:middle line:84%
Because we already
allowed for non-linear,

00:32:47.800 --> 00:32:52.202 align:middle line:84%
like, time reversal
operators [INAUDIBLE]

00:32:52.202 --> 00:32:52.910 align:middle line:90%
HONG LIU: No, no.

00:32:52.910 --> 00:32:55.580 align:middle line:84%
The time reversal
is linear, right?

00:32:55.580 --> 00:32:56.780 align:middle line:90%
Time reversal is linear.

00:32:56.780 --> 00:32:58.355 align:middle line:90%
AUDIENCE: Antilinear.

00:32:58.355 --> 00:32:58.980 align:middle line:90%
HONG LIU: Yeah.

00:32:58.980 --> 00:33:02.270 align:middle line:90%


00:33:02.270 --> 00:33:04.020 align:middle line:90%
The action of a--

00:33:04.020 --> 00:33:07.620 align:middle line:90%
normally, action of a--

00:33:07.620 --> 00:33:11.070 align:middle line:90%
yeah, this kind of--

00:33:11.070 --> 00:33:13.180 align:middle line:90%
just go to psi itself.

00:33:13.180 --> 00:33:16.355 align:middle line:90%
But you can-- yeah.

00:33:16.355 --> 00:33:19.460 align:middle line:90%


00:33:19.460 --> 00:33:21.565 align:middle line:90%
You're asking why not--

00:33:21.565 --> 00:33:25.840 align:middle line:84%
why we don't use a
psi squared term?

00:33:25.840 --> 00:33:31.908 align:middle line:84%
Or you are asking whether the
D can depend on spacetime?

00:33:31.908 --> 00:33:32.950 align:middle line:90%
Which one are you asking?

00:33:32.950 --> 00:33:33.575 align:middle line:90%
AUDIENCE: Yeah.

00:33:33.575 --> 00:33:36.705 align:middle line:84%
I mean, psi-- like, some
arbitrary function of psi.

00:33:36.705 --> 00:33:37.330 align:middle line:90%
HONG LIU: Yeah.

00:33:37.330 --> 00:33:38.740 align:middle line:84%
Yeah, but this is
the simplest one.

00:33:38.740 --> 00:33:39.240 align:middle line:90%
Yeah.

00:33:39.240 --> 00:33:43.660 align:middle line:84%
Yeah, if we can make this work,
then you don't have to do that.

00:33:43.660 --> 00:33:44.160 align:middle line:90%
Yeah.

00:33:44.160 --> 00:33:45.810 align:middle line:90%
You do the simplest one first.

00:33:45.810 --> 00:33:51.690 align:middle line:90%


00:33:51.690 --> 00:33:55.290 align:middle line:84%
Also, we do believe that
this kind of parity operation

00:33:55.290 --> 00:33:57.550 align:middle line:90%
should be a linear operation.

00:33:57.550 --> 00:33:58.050 align:middle line:90%
Yeah.

00:33:58.050 --> 00:34:01.230 align:middle line:84%
Yeah, if you add the
two operators together,

00:34:01.230 --> 00:34:05.040 align:middle line:84%
you do the parity,
and you should--

00:34:05.040 --> 00:34:08.290 align:middle line:84%
after you do the parity should
be still summed together.

00:34:08.290 --> 00:34:10.170 align:middle line:84%
If it's a non-linear
function, then you

00:34:10.170 --> 00:34:11.820 align:middle line:90%
won't satisfy that property.

00:34:11.820 --> 00:34:12.480 align:middle line:90%
Yeah.

00:34:12.480 --> 00:34:14.909 align:middle line:84%
Yeah, just from our
experience, the--

00:34:14.909 --> 00:34:17.010 align:middle line:84%
yeah, just physically,
we expect the parity

00:34:17.010 --> 00:34:18.337 align:middle line:90%
to be a linear operation.

00:34:18.337 --> 00:34:21.080 align:middle line:90%


00:34:21.080 --> 00:34:22.330 align:middle line:90%
Good?

00:34:22.330 --> 00:34:22.960 align:middle line:90%
OK.

00:34:22.960 --> 00:34:26.150 align:middle line:84%
So we want to find the D
so that this is satisfied.

00:34:26.150 --> 00:34:26.650 align:middle line:90%
OK?

00:34:26.650 --> 00:34:29.650 align:middle line:90%
So that means that the--

00:34:29.650 --> 00:34:30.580 align:middle line:90%
so this is easy.

00:34:30.580 --> 00:34:33.290 align:middle line:84%
So now let's just look
at this equation a bit.

00:34:33.290 --> 00:34:33.790 align:middle line:90%
OK?

00:34:33.790 --> 00:34:36.919 align:middle line:84%
So let's write it
more explicitly.

00:34:36.919 --> 00:34:39.909 align:middle line:90%


00:34:39.909 --> 00:34:42.469 align:middle line:90%
So partial mu prime--

00:34:42.469 --> 00:34:42.969 align:middle line:90%
yeah.

00:34:42.969 --> 00:34:44.900 align:middle line:90%
So we write this explicitly.

00:34:44.900 --> 00:34:49.239 align:middle line:84%
So partial 0 prime is the same
as the original partial prime

00:34:49.239 --> 00:34:52.540 align:middle line:84%
because you don't flip
the time direction.

00:34:52.540 --> 00:34:53.080 align:middle line:90%
OK?

00:34:53.080 --> 00:34:55.580 align:middle line:84%
And then you flip the
spatial direction.

00:34:55.580 --> 00:34:58.040 align:middle line:84%
So that means this becomes
a partial i, partial i --

00:34:58.040 --> 00:34:59.170 align:middle line:90%
gamma i partial i.

00:34:59.170 --> 00:35:03.100 align:middle line:84%
And so I remove the prime, but
I have a minus sign here, OK,

00:35:03.100 --> 00:35:05.470 align:middle line:90%
because you've changed signs.

00:35:05.470 --> 00:35:08.680 align:middle line:90%
And the minus n--

00:35:08.680 --> 00:35:12.800 align:middle line:84%
and this one from our definition
should be equal to this.

00:35:12.800 --> 00:35:18.910 align:middle line:84%
So this is just equal
to D psi x equal to 0.

00:35:18.910 --> 00:35:19.410 align:middle line:90%
OK?

00:35:19.410 --> 00:35:25.680 align:middle line:90%


00:35:25.680 --> 00:35:26.180 align:middle line:90%
OK?

00:35:26.180 --> 00:35:29.150 align:middle line:90%


00:35:29.150 --> 00:35:32.840 align:middle line:84%
So now if you compare
with this equation--

00:35:32.840 --> 00:35:37.580 align:middle line:84%
so the-- so now if you
compare with this equation,

00:35:37.580 --> 00:35:41.810 align:middle line:84%
now let's add the
D on this equation.

00:35:41.810 --> 00:35:46.090 align:middle line:84%
So that corresponding
to D-- gamma 0 partial 0

00:35:46.090 --> 00:35:51.880 align:middle line:84%
plus gamma i partial i
minus m psi x equal to 0.

00:35:51.880 --> 00:35:52.380 align:middle line:90%
OK?

00:35:52.380 --> 00:35:54.460 align:middle line:84%
So this is the
original equation.

00:35:54.460 --> 00:35:57.310 align:middle line:84%
And now we want these two
equations to be equivalent.

00:35:57.310 --> 00:35:57.810 align:middle line:90%
OK?

00:35:57.810 --> 00:36:00.227 align:middle line:84%
Because given this equation,
we should have that equation.

00:36:00.227 --> 00:36:00.750 align:middle line:90%
OK?

00:36:00.750 --> 00:36:03.850 align:middle line:84%
And so we want these two
equation to be equivalent.

00:36:03.850 --> 00:36:06.113 align:middle line:84%
So in order for them
to be equivalent,

00:36:06.113 --> 00:36:07.530 align:middle line:84%
you see the last
term is the same.

00:36:07.530 --> 00:36:10.545 align:middle line:84%
Because the m is just a
number that commutes with D.

00:36:10.545 --> 00:36:12.920 align:middle line:84%
And then in order for these
two equations to be the same,

00:36:12.920 --> 00:36:14.630 align:middle line:90%
we want D to commute--

00:36:14.630 --> 00:36:17.370 align:middle line:90%


00:36:17.370 --> 00:36:25.270 align:middle line:84%
it means that D should
commute with gamma 0

00:36:25.270 --> 00:36:34.060 align:middle line:90%
but anti-commute with gamma i.

00:36:34.060 --> 00:36:35.760 align:middle line:90%
OK?

00:36:35.760 --> 00:36:38.040 align:middle line:84%
Because when you
pass through here

00:36:38.040 --> 00:36:41.025 align:middle line:84%
to bring it to that
side of gamma i,

00:36:41.025 --> 00:36:42.880 align:middle line:90%
you should change the sign.

00:36:42.880 --> 00:36:45.570 align:middle line:90%
OK?

00:36:45.570 --> 00:36:59.610 align:middle line:84%
So now you can immediately
write down such a matrix.

00:36:59.610 --> 00:37:01.680 align:middle line:90%
Then D has to be--

00:37:01.680 --> 00:37:04.320 align:middle line:84%
it commutes with gamma 0,
anti-commutes with gamma i.

00:37:04.320 --> 00:37:07.800 align:middle line:84%
So it has to be
proportional to gamma 0.

00:37:07.800 --> 00:37:08.300 align:middle line:90%
OK?

00:37:08.300 --> 00:37:09.920 align:middle line:84%
And then you can
put the phase here.

00:37:09.920 --> 00:37:13.700 align:middle line:90%


00:37:13.700 --> 00:37:15.018 align:middle line:90%
And you can put the phase here.

00:37:15.018 --> 00:37:20.010 align:middle line:90%


00:37:20.010 --> 00:37:24.850 align:middle line:84%
And so this is
how it transforms.

00:37:24.850 --> 00:37:28.780 align:middle line:84%
So now you can further
constrain eta p.

00:37:28.780 --> 00:37:30.880 align:middle line:84%
Now you can further
constrain eta p.

00:37:30.880 --> 00:37:34.570 align:middle line:84%
But here, the story is
a little bit subtle.

00:37:34.570 --> 00:37:35.960 align:middle line:90%
We are not going through there.

00:37:35.960 --> 00:37:38.600 align:middle line:90%
So you said the fermion--

00:37:38.600 --> 00:37:40.780 align:middle line:84%
remember when we already
discussed in your quantum

00:37:40.780 --> 00:37:43.870 align:middle line:84%
mechanics, when you have
a spin-1/2 particle,

00:37:43.870 --> 00:37:48.770 align:middle line:84%
when you rotate 360 degree, we
actually don't have to go back

00:37:48.770 --> 00:37:49.270 align:middle line:90%
to itself.

00:37:49.270 --> 00:37:51.290 align:middle line:84%
You can go back
with a minus sign.

00:37:51.290 --> 00:37:51.790 align:middle line:90%
OK?

00:37:51.790 --> 00:37:54.940 align:middle line:90%
And so here-- and so yeah.

00:37:54.940 --> 00:37:57.850 align:middle line:84%
And so here, there's
an intricate story

00:37:57.850 --> 00:38:00.307 align:middle line:84%
you can discuss what
are allowed eta p etc.

00:38:00.307 --> 00:38:01.640 align:middle line:90%
And so we will not go into that.

00:38:01.640 --> 00:38:03.400 align:middle line:84%
But I think you
know the basic idea.

00:38:03.400 --> 00:38:05.060 align:middle line:90%
OK?

00:38:05.060 --> 00:38:05.620 align:middle line:90%
Good.

00:38:05.620 --> 00:38:06.220 align:middle line:90%
Any questions?

00:38:06.220 --> 00:38:06.760 align:middle line:90%
Yes?

