WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:04.428 align:middle line:90% [RUSTLING] 00:00:04.428 --> 00:00:06.396 align:middle line:90% [CLICKING] 00:00:06.396 --> 00:00:10.340 align:middle line:90% 00:00:10.340 --> 00:00:14.255 align:middle line:84% HONG LIU: Yeah, so, today, we are going to start a new topic. 00:00:14.255 --> 00:00:30.790 align:middle line:90% 00:00:30.790 --> 00:00:32.290 align:middle line:90% OK? 00:00:32.290 --> 00:00:34.210 align:middle line:84% So, first, we talk about chiral fermions. 00:00:34.210 --> 00:00:41.560 align:middle line:90% 00:00:41.560 --> 00:00:53.050 align:middle line:84% So remember, say, under Lorentz transformation, 00:00:53.050 --> 00:01:04.840 align:middle line:84% lambda, the Dirac spinor fields transform as S lambda psi x. 00:01:04.840 --> 00:01:08.110 align:middle line:84% And the x prime is a Lorentz transformation of x. 00:01:08.110 --> 00:01:13.500 align:middle line:84% So x prime is lambda act on x, OK? 00:01:13.500 --> 00:01:16.060 align:middle line:90% 00:01:16.060 --> 00:01:31.790 align:middle line:84% And then the S is given by, say, omega mu nu, sigma, mu, nu, OK? 00:01:31.790 --> 00:01:35.300 align:middle line:84% And the sigma mu, nu is driven by the commutator of the gamma 00:01:35.300 --> 00:01:37.100 align:middle line:90% matrices. 00:01:37.100 --> 00:01:38.390 align:middle line:90% So let me just write it down. 00:01:38.390 --> 00:01:40.400 align:middle line:90% So sigma mu, nu-- 00:01:40.400 --> 00:01:43.400 align:middle line:84% just remind you-- i divided by 4. 00:01:43.400 --> 00:01:47.040 align:middle line:90% 00:01:47.040 --> 00:01:47.540 align:middle line:90% OK. 00:01:47.540 --> 00:01:53.030 align:middle line:90% 00:01:53.030 --> 00:01:56.610 align:middle line:90% So one natural question-- 00:01:56.610 --> 00:02:01.600 align:middle line:84% so, previously, when we derived the Dirac equation, 00:02:01.600 --> 00:02:04.120 align:middle line:84% we showed that the Dirac equation requires, actually, 00:02:04.120 --> 00:02:06.835 align:middle line:90% psi to have four components, OK? 00:02:06.835 --> 00:02:11.990 align:middle line:90% 00:02:11.990 --> 00:02:14.570 align:middle line:84% But, then, we showed that the Dirac equation is covariant 00:02:14.570 --> 00:02:18.920 align:middle line:90% if the psi transform this way. 00:02:18.920 --> 00:02:21.590 align:middle line:84% So natural question is that whether you can actually 00:02:21.590 --> 00:02:25.010 align:middle line:90% restrict to a smaller part-- 00:02:25.010 --> 00:02:27.690 align:middle line:90% say, to a subset of psi-- 00:02:27.690 --> 00:02:29.580 align:middle line:84% whether they still have well-defined Lorentz 00:02:29.580 --> 00:02:34.290 align:middle line:84% transformation, whether we actually 00:02:34.290 --> 00:02:37.980 align:middle line:84% need to have four components to have well-defined-- 00:02:37.980 --> 00:02:40.440 align:middle line:84% four complex component to have well-defined Lorentz 00:02:40.440 --> 00:02:42.660 align:middle line:90% transformation, OK? 00:02:42.660 --> 00:02:46.470 align:middle line:84% And the answer turns out to be, no, you actually 00:02:46.470 --> 00:02:49.080 align:middle line:84% don't need to have four complex components 00:02:49.080 --> 00:02:51.360 align:middle line:84% to have well-defined Lorentz transformation. 00:02:51.360 --> 00:02:55.650 align:middle line:90% Actually, you can reduce it, OK? 00:02:55.650 --> 00:02:57.780 align:middle line:84% And so there are two ways to reduce it, 00:02:57.780 --> 00:02:59.610 align:middle line:84% and one is called chiral fermion, 00:02:59.610 --> 00:03:02.730 align:middle line:84% and the one is called the Majorana fermion, OK? 00:03:02.730 --> 00:03:06.830 align:middle line:84% So one is called the Majorana fermion. 00:03:06.830 --> 00:03:10.070 align:middle line:84% So we first talk about the chiral fermion and one way 00:03:10.070 --> 00:03:11.700 align:middle line:90% to do it. 00:03:11.700 --> 00:03:13.310 align:middle line:84% So, for this purpose, we will look 00:03:13.310 --> 00:03:17.120 align:middle line:84% at the specific representation of gamma matrices, OK? 00:03:17.120 --> 00:03:20.300 align:middle line:84% Consider-- so now I will use a representation which is 00:03:20.300 --> 00:03:23.930 align:middle line:90% different from what you-- 00:03:23.930 --> 00:03:27.350 align:middle line:84% so we consider the following one. 00:03:27.350 --> 00:03:32.660 align:middle line:90% Gamma 0 equal to 0, i, i, 0. 00:03:32.660 --> 00:03:40.250 align:middle line:84% Gamma i equal to 0 minus i sigma i, i sigma i. 00:03:40.250 --> 00:03:45.010 align:middle line:90% 00:03:45.010 --> 00:03:48.360 align:middle line:84% So let's look at this choice of gamma matrices, OK? 00:03:48.360 --> 00:03:55.500 align:middle line:84% So I will call this choice to be star, OK? 00:03:55.500 --> 00:03:58.410 align:middle line:90% And so you can also work it out. 00:03:58.410 --> 00:04:01.450 align:middle line:90% You find the sigma. 00:04:01.450 --> 00:04:05.340 align:middle line:90% So sigma 0, i-- 00:04:05.340 --> 00:04:08.560 align:middle line:84% so this is the sigma corresponding to the boost. 00:04:08.560 --> 00:04:11.940 align:middle line:84% So you find the sigma i, you can just do the commutator. 00:04:11.940 --> 00:04:16.790 align:middle line:84% From the sigma i is equal to minus i divided by 2, 00:04:16.790 --> 00:04:19.170 align:middle line:90% have this block diagonal form. 00:04:19.170 --> 00:04:23.160 align:middle line:84% Sigma i 0, 0 sigma i-- again, the small sigma i 00:04:23.160 --> 00:04:26.130 align:middle line:90% is always the Pauli matrices. 00:04:26.130 --> 00:04:30.930 align:middle line:84% And then you can also work out the sigma ij. 00:04:30.930 --> 00:04:35.040 align:middle line:84% Then you find the is given by minus 1/2, 00:04:35.040 --> 00:04:46.590 align:middle line:84% epsilon ijk, then sigma k, 0, 0, sigma k, OK? 00:04:46.590 --> 00:04:47.110 align:middle line:90% Oh, sorry. 00:04:47.110 --> 00:04:50.140 align:middle line:90% Here, there's a minus sign. 00:04:50.140 --> 00:04:53.190 align:middle line:90% OK, here, there's a minus sign. 00:04:53.190 --> 00:04:59.890 align:middle line:84% OK, so you find that they have the following form. 00:04:59.890 --> 00:05:01.620 align:middle line:84% So what do you observe about this? 00:05:01.620 --> 00:05:05.320 align:middle line:90% 00:05:05.320 --> 00:05:08.570 align:middle line:90% So do you see something? 00:05:08.570 --> 00:05:09.100 align:middle line:90% Yes? 00:05:09.100 --> 00:05:10.475 align:middle line:84% AUDIENCE: They're block diagonal. 00:05:10.475 --> 00:05:12.290 align:middle line:84% HONG LIU: Yes, they are block diagonal. 00:05:12.290 --> 00:05:15.100 align:middle line:84% So if sigma is block diagonal, then that 00:05:15.100 --> 00:05:20.640 align:middle line:84% means this S is also block diagonal, OK? 00:05:20.640 --> 00:05:23.370 align:middle line:84% So when S is block diagonal, what does that mean? 00:05:23.370 --> 00:05:30.350 align:middle line:90% 00:05:30.350 --> 00:05:31.557 align:middle line:90% Yes? 00:05:31.557 --> 00:05:33.409 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:05:33.409 --> 00:05:33.992 align:middle line:90% HONG LIU: Hmm? 00:05:33.992 --> 00:05:34.970 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:05:34.970 --> 00:05:36.790 align:middle line:90% HONG LIU: Exactly. 00:05:36.790 --> 00:05:41.740 align:middle line:84% So when the S is block diagonal, that means, when I write psi-- 00:05:41.740 --> 00:05:43.750 align:middle line:90% so, psi-- have four components. 00:05:43.750 --> 00:05:46.720 align:middle line:84% That means that the upper two component and the lower two 00:05:46.720 --> 00:05:49.940 align:middle line:84% components-- they don't transform to each other, OK? 00:05:49.940 --> 00:05:51.590 align:middle line:84% They only transform within themselves. 00:05:51.590 --> 00:05:54.080 align:middle line:84% They don't transform to each other, OK? 00:05:54.080 --> 00:06:02.135 align:middle line:84% So if I choose, that means this lambda is block diagonal. 00:06:02.135 --> 00:06:06.290 align:middle line:90% 00:06:06.290 --> 00:06:16.751 align:middle line:84% So that means, if I write psi x into two component vector, psi 00:06:16.751 --> 00:06:21.560 align:middle line:84% L and psi R, so I denote the upper two components by psi L 00:06:21.560 --> 00:06:24.956 align:middle line:84% and lower two components by psi R-- 00:06:24.956 --> 00:06:31.900 align:middle line:84% are two component complex vector. 00:06:31.900 --> 00:06:37.980 align:middle line:90% 00:06:37.980 --> 00:06:47.010 align:middle line:84% That means, under Lorentz transformation, under S lambda, 00:06:47.010 --> 00:06:50.760 align:middle line:90% psi L and psi R do not mix. 00:06:50.760 --> 00:06:54.690 align:middle line:90% 00:06:54.690 --> 00:06:58.540 align:middle line:84% So they just transform among themselves. 00:06:58.540 --> 00:07:00.190 align:middle line:84% They just transform among themselves. 00:07:00.190 --> 00:07:02.460 align:middle line:84% So they actually have well-defined Lorentz 00:07:02.460 --> 00:07:05.610 align:middle line:84% transformation as a smaller unit, OK? 00:07:05.610 --> 00:07:07.860 align:middle line:84% You don't need four components to be able to transform 00:07:07.860 --> 00:07:09.068 align:middle line:90% under Lorentz transformation. 00:07:09.068 --> 00:07:10.480 align:middle line:84% Actually, at least two components 00:07:10.480 --> 00:07:13.380 align:middle line:90% can already transform. 00:07:13.380 --> 00:07:15.410 align:middle line:90% OK. 00:07:15.410 --> 00:07:32.500 align:middle line:84% So this tells you, in a sense that the Lorentz covariance 00:07:32.500 --> 00:07:39.850 align:middle line:84% only requires two component spinors. 00:07:39.850 --> 00:07:43.690 align:middle line:90% 00:07:43.690 --> 00:07:45.580 align:middle line:84% Just by Lorentz transformation itself, 00:07:45.580 --> 00:07:47.410 align:middle line:84% you don't actually need four components. 00:07:47.410 --> 00:07:50.390 align:middle line:90% 00:07:50.390 --> 00:07:50.890 align:middle line:90% OK. 00:07:50.890 --> 00:07:54.108 align:middle line:90% 00:07:54.108 --> 00:07:55.900 align:middle line:84% So, now, I'm going to tell you, we actually 00:07:55.900 --> 00:07:59.000 align:middle line:90% knew this all along. 00:07:59.000 --> 00:08:04.060 align:middle line:84% So how did we know this all along? 00:08:04.060 --> 00:08:04.690 align:middle line:90% Yes. 00:08:04.690 --> 00:08:07.030 align:middle line:84% AUDIENCE: Why is there no [INAUDIBLE]?? 00:08:07.030 --> 00:08:15.883 align:middle line:90% 00:08:15.883 --> 00:08:16.550 align:middle line:90% HONG LIU: Sorry. 00:08:16.550 --> 00:08:17.745 align:middle line:90% Say it again. 00:08:17.745 --> 00:08:21.511 align:middle line:84% AUDIENCE: Didn't we have to go to four components in there 00:08:21.511 --> 00:08:26.435 align:middle line:84% because there are no representation [INAUDIBLE]?? 00:08:26.435 --> 00:08:27.310 align:middle line:90% HONG LIU: Yeah, yeah. 00:08:27.310 --> 00:08:30.760 align:middle line:84% We have to go to four components because there's 00:08:30.760 --> 00:08:36.730 align:middle line:84% no two component representation of the gamma matrices. 00:08:36.730 --> 00:08:39.610 align:middle line:84% But that doesn't say it's a Lorentz transformation. 00:08:39.610 --> 00:08:40.299 align:middle line:90% Yeah. 00:08:40.299 --> 00:08:43.330 align:middle line:84% To write down the algebra for the gamma matrices, 00:08:43.330 --> 00:08:46.040 align:middle line:90% you need four components. 00:08:46.040 --> 00:08:50.060 align:middle line:84% But we actually knew all along that two components 00:08:50.060 --> 00:08:54.340 align:middle line:84% is enough for Lorentz transformation. 00:08:54.340 --> 00:08:57.440 align:middle line:84% How do we actually know all along? 00:08:57.440 --> 00:09:00.495 align:middle line:84% AUDIENCE: Lorentz assumption isn't just [INAUDIBLE] spin 00:09:00.495 --> 00:09:03.710 align:middle line:84% like difference between the [INAUDIBLE]?? 00:09:03.710 --> 00:09:06.380 align:middle line:84% HONG LIU: Yeah, this is maybe more complicated. 00:09:06.380 --> 00:09:08.240 align:middle line:90% We have something much simpler. 00:09:08.240 --> 00:09:09.407 align:middle line:90% Yes. 00:09:09.407 --> 00:09:10.990 align:middle line:84% AUDIENCE: Is it the massless particle? 00:09:10.990 --> 00:09:12.460 align:middle line:90% HONG LIU: Yeah, exactly. 00:09:12.460 --> 00:09:17.020 align:middle line:84% So we already said before, if you have a massless case when 00:09:17.020 --> 00:09:19.870 align:middle line:84% m equal to 0, and the Dirac equation 00:09:19.870 --> 00:09:23.790 align:middle line:84% reduces to the two components, and it's 00:09:23.790 --> 00:09:26.460 align:middle line:84% enough to do two components, and the Dirac equation 00:09:26.460 --> 00:09:27.810 align:middle line:90% is covariant, OK? 00:09:27.810 --> 00:09:29.460 align:middle line:84% And so that means that, actually, you 00:09:29.460 --> 00:09:31.920 align:middle line:84% should be able to do it with two components, OK? 00:09:31.920 --> 00:09:33.503 align:middle line:84% Because the massless particle-- they 00:09:33.503 --> 00:09:35.920 align:middle line:84% should be able to transform on the Lorentz transformation. 00:09:35.920 --> 00:09:37.530 align:middle line:90% And so, yeah. 00:09:37.530 --> 00:09:44.660 align:middle line:90% 00:09:44.660 --> 00:09:49.100 align:middle line:84% So we already saw this because the massless. 00:09:49.100 --> 00:09:57.590 align:middle line:84% So the hint from before is that the massless only 00:09:57.590 --> 00:10:06.470 align:middle line:90% require two components. 00:10:06.470 --> 00:10:09.690 align:middle line:90% 00:10:09.690 --> 00:10:11.390 align:middle line:84% Since the massless case must also be 00:10:11.390 --> 00:10:13.295 align:middle line:90% Lorentz covariant so that-- 00:10:13.295 --> 00:10:14.930 align:middle line:84% and Lorentz symmetry itself should only 00:10:14.930 --> 00:10:16.830 align:middle line:90% require two components. 00:10:16.830 --> 00:10:20.660 align:middle line:84% So it is not a Lorentz symmetry, actually. 00:10:20.660 --> 00:10:23.300 align:middle line:84% Lorentz covariance requires Dirac theory 00:10:23.300 --> 00:10:24.440 align:middle line:90% to have four component. 