WEBVTT 00:00:00.000 --> 00:00:01.437 align:middle line:90% [SQUEAKING] 00:00:01.437 --> 00:00:04.311 align:middle line:90% [RUSTLING] 00:00:04.311 --> 00:00:05.269 align:middle line:90% [CLICKING] 00:00:05.269 --> 00:00:10.840 align:middle line:90% 00:00:10.840 --> 00:00:13.700 align:middle line:84% HONG LIU: So at the end of last lecture, 00:00:13.700 --> 00:00:19.420 align:middle line:84% so we discussed this LSZ theorem, which tells you 00:00:19.420 --> 00:00:26.740 align:middle line:84% how to obtain scattering amplitude from correlation 00:00:26.740 --> 00:00:30.170 align:middle line:84% functions, from time-ordered correlation functions. 00:00:30.170 --> 00:00:33.670 align:middle line:84% OK, so if you want to compute, say, 00:00:33.670 --> 00:00:37.870 align:middle line:84% some scattering amplitude from alpha to beta-- so alpha's 00:00:37.870 --> 00:00:41.470 align:middle line:84% some initial state and beta's some final state. 00:00:41.470 --> 00:00:48.130 align:middle line:84% Say alpha consists of momentum p1 and PN-- 00:00:48.130 --> 00:00:58.380 align:middle line:84% or pm, and beta, say momentum p m plus 1 and pn. 00:00:58.380 --> 00:01:03.390 align:middle line:84% And then you can get this scattering amplitude 00:01:03.390 --> 00:01:05.880 align:middle line:84% just by taking your momentum-space correlation 00:01:05.880 --> 00:01:12.520 align:middle line:90% function, OK, for the n points. 00:01:12.520 --> 00:01:14.430 align:middle line:84% So all together, you have an n points 00:01:14.430 --> 00:01:17.730 align:middle line:84% for the external momentum, OK, and then 00:01:17.730 --> 00:01:21.150 align:middle line:90% you take the on-shell limit. 00:01:21.150 --> 00:01:23.880 align:middle line:84% You take the on-shell limit, and then that 00:01:23.880 --> 00:01:36.950 align:middle line:84% gives you the product of the external propagators, then 00:01:36.950 --> 00:01:40.830 align:middle line:90% times the scattering amplitude. 00:01:40.830 --> 00:01:42.630 align:middle line:90% OK. 00:01:42.630 --> 00:01:46.080 align:middle line:84% So this is the relation, OK, so here I 00:01:46.080 --> 00:01:48.960 align:middle line:84% have stripped out the momentum conservation on both sides. 00:01:48.960 --> 00:01:51.390 align:middle line:84% Of course, momentum has to be conserved. 00:01:51.390 --> 00:01:58.410 align:middle line:84% Inside, the total momentum has to be zero. 00:01:58.410 --> 00:02:05.010 align:middle line:84% And so this limit is the on-shell limit, 00:02:05.010 --> 00:02:13.010 align:middle line:84% and in the on-shell limit, the initial momentum-- 00:02:13.010 --> 00:02:16.400 align:middle line:84% OK, actually, I should call it minus p. 00:02:16.400 --> 00:02:19.550 align:middle line:90% Sorry, it's minus pm. 00:02:19.550 --> 00:02:23.750 align:middle line:84% So the initial momentum, for those 00:02:23.750 --> 00:02:28.280 align:middle line:84% with the initial momentum, you take p1 to be-- 00:02:28.280 --> 00:02:34.130 align:middle line:90% p1 0 to have minus omega p1, OK? 00:02:34.130 --> 00:02:36.590 align:middle line:90% For the final momentum-- 00:02:36.590 --> 00:02:37.505 align:middle line:90% so p alpha. 00:02:37.505 --> 00:02:40.150 align:middle line:90% 00:02:40.150 --> 00:02:43.300 align:middle line:84% For the final momentum, you take the p beta 00:02:43.300 --> 00:02:52.120 align:middle line:84% 0 goes to omega p beta, the final, OK, with the plus sign. 00:02:52.120 --> 00:02:55.180 align:middle line:84% OK, so that's how you distinguish the initial state 00:02:55.180 --> 00:02:59.890 align:middle line:84% from the final state, because when you obtain a correlation 00:02:59.890 --> 00:03:02.890 align:middle line:84% function, you don't distinguish what 00:03:02.890 --> 00:03:04.680 align:middle line:84% is the initial and the final state. 00:03:04.680 --> 00:03:07.400 align:middle line:90% You just have some momentum. 00:03:07.400 --> 00:03:10.930 align:middle line:84% This is a function of some arbitrary momentum. 00:03:10.930 --> 00:03:13.690 align:middle line:84% But the scattering amplitude, of course, 00:03:13.690 --> 00:03:18.850 align:middle line:84% those momentum are on shell, and so the way you distinguish 00:03:18.850 --> 00:03:21.670 align:middle line:84% the initial momentum and the final momentum 00:03:21.670 --> 00:03:23.900 align:middle line:84% is by taking the initial momentum, 00:03:23.900 --> 00:03:27.100 align:middle line:84% say to go to the negative root, and the final momentum 00:03:27.100 --> 00:03:28.960 align:middle line:90% to take the positive root. 00:03:28.960 --> 00:03:33.430 align:middle line:84% But since the alpha, it consists of minus p1, and then 00:03:33.430 --> 00:03:37.090 align:middle line:84% the initial state, you have positive energy, OK, 00:03:37.090 --> 00:03:40.130 align:middle line:90% you have positive energy. 00:03:40.130 --> 00:03:45.600 align:middle line:84% So this tells you that when we compute the scattering 00:03:45.600 --> 00:03:46.280 align:middle line:90% amplitude-- 00:03:46.280 --> 00:03:51.180 align:middle line:90% 00:03:51.180 --> 00:03:56.815 align:middle line:84% when we compute the scattering amplitude, we should take the-- 00:03:56.815 --> 00:04:00.250 align:middle line:90% 00:04:00.250 --> 00:04:08.070 align:middle line:84% we just take all the Feynman diagrams 00:04:08.070 --> 00:04:11.280 align:middle line:84% which you used to calculate these correlation functions, 00:04:11.280 --> 00:04:13.245 align:middle line:90% OK, and then you sum over-- 00:04:13.245 --> 00:04:18.180 align:middle line:90% 00:04:18.180 --> 00:04:22.300 align:middle line:84% you sum over, say, the truncated-- 00:04:22.300 --> 00:04:27.270 align:middle line:90% 00:04:27.270 --> 00:04:31.410 align:middle line:84% so you see the relation between these correlation functions 00:04:31.410 --> 00:04:33.100 align:middle line:90% and the scattering amplitudes. 00:04:33.100 --> 00:04:37.030 align:middle line:90% So they differ by this product. 00:04:37.030 --> 00:04:40.630 align:middle line:84% They differ by this product of the external propagators, OK, 00:04:40.630 --> 00:04:45.850 align:middle line:84% so for each external momentum-- so here there's a propagator, 00:04:45.850 --> 00:04:50.122 align:middle line:84% and just as if that-- when you get-- 00:04:50.122 --> 00:04:53.830 align:middle line:84% so this scattering amplitude was corresponding to this one 00:04:53.830 --> 00:04:56.710 align:middle line:84% with all external propagators stripped. 00:04:56.710 --> 00:05:00.220 align:middle line:84% OK, so that's why you consider the truncated diagram 00:05:00.220 --> 00:05:02.200 align:middle line:84% not including the external propagators. 00:05:02.200 --> 00:05:04.900 align:middle line:90% 00:05:04.900 --> 00:05:06.340 align:middle line:90% And you take it on shell. 00:05:06.340 --> 00:05:08.890 align:middle line:90% 00:05:08.890 --> 00:05:12.640 align:middle line:84% OK, and also, since we are interested only 00:05:12.640 --> 00:05:14.480 align:middle line:90% in the process-- 00:05:14.480 --> 00:05:17.900 align:middle line:84% which, all particles participate in the scattering process-- 00:05:17.900 --> 00:05:21.585 align:middle line:84% so we also only consider the connected diagrams. 00:05:21.585 --> 00:05:24.600 align:middle line:90% 00:05:24.600 --> 00:05:30.240 align:middle line:84% OK, consider the connected diagrams. 00:05:30.240 --> 00:05:34.930 align:middle line:84% OK, so this provides a simplification, 00:05:34.930 --> 00:05:38.490 align:middle line:84% so you get fewer diagrams and simpler expression 00:05:38.490 --> 00:05:41.970 align:middle line:84% than you would have got from calculating these correlation 00:05:41.970 --> 00:05:45.460 align:middle line:84% functions, OK, correlation functions. 00:05:45.460 --> 00:05:47.460 align:middle line:90% So any questions on this? 00:05:47.460 --> 00:05:51.590 align:middle line:90% 00:05:51.590 --> 00:05:52.183 align:middle line:90% Yes? 00:05:52.183 --> 00:05:53.600 align:middle line:84% AUDIENCE: So can you explain again 00:05:53.600 --> 00:05:56.890 align:middle line:84% why this diagram, like, you have one branch 00:05:56.890 --> 00:05:57.890 align:middle line:90% and then there's a loop? 00:05:57.890 --> 00:05:59.360 align:middle line:90% HONG LIU: Oh, yeah, yeah. 00:05:59.360 --> 00:06:02.900 align:middle line:84% Yeah, I will explain that, but before that 00:06:02.900 --> 00:06:04.220 align:middle line:90% do you have other questions? 00:06:04.220 --> 00:06:08.470 align:middle line:90% 00:06:08.470 --> 00:06:09.360 align:middle line:90% OK. 00:06:09.360 --> 00:06:12.120 align:middle line:84% So now I will explain a few things. 00:06:12.120 --> 00:06:17.170 align:middle line:84% OK, the first thing is this sign convention, 00:06:17.170 --> 00:06:18.813 align:middle line:90% OK, this sign convention. 00:06:18.813 --> 00:06:25.920 align:middle line:90% 00:06:25.920 --> 00:06:34.980 align:middle line:84% OK, so remember this Gn, so let's go back 00:06:34.980 --> 00:06:37.770 align:middle line:84% to the definition of this Gn, these momentum-space 00:06:37.770 --> 00:06:38.970 align:middle line:90% correlation functions. 00:06:38.970 --> 00:06:42.760 align:middle line:90% 00:06:42.760 --> 00:06:45.450 align:middle line:84% So this is obtained by doing a Fourier transform. 00:06:45.450 --> 00:06:52.130 align:middle line:90% 00:06:52.130 --> 00:07:03.880 align:middle line:84% Say-- I think it would be minus sign. 00:07:03.880 --> 00:07:06.370 align:middle line:84% OK, by doing the Fourier transform-- 00:07:06.370 --> 00:07:20.433 align:middle line:90% 00:07:20.433 --> 00:07:21.850 align:middle line:84% yeah, by doing a Fourier transform 00:07:21.850 --> 00:07:24.972 align:middle line:84% of your coordinate-space correlation function, 00:07:24.972 --> 00:07:26.680 align:middle line:84% the coordinate-space correlation function 00:07:26.680 --> 00:07:29.160 align:middle line:84% can be written as the following, OK, 00:07:29.160 --> 00:07:33.540 align:middle line:90% so you have phi x1 and phi xn. 00:07:33.540 --> 00:07:36.020 align:middle line:90% 00:07:36.020 --> 00:07:36.520 align:middle line:90% OK. 00:07:36.520 --> 00:07:42.110 align:middle line:90% 00:07:42.110 --> 00:07:52.530 align:middle line:84% So now, for those to go to the initial state, 00:07:52.530 --> 00:07:54.090 align:middle line:84% OK, for those to go to initial state, 00:07:54.090 --> 00:07:58.380 align:middle line:84% we start from alpha to beta, so then you 00:07:58.380 --> 00:08:01.320 align:middle line:84% want those phi corresponding to the initial state 00:08:01.320 --> 00:08:05.460 align:middle line:84% to act on the right, OK, to act on the right. 00:08:05.460 --> 00:08:08.160 align:middle line:84% And then you do the Fourier transform, OK, and then 00:08:08.160 --> 00:08:09.940 align:middle line:90% you do the Fourier transform. 00:08:09.940 --> 00:08:12.000 align:middle line:84% So now let's just consider one of them. 00:08:12.000 --> 00:08:13.500 align:middle line:84% Let's just say consider you have phi 00:08:13.500 --> 00:08:17.040 align:middle line:84% x for the initial state acting on the right, 00:08:17.040 --> 00:08:18.565 align:middle line:84% and then you do a Fourier transform. 00:08:18.565 --> 00:08:19.065 align:middle line:90% OK. 00:08:19.065 --> 00:08:23.700 align:middle line:90% 00:08:23.700 --> 00:08:27.710 align:middle line:90% OK, you do a Fourier transform. 00:08:27.710 --> 00:08:35.240 align:middle line:84% And so this, if you just record the mode expansion 00:08:35.240 --> 00:08:39.860 align:middle line:84% for phi and the phi contains a and a dagger-- 00:08:39.860 --> 00:08:42.650 align:middle line:84% and the a pieces acting on the zero will just give you zero, 00:08:42.650 --> 00:08:44.900 align:middle line:84% so only a dagger piece will survive. 00:08:44.900 --> 00:08:48.590 align:middle line:84% OK, and the a dagger piece is multiplied by-- 00:08:48.590 --> 00:08:56.870 align:middle line:84% say you will have, say, some k, and then 00:08:56.870 --> 00:09:03.860 align:middle line:90% have i omega k t plus minus. 00:09:03.860 --> 00:09:10.197 align:middle line:84% Yeah, essentially-- yeah, let me just write it in a simple way. 00:09:10.197 --> 00:09:11.530 align:middle line:90% You just have exponential i k x. 00:09:11.530 --> 00:09:14.470 align:middle line:90% 00:09:14.470 --> 00:09:19.620 align:middle line:84% OK, I think it's the exponential minus i k x, so you just get 00:09:19.620 --> 00:09:22.620 align:middle line:90% the exponential minus i k x. 00:09:22.620 --> 00:09:24.600 align:middle line:84% OK, with k it's the on-shell momentum, 00:09:24.600 --> 00:09:28.680 align:middle line:90% so k is given by omega k and k. 00:09:28.680 --> 00:09:30.895 align:middle line:90% OK. 00:09:30.895 --> 00:09:33.200 align:middle line:84% So when phi x acting on the zero, 00:09:33.200 --> 00:09:36.420 align:middle line:84% you keep the part which is corresponding to the a dagger, 00:09:36.420 --> 00:09:40.010 align:middle line:84% and then you get a piece proportional like this. 00:09:40.010 --> 00:09:44.030 align:middle line:84% OK, and now when you do the Fourier transform 00:09:44.030 --> 00:09:51.900 align:middle line:84% and then you find just p, then your p just equals to minus k. 00:09:51.900 --> 00:09:54.570 align:middle line:84% OK, so that's related to the minus sign 00:09:54.570 --> 00:09:56.850 align:middle line:84% there, OK, related to the minus sign there, 00:09:56.850 --> 00:10:02.100 align:middle line:84% and also, when your p equal to minus k, then that means p0 00:10:02.100 --> 00:10:03.960 align:middle line:90% is equal to minus k. 00:10:03.960 --> 00:10:07.770 align:middle line:84% OK, so that's where that sign in the initial state come from. 00:10:07.770 --> 00:10:10.360 align:middle line:84% And the same thing with the final state, 00:10:10.360 --> 00:10:12.930 align:middle line:84% so for the final state, then you need to look at the phi 00:10:12.930 --> 00:10:17.850 align:middle line:84% x acting on the left to this, and then you 00:10:17.850 --> 00:10:20.920 align:middle line:90% do the Fourier transform. 