00:38:06.760 --> 00:38:09.571 align:middle line:84%
AUDIENCE: So isn't
this just be, if you

00:38:09.571 --> 00:38:11.560 align:middle line:84%
want these equations
to be equivalent,

00:38:11.560 --> 00:38:15.610 align:middle line:84%
that could be satisfied
with gamma 0 is real gamma i

00:38:15.610 --> 00:38:19.376 align:middle line:84%
is pure imaginary because
of this antilinear?

00:38:19.376 --> 00:38:22.225 align:middle line:90%


00:38:22.225 --> 00:38:22.850 align:middle line:90%
HONG LIU: Wait.

00:38:22.850 --> 00:38:23.930 align:middle line:90%
What do you mean by--

00:38:23.930 --> 00:38:24.430 align:middle line:90%
no, no, no.

00:38:24.430 --> 00:38:26.740 align:middle line:84%
These equations should be
applied for any gamma 0,

00:38:26.740 --> 00:38:28.129 align:middle line:90%
for any gamma i.

00:38:28.129 --> 00:38:28.796 align:middle line:90%
AUDIENCE: Right.

00:38:28.796 --> 00:38:29.988 align:middle line:90%
[INAUDIBLE]

00:38:29.988 --> 00:38:31.030 align:middle line:90%
HONG LIU: No, you cannot.

00:38:31.030 --> 00:38:31.880 align:middle line:90%
Yeah, yeah.

00:38:31.880 --> 00:38:35.114 align:middle line:84%
Because in parity, you should
work in any representation.

00:38:35.114 --> 00:38:36.890 align:middle line:90%
AUDIENCE: OK.

00:38:36.890 --> 00:38:38.690 align:middle line:90%
HONG LIU: OK?

00:38:38.690 --> 00:38:42.740 align:middle line:84%
So this is the story
for the parity.

00:38:42.740 --> 00:38:49.130 align:middle line:84%
So completely similar idea
for charge conjugation

00:38:49.130 --> 00:38:55.280 align:middle line:84%
and the time reversal
for fermions.

00:38:55.280 --> 00:38:58.030 align:middle line:84%
And so I will do it
a little bit faster.

00:38:58.030 --> 00:38:58.530 align:middle line:90%
OK?

00:38:58.530 --> 00:39:00.240 align:middle line:84%
So I will do it a
little bit faster--

00:39:00.240 --> 00:39:02.520 align:middle line:84%
mostly, again, just
outlining the same idea,

00:39:02.520 --> 00:39:04.835 align:middle line:84%
but then just write
down the results.

00:39:04.835 --> 00:39:18.884 align:middle line:90%


00:39:18.884 --> 00:39:23.920 align:middle line:90%
So for the charge conjugation--

00:39:23.920 --> 00:39:27.120 align:middle line:84%
now let's do the
charge conjugation.

00:39:27.120 --> 00:39:29.800 align:middle line:90%
Did I say-- yeah, sorry.

00:39:29.800 --> 00:39:35.162 align:middle line:84%
Here, I should put a
thing here with parity.

00:39:35.162 --> 00:39:37.910 align:middle line:84%
Because we are
doing parity here.

00:39:37.910 --> 00:39:39.820 align:middle line:84%
So now let's do the
charge conjugation.

00:39:39.820 --> 00:39:47.590 align:middle line:90%


00:39:47.590 --> 00:39:51.760 align:middle line:84%
So for the scalar case for
the charge conservation,

00:39:51.760 --> 00:39:55.380 align:middle line:90%
just take the phi star.

00:39:55.380 --> 00:40:02.070 align:middle line:84%
Again, so for the
Dirac field for psi

00:40:02.070 --> 00:40:07.290 align:middle line:84%
prime, again, this should
be proportional to phi star,

00:40:07.290 --> 00:40:10.590 align:middle line:90%
OK, to phi star.

00:40:10.590 --> 00:40:14.760 align:middle line:84%
But so a simple way
to construct phi star

00:40:14.760 --> 00:40:17.670 align:middle line:90%
is you take the psi-bar.

00:40:17.670 --> 00:40:18.210 align:middle line:90%
OK?

00:40:18.210 --> 00:40:20.100 align:middle line:90%
And then you take a--

00:40:20.100 --> 00:40:20.850 align:middle line:90%
transpose.

00:40:20.850 --> 00:40:21.750 align:middle line:90%
OK?

00:40:21.750 --> 00:40:25.470 align:middle line:84%
And so this is some linear
combination of the phi star.

00:40:25.470 --> 00:40:28.350 align:middle line:84%
And also, from what
we discussed before,

00:40:28.350 --> 00:40:30.920 align:middle line:90%
you should allow a matrix here.

00:40:30.920 --> 00:40:31.420 align:middle line:90%
OK?

00:40:31.420 --> 00:40:32.712 align:middle line:90%
You should allow a matrix here.

00:40:32.712 --> 00:40:35.660 align:middle line:84%
And the C, again, is a
matrix in the spinor space.

00:40:35.660 --> 00:40:36.160 align:middle line:90%
OK?

00:40:36.160 --> 00:40:39.280 align:middle line:90%


00:40:39.280 --> 00:40:42.320 align:middle line:90%
And now, again, we need--

00:40:42.320 --> 00:40:47.090 align:middle line:84%
so the statement of
the charge conjugation

00:40:47.090 --> 00:40:53.450 align:middle line:84%
is the statement that,
given there exists C,

00:40:53.450 --> 00:41:04.730 align:middle line:84%
there exists C such
that given equation star

00:41:04.730 --> 00:41:07.010 align:middle line:84%
that phi prime will
satisfy the same equation.

00:41:07.010 --> 00:41:14.860 align:middle line:90%


00:41:14.860 --> 00:41:17.230 align:middle line:84%
OK? psi prime satisfies
the same equation.

00:41:17.230 --> 00:41:17.730 align:middle line:90%
OK?

00:41:17.730 --> 00:41:23.020 align:middle line:90%


00:41:23.020 --> 00:41:27.410 align:middle line:90%
So yeah.

00:41:27.410 --> 00:41:28.520 align:middle line:90%
So we just do it.

00:41:28.520 --> 00:41:31.073 align:middle line:90%


00:41:31.073 --> 00:41:31.573 align:middle line:90%
OK?

00:41:31.573 --> 00:41:36.520 align:middle line:90%
So we just start from here.

00:41:36.520 --> 00:41:43.650 align:middle line:84%
So since the psi prime are
related by taking the bar,

00:41:43.650 --> 00:41:46.660 align:middle line:84%
and the complex
conj-- and transpose--

00:41:46.660 --> 00:41:50.310 align:middle line:84%
so let's then first do
the bar for this equation

00:41:50.310 --> 00:41:51.630 align:middle line:90%
and then do the transpose.

00:41:51.630 --> 00:41:52.260 align:middle line:90%
OK?

00:41:52.260 --> 00:41:57.045 align:middle line:84%
So when you do the bar
operation from that equation

00:41:57.045 --> 00:42:03.826 align:middle line:84%
for the star, then give the bar
equation [INAUDIBLE] psi bar--

00:42:03.826 --> 00:42:07.360 align:middle line:84%
yeah, you can easily
check yourself--

00:42:07.360 --> 00:42:09.350 align:middle line:90%
equal to 0.

00:42:09.350 --> 00:42:09.850 align:middle line:90%
OK?

00:42:09.850 --> 00:42:11.980 align:middle line:90%
Now you do a transpose.

00:42:11.980 --> 00:42:21.575 align:middle line:84%
And then you get gamma mu
t partial mu plus m psi t.

00:42:21.575 --> 00:42:22.075 align:middle line:90%
OK.

00:42:22.075 --> 00:42:29.010 align:middle line:90%


00:42:29.010 --> 00:42:33.810 align:middle line:84%
So then the psi-bar
t is just equal to C

00:42:33.810 --> 00:42:38.430 align:middle line:90%
minus 1 psi prime x equal to 0.

00:42:38.430 --> 00:42:40.800 align:middle line:84%
OK-- because from
here, by definition.

00:42:40.800 --> 00:42:42.308 align:middle line:90%
OK?

00:42:42.308 --> 00:42:44.975 align:middle line:84%
So now you want this equation to
be equivalent to that equation.

00:42:44.975 --> 00:42:48.930 align:middle line:90%


00:42:48.930 --> 00:42:49.430 align:middle line:90%
OK?

00:42:49.430 --> 00:42:51.290 align:middle line:90%
So as we do before--

00:42:51.290 --> 00:42:52.910 align:middle line:90%
a similar idea from that.

00:42:52.910 --> 00:42:55.190 align:middle line:90%
OK?

00:42:55.190 --> 00:42:58.370 align:middle line:90%
You just act C minus prime here.

00:42:58.370 --> 00:42:58.880 align:middle line:90%
OK?

00:42:58.880 --> 00:43:03.620 align:middle line:84%
So in the end, you conclude
that the gamma UT--

00:43:03.620 --> 00:43:08.440 align:middle line:84%
so from here, the equivalence
of these two equations,

00:43:08.440 --> 00:43:18.930 align:middle line:84%
you conclude that gamma mu t
should be related to gamma mu

00:43:18.930 --> 00:43:19.995 align:middle line:90%
by this C matrix.

00:43:19.995 --> 00:43:23.210 align:middle line:90%


00:43:23.210 --> 00:43:25.470 align:middle line:90%
OK, negative.

00:43:25.470 --> 00:43:28.430 align:middle line:84%
So if you can find the C,
satisfy it to this equation,

00:43:28.430 --> 00:43:31.315 align:middle line:90%
and then they are equivalent.

00:43:31.315 --> 00:43:31.815 align:middle line:90%
OK?

00:43:31.815 --> 00:43:40.160 align:middle line:90%


00:43:40.160 --> 00:43:42.790 align:middle line:90%
OK?

00:43:42.790 --> 00:43:46.070 align:middle line:90%
So does such a matrix C exist?

00:43:46.070 --> 00:43:49.270 align:middle line:90%


00:43:49.270 --> 00:43:53.340 align:middle line:84%
So we can say actually such
a matrix C always exists

00:43:53.340 --> 00:43:56.260 align:middle line:90%
for the following reason.

00:43:56.260 --> 00:44:10.435 align:middle line:84%
Because you can check given
gamma mu satisfy this Clifford

00:44:10.435 --> 00:44:15.830 align:middle line:84%
algebra for gamma matrices
and the minus gamma mu

00:44:15.830 --> 00:44:23.810 align:middle line:90%
t also satisfy the same algebra.

00:44:23.810 --> 00:44:24.310 align:middle line:90%
OK?

00:44:24.310 --> 00:44:26.990 align:middle line:90%
So this will do that.

00:44:26.990 --> 00:44:31.420 align:middle line:84%
So that means the
minus gamma mu t,

00:44:31.420 --> 00:44:36.990 align:middle line:84%
OK, is also another allowed
set of gamma matrices.

00:44:36.990 --> 00:44:37.490 align:middle line:90%
OK?

00:44:37.490 --> 00:44:39.860 align:middle line:84%
We got to satisfy
the same algebra.

00:44:39.860 --> 00:44:42.650 align:middle line:84%
And now we discussed
before, mathematically, we

00:44:42.650 --> 00:44:47.810 align:middle line:84%
can prove all representations
of the matrix are equivalent.

00:44:47.810 --> 00:44:51.290 align:middle line:84%
And that means there must
exist some matrix c which

00:44:51.290 --> 00:44:53.040 align:middle line:90%
relates to these two.

00:44:53.040 --> 00:44:53.540 align:middle line:90%
OK?

00:44:53.540 --> 00:44:54.890 align:middle line:90%
And c must exist.