00:10:24.440 --> 00:10:26.990 align:middle line:90% It is the mass, OK? 00:10:26.990 --> 00:10:28.310 align:middle line:90% So it is the mass-- 00:10:28.310 --> 00:10:33.620 align:middle line:90% 00:10:33.620 --> 00:10:35.090 align:middle line:90% mass m. 00:10:35.090 --> 00:10:39.170 align:middle line:84% If you want to describe a massive particle, 00:10:39.170 --> 00:10:42.170 align:middle line:84% then you must have four components, OK? 00:10:42.170 --> 00:10:44.690 align:middle line:84% So it is the mass which is the key. 00:10:44.690 --> 00:11:03.650 align:middle line:90% 00:11:03.650 --> 00:11:04.710 align:middle line:90% OK, good. 00:11:04.710 --> 00:11:05.735 align:middle line:90% Any questions on this? 00:11:05.735 --> 00:11:13.220 align:middle line:90% 00:11:13.220 --> 00:11:18.590 align:middle line:84% So, now, we have shown in this particular representation 00:11:18.590 --> 00:11:20.780 align:middle line:84% of gamma matrices, for this particular choice 00:11:20.780 --> 00:11:28.330 align:middle line:84% of gamma matrices, and then the psi transform block diagonally. 00:11:28.330 --> 00:11:34.110 align:middle line:84% But then we also said, but, now, consider a different choice 00:11:34.110 --> 00:11:36.450 align:middle line:90% of gamma matrices. 00:11:36.450 --> 00:11:39.310 align:middle line:84% And then this property will not hold, OK? 00:11:39.310 --> 00:11:40.780 align:middle line:90% This property will not hold. 00:11:40.780 --> 00:11:43.960 align:middle line:84% Now, the question is that, does this-- 00:11:43.960 --> 00:11:47.440 align:middle line:84% even with other gamma matrices, can we actually 00:11:47.440 --> 00:11:51.450 align:middle line:84% reduce psi to some smaller components? 00:11:51.450 --> 00:11:54.390 align:middle line:84% The answer still should still be yes 00:11:54.390 --> 00:11:58.410 align:middle line:84% because we said that all representation of gamma 00:11:58.410 --> 00:12:00.370 align:middle line:90% matrices should be equivalent. 00:12:00.370 --> 00:12:03.960 align:middle line:84% So if we can do it in this choice of gamma matrices, 00:12:03.960 --> 00:12:06.870 align:middle line:84% then we should be able to do it in any choice of gamma 00:12:06.870 --> 00:12:08.720 align:middle line:90% matrices, OK? 00:12:08.720 --> 00:12:12.580 align:middle line:84% So, now, let me tell you how to do it for the general gamma 00:12:12.580 --> 00:12:13.080 align:middle line:90% matrices. 00:12:13.080 --> 00:12:17.120 align:middle line:90% 00:12:17.120 --> 00:12:26.030 align:middle line:84% So this property that you can reduce to two components 00:12:26.030 --> 00:12:39.360 align:middle line:84% should exist for all choice of gamma matrices. 00:12:39.360 --> 00:12:43.230 align:middle line:90% 00:12:43.230 --> 00:12:47.600 align:middle line:84% Just for other choice of gamma matrices, 00:12:47.600 --> 00:12:53.640 align:middle line:84% to separate the psi into psi L and psi R is more subtle. 00:12:53.640 --> 00:12:56.670 align:middle line:84% You no longer-- is just simple the upper two component 00:12:56.670 --> 00:12:57.970 align:middle line:90% or lower two component. 00:12:57.970 --> 00:13:02.040 align:middle line:84% So we have to do a little bit of work, OK? 00:13:02.040 --> 00:13:05.610 align:middle line:84% Actually, we don't need to do much work if you actually 00:13:05.610 --> 00:13:09.630 align:middle line:90% find the right trick, OK? 00:13:09.630 --> 00:13:12.960 align:middle line:84% And so the beautiful trick to do this 00:13:12.960 --> 00:13:16.320 align:middle line:84% for any choice of gamma matrices is 00:13:16.320 --> 00:13:18.195 align:middle line:84% that you can introduce the following object-- 00:13:18.195 --> 00:13:23.610 align:middle line:90% 00:13:23.610 --> 00:13:24.855 align:middle line:90% what is called gamma 5. 00:13:24.855 --> 00:13:28.840 align:middle line:90% 00:13:28.840 --> 00:13:35.170 align:middle line:84% So gamma 5 is defined to be i gamma 0, gamma 1, gamma 00:13:35.170 --> 00:13:37.180 align:middle line:90% 2, gamma 3, OK? 00:13:37.180 --> 00:13:40.210 align:middle line:84% So you take the product of all the gamma 00:13:40.210 --> 00:13:46.860 align:middle line:84% matrices together and then with a factor of i, OK? 00:13:46.860 --> 00:13:50.490 align:middle line:84% So the i there is for the purpose that if you-- 00:13:50.490 --> 00:13:52.980 align:middle line:90% you can check yourself-- 00:13:52.980 --> 00:13:56.760 align:middle line:84% that the gamma 5 is actually Hermitian. 00:13:56.760 --> 00:14:01.190 align:middle line:84% So the i is there for this purpose, OK? 00:14:01.190 --> 00:14:04.020 align:middle line:90% You need i for this to be true. 00:14:04.020 --> 00:14:13.280 align:middle line:84% You also can check yourself that gamma 5 squared is equal to 1, 00:14:13.280 --> 00:14:15.620 align:middle line:90% OK? 00:14:15.620 --> 00:14:20.000 align:middle line:84% So, this, you can almost easily understand because it's 00:14:20.000 --> 00:14:22.460 align:middle line:90% all gamma 0, gamma 1, gamma 3. 00:14:22.460 --> 00:14:26.990 align:middle line:84% So you multiply itself again because any same gamma matrix, 00:14:26.990 --> 00:14:30.020 align:middle line:84% they multiply either 1 or minus 1. 00:14:30.020 --> 00:14:32.360 align:middle line:84% So you multiply them together, in the end, 00:14:32.360 --> 00:14:33.950 align:middle line:90% they can be either 1 or minus 1. 00:14:33.950 --> 00:14:37.370 align:middle line:84% Just turns out, for this choice of i, it's 1, OK? 00:14:37.370 --> 00:14:39.470 align:middle line:90% So yeah. 00:14:39.470 --> 00:14:44.960 align:middle line:84% And then you can also check the gamma 5 anticommute 00:14:44.960 --> 00:14:46.115 align:middle line:90% with any gamma matrices. 00:14:46.115 --> 00:14:49.070 align:middle line:90% 00:14:49.070 --> 00:14:51.620 align:middle line:84% So mu here is, of course, from 0 to 3. 00:14:51.620 --> 00:14:54.550 align:middle line:90% 00:14:54.550 --> 00:14:59.120 align:middle line:90% And so this is of-- 00:14:59.120 --> 00:15:02.668 align:middle line:84% you can see immediately from here-- 00:15:02.668 --> 00:15:03.960 align:middle line:90% you can immediately from here-- 00:15:03.960 --> 00:15:10.630 align:middle line:84% because gamma matrices whose indices are not the same, 00:15:10.630 --> 00:15:12.660 align:middle line:90% they anticommute, OK? 00:15:12.660 --> 00:15:17.490 align:middle line:84% So if you try to commute this with any gamma matrices, 00:15:17.490 --> 00:15:18.870 align:middle line:90% you have three of them. 00:15:18.870 --> 00:15:21.600 align:middle line:90% 00:15:21.600 --> 00:15:25.440 align:middle line:84% Yeah, because this runs over all gamma matrices. 00:15:25.440 --> 00:15:29.810 align:middle line:84% So the one in which-- yeah, so if you take with some gamma mu, 00:15:29.810 --> 00:15:31.220 align:middle line:84% and that particular one, which is 00:15:31.220 --> 00:15:33.590 align:middle line:84% the same as gamma mu, of course, commutes with gamma mu. 00:15:33.590 --> 00:15:35.270 align:middle line:90% But then you have three others. 00:15:35.270 --> 00:15:39.140 align:middle line:84% But three others will give you minus sign, OK? 00:15:39.140 --> 00:15:40.730 align:middle line:90% And you can also check yourself. 00:15:40.730 --> 00:15:43.590 align:middle line:90% 00:15:43.590 --> 00:15:46.248 align:middle line:84% Gamma 5 actually have 0 trace, OK? 00:15:46.248 --> 00:15:47.790 align:middle line:84% So, this, I will leave as an exercise 00:15:47.790 --> 00:15:54.215 align:middle line:84% for yourself, what you can do is you did before with other-- 00:15:54.215 --> 00:15:56.990 align:middle line:90% 00:15:56.990 --> 00:16:00.230 align:middle line:90% yeah, in your homework-- 00:16:00.230 --> 00:16:03.240 align:middle line:84% yeah, similar to the exercise you have done in your homework. 00:16:03.240 --> 00:16:06.590 align:middle line:90% So, now, from this properties-- 00:16:06.590 --> 00:16:09.410 align:middle line:84% now we can say the following things about the gamma 5 00:16:09.410 --> 00:16:10.610 align:middle line:90% matrix. 00:16:10.610 --> 00:16:17.900 align:middle line:84% First, because gamma 5 squared, squared to 1. 00:16:17.900 --> 00:16:19.820 align:middle line:90% And, also, this is Hermitian. 00:16:19.820 --> 00:16:25.300 align:middle line:84% So it is Hermitian means its eigenvalue is all real, OK? 00:16:25.300 --> 00:16:28.510 align:middle line:84% So its eigenvalues are all real, and gamma 5 00:16:28.510 --> 00:16:31.210 align:middle line:84% squared equal to 1-- that means its eigenvalue is either 00:16:31.210 --> 00:16:33.190 align:middle line:90% plus or minus 1, OK? 00:16:33.190 --> 00:16:40.190 align:middle line:84% So have eigenvalues plus, minus 1. 00:16:40.190 --> 00:16:44.070 align:middle line:90% 00:16:44.070 --> 00:16:47.770 align:middle line:84% And then from the property that this is traceless, 00:16:47.770 --> 00:16:51.220 align:middle line:84% they tell you the number of the eigenvalues, which is plus 1, 00:16:51.220 --> 00:16:52.970 align:middle line:90% and the number of minus 1. 00:16:52.970 --> 00:16:54.740 align:middle line:90% They should be the same. 00:16:54.740 --> 00:16:56.110 align:middle line:90% Otherwise, they won't cancel. 00:16:56.110 --> 00:16:57.370 align:middle line:90% It won't be traceless. 00:16:57.370 --> 00:17:01.838 align:middle line:84% And so each eigenspace is two-dimensional. 00:17:01.838 --> 00:17:08.530 align:middle line:90% 00:17:08.530 --> 00:17:09.210 align:middle line:90% OK? 00:17:09.210 --> 00:17:12.160 align:middle line:90% So you have four eigenvalues. 00:17:12.160 --> 00:17:13.980 align:middle line:90% So there's 2 plus 1, 2 minus 1. 00:17:13.980 --> 00:17:14.760 align:middle line:90% It must be. 00:17:14.760 --> 00:17:18.859 align:middle line:90% 00:17:18.859 --> 00:17:22.040 align:middle line:84% So since you have eigenvalues 2 plus 1, 2 minus 1, 00:17:22.040 --> 00:17:26.180 align:middle line:84% and then we can introduce a projector to project 00:17:26.180 --> 00:17:28.970 align:middle line:84% into the eigenspace, say, with plus 1-- 00:17:28.970 --> 00:17:32.420 align:middle line:84% with eigenvalue plus 1 or the eigenvalue minus 1, OK? 00:17:32.420 --> 00:17:37.400 align:middle line:84% So I can introduce a projector which, for historical reasons, 00:17:37.400 --> 00:17:39.760 align:middle line:90% is called PL. 00:17:39.760 --> 00:17:44.530 align:middle line:84% It's defined to be 1/2 1 plus gamma 5. 00:17:44.530 --> 00:17:53.530 align:middle line:84% And the PR is 1/2 1 minus gamma 5, OK? 00:17:53.530 --> 00:18:00.280 align:middle line:84% So this will project into eigenspace with the eigenvalue 00:18:00.280 --> 00:18:01.900 align:middle line:90% plus 1 squared. 00:18:01.900 --> 00:18:06.200 align:middle line:84% And this will project into an eigenspace with a minus 1, OK? 00:18:06.200 --> 00:18:08.630 align:middle line:90% Because when 1 minus 1 plus-- 00:18:08.630 --> 00:18:10.970 align:middle line:90% yeah, anyway. 00:18:10.970 --> 00:18:12.530 align:middle line:90% So you can check, OK? 00:18:12.530 --> 00:18:17.100 align:middle line:90% 00:18:17.100 --> 00:18:18.990 align:middle line:84% So you can check they are really projectors. 00:18:18.990 --> 00:18:24.060 align:middle line:84% So PL squared equal to PR squared equal to 1 00:18:24.060 --> 00:18:30.990 align:middle line:84% and PLPR equal to 0 and PL plus PR equal to identity, OK? 00:18:30.990 --> 00:18:34.460 align:middle line:90% 00:18:34.460 --> 00:18:39.220 align:middle line:90% And then-- OK. 00:18:39.220 --> 00:18:43.340 align:middle line:90% 00:18:43.340 --> 00:18:45.200 align:middle line:90% So, now, I introduce-- 00:18:45.200 --> 00:18:51.020 align:middle line:90% 00:18:51.020 --> 00:18:53.810 align:middle line:90% now I can project-- 00:18:53.810 --> 00:18:59.510 align:middle line:84% define the projection of psi L of the projection of PL psi-- 00:18:59.510 --> 00:19:03.500 align:middle line:90% project to the left space, OK? 00:19:03.500 --> 00:19:06.950 align:middle line:84% And psi R to be the projection to the other space. 00:19:06.950 --> 00:19:10.770 align:middle line:90% 00:19:10.770 --> 00:19:11.270 align:middle line:90% OK. 00:19:11.270 --> 00:19:13.890 align:middle line:90% I define them this way. 00:19:13.890 --> 00:19:17.360 align:middle line:84% And then you can easily see, by definition, 00:19:17.360 --> 00:19:20.870 align:middle line:84% you can easily convince yourself gamma 5 acting on psi 00:19:20.870 --> 00:19:23.990 align:middle line:84% then just equal to psi L. And the gamma 5 psi 00:19:23.990 --> 00:19:28.940 align:middle line:90% R is equal to minus psi R, OK? 00:19:28.940 --> 00:19:32.240 align:middle line:84% So they project into the eigenvalues of plus, minus 1. 00:19:32.240 --> 00:19:32.900 align:middle line:90% Yes? 00:19:32.900 --> 00:19:33.890 align:middle line:84% AUDIENCE: Wait, why is PL squared 00:19:33.890 --> 00:19:35.170 align:middle line:90% equal to PR squared plus 1? 00:19:35.170 --> 00:19:37.720 align:middle line:90% 00:19:37.720 --> 00:19:38.830 align:middle line:90% HONG LIU: Oh, sorry. 00:19:38.830 --> 00:19:40.700 align:middle line:90% Sorry, sorry, sorry. 00:19:40.700 --> 00:19:41.530 align:middle line:90% No, no. 00:19:41.530 --> 00:19:42.586 align:middle line:90% This is good. 00:19:42.586 --> 00:19:45.250 align:middle line:90% 00:19:45.250 --> 00:19:47.200 align:middle line:90% This is completely wrong, OK? 00:19:47.200 --> 00:19:48.250 align:middle line:90% This is completely wrong. 00:19:48.250 --> 00:19:50.350 align:middle line:90% I was dreaming. 00:19:50.350 --> 00:19:56.450 align:middle line:84% So PL squared equal to PL, PR squared. 00:19:56.450 --> 00:19:57.020 align:middle line:90% Sorry. 00:19:57.020 --> 00:19:57.850 align:middle line:90% Yeah, thank you. 00:19:57.850 --> 00:20:02.160 align:middle line:90% 00:20:02.160 --> 00:20:04.455 align:middle line:84% Yeah, so you can check their projectors, OK? 00:20:04.455 --> 00:20:09.550 align:middle line:90% 00:20:09.550 --> 00:20:11.830 align:middle line:90% So, indeed, you see-- 00:20:11.830 --> 00:20:13.610 align:middle line:84% so, from here, from this definition, 00:20:13.610 --> 00:20:15.190 align:middle line:90% you can check this is true, OK? 00:20:15.190 --> 00:20:17.530 align:middle line:90% This is a one-second check. 00:20:17.530 --> 00:20:22.000 align:middle line:84% And so, indeed, they project to the eigenspace of gamma 5 00:20:22.000 --> 00:20:24.910 align:middle line:90% or plus, minus 1. 