00:10:20.920 --> 00:10:21.420 align:middle line:90% OK. 00:10:21.420 --> 00:10:24.370 align:middle line:90% 00:10:24.370 --> 00:10:27.500 align:middle line:84% You do the Fourier transform, and then in this case, 00:10:27.500 --> 00:10:29.500 align:middle line:90% this becomes-- 00:10:29.500 --> 00:10:32.720 align:middle line:84% on the left, it's the A part acting on the left, 00:10:32.720 --> 00:10:35.690 align:middle line:84% so this gives you k exponential i 00:10:35.690 --> 00:10:42.710 align:middle line:84% k x, OK, is the a acting on the left. 00:10:42.710 --> 00:10:45.430 align:middle line:84% And then when you do the Fourier transform, then here it 00:10:45.430 --> 00:10:47.350 align:middle line:90% gives p equal to k. 00:10:47.350 --> 00:10:51.460 align:middle line:84% OK, so that's why in the final state, 00:10:51.460 --> 00:10:55.690 align:middle line:84% you just have p equal to k, and then p0 00:10:55.690 --> 00:10:57.625 align:middle line:90% is equal to just omega k. 00:10:57.625 --> 00:11:01.290 align:middle line:90% OK, so this explains the sign. 00:11:01.290 --> 00:11:03.050 align:middle line:90% So this explains the sign. 00:11:03.050 --> 00:11:06.080 align:middle line:84% It's just from whether you act on the initial state 00:11:06.080 --> 00:11:07.400 align:middle line:90% or act on the final state. 00:11:07.400 --> 00:11:07.900 align:middle line:90% OK. 00:11:07.900 --> 00:11:12.360 align:middle line:90% 00:11:12.360 --> 00:11:13.380 align:middle line:90% Good? 00:11:13.380 --> 00:11:14.930 align:middle line:90% So-- yes? 00:11:14.930 --> 00:11:17.028 align:middle line:84% AUDIENCE: Right, so time-ordering of x1 00:11:17.028 --> 00:11:19.870 align:middle line:90% to xn right here? 00:11:19.870 --> 00:11:20.560 align:middle line:90% HONG LIU: Yeah. 00:11:20.560 --> 00:11:24.940 align:middle line:84% It-- of course, when you derive that, it matters, but here, 00:11:24.940 --> 00:11:27.200 align:middle line:84% for this argument, it doesn't matter. 00:11:27.200 --> 00:11:30.200 align:middle line:90% It just has to act to the right. 00:11:30.200 --> 00:11:34.250 align:middle line:84% Yeah, for the initial state, you have to act to the right. 00:11:34.250 --> 00:11:36.530 align:middle line:84% Yeah, of course, to derive that theorem, 00:11:36.530 --> 00:11:37.991 align:middle line:90% the time-order matters. 00:11:37.991 --> 00:11:38.491 align:middle line:90% OK. 00:11:38.491 --> 00:11:42.640 align:middle line:90% 00:11:42.640 --> 00:11:43.277 align:middle line:90% Yes? 00:11:43.277 --> 00:11:45.610 align:middle line:84% AUDIENCE: So this was just for free scalar field theory, 00:11:45.610 --> 00:11:47.400 align:middle line:84% that you showed that this is true? 00:11:47.400 --> 00:11:49.858 align:middle line:84% HONG LIU: No, this is not for the free scalar field theory. 00:11:49.858 --> 00:11:54.580 align:middle line:84% Yeah, it's the-- here I just detail you the sign convention. 00:11:54.580 --> 00:11:58.600 align:middle line:84% OK, you can do this for the full interacting theory. 00:11:58.600 --> 00:12:02.170 align:middle line:84% Yeah, just use the free scalar as the-- yeah, 00:12:02.170 --> 00:12:05.500 align:middle line:84% because when you go to plus or minus infinity, 00:12:05.500 --> 00:12:08.470 align:middle line:84% you can just reduce to the free particles. 00:12:08.470 --> 00:12:14.380 align:middle line:90% 00:12:14.380 --> 00:12:15.700 align:middle line:90% OK. 00:12:15.700 --> 00:12:16.630 align:middle line:90% Other questions? 00:12:16.630 --> 00:12:20.540 align:middle line:90% 00:12:20.540 --> 00:12:22.350 align:middle line:90% OK, good. 00:12:22.350 --> 00:12:23.900 align:middle line:90% So this is the first comment. 00:12:23.900 --> 00:12:30.330 align:middle line:84% The second comment, each side, we need to here-- 00:12:30.330 --> 00:12:34.200 align:middle line:84% from here, we need to truncate the external propagator. 00:12:34.200 --> 00:12:37.740 align:middle line:84% So we mentioned that if you have a diagram like this, 00:12:37.740 --> 00:12:45.348 align:middle line:84% OK, if you have a diagram like this, say minus P1, minus P2, 00:12:45.348 --> 00:12:50.302 align:middle line:84% to P3, P4, for a diagram like this, OK, 00:12:50.302 --> 00:12:52.260 align:middle line:84% and then the square root and amplitude are just 00:12:52.260 --> 00:12:57.150 align:middle line:84% given by minus i lambda, because you need to throw away 00:12:57.150 --> 00:12:58.560 align:middle line:90% all the external propagators. 00:12:58.560 --> 00:13:00.852 align:middle line:84% You don't have to worry about the external propagators. 00:13:00.852 --> 00:13:03.570 align:middle line:84% OK, you just need to-- you have to truncate 00:13:03.570 --> 00:13:08.590 align:middle line:84% all the external propagators, so that means you also 00:13:08.590 --> 00:13:10.380 align:middle line:90% throw away diagrams like this. 00:13:10.380 --> 00:13:18.770 align:middle line:90% 00:13:18.770 --> 00:13:21.230 align:middle line:84% You can consider arbitrary, complicated diagram, 00:13:21.230 --> 00:13:24.830 align:middle line:84% OK, as far as that only happens-- 00:13:24.830 --> 00:13:25.640 align:middle line:90% yeah. 00:13:25.640 --> 00:13:27.180 align:middle line:90% Yeah, yeah, here, it should-- 00:13:27.180 --> 00:13:32.450 align:middle line:84% I don't draw very well, so this just touch at the one point, 00:13:32.450 --> 00:13:34.990 align:middle line:90% OK? 00:13:34.990 --> 00:13:37.510 align:middle line:84% So all these diagrams, you can ignore them, OK, 00:13:37.510 --> 00:13:40.660 align:middle line:84% because they're just corresponding to the correction 00:13:40.660 --> 00:13:43.780 align:middle line:84% to the external propagator, OK, because they 00:13:43.780 --> 00:13:46.870 align:middle line:90% don't touch with the-- 00:13:46.870 --> 00:13:48.730 align:middle line:84% they only-- yeah, you can also-- yeah, 00:13:48.730 --> 00:13:51.670 align:middle line:84% you can do it on any of them, OK? 00:13:51.670 --> 00:13:53.500 align:middle line:84% So such kind of diagrams, they only 00:13:53.500 --> 00:13:56.260 align:middle line:84% change the external propagator, but since we truncate 00:13:56.260 --> 00:13:59.050 align:middle line:84% the external propagator, and then 00:13:59.050 --> 00:14:02.470 align:middle line:84% they don't matter at all, OK, and it's all 00:14:02.470 --> 00:14:04.590 align:middle line:90% included in this diagram. 00:14:04.590 --> 00:14:05.640 align:middle line:90% OK. 00:14:05.640 --> 00:14:09.090 align:middle line:90% So the reason is the following. 00:14:09.090 --> 00:14:10.500 align:middle line:90% The reason is the following. 00:14:10.500 --> 00:14:15.060 align:middle line:84% All this diagram do is just modify 00:14:15.060 --> 00:14:18.450 align:middle line:84% the properties of the external propagator, 00:14:18.450 --> 00:14:22.500 align:middle line:84% OK, modify the properties of the external propagator. 00:14:22.500 --> 00:14:25.980 align:middle line:84% And the only way all these diagrams 00:14:25.980 --> 00:14:29.050 align:middle line:84% can modify the external propagators 00:14:29.050 --> 00:14:32.630 align:middle line:84% is to give you an overall constant, 00:14:32.630 --> 00:14:34.870 align:middle line:84% OK, so that's what this Z is corresponding to. 00:14:34.870 --> 00:14:37.690 align:middle line:84% So this Z essentially just captures 00:14:37.690 --> 00:14:40.540 align:middle line:84% all these different corrections, OK, 00:14:40.540 --> 00:14:42.850 align:middle line:90% all these different corrections. 00:14:42.850 --> 00:14:45.850 align:middle line:84% And now you have truncated them, and so you 00:14:45.850 --> 00:14:47.350 align:middle line:90% don't need to worry about them. 00:14:47.350 --> 00:14:48.890 align:middle line:90% OK. 00:14:48.890 --> 00:14:52.640 align:middle line:84% So Z also has an expansion, just 1 plus order lambda, 00:14:52.640 --> 00:14:55.700 align:middle line:84% et cetera, so the leading order, so Z does not contribute. 00:14:55.700 --> 00:14:58.190 align:middle line:84% OK, you can just set it to 1, and when 00:14:58.190 --> 00:15:03.730 align:middle line:84% you go to the higher order, then the Z can make a contribution. 00:15:03.730 --> 00:15:05.800 align:middle line:84% OK, so you just need to separately take 00:15:05.800 --> 00:15:09.310 align:middle line:84% into account the Z, and there's no need to contribute-- 00:15:09.310 --> 00:15:11.110 align:middle line:84% to calculate those things separately. 00:15:11.110 --> 00:15:11.872 align:middle line:90% Yes? 00:15:11.872 --> 00:15:13.330 align:middle line:84% AUDIENCE: We also truncate diagrams 00:15:13.330 --> 00:15:15.725 align:middle line:84% with loops in two different legs? 00:15:15.725 --> 00:15:16.350 align:middle line:90% HONG LIU: Yeah. 00:15:16.350 --> 00:15:19.530 align:middle line:84% Yeah, yeah, any of them, you can do any number of-- 00:15:19.530 --> 00:15:23.190 align:middle line:84% as far as they only concern external legs, 00:15:23.190 --> 00:15:26.310 align:middle line:84% it's fine, yeah, because this only-- 00:15:26.310 --> 00:15:29.370 align:middle line:84% any of those corrections only concerns one leg, yeah. 00:15:29.370 --> 00:15:30.330 align:middle line:90% Yes? 00:15:30.330 --> 00:15:32.890 align:middle line:84% AUDIENCE: But when calculating the actual, let's say, 00:15:32.890 --> 00:15:34.650 align:middle line:84% n-point function, we have-- we're supposed 00:15:34.650 --> 00:15:35.550 align:middle line:90% to include all of this? 00:15:35.550 --> 00:15:36.450 align:middle line:90% HONG LIU: That's right, yeah. 00:15:36.450 --> 00:15:38.117 align:middle line:84% When you calculate the n-point function, 00:15:38.117 --> 00:15:40.215 align:middle line:84% you have to include all this, but when 00:15:40.215 --> 00:15:42.465 align:middle line:84% you calculate scattering amplitude, you don't need to. 00:15:42.465 --> 00:15:43.170 align:middle line:84% AUDIENCE: Don't need to, all right. 00:15:43.170 --> 00:15:44.040 align:middle line:84% HONG LIU: Because in the scattering amplitude, 00:15:44.040 --> 00:15:46.350 align:middle line:84% you literally divide by the external propagator, 00:15:46.350 --> 00:15:50.940 align:middle line:90% and all of these things, they-- 00:15:50.940 --> 00:15:54.930 align:middle line:84% yeah, all these things, essentially, they 00:15:54.930 --> 00:15:56.880 align:middle line:90% just modify the-- 00:15:56.880 --> 00:15:59.040 align:middle line:84% give you the correction to the external propagator 00:15:59.040 --> 00:16:01.350 align:middle line:84% and include it in that constant Z, yeah. 00:16:01.350 --> 00:16:04.080 align:middle line:84% AUDIENCE: So is Z the same or different 00:16:04.080 --> 00:16:06.870 align:middle line:90% for different processes? 00:16:06.870 --> 00:16:08.870 align:middle line:90% HONG LIU: Yeah, Z is the same. 00:16:08.870 --> 00:16:12.250 align:middle line:84% No, Z is the same for different processes, 00:16:12.250 --> 00:16:14.320 align:middle line:84% but it's different for different particles. 00:16:14.320 --> 00:16:16.450 align:middle line:84% So here we only have one type of particles, 00:16:16.450 --> 00:16:18.660 align:middle line:84% so we only have one Z. But if you 00:16:18.660 --> 00:16:20.193 align:middle line:84% have two kinds of particles, then z 00:16:20.193 --> 00:16:21.735 align:middle line:84% is different for different particles. 00:16:21.735 --> 00:16:22.443 align:middle line:90% AUDIENCE: Oh, OK. 00:16:22.443 --> 00:16:26.490 align:middle line:84% So for example, for, let's say, a fixed process, 00:16:26.490 --> 00:16:29.670 align:middle line:84% let's say, like, 3 going to 3, Z would be constant 00:16:29.670 --> 00:16:31.230 align:middle line:84% for all choices of momentum, right? 00:16:31.230 --> 00:16:32.010 align:middle line:90% Am I correct to say that? 00:16:32.010 --> 00:16:33.630 align:middle line:84% HONG LIU: Yeah, yeah, yeah, it's all-- yeah, 00:16:33.630 --> 00:16:34.830 align:middle line:84% a constant for all of them, yeah. 00:16:34.830 --> 00:16:36.450 align:middle line:84% AUDIENCE: OK, yeah, because if it depended on momentum, then 00:16:36.450 --> 00:16:37.440 align:middle line:84% it would be kind of useless, right? 00:16:37.440 --> 00:16:39.690 align:middle line:84% HONG LIU: No, no, no, it does not depend on momentum. 00:16:39.690 --> 00:16:43.920 align:middle line:90% Yeah, this just-- correct. 00:16:43.920 --> 00:16:47.010 align:middle line:84% You see, all such things don't change the momentum. 00:16:47.010 --> 00:16:48.330 align:middle line:90% OK, the momentum don't change. 00:16:48.330 --> 00:16:49.750 align:middle line:90% Yes. 00:16:49.750 --> 00:16:50.560 align:middle line:90% Yeah. 00:16:50.560 --> 00:16:52.990 align:middle line:84% So we will not go into details of the Z. 00:16:52.990 --> 00:16:56.650 align:middle line:84% And that is in the QFT2, and so we 00:16:56.650 --> 00:17:01.600 align:middle line:84% will discuss how to calculate this z in QFT2. 00:17:01.600 --> 00:17:03.530 align:middle line:84% But the leading order, they don't matter, 00:17:03.530 --> 00:17:04.960 align:middle line:90% and so we will start with 1. 00:17:04.960 --> 00:17:09.609 align:middle line:84% So for our purpose, actually, it's not important. 00:17:09.609 --> 00:17:10.540 align:middle line:90% Yes? 00:17:10.540 --> 00:17:12.880 align:middle line:84% AUDIENCE: Is there a physical interpretation for what 00:17:12.880 --> 00:17:14.765 align:middle line:90% interactions are included in Z? 00:17:14.765 --> 00:17:15.390 align:middle line:90% HONG LIU: Yeah. 00:17:15.390 --> 00:17:17.040 align:middle line:84% Yeah, this is the self-interact-- yeah, 00:17:17.040 --> 00:17:20.190 align:middle line:84% just when the-- when you have an interacting theory, 00:17:20.190 --> 00:17:23.200 align:middle line:84% so the particle can interact with itself, when 00:17:23.200 --> 00:17:26.325 align:middle line:84% the particle propagates, it actually 00:17:26.325 --> 00:17:27.950 align:middle line:84% can interact with the virtual particle. 00:17:27.950 --> 00:17:29.660 align:middle line:84% It just all comes from this kind of diagram. 00:17:29.660 --> 00:17:31.310 align:middle line:84% You can have single particle, and you 00:17:31.310 --> 00:17:34.070 align:middle line:84% can have such a diagram like this and all these diagrams. 00:17:34.070 --> 00:17:39.410 align:middle line:90% 00:17:39.410 --> 00:17:42.050 align:middle line:84% Correspondingly, you have a particle propagating, 00:17:42.050 --> 00:17:44.900 align:middle line:84% but that particle can interact with the virtual particle, 00:17:44.900 --> 00:17:47.190 align:middle line:90% its own virtual particle. 