00:44:54.890 --> 00:44:56.590 align:middle line:90%
OK?

00:44:56.590 --> 00:44:59.930 align:middle line:90%
So this tells you c must exist.

00:44:59.930 --> 00:45:05.590 align:middle line:90%


00:45:05.590 --> 00:45:10.720 align:middle line:84%
So now it's not so easy to
write down the C explicitly

00:45:10.720 --> 00:45:12.370 align:middle line:90%
in this case.

00:45:12.370 --> 00:45:12.880 align:middle line:90%
OK?

00:45:12.880 --> 00:45:16.630 align:middle line:84%
So here, we can
write the expression

00:45:16.630 --> 00:45:20.020 align:middle line:84%
for D regardless of the
choice of gamma matrices.

00:45:20.020 --> 00:45:23.200 align:middle line:84%
And here, it's actually
not so easy to write

00:45:23.200 --> 00:45:25.150 align:middle line:90%
the explicit form of C. OK?

00:45:25.150 --> 00:45:27.250 align:middle line:84%
So of course, for
specific choice

00:45:27.250 --> 00:45:30.160 align:middle line:84%
of representation of gamma,
you can write down C. OK?

00:45:30.160 --> 00:45:36.880 align:middle line:84%
But it's enough for us
to know that C exists.

00:45:36.880 --> 00:45:38.040 align:middle line:90%
OK.

00:45:38.040 --> 00:45:40.830 align:middle line:84%
And then this is
the transformation

00:45:40.830 --> 00:45:43.730 align:middle line:90%
for the charge conjugation.

00:45:43.730 --> 00:45:44.240 align:middle line:90%
OK?

00:45:44.240 --> 00:45:46.760 align:middle line:84%
It's the-- it transforms
this way with C

00:45:46.760 --> 00:45:48.420 align:middle line:90%
satisfy this condition.

00:45:48.420 --> 00:45:49.340 align:middle line:90%
OK?

00:45:49.340 --> 00:45:51.967 align:middle line:90%
C satisfy this condition.

00:45:51.967 --> 00:45:53.300 align:middle line:90%
Are there any questions on this?

00:45:53.300 --> 00:45:56.970 align:middle line:90%


00:45:56.970 --> 00:45:57.470 align:middle line:90%
OK.

00:45:57.470 --> 00:45:57.790 align:middle line:90%
Good.

00:45:57.790 --> 00:45:59.485 align:middle line:84%
And then we'll talk
about time reversal.

00:45:59.485 --> 00:46:08.420 align:middle line:90%


00:46:08.420 --> 00:46:09.680 align:middle line:90%
The idea is, again, similar.

00:46:09.680 --> 00:46:14.730 align:middle line:90%


00:46:14.730 --> 00:46:16.340 align:middle line:90%
So now we use UT.

00:46:16.340 --> 00:46:19.750 align:middle line:90%


00:46:19.750 --> 00:46:22.260 align:middle line:90%
OK?

00:46:22.260 --> 00:46:31.260 align:middle line:84%
So now suppose the psi
prime is equal to, again,

00:46:31.260 --> 00:46:36.360 align:middle line:90%
some matrix and then psi tx.

00:46:36.360 --> 00:46:36.860 align:middle line:90%
OK?

00:46:36.860 --> 00:46:41.230 align:middle line:84%
This is the direct analog
of this transformation.

00:46:41.230 --> 00:46:42.920 align:middle line:90%
OK?

00:46:42.920 --> 00:46:49.210 align:middle line:84%
And they're the same as the
psi prime x prime equal to DT

00:46:49.210 --> 00:46:50.710 align:middle line:90%
psi x.

00:46:50.710 --> 00:46:53.890 align:middle line:90%
And now x prime is equal to Tx.

00:46:53.890 --> 00:46:54.550 align:middle line:90%
OK?

00:46:54.550 --> 00:46:57.440 align:middle line:84%
You just flip the
time direction.

00:46:57.440 --> 00:47:01.740 align:middle line:84%
You can just flip
the time direction.

00:47:01.740 --> 00:47:06.830 align:middle line:84%
And now, again, the statement
of the time reversal--

00:47:06.830 --> 00:47:15.400 align:middle line:90%


00:47:15.400 --> 00:47:17.350 align:middle line:84%
again, the statement
of the time reversal--

00:47:17.350 --> 00:47:46.550 align:middle line:90%


00:47:46.550 --> 00:47:48.920 align:middle line:84%
Now the statement of the
time reversal is that,

00:47:48.920 --> 00:47:57.990 align:middle line:84%
given star, then it's
again that the gamma

00:47:57.990 --> 00:48:04.420 align:middle line:90%
mu partial mu prime minus m--

00:48:04.420 --> 00:48:07.210 align:middle line:84%
then you equal to psi
prime x prime equal to 0.

00:48:07.210 --> 00:48:09.240 align:middle line:90%
OK?

00:48:09.240 --> 00:48:12.440 align:middle line:84%
Again, this new
frame equal to 0.

00:48:12.440 --> 00:48:13.890 align:middle line:90%
OK.

00:48:13.890 --> 00:48:16.360 align:middle line:84%
And then the story
will be similar.

00:48:16.360 --> 00:48:16.860 align:middle line:90%
OK?

00:48:16.860 --> 00:48:19.020 align:middle line:84%
The story will be
similar what you--

00:48:19.020 --> 00:48:21.960 align:middle line:84%
and then, yeah,
you just again try

00:48:21.960 --> 00:48:24.170 align:middle line:84%
to match starting
from that equation.

00:48:24.170 --> 00:48:25.920 align:middle line:84%
And then you try to
match those equations.

00:48:25.920 --> 00:48:26.420 align:middle line:90%
OK?

00:48:26.420 --> 00:48:28.170 align:middle line:84%
So as I said, we'll
not go into detail.

00:48:28.170 --> 00:48:33.360 align:middle line:84%
And in the end, you find that
DT should satisfy this equation.

00:48:33.360 --> 00:48:34.020 align:middle line:90%
OK?

00:48:34.020 --> 00:48:36.135 align:middle line:84%
So you find DT should
satisfy this equation.

00:48:36.135 --> 00:48:37.230 align:middle line:90%
Our gamma mu t--

00:48:37.230 --> 00:48:40.590 align:middle line:90%


00:48:40.590 --> 00:48:50.440 align:middle line:84%
again, gamma mu transpose is
equal to DT transpose gamma mu

00:48:50.440 --> 00:48:55.930 align:middle line:90%
DT minus 1 transpose.

00:48:55.930 --> 00:48:56.860 align:middle line:90%
OK?

00:48:56.860 --> 00:48:59.620 align:middle line:84%
So gamma mu t has to
satisfy this equation.

00:48:59.620 --> 00:49:13.424 align:middle line:90%


00:49:13.424 --> 00:49:13.930 align:middle line:90%
OK?

00:49:13.930 --> 00:49:18.310 align:middle line:84%
So now if you compare
these two equations--

00:49:18.310 --> 00:49:20.650 align:middle line:90%
that and this equation--

00:49:20.650 --> 00:49:23.030 align:middle line:84%
they just differ
by a minus sign.

00:49:23.030 --> 00:49:23.530 align:middle line:90%
OK?

00:49:23.530 --> 00:49:25.270 align:middle line:84%
They just differ
by a minus sign.

00:49:25.270 --> 00:49:29.520 align:middle line:84%
So we conclude
that we can do this

00:49:29.520 --> 00:49:44.920 align:middle line:84%
by taking DT transpose
equal to C minus 1 gamma 5.

00:49:44.920 --> 00:49:47.290 align:middle line:84%
So the gamma 5 will
generate this minus sign.

00:49:47.290 --> 00:49:53.740 align:middle line:84%
But when you commute
with gamma mu, and then

00:49:53.740 --> 00:49:57.106 align:middle line:84%
that will account for
the extra minus sign.

00:49:57.106 --> 00:49:58.800 align:middle line:90%
OK?

00:49:58.800 --> 00:50:01.140 align:middle line:84%
That will account
for the extra minus.

00:50:01.140 --> 00:50:05.700 align:middle line:84%
So if you know C, then you
also know this time translation

00:50:05.700 --> 00:50:06.865 align:middle line:90%
symmetry matrix.

00:50:06.865 --> 00:50:07.365 align:middle line:90%
OK?

00:50:07.365 --> 00:50:12.570 align:middle line:90%


00:50:12.570 --> 00:50:13.290 align:middle line:90%
Yes?

00:50:13.290 --> 00:50:16.435 align:middle line:90%
AUDIENCE: [INAUDIBLE]

00:50:16.435 --> 00:50:17.310 align:middle line:90%
HONG LIU: Yeah, yeah.

00:50:17.310 --> 00:50:21.030 align:middle line:84%
Yeah, because of the--
only the operator itself,

00:50:21.030 --> 00:50:23.340 align:middle line:90%
only U is anti-linear.

00:50:23.340 --> 00:50:24.400 align:middle line:90%
Right?

00:50:24.400 --> 00:50:26.700 align:middle line:84%
t just goes right into
the specific matrix

00:50:26.700 --> 00:50:28.310 align:middle line:90%
acting on the spinner space.

00:50:28.310 --> 00:50:30.508 align:middle line:84%
And so that's just
an ordinary matrix.

00:50:30.508 --> 00:50:32.420 align:middle line:90%
Yeah.

00:50:32.420 --> 00:50:34.760 align:middle line:84%
Yeah, DT is just
ordinary numbers.

00:50:34.760 --> 00:50:36.520 align:middle line:84%
They're made up from
the gamma matrices.

00:50:36.520 --> 00:50:37.020 align:middle line:90%
Yeah.

00:50:37.020 --> 00:50:39.780 align:middle line:90%


00:50:39.780 --> 00:50:42.610 align:middle line:90%
Other questions?

00:50:42.610 --> 00:50:43.300 align:middle line:90%
Yes?

00:50:43.300 --> 00:50:44.770 align:middle line:84%
AUDIENCE: So is
parity a response

00:50:44.770 --> 00:50:46.245 align:middle line:90%
to the reflections in space?

00:50:46.245 --> 00:50:46.870 align:middle line:90%
HONG LIU: Yeah.

00:50:46.870 --> 00:50:49.030 align:middle line:84%
AUDIENCE: And time reversal--
is that a reflection in time?

00:50:49.030 --> 00:50:49.655 align:middle line:90%
HONG LIU: Yeah.

00:50:49.655 --> 00:50:57.090 align:middle line:84%
AUDIENCE: So can you,
like, boost into spacetime?

00:50:57.090 --> 00:51:00.960 align:middle line:84%
If you make a Lorentz boost,
could space reflections then--

00:51:00.960 --> 00:51:01.600 align:middle line:90%
HONG LIU: No.

00:51:01.600 --> 00:51:03.630 align:middle line:84%
No, they cannot
relate these two.

00:51:03.630 --> 00:51:06.568 align:middle line:84%
They are independent,
discrete transformations.

00:51:06.568 --> 00:51:07.860 align:middle line:90%
They're not related to a boost.

00:51:07.860 --> 00:51:09.047 align:middle line:90%
Yeah.

00:51:09.047 --> 00:51:10.380 align:middle line:90%
They are not related by a boost.

00:51:10.380 --> 00:51:12.750 align:middle line:90%


00:51:12.750 --> 00:51:14.250 align:middle line:84%
Yeah, if they're
related by a boost,

00:51:14.250 --> 00:51:19.620 align:middle line:84%
then we should only need
to worry about one of them.

00:51:19.620 --> 00:51:22.530 align:middle line:84%
Because the boost we
already understood.

00:51:22.530 --> 00:51:24.070 align:middle line:90%
Yeah.

00:51:24.070 --> 00:51:24.570 align:middle line:90%
Yeah.