00:20:24.910 --> 00:20:26.790 align:middle line:90% OK. 00:20:26.790 --> 00:20:31.230 align:middle line:90% So then, by definition-- 00:20:31.230 --> 00:20:33.420 align:middle line:84% OK, so now this psi L, psi R, which 00:20:33.420 --> 00:20:37.180 align:middle line:84% is now defined for any choice of gamma matrices-- 00:20:37.180 --> 00:20:56.155 align:middle line:84% so, again, they have to two independent complex component, 00:20:56.155 --> 00:20:56.655 align:middle line:90% OK? 00:20:56.655 --> 00:21:08.270 align:middle line:90% 00:21:08.270 --> 00:21:14.440 align:middle line:84% And so they call the chiral spinors-- 00:21:14.440 --> 00:21:17.590 align:middle line:90% 00:21:17.590 --> 00:21:19.480 align:middle line:84% sometimes also called Weyl spinors. 00:21:19.480 --> 00:21:23.830 align:middle line:90% 00:21:23.830 --> 00:21:27.910 align:middle line:84% And so this is the analog of psi L and psi R 00:21:27.910 --> 00:21:32.790 align:middle line:84% here for the general choice of gamma matrices. 00:21:32.790 --> 00:21:36.820 align:middle line:84% So, now, we will check this actually, indeed. 00:21:36.820 --> 00:21:39.400 align:middle line:84% So now the claim is that psi L and psi R 00:21:39.400 --> 00:21:42.555 align:middle line:84% defined this way will transform under themselves 00:21:42.555 --> 00:21:43.930 align:middle line:84% under the Lorentz transformation. 00:21:43.930 --> 00:21:46.660 align:middle line:84% They will not mix with each other, OK? 00:21:46.660 --> 00:21:52.970 align:middle line:84% So, again, psi L and psi R here each have four components, OK? 00:21:52.970 --> 00:21:57.080 align:middle line:84% They just have only two independent compact components. 00:21:57.080 --> 00:21:59.860 align:middle line:84% So they still have four complex components, OK? 00:21:59.860 --> 00:22:02.020 align:middle line:84% And there's still four component spinors 00:22:02.020 --> 00:22:06.850 align:middle line:84% because just there's only two independent ones. 00:22:06.850 --> 00:22:10.300 align:middle line:84% There's only two independent ones. 00:22:10.300 --> 00:22:15.250 align:middle line:84% OK, so, now, it's easy to check they actually transform 00:22:15.250 --> 00:22:16.790 align:middle line:90% among themselves. 00:22:16.790 --> 00:22:27.820 align:middle line:84% So you can check the gamma 5 actually commutes 00:22:27.820 --> 00:22:29.355 align:middle line:90% with sigma mu, nu. 00:22:29.355 --> 00:22:34.650 align:middle line:90% 00:22:34.650 --> 00:22:35.150 align:middle line:90% OK. 00:22:35.150 --> 00:22:39.660 align:middle line:90% 00:22:39.660 --> 00:22:41.475 align:middle line:90% So this is very easy to see. 00:22:41.475 --> 00:22:44.470 align:middle line:90% 00:22:44.470 --> 00:22:48.510 align:middle line:84% So, from here, gamma 5 commutes with any gamma 00:22:48.510 --> 00:22:54.890 align:middle line:84% mu or anticommute with any gamma mu. 00:22:54.890 --> 00:22:57.920 align:middle line:84% And the sigma mu, nu is just the sum of two-- 00:22:57.920 --> 00:23:00.260 align:middle line:90% the product of two gamma mus-- 00:23:00.260 --> 00:23:01.220 align:middle line:90% have even gamma mus. 00:23:01.220 --> 00:23:06.460 align:middle line:84% So the gamma 5 will compete with them, OK? 00:23:06.460 --> 00:23:08.120 align:middle line:84% So gamma 5 will commute with them. 00:23:08.120 --> 00:23:10.210 align:middle line:84% So if gamma 5 commutes with sigma mu, nu, 00:23:10.210 --> 00:23:13.480 align:middle line:84% then gamma 5 commutes with S lambda 00:23:13.480 --> 00:23:18.530 align:middle line:84% because S lambda just generated by sigma mu, nu. 00:23:18.530 --> 00:23:19.535 align:middle line:90% And then that means-- 00:23:19.535 --> 00:23:22.060 align:middle line:90% 00:23:22.060 --> 00:23:24.360 align:middle line:84% so we commute with S. That means, 00:23:24.360 --> 00:23:28.823 align:middle line:84% under transformation, S will not-- 00:23:28.823 --> 00:23:30.240 align:middle line:84% under transformation by S will not 00:23:30.240 --> 00:23:32.550 align:middle line:84% change the eigenvalues of gamma 5, OK? 00:23:32.550 --> 00:23:41.640 align:middle line:84% So that means that psi L prime, S lambda, psi L, and gamma 00:23:41.640 --> 00:23:47.400 align:middle line:84% 5 acting on psi L prime is still gamma L. 00:23:47.400 --> 00:23:48.900 align:middle line:84% So it's still within the same space. 00:23:48.900 --> 00:23:51.880 align:middle line:90% 00:23:51.880 --> 00:23:53.960 align:middle line:90% And, similarly, we say-- 00:23:53.960 --> 00:23:57.340 align:middle line:84% OK, so that tells you that psi L and psi 00:23:57.340 --> 00:24:00.910 align:middle line:84% R-- they transform separately because the gamma 5 commutes 00:24:00.910 --> 00:24:05.350 align:middle line:90% with Lorentz transformation. 00:24:05.350 --> 00:24:07.830 align:middle line:84% And so each eigenspace, they transform separately 00:24:07.830 --> 00:24:09.520 align:middle line:90% from each other. 00:24:09.520 --> 00:24:10.020 align:middle line:90% OK. 00:24:10.020 --> 00:24:15.420 align:middle line:90% 00:24:15.420 --> 00:24:15.920 align:middle line:90% Good. 00:24:15.920 --> 00:24:16.837 align:middle line:90% Any questions on this? 00:24:16.837 --> 00:24:19.220 align:middle line:90% 00:24:19.220 --> 00:24:23.360 align:middle line:90% So you can also find in the-- 00:24:23.360 --> 00:24:34.180 align:middle line:84% so in the chiral representation, you can check in this star-- 00:24:34.180 --> 00:24:37.090 align:middle line:84% so the star-- this particular choice of gamma 00:24:37.090 --> 00:24:40.330 align:middle line:84% is called the chiral representation, OK? 00:24:40.330 --> 00:24:43.810 align:middle line:84% Because in that choice of gamma-- 00:24:43.810 --> 00:24:46.060 align:middle line:84% things simplify, we just have upper two components 00:24:46.060 --> 00:24:47.570 align:middle line:90% and lower two components. 00:24:47.570 --> 00:24:54.280 align:middle line:84% So you can check yourself, just by working it out, 00:24:54.280 --> 00:25:04.510 align:middle line:84% that the gamma 5 indeed just have block diagonal form-- 00:25:04.510 --> 00:25:07.750 align:middle line:90% 1, 0, 0 minus 1, OK? 00:25:07.750 --> 00:25:08.830 align:middle line:90% So that's why. 00:25:08.830 --> 00:25:11.320 align:middle line:84% In that case, it's very simple, OK? 00:25:11.320 --> 00:25:13.570 align:middle line:84% But in other representation, gamma 5 00:25:13.570 --> 00:25:14.680 align:middle line:90% can be more complicated. 00:25:14.680 --> 00:25:23.120 align:middle line:90% 00:25:23.120 --> 00:25:23.620 align:middle line:90% Good. 00:25:23.620 --> 00:25:24.537 align:middle line:90% Any questions on this? 00:25:24.537 --> 00:25:31.760 align:middle line:90% 00:25:31.760 --> 00:25:34.100 align:middle line:90% Yes, you have a question? 00:25:34.100 --> 00:25:35.480 align:middle line:90% OK. 00:25:35.480 --> 00:25:36.350 align:middle line:90% Yes? 00:25:36.350 --> 00:25:38.267 align:middle line:84% AUDIENCE: Another way to do that, as you said, 00:25:38.267 --> 00:25:41.910 align:middle line:84% would be to try to find the unitary matrix that shows-- 00:25:41.910 --> 00:25:43.620 align:middle line:84% under which this arbitrary representation 00:25:43.620 --> 00:25:45.880 align:middle line:84% is equivalent to the chiral representation. 00:25:45.880 --> 00:25:49.420 align:middle line:84% From this argument, can we figure out what 00:25:49.420 --> 00:25:50.945 align:middle line:84% that unitary [INAUDIBLE] looks like? 00:25:50.945 --> 00:25:51.820 align:middle line:90% HONG LIU: Yeah, yeah. 00:25:51.820 --> 00:25:52.510 align:middle line:90% Yeah, you can. 00:25:52.510 --> 00:25:53.580 align:middle line:90% Yeah, yeah. 00:25:53.580 --> 00:25:55.330 align:middle line:84% No it's not a unitary transformation, just 00:25:55.330 --> 00:25:57.760 align:middle line:90% a similar transformation, yeah. 00:25:57.760 --> 00:26:00.370 align:middle line:84% So each of them are related by a similar transformation. 00:26:00.370 --> 00:26:04.000 align:middle line:90% And, indeed, the gamma 5-- 00:26:04.000 --> 00:26:05.890 align:middle line:84% gamma 5 in the other representation 00:26:05.890 --> 00:26:08.473 align:middle line:84% are related to this one just by a similar transformation, too. 00:26:08.473 --> 00:26:11.650 align:middle line:90% 00:26:11.650 --> 00:26:13.690 align:middle line:84% Yeah, so I will use that language 00:26:13.690 --> 00:26:15.930 align:middle line:84% when I talk about Majorana spinor. 00:26:15.930 --> 00:26:17.800 align:middle line:84% So, in this case, it's sufficiently simple. 00:26:17.800 --> 00:26:20.150 align:middle line:84% I don't need to use that language. 00:26:20.150 --> 00:26:22.800 align:middle line:84% Yeah, but you can use that language. 00:26:22.800 --> 00:26:26.970 align:middle line:84% OK, so let's go back to this chiral representation 00:26:26.970 --> 00:26:30.270 align:middle line:84% and write the Dirac equation into 00:26:30.270 --> 00:26:32.027 align:middle line:90% this chiral representation. 00:26:32.027 --> 00:26:36.950 align:middle line:90% 00:26:36.950 --> 00:26:38.705 align:middle line:90% Yeah, one second. 00:26:38.705 --> 00:26:43.080 align:middle line:90% 00:26:43.080 --> 00:26:44.820 align:middle line:84% So, now, if you write the Dirac equation 00:26:44.820 --> 00:26:48.820 align:middle line:90% in terms of psi L and psi R-- 00:26:48.820 --> 00:26:51.850 align:middle line:84% so, remember, the Dirac equation have the following form 00:26:51.850 --> 00:26:53.950 align:middle line:90% of Dirac Lagrangian density. 00:26:53.950 --> 00:26:59.830 align:middle line:90% 00:26:59.830 --> 00:27:02.310 align:middle line:90% OK? 00:27:02.310 --> 00:27:06.720 align:middle line:84% And then since psi is just equal to the sum of the-- 00:27:06.720 --> 00:27:14.820 align:middle line:90% so psi-- psi R, OK? 00:27:14.820 --> 00:27:17.790 align:middle line:84% And then you can just write this in terms of psi L 00:27:17.790 --> 00:27:21.450 align:middle line:84% and psi R. Write this in terms psi L and psi R. 00:27:21.450 --> 00:27:24.150 align:middle line:84% And then you find that the cross term vanish. 00:27:24.150 --> 00:27:26.070 align:middle line:84% You can also check this explicitly 00:27:26.070 --> 00:27:29.130 align:middle line:84% in the chiral representation, but the expression 00:27:29.130 --> 00:27:31.230 align:middle line:90% I'm writing down is general, OK? 00:27:31.230 --> 00:27:35.710 align:middle line:84% So you can write it as psi dagger 00:27:35.710 --> 00:27:48.945 align:middle line:84% partial sub 0, plus i sigma i, partial i, psi L. Yeah, sorry. 00:27:48.945 --> 00:27:53.470 align:middle line:90% 00:27:53.470 --> 00:27:55.210 align:middle line:90% Yeah. 00:27:55.210 --> 00:27:59.680 align:middle line:90% Yeah, actually, psi R dagger. 00:27:59.680 --> 00:28:00.978 align:middle line:90% Yeah, let me first write. 00:28:00.978 --> 00:28:02.270 align:middle line:90% I think I said something wrong. 00:28:02.270 --> 00:28:02.830 align:middle line:90% Yeah, OK. 00:28:02.830 --> 00:28:40.850 align:middle line:90% 00:28:40.850 --> 00:28:41.350 align:middle line:90% OK? 00:28:41.350 --> 00:28:44.620 align:middle line:90% 00:28:44.620 --> 00:28:47.410 align:middle line:84% So, yeah, as I said, so this expression 00:28:47.410 --> 00:28:49.750 align:middle line:84% only applies to the chiral representation, OK? 00:28:49.750 --> 00:28:58.570 align:middle line:84% So in the chiral representation of gamma star-- 00:28:58.570 --> 00:29:01.630 align:middle line:84% in this space of star, and then we have two components. 00:29:01.630 --> 00:29:04.090 align:middle line:84% Then I can write this psi and the psi L and psi R 00:29:04.090 --> 00:29:05.890 align:middle line:90% into two components. 00:29:05.890 --> 00:29:07.720 align:middle line:90% And then, yeah. 00:29:07.720 --> 00:29:10.390 align:middle line:84% So this is just ordinary sigma matrices, OK? 00:29:10.390 --> 00:29:14.470 align:middle line:90% 00:29:14.470 --> 00:29:17.140 align:middle line:84% And so this is the expression you get, OK? 00:29:17.140 --> 00:29:20.170 align:middle line:90% 00:29:20.170 --> 00:29:21.265 align:middle line:90% So what you notice-- 00:29:21.265 --> 00:29:26.110 align:middle line:90% 00:29:26.110 --> 00:29:27.550 align:middle line:90% is that for m equal to 0-- 00:29:27.550 --> 00:29:30.760 align:middle line:90% 00:29:30.760 --> 00:29:44.620 align:middle line:84% so there's no coupling between psi L and psi R, OK? 00:29:44.620 --> 00:29:48.700 align:middle line:84% So it's the mass term which coupled them together, OK? 00:29:48.700 --> 00:29:50.860 align:middle line:90% Kinetic term-- psi L-- 00:29:50.860 --> 00:29:56.380 align:middle line:84% there's no cross term between the psi L and psi R, OK? 00:29:56.380 --> 00:29:58.090 align:middle line:84% And this behavior is actually general. 00:29:58.090 --> 00:30:00.277 align:middle line:84% You can write it in arbitrary representations. 00:30:00.277 --> 00:30:02.110 align:middle line:84% But, of course, in arbitrary representation, 00:30:02.110 --> 00:30:04.510 align:middle line:84% I can no longer use this sigma i, OK? 00:30:04.510 --> 00:30:10.630 align:middle line:90% And so in this particular form-- 00:30:10.630 --> 00:30:12.610 align:middle line:84% even though this feature is general, 00:30:12.610 --> 00:30:15.340 align:middle line:84% but this particular form of the kinetic term 00:30:15.340 --> 00:30:18.630 align:middle line:84% only applies for the chiral representation. 00:30:18.630 --> 00:30:19.130 align:middle line:90% OK. 00:30:19.130 --> 00:30:22.460 align:middle line:90% 00:30:22.460 --> 00:30:26.660 align:middle line:84% So for m equal to 0, you don't have coupling between the psi L 00:30:26.660 --> 00:30:31.360 align:middle line:84% and psi R. And then you only have to say, oh, diagonal 00:30:31.360 --> 00:30:33.560 align:middle line:90% term and psi R term. 00:30:33.560 --> 00:30:38.320 align:middle line:84% And then that gives you something else, OK? 00:30:38.320 --> 00:30:41.200 align:middle line:84% So, again, this is reduced to our previous statement 00:30:41.200 --> 00:30:43.420 align:middle line:84% that if you have a massless case, 00:30:43.420 --> 00:30:46.000 align:middle line:84% you can describe using a two-component spinor, OK? 00:30:46.000 --> 00:30:48.160 align:middle line:90% So here, indeed. 00:30:48.160 --> 00:30:51.290 align:middle line:84% But, here, there's also something extra. 