00:17:47.190 --> 00:17:50.100 align:middle line:84% OK, and so the rule corresponding to the particle 00:17:50.100 --> 00:17:52.390 align:middle line:90% loop, it's-- 00:17:52.390 --> 00:17:54.670 align:middle line:84% so this is the real external particle, 00:17:54.670 --> 00:17:57.940 align:middle line:84% but anything coming in the loop, you 00:17:57.940 --> 00:18:01.060 align:middle line:84% can imagine how the virtual particle which come out 00:18:01.060 --> 00:18:03.820 align:middle line:84% from the vacuum, and then this can be interpreted 00:18:03.820 --> 00:18:06.790 align:middle line:84% as the particle interacting with its own virtual particle, yes, 00:18:06.790 --> 00:18:08.750 align:middle line:84% a virtual particle coming out from the back. 00:18:08.750 --> 00:18:09.520 align:middle line:90% Yeah. 00:18:09.520 --> 00:18:14.080 align:middle line:84% And so that, this kind of interaction, 00:18:14.080 --> 00:18:17.520 align:middle line:84% will affect the property of the propagation 00:18:17.520 --> 00:18:22.740 align:middle line:84% but can have the most effect by prefactor, 00:18:22.740 --> 00:18:25.050 align:middle line:84% but actually can change the mass, too. 00:18:25.050 --> 00:18:28.200 align:middle line:84% But for the-- oh, it can correct for the mass-- 00:18:28.200 --> 00:18:32.790 align:middle line:84% and also, that's the subject of the QFT2-- 00:18:32.790 --> 00:18:34.560 align:middle line:84% can correct for the mass, I think 00:18:34.560 --> 00:18:38.171 align:middle line:84% at most can change the overall factor by z. 00:18:38.171 --> 00:18:42.890 align:middle line:90% 00:18:42.890 --> 00:18:43.760 align:middle line:90% Other questions? 00:18:43.760 --> 00:18:47.420 align:middle line:90% 00:18:47.420 --> 00:18:47.920 align:middle line:90% Good. 00:18:47.920 --> 00:18:48.220 align:middle line:90% Good. 00:18:48.220 --> 00:18:48.720 align:middle line:90% OK. 00:18:48.720 --> 00:18:51.320 align:middle line:90% 00:18:51.320 --> 00:18:51.850 align:middle line:90% Good. 00:18:51.850 --> 00:18:54.230 align:middle line:84% If you don't have other questions, 00:18:54.230 --> 00:18:59.920 align:middle line:84% so let's conclude our discussion of chapter 3 on interacting 00:18:59.920 --> 00:19:01.350 align:middle line:90% theories. 00:19:01.350 --> 00:19:04.640 align:middle line:84% So as I said before, as I mentioned before, 00:19:04.640 --> 00:19:09.580 align:middle line:84% and now you are really equipped with the technique, 00:19:09.580 --> 00:19:14.530 align:middle line:84% now in principle you can treat any interacting theory. 00:19:14.530 --> 00:19:17.590 align:middle line:84% So the technique, even though we just used the scalar theory, 00:19:17.590 --> 00:19:19.390 align:middle line:90% but the technique is the same. 00:19:19.390 --> 00:19:23.350 align:middle line:84% OK, just for different theory, you have different details, OK, 00:19:23.350 --> 00:19:25.870 align:middle line:84% and now you have already equipped 00:19:25.870 --> 00:19:29.710 align:middle line:84% the foundation, the basic tools for dealing 00:19:29.710 --> 00:19:31.600 align:middle line:90% with any interacting theory. 00:19:31.600 --> 00:19:38.270 align:middle line:84% And for the goal of this course, we 00:19:38.270 --> 00:19:41.060 align:middle line:84% want in the end to be able to calculate the, say, 00:19:41.060 --> 00:19:45.230 align:middle line:84% interactions in quantum electrodynamics, OK, 00:19:45.230 --> 00:19:47.660 align:middle line:84% and for that purpose, we still need 00:19:47.660 --> 00:19:50.600 align:middle line:90% to have some other preparations. 00:19:50.600 --> 00:19:54.110 align:middle line:84% So now let's discuss how to describe fermions. 00:19:54.110 --> 00:19:57.630 align:middle line:84% OK, and now we've described the scalar 00:19:57.630 --> 00:20:00.570 align:middle line:84% and how the scalars interact, and now we'll 00:20:00.570 --> 00:20:01.530 align:middle line:90% talk about fermions. 00:20:01.530 --> 00:20:08.700 align:middle line:90% 00:20:08.700 --> 00:20:10.950 align:middle line:90% OK. 00:20:10.950 --> 00:20:13.080 align:middle line:84% So let me say a little bit of history. 00:20:13.080 --> 00:20:16.770 align:middle line:90% 00:20:16.770 --> 00:20:20.790 align:middle line:84% So, soon after the quantum mechanics 00:20:20.790 --> 00:20:23.700 align:middle line:84% was proposed by Schrodinger and Heisenberg, 00:20:23.700 --> 00:20:27.870 align:middle line:84% et cetera, and then people tried to generalize 00:20:27.870 --> 00:20:29.530 align:middle line:90% to relativistic situation-- 00:20:29.530 --> 00:20:32.350 align:middle line:84% OK, so that came from this Klein-Gordon equation, 00:20:32.350 --> 00:20:35.490 align:middle line:84% which we discussed before, which in-- 00:20:35.490 --> 00:20:41.400 align:middle line:84% this was the first attempt to write down a wave equation 00:20:41.400 --> 00:20:43.020 align:middle line:90% for relativistic particles. 00:20:43.020 --> 00:20:45.540 align:middle line:84% OK, and we discussed before, this 00:20:45.540 --> 00:20:49.980 align:middle line:84% does not really make sense as a relativistic quantum mechanics. 00:20:49.980 --> 00:20:58.160 align:middle line:84% And yeah, yeah, but at the time, this Klein-Gordon equation, 00:20:58.160 --> 00:21:02.150 align:middle line:84% if you interpret it as a wave equation, suffers some-- 00:21:02.150 --> 00:21:03.740 align:middle line:90% suffers from some difficulties. 00:21:03.740 --> 00:21:08.770 align:middle line:84% One is that it does not have a positive probability. 00:21:08.770 --> 00:21:13.600 align:middle line:90% 00:21:13.600 --> 00:21:15.040 align:middle line:90% You cannot define-- 00:21:15.040 --> 00:21:28.790 align:middle line:84% I should say cannot define positive definite probability. 00:21:28.790 --> 00:21:41.410 align:middle line:90% 00:21:41.410 --> 00:21:44.350 align:middle line:84% And the second difficulty is that it has negative energy 00:21:44.350 --> 00:21:46.610 align:middle line:90% state. 00:21:46.610 --> 00:21:47.110 align:middle line:90% OK. 00:21:47.110 --> 00:21:56.090 align:middle line:90% 00:21:56.090 --> 00:22:00.200 align:middle line:84% So yeah, of course, as I mentioned before, 00:22:00.200 --> 00:22:02.030 align:middle line:90% that this was-- 00:22:02.030 --> 00:22:05.420 align:middle line:84% there's more fundamental reason that actually 00:22:05.420 --> 00:22:09.390 align:middle line:84% relativistic quantum mechanics does not make sense, 00:22:09.390 --> 00:22:12.710 align:middle line:84% but at the time, in the late 1920s, 00:22:12.710 --> 00:22:14.300 align:middle line:90% people didn't realize that. 00:22:14.300 --> 00:22:17.480 align:middle line:84% OK, people just looked at those difficulties, 00:22:17.480 --> 00:22:19.410 align:middle line:84% they thought it's a technical difficulty. 00:22:19.410 --> 00:22:23.540 align:middle line:84% So Dirac proceeded trying to correct those difficulties, 00:22:23.540 --> 00:22:25.040 align:middle line:90% to overcome those difficulties. 00:22:25.040 --> 00:22:28.650 align:middle line:90% OK, so Dirac then soon came up-- 00:22:28.650 --> 00:22:35.420 align:middle line:84% so I think this is around 1926, and then 1928, then Dirac 00:22:35.420 --> 00:22:41.590 align:middle line:84% came up with this Dirac theory, this Dirac equation. 00:22:41.590 --> 00:22:46.650 align:middle line:84% OK, so Dirac equation was aimed to cure those problems, 00:22:46.650 --> 00:22:51.790 align:middle line:84% OK, so Dirac concluded that the Klein-Gordon equation, 00:22:51.790 --> 00:22:54.960 align:middle line:84% the reason it had those problems was 00:22:54.960 --> 00:22:56.640 align:middle line:84% because the Klein-Gordon equation has 00:22:56.640 --> 00:22:59.740 align:middle line:90% a second-order derivative. 00:22:59.740 --> 00:23:03.000 align:middle line:84% And then he said, if we have a first-order-- 00:23:03.000 --> 00:23:05.240 align:middle line:84% have an equation with first-order-- yeah, 00:23:05.240 --> 00:23:08.365 align:middle line:84% he speculated, if we had the equation with 00:23:08.365 --> 00:23:10.740 align:middle line:84% first-order-in-time derivative, just like the Schrodinger 00:23:10.740 --> 00:23:14.940 align:middle line:84% equation, and then both problems maybe can be solved, OK, 00:23:14.940 --> 00:23:15.810 align:middle line:90% can be solved. 00:23:15.810 --> 00:23:22.540 align:middle line:84% And then Dirac came up with the Dirac equation. 00:23:22.540 --> 00:23:24.180 align:middle line:84% So it turns out that the Dirac equation 00:23:24.180 --> 00:23:28.390 align:middle line:84% solved the first problem, OK, but didn't really solve 00:23:28.390 --> 00:23:29.350 align:middle line:90% the second problem. 00:23:29.350 --> 00:23:34.420 align:middle line:84% OK, but the level is, again, due to more fundamental reasons, 00:23:34.420 --> 00:23:37.570 align:middle line:84% you cannot really interpret the Dirac equation 00:23:37.570 --> 00:23:40.695 align:middle line:84% as a relativistic quantum mechanics equation, 00:23:40.695 --> 00:23:42.820 align:middle line:84% a wave equation for relativistic quantum mechanics. 00:23:42.820 --> 00:23:45.040 align:middle line:84% Actually, the Dirac equation should be 00:23:45.040 --> 00:23:46.720 align:middle line:90% interpreted as a field theory. 00:23:46.720 --> 00:23:50.290 align:middle line:84% So nowadays, we interpret this this is the-- gives the field 00:23:50.290 --> 00:23:59.710 align:middle line:90% theory for-- 00:23:59.710 --> 00:24:07.960 align:middle line:90% 00:24:07.960 --> 00:24:11.300 align:middle line:84% OK, so of course, Dirac didn't know this, 00:24:11.300 --> 00:24:13.900 align:middle line:84% so essentially, he discovered this beautiful theory 00:24:13.900 --> 00:24:17.320 align:middle line:84% for the wrong motivation, OK, for the wrong motivation. 00:24:17.320 --> 00:24:22.870 align:middle line:84% And yeah, this happens over and over again in physics. 00:24:22.870 --> 00:24:24.850 align:middle line:90% OK, people make great-- 00:24:24.850 --> 00:24:30.610 align:middle line:84% made great discoveries often for the wrong motivations. 00:24:30.610 --> 00:24:33.920 align:middle line:84% But the key is that if you are good enough, 00:24:33.920 --> 00:24:36.250 align:middle line:84% you will find something new and that something new 00:24:36.250 --> 00:24:36.910 align:middle line:90% will be useful. 00:24:36.910 --> 00:24:44.410 align:middle line:84% [LAUGHS] OK, and this Dirac theory is a prime example which 00:24:44.410 --> 00:24:47.410 align:middle line:84% is actually one of the most beautiful-- 00:24:47.410 --> 00:24:50.740 align:middle line:84% we will see, this is one of the most beautiful equations 00:24:50.740 --> 00:24:53.740 align:middle line:90% in mathematical physics. 00:24:53.740 --> 00:24:56.780 align:middle line:84% Yeah, but also, it's actually describing electrons, 00:24:56.780 --> 00:25:00.810 align:middle line:84% so it's not only beautiful, but it's actually useful. 00:25:00.810 --> 00:25:06.900 align:middle line:84% OK, so this, first, I'll talk about, 00:25:06.900 --> 00:25:15.840 align:middle line:84% introduce the Dirac equation, OK, and its covariance. 00:25:15.840 --> 00:25:21.730 align:middle line:90% 00:25:21.730 --> 00:25:26.860 align:middle line:84% OK, so the best way to introduce the Dirac equation 00:25:26.860 --> 00:25:30.850 align:middle line:84% was still his original motivation, 00:25:30.850 --> 00:25:36.310 align:middle line:84% is that we want to find the first-order equation which 00:25:36.310 --> 00:25:37.670 align:middle line:90% is Lorentz invariant. 00:25:37.670 --> 00:25:42.380 align:middle line:84% OK, so the goal is that you write down the equation, 00:25:42.380 --> 00:25:45.110 align:middle line:84% like a Schrodinger equation, which is first-order-in-time 00:25:45.110 --> 00:25:54.890 align:middle line:84% derivative, OK, but is Lorentz covariant. 00:25:54.890 --> 00:25:59.350 align:middle line:90% 00:25:59.350 --> 00:26:01.990 align:middle line:90% OK, but it's Lorentz covariant. 00:26:01.990 --> 00:26:03.970 align:middle line:90% OK. 00:26:03.970 --> 00:26:09.420 align:middle line:84% So but Lorentz covariant means that this equation 00:26:09.420 --> 00:26:12.760 align:middle line:84% has the same form when you go to a different frame. 00:26:12.760 --> 00:26:15.540 align:middle line:84% OK, yeah, that's what we mean by-- 00:26:15.540 --> 00:26:18.060 align:middle line:84% just when you go to a different Lorentz frame, the equation, 00:26:18.060 --> 00:26:20.340 align:middle line:84% the form of the equation looks the same. 00:26:20.340 --> 00:26:22.890 align:middle line:84% Just different observers in different laboratory, 00:26:22.890 --> 00:26:25.200 align:middle line:84% they see the same equation, OK, so that's what 00:26:25.200 --> 00:26:28.410 align:middle line:90% we mean by Lorentz covariant. 00:26:28.410 --> 00:26:32.550 align:middle line:84% OK, so but for this to be Lorentz covariant, 00:26:32.550 --> 00:26:36.180 align:middle line:84% remember, Lorentz transformation transform t to x, 00:26:36.180 --> 00:26:40.410 align:middle line:84% so immediately, you conclude that H 00:26:40.410 --> 00:26:49.440 align:middle line:84% must be first-order in spatial derivatives. 00:26:49.440 --> 00:26:59.230 align:middle line:90% 00:26:59.230 --> 00:27:01.890 align:middle line:90% OK. 00:27:01.890 --> 00:27:08.740 align:middle line:84% So the only-- then the most general way you can write it, 00:27:08.740 --> 00:27:12.540 align:middle line:84% so let's try to write the most general-- 00:27:12.540 --> 00:27:13.170 align:middle line:90% yeah. 00:27:13.170 --> 00:27:15.820 align:middle line:84% Yeah, let's try to write something like that. 00:27:15.820 --> 00:27:18.210 align:middle line:84% So your first-order derivative, then, 00:27:18.210 --> 00:27:25.350 align:middle line:84% has to have the following form, alpha minus i-- 00:27:25.350 --> 00:27:30.390 align:middle line:84% yeah, so this is the gradient operator, yeah, 00:27:30.390 --> 00:27:33.790 align:middle line:84% spatial derivatives, and yeah, that's i just for convenience. 00:27:33.790 --> 00:27:36.600 align:middle line:84% And then this is a vector, have three components, 00:27:36.600 --> 00:27:38.580 align:middle line:84% and then have to contract with something, 00:27:38.580 --> 00:27:40.140 align:middle line:90% and then we include the alpha. 