00:51:24.570 --> 00:51:27.960 align:middle line:84%
So they are independent
discrete transformations.

00:51:27.960 --> 00:51:32.220 align:middle line:84%
They are not related
by any other continuous

00:51:32.220 --> 00:51:33.388 align:middle line:90%
transformation.

00:51:33.388 --> 00:51:36.980 align:middle line:90%


00:51:36.980 --> 00:51:39.480 align:middle line:90%
Other questions?

00:51:39.480 --> 00:51:40.140 align:middle line:90%
Yes?

00:51:40.140 --> 00:51:42.515 align:middle line:84%
AUDIENCE: There is only
one parity transformation

00:51:42.515 --> 00:51:44.522 align:middle line:90%
[INAUDIBLE].

00:51:44.522 --> 00:51:45.730 align:middle line:90%
HONG LIU: Yeah, that's right.

00:51:45.730 --> 00:51:46.930 align:middle line:90%
Yeah, yeah, yeah, yeah.

00:51:46.930 --> 00:51:48.760 align:middle line:84%
Just in any dimension,
you have just one.

00:51:48.760 --> 00:51:49.260 align:middle line:90%
Yeah.

00:51:49.260 --> 00:51:54.133 align:middle line:90%


00:51:54.133 --> 00:51:54.880 align:middle line:90%
Good.

00:51:54.880 --> 00:51:56.980 align:middle line:90%
Good.

00:51:56.980 --> 00:51:58.210 align:middle line:90%
OK.

00:51:58.210 --> 00:51:59.050 align:middle line:90%
Yes?

00:51:59.050 --> 00:52:01.092 align:middle line:84%
AUDIENCE: Yeah, I don't
know if this makes sense.

00:52:01.092 --> 00:52:04.945 align:middle line:84%
But is it possible to have a
situation where in one frame

00:52:04.945 --> 00:52:09.612 align:middle line:84%
a quantity is conserved
through parity or time reversal

00:52:09.612 --> 00:52:12.550 align:middle line:84%
transform, but then in
a Lorentz boosted frame

00:52:12.550 --> 00:52:15.704 align:middle line:84%
it's different than
what [INAUDIBLE]

00:52:15.704 --> 00:52:18.590 align:middle line:84%
or if it's conserved in
one, it's conserved in all?

00:52:18.590 --> 00:52:23.255 align:middle line:90%


00:52:23.255 --> 00:52:23.880 align:middle line:90%
HONG LIU: Yeah.

00:52:23.880 --> 00:52:24.380 align:middle line:90%
Yeah.

00:52:24.380 --> 00:52:27.270 align:middle line:90%


00:52:27.270 --> 00:52:29.840 align:middle line:84%
If it's conserved in one,
it's conserved in all of them.

00:52:29.840 --> 00:52:32.090 align:middle line:84%
Yeah, it's because the
conservation equation

00:52:32.090 --> 00:52:34.370 align:middle line:90%
is a covariant equation.

00:52:34.370 --> 00:52:37.871 align:middle line:84%
Yeah, because the conservation
equation is partial mu,

00:52:37.871 --> 00:52:39.980 align:middle line:90%
J mu mu equal to 0.

00:52:39.980 --> 00:52:42.530 align:middle line:84%
So this equation have the
same form in any frame.

00:52:42.530 --> 00:52:43.460 align:middle line:90%
Yeah.

00:52:43.460 --> 00:52:47.330 align:middle line:84%
Yeah, but the charge is
different in different frame.

00:52:47.330 --> 00:52:49.610 align:middle line:84%
Yeah, they're related
by the transformation.

00:52:49.610 --> 00:52:52.463 align:middle line:84%
Just in any frame, you can
define a conserved charge.

00:52:52.463 --> 00:52:52.963 align:middle line:90%
Yeah.

00:52:52.963 --> 00:52:57.220 align:middle line:90%


00:52:57.220 --> 00:52:58.980 align:middle line:90%
Good?

00:52:58.980 --> 00:53:00.270 align:middle line:90%
OK.

00:53:00.270 --> 00:53:03.210 align:middle line:84%
So now let's move on
to the next topic.

00:53:03.210 --> 00:53:07.060 align:middle line:90%


00:53:07.060 --> 00:53:10.738 align:middle line:84%
So now, finally, we'll
talk about path integrals

00:53:10.738 --> 00:53:11.280 align:middle line:90%
for fermions.

00:53:11.280 --> 00:53:15.000 align:middle line:90%


00:53:15.000 --> 00:53:17.830 align:middle line:84%
And with path integrals,
then we can do interactions.

00:53:17.830 --> 00:53:18.330 align:middle line:90%
OK?

00:53:18.330 --> 00:53:21.750 align:middle line:84%
Remember, when we know how to
do path integrals for scalar,

00:53:21.750 --> 00:53:23.520 align:middle line:84%
then we can easily
do interactions.

00:53:23.520 --> 00:53:26.070 align:middle line:84%
And now let's do the path
integral for fermion.

00:53:26.070 --> 00:53:30.510 align:middle line:84%
And then that will give
us a very simple way

00:53:30.510 --> 00:53:32.695 align:middle line:90%
to treat interacting theories.

00:53:32.695 --> 00:53:33.195 align:middle line:90%
OK?

00:53:33.195 --> 00:53:38.210 align:middle line:90%


00:53:38.210 --> 00:53:48.140 align:middle line:84%
So let's just recall in the path
integral we considered before--

00:53:48.140 --> 00:53:50.410 align:middle line:84%
so if you have ordinary
quantum mechanics--

00:53:50.410 --> 00:53:53.600 align:middle line:90%


00:53:53.600 --> 00:54:01.690 align:middle line:84%
so you have x and the p
commutator equal to i.

00:54:01.690 --> 00:54:04.900 align:middle line:84%
And then the path integral,
you just integrate over

00:54:04.900 --> 00:54:08.980 align:middle line:90%
all configurations of xt.

00:54:08.980 --> 00:54:09.640 align:middle line:90%
OK?

00:54:09.640 --> 00:54:12.140 align:middle line:84%
So that's what you do
in quantum mechanics.

00:54:12.140 --> 00:54:15.590 align:middle line:84%
So now xt is just an
ordinary function.

00:54:15.590 --> 00:54:16.090 align:middle line:90%
OK?

00:54:16.090 --> 00:54:18.730 align:middle line:84%
You just integrate over
all possible trajectories.

00:54:18.730 --> 00:54:22.570 align:middle line:84%
And xt is just an
ordinary function.

00:54:22.570 --> 00:54:27.160 align:middle line:84%
So now when we go to
quantum field theory,

00:54:27.160 --> 00:54:32.550 align:middle line:84%
then we have phi
then commute with--

00:54:32.550 --> 00:54:34.170 align:middle line:90%
yeah, yeah, yeah.

00:54:34.170 --> 00:54:37.498 align:middle line:84%
So let me just put a hat
just to show, to emphasize

00:54:37.498 --> 00:54:38.790 align:middle line:90%
that this is our operator here.

00:54:38.790 --> 00:54:42.270 align:middle line:90%
And the phi-- we have tx.

00:54:42.270 --> 00:54:46.305 align:middle line:84%
And then we have pi phi,
the conjugate momentum.

00:54:46.305 --> 00:54:48.940 align:middle line:90%


00:54:48.940 --> 00:54:51.730 align:middle line:84%
And again, this is equal
to i something, OK--

00:54:51.730 --> 00:54:52.565 align:middle line:90%
the delta function.

00:54:52.565 --> 00:54:53.065 align:middle line:90%
OK?

00:54:53.065 --> 00:54:57.210 align:middle line:90%


00:54:57.210 --> 00:54:57.780 align:middle line:90%
OK.

00:54:57.780 --> 00:54:58.660 align:middle line:90%
This is i.

00:54:58.660 --> 00:55:01.565 align:middle line:84%
And then you have
delta function.

00:55:01.565 --> 00:55:03.190 align:middle line:84%
And then when you do
the path integral,

00:55:03.190 --> 00:55:06.060 align:middle line:84%
you just integrate over all
configurations of phi xt.

00:55:06.060 --> 00:55:08.920 align:middle line:90%


00:55:08.920 --> 00:55:12.380 align:middle line:84%
And now this is just
a classical field.

00:55:12.380 --> 00:55:14.930 align:middle line:84%
Now, essentially, you view this
just as an ordinary function.

00:55:14.930 --> 00:55:15.470 align:middle line:90%
OK?

00:55:15.470 --> 00:55:18.260 align:middle line:84%
You integrate just all
possible values of it,

00:55:18.260 --> 00:55:19.880 align:middle line:84%
OK, all possible
functions of it,

00:55:19.880 --> 00:55:22.070 align:middle line:84%
OK, all possible
functions of phi.

00:55:22.070 --> 00:55:24.860 align:middle line:90%


00:55:24.860 --> 00:55:27.930 align:middle line:90%
So now we have to do fermions.

00:55:27.930 --> 00:55:30.420 align:middle line:90%
We have to do psi.

00:55:30.420 --> 00:55:33.390 align:middle line:90%
So psi is a little bit weird.

00:55:33.390 --> 00:55:36.030 align:middle line:84%
Because as we discussed
before, the psi--

00:55:36.030 --> 00:55:37.860 align:middle line:84%
we no longer have
the commutator.

00:55:37.860 --> 00:55:40.320 align:middle line:84%
You actually have
an anti-commutator.

00:55:40.320 --> 00:55:41.400 align:middle line:90%
OK?

00:55:41.400 --> 00:55:43.480 align:middle line:90%
So what do you do?

00:55:43.480 --> 00:55:43.980 align:middle line:90%
OK?

00:55:43.980 --> 00:55:46.410 align:middle line:84%
Do you still just
follow the same rule,

00:55:46.410 --> 00:55:49.080 align:middle line:84%
or do you actually have
to change the rule?

00:55:49.080 --> 00:55:50.100 align:middle line:90%
OK?

00:55:50.100 --> 00:55:54.510 align:middle line:84%
So that actually was
a question which--

00:55:54.510 --> 00:55:58.950 align:middle line:84%
so Feynman actually-- he came
up with this idea of the path

00:55:58.950 --> 00:56:02.870 align:middle line:90%
integral in 1948.

00:56:02.870 --> 00:56:05.030 align:middle line:84%
And then, of course,
generalization to scalar

00:56:05.030 --> 00:56:05.940 align:middle line:90%
will be immediate.

00:56:05.940 --> 00:56:07.380 align:middle line:90%
OK?

00:56:07.380 --> 00:56:11.180 align:middle line:84%
But then he wanted to
generalize to fermions.

00:56:11.180 --> 00:56:13.890 align:middle line:84%
But then for a long
time, he couldn't do it.

00:56:13.890 --> 00:56:17.040 align:middle line:84%
And if you just use this
method to do for fermions--

00:56:17.040 --> 00:56:18.910 align:middle line:90%
and they just didn't work.

00:56:18.910 --> 00:56:19.410 align:middle line:90%
OK?

00:56:19.410 --> 00:56:23.150 align:middle line:84%
We will not go
through that exercise.

00:56:23.150 --> 00:56:26.230 align:middle line:84%
If you does this for fermion,
it just does not work.

00:56:26.230 --> 00:56:29.130 align:middle line:84%
So for many years, he
couldn't find the answer.

00:56:29.130 --> 00:56:32.100 align:middle line:90%
But then around the 1962--

00:56:32.100 --> 00:56:34.530 align:middle line:90%
I think it's 1962 or 1965.