00:30:51.290 --> 00:30:52.420 align:middle line:90% So what do you see-- 00:30:52.420 --> 00:30:53.590 align:middle line:90% something extra here? 00:30:53.590 --> 00:30:59.970 align:middle line:90% 00:30:59.970 --> 00:31:01.110 align:middle line:90% Yeah. 00:31:01.110 --> 00:31:03.210 align:middle line:90% Did somebody raise your-- yes. 00:31:03.210 --> 00:31:05.080 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:31:05.080 --> 00:31:06.104 align:middle line:90% HONG LIU: Yeah. 00:31:06.104 --> 00:31:10.944 align:middle line:84% AUDIENCE: Why are we now taking the psi equals psi L plus psi 00:31:10.944 --> 00:31:15.532 align:middle line:84% R instead of before we had it as psi L and Psi R 00:31:15.532 --> 00:31:16.490 align:middle line:90% HONG LIU: Right, right. 00:31:16.490 --> 00:31:17.120 align:middle line:90% Yeah, yeah. 00:31:17.120 --> 00:31:18.850 align:middle line:90% Sorry. 00:31:18.850 --> 00:31:19.998 align:middle line:90% A good question. 00:31:19.998 --> 00:31:21.290 align:middle line:90% Yeah, this expression is wrong. 00:31:21.290 --> 00:31:23.600 align:middle line:90% OK. 00:31:23.600 --> 00:31:25.810 align:middle line:90% Somehow, I was doing a-- 00:31:25.810 --> 00:31:27.330 align:middle line:90% I was trying to-- yeah. 00:31:27.330 --> 00:31:30.210 align:middle line:90% 00:31:30.210 --> 00:31:32.490 align:middle line:84% I remembered I wrote this in the general basis, 00:31:32.490 --> 00:31:34.650 align:middle line:84% but then I realized I only wrote it in that basis. 00:31:34.650 --> 00:31:36.240 align:middle line:84% Yeah, in the general basis, I would 00:31:36.240 --> 00:31:39.480 align:middle line:84% have psi equal to psi L plus psi R. Yeah. 00:31:39.480 --> 00:31:41.610 align:middle line:84% But then I realized, I only write the kinetic term 00:31:41.610 --> 00:31:43.590 align:middle line:90% in this specific basis. 00:31:43.590 --> 00:31:46.410 align:middle line:90% 00:31:46.410 --> 00:31:46.910 align:middle line:90% Good. 00:31:46.910 --> 00:31:47.410 align:middle line:90% Yeah. 00:31:47.410 --> 00:31:50.792 align:middle line:84% Yeah, so that expression does not apply for the chiral basis 00:31:50.792 --> 00:31:51.875 align:middle line:90% but apply for the general. 00:31:51.875 --> 00:31:56.710 align:middle line:90% 00:31:56.710 --> 00:32:00.590 align:middle line:84% So here, actually, something profound 00:32:00.590 --> 00:32:06.430 align:middle line:84% happens because, when m equal to 0, when you don't have coupling 00:32:06.430 --> 00:32:09.490 align:middle line:84% between psi L and psi R, you actually 00:32:09.490 --> 00:32:11.980 align:middle line:90% get the extra symmetry, OK? 00:32:11.980 --> 00:32:24.890 align:middle line:84% So in the Dirac Lagrangian, as we discussed earlier, 00:32:24.890 --> 00:32:28.150 align:middle line:90% so we have a U1 symmetry. 00:32:28.150 --> 00:32:33.290 align:middle line:84% Psi goes to exponential i alpha psi, OK? 00:32:33.290 --> 00:32:35.810 align:middle line:84% So this Dirac Lagrangian is invariant under that 00:32:35.810 --> 00:32:37.070 align:middle line:90% because the psi is complex. 00:32:37.070 --> 00:32:43.830 align:middle line:90% 00:32:43.830 --> 00:32:48.340 align:middle line:84% But, now, psi L and psi R-- they are separate. 00:32:48.340 --> 00:32:51.330 align:middle line:84% So, now, I can actually transform psi L and psi R 00:32:51.330 --> 00:32:52.740 align:middle line:90% separately, OK? 00:32:52.740 --> 00:32:57.030 align:middle line:84% So under this transformation, psi L and psi R 00:32:57.030 --> 00:32:59.760 align:middle line:90% transform the same, OK? 00:32:59.760 --> 00:33:05.410 align:middle line:90% But, now, m equal to 0-- 00:33:05.410 --> 00:33:09.760 align:middle line:84% I can have psi L goes to exponential i alpha L, 00:33:09.760 --> 00:33:22.710 align:middle line:84% psi L. Psi R goes to exponential i alpha R, psi R because they 00:33:22.710 --> 00:33:26.640 align:middle line:90% only couple to themselves, OK? 00:33:26.640 --> 00:33:32.390 align:middle line:84% And, now, I have this symmetry, OK? 00:33:32.390 --> 00:33:36.770 align:middle line:84% And so, now-- so, here, you have U1, 00:33:36.770 --> 00:33:43.340 align:middle line:84% and now you have U1 times U1 called U1 times L and U1R. 00:33:43.340 --> 00:33:48.430 align:middle line:84% So these are called chiral symmetries 00:33:48.430 --> 00:33:52.240 align:middle line:84% because they transform the left and the right separately. 00:33:52.240 --> 00:33:55.335 align:middle line:90% 00:33:55.335 --> 00:33:55.835 align:middle line:90% OK. 00:33:55.835 --> 00:34:01.090 align:middle line:90% 00:34:01.090 --> 00:34:01.750 align:middle line:90% Yes? 00:34:01.750 --> 00:34:03.560 align:middle line:84% AUDIENCE: I'm a little bit confused 00:34:03.560 --> 00:34:06.430 align:middle line:84% why you can't write psi as the sum of the two projections? 00:34:06.430 --> 00:34:07.120 align:middle line:90% HONG LIU: Sorry? 00:34:07.120 --> 00:34:08.643 align:middle line:84% AUDIENCE: Why you can't write psi 00:34:08.643 --> 00:34:10.060 align:middle line:84% as the sum of the two projections? 00:34:10.060 --> 00:34:10.601 align:middle line:90% HONG LIU: No. 00:34:10.601 --> 00:34:12.810 align:middle line:84% No, I can write it-- no, I can write it that way. 00:34:12.810 --> 00:34:16.710 align:middle line:84% Just, now, I'm using the two-component form. 00:34:16.710 --> 00:34:19.739 align:middle line:84% When I write two-component form, then I write psi that way, 00:34:19.739 --> 00:34:24.960 align:middle line:84% it doesn't make sense because psi L is the upper two 00:34:24.960 --> 00:34:27.030 align:middle line:84% component, and psi R is the lower two component. 00:34:27.030 --> 00:34:27.800 align:middle line:90% Yeah. 00:34:27.800 --> 00:34:28.800 align:middle line:90% Yeah, yeah. 00:34:28.800 --> 00:34:32.550 align:middle line:84% I'm using the same notation for this spaces. 00:34:32.550 --> 00:34:35.310 align:middle line:84% And so in these spaces, psi L and psi R, 00:34:35.310 --> 00:34:36.630 align:middle line:90% they only have two components. 00:34:36.630 --> 00:34:39.780 align:middle line:84% But in the general case, there have four components, OK? 00:34:39.780 --> 00:34:42.429 align:middle line:84% So in the general case, I can write psi equal to psi L 00:34:42.429 --> 00:34:45.190 align:middle line:84% and psi R. But in these spaces, I cannot. 00:34:45.190 --> 00:34:47.651 align:middle line:84% Yeah, using this notation, I cannot. 00:34:47.651 --> 00:34:55.719 align:middle line:90% 00:34:55.719 --> 00:35:00.160 align:middle line:84% So, now, you have a new symmetry equal to-- 00:35:00.160 --> 00:35:09.410 align:middle line:84% which you can transform them separately, OK? 00:35:09.410 --> 00:35:12.110 align:middle line:84% And the symmetries are one of the most important aspect 00:35:12.110 --> 00:35:15.930 align:middle line:84% of physics, and they have very important implications, 00:35:15.930 --> 00:35:16.920 align:middle line:90% et cetera. 00:35:16.920 --> 00:35:23.990 align:middle line:84% And the chiral symmetry actually has also very important effect 00:35:23.990 --> 00:35:26.780 align:middle line:90% in particle physics-- 00:35:26.780 --> 00:35:30.020 align:middle line:90% for example, the pions. 00:35:30.020 --> 00:35:33.290 align:middle line:90% The pions has to do with-- 00:35:33.290 --> 00:35:35.180 align:middle line:90% I will not go into detail. 00:35:35.180 --> 00:35:39.890 align:middle line:84% The pions-- they essentially come from the chiral 00:35:39.890 --> 00:35:41.360 align:middle line:90% symmetries. 00:35:41.360 --> 00:35:43.700 align:middle line:84% Without the chiral symmetries, there's no pion. 00:35:43.700 --> 00:35:46.070 align:middle line:90% There's no pions, OK? 00:35:46.070 --> 00:35:51.620 align:middle line:84% And actually understanding how the pions 00:35:51.620 --> 00:35:53.920 align:middle line:84% come from the chiral symmetries, et cetera-- 00:35:53.920 --> 00:35:58.110 align:middle line:84% there was a Nobel Prize to Nambu a number of years ago, 00:35:58.110 --> 00:35:59.930 align:middle line:84% which our colleague Goldstone also 00:35:59.930 --> 00:36:04.560 align:middle line:84% made a very important contribution to that. 00:36:04.560 --> 00:36:08.750 align:middle line:84% And, also, nature-- also interesting 00:36:08.750 --> 00:36:12.540 align:middle line:84% about this chiral symmetry is that they are there. 00:36:12.540 --> 00:36:14.990 align:middle line:84% Say, if you have a classical massless-- say, 00:36:14.990 --> 00:36:18.920 align:middle line:84% if you have a massless Lagrangian, 00:36:18.920 --> 00:36:21.890 align:middle line:84% then you can have the symmetry in the classical level, 00:36:21.890 --> 00:36:23.430 align:middle line:90% in the Lagrangian level. 00:36:23.430 --> 00:36:24.980 align:middle line:84% But once you quantize this theory, 00:36:24.980 --> 00:36:28.140 align:middle line:90% you find the symmetry goes away. 00:36:28.140 --> 00:36:29.130 align:middle line:90% It becomes anomalous. 00:36:29.130 --> 00:36:30.840 align:middle line:90% It could become anomalous. 00:36:30.840 --> 00:36:33.090 align:middle line:84% Symmetry is only present in the classical level 00:36:33.090 --> 00:36:36.777 align:middle line:90% but not in the quantum level. 00:36:36.777 --> 00:36:38.610 align:middle line:84% And, again, that plays a very important role 00:36:38.610 --> 00:36:40.130 align:middle line:90% in particle physics, actually. 00:36:40.130 --> 00:36:40.630 align:middle line:90% OK. 00:36:40.630 --> 00:36:44.280 align:middle line:84% Yeah, the bottom line is that the chiral symmetry 00:36:44.280 --> 00:36:49.470 align:middle line:84% is very important in many aspects of physics. 00:36:49.470 --> 00:36:52.200 align:middle line:84% It's also important in many condensed matter systems, 00:36:52.200 --> 00:36:54.880 align:middle line:90% like liquid helium, et cetera. 00:36:54.880 --> 00:36:58.275 align:middle line:84% So you can also write this symmetry for general gamma. 00:36:58.275 --> 00:37:04.470 align:middle line:90% 00:37:04.470 --> 00:37:08.530 align:middle line:84% So for general gamma, you can have your previous symmetry. 00:37:08.530 --> 00:37:11.620 align:middle line:84% So, now, I'm using the four-component notation, 00:37:11.620 --> 00:37:15.490 align:middle line:84% which psi L and psi R transform the same way. 00:37:15.490 --> 00:37:17.205 align:middle line:84% And then, now, you have a new symmetry. 00:37:17.205 --> 00:37:22.280 align:middle line:90% 00:37:22.280 --> 00:37:25.760 align:middle line:90% Gamma 5-- OK. 00:37:25.760 --> 00:37:28.430 align:middle line:84% And now you can put the gamma 5 in the exponents. 00:37:28.430 --> 00:37:37.510 align:middle line:90% 00:37:37.510 --> 00:37:38.770 align:middle line:90% Good. 00:37:38.770 --> 00:37:40.230 align:middle line:90% Any questions on this? 00:37:40.230 --> 00:37:40.890 align:middle line:90% Yes. 00:37:40.890 --> 00:37:42.365 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:37:42.365 --> 00:37:43.240 align:middle line:90% HONG LIU: Yeah, yeah. 00:37:43.240 --> 00:37:44.655 align:middle line:84% Alpha tilde is just some other constant-- 00:37:44.655 --> 00:37:45.360 align:middle line:90% AUDIENCE: Oh, alpha tilde. 00:37:45.360 --> 00:37:46.810 align:middle line:84% HONG LIU: If tilde is just some other constant, 00:37:46.810 --> 00:37:47.893 align:middle line:90% then you multiply gamma 5. 00:37:47.893 --> 00:37:50.780 align:middle line:90% 00:37:50.780 --> 00:37:55.150 align:middle line:84% So the way to understand that these two are related-- 00:37:55.150 --> 00:37:58.850 align:middle line:84% so think about the transformation here. 00:37:58.850 --> 00:38:02.330 align:middle line:84% So, here, we can rewrite a little bit differently. 00:38:02.330 --> 00:38:06.370 align:middle line:84% We can consider rewrite this alpha L and alpha R 00:38:06.370 --> 00:38:09.120 align:middle line:90% in terms of the following. 00:38:09.120 --> 00:38:11.900 align:middle line:84% Let's consider the two transformation-- one 00:38:11.900 --> 00:38:16.350 align:middle line:84% transformation, psi L, and psi R transform the same. 00:38:16.350 --> 00:38:21.090 align:middle line:84% And the other transformation is that they transform oppositely. 00:38:21.090 --> 00:38:23.460 align:middle line:84% They transform in the opposite phase, OK? 00:38:23.460 --> 00:38:26.325 align:middle line:90% So I write the alpha-- 00:38:26.325 --> 00:38:28.350 align:middle line:84% and this writing is like that, OK? 00:38:28.350 --> 00:38:33.110 align:middle line:84% So, in this way, psi L and psi R transform the same, but-- 00:38:33.110 --> 00:38:35.370 align:middle line:84% psi L and psi R, they transform opposite 00:38:35.370 --> 00:38:39.080 align:middle line:84% because they have opposite eigenvalue on the gamma 5. 00:38:39.080 --> 00:38:40.700 align:middle line:84% And so this is equivalent to that. 00:38:40.700 --> 00:38:46.660 align:middle line:90% 00:38:46.660 --> 00:38:47.210 align:middle line:90% Good. 00:38:47.210 --> 00:38:49.870 align:middle line:90% Any questions on this? 00:38:49.870 --> 00:38:50.650 align:middle line:90% Yes. 00:38:50.650 --> 00:38:54.500 align:middle line:84% AUDIENCE: Why is it called gamma 5 rather than, say, gamma 4? 00:38:54.500 --> 00:38:56.870 align:middle line:84% HONG LIU: I think, again, it's a historical reason. 00:38:56.870 --> 00:38:59.870 align:middle line:90% 00:38:59.870 --> 00:39:06.280 align:middle line:84% So people often like to go to Euclidean space. 00:39:06.280 --> 00:39:07.890 align:middle line:84% So when you go to Euclidean space, 00:39:07.890 --> 00:39:11.610 align:middle line:90% you continue gamma 0 to gamma 4. 00:39:11.610 --> 00:39:14.310 align:middle line:90% Yeah, just-- yeah, to gamma 4. 00:39:14.310 --> 00:39:17.580 align:middle line:84% And then you reserve gamma 4 for that. 00:39:17.580 --> 00:39:19.530 align:middle line:84% Then the gamma 5 is the next one you take. 00:39:19.530 --> 00:39:22.133 align:middle line:90% 00:39:22.133 --> 00:39:22.800 align:middle line:90% Other questions? 00:39:22.800 --> 00:39:23.820 align:middle line:90% Yes. 00:39:23.820 --> 00:39:27.030 align:middle line:84% AUDIENCE: Is there a physical reason why 00:39:27.030 --> 00:39:29.190 align:middle line:90% the massless case is special? 00:39:29.190 --> 00:39:31.050 align:middle line:84% Is it because it becomes scale-free, 00:39:31.050 --> 00:39:32.370 align:middle line:90% or something like that? 00:39:32.370 --> 00:39:33.870 align:middle line:90% HONG LIU: Yeah, yeah. 00:39:33.870 --> 00:39:36.510 align:middle line:84% So massless case-- it's always special. 00:39:36.510 --> 00:39:39.810 align:middle line:84% And you will see, in physics, actually, 00:39:39.810 --> 00:39:43.260 align:middle line:84% the massless case actually gives you 00:39:43.260 --> 00:39:45.990 align:middle line:84% very much richer structure, normally, 00:39:45.990 --> 00:39:51.570 align:middle line:90% than the massive particle. 