00:27:40.140 --> 00:27:44.700 align:middle line:84% And then you can at most add the constant, OK, so let me just 00:27:44.700 --> 00:27:47.100 align:middle line:84% write this-- for historical reasons, 00:27:47.100 --> 00:27:49.150 align:middle line:84% write this constant as m times beta. 00:27:49.150 --> 00:27:49.650 align:middle line:90% OK. 00:27:49.650 --> 00:27:53.860 align:middle line:90% 00:27:53.860 --> 00:27:59.390 align:middle line:84% So if you look at this equation form, 00:27:59.390 --> 00:28:01.790 align:middle line:84% you say this doesn't make any sense, OK? 00:28:01.790 --> 00:28:08.660 align:middle line:84% So alpha and beta, they have to be some kind of constant, OK, 00:28:08.660 --> 00:28:12.000 align:middle line:84% but if alpha and beta are constant, 00:28:12.000 --> 00:28:15.520 align:middle line:84% then this is not even a rotational invariant, not 00:28:15.520 --> 00:28:18.130 align:middle line:84% to mention Lorentz environment, OK, 00:28:18.130 --> 00:28:23.980 align:middle line:84% because this derivative is not contracted by any other Lorentz 00:28:23.980 --> 00:28:24.910 align:middle line:90% indices. 00:28:24.910 --> 00:28:26.590 align:middle line:84% So alpha is just some constant, OK, 00:28:26.590 --> 00:28:39.430 align:middle line:84% so this cannot be Lorentz covariant, 00:28:39.430 --> 00:28:47.750 align:middle line:84% even cannot be rotational environment, OK, 00:28:47.750 --> 00:28:51.605 align:middle line:90% if alpha or beta are constant. 00:28:51.605 --> 00:29:01.310 align:middle line:90% 00:29:01.310 --> 00:29:09.710 align:middle line:84% OK, and psi is the ordinary function. 00:29:09.710 --> 00:29:18.953 align:middle line:90% 00:29:18.953 --> 00:29:19.453 align:middle line:90% OK. 00:29:19.453 --> 00:29:21.960 align:middle line:90% 00:29:21.960 --> 00:29:28.500 align:middle line:84% So yeah, so you can have alpha x, partial x, alpha 00:29:28.500 --> 00:29:30.570 align:middle line:84% y, partial y, alpha z, partial z, 00:29:30.570 --> 00:29:34.620 align:middle line:84% OK, so you can easily convince yourself, when you rotate xyz, 00:29:34.620 --> 00:29:39.340 align:middle line:84% this is not symmetric because alpha is some constant. 00:29:39.340 --> 00:29:43.540 align:middle line:84% And so I'm sure this idea came to many people trying 00:29:43.540 --> 00:29:47.500 align:middle line:84% to look for some first-order equation which are Lorentz 00:29:47.500 --> 00:29:51.860 align:middle line:84% covariant, then after five minutes, 00:29:51.860 --> 00:29:54.860 align:middle line:84% you realize this is not possible. 00:29:54.860 --> 00:29:58.730 align:middle line:84% OK, this is just so simple, it's just not possible. 00:29:58.730 --> 00:30:00.780 align:middle line:90% OK. 00:30:00.780 --> 00:30:03.870 align:middle line:90% But then Dirac made it work. 00:30:03.870 --> 00:30:06.747 align:middle line:84% OK, it's really, say, a stroke of genius. 00:30:06.747 --> 00:30:08.580 align:middle line:84% It's really a stroke of genius because there 00:30:08.580 --> 00:30:11.700 align:middle line:90% was nothing like this before. 00:30:11.700 --> 00:30:13.740 align:middle line:84% Just even from a mathematical point of view, 00:30:13.740 --> 00:30:16.320 align:middle line:84% it's purely, purely imaginative, OK, 00:30:16.320 --> 00:30:18.240 align:middle line:90% just nothing like this before. 00:30:18.240 --> 00:30:20.850 align:middle line:84% Like when Einstein wrote down his theory, et cetera, 00:30:20.850 --> 00:30:22.230 align:middle line:90% you can still trace-- 00:30:22.230 --> 00:30:25.470 align:middle line:84% there some clues, OK, but when Dirac-- 00:30:25.470 --> 00:30:26.970 align:middle line:90% this one just really-- 00:30:26.970 --> 00:30:28.020 align:middle line:90% [LAUGHTER] 00:30:28.020 --> 00:30:31.950 align:middle line:84% --like music, just came out from his mind, OK? 00:30:31.950 --> 00:30:37.050 align:middle line:84% It just-- and then he reasoned, OK, 00:30:37.050 --> 00:30:40.890 align:middle line:84% if this is constant which does not work, 00:30:40.890 --> 00:30:48.200 align:middle line:84% and then let's make alpha and beta to be matrices. 00:30:48.200 --> 00:30:51.660 align:middle line:90% 00:30:51.660 --> 00:31:02.840 align:middle line:84% Take them-- OK, and then, so let's 00:31:02.840 --> 00:31:04.730 align:middle line:90% say they are n-by-n matrices. 00:31:04.730 --> 00:31:07.930 align:middle line:90% 00:31:07.930 --> 00:31:11.410 align:middle line:84% Then in order for that equation to make sense, 00:31:11.410 --> 00:31:14.320 align:middle line:84% then psi has to be a n-component vector. 00:31:14.320 --> 00:31:23.740 align:middle line:90% 00:31:23.740 --> 00:31:26.210 align:middle line:90% OK. 00:31:26.210 --> 00:31:32.890 align:middle line:84% So even for some people, you come up with this idea, 00:31:32.890 --> 00:31:34.390 align:middle line:84% you will not imagine this will work. 00:31:34.390 --> 00:31:36.860 align:middle line:84% OK, you just say, oh, this will be a mess, 00:31:36.860 --> 00:31:38.400 align:middle line:90% but somehow, he made it work. 00:31:38.400 --> 00:31:41.370 align:middle line:84% OK, so we will see how to make this work. 00:31:41.370 --> 00:31:49.760 align:middle line:84% And so now if you want H to be Hermitian, 00:31:49.760 --> 00:31:51.530 align:middle line:84% and then you can immediately conclude-- 00:31:51.530 --> 00:31:54.530 align:middle line:84% so that's why I put the minus i thing here, 00:31:54.530 --> 00:31:56.225 align:middle line:90% is the alpha and the beta. 00:31:56.225 --> 00:31:58.820 align:middle line:90% 00:31:58.820 --> 00:32:03.320 align:middle line:84% So m will just be some constant, OK, and here is a matrix. 00:32:03.320 --> 00:32:06.350 align:middle line:84% You can always take a constant out, OK, so alpha, 00:32:06.350 --> 00:32:08.345 align:middle line:90% beta are Hermitian matrices. 00:32:08.345 --> 00:32:13.958 align:middle line:90% 00:32:13.958 --> 00:32:15.875 align:middle line:84% They're just some constant Hermitian matrices. 00:32:15.875 --> 00:32:21.630 align:middle line:90% 00:32:21.630 --> 00:32:34.890 align:middle line:84% And then, so then he reasoned that for this equation, 00:32:34.890 --> 00:32:39.290 align:middle line:84% if we want this equation to be Lorentz covariant, 00:32:39.290 --> 00:32:43.520 align:middle line:84% then at least it should have the relativistic plane 00:32:43.520 --> 00:32:46.130 align:middle line:90% wave as its solution. 00:32:46.130 --> 00:32:48.920 align:middle line:84% OK, if you have a covariant equation, and at least 00:32:48.920 --> 00:32:53.570 align:middle line:84% you should have the relativistic plane wave as its solution. 00:32:53.570 --> 00:32:57.380 align:middle line:84% If it does not even have that type of solution, then you, 00:32:57.380 --> 00:32:59.660 align:middle line:84% yeah, of course cannot be covariant. 00:32:59.660 --> 00:33:01.850 align:middle line:90% OK, so the minimal requirement-- 00:33:01.850 --> 00:33:06.010 align:middle line:90% 00:33:06.010 --> 00:33:09.330 align:middle line:84% so before we really try to see how can we make this 00:33:09.330 --> 00:33:12.810 align:middle line:84% into a covariant equation, we say let's consider 00:33:12.810 --> 00:33:14.040 align:middle line:90% minimal requirement. 00:33:14.040 --> 00:33:20.950 align:middle line:90% 00:33:20.950 --> 00:33:24.285 align:middle line:84% So we'd like to-- here, let me call this equation star. 00:33:24.285 --> 00:33:31.030 align:middle line:90% 00:33:31.030 --> 00:33:47.962 align:middle line:84% Star should have plane wave solutions 00:33:47.962 --> 00:34:02.630 align:middle line:84% with the standard relativistic dispersion relation, OK? 00:34:02.630 --> 00:34:13.159 align:middle line:90% 00:34:13.159 --> 00:34:15.560 align:middle line:84% Now, these, you should have p squared equal to minus m 00:34:15.560 --> 00:34:16.850 align:middle line:90% squared. 00:34:16.850 --> 00:34:18.110 align:middle line:90% OK, you have a plane wave. 00:34:18.110 --> 00:34:20.989 align:middle line:84% The plane wave will be labeled by p, 00:34:20.989 --> 00:34:22.460 align:middle line:84% and then you should have p squared 00:34:22.460 --> 00:34:23.510 align:middle line:90% equal to minus m squared. 00:34:23.510 --> 00:34:29.980 align:middle line:84% Then the m will be its mass, OK, its mass. 00:34:29.980 --> 00:34:34.239 align:middle line:84% And to do this, the simplest way to do this, 00:34:34.239 --> 00:34:36.449 align:middle line:84% so we know that the Klein-Gordon equation-- 00:34:36.449 --> 00:34:39.139 align:middle line:90% 00:34:39.139 --> 00:34:43.179 align:middle line:84% so we know the Klein-Gordon equation have such solutions. 00:34:43.179 --> 00:34:46.630 align:middle line:84% OK, Klein-Gordon equation have such solutions. 00:34:46.630 --> 00:34:51.370 align:middle line:84% And then the simplest way to do this is-- 00:34:51.370 --> 00:34:53.380 align:middle line:84% this is the first-order equation. 00:34:53.380 --> 00:34:56.750 align:middle line:84% And now imagine if we square this equation, 00:34:56.750 --> 00:35:00.200 align:middle line:84% and then this should reduce to the Klein-Gordon equation. 00:35:00.200 --> 00:35:03.320 align:middle line:84% If you can make that work, then this property 00:35:03.320 --> 00:35:05.270 align:middle line:90% will be satisfied, OK? 00:35:05.270 --> 00:35:10.190 align:middle line:90% So this will be satisfied-- 00:35:10.190 --> 00:35:17.450 align:middle line:84% can be satisfied if square of star, 00:35:17.450 --> 00:35:28.375 align:middle line:84% OK, satisfies reduced to the Klein-Gordon equation. 00:35:28.375 --> 00:35:31.440 align:middle line:90% 00:35:31.440 --> 00:35:34.830 align:middle line:90% OK, so now let's try to do this. 00:35:34.830 --> 00:35:37.010 align:middle line:84% So when we square this star, we just 00:35:37.010 --> 00:35:40.390 align:middle line:84% act twice, so essentially, you have-- 00:35:40.390 --> 00:36:05.880 align:middle line:90% 00:36:05.880 --> 00:36:09.780 align:middle line:84% so you have-- so when you square that equation, 00:36:09.780 --> 00:36:14.070 align:middle line:84% then you just get partial partial t square psi 00:36:14.070 --> 00:36:16.710 align:middle line:90% equal to H square psi. 00:36:16.710 --> 00:36:22.140 align:middle line:84% OK, and then we try to make this of this Klein-Gordon form. 00:36:22.140 --> 00:36:29.070 align:middle line:84% OK, so the right-hand side, we just have the form minus i 00:36:29.070 --> 00:36:38.820 align:middle line:84% with alpha dot with this, and then plus beta m square psi. 00:36:38.820 --> 00:36:45.410 align:middle line:84% OK, we have these, and then you can just expand this 00:36:45.410 --> 00:36:50.015 align:middle line:84% explicitly, so the right-hand side, so then we have minus-- 00:36:50.015 --> 00:36:53.100 align:middle line:90% 00:36:53.100 --> 00:36:59.130 align:middle line:84% let's square this first-- then you have minus alpha i alpha j. 00:36:59.130 --> 00:37:00.930 align:middle line:90% OK. 00:37:00.930 --> 00:37:07.220 align:middle line:84% And then you have partial partial xi partial partial xj 00:37:07.220 --> 00:37:10.430 align:middle line:90% psi, and then let's square this. 00:37:10.430 --> 00:37:15.530 align:middle line:84% Then you have beta square m square psi, 00:37:15.530 --> 00:37:18.230 align:middle line:90% and then you have cross term. 00:37:18.230 --> 00:37:25.460 align:middle line:84% So cross term now has the form beta alpha i plus alpha i beta. 00:37:25.460 --> 00:37:28.250 align:middle line:84% But now remember, beta and alpha, they are not constant. 00:37:28.250 --> 00:37:31.460 align:middle line:84% They are matrices, OK, so they don't necessarily commute. 00:37:31.460 --> 00:37:34.550 align:middle line:84% So you have to be careful about the orders. 00:37:34.550 --> 00:37:40.730 align:middle line:84% OK, and m partial psi partial xi. 00:37:40.730 --> 00:37:42.560 align:middle line:84% OK, so the right-hand side is just 00:37:42.560 --> 00:37:44.120 align:middle line:90% like this when you square it. 00:37:44.120 --> 00:37:50.030 align:middle line:90% 00:37:50.030 --> 00:37:52.670 align:middle line:84% And now we wanted to look at the Klein-Gordon equation. 00:37:52.670 --> 00:37:55.410 align:middle line:84% The Klein-Gordon equation has the following form. 00:37:55.410 --> 00:37:58.980 align:middle line:90% So now we want it to be-- 00:37:58.980 --> 00:38:07.080 align:middle line:84% OK, hopefully to be given by minus partial x square psi plus 00:38:07.080 --> 00:38:08.280 align:middle line:90% m square psi. 00:38:08.280 --> 00:38:12.060 align:middle line:84% OK, so this one would be the Klein-Gordon equation, OK, 00:38:12.060 --> 00:38:14.420 align:middle line:90% which we wrote before. 00:38:14.420 --> 00:38:20.810 align:middle line:84% OK, so by x-- yeah, here, sorry, I should say xi square. 00:38:20.810 --> 00:38:25.170 align:middle line:90% 00:38:25.170 --> 00:38:26.070 align:middle line:90% Yeah. 00:38:26.070 --> 00:38:28.290 align:middle line:90% OK? 00:38:28.290 --> 00:38:29.905 align:middle line:90% So i should be summed, OK? 00:38:29.905 --> 00:38:35.680 align:middle line:90% 00:38:35.680 --> 00:38:39.970 align:middle line:84% So now we want this equation, so we want the right-hand side 00:38:39.970 --> 00:38:42.440 align:middle line:90% to be equal to this equation. 00:38:42.440 --> 00:38:46.220 align:middle line:84% If that works, then we're guaranteed to have a plane-wave 00:38:46.220 --> 00:38:49.910 align:middle line:84% solution of such a dispersion relation because this equation 00:38:49.910 --> 00:38:51.110 align:middle line:90% has-- 00:38:51.110 --> 00:38:55.258 align:middle line:84% OK, because this equation has the plane-wave equation 00:38:55.258 --> 00:38:57.800 align:middle line:84% with-- yeah, plane-wave solution with that kind of dispersion 00:38:57.800 --> 00:38:59.010 align:middle line:90% relation. 00:38:59.010 --> 00:39:03.290 align:middle line:84% So now we just compare the both sides, so for this 00:39:03.290 --> 00:39:14.650 align:middle line:84% to be true, so we just need to have-- 00:39:14.650 --> 00:39:17.800 align:middle line:84% so let's compare the second-order derivative term. 00:39:17.800 --> 00:39:22.700 align:middle line:84% And then we need alpha i, so we need-- 00:39:22.700 --> 00:39:27.300 align:middle line:90% 00:39:27.300 --> 00:39:33.530 align:middle line:84% first, when i not equal to j, the off-diagonal terms, 00:39:33.530 --> 00:39:36.250 align:middle line:84% they all should vanish, OK, because here there's 00:39:36.250 --> 00:39:37.840 align:middle line:90% only diagonal terms. 