00:56:34.530 --> 00:56:38.985 align:middle line:84%
Anyway, just around
some early 1960s,

00:56:38.985 --> 00:56:42.030 align:middle line:84%
a former Soviet
physicist actually

00:56:42.030 --> 00:56:45.830 align:middle line:84%
came up with a brilliant idea
to solve this problem, OK--

00:56:45.830 --> 00:56:50.370 align:middle line:84%
and, like, 15 years later after
the discovery of path integral.

00:56:50.370 --> 00:56:55.640 align:middle line:84%
And so the basic idea is that
when you have a Dirac field--

00:56:55.640 --> 00:57:27.240 align:middle line:90%


00:57:27.240 --> 00:57:33.510 align:middle line:84%
so when you have a
Dirac field, then you

00:57:33.510 --> 00:57:35.250 align:middle line:90%
have anti-commutation relation.

00:57:35.250 --> 00:57:40.825 align:middle line:84%
psi and pi psi is an
anti-commutation relation.

00:57:40.825 --> 00:57:41.325 align:middle line:90%
OK?

00:57:41.325 --> 00:57:43.920 align:middle line:90%


00:57:43.920 --> 00:57:47.320 align:middle line:90%
It's an anti-commutator.

00:57:47.320 --> 00:57:50.530 align:middle line:84%
And then this guy
called Berezin.

00:57:50.530 --> 00:57:53.800 align:middle line:84%
Berezin then postulates
that, when you

00:57:53.800 --> 00:57:55.360 align:middle line:90%
do the path integral, what is--

00:57:55.360 --> 00:57:58.570 align:middle line:84%
in the path integral will be an
analog of the classical version

00:57:58.570 --> 00:58:02.660 align:middle line:84%
of this field, OK, the classical
version of this operator.

00:58:02.660 --> 00:58:04.540 align:middle line:84%
So then what would be
the classical version

00:58:04.540 --> 00:58:07.530 align:middle line:90%
of this anti-commutator?

00:58:07.530 --> 00:58:11.310 align:middle line:84%
And he had just written it just
should be some quantities which

00:58:11.310 --> 00:58:12.790 align:middle line:90%
anti-commute.

00:58:12.790 --> 00:58:13.290 align:middle line:90%
OK?

00:58:13.290 --> 00:58:19.920 align:middle line:84%
So when we say classical,
we put it in quotes

00:58:19.920 --> 00:58:23.083 align:middle line:84%
because we only do this
in the path integral.

00:58:23.083 --> 00:58:24.750 align:middle line:84%
And we're then, of
course, corresponding

00:58:24.750 --> 00:58:33.200 align:middle line:84%
to anti-commuting
fields, objects.

00:58:33.200 --> 00:58:34.405 align:middle line:90%
OK?

00:58:34.405 --> 00:58:36.030 align:middle line:84%
And then when you do
the path integral,

00:58:36.030 --> 00:58:39.780 align:middle line:90%
you just integrate over psi.

00:58:39.780 --> 00:58:41.800 align:middle line:84%
But psi is an
anti-commuting object.

00:58:41.800 --> 00:58:42.300 align:middle line:90%
OK?

00:58:42.300 --> 00:58:44.717 align:middle line:84%
It turns out that this just
brilliantly solve the problem.

00:58:44.717 --> 00:58:45.440 align:middle line:90%
OK?

00:58:45.440 --> 00:58:46.530 align:middle line:90%
It just immediately works.

00:58:46.530 --> 00:58:47.238 align:middle line:90%
It's very simple.

00:58:47.238 --> 00:58:49.900 align:middle line:90%


00:58:49.900 --> 00:58:52.090 align:middle line:84%
So in hindsight, it's
extremely simple.

00:58:52.090 --> 00:58:55.240 align:middle line:84%
But actually, Feynman couldn't
figure it out himself.

00:58:55.240 --> 00:58:56.130 align:middle line:90%
OK.

00:58:56.130 --> 00:58:58.790 align:middle line:90%
[LIGHT CHUCKLE] Right.

00:58:58.790 --> 00:59:01.730 align:middle line:84%
Anyway, so now let's
talk a little bit

00:59:01.730 --> 00:59:04.950 align:middle line:84%
about this
anti-commuting object.

00:59:04.950 --> 00:59:05.450 align:middle line:90%
OK?

00:59:05.450 --> 00:59:08.450 align:middle line:84%
So these are called the
Grassmann variables or numbers.

00:59:08.450 --> 00:59:22.770 align:middle line:90%


00:59:22.770 --> 00:59:28.720 align:middle line:84%
So this Grassmann object,
they just anti-commutes.

00:59:28.720 --> 00:59:30.380 align:middle line:90%
OK?

00:59:30.380 --> 00:59:35.760 align:middle line:84%
So they even anti-commute
with themself.

00:59:35.760 --> 00:59:39.520 align:middle line:84%
So if you anti-commute with
yourself, then the only thing

00:59:39.520 --> 00:59:42.560 align:middle line:84%
you can have is a theta
square that must be 0.

00:59:42.560 --> 00:59:43.060 align:middle line:90%
OK?

00:59:43.060 --> 00:59:45.160 align:middle line:84%
If you have anti-commuting
object theta,

00:59:45.160 --> 00:59:46.600 align:middle line:84%
and you can
anti-commute yourself,

00:59:46.600 --> 00:59:47.688 align:middle line:90%
it will be equal to 0.

00:59:47.688 --> 00:59:50.230 align:middle line:84%
Because theta-squared should be
equal to minus theta-squared.

00:59:50.230 --> 00:59:53.110 align:middle line:84%
And then, of course,
it can only be 0.

00:59:53.110 --> 00:59:56.800 align:middle line:84%
And so if you have two such
objects, theta and eta--

00:59:56.800 --> 01:00:00.750 align:middle line:90%
and then equal to minus eta.

01:00:00.750 --> 01:00:03.880 align:middle line:84%
And of course, eta-squared
should also be 0.

01:00:03.880 --> 01:00:04.380 align:middle line:90%
OK?

01:00:04.380 --> 01:00:08.300 align:middle line:90%


01:00:08.300 --> 01:00:12.110 align:middle line:84%
So this kind of property
make this kind of object

01:00:12.110 --> 01:00:13.690 align:middle line:90%
very simple.

01:00:13.690 --> 01:00:17.240 align:middle line:84%
So now we can also talk about
the functions of such objects.

01:00:17.240 --> 01:00:17.740 align:middle line:90%
OK?

01:00:17.740 --> 01:00:20.060 align:middle line:84%
We can also talk about the
function of such objects.

01:00:20.060 --> 01:00:26.140 align:middle line:84%
Say, if we have a
function f of theta--

01:00:26.140 --> 01:00:26.890 align:middle line:90%
OK?

01:00:26.890 --> 01:00:34.210 align:middle line:84%
So we define the function in
terms of the power series.

01:00:34.210 --> 01:00:36.650 align:middle line:90%
So suppose that we have an x--

01:00:36.650 --> 01:00:39.450 align:middle line:84%
say, if you write down a
Gaussian function or whatever

01:00:39.450 --> 01:00:44.200 align:middle line:84%
function, we define this
function by the power series.

01:00:44.200 --> 01:00:47.310 align:middle line:84%
So power series means
that to zeroth order,

01:00:47.310 --> 01:00:53.250 align:middle line:84%
we have f0 corresponding
to theta equal to 0,

01:00:53.250 --> 01:00:57.510 align:middle line:90%
and then you have f1 theta.

01:00:57.510 --> 01:00:58.500 align:middle line:90%
OK?

01:00:58.500 --> 01:01:01.920 align:middle line:84%
But now if you look at
f2 theta, theta-squared--

01:01:01.920 --> 01:01:03.180 align:middle line:90%
that would be 0.

01:01:03.180 --> 01:01:05.520 align:middle line:84%
So all higher-order
terms can be 0.

01:01:05.520 --> 01:01:08.490 align:middle line:90%
So you only have two terms.

01:01:08.490 --> 01:01:09.000 align:middle line:90%
OK?

01:01:09.000 --> 01:01:10.620 align:middle line:84%
So the only function
of this variable

01:01:10.620 --> 01:01:13.380 align:middle line:84%
will have two, terms, the
constant term and the term

01:01:13.380 --> 01:01:15.960 align:middle line:90%
proportional to theta.

01:01:15.960 --> 01:01:19.180 align:middle line:84%
And then the functions of such
variables are very simple.

01:01:19.180 --> 01:01:21.230 align:middle line:90%
OK.

01:01:21.230 --> 01:01:27.200 align:middle line:84%
And if you have two variables,
theta 1 and theta 2,

01:01:27.200 --> 01:01:30.680 align:middle line:84%
then you just expand
until you encounter--

01:01:30.680 --> 01:01:37.805 align:middle line:84%
you have f0 plus f1
theta 1 plus f2 theta

01:01:37.805 --> 01:01:43.670 align:middle line:90%
2 then plus f12 theta 1 theta 2.

01:01:43.670 --> 01:01:44.690 align:middle line:90%
That's it.

01:01:44.690 --> 01:01:46.280 align:middle line:90%
OK?

01:01:46.280 --> 01:01:46.780 align:middle line:90%
That's it.

01:01:46.780 --> 01:01:48.155 align:middle line:84%
Because for any
other term we are

01:01:48.155 --> 01:01:50.620 align:middle line:84%
involving either theta
1-squared or theta 2-squared,

01:01:50.620 --> 01:01:51.280 align:middle line:90%
that will be 0.

01:01:51.280 --> 01:01:55.110 align:middle line:90%


01:01:55.110 --> 01:01:59.540 align:middle line:84%
So the differentiation
will be extremely simple.

01:01:59.540 --> 01:02:01.690 align:middle line:84%
So if you're taking
d f theta d theta--

01:02:01.690 --> 01:02:04.510 align:middle line:90%


01:02:04.510 --> 01:02:06.987 align:middle line:84%
so you just-- because
this is the constant.

01:02:06.987 --> 01:02:08.320 align:middle line:90%
And so this is just equal to f1.

01:02:08.320 --> 01:02:12.010 align:middle line:90%


01:02:12.010 --> 01:02:13.000 align:middle line:90%
OK.

01:02:13.000 --> 01:02:18.340 align:middle line:84%
But the one thing-- you
do have to be careful

01:02:18.340 --> 01:02:20.485 align:middle line:84%
when you have this
multiple-variable function.

01:02:20.485 --> 01:02:23.000 align:middle line:90%


01:02:23.000 --> 01:02:24.740 align:middle line:84%
And then it actually
matters whether you

01:02:24.740 --> 01:02:26.390 align:middle line:84%
take the derivative
from the left

01:02:26.390 --> 01:02:29.060 align:middle line:84%
or take the derivative
from the right.

01:02:29.060 --> 01:02:30.090 align:middle line:90%
OK?

01:02:30.090 --> 01:02:34.660 align:middle line:84%
The direction you take the
derivative becomes important.

01:02:34.660 --> 01:02:35.160 align:middle line:90%
OK?

01:02:35.160 --> 01:02:42.970 align:middle line:84%
So for example, if we take
the derivative-- yeah.

01:02:42.970 --> 01:02:44.240 align:middle line:90%
Let me do it here.

01:02:44.240 --> 01:02:47.980 align:middle line:84%
So if you, from the left,
take the derivative f theta

01:02:47.980 --> 01:02:55.980 align:middle line:84%
1 on this f theta 1, theta
2, and then this term is 0.

01:02:55.980 --> 01:02:58.890 align:middle line:90%
This term gives us f1.

01:02:58.890 --> 01:03:01.378 align:middle line:84%
This term does not
depend on theta 1.

01:03:01.378 --> 01:03:03.670 align:middle line:84%
And so this term is also 0
when we take the derivative.

01:03:03.670 --> 01:03:05.220 align:middle line:90%
And here, we have theta 1.

01:03:05.220 --> 01:03:08.920 align:middle line:90%
So we just get f12 theta 2.