00:39:51.570 --> 00:39:54.060 align:middle line:84% Mathematically, it's because the massless case-- 00:39:54.060 --> 00:39:57.290 align:middle line:90% 00:39:57.290 --> 00:39:59.180 align:middle line:84% in the massless case, the representation 00:39:59.180 --> 00:40:01.490 align:middle line:84% of the Lorentz group is very different 00:40:01.490 --> 00:40:03.440 align:middle line:90% from the massive case. 00:40:03.440 --> 00:40:05.390 align:middle line:84% For example, if you have a vector field, 00:40:05.390 --> 00:40:08.330 align:middle line:84% say, for Maxwell field, for the photon, 00:40:08.330 --> 00:40:10.910 align:middle line:84% massless photon have two polarizations. 00:40:10.910 --> 00:40:13.040 align:middle line:84% But if you have a massive vector-- 00:40:13.040 --> 00:40:14.690 align:middle line:90% say, for the photon is massive-- 00:40:14.690 --> 00:40:16.850 align:middle line:84% then we'll have three polarizations. 00:40:16.850 --> 00:40:20.180 align:middle line:84% And so the massless case and the massive case 00:40:20.180 --> 00:40:22.400 align:middle line:90% are very, very different. 00:40:22.400 --> 00:40:24.830 align:middle line:90% The fermion case is the same. 00:40:24.830 --> 00:40:26.810 align:middle line:84% So if you have a massive fermion, 00:40:26.810 --> 00:40:28.432 align:middle line:84% you have four complex components. 00:40:28.432 --> 00:40:30.140 align:middle line:84% But if you have a massless case, then you 00:40:30.140 --> 00:40:32.770 align:middle line:90% have two complex components. 00:40:32.770 --> 00:40:34.270 align:middle line:84% AUDIENCE: How about very small mass? 00:40:34.270 --> 00:40:39.270 align:middle line:84% Doesn't that cause a problem if you go from x [INAUDIBLE]?? 00:40:39.270 --> 00:40:40.242 align:middle line:90% HONG LIU: Yeah, yeah. 00:40:40.242 --> 00:40:41.700 align:middle line:84% That's a very interesting question. 00:40:41.700 --> 00:40:43.075 align:middle line:84% And there are a lot of subtleties 00:40:43.075 --> 00:40:47.190 align:middle line:84% associated with that kind of questions, indeed. 00:40:47.190 --> 00:40:47.850 align:middle line:90% Yes? 00:40:47.850 --> 00:40:49.516 align:middle line:84% AUDIENCE: Is there any sense in which 00:40:49.516 --> 00:40:53.530 align:middle line:84% we can treat psi L and psi R as different dynamical fields, 00:40:53.530 --> 00:40:59.000 align:middle line:84% like a complex scalar field the case phi and phi star? 00:40:59.000 --> 00:41:02.920 align:middle line:84% HONG LIU: Oh, psi L and psi R certainly are independent-- 00:41:02.920 --> 00:41:07.980 align:middle line:84% you can certainly treat them as independent dynamical, yeah. 00:41:07.980 --> 00:41:08.940 align:middle line:90% Other questions? 00:41:08.940 --> 00:41:12.890 align:middle line:90% 00:41:12.890 --> 00:41:15.100 align:middle line:90% OK. 00:41:15.100 --> 00:41:19.690 align:middle line:84% So let's conclude our discussion of the chiral spinors. 00:41:19.690 --> 00:41:22.000 align:middle line:84% And now we can talk about the Majorana spinors. 00:41:22.000 --> 00:41:31.840 align:middle line:90% 00:41:31.840 --> 00:41:40.590 align:middle line:84% OK, so the so the Dirac spinor, which 00:41:40.590 --> 00:41:44.320 align:middle line:84% we have talked about so far, is four components. 00:41:44.320 --> 00:41:48.810 align:middle line:84% So you have four times two real components, OK? 00:41:48.810 --> 00:41:51.300 align:middle line:84% And then the chiral spinor we talked about-- 00:41:51.300 --> 00:41:54.190 align:middle line:90% 00:41:54.190 --> 00:41:57.800 align:middle line:84% essentially, you have two complex components. 00:41:57.800 --> 00:42:02.150 align:middle line:84% It's, say, 2 times 2 real components, OK? 00:42:02.150 --> 00:42:03.990 align:middle line:90% 2 times 2 real components. 00:42:03.990 --> 00:42:05.870 align:middle line:84% So the next one I'm going to talk about 00:42:05.870 --> 00:42:14.380 align:middle line:84% is the Majorana, in which case, I would argue, 00:42:14.380 --> 00:42:17.770 align:middle line:84% we have 4 times 1 real component. 00:42:17.770 --> 00:42:21.920 align:middle line:84% So it has four independent real components, OK? 00:42:21.920 --> 00:42:22.420 align:middle line:90% Yes? 00:42:22.420 --> 00:42:24.600 align:middle line:84% AUDIENCE: Sorry going back real quick. 00:42:24.600 --> 00:42:27.630 align:middle line:84% So I'm assuming R and L are for right-handed and left-handed 00:42:27.630 --> 00:42:28.480 align:middle line:90% in this [INAUDIBLE]. 00:42:28.480 --> 00:42:30.990 align:middle line:84% HONG LIU: Yeah, yeah, yeah, yeah. 00:42:30.990 --> 00:42:34.960 align:middle line:84% AUDIENCE: So can you ever have a spinor where-- 00:42:34.960 --> 00:42:37.780 align:middle line:84% what does it mean to have one of [INAUDIBLE] non-zero? 00:42:37.780 --> 00:42:40.140 align:middle line:84% So both of them are non-zero, I guess. 00:42:40.140 --> 00:42:41.910 align:middle line:84% What is that-- because, in my mind, 00:42:41.910 --> 00:42:43.690 align:middle line:84% it is either right-handed or right handed. 00:42:43.690 --> 00:42:44.860 align:middle line:90% What is the mixture of-- 00:42:44.860 --> 00:42:46.090 align:middle line:90% HONG LIU: No, no, no, no. 00:42:46.090 --> 00:42:49.120 align:middle line:84% Physically, electron contains both left and right. 00:42:49.120 --> 00:42:51.490 align:middle line:84% Yeah, so for massive particle, because they always 00:42:51.490 --> 00:42:53.610 align:middle line:90% couple together-- 00:42:53.610 --> 00:42:56.930 align:middle line:84% so the left hand, so psi L turning into psi R, 00:42:56.930 --> 00:42:59.370 align:middle line:84% psi R turning to psi L all the time. 00:42:59.370 --> 00:43:03.430 align:middle line:84% And for massive particle, you cannot really separate them. 00:43:03.430 --> 00:43:09.520 align:middle line:84% But, now, for massless particle, then psi L and psi R-- yeah, 00:43:09.520 --> 00:43:12.710 align:middle line:84% this is a very good question-- psi L and psi R are preserved. 00:43:12.710 --> 00:43:16.030 align:middle line:84% So massless particle is either psi L or psi R. 00:43:16.030 --> 00:43:20.140 align:middle line:84% And then if you look at their so-called helicity, 00:43:20.140 --> 00:43:22.730 align:middle line:84% it's either left-handed or right-handed. 00:43:22.730 --> 00:43:26.620 align:middle line:84% So that's where the name psi L and psi R come from. 00:43:26.620 --> 00:43:29.180 align:middle line:84% But, for the massive particle, you cannot make this 00:43:29.180 --> 00:43:30.310 align:middle line:90% separation. 00:43:30.310 --> 00:43:32.440 align:middle line:84% But for the massless case, then it's preserved. 00:43:32.440 --> 00:43:37.820 align:middle line:84% And then it's generally left-handed or right-handed. 00:43:37.820 --> 00:43:38.580 align:middle line:90% OK, good. 00:43:38.580 --> 00:43:41.450 align:middle line:84% And, now, let's talk about the last case-- 00:43:41.450 --> 00:43:43.265 align:middle line:90% this case, which is-- 00:43:43.265 --> 00:43:45.645 align:middle line:84% you have four real components, OK? 00:43:45.645 --> 00:43:49.990 align:middle line:90% 00:43:49.990 --> 00:43:50.920 align:middle line:90% So what do you do? 00:43:50.920 --> 00:43:53.440 align:middle line:84% Again, we follow the similar strategy 00:43:53.440 --> 00:43:58.780 align:middle line:84% to see whether it's possible to have four real components. 00:43:58.780 --> 00:44:01.900 align:middle line:84% Again, we first-- the idea is that you first 00:44:01.900 --> 00:44:05.230 align:middle line:84% try to find the special representation of gamma 00:44:05.230 --> 00:44:10.200 align:middle line:84% matrices so that a spinor that can be real, OK? 00:44:10.200 --> 00:44:13.260 align:middle line:84% And then you try to generalize to any representation of gamma 00:44:13.260 --> 00:44:14.470 align:middle line:90% matrices. 00:44:14.470 --> 00:44:14.970 align:middle line:90% OK. 00:44:14.970 --> 00:44:18.100 align:middle line:90% 00:44:18.100 --> 00:44:26.050 align:middle line:84% So, now, if you look at the Dirac equation-- 00:44:26.050 --> 00:44:47.392 align:middle line:90% 00:44:47.392 --> 00:44:48.725 align:middle line:90% now, look at the Dirac equation. 00:44:48.725 --> 00:44:58.310 align:middle line:90% 00:44:58.310 --> 00:45:05.360 align:middle line:84% So if I want to find the real spinor, 00:45:05.360 --> 00:45:07.550 align:middle line:84% then I ask myself whether real spinor 00:45:07.550 --> 00:45:10.700 align:middle line:84% is compatible with this equation, OK? 00:45:10.700 --> 00:45:14.300 align:middle line:84% So if the real scalar is not compatible with this equation-- 00:45:14.300 --> 00:45:17.300 align:middle line:84% because I take the complex conjugate, 00:45:17.300 --> 00:45:18.635 align:middle line:90% I then take a gamma mu star. 00:45:18.635 --> 00:45:23.480 align:middle line:90% 00:45:23.480 --> 00:45:30.120 align:middle line:84% If psi is real, then psi also have to satisfy this equation, 00:45:30.120 --> 00:45:30.630 align:middle line:90% OK? 00:45:30.630 --> 00:45:34.400 align:middle line:84% But, in general, gamma mu is complex, as we wrote before-- 00:45:34.400 --> 00:45:35.450 align:middle line:90% yeah, I just erased it. 00:45:35.450 --> 00:45:39.270 align:middle line:84% For example, in these spaces, it's complex. 00:45:39.270 --> 00:45:42.670 align:middle line:84% With complex, then these two equations are not compatible, 00:45:42.670 --> 00:45:45.600 align:middle line:90% and then psi cannot be real, OK? 00:45:45.600 --> 00:45:46.290 align:middle line:90% Simple as that. 00:45:46.290 --> 00:45:49.110 align:middle line:90% 00:45:49.110 --> 00:45:51.570 align:middle line:90% But there's a way out. 00:45:51.570 --> 00:45:53.790 align:middle line:84% The way out is that, if they exist, 00:45:53.790 --> 00:45:58.690 align:middle line:84% the representation of gamma mu, the question 00:45:58.690 --> 00:46:06.720 align:middle line:84% is whether there exists a representation of gamma mu 00:46:06.720 --> 00:46:07.995 align:middle line:90% so that gamma mu is real. 00:46:07.995 --> 00:46:13.680 align:middle line:90% 00:46:13.680 --> 00:46:18.570 align:middle line:84% So if this is real, and then these two equations become 00:46:18.570 --> 00:46:27.370 align:middle line:84% the same, and then for psi being real 00:46:27.370 --> 00:46:30.980 align:middle line:84% is compatible with the Dirac equation, OK? 00:46:30.980 --> 00:46:34.940 align:middle line:84% It's compatible with the Dirac equation. 00:46:34.940 --> 00:46:41.240 align:middle line:84% So, now, let's-- so then this becomes a question of trial 00:46:41.240 --> 00:46:42.710 align:middle line:90% and error, OK? 00:46:42.710 --> 00:46:44.990 align:middle line:84% So you try to find the representation of gamma 00:46:44.990 --> 00:46:48.960 align:middle line:90% matrices so that it's real OK. 00:46:48.960 --> 00:46:54.990 align:middle line:84% And then it turns out, you can find it, 00:46:54.990 --> 00:46:56.860 align:middle line:90% and here is the answer. 00:46:56.860 --> 00:46:59.940 align:middle line:84% So let me just write down the answer. 00:46:59.940 --> 00:47:04.070 align:middle line:84% I don't know how he originally found it-- 00:47:04.070 --> 00:47:07.895 align:middle line:84% Majorana originally found it, but here is the answer. 00:47:07.895 --> 00:47:32.290 align:middle line:90% 00:47:32.290 --> 00:47:35.600 align:middle line:90% So this is four gamma matrices. 00:47:35.600 --> 00:47:37.900 align:middle line:84% And you can see there, each of them 00:47:37.900 --> 00:47:43.530 align:middle line:84% is real because sigma 2 is pure imaginary, and sigma 1, sigma 3 00:47:43.530 --> 00:47:44.920 align:middle line:90% are real. 00:47:44.920 --> 00:47:46.630 align:middle line:90% And so this is purely real. 00:47:46.630 --> 00:47:47.400 align:middle line:90% And you can check. 00:47:47.400 --> 00:47:51.258 align:middle line:84% This satisfied the algebra of the government matrices. 00:47:51.258 --> 00:47:53.050 align:middle line:84% They satisfy the algebra of gamma matrices. 00:47:53.050 --> 00:47:57.990 align:middle line:84% They anticommute with each other, and each of them 00:47:57.990 --> 00:47:58.740 align:middle line:90% square-- 00:47:58.740 --> 00:48:00.653 align:middle line:90% so each of them squaring 1. 00:48:00.653 --> 00:48:02.070 align:middle line:84% So these three square them into 1. 00:48:02.070 --> 00:48:03.300 align:middle line:90% This square equal minus 1. 00:48:03.300 --> 00:48:12.480 align:middle line:90% 00:48:12.480 --> 00:48:15.210 align:middle line:84% So, now, this is compatible with Dirac equation, 00:48:15.210 --> 00:48:18.370 align:middle line:90% but this is actually not enough. 00:48:18.370 --> 00:48:20.910 align:middle line:84% We also have to be sure this is compatible with the Lorentz 00:48:20.910 --> 00:48:24.430 align:middle line:90% transformation, OK? 00:48:24.430 --> 00:48:26.930 align:middle line:84% So, now, let's check whether this is compatible with Lorentz 00:48:26.930 --> 00:48:29.130 align:middle line:90% transformation. 00:48:29.130 --> 00:48:32.750 align:middle line:90% So, now, we have this sigma mu-- 00:48:32.750 --> 00:48:33.590 align:middle line:90% gamma mu. 00:48:33.590 --> 00:48:36.800 align:middle line:84% And, remember, the sigma mu, nu-- 00:48:36.800 --> 00:48:38.030 align:middle line:90% I just erased it. 00:48:38.030 --> 00:48:43.290 align:middle line:90% 00:48:43.290 --> 00:48:47.810 align:middle line:84% So now this is pure imaginary because if gamma mu and gamma 00:48:47.810 --> 00:48:50.030 align:middle line:84% nu are real, their commutator is also real. 00:48:50.030 --> 00:48:53.750 align:middle line:84% And then sigma mu, nu will be pure imaginary. 00:48:53.750 --> 00:48:55.295 align:middle line:90% And then that means S lambda-- 00:48:55.295 --> 00:49:04.640 align:middle line:90% 00:49:04.640 --> 00:49:06.245 align:middle line:90% so this is now purely real. 00:49:06.245 --> 00:49:10.260 align:middle line:90% 00:49:10.260 --> 00:49:12.610 align:middle line:90% Now, this means this is real. 00:49:12.610 --> 00:49:14.800 align:middle line:90% And, now, we are done, OK? 00:49:14.800 --> 00:49:19.180 align:middle line:84% That means that if we take the psi is real, then 00:49:19.180 --> 00:49:24.010 align:middle line:84% after Lorentz transformation, this remains to be real, OK? 00:49:24.010 --> 00:49:27.550 align:middle line:84% And so that means that it's compatible with Lorentz 00:49:27.550 --> 00:49:29.260 align:middle line:90% transformation. 