00:39:37.840 --> 00:39:46.960 align:middle line:84% So that means that the alpha i alpha j plus alpha j alpha i 00:39:46.960 --> 00:39:48.790 align:middle line:90% should equal to 0. 00:39:48.790 --> 00:39:51.280 align:middle line:84% OK, remember, the matrices, they don't 00:39:51.280 --> 00:39:53.770 align:middle line:84% commute, so we should be careful about the ordering. 00:39:53.770 --> 00:40:04.380 align:middle line:84% OK, so when i not equal to j and when i equal to j, 00:40:04.380 --> 00:40:06.360 align:middle line:84% then you should just reduce to here. 00:40:06.360 --> 00:40:11.950 align:middle line:84% OK, and that means that the alpha i square should 00:40:11.950 --> 00:40:12.940 align:middle line:90% equal to-- 00:40:12.940 --> 00:40:14.770 align:middle line:90% and here you should just-- 00:40:14.770 --> 00:40:16.060 align:middle line:90% here you should just-- 00:40:16.060 --> 00:40:20.200 align:middle line:84% this term just will give that, and the-- 00:40:20.200 --> 00:40:22.600 align:middle line:84% should equal to beta square equal to 1. 00:40:22.600 --> 00:40:24.520 align:middle line:84% OK, so here, there's no summation, OK, 00:40:24.520 --> 00:40:25.840 align:middle line:90% no summation over i. 00:40:25.840 --> 00:40:31.280 align:middle line:90% 00:40:31.280 --> 00:40:37.230 align:middle line:84% OK, so each alpha i square has to be 0, 00:40:37.230 --> 00:40:39.540 align:middle line:84% and now we want this linear term to be 0. 00:40:39.540 --> 00:40:42.330 align:middle line:90% 00:40:42.330 --> 00:40:55.040 align:middle line:84% And then you want beta alpha i plus alpha i beta to be 0, OK? 00:40:55.040 --> 00:40:58.160 align:middle line:84% And also, just let me put it together. 00:40:58.160 --> 00:41:02.360 align:middle line:84% We said the alpha and the beta, they must be Hermitian, 00:41:02.360 --> 00:41:09.380 align:middle line:84% so that means that the alpha i dagger is equal to alpha i 00:41:09.380 --> 00:41:12.590 align:middle line:84% and the beta dagger equal to beta. 00:41:12.590 --> 00:41:17.030 align:middle line:90% 00:41:17.030 --> 00:41:21.900 align:middle line:84% So if we satisfy all these four conditions, 00:41:21.900 --> 00:41:26.660 align:middle line:84% and then we will guarantee that that equation star should 00:41:26.660 --> 00:41:28.070 align:middle line:90% have the plane-wave solution. 00:41:28.070 --> 00:41:32.190 align:middle line:84% OK, that should have the plane-wave solution. 00:41:32.190 --> 00:41:38.960 align:middle line:84% And so now you just try to find matrices, 00:41:38.960 --> 00:41:40.430 align:middle line:90% satisfy those conditions. 00:41:40.430 --> 00:41:42.450 align:middle line:90% OK. 00:41:42.450 --> 00:41:46.065 align:middle line:84% So we already said that the 1-by-1 matrix don't work, 00:41:46.065 --> 00:41:48.760 align:middle line:84% that if they are constant, of course, they don't work. 00:41:48.760 --> 00:41:53.840 align:middle line:84% And so you can also check 2-by-2 matrices. 00:41:53.840 --> 00:41:55.790 align:middle line:90% It's not enough to do this. 00:41:55.790 --> 00:41:57.530 align:middle line:90% 3-by-3 is not enough. 00:41:57.530 --> 00:42:00.830 align:middle line:84% When you go to 4-by-4, then you finally find the solutions. 00:42:00.830 --> 00:42:03.300 align:middle line:84% OK, you actually find the infinite number of solutions. 00:42:03.300 --> 00:42:15.920 align:middle line:84% So you see that to satisfy them needs at least a 4-by-4 matrix, 00:42:15.920 --> 00:42:17.630 align:middle line:90% so n has to be 4. 00:42:17.630 --> 00:42:20.090 align:middle line:90% OK, so this n has to be 4. 00:42:20.090 --> 00:42:23.160 align:middle line:90% 00:42:23.160 --> 00:42:28.130 align:middle line:84% And so let me give you some possible solutions. 00:42:28.130 --> 00:42:31.420 align:middle line:84% So for example, here is one solution. 00:42:31.420 --> 00:42:38.470 align:middle line:84% Say you take beta to be 1, so all matrices here 00:42:38.470 --> 00:42:40.730 align:middle line:84% should be understood as 2-by-2 matrices, 00:42:40.730 --> 00:42:47.560 align:middle line:84% OK, so all together, it's 4-by-4 matrix, so 1 and minus 2. 00:42:47.560 --> 00:42:49.480 align:middle line:84% So beta's given by this, so this is 00:42:49.480 --> 00:42:58.000 align:middle line:84% one solution, and alpha i given by 0, sigma i, sigma i, 0. 00:42:58.000 --> 00:43:05.930 align:middle line:84% And sigma are Pauli matrices, OK, sigma i, Pauli matrices. 00:43:05.930 --> 00:43:09.250 align:middle line:90% 00:43:09.250 --> 00:43:12.890 align:middle line:84% OK, so this is one solution you can check yourself. 00:43:12.890 --> 00:43:14.410 align:middle line:90% OK, I will not do it here. 00:43:14.410 --> 00:43:18.610 align:middle line:84% And this actually is a solution that 00:43:18.610 --> 00:43:22.700 align:middle line:84% satisfies all these four conditions, 00:43:22.700 --> 00:43:26.170 align:middle line:90% and here is another solution. 00:43:26.170 --> 00:43:30.180 align:middle line:84% So now let me just save time in not writing down 00:43:30.180 --> 00:43:31.890 align:middle line:90% this two-by-two. 00:43:31.890 --> 00:43:35.970 align:middle line:84% So for example, beta can be 0, 1, 1, 0. 00:43:35.970 --> 00:43:38.325 align:middle line:84% OK, this one here is a two-by-two matrix, 00:43:38.325 --> 00:43:48.210 align:middle line:84% OK, and alpha i equal to sigma i, 0, 0, minus sigma i. 00:43:48.210 --> 00:43:50.490 align:middle line:90% OK. 00:43:50.490 --> 00:43:54.390 align:middle line:84% So you can check both of them satisfy those conditions. 00:43:54.390 --> 00:43:55.380 align:middle line:90% Yes? 00:43:55.380 --> 00:43:58.710 align:middle line:84% AUDIENCE: Sorry, so when you have alpha times grad psi, 00:43:58.710 --> 00:44:01.785 align:middle line:84% is it-- do you act the grad on each element of psi 00:44:01.785 --> 00:44:02.910 align:middle line:90% and then multiply by alpha? 00:44:02.910 --> 00:44:06.030 align:middle line:84% Or do you-- like, what's the order of operations? 00:44:06.030 --> 00:44:09.720 align:middle line:84% HONG LIU: Oh, it doesn't matter because alpha is just 00:44:09.720 --> 00:44:12.144 align:middle line:90% some constant, right? 00:44:12.144 --> 00:44:16.810 align:middle line:84% Alpha and the grad, they don't act on the same space. 00:44:16.810 --> 00:44:19.290 align:middle line:84% So that derivative just acts on derivatives, 00:44:19.290 --> 00:44:24.230 align:middle line:84% and alpha acts on a different component of psi. 00:44:24.230 --> 00:44:26.230 align:middle line:84% AUDIENCE: I see, but if you took the grad of psi 00:44:26.230 --> 00:44:28.360 align:middle line:84% and then multiplied by alpha, then you'd 00:44:28.360 --> 00:44:30.285 align:middle line:90% mix up the components of it. 00:44:30.285 --> 00:44:30.910 align:middle line:90% HONG LIU: Yeah. 00:44:30.910 --> 00:44:31.700 align:middle line:90% Yeah, it's fine. 00:44:31.700 --> 00:44:32.200 align:middle line:90% Yeah. 00:44:32.200 --> 00:44:34.690 align:middle line:84% Yeah, you can do it either before-- yeah, they commute, 00:44:34.690 --> 00:44:37.715 align:middle line:84% so the operation of the alpha and the operation of grad, they 00:44:37.715 --> 00:44:38.215 align:middle line:90% commute. 00:44:38.215 --> 00:44:38.715 align:middle line:90% Yeah. 00:44:38.715 --> 00:44:41.180 align:middle line:90% 00:44:41.180 --> 00:44:42.165 align:middle line:90% Yeah? 00:44:42.165 --> 00:44:42.790 align:middle line:90% AUDIENCE: Yeah. 00:44:42.790 --> 00:44:47.710 align:middle line:84% So you'd have a-- so alpha x because alpha x would just 00:44:47.710 --> 00:44:52.980 align:middle line:84% be the Pauli x matrices times the grad x, 00:44:52.980 --> 00:44:54.980 align:middle line:84% and then you'd need another one for y and for z? 00:44:54.980 --> 00:44:55.605 align:middle line:90% HONG LIU: Yeah. 00:44:55.605 --> 00:44:56.180 align:middle line:90% Yeah. 00:44:56.180 --> 00:44:57.110 align:middle line:90% Yeah. 00:44:57.110 --> 00:45:01.310 align:middle line:84% So yeah, so I urge you-- so here I don't have time to write 00:45:01.310 --> 00:45:03.720 align:middle line:90% everything very explicitly. 00:45:03.720 --> 00:45:06.710 align:middle line:84% So here you just write it as alpha 00:45:06.710 --> 00:45:11.990 align:middle line:84% x partial x plus alpha y partial y plus alpha z partial z, 00:45:11.990 --> 00:45:13.820 align:middle line:90% and each alpha is a matrix. 00:45:13.820 --> 00:45:18.950 align:middle line:84% And alpha acts on different components of psi, 00:45:18.950 --> 00:45:21.200 align:middle line:84% and this just acts on-- the derivative acts 00:45:21.200 --> 00:45:23.492 align:middle line:90% on all components of psi. 00:45:23.492 --> 00:45:25.200 align:middle line:90% Yeah. 00:45:25.200 --> 00:45:27.110 align:middle line:90% OK? 00:45:27.110 --> 00:45:28.280 align:middle line:90% Good? 00:45:28.280 --> 00:45:34.540 align:middle line:84% So now, good, you say we have an equation, OK, so far, so psi-- 00:45:34.540 --> 00:45:51.310 align:middle line:90% 00:45:51.310 --> 00:45:54.360 align:middle line:90% so alpha is a 4-by-4 matrix. 00:45:54.360 --> 00:45:58.350 align:middle line:84% Yeah, alpha and beta are 4-by-4 matrices, 00:45:58.350 --> 00:46:02.600 align:middle line:84% so that means that psi should be a four-component vector. 00:46:02.600 --> 00:46:11.430 align:middle line:90% 00:46:11.430 --> 00:46:16.680 align:middle line:84% OK, it's a four-component vector, OK, so we will take-- 00:46:16.680 --> 00:46:19.740 align:middle line:84% so we will denote it as psi alpha. 00:46:19.740 --> 00:46:27.962 align:middle line:84% So alpha equal to 1, 2, 3, 4, OK, and we-- 00:46:27.962 --> 00:46:30.420 align:middle line:84% for the moment, let's just take the most general situation. 00:46:30.420 --> 00:46:35.760 align:middle line:84% We'll take alpha to be complex, OK, each of them to be complex. 00:46:35.760 --> 00:46:38.550 align:middle line:90% OK. 00:46:38.550 --> 00:46:39.620 align:middle line:90% Yes? 00:46:39.620 --> 00:46:43.365 align:middle line:84% AUDIENCE: Is alpha, like, a vector of matrices? 00:46:43.365 --> 00:46:43.990 align:middle line:90% HONG LIU: Yeah. 00:46:43.990 --> 00:46:46.545 align:middle line:84% AUDIENCE: Which element of alpha is a matrix? 00:46:46.545 --> 00:46:47.670 align:middle line:90% HONG LIU: What do you mean? 00:46:47.670 --> 00:46:49.337 align:middle line:84% AUDIENCE: So, like, alpha i is a matrix. 00:46:49.337 --> 00:46:52.650 align:middle line:84% HONG LIU: Yeah, alpha 1, alpha 2, alpha 3, they are all-- 00:46:52.650 --> 00:46:54.000 align:middle line:90% they are three matrices. 00:46:54.000 --> 00:47:00.700 align:middle line:84% AUDIENCE: And you said we'd need the first on the list. 00:47:00.700 --> 00:47:03.310 align:middle line:84% Wouldn't we want each alpha i to be diagonal? 00:47:03.310 --> 00:47:06.640 align:middle line:84% Is that what you're saying, that each matrix in alpha 00:47:06.640 --> 00:47:07.807 align:middle line:90% should be a diagonal matrix? 00:47:07.807 --> 00:47:08.348 align:middle line:90% HONG LIU: No. 00:47:08.348 --> 00:47:09.490 align:middle line:90% No, no, no, no, no, no, no. 00:47:09.490 --> 00:47:14.180 align:middle line:84% No, alpha, no, we don't know the form of the alpha, right? 00:47:14.180 --> 00:47:17.480 align:middle line:84% No, alpha are just matrices, so this 00:47:17.480 --> 00:47:20.840 align:middle line:84% means that alpha 1 alpha 2 plus alpha 2 alpha 1, 00:47:20.840 --> 00:47:24.350 align:middle line:84% as a matrix product, should give you zero. 00:47:24.350 --> 00:47:26.140 align:middle line:90% Yeah. 00:47:26.140 --> 00:47:27.565 align:middle line:90% Yeah, alpha itself is a matrix. 00:47:27.565 --> 00:47:33.870 align:middle line:90% 00:47:33.870 --> 00:47:35.030 align:middle line:90% AUDIENCE: Of matrices. 00:47:35.030 --> 00:47:35.572 align:middle line:90% HONG LIU: Hm? 00:47:35.572 --> 00:47:36.893 align:middle line:90% AUDIENCE: Of matrices. 00:47:36.893 --> 00:47:37.560 align:middle line:90% HONG LIU: Sorry? 00:47:37.560 --> 00:47:38.910 align:middle line:90% AUDIENCE: Each alpha i-- 00:47:38.910 --> 00:47:41.700 align:middle line:84% each component of alpha is a matrix. 00:47:41.700 --> 00:47:42.200 align:middle line:90% Right? 00:47:42.200 --> 00:47:42.825 align:middle line:90% HONG LIU: Yeah. 00:47:42.825 --> 00:47:43.610 align:middle line:90% Yeah, yeah, yeah. 00:47:43.610 --> 00:47:45.115 align:middle line:84% Yeah, just, alpha have three components. 00:47:45.115 --> 00:47:45.782 align:middle line:90% AUDIENCE: Right. 00:47:45.782 --> 00:47:48.365 align:middle line:84% HONG LIU: And each component is a matrix. 00:47:48.365 --> 00:47:49.490 align:middle line:90% Each component is a matrix. 00:47:49.490 --> 00:47:51.710 align:middle line:84% Just, if you have alpha 1, alpha 2, alpha 3, 00:47:51.710 --> 00:47:55.100 align:middle line:84% so here, here you see explicitly, alpha 1 00:47:55.100 --> 00:48:00.140 align:middle line:84% is equal to sigma 1, sigma 1, alpha 2 is sigma 2, sigma 2, 00:48:00.140 --> 00:48:02.910 align:middle line:90% et cetera. 00:48:02.910 --> 00:48:05.090 align:middle line:90% Other questions? 00:48:05.090 --> 00:48:08.230 align:middle line:84% OK, yeah, at the beginning, this may a little bit more-- 00:48:08.230 --> 00:48:12.580 align:middle line:84% not very intuitive, OK, but if you just work through it, 00:48:12.580 --> 00:48:14.407 align:middle line:84% then you will get a feeling about it, OK, 00:48:14.407 --> 00:48:15.740 align:middle line:90% you will get a feeling about it. 00:48:15.740 --> 00:48:17.800 align:middle line:84% So that's why I say this was really genius, 00:48:17.800 --> 00:48:20.560 align:middle line:84% because just nobody could have thought of this. 00:48:20.560 --> 00:48:24.370 align:middle line:90% OK, it just came from nowhere. 00:48:24.370 --> 00:48:25.960 align:middle line:90% Really, there was no clue, OK? 00:48:25.960 --> 00:48:30.190 align:middle line:84% There was no clue of such a structure. 00:48:30.190 --> 00:48:31.300 align:middle line:90% Yeah. 00:48:31.300 --> 00:48:38.200 align:middle line:84% OK, so this is a new object, so we call it spinor. 00:48:38.200 --> 00:48:40.870 align:middle line:84% OK, we call it spinor because it-- 00:48:40.870 --> 00:48:44.350 align:middle line:84% later we will see that this describes spin-half particles, 00:48:44.350 --> 00:48:45.830 align:middle line:90% so that's why we call it spinor. 