01:03:08.920 --> 01:03:09.420 align:middle line:90%
OK?

01:03:09.420 --> 01:03:12.780 align:middle line:84%
So this is derivative
from the left.

01:03:12.780 --> 01:03:24.615 align:middle line:84%
But now if I take the derivative
from the right, theta 1, OK--

01:03:24.615 --> 01:03:29.240 align:middle line:90%


01:03:29.240 --> 01:03:33.250 align:middle line:84%
so this arrow means you take
the derivative from the right.

01:03:33.250 --> 01:03:35.220 align:middle line:84%
And again, the same
thing for this one.

01:03:35.220 --> 01:03:35.970 align:middle line:90%
It doesn't matter.

01:03:35.970 --> 01:03:37.500 align:middle line:90%
There's only one theta 1.

01:03:37.500 --> 01:03:39.300 align:middle line:90%
So you just get f1.

01:03:39.300 --> 01:03:41.130 align:middle line:90%
Again, these two will be 0.

01:03:41.130 --> 01:03:44.430 align:middle line:84%
But now if we take the
derivative from the right,

01:03:44.430 --> 01:03:47.430 align:middle line:84%
then you have to move theta
1 to the right of theta 2

01:03:47.430 --> 01:03:49.000 align:middle line:84%
first because there's
a theta 2 here.

01:03:49.000 --> 01:03:50.542 align:middle line:84%
And then that gives
you a minus sign.

01:03:50.542 --> 01:03:54.000 align:middle line:90%
So it will give you f12 theta 2.

01:03:54.000 --> 01:03:56.160 align:middle line:90%
So you differ by a minus sign.

01:03:56.160 --> 01:03:57.330 align:middle line:90%
OK.

01:03:57.330 --> 01:04:01.620 align:middle line:84%
So from now on, without
specifically mentioning,

01:04:01.620 --> 01:04:03.340 align:middle line:84%
we always take
derivative from the left.

01:04:03.340 --> 01:04:04.042 align:middle line:90%
OK?

01:04:04.042 --> 01:04:05.500 align:middle line:84%
So the convention
is we always take

01:04:05.500 --> 01:04:06.625 align:middle line:90%
the derivative of the left.

01:04:06.625 --> 01:04:07.499 align:middle line:90%
Yes?

01:04:07.499 --> 01:04:11.563 align:middle line:90%
AUDIENCE: [INAUDIBLE]

01:04:11.563 --> 01:04:12.230 align:middle line:90%
HONG LIU: Sorry?

01:04:12.230 --> 01:04:15.140 align:middle line:84%
AUDIENCE: [INAUDIBLE]
coefficient [INAUDIBLE] f0

01:04:15.140 --> 01:04:16.600 align:middle line:90%
[INAUDIBLE]

01:04:16.600 --> 01:04:17.555 align:middle line:90%


01:04:17.555 --> 01:04:18.680 align:middle line:90%
HONG LIU: What do you mean?

01:04:18.680 --> 01:04:20.120 align:middle line:90%
Like, what type of object?

01:04:20.120 --> 01:04:22.340 align:middle line:90%
AUDIENCE: Like, [INAUDIBLE]

01:04:22.340 --> 01:04:23.450 align:middle line:90%
HONG LIU: Oh, yeah, yeah.

01:04:23.450 --> 01:04:24.688 align:middle line:90%
They can be complex numbers.

01:04:24.688 --> 01:04:25.730 align:middle line:90%
They can be real numbers.

01:04:25.730 --> 01:04:28.415 align:middle line:84%
It's just-- f0, f1
is just some numbers.

01:04:28.415 --> 01:04:30.920 align:middle line:90%


01:04:30.920 --> 01:04:31.420 align:middle line:90%
Yeah.

01:04:31.420 --> 01:04:33.400 align:middle line:90%
For example-- yeah.

01:04:33.400 --> 01:04:36.715 align:middle line:84%
So suppose f is a
Gaussian function.

01:04:36.715 --> 01:04:41.290 align:middle line:90%


01:04:41.290 --> 01:04:42.370 align:middle line:90%
OK?

01:04:42.370 --> 01:04:44.890 align:middle line:90%
And then you just write 1--

01:04:44.890 --> 01:04:46.090 align:middle line:90%
yeah, yeah.

01:04:46.090 --> 01:04:47.260 align:middle line:90%
Say exponential theta.

01:04:47.260 --> 01:04:48.630 align:middle line:90%
1 plus theta.

01:04:48.630 --> 01:04:49.130 align:middle line:90%
OK.

01:04:49.130 --> 01:04:50.710 align:middle line:84%
Is exponential from
just one-- just

01:04:50.710 --> 01:04:52.690 align:middle line:90%
do Taylor expansion of theta.

01:04:52.690 --> 01:04:55.420 align:middle line:84%
And the f0 and 1 just come
from the Taylor expansion

01:04:55.420 --> 01:04:57.760 align:middle line:84%
of whatever function
you're looking at here.

01:04:57.760 --> 01:04:58.540 align:middle line:90%
OK.

01:04:58.540 --> 01:05:00.340 align:middle line:84%
Yeah, similarly--
the same thing.

01:05:00.340 --> 01:05:02.000 align:middle line:90%
1 plus theta.

01:05:02.000 --> 01:05:03.890 align:middle line:84%
So you have a
function like that.

01:05:03.890 --> 01:05:06.552 align:middle line:90%
So it's just 1 minus theta.

01:05:06.552 --> 01:05:07.510 align:middle line:90%
Yeah, it's very simple.

01:05:07.510 --> 01:05:09.280 align:middle line:90%
OK?

01:05:09.280 --> 01:05:12.650 align:middle line:84%
So the functions of these
variables are very simple.

01:05:12.650 --> 01:05:13.150 align:middle line:90%
OK.

01:05:13.150 --> 01:05:16.940 align:middle line:90%


01:05:16.940 --> 01:05:17.720 align:middle line:90%
Good.

01:05:17.720 --> 01:05:20.330 align:middle line:84%
So we talked about
the functions.

01:05:20.330 --> 01:05:22.890 align:middle line:84%
So we talked about
the functions.

01:05:22.890 --> 01:05:24.720 align:middle line:90%
Also, we talked about the--

01:05:24.720 --> 01:05:26.130 align:middle line:90%
yeah, the functions of them.

01:05:26.130 --> 01:05:27.390 align:middle line:90%
We talked about derivatives.

01:05:27.390 --> 01:05:30.660 align:middle line:84%
And then we need to talk about
the integrals of them, OK--

01:05:30.660 --> 01:05:32.650 align:middle line:90%
so how to define integration.

01:05:32.650 --> 01:05:36.080 align:middle line:90%


01:05:36.080 --> 01:05:39.200 align:middle line:84%
So now let's talk about
how to do integration.

01:05:39.200 --> 01:05:41.540 align:middle line:90%
So how do you do integrals?

01:05:41.540 --> 01:05:44.612 align:middle line:90%
Because we need to do the--

01:05:44.612 --> 01:05:46.070 align:middle line:84%
to be able to do
path integrals, we

01:05:46.070 --> 01:05:48.350 align:middle line:84%
need to integrate
over such objects.

01:05:48.350 --> 01:05:50.280 align:middle line:90%
OK?

01:05:50.280 --> 01:05:53.040 align:middle line:84%
So to do integration,
then we have

01:05:53.040 --> 01:05:56.755 align:middle line:84%
to specify the rule
for integration

01:05:56.755 --> 01:05:58.880 align:middle line:84%
because it's no longer
intuitive like a derivative.

01:05:58.880 --> 01:06:00.380 align:middle line:84%
You can just take
the derivative.

01:06:00.380 --> 01:06:01.100 align:middle line:90%
OK?

01:06:01.100 --> 01:06:05.270 align:middle line:84%
So integration-- we have
to specify the rule.

01:06:05.270 --> 01:06:09.230 align:middle line:84%
And then the idea is
you use the standard,

01:06:09.230 --> 01:06:12.140 align:middle line:84%
the rule for integration,
and then translate--

01:06:12.140 --> 01:06:13.800 align:middle line:90%
apply to this case.

01:06:13.800 --> 01:06:14.300 align:middle line:90%
OK?

01:06:14.300 --> 01:06:20.030 align:middle line:84%
So the rules we are going to
require for the integration

01:06:20.030 --> 01:06:21.219 align:middle line:90%
are the following.

01:06:21.219 --> 01:06:56.190 align:middle line:90%


01:06:56.190 --> 01:07:03.060 align:middle line:84%
So the first rule is that the
integration should be linear.

01:07:03.060 --> 01:07:03.560 align:middle line:90%
OK?

01:07:03.560 --> 01:07:07.160 align:middle line:84%
So if I have, say, an
integration of f theta--

01:07:07.160 --> 01:07:13.772 align:middle line:84%
I have two functions, f
theta a plus g theta b.

01:07:13.772 --> 01:07:15.540 align:middle line:90%
And a and b are some numbers.

01:07:15.540 --> 01:07:18.120 align:middle line:84%
They can either be
ordinary C-numbers.

01:07:18.120 --> 01:07:21.770 align:middle line:84%
They also can be Grassmann
numbers, which anti-commute.

01:07:21.770 --> 01:07:23.090 align:middle line:90%
OK?

01:07:23.090 --> 01:07:25.140 align:middle line:90%
And then by linear--

01:07:25.140 --> 01:07:29.090 align:middle line:90%


01:07:29.090 --> 01:07:31.070 align:middle line:84%
but the integration
operation is linear.

01:07:31.070 --> 01:07:36.430 align:middle line:84%
This means that this is
the same as the d theta f

01:07:36.430 --> 01:07:42.030 align:middle line:84%
theta times a, where
a just constant

01:07:42.030 --> 01:07:44.730 align:middle line:84%
so you can just take it
outside of the integral.

01:07:44.730 --> 01:07:50.610 align:middle line:90%
d theta, g theta b.

01:07:50.610 --> 01:07:52.000 align:middle line:90%
OK?

01:07:52.000 --> 01:07:55.330 align:middle line:84%
So it should be
linear, the operation.

01:07:55.330 --> 01:07:58.120 align:middle line:90%


01:07:58.120 --> 01:08:05.260 align:middle line:84%
And the second rule which we
require for such integration

01:08:05.260 --> 01:08:07.980 align:middle line:84%
is that the total
derivative should be 0.

01:08:07.980 --> 01:08:09.860 align:middle line:90%
OK?

01:08:09.860 --> 01:08:20.160 align:middle line:84%
So the total derivative
should be equal to 0.

01:08:20.160 --> 01:08:22.750 align:middle line:90%


01:08:22.750 --> 01:08:24.229 align:middle line:90%
OK?

01:08:24.229 --> 01:08:26.359 align:middle line:84%
And turns out that
these two conditions

01:08:26.359 --> 01:08:31.069 align:middle line:84%
can be uniquely then it
is enough to actually fix

01:08:31.069 --> 01:08:36.390 align:middle line:84%
the other properties
of the integral.

01:08:36.390 --> 01:08:36.890 align:middle line:90%
OK?

01:08:36.890 --> 01:08:40.569 align:middle line:84%
Yeah, it just can be
used to fix the integral.

01:08:40.569 --> 01:08:41.290 align:middle line:90%
Yes?

01:08:41.290 --> 01:08:47.750 align:middle line:84%
AUDIENCE: [INAUDIBLE] in not
all derivatives [INAUDIBLE]

01:08:47.750 --> 01:08:49.435 align:middle line:90%
when the boundary [INAUDIBLE]?

01:08:49.435 --> 01:08:50.060 align:middle line:90%
HONG LIU: Yeah.