00:49:29.260 --> 00:49:32.250 align:middle line:84% So if we were not compatible with Lorentz transmission, 00:49:32.250 --> 00:49:35.010 align:middle line:90% then we were finished, OK? 00:49:35.010 --> 00:49:36.720 align:middle line:84% So this shows that this remains real. 00:49:36.720 --> 00:49:52.738 align:middle line:90% 00:49:52.738 --> 00:49:56.350 align:middle line:84% So such a spinor is called a Majorana spinor, OK? 00:49:56.350 --> 00:49:59.140 align:middle line:90% So it has four real components. 00:49:59.140 --> 00:50:02.080 align:middle line:84% Yeah, we wrote there-- it has four real components. 00:50:02.080 --> 00:50:06.940 align:middle line:90% 00:50:06.940 --> 00:50:11.230 align:middle line:84% So you can quantize it, which, I think 00:50:11.230 --> 00:50:14.140 align:middle line:84% I will give it as an exercise for you to do, OK? 00:50:14.140 --> 00:50:15.430 align:middle line:90% You can quantize it. 00:50:15.430 --> 00:50:21.910 align:middle line:84% And then, in this case, then the fermions 00:50:21.910 --> 00:50:31.000 align:middle line:84% are its own antiparticle, rather than for the Dirac spinor. 00:50:31.000 --> 00:50:32.950 align:middle line:84% You have particle and antiparticle, 00:50:32.950 --> 00:50:39.374 align:middle line:84% so this is the analog of the real scalar in the spinor case. 00:50:39.374 --> 00:50:42.100 align:middle line:90% 00:50:42.100 --> 00:50:48.490 align:middle line:84% So this was discovered by Majorana in 1937, OK? 00:50:48.490 --> 00:50:49.660 align:middle line:90% And he was very young. 00:50:49.660 --> 00:50:52.240 align:middle line:90% At the time, he was 31. 00:50:52.240 --> 00:50:57.670 align:middle line:84% And, yeah, a brilliant physicist, complete genius. 00:50:57.670 --> 00:51:01.070 align:middle line:90% And then, in 1938-- 00:51:01.070 --> 00:51:03.860 align:middle line:90% so he lived in Sicily, OK? 00:51:03.860 --> 00:51:05.600 align:middle line:90% So his hometown was in Sicily. 00:51:05.600 --> 00:51:13.220 align:middle line:84% So he boarded a ship from Naples to Sicily. 00:51:13.220 --> 00:51:17.150 align:middle line:84% And then he just disappeared on the ship, never seen again. 00:51:17.150 --> 00:51:20.110 align:middle line:90% 00:51:20.110 --> 00:51:25.760 align:middle line:84% At the age of 32, he just disappeared. 00:51:25.760 --> 00:51:29.300 align:middle line:90% Yeah, it's a quite-- 00:51:29.300 --> 00:51:32.690 align:middle line:90% 00:51:32.690 --> 00:51:34.790 align:middle line:84% yeah, extremely brilliant physicist. 00:51:34.790 --> 00:51:35.900 align:middle line:90% Yeah. 00:51:35.900 --> 00:51:37.940 align:middle line:84% And there are all kinds of stories 00:51:37.940 --> 00:51:41.630 align:middle line:84% about his disappearance-- that he may be killed by Mafia 00:51:41.630 --> 00:51:44.360 align:middle line:90% or maybe suicide, et cetera. 00:51:44.360 --> 00:51:46.600 align:middle line:90% But just nobody knows. 00:51:46.600 --> 00:51:47.925 align:middle line:90% Yes? 00:51:47.925 --> 00:51:50.300 align:middle line:84% AUDIENCE: You said that all the choices of gamma matrices 00:51:50.300 --> 00:51:51.260 align:middle line:90% were equivalent. 00:51:51.260 --> 00:51:52.190 align:middle line:90% HONG LIU: Yeah. 00:51:52.190 --> 00:51:54.410 align:middle line:84% AUDIENCE: This doesn't really feel equivalent. 00:51:54.410 --> 00:51:56.650 align:middle line:84% It is still equivalent to the other representations? 00:51:56.650 --> 00:51:57.250 align:middle line:90% HONG LIU: Yeah, yeah. 00:51:57.250 --> 00:51:58.270 align:middle line:90% We will talk about that. 00:51:58.270 --> 00:52:02.070 align:middle line:90% 00:52:02.070 --> 00:52:07.940 align:middle line:84% So now we have chosen a very specific representation 00:52:07.940 --> 00:52:14.450 align:middle line:84% for gamma matrices, which psi can be chosen to be real, OK? 00:52:14.450 --> 00:52:21.170 align:middle line:84% But how about for the general representation? 00:52:21.170 --> 00:52:23.225 align:middle line:84% So now we talk about the general. 00:52:23.225 --> 00:52:29.240 align:middle line:90% 00:52:29.240 --> 00:52:34.360 align:middle line:90% Majorana spinor-- yeah. 00:52:34.360 --> 00:52:36.060 align:middle line:84% Yeah, also, let me just make a remark. 00:52:36.060 --> 00:52:39.750 align:middle line:84% Majorana spinor, of course, also plays a very important role 00:52:39.750 --> 00:52:41.550 align:middle line:90% in modern-day physics. 00:52:41.550 --> 00:52:45.720 align:middle line:90% 00:52:45.720 --> 00:52:51.810 align:middle line:84% Say, for example, people suspect a neutrino could be a Majorana 00:52:51.810 --> 00:52:53.310 align:middle line:90% spinner, OK? 00:52:53.310 --> 00:52:57.330 align:middle line:84% So to check whether a neutrino is a Majorana spinor-- 00:52:57.330 --> 00:53:00.600 align:middle line:90% 00:53:00.600 --> 00:53:03.990 align:middle line:84% yeah, it's a forefront experimental program-- 00:53:03.990 --> 00:53:06.600 align:middle line:90% has been pursued by many years. 00:53:06.600 --> 00:53:12.990 align:middle line:84% And, also, in condensed matter, in quantum information, 00:53:12.990 --> 00:53:16.090 align:middle line:84% and Majorana spinor play a very important role. 00:53:16.090 --> 00:53:19.210 align:middle line:84% And so, in condensed matter, you only have electron. 00:53:19.210 --> 00:53:23.440 align:middle line:84% So electron and the Majorana spinor is, like, half electron, 00:53:23.440 --> 00:53:23.940 align:middle line:90% OK? 00:53:23.940 --> 00:53:26.810 align:middle line:84% Because the electron have eight components, right? 00:53:26.810 --> 00:53:27.390 align:middle line:90% Remember. 00:53:27.390 --> 00:53:29.170 align:middle line:84% And Majorana only have four components. 00:53:29.170 --> 00:53:32.190 align:middle line:84% So Majorana is, like, half electron. 00:53:32.190 --> 00:53:36.280 align:middle line:84% So precisely because it's heuristically half electron, 00:53:36.280 --> 00:53:40.210 align:middle line:84% it has very stable topological properties, 00:53:40.210 --> 00:53:43.150 align:middle line:84% which a single electron does not have. 00:53:43.150 --> 00:53:48.280 align:middle line:84% And whether you can engineer in your condensed matter systems, 00:53:48.280 --> 00:53:50.620 align:middle line:84% Majorana spinor then became a Holy Grail. 00:53:50.620 --> 00:53:54.220 align:middle line:84% Because if you can do it, and then you can do lots of-- 00:53:54.220 --> 00:53:58.840 align:middle line:84% yeah, you can achieve more stable quantum computation, 00:53:58.840 --> 00:54:00.760 align:middle line:90% et cetera. 00:54:00.760 --> 00:54:01.510 align:middle line:90% Yeah. 00:54:01.510 --> 00:54:04.090 align:middle line:84% During the last number of years, there 00:54:04.090 --> 00:54:06.580 align:middle line:84% have been various experimental reports. 00:54:06.580 --> 00:54:10.480 align:middle line:84% People say they have engineered Majorana spinor 00:54:10.480 --> 00:54:14.920 align:middle line:84% in the lab, which I think has never been fully confirmed, 00:54:14.920 --> 00:54:15.880 align:middle line:90% I think. 00:54:15.880 --> 00:54:18.130 align:middle line:84% None of them has been fully confirmed. 00:54:18.130 --> 00:54:23.310 align:middle line:84% Anyway, so, yeah, so, now, let's talk about Majorana spinor 00:54:23.310 --> 00:54:33.415 align:middle line:84% in general basis, for general gamma mu. 00:54:33.415 --> 00:54:36.370 align:middle line:90% 00:54:36.370 --> 00:54:43.570 align:middle line:84% So the idea would be similar to the case of the chiral spinor. 00:54:43.570 --> 00:54:46.180 align:middle line:90% For the chiral spinor-- 00:54:46.180 --> 00:54:48.220 align:middle line:84% in the chiral basis, it's very simple, just 00:54:48.220 --> 00:54:51.780 align:middle line:90% upper and lower components. 00:54:51.780 --> 00:54:54.020 align:middle line:84% So, for the chiral spinor, you have 00:54:54.020 --> 00:54:55.430 align:middle line:84% to introduce some other structure 00:54:55.430 --> 00:54:59.780 align:middle line:84% to isolate psi L and psi R. So you have to-- 00:54:59.780 --> 00:55:02.000 align:middle line:84% now, you have this nontrivial condition, OK? 00:55:02.000 --> 00:55:05.177 align:middle line:84% And the chiral fermions come from this nontrivial condition. 00:55:05.177 --> 00:55:09.280 align:middle line:90% 00:55:09.280 --> 00:55:10.900 align:middle line:84% So, now, the key is that, how do you 00:55:10.900 --> 00:55:16.070 align:middle line:84% find the analogous condition to the psi to be real 00:55:16.070 --> 00:55:18.560 align:middle line:90% in the general basis, OK? 00:55:18.560 --> 00:55:23.370 align:middle line:84% Because when gamma mu is generally complex, clearly, 00:55:23.370 --> 00:55:25.730 align:middle line:84% you cannot set the psi equal to psi star, OK? 00:55:25.730 --> 00:55:27.290 align:middle line:90% That does not make sense. 00:55:27.290 --> 00:55:30.530 align:middle line:84% You have to find another equivalent equation 00:55:30.530 --> 00:55:32.420 align:middle line:84% to do, essentially, the same thing, OK? 00:55:32.420 --> 00:55:35.180 align:middle line:90% So that's the basic idea. 00:55:35.180 --> 00:55:41.430 align:middle line:84% OK, so for this purpose, we want to-- 00:55:41.430 --> 00:55:43.950 align:middle line:84% now, we use any gamma matrices that 00:55:43.950 --> 00:55:47.090 align:middle line:84% are equivalent to each other up to a similar transformation, 00:55:47.090 --> 00:55:48.820 align:middle line:90% OK? 00:55:48.820 --> 00:55:59.840 align:middle line:84% So let's denote-- this basis by gamma m, OK? 00:55:59.840 --> 00:56:05.870 align:middle line:84% And then we have gamma m, which is this called Majorana basis. 00:56:05.870 --> 00:56:09.860 align:middle line:84% And then any choice of gamma mu then related to gamma m 00:56:09.860 --> 00:56:11.690 align:middle line:84% by a similar transformation-- that 00:56:11.690 --> 00:56:22.460 align:middle line:84% means there exists some matrix C that the C can take any gamma 00:56:22.460 --> 00:56:26.390 align:middle line:90% mu into gamma mu m, OK? 00:56:26.390 --> 00:56:29.690 align:middle line:84% So there must exist C, and this equation is satisfied. 00:56:29.690 --> 00:56:37.265 align:middle line:90% 00:56:37.265 --> 00:56:37.765 align:middle line:90% Good. 00:56:37.765 --> 00:56:45.010 align:middle line:90% 00:56:45.010 --> 00:56:47.530 align:middle line:84% So, now, given this C, then we can easily 00:56:47.530 --> 00:56:59.260 align:middle line:84% write down the condition for the general basis 00:56:59.260 --> 00:57:09.190 align:middle line:84% because under such a change of basis, 00:57:09.190 --> 00:57:11.720 align:middle line:84% the spinor in the Majorana basis, which is real, 00:57:11.720 --> 00:57:14.900 align:middle line:84% is related to the spinor in the gamma mu basis 00:57:14.900 --> 00:57:18.840 align:middle line:90% by this transformation C, OK? 00:57:18.840 --> 00:57:21.270 align:middle line:84% So the C relates to the gamma matrices. 00:57:21.270 --> 00:57:23.400 align:middle line:84% But the C, of course, also relates to the spinor. 00:57:23.400 --> 00:57:25.110 align:middle line:90% It's just a change of basis, OK? 00:57:25.110 --> 00:57:29.840 align:middle line:84% And so psi M will be related to psi by C. 00:57:29.840 --> 00:57:36.620 align:middle line:84% And now, since the psi M is equal to psi M star, 00:57:36.620 --> 00:57:43.920 align:middle line:84% so that means that C star psi star should be equal to C psi, 00:57:43.920 --> 00:57:44.420 align:middle line:90% OK? 00:57:44.420 --> 00:57:50.990 align:middle line:84% So that means the psi star should be equal to B psi 00:57:50.990 --> 00:57:59.390 align:middle line:84% with B equal to C minus 1 star C, OK? 00:57:59.390 --> 00:58:01.340 align:middle line:84% So that should be the condition which you 00:58:01.340 --> 00:58:02.705 align:middle line:90% impose in the general basis. 00:58:02.705 --> 00:58:13.282 align:middle line:90% 00:58:13.282 --> 00:58:14.740 align:middle line:84% OK, so that should be the condition 00:58:14.740 --> 00:58:17.240 align:middle line:84% you should impose in the general basis. 00:58:17.240 --> 00:58:20.350 align:middle line:84% So you have to introduce this C. So if you find that 00:58:20.350 --> 00:58:28.410 align:middle line:84% the transformation between the general gamma mu and the gamma 00:58:28.410 --> 00:58:32.000 align:middle line:84% mu m, and then you can use that to find the B. 00:58:32.000 --> 00:58:35.580 align:middle line:84% And once you find the B, and then you can find the-- you can 00:58:35.580 --> 00:58:36.540 align:middle line:90% impose the-- 00:58:36.540 --> 00:58:38.760 align:middle line:84% yeah, so this is called a Majorana condition 00:58:38.760 --> 00:58:39.840 align:middle line:90% in the general basis. 00:58:39.840 --> 00:58:53.170 align:middle line:90% 00:58:53.170 --> 00:58:56.770 align:middle line:84% So let's understand a little bit. 00:58:56.770 --> 00:59:00.435 align:middle line:84% So, actually, we can understand the B more directly, OK? 00:59:00.435 --> 00:59:02.310 align:middle line:84% So, here, we expanded from C. But we actually 00:59:02.310 --> 00:59:05.220 align:middle line:90% can find the B more directly. 00:59:05.220 --> 00:59:13.920 align:middle line:84% We can just take the complex conjugate of this equation 00:59:13.920 --> 00:59:16.830 align:middle line:90% because gamma mu m is real. 00:59:16.830 --> 00:59:19.690 align:middle line:84% So we can also take the complex conjugate of this equation. 00:59:19.690 --> 00:59:28.997 align:middle line:84% So that means that, since gamma mu m equal to gamma mu m star, 00:59:28.997 --> 00:59:31.080 align:middle line:84% but if we take complex conjugate of that equation, 00:59:31.080 --> 00:59:35.370 align:middle line:84% means that the C star gamma mu, C minus 1 star 00:59:35.370 --> 00:59:40.770 align:middle line:84% is equal to C gamma mu, C minus 1. 00:59:40.770 --> 00:59:42.960 align:middle line:90% And now, again, you just-- 00:59:42.960 --> 00:59:43.815 align:middle line:90% sorry. 