00:48:45.830 --> 00:48:46.330 align:middle line:90% OK. 00:48:46.330 --> 00:48:51.290 align:middle line:90% 00:48:51.290 --> 00:48:52.600 align:middle line:90% Good. 00:48:52.600 --> 00:48:53.380 align:middle line:90% AUDIENCE: Dr. Liu? 00:48:53.380 --> 00:48:54.040 align:middle line:90% HONG LIU: Yeah? 00:48:54.040 --> 00:48:57.040 align:middle line:84% AUDIENCE: So if we take n to be larger, 00:48:57.040 --> 00:48:59.420 align:middle line:90% then we describe all the spins? 00:48:59.420 --> 00:49:00.235 align:middle line:90% HONG LIU: Hm? 00:49:00.235 --> 00:49:03.860 align:middle line:84% AUDIENCE: If we take a n by n matrix where n is not so small, 00:49:03.860 --> 00:49:05.080 align:middle line:90% then we can use that --? 00:49:05.080 --> 00:49:06.970 align:middle line:90% HONG LIU: No, you get-- 00:49:06.970 --> 00:49:10.720 align:middle line:84% just we are not using those matrices in the efficient way. 00:49:10.720 --> 00:49:12.250 align:middle line:90% Yeah. 00:49:12.250 --> 00:49:14.045 align:middle line:84% Yeah, this become-- you can reduce always-- 00:49:14.045 --> 00:49:15.420 align:middle line:84% yeah, just from physical purpose, 00:49:15.420 --> 00:49:17.840 align:middle line:84% you can always reduce it to 4, yeah. 00:49:17.840 --> 00:49:18.420 align:middle line:90% Yes? 00:49:18.420 --> 00:49:20.420 align:middle line:84% AUDIENCE: Well, what if I wanted a wave equation 00:49:20.420 --> 00:49:22.567 align:middle line:90% for higher spin? 00:49:22.567 --> 00:49:23.650 align:middle line:90% HONG LIU: For higher spin? 00:49:23.650 --> 00:49:25.210 align:middle line:90% AUDIENCE: Yeah, like 3/2 or 5/2. 00:49:25.210 --> 00:49:25.835 align:middle line:90% HONG LIU: Yeah. 00:49:25.835 --> 00:49:28.470 align:middle line:84% Yeah, if you know how to do the two halves, 00:49:28.470 --> 00:49:29.470 align:middle line:90% then you can generalize. 00:49:29.470 --> 00:49:30.010 align:middle line:90% Yeah. 00:49:30.010 --> 00:49:32.860 align:middle line:84% Yeah, so one half, essentially, you 00:49:32.860 --> 00:49:36.150 align:middle line:84% can-- yeah, based on one half, you can generalize it. 00:49:36.150 --> 00:49:38.110 align:middle line:90% Yeah. 00:49:38.110 --> 00:49:39.110 align:middle line:90% AUDIENCE: Thank you. 00:49:39.110 --> 00:49:39.860 align:middle line:90% HONG LIU: Yes? 00:49:39.860 --> 00:49:42.590 align:middle line:84% AUDIENCE: Just to clarify, so alpha i, 00:49:42.590 --> 00:49:46.490 align:middle line:84% is it a 2-by-2 matrix of 2-by-2 matrices? 00:49:46.490 --> 00:49:48.492 align:middle line:84% Or is it just a 4-by-4 matrix and that's just 00:49:48.492 --> 00:49:49.700 align:middle line:90% a convenient way to write it? 00:49:49.700 --> 00:49:51.433 align:middle line:84% HONG LIU: Yeah, this is a 4-by-4 matrix. 00:49:51.433 --> 00:49:53.850 align:middle line:84% It's just a convenient way to write these 4-by-4 matrices. 00:49:53.850 --> 00:49:54.160 align:middle line:90% AUDIENCE: Sure. 00:49:54.160 --> 00:49:54.660 align:middle line:90% OK. 00:49:54.660 --> 00:49:58.790 align:middle line:84% HONG LIU: So that I don't have to write all four components. 00:49:58.790 --> 00:49:59.520 align:middle line:90% I just-- yeah. 00:49:59.520 --> 00:49:59.810 align:middle line:90% AUDIENCE: Yeah, yeah, yeah. 00:49:59.810 --> 00:50:00.310 align:middle line:90% OK. 00:50:00.310 --> 00:50:02.750 align:middle line:84% HONG LIU: So I divided these 4-by-4 matrices 00:50:02.750 --> 00:50:06.338 align:middle line:84% into four 2-by-2 blocks, and then I specify each block. 00:50:06.338 --> 00:50:07.880 align:middle line:84% AUDIENCE: Yeah, it's just blocks, not 00:50:07.880 --> 00:50:08.945 align:middle line:90% matrices in the matrix. 00:50:08.945 --> 00:50:09.820 align:middle line:90% HONG LIU: Yeah, yeah. 00:50:09.820 --> 00:50:13.490 align:middle line:84% No, no, just the blocks of that 4-by-4 matrix, yeah, 00:50:13.490 --> 00:50:18.080 align:middle line:84% separate the single 4-by-4 matrix into four 2-by-2 blocks. 00:50:18.080 --> 00:50:19.420 align:middle line:90% AUDIENCE: Yeah. 00:50:19.420 --> 00:50:21.520 align:middle line:90% HONG LIU: OK? 00:50:21.520 --> 00:50:22.020 align:middle line:90% Good? 00:50:22.020 --> 00:50:26.380 align:middle line:90% 00:50:26.380 --> 00:50:30.100 align:middle line:84% OK, so for later convenience, let's introduce 00:50:30.100 --> 00:50:32.290 align:middle line:90% a slightly different notation. 00:50:32.290 --> 00:50:35.960 align:middle line:84% OK, so now we have this equation, 00:50:35.960 --> 00:50:37.615 align:middle line:90% so now we have the form of this. 00:50:37.615 --> 00:50:40.470 align:middle line:90% 00:50:40.470 --> 00:50:44.750 align:middle line:84% So now we have the form of this partial t 00:50:44.750 --> 00:50:56.840 align:middle line:84% psi equal to minus alpha plus beta m psi, so now 00:50:56.840 --> 00:51:01.040 align:middle line:84% let's multiply both sides by a factor-- 00:51:01.040 --> 00:51:01.720 align:middle line:90% by beta. 00:51:01.720 --> 00:51:06.350 align:middle line:84% OK, so beta is a matrix, OK, so this is a matrix equation. 00:51:06.350 --> 00:51:08.510 align:middle line:84% OK, so this is a matrix equation. 00:51:08.510 --> 00:51:13.280 align:middle line:84% Let's multiply both sides by beta, OK, 00:51:13.280 --> 00:51:22.010 align:middle line:84% and then we get i beta partial t psi equal to minus i beta times 00:51:22.010 --> 00:51:22.510 align:middle line:90% alpha. 00:51:22.510 --> 00:51:27.500 align:middle line:90% 00:51:27.500 --> 00:51:31.930 align:middle line:84% Yeah, and let me just write it maybe this way, 00:51:31.930 --> 00:51:42.120 align:middle line:84% so alpha i and partial xi psi, OK, then plus m psi. 00:51:42.120 --> 00:51:44.880 align:middle line:84% OK, so you'd get an equation like this. 00:51:44.880 --> 00:51:47.400 align:middle line:90% 00:51:47.400 --> 00:51:48.850 align:middle line:90% OK. 00:51:48.850 --> 00:51:54.470 align:middle line:90% So now I will denote-- 00:51:54.470 --> 00:51:59.970 align:middle line:84% introduce a new notation so that it looks nicer. 00:51:59.970 --> 00:52:02.610 align:middle line:84% I'll denote the gamma 0 equal to i 00:52:02.610 --> 00:52:13.870 align:middle line:84% beta and then the gamma i equal to i times beta alpha i. 00:52:13.870 --> 00:52:17.950 align:middle line:84% OK, and then let's all pull it to the same side, 00:52:17.950 --> 00:52:21.020 align:middle line:84% and then this becomes the following equation, 00:52:21.020 --> 00:52:23.810 align:middle line:84% then the equation has the following form, gamma mu 00:52:23.810 --> 00:52:28.710 align:middle line:84% partial mu minus m psi equal to 0. 00:52:28.710 --> 00:52:31.940 align:middle line:84% OK, so this is the form of our Dirac equation. 00:52:31.940 --> 00:52:37.850 align:middle line:90% 00:52:37.850 --> 00:52:43.690 align:middle line:84% OK, so this becomes gamma 0 times partial t, 00:52:43.690 --> 00:52:47.480 align:middle line:84% so this becomes gamma i times partial xi. 00:52:47.480 --> 00:52:51.160 align:middle line:84% When they come together, then become gamma mu partial mu, 00:52:51.160 --> 00:52:54.100 align:middle line:84% and then m will move to the other side. 00:52:54.100 --> 00:52:55.780 align:middle line:90% OK. 00:52:55.780 --> 00:53:01.150 align:middle line:84% So due to different conventions of the Minkowski 00:53:01.150 --> 00:53:04.780 align:middle line:84% metric signature you use, et cetera, so different books 00:53:04.780 --> 00:53:08.990 align:middle line:84% or different places, you will see i in different places. 00:53:08.990 --> 00:53:10.780 align:middle line:90% Some equations have i-- 00:53:10.780 --> 00:53:12.430 align:middle line:90% yeah, some books have i here. 00:53:12.430 --> 00:53:14.710 align:middle line:90% Some books have i here. 00:53:14.710 --> 00:53:16.855 align:middle line:90% In my version, there's no i, OK? 00:53:16.855 --> 00:53:17.410 align:middle line:90% [LAUGHTER] 00:53:17.410 --> 00:53:22.330 align:middle line:84% And so I know this is annoying, but yeah, 00:53:22.330 --> 00:53:24.530 align:middle line:90% but this is just a fact of life. 00:53:24.530 --> 00:53:25.030 align:middle line:90% Yeah. 00:53:25.030 --> 00:53:27.160 align:middle line:84% Yeah, so I find that this version 00:53:27.160 --> 00:53:30.430 align:middle line:84% is most simplest in terms of notation 00:53:30.430 --> 00:53:33.340 align:middle line:84% anyway, so that's the convention we use. 00:53:33.340 --> 00:53:36.810 align:middle line:90% 00:53:36.810 --> 00:53:39.010 align:middle line:90% Good. 00:53:39.010 --> 00:53:41.510 align:middle line:90% Good. 00:53:41.510 --> 00:53:44.360 align:middle line:84% So now, again, this is a matrix equation. 00:53:44.360 --> 00:53:47.390 align:middle line:84% Now let me just write it in the component form, 00:53:47.390 --> 00:53:50.390 align:middle line:84% OK, so this is in the component form. 00:53:50.390 --> 00:53:54.180 align:middle line:90% I have gamma mu, which-- 00:53:54.180 --> 00:53:58.080 align:middle line:84% each of them is a matrix, alpha, beta, so alpha and beta, 00:53:58.080 --> 00:53:58.980 align:middle line:90% they're always run. 00:53:58.980 --> 00:54:01.320 align:middle line:90% Yeah, sorry for this. 00:54:01.320 --> 00:54:04.560 align:middle line:84% Now the alpha, beta just means the-- 00:54:04.560 --> 00:54:09.910 align:middle line:84% yeah, means the indices, OK, not the matrices. 00:54:09.910 --> 00:54:11.970 align:middle line:90% So alpha and the beta-- 00:54:11.970 --> 00:54:15.420 align:middle line:84% actually, convention is to write this like that. 00:54:15.420 --> 00:54:16.980 align:middle line:90% OK, it doesn't matter. 00:54:16.980 --> 00:54:25.400 align:middle line:84% So partial mu psi beta minus m psi alpha equal to 0, OK, 00:54:25.400 --> 00:54:28.160 align:middle line:90% so this is a matrix equation. 00:54:28.160 --> 00:54:30.150 align:middle line:84% There are all together four equations, 00:54:30.150 --> 00:54:35.180 align:middle line:84% so the beta is summed because beta is repeated. 00:54:35.180 --> 00:54:40.460 align:middle line:84% So beta is summed, and then, yeah, 00:54:40.460 --> 00:54:42.770 align:middle line:90% and the mu is also summed. 00:54:42.770 --> 00:54:44.810 align:middle line:84% So this is a little bit intricate equation. 00:54:44.810 --> 00:54:46.768 align:middle line:84% OK, so this is a little bit intricate equation, 00:54:46.768 --> 00:54:51.280 align:middle line:84% but once you get used to it, it's not that difficult. Yes? 00:54:51.280 --> 00:54:53.280 align:middle line:84% AUDIENCE: There's no meaning to the upstairsness 00:54:53.280 --> 00:54:56.553 align:middle line:84% or downstairsness of alpha and beta, right? 00:54:56.553 --> 00:54:58.470 align:middle line:84% HONG LIU: There's a meaning of upstairs, yeah, 00:54:58.470 --> 00:55:03.790 align:middle line:84% because these two index are not symmetric, 00:55:03.790 --> 00:55:07.350 align:middle line:84% so it's easier to put one upstairs, one downstairs. 00:55:07.350 --> 00:55:11.520 align:middle line:84% Yeah, these two indices are not symmetric. 00:55:11.520 --> 00:55:12.020 align:middle line:90% Yeah. 00:55:12.020 --> 00:55:12.520 align:middle line:90% OK. 00:55:12.520 --> 00:55:17.360 align:middle line:84% So yeah, it takes a little bit time to get used to it, OK, 00:55:17.360 --> 00:55:22.250 align:middle line:84% and I know some people develop psychological fears 00:55:22.250 --> 00:55:26.510 align:middle line:84% for fermions because you have to deal with those gamma matrices, 00:55:26.510 --> 00:55:28.850 align:middle line:90% OK? 00:55:28.850 --> 00:55:31.280 align:middle line:84% For a long while, actually, I have this psychological fear 00:55:31.280 --> 00:55:31.780 align:middle line:90% myself. 00:55:31.780 --> 00:55:32.850 align:middle line:90% [LAUGHTER] 00:55:32.850 --> 00:55:35.780 align:middle line:84% When I look at fermions, I want to be away from it 00:55:35.780 --> 00:55:38.330 align:middle line:84% because I don't want to deal with those gamma matrices, 00:55:38.330 --> 00:55:43.370 align:middle line:84% but these are beautiful objects if you get used to them. 00:55:43.370 --> 00:55:46.220 align:middle line:90% 00:55:46.220 --> 00:55:50.420 align:middle line:84% OK, so now those conditions, you can also 00:55:50.420 --> 00:55:54.200 align:middle line:84% write them in the compact way in terms of the gamma-- 00:55:54.200 --> 00:55:56.150 align:middle line:90% in terms of gamma matrices. 00:55:56.150 --> 00:56:02.960 align:middle line:84% OK, so 1, 2, 3 now can be written as-- 00:56:02.960 --> 00:56:07.010 align:middle line:84% in terms of gamma matrices gamma mu gamma 00:56:07.010 --> 00:56:10.620 align:middle line:90% nu plus gamma nu gamma mu. 00:56:10.620 --> 00:56:18.980 align:middle line:84% So mu nu is always from 0 to 3, OK, equal to 0, or mu not equal 00:56:18.980 --> 00:56:25.180 align:middle line:84% to nu, and then gamma 0 squared is equal to minus 1. 00:56:25.180 --> 00:56:28.150 align:middle line:84% It's because this gamma 0 is i beta, 00:56:28.150 --> 00:56:30.100 align:middle line:84% and the beta squared is equal to 1. 00:56:30.100 --> 00:56:34.630 align:middle line:84% And then the gamma i square, so there's no summation over i 00:56:34.630 --> 00:56:38.480 align:middle line:84% here, OK, because each matrix gamma i square equal to 1. 00:56:38.480 --> 00:56:38.980 align:middle line:90% OK. 00:56:38.980 --> 00:56:41.690 align:middle line:90% 00:56:41.690 --> 00:56:47.107 align:middle line:84% And you can write this further in a more compact form. 00:56:47.107 --> 00:56:49.190 align:middle line:84% You can write this further in a more compact form. 00:56:49.190 --> 00:57:09.380 align:middle line:90% 00:57:09.380 --> 00:57:14.050 align:middle line:84% So you can write this more compactly as gamma mu 00:57:14.050 --> 00:57:19.450 align:middle line:84% gamma nu anticommutator equal to 2 eta mu nu. 00:57:19.450 --> 00:57:22.180 align:middle line:84% So anti-commutator just means that if you 00:57:22.180 --> 00:57:28.990 align:middle line:84% have objects, two objects, curly bracket means a b plus b a, OK? 00:57:28.990 --> 00:57:31.615 align:middle line:90% So this is the key equation. 00:57:31.615 --> 00:57:34.300 align:middle line:90% 00:57:34.300 --> 00:57:36.580 align:middle line:84% OK, so all the gamma matrices, so you 00:57:36.580 --> 00:57:42.850 align:middle line:84% can check yourself, OK, all these are given by just this. 00:57:42.850 --> 00:57:46.280 align:middle line:84% You can easily see, when mu not equal to nu, of course, 00:57:46.280 --> 00:57:48.430 align:middle line:84% the right-hand side is 0, so that just 00:57:48.430 --> 00:57:50.320 align:middle line:90% is equal to that equation. 