01:08:50.060 --> 01:08:51.569 align:middle line:90%
Here, we don't have a boundary.

01:08:51.569 --> 01:08:53.430 align:middle line:90%
It's not required to be 0.

01:08:53.430 --> 01:08:55.080 align:middle line:84%
Yeah, so theta--
yeah, we don't know

01:08:55.080 --> 01:08:58.090 align:middle line:84%
how to specify the
boundaries for theta.

01:08:58.090 --> 01:08:59.744 align:middle line:90%
Yes?

01:08:59.744 --> 01:09:05.609 align:middle line:84%
AUDIENCE: How does [INAUDIBLE]
multiply by it [INAUDIBLE]

01:09:05.609 --> 01:09:07.890 align:middle line:90%
HONG LIU: Yeah.

01:09:07.890 --> 01:09:09.979 align:middle line:90%
You say this one?

01:09:09.979 --> 01:09:13.920 align:middle line:84%
AUDIENCE: No,
[INAUDIBLE] symbol.

01:09:13.920 --> 01:09:15.819 align:middle line:90%
HONG LIU: Yeah.

01:09:15.819 --> 01:09:16.319 align:middle line:90%
Sorry.

01:09:16.319 --> 01:09:16.950 align:middle line:90%
Say it again?

01:09:16.950 --> 01:09:19.615 align:middle line:84%
AUDIENCE: Like,
suppose [INAUDIBLE]

01:09:19.615 --> 01:09:21.240 align:middle line:84%
HONG LIU: Yeah, yeah,
yeah, yeah, yeah.

01:09:21.240 --> 01:09:23.535 align:middle line:84%
Indeed, it does not
determine-- yeah,

01:09:23.535 --> 01:09:25.359 align:middle line:84%
it only determines
up to a constant.

01:09:25.359 --> 01:09:25.859 align:middle line:90%
Yeah.

01:09:25.859 --> 01:09:28.955 align:middle line:84%
Yeah, but then you
can fix that constant.

01:09:28.955 --> 01:09:30.830 align:middle line:84%
Yeah, I'm going to talk
about it immediately.

01:09:30.830 --> 01:09:31.859 align:middle line:90%
Yeah.

01:09:31.859 --> 01:09:33.380 align:middle line:90%
Good?

01:09:33.380 --> 01:09:36.840 align:middle line:90%
So we have these two conditions.

01:09:36.840 --> 01:09:39.859 align:middle line:84%
So let's see what these
two conditions tell us.

01:09:39.859 --> 01:09:40.700 align:middle line:90%
OK?

01:09:40.700 --> 01:09:46.609 align:middle line:84%
So from this condition two,
we conclude that d theta f2

01:09:46.609 --> 01:09:50.880 align:middle line:90%
is equal to 0 because of the--

01:09:50.880 --> 01:09:52.770 align:middle line:90%
the derivative of f theta--

01:09:52.770 --> 01:09:54.805 align:middle line:90%
or sorry, just f1 equal to 0.

01:09:54.805 --> 01:09:55.305 align:middle line:90%
OK?

01:09:55.305 --> 01:09:58.050 align:middle line:90%


01:09:58.050 --> 01:10:01.360 align:middle line:84%
We write-- yeah,
just f1 equal to 0.

01:10:01.360 --> 01:10:06.708 align:middle line:84%
And since f1 is a C-number,
we can take this f1 outside

01:10:06.708 --> 01:10:07.375 align:middle line:90%
of the integral.

01:10:07.375 --> 01:10:11.390 align:middle line:90%


01:10:11.390 --> 01:10:13.490 align:middle line:84%
So that means this
is equal to 0.

01:10:13.490 --> 01:10:17.210 align:middle line:84%
So that means that the
theta 1 is equal to 0.

01:10:17.210 --> 01:10:18.260 align:middle line:90%
OK?

01:10:18.260 --> 01:10:23.305 align:middle line:84%
So just d theta any
constant, it should be 0.

01:10:23.305 --> 01:10:24.513 align:middle line:90%
OK.

01:10:24.513 --> 01:10:25.930 align:middle line:84%
Just any constant,
it should be 0.

01:10:25.930 --> 01:10:28.270 align:middle line:90%
OK?

01:10:28.270 --> 01:10:33.785 align:middle line:84%
So this is one condition--
so one thing we deduce.

01:10:33.785 --> 01:10:36.440 align:middle line:90%


01:10:36.440 --> 01:10:38.510 align:middle line:90%
OK?

01:10:38.510 --> 01:10:40.550 align:middle line:84%
So now let's call
this property 3 --

01:10:40.550 --> 01:10:44.160 align:middle line:84%
equation three this one
follows from the one and two.

01:10:44.160 --> 01:10:50.032 align:middle line:90%


01:10:50.032 --> 01:10:55.460 align:middle line:84%
So now remember from the
property of f theta equal

01:10:55.460 --> 01:10:57.740 align:middle line:90%
to f0 plus f1 theta--

01:10:57.740 --> 01:11:04.961 align:middle line:84%
so that means d theta f
theta is equal to d theta

01:11:04.961 --> 01:11:07.680 align:middle line:90%
f0 plus f1 theta.

01:11:07.680 --> 01:11:10.450 align:middle line:90%


01:11:10.450 --> 01:11:10.950 align:middle line:90%
OK?

01:11:10.950 --> 01:11:14.680 align:middle line:84%
So that means-- so if we
use this property one,

01:11:14.680 --> 01:11:18.280 align:middle line:84%
you can take this
constant outside it.

01:11:18.280 --> 01:11:18.780 align:middle line:90%
OK?

01:11:18.780 --> 01:11:21.390 align:middle line:90%
And this term is 0.

01:11:21.390 --> 01:11:24.550 align:middle line:90%
So you just get the second term.

01:11:24.550 --> 01:11:27.670 align:middle line:84%
You just get f1, d
theta and the theta.

01:11:27.670 --> 01:11:31.300 align:middle line:90%


01:11:31.300 --> 01:11:33.800 align:middle line:84%
So now if we just fix
the value of this object,

01:11:33.800 --> 01:11:36.570 align:middle line:90%
then we fix the full integral.

01:11:36.570 --> 01:11:38.370 align:middle line:90%
OK?

01:11:38.370 --> 01:11:42.000 align:middle line:84%
And so you just fix
this object to be 1.

01:11:42.000 --> 01:11:43.800 align:middle line:90%
So we just fix--

01:11:43.800 --> 01:11:46.320 align:middle line:90%


01:11:46.320 --> 01:11:48.650 align:middle line:90%
define it to be 1.

01:11:48.650 --> 01:11:49.150 align:middle line:90%
OK?

01:11:49.150 --> 01:11:50.640 align:middle line:90%
Define up to a constant.

01:11:50.640 --> 01:11:52.210 align:middle line:90%
We just define it be 1.

01:11:52.210 --> 01:11:52.710 align:middle line:90%
OK.

01:11:52.710 --> 01:11:54.640 align:middle line:90%
So that's it.

01:11:54.640 --> 01:11:57.640 align:middle line:84%
So that is fully
specified to the rule.

01:11:57.640 --> 01:12:07.465 align:middle line:84%
And then we conclude that d
theta f theta is equal to f1.

01:12:07.465 --> 01:12:13.370 align:middle line:90%


01:12:13.370 --> 01:12:15.005 align:middle line:90%
So now you notice--

01:12:15.005 --> 01:12:17.600 align:middle line:90%


01:12:17.600 --> 01:12:20.310 align:middle line:84%
yeah, so this is the
rule you should remember.

01:12:20.310 --> 01:12:24.690 align:middle line:84%
So now you notice that
this is the same as you

01:12:24.690 --> 01:12:26.100 align:middle line:90%
take the derivative of theta.

01:12:26.100 --> 01:12:28.590 align:middle line:90%


01:12:28.590 --> 01:12:29.090 align:middle line:90%
OK?

01:12:29.090 --> 01:12:33.780 align:middle line:84%
So this story is a
little bit funny.

01:12:33.780 --> 01:12:34.280 align:middle line:90%
OK?

01:12:34.280 --> 01:12:35.870 align:middle line:84%
So the integration
of a function is

01:12:35.870 --> 01:12:38.500 align:middle line:84%
the same as the derivative
of the function.

01:12:38.500 --> 01:12:39.000 align:middle line:90%
OK.

01:12:39.000 --> 01:12:41.450 align:middle line:90%


01:12:41.450 --> 01:12:41.950 align:middle line:90%
OK.

01:12:41.950 --> 01:12:43.120 align:middle line:90%
That's it.

01:12:43.120 --> 01:12:49.210 align:middle line:84%
So if the calculus for ordinary
variables are such simple of--

01:12:49.210 --> 01:12:53.330 align:middle line:84%
[CHUCKLES]---- when we learn
calculus, it will be much,

01:12:53.330 --> 01:12:55.330 align:middle line:90%
much easier.

01:12:55.330 --> 01:12:55.850 align:middle line:90%
But yeah.

01:12:55.850 --> 01:12:56.350 align:middle line:90%
Yeah.

01:12:56.350 --> 01:12:58.625 align:middle line:84%
So these Grassmann
variables are very simple.

01:12:58.625 --> 01:12:59.125 align:middle line:90%
OK.

01:12:59.125 --> 01:13:02.830 align:middle line:90%


01:13:02.830 --> 01:13:05.440 align:middle line:90%
Any questions on this?

01:13:05.440 --> 01:13:06.550 align:middle line:90%
Yes?

01:13:06.550 --> 01:13:09.878 align:middle line:84%
AUDIENCE: So what's the
point in defining integration

01:13:09.878 --> 01:13:11.630 align:middle line:84%
in this way if it
ends up just being

01:13:11.630 --> 01:13:13.702 align:middle line:90%
the same as the derivative one?

01:13:13.702 --> 01:13:17.264 align:middle line:84%
Is it useful to have both
integration and derivatives

01:13:17.264 --> 01:13:20.140 align:middle line:84%
in [INAUDIBLE] if they
did the same thing?

01:13:20.140 --> 01:13:21.970 align:middle line:90%
HONG LIU: Yeah.

01:13:21.970 --> 01:13:24.490 align:middle line:84%
So this is for the
single variable, right?

01:13:24.490 --> 01:13:26.050 align:middle line:84%
When you go to
the multivariable,

01:13:26.050 --> 01:13:28.430 align:middle line:84%
then things are a little
bit more intricate.

01:13:28.430 --> 01:13:28.930 align:middle line:90%
Yeah.

01:13:28.930 --> 01:13:31.360 align:middle line:90%
Yeah, but this rule will apply.

01:13:31.360 --> 01:13:34.930 align:middle line:84%
This integration
rule is general.

01:13:34.930 --> 01:13:35.946 align:middle line:90%
Yes?

01:13:35.946 --> 01:13:37.696 align:middle line:84%
AUDIENCE: Why dont you
define the integral

01:13:37.696 --> 01:13:41.010 align:middle line:84%
as the opposite of the-- like,
the reverse the derivative?

01:13:41.010 --> 01:13:43.176 align:middle line:84%
Because now, for example,
integrating and taking

01:13:43.176 --> 01:13:46.722 align:middle line:90%
the derivative [INAUDIBLE]

01:13:46.722 --> 01:13:47.680 align:middle line:90%
HONG LIU: That's right.

01:13:47.680 --> 01:13:50.290 align:middle line:84%
Yeah, just you don't know how
to do that inverse operation

01:13:50.290 --> 01:13:50.948 align:middle line:90%
anymore.

01:13:50.948 --> 01:13:51.490 align:middle line:90%
AUDIENCE: OK.

01:13:51.490 --> 01:13:51.760 align:middle line:90%
HONG LIU: Yeah.

01:13:51.760 --> 01:13:54.100 align:middle line:84%
Just those kind of objects
are sufficiently weird.