00:59:43.815 --> 00:59:46.020 align:middle line:90% This would be star, OK? 00:59:46.020 --> 00:59:48.750 align:middle line:84% So, now, if we put all the C to this side, 00:59:48.750 --> 00:59:56.280 align:middle line:84% and then we find gamma mu star is 00:59:56.280 --> 01:00:01.050 align:middle line:84% just equal to B gamma mu, B minus 1, OK? 01:00:01.050 --> 01:00:03.180 align:middle line:84% So you just put this to this side. 01:00:03.180 --> 01:00:05.280 align:middle line:84% Then you just-- so this becomes B, 01:00:05.280 --> 01:00:09.300 align:middle line:84% and then this becomes B minus 1, OK? 01:00:09.300 --> 01:00:13.950 align:middle line:90% So, now, we see that B-- 01:00:13.950 --> 01:00:15.740 align:middle line:90% this actually makes sense. 01:00:15.740 --> 01:00:20.530 align:middle line:84% This is actually the matrix we take gamma mu to gamma mu star, 01:00:20.530 --> 01:00:22.430 align:middle line:90% OK? 01:00:22.430 --> 01:00:25.735 align:middle line:84% So B is the matrix to take star, OK? 01:00:25.735 --> 01:00:31.830 align:middle line:90% 01:00:31.830 --> 01:00:32.330 align:middle line:90% Good. 01:00:32.330 --> 01:00:33.970 align:middle line:90% Any questions on this? 01:00:33.970 --> 01:00:35.432 align:middle line:90% Yes? 01:00:35.432 --> 01:00:38.610 align:middle line:84% AUDIENCE: [INAUDIBLE] this is [INAUDIBLE].. 01:00:38.610 --> 01:00:39.870 align:middle line:90% What group to they belong to? 01:00:39.870 --> 01:00:44.400 align:middle line:84% HONG LIU: Well, they're just general, nonsingular matrices. 01:00:44.400 --> 01:00:45.890 align:middle line:90% Yeah. 01:00:45.890 --> 01:00:49.295 align:middle line:84% Yeah, just 4-by-4 nonsingular matrices. 01:00:49.295 --> 01:00:53.480 align:middle line:90% 01:00:53.480 --> 01:00:57.225 align:middle line:84% They often can be chosen to be unitary. 01:00:57.225 --> 01:00:58.600 align:middle line:84% But, in principle, you don't have 01:00:58.600 --> 01:01:00.324 align:middle line:90% to choose them to be unitary. 01:01:00.324 --> 01:01:05.100 align:middle line:90% 01:01:05.100 --> 01:01:08.790 align:middle line:90% OK, so, now, let's double check. 01:01:08.790 --> 01:01:12.450 align:middle line:84% So let's call this equation star star. 01:01:12.450 --> 01:01:17.550 align:middle line:90% So let's check that star star-- 01:01:17.550 --> 01:01:21.030 align:middle line:84% so we show that, here, in this representation, 01:01:21.030 --> 01:01:23.520 align:middle line:84% this is compatible with Lorentz transformation, OK? 01:01:23.520 --> 01:01:27.750 align:middle line:84% So we still need to check star, star 01:01:27.750 --> 01:01:36.150 align:middle line:84% is compatible with Lorentz transformation. 01:01:36.150 --> 01:01:41.340 align:middle line:90% 01:01:41.340 --> 01:01:45.720 align:middle line:84% So what do we mean by this is compatible with Lorentz 01:01:45.720 --> 01:01:47.740 align:middle line:90% transformation? 01:01:47.740 --> 01:01:51.220 align:middle line:84% We mean that, if we take a psi which 01:01:51.220 --> 01:01:52.630 align:middle line:90% satisfies this condition-- 01:01:52.630 --> 01:01:58.860 align:middle line:90% 01:01:58.860 --> 01:02:04.070 align:middle line:84% take a psi which satisfies star star, 01:02:04.070 --> 01:02:06.560 align:middle line:84% and then we make a Lorentz transformation psi 01:02:06.560 --> 01:02:15.860 align:middle line:84% prime equal to S lambda psi, and then psi prime should also 01:02:15.860 --> 01:02:17.614 align:middle line:90% satisfy that condition-- 01:02:17.614 --> 01:02:21.570 align:middle line:90% 01:02:21.570 --> 01:02:22.440 align:middle line:90% star star, OK? 01:02:22.440 --> 01:02:24.982 align:middle line:84% So that means this is compatible with Lorentz transformation. 01:02:24.982 --> 01:02:29.070 align:middle line:84% It means that the psi prime star should 01:02:29.070 --> 01:02:34.740 align:middle line:84% be equal to the same as B psi prime, OK? 01:02:34.740 --> 01:02:42.080 align:middle line:84% So it means that psi prime star should satisfy that equation. 01:02:42.080 --> 01:02:43.910 align:middle line:90% So, now, let's check that. 01:02:43.910 --> 01:02:44.870 align:middle line:90% Now, let's check this. 01:02:44.870 --> 01:02:49.110 align:middle line:90% 01:02:49.110 --> 01:02:53.283 align:middle line:84% So from here-- so before checking that, 01:02:53.283 --> 01:02:54.700 align:middle line:84% do you have any questions on this? 01:02:54.700 --> 01:02:58.750 align:middle line:90% 01:02:58.750 --> 01:02:59.250 align:middle line:90% OK. 01:02:59.250 --> 01:03:30.450 align:middle line:90% 01:03:30.450 --> 01:03:33.080 align:middle line:90% Good. 01:03:33.080 --> 01:03:34.850 align:middle line:84% So, first, from this equation-- let's 01:03:34.850 --> 01:03:36.095 align:middle line:90% call this star, star, star. 01:03:36.095 --> 01:03:39.740 align:middle line:90% 01:03:39.740 --> 01:03:42.830 align:middle line:84% From this star cube equation, so we 01:03:42.830 --> 01:03:56.780 align:middle line:84% can find the B when you act on sigma mu B minus 1. 01:03:56.780 --> 01:04:01.790 align:middle line:84% So that gives you minus sigma mu, nu, star, OK? 01:04:01.790 --> 01:04:08.030 align:middle line:84% So this is obvious because sigma mu, nu has i there. 01:04:08.030 --> 01:04:10.010 align:middle line:84% So the minus sign comes from the i. 01:04:10.010 --> 01:04:15.320 align:middle line:84% And, otherwise, the B takes each gamma matrices there into star, 01:04:15.320 --> 01:04:18.080 align:middle line:90% OK? 01:04:18.080 --> 01:04:24.660 align:middle line:84% And then that means that the S star, lambda, 01:04:24.660 --> 01:04:30.635 align:middle line:84% which is given by exponential 1/2 omega mu, nu, sigma star-- 01:04:30.635 --> 01:04:33.930 align:middle line:90% 01:04:33.930 --> 01:04:38.675 align:middle line:84% sigma mu, nu star, now this is equal to-- 01:04:38.675 --> 01:04:45.310 align:middle line:90% 01:04:45.310 --> 01:04:47.815 align:middle line:90% yeah, you can just plug this in. 01:04:47.815 --> 01:04:50.910 align:middle line:90% 01:04:50.910 --> 01:04:53.620 align:middle line:84% It just becomes exponential minus i 01:04:53.620 --> 01:05:06.480 align:middle line:84% omega mu, nu, B sigma B minus 1, OK? 01:05:06.480 --> 01:05:07.850 align:middle line:90% So let me just-- 01:05:07.850 --> 01:05:11.850 align:middle line:84% yeah-- B sigma mu, nu, B minus 1, OK? 01:05:11.850 --> 01:05:14.640 align:middle line:84% So I just inserted the sigma mu, nu, star. 01:05:14.640 --> 01:05:16.440 align:middle line:84% It's equal to minus that here, OK? 01:05:16.440 --> 01:05:19.740 align:middle line:90% 01:05:19.740 --> 01:05:23.670 align:middle line:84% So, now, you can see that this B minus 1 is in the exponential. 01:05:23.670 --> 01:05:25.980 align:middle line:84% You can immediately take it down. 01:05:25.980 --> 01:05:33.295 align:middle line:84% So that is just equal to B S lambda B minus 1, OK? 01:05:33.295 --> 01:05:36.970 align:middle line:90% 01:05:36.970 --> 01:05:39.750 align:middle line:84% So because when you expand this in power series, B and B 01:05:39.750 --> 01:05:43.710 align:middle line:84% minus 1 always cancel, except the first one and the last one. 01:05:43.710 --> 01:05:46.830 align:middle line:84% So we have used this trick many times. 01:05:46.830 --> 01:05:52.380 align:middle line:84% Yeah, and then so we get a very nice relation that, 01:05:52.380 --> 01:05:56.220 align:middle line:84% under Lorentz transformation-- so the Lorentz transformation 01:05:56.220 --> 01:05:58.230 align:middle line:84% matrix on the complex conjugation, 01:05:58.230 --> 01:06:03.660 align:middle line:84% again, is generated by this B, related by this B matrix. 01:06:03.660 --> 01:06:05.415 align:middle line:84% And, now, it's just immediate, OK? 01:06:05.415 --> 01:06:09.360 align:middle line:90% 01:06:09.360 --> 01:06:15.080 align:middle line:84% And now just immediate, so when you have the psi prime equal 01:06:15.080 --> 01:06:20.560 align:middle line:84% to that, let's just take the star of this equation, OK? 01:06:20.560 --> 01:06:36.770 align:middle line:84% So psi prime star just equal to S star, lambda psi star, OK? 01:06:36.770 --> 01:06:44.300 align:middle line:84% So that is equal to B S lambda B minus 1 and B S-- 01:06:44.300 --> 01:06:48.590 align:middle line:90% B psi, OK? 01:06:48.590 --> 01:06:54.650 align:middle line:84% So this is equal to B S lambda psi, OK? 01:06:54.650 --> 01:06:58.190 align:middle line:84% So this is equal to B psi prime-- 01:06:58.190 --> 01:06:59.780 align:middle line:84% precisely what we were trying to show. 01:06:59.780 --> 01:07:09.420 align:middle line:90% 01:07:09.420 --> 01:07:09.920 align:middle line:90% Good. 01:07:09.920 --> 01:07:10.930 align:middle line:90% Any questions on this? 01:07:10.930 --> 01:07:17.110 align:middle line:90% 01:07:17.110 --> 01:07:19.710 align:middle line:84% OK, so, now, let me give you an explicit example 01:07:19.710 --> 01:07:28.320 align:middle line:90% of this matrix B in the-- 01:07:28.320 --> 01:07:30.480 align:middle line:84% so for this Majorana representation, 01:07:30.480 --> 01:07:32.760 align:middle line:84% the B is just equal to identity, OK? 01:07:32.760 --> 01:07:37.170 align:middle line:84% So B is just equal to identity in this representation. 01:07:37.170 --> 01:07:39.772 align:middle line:84% And, now, let's try to give you an example of the B 01:07:39.772 --> 01:07:42.330 align:middle line:90% in the other representation. 01:07:42.330 --> 01:07:46.240 align:middle line:84% So suppose, in the chiral representation, 01:07:46.240 --> 01:07:50.160 align:middle line:90% which I wrote down before-- 01:07:50.160 --> 01:07:54.630 align:middle line:84% so, yeah, I should not have erased it. 01:07:54.630 --> 01:07:56.220 align:middle line:90% Yeah, anyway. 01:07:56.220 --> 01:07:59.320 align:middle line:84% So in the chiral representation I wrote down before, 01:07:59.320 --> 01:08:01.350 align:middle line:84% so if you stare at that expression, 01:08:01.350 --> 01:08:07.185 align:middle line:84% you find that gamma 0, gamma 1, and gamma 2-- 01:08:07.185 --> 01:08:12.330 align:middle line:84% or gamma 0, gamma 1, and gamma 3 are imaginary, pure imaginary. 01:08:12.330 --> 01:08:17.880 align:middle line:90% 01:08:17.880 --> 01:08:20.040 align:middle line:90% And the gamma 2 is real. 01:08:20.040 --> 01:08:22.800 align:middle line:90% 01:08:22.800 --> 01:08:27.720 align:middle line:90% Gamma 2 is real, OK? 01:08:27.720 --> 01:08:32.370 align:middle line:84% So this pure imaginary means, when you take the star of them, 01:08:32.370 --> 01:08:34.210 align:middle line:90% you get the minus sign. 01:08:34.210 --> 01:08:35.850 align:middle line:84% So this one, you take the star of them, 01:08:35.850 --> 01:08:40.160 align:middle line:90% you just get back to itself. 01:08:40.160 --> 01:08:48.790 align:middle line:84% So, now, if we look at this equation, 01:08:48.790 --> 01:08:52.840 align:middle line:84% so if this is pure imaginary, you get the star. 01:08:52.840 --> 01:08:55.290 align:middle line:90% You get the minus itself. 01:08:55.290 --> 01:08:57.420 align:middle line:84% And then, essentially, you get the minus self 01:08:57.420 --> 01:09:02.680 align:middle line:84% means the B actually anticommutes with gamma mu, OK? 01:09:02.680 --> 01:09:05.590 align:middle line:84% Because you can just bring B minus 1 to this side, 01:09:05.590 --> 01:09:10.600 align:middle line:84% so it just becomes gamma mu star B equal to B gamma mu. 01:09:10.600 --> 01:09:12.340 align:middle line:84% So if this is minus gamma mu, that 01:09:12.340 --> 01:09:16.270 align:middle line:84% means B should anticommute with gamma mu, OK? 01:09:16.270 --> 01:09:22.109 align:middle line:84% But if gamma is real, that means B will commute with mu, OK? 01:09:22.109 --> 01:09:25.560 align:middle line:84% If mu is real, then that means B should commute with mu. 01:09:25.560 --> 01:09:30.990 align:middle line:84% So, now, in this chiral basis, these three are pure imaginary. 01:09:30.990 --> 01:09:34.580 align:middle line:84% That means B needs to anticommute with them. 01:09:34.580 --> 01:09:35.399 align:middle line:90% But this is real. 01:09:35.399 --> 01:09:37.770 align:middle line:84% It means B needs to commute with this guy. 01:09:37.770 --> 01:09:39.410 align:middle line:90% Then what is B? 01:09:39.410 --> 01:09:40.400 align:middle line:90% AUDIENCE: Gamma 2. 01:09:40.400 --> 01:09:41.346 align:middle line:90% HONG LIU: Hmm? 01:09:41.346 --> 01:09:42.240 align:middle line:90% AUDIENCE: Gamma 2. 01:09:42.240 --> 01:09:43.050 align:middle line:90% HONG LIU: Exactly. 01:09:43.050 --> 01:09:47.005 align:middle line:84% So B, in this case, can only be gamma 2, OK? 01:09:47.005 --> 01:09:50.380 align:middle line:90% 01:09:50.380 --> 01:09:53.260 align:middle line:84% And then we can work out, what is the Majorana 01:09:53.260 --> 01:09:54.384 align:middle line:90% condition in this basis-- 01:09:54.384 --> 01:09:59.600 align:middle line:90% 01:09:59.600 --> 01:10:01.280 align:middle line:90% so, essentially, this condition. 01:10:01.280 --> 01:10:04.710 align:middle line:90% 01:10:04.710 --> 01:10:15.890 align:middle line:84% So that means that the psi star should be equal to gamma 2 psi, 01:10:15.890 --> 01:10:16.910 align:middle line:90% OK? 01:10:16.910 --> 01:10:20.040 align:middle line:84% So that's the Majorana condition here. 01:10:20.040 --> 01:10:22.400 align:middle line:84% So, now, remember, in the chiral basis, 01:10:22.400 --> 01:10:26.870 align:middle line:84% we can write psi in terms of the psi L and psi R. 01:10:26.870 --> 01:10:28.910 align:middle line:84% So, essentially, we have this condition. 01:10:28.910 --> 01:10:35.860 align:middle line:90% 01:10:35.860 --> 01:10:38.210 align:middle line:90% So I erased my gamma 2. 01:10:38.210 --> 01:10:40.160 align:middle line:90% So I saw the gamma 2-- 01:10:40.160 --> 01:10:42.290 align:middle line:90% let me write it here explicitly. 01:10:42.290 --> 01:10:54.670 align:middle line:84% It's minus i gamma 2, 0 minus i sigma 2 and sigma 2, here, OK? 01:10:54.670 --> 01:10:56.440 align:middle line:84% So, now, if you look at this condition, 01:10:56.440 --> 01:11:00.460 align:middle line:84% this means that no longer psi L and psi R are no longer 01:11:00.460 --> 01:11:02.060 align:middle line:90% independent of each other. 