00:57:50.320 --> 00:57:54.940 align:middle line:84% When mu equal to nu, when there is-- when they're both 0, 00:57:54.940 --> 00:57:56.680 align:middle line:90% then this gives you minus 1. 00:57:56.680 --> 00:57:59.080 align:middle line:84% That's corresponding to the gamma 0 case, 00:57:59.080 --> 00:58:03.081 align:middle line:84% and then when they're ij, then corresponding to gamma x. 00:58:03.081 --> 00:58:03.581 align:middle line:90% OK. 00:58:03.581 --> 00:58:06.738 align:middle line:90% 00:58:06.738 --> 00:58:09.920 align:middle line:84% And so this equation, when the later mathematicians, 00:58:09.920 --> 00:58:13.680 align:middle line:84% of course, studied this, a mathematician 00:58:13.680 --> 00:58:15.930 align:middle line:84% would say this is a beautiful object. 00:58:15.930 --> 00:58:18.060 align:middle line:84% And then they studied this, so now it's 00:58:18.060 --> 00:58:19.065 align:middle line:90% called Clifford algebra. 00:58:19.065 --> 00:58:21.600 align:middle line:90% 00:58:21.600 --> 00:58:24.135 align:middle line:84% So this object is called Clifford algebra. 00:58:24.135 --> 00:58:27.645 align:middle line:90% 00:58:27.645 --> 00:58:29.970 align:middle line:90% OK. 00:58:29.970 --> 00:58:38.370 align:middle line:84% And so then any sets of gamma mu-- 00:58:38.370 --> 00:58:45.540 align:middle line:84% so gamma mu are a set of four matrices satisfying this 00:58:45.540 --> 00:58:47.920 align:middle line:90% equation-- 00:58:47.920 --> 00:58:51.280 align:middle line:90% OK, so let me call this star-- 00:58:51.280 --> 00:58:53.800 align:middle line:90% 00:58:53.800 --> 00:59:07.340 align:middle line:84% is called a representation of the algebra, 00:59:07.340 --> 00:59:10.820 align:middle line:90% of the Clifford algebra. 00:59:10.820 --> 00:59:11.320 align:middle line:90% OK. 00:59:11.320 --> 00:59:17.970 align:middle line:90% 00:59:17.970 --> 00:59:20.440 align:middle line:90% OK. 00:59:20.440 --> 00:59:30.720 align:middle line:84% So from these two solutions of alpha and beta, 00:59:30.720 --> 00:59:32.940 align:middle line:84% we can easily-- from here, we can work out 00:59:32.940 --> 00:59:40.740 align:middle line:84% what is the gamma nu and the gamma i, so yeah, 00:59:40.740 --> 00:59:44.400 align:middle line:90% so here are two examples. 00:59:44.400 --> 00:59:46.920 align:middle line:84% Also, yeah, before-- talking about examples, also, 00:59:46.920 --> 00:59:59.580 align:middle line:84% from number 4, you also find that the gamma 0 00:59:59.580 --> 01:00:03.810 align:middle line:84% dagger equal to minus gamma 0, and gamma-- 01:00:03.810 --> 01:00:09.180 align:middle line:84% so gamma 0 is anti-Hermitian and the gamma i is Hermitian. 01:00:09.180 --> 01:00:10.350 align:middle line:90% OK. 01:00:10.350 --> 01:00:15.000 align:middle line:84% And you can also write this together in the more-- 01:00:15.000 --> 01:00:21.690 align:middle line:84% or equivalently write it as gamma mu dagger gamma 01:00:21.690 --> 01:00:24.120 align:middle line:90% 0 gamma mu gamma 0. 01:00:24.120 --> 01:00:28.330 align:middle line:84% OK, so you can check that this equation is the same 01:00:28.330 --> 01:00:29.490 align:middle line:90% as these two equations. 01:00:29.490 --> 01:00:32.245 align:middle line:90% OK. 01:00:32.245 --> 01:00:32.745 align:middle line:90% Yes? 01:00:32.745 --> 01:00:34.530 align:middle line:84% AUDIENCE: Do these representations 01:00:34.530 --> 01:00:38.520 align:middle line:84% generate a certain transformation? 01:00:38.520 --> 01:00:40.640 align:middle line:90% HONG LIU: Yeah, yeah. 01:00:40.640 --> 01:00:43.140 align:middle line:84% I will talk about things related to this a little bit later. 01:00:43.140 --> 01:00:43.640 align:middle line:90% Yeah. 01:00:43.640 --> 01:00:47.880 align:middle line:90% 01:00:47.880 --> 01:00:48.870 align:middle line:90% Good? 01:00:48.870 --> 01:00:51.390 align:middle line:84% And then we can write down explicit solutions 01:00:51.390 --> 01:00:54.360 align:middle line:84% for those gammas, so that's two representations. 01:00:54.360 --> 01:00:57.060 align:middle line:84% From those solutions of beta and alpha, 01:00:57.060 --> 01:01:01.030 align:middle line:84% we can write down different solutions of gamma. 01:01:01.030 --> 01:01:08.650 align:middle line:84% So for example, for 1, you're corresponding to gamma 0 equal 01:01:08.650 --> 01:01:10.330 align:middle line:90% to minus i. 01:01:10.330 --> 01:01:16.420 align:middle line:84% So now this is minus i times a 2-by-2 matrix, OK, or i, 0, 0, 01:01:16.420 --> 01:01:24.660 align:middle line:84% minus i, OK, and the gamma i is equal to 0, minus sigma i, 01:01:24.660 --> 01:01:29.250 align:middle line:90% minus i sigma i, 0. 01:01:29.250 --> 01:01:31.170 align:middle line:84% And the second solution there corresponding 01:01:31.170 --> 01:01:42.040 align:middle line:84% to gamma 0 equal to 0, i, i, 0, and then gamma i equal to 0-- 01:01:42.040 --> 01:01:45.200 align:middle line:90% 01:01:45.200 --> 01:01:52.170 align:middle line:84% I think it's also minus i sigma i, i sigma i, 0. 01:01:52.170 --> 01:01:52.670 align:middle line:90% OK. 01:01:52.670 --> 01:01:56.880 align:middle line:90% 01:01:56.880 --> 01:01:59.940 align:middle line:84% Good, so these are just-- again, these are just good-- 01:01:59.940 --> 01:02:02.070 align:middle line:84% so these are two different representations 01:02:02.070 --> 01:02:03.316 align:middle line:90% of this algebra. 01:02:03.316 --> 01:02:03.816 align:middle line:90% OK. 01:02:03.816 --> 01:02:06.990 align:middle line:90% 01:02:06.990 --> 01:02:08.940 align:middle line:90% So now let me make some remarks. 01:02:08.940 --> 01:02:11.530 align:middle line:84% So before I proceed further, do you have questions? 01:02:11.530 --> 01:02:13.290 align:middle line:90% Yes? 01:02:13.290 --> 01:02:16.950 align:middle line:84% AUDIENCE: For 2, shouldn't the Pauli matrices be-- 01:02:16.950 --> 01:02:17.689 align:middle line:90% oh, nevermind. 01:02:17.689 --> 01:02:18.231 align:middle line:90% HONG LIU: Oh. 01:02:18.231 --> 01:02:21.530 align:middle line:90% 01:02:21.530 --> 01:02:24.450 align:middle line:90% Other questions? 01:02:24.450 --> 01:02:25.257 align:middle line:90% Yes? 01:02:25.257 --> 01:02:27.090 align:middle line:84% AUDIENCE: So I guess that I don't understand 01:02:27.090 --> 01:02:30.510 align:middle line:84% what space this side vector, the four entries-- 01:02:30.510 --> 01:02:31.680 align:middle line:90% like, what is that? 01:02:31.680 --> 01:02:33.510 align:middle line:90% Is this a Lorentz four-vector? 01:02:33.510 --> 01:02:35.390 align:middle line:84% HONG LIU: No, it's not a Lorentz four-vector. 01:02:35.390 --> 01:02:36.570 align:middle line:84% AUDIENCE: So in what sense is it a vector? 01:02:36.570 --> 01:02:37.560 align:middle line:90% How does it transform? 01:02:37.560 --> 01:02:39.185 align:middle line:84% HONG LIU: Yeah, so this is a new space. 01:02:39.185 --> 01:02:40.390 align:middle line:90% We will talk about that. 01:02:40.390 --> 01:02:40.890 align:middle line:90% Yeah. 01:02:40.890 --> 01:02:44.760 align:middle line:84% Yeah, so this is a new space, and so 01:02:44.760 --> 01:02:48.530 align:middle line:84% that's called-- this is called spinor space, yeah. 01:02:48.530 --> 01:02:49.460 align:middle line:90% Yes? 01:02:49.460 --> 01:02:51.050 align:middle line:84% AUDIENCE: How do we know that-- are these the only two 01:02:51.050 --> 01:02:51.830 align:middle line:90% representations? 01:02:51.830 --> 01:02:52.415 align:middle line:90% HONG LIU: Oh, no, no, no. 01:02:52.415 --> 01:02:53.870 align:middle line:84% There are infinite number of them. 01:02:53.870 --> 01:02:56.150 align:middle line:84% Yeah, I mentioned there are infinite number 01:02:56.150 --> 01:02:58.970 align:middle line:84% of such solutions, and this is just two of them. 01:02:58.970 --> 01:03:02.780 align:middle line:84% Yeah, I will comment on all those different solutions. 01:03:02.780 --> 01:03:03.620 align:middle line:90% Other questions? 01:03:03.620 --> 01:03:06.710 align:middle line:90% 01:03:06.710 --> 01:03:07.210 align:middle line:90% OK? 01:03:07.210 --> 01:03:07.710 align:middle line:90% Good? 01:03:07.710 --> 01:03:14.240 align:middle line:90% 01:03:14.240 --> 01:03:20.635 align:middle line:90% So now let me make some remarks. 01:03:20.635 --> 01:03:28.350 align:middle line:90% 01:03:28.350 --> 01:03:39.300 align:middle line:84% OK, so first, if you consider the case m equal to 0, 01:03:39.300 --> 01:03:43.950 align:middle line:84% OK, if you can see the case m equal to 0, and then this is, 01:03:43.950 --> 01:03:49.410 align:middle line:84% like, for massless, so when m equal to zero, then 01:03:49.410 --> 01:03:51.870 align:middle line:84% when you reduce to this Klein-Gordon equation, what you 01:03:51.870 --> 01:03:53.340 align:middle line:90% get is the massless equation. 01:03:53.340 --> 01:03:54.710 align:middle line:90% OK. 01:03:54.710 --> 01:03:59.010 align:middle line:84% OK, it's a massless equation, and then you 01:03:59.010 --> 01:04:04.140 align:middle line:84% will have dispersion relation p squared equal to 0. 01:04:04.140 --> 01:04:09.370 align:middle line:84% OK, so in this case, the original equation 01:04:09.370 --> 01:04:13.960 align:middle line:84% just becomes partial t equal to minus i alpha 01:04:13.960 --> 01:04:17.860 align:middle line:90% i partial partial xi psi. 01:04:17.860 --> 01:04:19.570 align:middle line:90% OK, so that's your equation. 01:04:19.570 --> 01:04:23.140 align:middle line:90% There's no this m beta term. 01:04:23.140 --> 01:04:24.220 align:middle line:90% There's no m beta term. 01:04:24.220 --> 01:04:27.060 align:middle line:90% 01:04:27.060 --> 01:04:30.100 align:middle line:90% So in this case, you just-- 01:04:30.100 --> 01:04:33.950 align:middle line:84% the same story just here, you just forget about beta. 01:04:33.950 --> 01:04:35.800 align:middle line:84% OK, the same story, you forget the beta, 01:04:35.800 --> 01:04:41.590 align:middle line:84% and then you only need alpha i alpha 01:04:41.590 --> 01:04:49.060 align:middle line:84% j the commutator to be 0 for i not equal to j, essentially 01:04:49.060 --> 01:04:51.310 align:middle line:90% just that equation 1 there. 01:04:51.310 --> 01:04:56.320 align:middle line:84% And then also, you'd want the alpha i square equal to 1, OK, 01:04:56.320 --> 01:04:58.240 align:middle line:90% so for any i. 01:04:58.240 --> 01:05:05.990 align:middle line:84% OK, so now these are the conditions for the alphas, 01:05:05.990 --> 01:05:09.300 align:middle line:84% and now you can actually satisfy by 2-by-2 matrices. 01:05:09.300 --> 01:05:12.523 align:middle line:84% So what matrix satisfies this kind of relation? 01:05:12.523 --> 01:05:13.190 align:middle line:90% AUDIENCE: Pauli. 01:05:13.190 --> 01:05:16.490 align:middle line:84% HONG LIU: Yeah, so the Pauli matrices 01:05:16.490 --> 01:05:19.940 align:middle line:84% anticommute among themselves, and their square is equal to 1. 01:05:19.940 --> 01:05:24.800 align:middle line:84% So you can just take off i in order to simplify here. 01:05:24.800 --> 01:05:29.630 align:middle line:84% So this tells you something important, tells you actually 01:05:29.630 --> 01:05:31.610 align:middle line:84% the equation for a massless particle 01:05:31.610 --> 01:05:34.550 align:middle line:84% and a massive particle are very different. 01:05:34.550 --> 01:05:36.470 align:middle line:84% So massive particle, you actually need 01:05:36.470 --> 01:05:40.270 align:middle line:84% the four components, but for the massive-- 01:05:40.270 --> 01:05:43.810 align:middle line:84% but for the massless particle, you 01:05:43.810 --> 01:05:47.080 align:middle line:84% can describe by a sigma matrix here, only 2 by 2. 01:05:47.080 --> 01:05:58.630 align:middle line:84% OK, so that means for massless psi, 01:05:58.630 --> 01:06:12.520 align:middle line:84% it can be described using a two-component vector, 01:06:12.520 --> 01:06:20.400 align:middle line:84% OK, so that in this case, psi equal to psi 1 and psi 2. 01:06:20.400 --> 01:06:23.800 align:middle line:84% Again, they are all just complex. 01:06:23.800 --> 01:06:24.300 align:middle line:90% OK. 01:06:24.300 --> 01:06:48.720 align:middle line:90% 01:06:48.720 --> 01:06:49.220 align:middle line:90% Good? 01:06:49.220 --> 01:06:56.660 align:middle line:84% So this actually-- yeah, we will elaborate on this later. 01:06:56.660 --> 01:07:00.610 align:middle line:84% So this actually tells you that actually the massive particle 01:07:00.610 --> 01:07:02.980 align:middle line:84% have more degrees of freedom than the massless particle, 01:07:02.980 --> 01:07:08.030 align:middle line:84% OK, more degrees than the massless particle. 01:07:08.030 --> 01:07:09.650 align:middle line:90% Good? 01:07:09.650 --> 01:07:18.250 align:middle line:84% And then the second thing is what Dirac essentially was-- 01:07:18.250 --> 01:07:22.070 align:middle line:90% Dirac's original motivation. 01:07:22.070 --> 01:07:31.080 align:middle line:84% So from Dirac equation, you can show you can derive a current. 01:07:31.080 --> 01:07:35.550 align:middle line:84% Just as you can do for the Schrodinger equation, 01:07:35.550 --> 01:07:40.210 align:middle line:84% you can derive an equation like this for some j mu 01:07:40.210 --> 01:07:46.990 align:middle line:84% and with j 0, with the 0-th component of this thing, 01:07:46.990 --> 01:07:57.050 align:middle line:90% positive definite classically. 01:07:57.050 --> 01:07:59.790 align:middle line:90% 01:07:59.790 --> 01:08:01.500 align:middle line:84% So I emphasize that this is classically. 01:08:01.500 --> 01:08:03.360 align:middle line:84% You will-- later, you will see why, OK? 01:08:03.360 --> 01:08:08.640 align:middle line:90% 01:08:08.640 --> 01:08:12.780 align:middle line:84% And this, I will leave it to your pset, 01:08:12.780 --> 01:08:16.050 align:middle line:84% OK, so this is very similar to the derivation 01:08:16.050 --> 01:08:18.840 align:middle line:84% of such occurrence in the case of the just 01:08:18.840 --> 01:08:20.399 align:middle line:84% non-relativistic Schrodinger equation 01:08:20.399 --> 01:08:22.770 align:middle line:84% because this has the same structure. 