01:13:54.100 --> 01:13:55.630 align:middle line:90%
Yeah, it's not easy.

01:13:55.630 --> 01:13:56.680 align:middle line:90%
Yeah.

01:13:56.680 --> 01:14:00.940 align:middle line:90%
So the basic idea, I think--

01:14:00.940 --> 01:14:04.680 align:middle line:84%
so you say, why Feynman
didn't come up with this?

01:14:04.680 --> 01:14:06.270 align:middle line:84%
Because if you just
say, oh, I have

01:14:06.270 --> 01:14:08.890 align:middle line:84%
to do anti-commuting,
that's not enough.

01:14:08.890 --> 01:14:11.960 align:middle line:84%
You have to invent the
whole calculus for this kind

01:14:11.960 --> 01:14:13.550 align:middle line:90%
of anti-commuting object.

01:14:13.550 --> 01:14:15.470 align:middle line:84%
And now the question
is that, how

01:14:15.470 --> 01:14:17.960 align:middle line:84%
do you invent such a
calculus so that it's

01:14:17.960 --> 01:14:21.800 align:middle line:84%
as natural as possible
compared to the standard way?

01:14:21.800 --> 01:14:22.550 align:middle line:90%
Yeah, yeah.

01:14:22.550 --> 01:14:24.830 align:middle line:84%
And so I think this
is a set of rule

01:14:24.830 --> 01:14:27.440 align:middle line:90%
which seems to work very well.

01:14:27.440 --> 01:14:31.100 align:middle line:90%


01:14:31.100 --> 01:14:32.890 align:middle line:90%
OK.

01:14:32.890 --> 01:14:33.830 align:middle line:90%
So yeah.

01:14:33.830 --> 01:14:34.330 align:middle line:90%
Yeah.

01:14:34.330 --> 01:14:36.495 align:middle line:84%
So when we say that
Feynman missed it,

01:14:36.495 --> 01:14:37.870 align:middle line:84%
it's not that he
missed something

01:14:37.870 --> 01:14:39.370 align:middle line:90%
that's just a trivial idea.

01:14:39.370 --> 01:14:41.200 align:middle line:84%
He actually missed
the whole calculus.

01:14:41.200 --> 01:14:41.700 align:middle line:90%
OK.

01:14:41.700 --> 01:14:52.198 align:middle line:90%


01:14:52.198 --> 01:14:53.990 align:middle line:84%
So now let's just make
a couple of remarks.

01:14:53.990 --> 01:15:06.950 align:middle line:90%


01:15:06.950 --> 01:15:19.910 align:middle line:84%
So suppose eta is another
Grassmann variable.

01:15:19.910 --> 01:15:20.410 align:middle line:90%
OK?

01:15:20.410 --> 01:15:24.196 align:middle line:90%


01:15:24.196 --> 01:15:27.320 align:middle line:84%
It means that it's
also anti-commute.

01:15:27.320 --> 01:15:34.600 align:middle line:84%
So now if we put
eta before f theta--

01:15:34.600 --> 01:15:37.420 align:middle line:90%


01:15:37.420 --> 01:15:39.500 align:middle line:90%
OK?

01:15:39.500 --> 01:15:43.930 align:middle line:84%
And now, because eta have
nothing to do with theta,

01:15:43.930 --> 01:15:47.880 align:middle line:84%
you should be able to take
it outside of the integral.

01:15:47.880 --> 01:15:50.310 align:middle line:84%
But different
variables anti-commute.

01:15:50.310 --> 01:15:54.660 align:middle line:84%
So when you take eta
outside the thing,

01:15:54.660 --> 01:15:56.690 align:middle line:84%
the rule is that you
can just take it out.

01:15:56.690 --> 01:15:58.290 align:middle line:84%
It commutes through
this d theta.

01:15:58.290 --> 01:16:00.180 align:middle line:90%
You get a minus sign.

01:16:00.180 --> 01:16:07.200 align:middle line:84%
So you just get the
eta d theta f theta.

01:16:07.200 --> 01:16:07.770 align:middle line:90%
OK?

01:16:07.770 --> 01:16:11.601 align:middle line:90%
So this becomes just f1.

01:16:11.601 --> 01:16:13.550 align:middle line:90%
OK.

01:16:13.550 --> 01:16:17.990 align:middle line:84%
So you can show that this
rule, which I'm saying here,

01:16:17.990 --> 01:16:21.430 align:middle line:84%
is compatible with
also this rule.

01:16:21.430 --> 01:16:21.930 align:middle line:90%
OK?

01:16:21.930 --> 01:16:23.580 align:middle line:84%
It's also compatible
with this rule

01:16:23.580 --> 01:16:26.580 align:middle line:84%
because you can
also commute the eta

01:16:26.580 --> 01:16:29.040 align:middle line:90%
with f inside of the integral.

01:16:29.040 --> 01:16:31.090 align:middle line:90%
And then you can use this rule.

01:16:31.090 --> 01:16:33.090 align:middle line:84%
And you can show these
two rules are compatible.

01:16:33.090 --> 01:16:33.590 align:middle line:90%
OK?

01:16:33.590 --> 01:16:36.230 align:middle line:90%


01:16:36.230 --> 01:16:37.475 align:middle line:90%
So this is the first remark.

01:16:37.475 --> 01:16:47.030 align:middle line:90%


01:16:47.030 --> 01:16:49.160 align:middle line:90%
The second remark is that the--

01:16:49.160 --> 01:16:50.900 align:middle line:84%
it's hard to do
integration by parts.

01:16:50.900 --> 01:16:55.100 align:middle line:90%


01:16:55.100 --> 01:17:01.320 align:middle line:84%
So now let's-- consider
integral like this.

01:17:01.320 --> 01:17:03.520 align:middle line:90%
a would be some ordinary--

01:17:03.520 --> 01:17:04.755 align:middle line:90%
a is just some number.

01:17:04.755 --> 01:17:07.330 align:middle line:90%


01:17:07.330 --> 01:17:09.580 align:middle line:90%
OK?

01:17:09.580 --> 01:17:14.380 align:middle line:84%
So you can just
expand this trivially.

01:17:14.380 --> 01:17:19.470 align:middle line:84%
So this just gives-- you
just find this f1 times a.

01:17:19.470 --> 01:17:20.190 align:middle line:90%
OK?

01:17:20.190 --> 01:17:22.080 align:middle line:90%
Just f1 times a.

01:17:22.080 --> 01:17:23.970 align:middle line:90%
OK.

01:17:23.970 --> 01:17:27.360 align:middle line:84%
And of course, f1
times a is also

01:17:27.360 --> 01:17:35.780 align:middle line:84%
the same as a times d
theta prime f theta prime.

01:17:35.780 --> 01:17:40.250 align:middle line:84%
If theta prime is a theta
because it's a dummy variable.

01:17:40.250 --> 01:17:41.000 align:middle line:90%
It doesn't matter.

01:17:41.000 --> 01:17:41.500 align:middle line:90%
OK?

01:17:41.500 --> 01:17:44.080 align:middle line:90%


01:17:44.080 --> 01:17:46.960 align:middle line:90%
So this tells you--

01:17:46.960 --> 01:17:49.570 align:middle line:90%


01:17:49.570 --> 01:17:51.580 align:middle line:84%
so then when
comparing these two--

01:17:51.580 --> 01:17:54.020 align:middle line:90%


01:17:54.020 --> 01:17:54.520 align:middle line:90%
OK.

01:17:54.520 --> 01:17:58.040 align:middle line:84%
So this is like a
change of variable.

01:17:58.040 --> 01:17:58.880 align:middle line:90%
OK?

01:17:58.880 --> 01:17:59.600 align:middle line:90%
Yes?

01:17:59.600 --> 01:18:03.056 align:middle line:84%
AUDIENCE: So if eta has to obey
anti-commutation with d theta,

01:18:03.056 --> 01:18:05.476 align:middle line:84%
why doesn't theta have to
obey anti-commutation with

01:18:05.476 --> 01:18:06.450 align:middle line:90%
[INAUDIBLE]?

01:18:06.450 --> 01:18:07.440 align:middle line:90%
HONG LIU: Yeah.

01:18:07.440 --> 01:18:08.440 align:middle line:90%
Theta does have two.

01:18:08.440 --> 01:18:08.940 align:middle line:90%
Yeah.

01:18:08.940 --> 01:18:10.804 align:middle line:84%
AUDIENCE: But then
why [INAUDIBLE]??

01:18:10.804 --> 01:18:14.680 align:middle line:90%


01:18:14.680 --> 01:18:16.550 align:middle line:90%
HONG LIU: OK.

01:18:16.550 --> 01:18:20.980 align:middle line:84%
But d theta and theta are
two different objects, right?

01:18:20.980 --> 01:18:22.840 align:middle line:84%
They anti-commute
against each other.

01:18:22.840 --> 01:18:26.650 align:middle line:84%
But this theta means the
infinitesimal variation

01:18:26.650 --> 01:18:31.020 align:middle line:84%
of theta, of course, even though
we cannot quantify what this

01:18:31.020 --> 01:18:33.420 align:middle line:90%
infinitesimal variation means.

01:18:33.420 --> 01:18:35.650 align:middle line:84%
But these two are
different object.

01:18:35.650 --> 01:18:36.150 align:middle line:90%
Yeah.

01:18:36.150 --> 01:18:37.442 align:middle line:90%
So their product is not-- yeah.

01:18:37.442 --> 01:18:39.870 align:middle line:90%


01:18:39.870 --> 01:18:40.640 align:middle line:90%
OK.

01:18:40.640 --> 01:18:41.140 align:middle line:90%
Good.

01:18:41.140 --> 01:18:43.190 align:middle line:90%
So this tells you--

01:18:43.190 --> 01:18:46.760 align:middle line:84%
so normally when we do
a change of variable--

01:18:46.760 --> 01:18:49.590 align:middle line:90%


01:18:49.590 --> 01:18:53.930 align:middle line:84%
so this means that when you
do the change of variable,

01:18:53.930 --> 01:18:57.640 align:middle line:90%
theta prime is equal to a theta.

01:18:57.640 --> 01:19:02.110 align:middle line:84%
And d theta prime is actually
equal to 1 over a d theta,

01:19:02.110 --> 01:19:04.600 align:middle line:90%
OK, if you compare these two.

01:19:04.600 --> 01:19:06.160 align:middle line:90%
So now this is a very--

01:19:06.160 --> 01:19:08.950 align:middle line:84%
so now, again, this is
opposite to the standard story.

01:19:08.950 --> 01:19:10.000 align:middle line:90%
OK?

01:19:10.000 --> 01:19:14.950 align:middle line:84%
So d theta prime is actually
equal to 1 over a d theta.

01:19:14.950 --> 01:19:15.450 align:middle line:90%
OK.

01:19:15.450 --> 01:19:18.340 align:middle line:90%


01:19:18.340 --> 01:19:18.840 align:middle line:90%
OK.

01:19:18.840 --> 01:19:22.530 align:middle line:90%
So let's conclude here.

01:19:22.530 --> 01:19:26.330 align:middle line:90%


01:19:26.330 --> 01:19:26.830 align:middle line:90%
Yeah.

01:19:26.830 --> 01:19:31.200 align:middle line:84%
So our next lecture
is now next Wednesday

01:19:31.200 --> 01:19:33.445 align:middle line:90%
because Monday is a holiday.

01:19:33.445 --> 01:19:35.070 align:middle line:84%
Yeah, it's going to
be-- unfortunately,

01:19:35.070 --> 01:19:39.950 align:middle line:84%
I hope you still remember this
when we get to next Wednesday.

01:19:39.950 --> 01:19:43.000 align:middle line:90%