01:11:02.060 --> 01:11:08.680 align:middle line:84% So psi L star should be equal to minus i sigma 2 psi R 01:11:08.680 --> 01:11:14.280 align:middle line:84% or, equivalently, psi R equal to i sigma 2 psi L-- 01:11:14.280 --> 01:11:17.340 align:middle line:90% 01:11:17.340 --> 01:11:20.550 align:middle line:90% or L star, OK? 01:11:20.550 --> 01:11:25.340 align:middle line:84% So, in this case, the psi, then, have the following form-- 01:11:25.340 --> 01:11:30.450 align:middle line:84% psi L i sigma 2 psi L. So sigma R 01:11:30.450 --> 01:11:35.010 align:middle line:84% just can be expressed in terms of sigma L. 01:11:35.010 --> 01:11:41.160 align:middle line:84% So this is the Majorana spinor in the chiral basis, OK? 01:11:41.160 --> 01:11:44.550 align:middle line:84% You see, there are only four independent, real components 01:11:44.550 --> 01:11:48.270 align:middle line:84% because each psi L is two complex components, OK? 01:11:48.270 --> 01:11:51.950 align:middle line:90% 01:11:51.950 --> 01:11:52.580 align:middle line:90% Yes? 01:11:52.580 --> 01:11:54.038 align:middle line:84% AUDIENCE: So why do you [INAUDIBLE] 01:11:54.038 --> 01:11:58.023 align:middle line:84% that psi and psi star are not independent [INAUDIBLE]?? 01:11:58.023 --> 01:11:58.606 align:middle line:90% HONG LIU: Hmm? 01:11:58.606 --> 01:11:59.380 align:middle line:90% AUDIENCE: Why are they-- 01:11:59.380 --> 01:12:00.255 align:middle line:90% HONG LIU: No, no, no. 01:12:00.255 --> 01:12:03.360 align:middle line:84% No, here, we are imposing this condition, right? 01:12:03.360 --> 01:12:05.610 align:middle line:90% We are imposing this condition. 01:12:05.610 --> 01:12:06.110 align:middle line:90% Yeah. 01:12:06.110 --> 01:12:07.568 align:middle line:84% Yeah, this is Majorana condition we 01:12:07.568 --> 01:12:09.040 align:middle line:90% want to impose in this basis. 01:12:09.040 --> 01:12:12.005 align:middle line:84% AUDIENCE: And this is now independent of massless 01:12:12.005 --> 01:12:12.880 align:middle line:90% or massive particles? 01:12:12.880 --> 01:12:14.030 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 01:12:14.030 --> 01:12:15.260 align:middle line:90% Yeah, this is-- yeah. 01:12:15.260 --> 01:12:18.730 align:middle line:90% 01:12:18.730 --> 01:12:20.140 align:middle line:90% Good? 01:12:20.140 --> 01:12:24.100 align:middle line:84% So this concludes our discussion of the Majorana spinor. 01:12:24.100 --> 01:12:26.670 align:middle line:84% Do you have any questions on this? 01:12:26.670 --> 01:12:27.390 align:middle line:90% Yes? 01:12:27.390 --> 01:12:33.200 align:middle line:84% AUDIENCE: So is the orthogonal component of psi-- 01:12:33.200 --> 01:12:36.240 align:middle line:84% the Majorana fermion [INAUDIBLE]?? 01:12:36.240 --> 01:12:37.020 align:middle line:90% HONG LIU: Sorry? 01:12:37.020 --> 01:12:38.437 align:middle line:84% AUDIENCE: The orthogonal component 01:12:38.437 --> 01:12:41.800 align:middle line:90% of this Majorana species-- 01:12:41.800 --> 01:12:44.740 align:middle line:84% like, possible-- in the chiral one, 01:12:44.740 --> 01:12:47.060 align:middle line:84% like, psi L, and then you [INAUDIBLE] 01:12:47.060 --> 01:12:50.265 align:middle line:84% and then psi R. [INAUDIBLE] two components of psi. 01:12:50.265 --> 01:12:50.890 align:middle line:90% HONG LIU: Yeah. 01:12:50.890 --> 01:12:52.180 align:middle line:90% AUDIENCE: So we are-- 01:12:52.180 --> 01:12:54.910 align:middle line:84% we have one component of psi, and there 01:12:54.910 --> 01:12:57.080 align:middle line:90% should be another one, right? 01:12:57.080 --> 01:12:57.705 align:middle line:90% So is that one? 01:12:57.705 --> 01:12:58.372 align:middle line:90% HONG LIU: Sorry. 01:12:58.372 --> 01:13:00.440 align:middle line:84% I don't quite understand your question. 01:13:00.440 --> 01:13:01.607 align:middle line:90% Say it again-- what? 01:13:01.607 --> 01:13:02.690 align:middle line:90% AUDIENCE: OK, [INAUDIBLE]. 01:13:02.690 --> 01:13:03.950 align:middle line:90% HONG LIU: OK, OK. 01:13:03.950 --> 01:13:04.670 align:middle line:90% Other questions? 01:13:04.670 --> 01:13:05.270 align:middle line:90% Yes? 01:13:05.270 --> 01:13:07.580 align:middle line:84% AUDIENCE: So, I guess, related to an earlier question, 01:13:07.580 --> 01:13:09.470 align:middle line:84% how do we consider handedness here 01:13:09.470 --> 01:13:13.792 align:middle line:84% for a massless particle that's also the Majorana, which psi L 01:13:13.792 --> 01:13:17.070 align:middle line:90% and psi R are [INAUDIBLE]? 01:13:17.070 --> 01:13:18.150 align:middle line:90% HONG LIU: Right. 01:13:18.150 --> 01:13:24.982 align:middle line:90% So for the-- yeah, if we-- 01:13:24.982 --> 01:13:26.440 align:middle line:84% for massless particle, you can just 01:13:26.440 --> 01:13:30.850 align:middle line:84% direct it because then you don't have to think about the psi R, 01:13:30.850 --> 01:13:34.510 align:middle line:84% and this, you just have the same number degrees of freedom 01:13:34.510 --> 01:13:35.620 align:middle line:90% as a massless particle. 01:13:35.620 --> 01:13:40.330 align:middle line:90% 01:13:40.330 --> 01:13:42.160 align:middle line:90% Other questions? 01:13:42.160 --> 01:13:42.660 align:middle line:90% Good. 01:13:42.660 --> 01:13:45.500 align:middle line:90% OK. 01:13:45.500 --> 01:13:46.320 align:middle line:90% OK, good. 01:13:46.320 --> 01:13:49.220 align:middle line:84% So let's now go to the next topic. 01:13:49.220 --> 01:13:52.160 align:middle line:84% We only have a few minutes, so we can only just 01:13:52.160 --> 01:13:53.285 align:middle line:90% make some general comments. 01:13:53.285 --> 01:14:01.260 align:middle line:90% 01:14:01.260 --> 01:14:03.450 align:middle line:84% So so far, mostly, we have been talking 01:14:03.450 --> 01:14:05.430 align:middle line:90% about continuous symmetries. 01:14:05.430 --> 01:14:09.060 align:middle line:90% 01:14:09.060 --> 01:14:14.635 align:middle line:84% But there are also discrete symmetries, OK? 01:14:14.635 --> 01:14:17.960 align:middle line:90% 01:14:17.960 --> 01:14:25.930 align:middle line:84% So, by definition, discrete symmetries 01:14:25.930 --> 01:14:33.505 align:middle line:84% are symmetries, which don't have continuous parameters, OK? 01:14:33.505 --> 01:14:37.780 align:middle line:90% 01:14:37.780 --> 01:14:40.300 align:middle line:84% So continuous symmetries are symmetries, which-- 01:14:40.300 --> 01:14:43.450 align:middle line:90% 01:14:43.450 --> 01:14:46.390 align:middle line:84% yeah, which the transformation dependence 01:14:46.390 --> 01:14:48.520 align:middle line:90% on continuous parameter. 01:14:48.520 --> 01:14:50.770 align:middle line:84% Discrete symmetries, you just don't, OK? 01:14:50.770 --> 01:14:53.740 align:middle line:84% You don't have continuous parameter. 01:14:53.740 --> 01:14:58.690 align:middle line:84% So simple example-- say, let's imagine we have-- 01:14:58.690 --> 01:15:01.555 align:middle line:84% so this real scalar theory-- we can see that before. 01:15:01.555 --> 01:15:11.410 align:middle line:90% 01:15:11.410 --> 01:15:15.150 align:middle line:84% So this theory has a discrete symmetry because this 01:15:15.150 --> 01:15:16.950 align:middle line:90% is invariant under phi. 01:15:16.950 --> 01:15:20.940 align:middle line:90% It goes to minus phi, OK? 01:15:20.940 --> 01:15:23.790 align:middle line:84% So because you see all the terms are even, 01:15:23.790 --> 01:15:27.150 align:middle line:84% so it's invariant under phi go to minus phi. 01:15:27.150 --> 01:15:29.880 align:middle line:84% And this transformation is no continuous parameter, OK? 01:15:29.880 --> 01:15:33.190 align:middle line:90% So this is a discrete symmetry. 01:15:33.190 --> 01:15:35.940 align:middle line:90% And so this is-- 01:15:35.940 --> 01:15:37.873 align:middle line:84% if you do it twice, you go back to itself. 01:15:37.873 --> 01:15:39.540 align:middle line:84% So this is often called the Z2 symmetry. 01:15:39.540 --> 01:15:43.500 align:middle line:90% 01:15:43.500 --> 01:15:45.330 align:middle line:84% OK, so this is called the Z2 symmetry. 01:15:45.330 --> 01:15:48.480 align:middle line:90% 01:15:48.480 --> 01:15:53.540 align:middle line:84% And there are also spacetime discrete symmetries, OK? 01:15:53.540 --> 01:15:55.940 align:middle line:84% So this is an internal discrete symmetry-- have nothing 01:15:55.940 --> 01:15:57.740 align:middle line:90% to do with spacetime, OK? 01:15:57.740 --> 01:16:00.170 align:middle line:84% There are also spacetime discrete symmetries. 01:16:00.170 --> 01:16:14.640 align:middle line:90% 01:16:14.640 --> 01:16:18.350 align:middle line:84% So spacetime discrete symmetries including, say, if we consider 01:16:18.350 --> 01:16:19.850 align:middle line:90% Minkowski spacetime-- 01:16:19.850 --> 01:16:27.610 align:middle line:90% 01:16:27.610 --> 01:16:30.820 align:middle line:84% so you can have t goes to minus t. 01:16:30.820 --> 01:16:38.300 align:middle line:84% So you can have so-called time reversal, 01:16:38.300 --> 01:16:44.110 align:middle line:84% which corresponding to your t, x goes to minus t, x, OK? 01:16:44.110 --> 01:16:47.050 align:middle line:90% You just transform the time. 01:16:47.050 --> 01:16:49.095 align:middle line:84% You can also have the so-called parity. 01:16:49.095 --> 01:16:52.500 align:middle line:90% 01:16:52.500 --> 01:16:53.640 align:middle line:90% You take t. 01:16:53.640 --> 01:17:00.315 align:middle line:84% Then you revert all the spatial direction, OK? 01:17:00.315 --> 01:17:09.000 align:middle line:90% 01:17:09.000 --> 01:17:13.110 align:middle line:84% So comment that you can ask why we actually 01:17:13.110 --> 01:17:16.230 align:middle line:90% reverse all three directions. 01:17:16.230 --> 01:17:18.690 align:middle line:84% How about if I just reverse one direction 01:17:18.690 --> 01:17:21.060 align:middle line:90% or reverse two directions, OK? 01:17:21.060 --> 01:17:23.910 align:middle line:84% That seems also to be a discrete symmetry. 01:17:23.910 --> 01:17:25.270 align:middle line:90% And indeed. 01:17:25.270 --> 01:17:28.310 align:middle line:84% So if you just change the directions, 01:17:28.310 --> 01:17:32.220 align:middle line:84% say, in the x direction, that's also a discrete symmetry. 01:17:32.220 --> 01:17:33.940 align:middle line:84% And if you only change the direction 01:17:33.940 --> 01:17:35.640 align:middle line:84% in both x and the y direction, that's 01:17:35.640 --> 01:17:37.690 align:middle line:90% also a discrete symmetry. 01:17:37.690 --> 01:17:42.510 align:middle line:90% But if you change-- 01:17:42.510 --> 01:17:46.870 align:middle line:84% if you do the reflection in two directions, 01:17:46.870 --> 01:17:49.840 align:middle line:84% that's equivalent to a 90-degree rotation-- 01:17:49.840 --> 01:17:57.670 align:middle line:84% a 180-degree rotation in that plane, OK? 01:17:57.670 --> 01:18:02.270 align:middle line:84% And so it's part of the continuous symmetries, 01:18:02.270 --> 01:18:06.200 align:middle line:84% so it's not independent discrete symmetry. 01:18:06.200 --> 01:18:10.430 align:middle line:84% And now, when you change all three directions compared 01:18:10.430 --> 01:18:13.580 align:middle line:84% to change one direction, you differ only 01:18:13.580 --> 01:18:15.390 align:middle line:90% by changing two directions. 01:18:15.390 --> 01:18:17.240 align:middle line:84% So that means changing all three directions 01:18:17.240 --> 01:18:18.907 align:middle line:84% and changing one direction-- they differ 01:18:18.907 --> 01:18:21.380 align:middle line:90% by 180-degree rotation, OK? 01:18:21.380 --> 01:18:23.060 align:middle line:84% So that means that when you change-- 01:18:23.060 --> 01:18:29.040 align:middle line:84% this is the only independent discrete symmetry 01:18:29.040 --> 01:18:33.150 align:middle line:84% from the spatial reflection point of view. 01:18:33.150 --> 01:18:35.160 align:middle line:90% OK. 01:18:35.160 --> 01:18:40.030 align:middle line:90% So for a complex scalar field-- 01:18:40.030 --> 01:18:42.060 align:middle line:84% so if you consider complex scalar field-- 01:18:42.060 --> 01:18:48.630 align:middle line:90% 01:18:48.630 --> 01:18:52.320 align:middle line:84% if you can see the complex scalar field, 01:18:52.320 --> 01:18:57.380 align:middle line:84% then this is no longer a discrete symmetry 01:18:57.380 --> 01:19:01.220 align:middle line:84% because, remember, we can rotate phi by a phase. 01:19:01.220 --> 01:19:02.900 align:middle line:84% When you rotate the phi by a phase, 01:19:02.900 --> 01:19:04.970 align:middle line:84% if you take that phase to be pi-- 01:19:04.970 --> 01:19:08.630 align:middle line:84% say, exponential i pi-- and then you take to be minus phi. 01:19:08.630 --> 01:19:11.000 align:middle line:84% And so, in that case, this is part 01:19:11.000 --> 01:19:12.890 align:middle line:84% of the continuous symmetry, so it's no longer 01:19:12.890 --> 01:19:15.500 align:middle line:90% independent discrete symmetry. 01:19:15.500 --> 01:19:17.090 align:middle line:84% But, here, there's, nevertheless, 01:19:17.090 --> 01:19:19.910 align:middle line:90% another discrete symmetry. 01:19:19.910 --> 01:19:22.400 align:middle line:84% Can you see what is the other independent discrete symmetry 01:19:22.400 --> 01:19:23.880 align:middle line:90% here? 01:19:23.880 --> 01:19:25.690 align:middle line:90% Yes? 01:19:25.690 --> 01:19:26.230 align:middle line:90% Good. 01:19:26.230 --> 01:19:30.430 align:middle line:84% You can take phi to phi star, OK? 01:19:30.430 --> 01:19:33.640 align:middle line:84% You can exchange phi to phi star, OK? 01:19:33.640 --> 01:19:35.020 align:middle line:90% It's a complex conjugation. 01:19:35.020 --> 01:19:38.460 align:middle line:84% And this is often called charge conjugation. 01:19:38.460 --> 01:19:44.360 align:middle line:90% 01:19:44.360 --> 01:19:44.860 align:middle line:90% OK. 01:19:44.860 --> 01:19:48.460 align:middle line:84% This is often called charge conjugation because, remember, 01:19:48.460 --> 01:19:53.270 align:middle line:84% heuristically, we can think of phi as create-- 01:19:53.270 --> 01:19:55.270 align:middle line:84% yeah, it's just one of them create the particle, 01:19:55.270 --> 01:19:57.350 align:middle line:84% and the other create the antiparticle. 01:19:57.350 --> 01:19:59.770 align:middle line:84% And they have opposite charge, OK? 01:19:59.770 --> 01:20:01.900 align:middle line:84% So it's called a charge conjugation. 01:20:01.900 --> 01:20:05.470 align:middle line:84% So, this, will give a symbol called T-- 01:20:05.470 --> 01:20:09.610 align:middle line:84% script T. And, this, we give a symbol called P-- script P. 01:20:09.610 --> 01:20:13.300 align:middle line:84% And, this, we give a symbol called script C. So, 01:20:13.300 --> 01:20:16.420 align:middle line:84% altogether, they are called CPT symmetry, OK? 01:20:16.420 --> 01:20:16.920 align:middle line:90% Yeah. 01:20:16.920 --> 01:20:19.140 align:middle line:90% Yeah, let's stop here. 01:20:19.140 --> 01:20:23.000 align:middle line:90%