01:08:22.770 --> 01:08:25.170 align:middle line:84% Yeah, the Dirac equation, when we start it, 01:08:25.170 --> 01:08:27.870 align:middle line:84% has the structure of the non-relativistic Schrodinger 01:08:27.870 --> 01:08:32.130 align:middle line:84% equation, and yeah, so you can show something 01:08:32.130 --> 01:08:32.970 align:middle line:90% like this exists. 01:08:32.970 --> 01:08:35.130 align:middle line:90% OK. 01:08:35.130 --> 01:08:38.580 align:middle line:84% And then the third point is related to the question 01:08:38.580 --> 01:08:39.839 align:middle line:90% many of you may have. 01:08:39.839 --> 01:08:42.450 align:middle line:90% 01:08:42.450 --> 01:08:45.830 align:middle line:84% So we said, what's the meaning of all 01:08:45.830 --> 01:08:49.160 align:middle line:84% these different solutions for alpha 01:08:49.160 --> 01:08:51.470 align:middle line:90% and beta or for gammas, OK? 01:08:51.470 --> 01:08:54.484 align:middle line:84% So as I mentioned, you can have infinite number of solutions. 01:08:54.484 --> 01:08:55.609 align:middle line:90% What's the meaning of them? 01:08:55.609 --> 01:08:58.450 align:middle line:90% 01:08:58.450 --> 01:09:06.090 align:middle line:84% So first, let's imagine when we look at this equation-- 01:09:06.090 --> 01:09:10.609 align:middle line:84% so as I said, this is a matrix equation, 01:09:10.609 --> 01:09:12.880 align:middle line:84% so in this matrix equation, then you 01:09:12.880 --> 01:09:16.450 align:middle line:84% have this psi which is a four-component vector, OK, 01:09:16.450 --> 01:09:18.890 align:middle line:90% some four-component vector. 01:09:18.890 --> 01:09:21.970 align:middle line:84% So now let's imagine we make a basis change 01:09:21.970 --> 01:09:29.260 align:middle line:84% in this four-component vector, OK, so imagine, just say, 01:09:29.260 --> 01:09:43.390 align:middle line:84% consider making a basis change in psi. 01:09:43.390 --> 01:09:46.120 align:middle line:90% 01:09:46.120 --> 01:09:50.050 align:middle line:84% OK, so a basis change in psi just means, in linear algebra, 01:09:50.050 --> 01:09:55.460 align:middle line:84% this is a vector, means we consider another psi prime, 01:09:55.460 --> 01:09:59.900 align:middle line:84% which is-- so we take psi to some other psi prime 01:09:59.900 --> 01:10:04.210 align:middle line:84% which is related to psi by invertible matrix, so 01:10:04.210 --> 01:10:16.700 align:middle line:84% B, just some constant, complex, invertible matrix. 01:10:16.700 --> 01:10:22.270 align:middle line:84% OK, so essentially, you just make a linear superposition 01:10:22.270 --> 01:10:23.270 align:middle line:90% of different components. 01:10:23.270 --> 01:10:31.190 align:middle line:84% OK, so yeah, so this corresponding 01:10:31.190 --> 01:10:32.135 align:middle line:90% to a basis change. 01:10:32.135 --> 01:10:34.720 align:middle line:90% 01:10:34.720 --> 01:10:40.900 align:middle line:84% So now if psi satisfies this equation, 01:10:40.900 --> 01:10:44.300 align:middle line:84% psi satisfies this equation, you can easily convince yourself 01:10:44.300 --> 01:10:54.110 align:middle line:84% that the psi prime satisfies the following equation, 01:10:54.110 --> 01:11:02.710 align:middle line:84% the gamma mu prime partial mu minus m psi prime equal to 0, 01:11:02.710 --> 01:11:10.850 align:middle line:84% and the gamma mu prime is equal to B gamma mu B minus 1. 01:11:10.850 --> 01:11:11.350 align:middle line:90% OK. 01:11:11.350 --> 01:11:18.560 align:middle line:90% 01:11:18.560 --> 01:11:20.990 align:middle line:84% So yeah, so this easily can be shown. 01:11:20.990 --> 01:11:23.570 align:middle line:90% You just multiply B from both-- 01:11:23.570 --> 01:11:26.570 align:middle line:84% just multiply B to this equation, 01:11:26.570 --> 01:11:29.270 align:middle line:84% OK, multiply B to this equation, and for this term, 01:11:29.270 --> 01:11:32.630 align:middle line:84% you can just directly give-- psi gives you psi prime. 01:11:32.630 --> 01:11:34.940 align:middle line:84% And for this term, then you just get the B, 01:11:34.940 --> 01:11:38.120 align:middle line:84% and then here you can insert B minus 1, B, 01:11:38.120 --> 01:11:39.680 align:middle line:90% and then, yeah, you get that. 01:11:39.680 --> 01:11:43.580 align:middle line:84% OK, so this, you can easily convince yourself, just 01:11:43.580 --> 01:11:46.350 align:middle line:90% a couple of lines. 01:11:46.350 --> 01:11:52.180 align:middle line:84% So now, so psi prime satisfies essentially the same equation 01:11:52.180 --> 01:11:57.280 align:middle line:84% but a different gamma matrix, OK, a gamma mu prime, 01:11:57.280 --> 01:11:59.325 align:middle line:84% so now you can easily check yourself, OK? 01:11:59.325 --> 01:12:09.100 align:middle line:90% 01:12:09.100 --> 01:12:11.740 align:middle line:84% So you can easily check yourself this gamma mu prime also 01:12:11.740 --> 01:12:13.570 align:middle line:90% satisfies that algebra. 01:12:13.570 --> 01:12:15.865 align:middle line:84% OK, so you can easily check yourself. 01:12:15.865 --> 01:12:50.590 align:middle line:90% 01:12:50.590 --> 01:12:57.400 align:middle line:84% So then we conclude, any sets, since we 01:12:57.400 --> 01:13:00.850 align:middle line:84% can make a basis transformation as you want, OK, that the-- 01:13:00.850 --> 01:13:03.850 align:middle line:84% except the basis transformation should not change physics, OK? 01:13:03.850 --> 01:13:30.820 align:middle line:84% So any sets of gamma mu related by a similarity transformation 01:13:30.820 --> 01:13:31.640 align:middle line:90% are equivalent. 01:13:31.640 --> 01:13:32.140 align:middle line:90% OK. 01:13:32.140 --> 01:13:37.984 align:middle line:90% 01:13:37.984 --> 01:13:42.540 align:middle line:84% OK, they're equivalent, and because they should not 01:13:42.540 --> 01:13:44.520 align:middle line:84% give you new physics just because one 01:13:44.520 --> 01:13:45.750 align:middle line:90% is going to change the basis. 01:13:45.750 --> 01:13:46.920 align:middle line:90% OK. 01:13:46.920 --> 01:13:49.050 align:middle line:84% So now, here is a highly nontrivial 01:13:49.050 --> 01:13:54.510 align:middle line:90% mathematical statement, so now-- 01:13:54.510 --> 01:13:59.160 align:middle line:84% which of course I will not prove here because it'd just 01:13:59.160 --> 01:14:01.590 align:middle line:90% take too much-- 01:14:01.590 --> 01:14:03.000 align:middle line:90% yeah. 01:14:03.000 --> 01:14:05.700 align:middle line:90% So you can show-- 01:14:05.700 --> 01:14:10.980 align:middle line:84% I'll just quote the result. So you can show, OK, 01:14:10.980 --> 01:14:17.490 align:middle line:84% with a little bit of effort that under such kind 01:14:17.490 --> 01:14:25.230 align:middle line:84% of equivalence relation, means that the similarity 01:14:25.230 --> 01:14:27.180 align:middle line:84% transformations, they are all equivalent, 01:14:27.180 --> 01:14:47.520 align:middle line:84% and such equivalence relation, the representations of star 01:14:47.520 --> 01:14:48.270 align:middle line:90% is unique. 01:14:48.270 --> 01:14:53.080 align:middle line:90% 01:14:53.080 --> 01:14:55.220 align:middle line:90% OK. 01:14:55.220 --> 01:15:01.190 align:middle line:84% So you can show any matrices which satisfy that equation, 01:15:01.190 --> 01:15:04.640 align:middle line:84% they're all related by similarity transformation, 01:15:04.640 --> 01:15:07.100 align:middle line:84% OK, so they're all physically equivalent. 01:15:07.100 --> 01:15:09.447 align:middle line:84% They're just corresponding to a change of basis. 01:15:09.447 --> 01:15:11.405 align:middle line:84% OK, they're corresponding to a change of basis. 01:15:11.405 --> 01:15:16.520 align:middle line:90% 01:15:16.520 --> 01:15:17.270 align:middle line:90% OK. 01:15:17.270 --> 01:15:27.860 align:middle line:84% So but different forms of the gamma, they may-- 01:15:27.860 --> 01:15:29.870 align:middle line:84% different forms of the gamma matrices, 01:15:29.870 --> 01:15:32.330 align:middle line:84% they may be useful for different purposes. 01:15:32.330 --> 01:15:34.340 align:middle line:84% OK, they may be useful for different purposes. 01:15:34.340 --> 01:15:38.840 align:middle line:84% For example, this I, solution I we 01:15:38.840 --> 01:15:45.860 align:middle line:84% wrote down before, it's convenient for if you 01:15:45.860 --> 01:15:47.600 align:middle line:84% want to take the nonrelativistic limit. 01:15:47.600 --> 01:15:52.450 align:middle line:90% 01:15:52.450 --> 01:15:55.942 align:middle line:84% For example, if you want to make a connection 01:15:55.942 --> 01:15:59.170 align:middle line:84% with a nonrelativistic quantum mechanics, 01:15:59.170 --> 01:16:03.670 align:middle line:84% and that's actually the most convenient form, the matrix 01:16:03.670 --> 01:16:04.870 align:middle line:90% to use. 01:16:04.870 --> 01:16:08.770 align:middle line:90% And II is actually in the-- 01:16:08.770 --> 01:16:12.190 align:middle line:84% opposite regime, it's convenient for 01:16:12.190 --> 01:16:14.320 align:middle line:90% the ultrarelativistic regime. 01:16:14.320 --> 01:16:18.510 align:middle line:90% 01:16:18.510 --> 01:16:21.360 align:middle line:84% OK, so it depends on which regime, 01:16:21.360 --> 01:16:24.660 align:middle line:84% sometimes you use different gamma matrices. 01:16:24.660 --> 01:16:30.240 align:middle line:84% OK, they make your algebra a bit more convenient. 01:16:30.240 --> 01:16:35.570 align:middle line:90% 01:16:35.570 --> 01:16:43.100 align:middle line:84% So now having introduced the Dirac equation and then 01:16:43.100 --> 01:16:45.740 align:middle line:84% the structure of the Dirac equation, 01:16:45.740 --> 01:16:47.780 align:middle line:84% but still we haven't showed that the Dirac 01:16:47.780 --> 01:16:49.390 align:middle line:90% equation is covariant. 01:16:49.390 --> 01:16:53.930 align:middle line:84% OK, we just showed that the Dirac equation can 01:16:53.930 --> 01:16:57.800 align:middle line:84% have the plane-wave solution and that the plane-wave solution 01:16:57.800 --> 01:17:00.320 align:middle line:84% will have a standard of nonrelativistic dispersion 01:17:00.320 --> 01:17:01.520 align:middle line:90% relation. 01:17:01.520 --> 01:17:18.820 align:middle line:84% OK, so in order to show that the Dirac equation is covariant, 01:17:18.820 --> 01:17:20.530 align:middle line:90% we have to show-- 01:17:20.530 --> 01:17:24.310 align:middle line:84% we have to make a Lorentz transform and show 01:17:24.310 --> 01:17:28.580 align:middle line:84% that the Dirac equations are the same in every Lorentz frame. 01:17:28.580 --> 01:17:34.120 align:middle line:84% OK, we have to show that, OK, and we are running out of time. 01:17:34.120 --> 01:17:38.290 align:middle line:84% So we, of course, won't have time to do that today, 01:17:38.290 --> 01:17:44.340 align:middle line:84% but let me just remind you how this Lorentz covariance works 01:17:44.340 --> 01:17:45.690 align:middle line:90% for the scalar case. 01:17:45.690 --> 01:17:49.620 align:middle line:84% OK, so remember, so recall, yeah, let's 01:17:49.620 --> 01:17:56.070 align:middle line:84% just quickly recall, for scalar, we have phi x. 01:17:56.070 --> 01:17:58.703 align:middle line:84% Then under Lorentz transformation-- 01:17:58.703 --> 01:18:00.120 align:middle line:84% so consider Lorentz transformation 01:18:00.120 --> 01:18:07.830 align:middle line:84% which x mu goes to x prime mu equal to lambda nu mu x nu. 01:18:07.830 --> 01:18:12.390 align:middle line:84% Consider such a Lorentz transformation, OK, so and then 01:18:12.390 --> 01:18:17.130 align:middle line:84% phi transforms as following, phi prime x prime phi prime. 01:18:17.130 --> 01:18:19.470 align:middle line:84% New phi evaluated at the new position 01:18:19.470 --> 01:18:22.770 align:middle line:84% should be the same as phi evaluated at the old position, 01:18:22.770 --> 01:18:24.360 align:middle line:90% OK? 01:18:24.360 --> 01:18:30.330 align:middle line:84% Or let me just write it here so that I don't have to erase. 01:18:30.330 --> 01:18:36.480 align:middle line:84% So the phi-- yeah, just equal to phi lambda minus 1 x. 01:18:36.480 --> 01:18:38.830 align:middle line:90% OK. 01:18:38.830 --> 01:18:44.430 align:middle line:84% So now if you look at the Klein-Gordon equation, 01:18:44.430 --> 01:18:47.490 align:middle line:84% let's see how this is covariant, OK, 01:18:47.490 --> 01:18:49.890 align:middle line:84% so now let's see in a different frame. 01:18:49.890 --> 01:18:51.720 align:middle line:90% OK. 01:18:51.720 --> 01:18:53.640 align:middle line:84% So covariance means that when we go 01:18:53.640 --> 01:18:59.190 align:middle line:84% to a new frame, partial prime square-- 01:18:59.190 --> 01:19:04.060 align:middle line:84% OK, so this means in the prime coordinates-- 01:19:04.060 --> 01:19:10.180 align:middle line:84% minus m square and phi prime evaluated at the x prime, 01:19:10.180 --> 01:19:13.000 align:middle line:90% there must be a-- 01:19:13.000 --> 01:19:17.470 align:middle line:84% yeah, so this is in another Lorentz frame, OK, 01:19:17.470 --> 01:19:19.540 align:middle line:90% so they have the same form. 01:19:19.540 --> 01:19:24.950 align:middle line:84% Indeed, you see these two are equivalent because this just 01:19:24.950 --> 01:19:26.480 align:middle line:90% equal to that. 01:19:26.480 --> 01:19:30.050 align:middle line:84% OK, trivially, you could do that just by definitions. 01:19:30.050 --> 01:19:35.880 align:middle line:84% And this is a Lorentz scalar, so this is also equal to that. 01:19:35.880 --> 01:19:39.810 align:middle line:84% OK, so you see that the Klein-Gordon equation indeed 01:19:39.810 --> 01:19:40.950 align:middle line:90% is Lorentz covariant. 01:19:40.950 --> 01:19:43.800 align:middle line:90% OK, it's the same in any frame. 01:19:43.800 --> 01:19:46.940 align:middle line:84% So now we want to show, OK, so now we 01:19:46.940 --> 01:19:49.280 align:middle line:84% want to show that the Dirac equation has 01:19:49.280 --> 01:19:52.477 align:middle line:84% the same property, OK, and that is much more nontrivial. 01:19:52.477 --> 01:19:53.810 align:middle line:90% OK, that's much more nontrivial. 01:19:53.810 --> 01:20:01.040 align:middle line:84% Again, it's really ingenious, ingenious, yeah, 01:20:01.040 --> 01:20:03.650 align:middle line:90% but we see, actually, it works. 01:20:03.650 --> 01:20:08.500 align:middle line:90% OK, so we will do it next time. 01:20:08.500 --> 01:20:12.000 align:middle line:90%