WEBVTT 00:00:00.000 --> 00:00:01.916 align:middle line:90% [SQUEAKING] 00:00:01.916 --> 00:00:04.311 align:middle line:90% [RUSTLING] 00:00:04.311 --> 00:00:07.664 align:middle line:90% [CLICKING] 00:00:07.664 --> 00:00:10.690 align:middle line:90% 00:00:10.690 --> 00:00:15.530 align:middle line:84% PROFESSOR: So last time, we talked about Dirac equation. 00:00:15.530 --> 00:00:23.225 align:middle line:84% So the Dirac equation has the following form: 00:00:23.225 --> 00:00:29.490 align:middle line:84% gamma mu partial mu minus m psi equal to 0. 00:00:29.490 --> 00:00:32.570 align:middle line:90% 00:00:32.570 --> 00:00:35.100 align:middle line:90% So here, we have two spaces. 00:00:35.100 --> 00:00:43.190 align:middle line:84% So one is your standard physical spacetime, so with x mu. 00:00:43.190 --> 00:00:45.680 align:middle line:84% And then we have a lot of internal space 00:00:45.680 --> 00:00:51.300 align:middle line:84% which is labeled by the index of the psi and the gamma, 00:00:51.300 --> 00:00:52.970 align:middle line:90% which has suppressed here. 00:00:52.970 --> 00:00:55.385 align:middle line:84% So psi should be considered as a four vector. 00:00:55.385 --> 00:00:58.220 align:middle line:90% 00:00:58.220 --> 00:01:01.840 align:middle line:84% So alpha equal to 1, 2, 3, 4, and the gamma 00:01:01.840 --> 00:01:07.900 align:middle line:84% mu should be considered as a matrix-- 00:01:07.900 --> 00:01:11.200 align:middle line:90% a 4x4 matrix. 00:01:11.200 --> 00:01:15.740 align:middle line:84% And so this space labeled by alpha-- 00:01:15.740 --> 00:01:17.365 align:middle line:84% so this is called the spinor space. 00:01:17.365 --> 00:01:23.320 align:middle line:90% 00:01:23.320 --> 00:01:24.770 align:middle line:90% This is called spinor space. 00:01:24.770 --> 00:01:27.850 align:middle line:84% So you have to be careful that now we 00:01:27.850 --> 00:01:31.510 align:middle line:84% have two spaces intertwined together. 00:01:31.510 --> 00:01:34.240 align:middle line:84% One is your ordinary spacetime, and so this 00:01:34.240 --> 00:01:39.130 align:middle line:84% is a function of the spacetime, but it's carried on index-- 00:01:39.130 --> 00:01:41.330 align:middle line:90% yeah, carries on indices. 00:01:41.330 --> 00:01:44.750 align:middle line:84% And so this is the matrix equation. 00:01:44.750 --> 00:01:48.970 align:middle line:84% So altogether, there are four equations here 00:01:48.970 --> 00:01:50.710 align:middle line:90% for each component of psi. 00:01:50.710 --> 00:01:54.730 align:middle line:90% 00:01:54.730 --> 00:01:57.210 align:middle line:90% And yeah, so let me pull-- 00:01:57.210 --> 00:01:59.120 align:middle line:84% let me just write this a little better. 00:01:59.120 --> 00:02:00.805 align:middle line:90% So this is x-- 00:02:00.805 --> 00:02:03.730 align:middle line:90% 00:02:03.730 --> 00:02:07.680 align:middle line:84% let me call this equation 1, which I will use later. 00:02:07.680 --> 00:02:10.185 align:middle line:84% And this gamma mu, they are not ordinary matrices. 00:02:10.185 --> 00:02:13.150 align:middle line:90% They satisfy this condition. 00:02:13.150 --> 00:02:20.125 align:middle line:84% Gamma mu, the anticommutator with nu equal to 2 eta mu nu. 00:02:20.125 --> 00:02:22.630 align:middle line:90% 00:02:22.630 --> 00:02:28.090 align:middle line:84% So that means different gamma matrices when they are indexed 00:02:28.090 --> 00:02:29.600 align:middle line:90% are not the same. 00:02:29.600 --> 00:02:32.380 align:middle line:84% They anticommute with each other. 00:02:32.380 --> 00:02:36.100 align:middle line:84% So if you pass them through, you get a minus sign. 00:02:36.100 --> 00:02:37.870 align:middle line:84% So they anticommute with each other 00:02:37.870 --> 00:02:40.930 align:middle line:84% because when mu not equal to nu, the right-hand side is 0. 00:02:40.930 --> 00:02:45.250 align:middle line:84% And when you pass through each other, you get the minus. 00:02:45.250 --> 00:02:47.240 align:middle line:90% And the square of them-- 00:02:47.240 --> 00:02:50.740 align:middle line:84% so gamma 0 square, you get minus 1 because this is 0, 0. 00:02:50.740 --> 00:02:51.550 align:middle line:90% It's minus 1. 00:02:51.550 --> 00:02:54.610 align:middle line:84% And the gamma i squared, you just get one. 00:02:54.610 --> 00:02:56.780 align:middle line:90% So this is very simple. 00:02:56.780 --> 00:02:57.690 align:middle line:90% Yes? 00:02:57.690 --> 00:02:59.482 align:middle line:84% STUDENT: When you say gamma i squared is 1, 00:02:59.482 --> 00:03:01.298 align:middle line:84% you mean like its the identity matrix? 00:03:01.298 --> 00:03:02.840 align:middle line:84% PROFESSOR: Yeah, its identity matrix. 00:03:02.840 --> 00:03:06.290 align:middle line:84% Yeah, when I say gamma 0 square is minus 1, 00:03:06.290 --> 00:03:08.370 align:middle line:84% it's also minus times identity matrix. 00:03:08.370 --> 00:03:10.990 align:middle line:90% Yeah, exactly. 00:03:10.990 --> 00:03:17.280 align:middle line:84% And also not all gamma mu are Hermitian 00:03:17.280 --> 00:03:19.650 align:middle line:84% So the relation is that the gamma mu 00:03:19.650 --> 00:03:25.605 align:middle line:84% dagger is equal to gamma 0 gamma mu gamma 0. 00:03:25.605 --> 00:03:28.290 align:middle line:90% 00:03:28.290 --> 00:03:31.070 align:middle line:90% So again, from here, you can-- 00:03:31.070 --> 00:03:35.910 align:middle line:84% yeah, so this is just a compact way to write its properties. 00:03:35.910 --> 00:03:39.800 align:middle line:84% When you have-- when mu is equal to 0-- 00:03:39.800 --> 00:03:44.960 align:middle line:84% and then you have gamma 0 times gamma 0 squared is minus 1. 00:03:44.960 --> 00:03:47.660 align:middle line:84% And then you get the minus gamma 0-- tells you that the gamma 00:03:47.660 --> 00:03:50.160 align:middle line:90% 0 is anti-Hermitian. 00:03:50.160 --> 00:03:52.980 align:middle line:84% And if you take the index to be i here, 00:03:52.980 --> 00:03:55.140 align:middle line:90% and then you have index i here-- 00:03:55.140 --> 00:03:58.140 align:middle line:84% and then when 0 and i-- they are not 00:03:58.140 --> 00:03:59.580 align:middle line:90% the same, so they anticommute. 00:03:59.580 --> 00:04:02.700 align:middle line:84% You can pass through this gamma 0 through this gamma i. 00:04:02.700 --> 00:04:06.450 align:middle line:84% You get the minus sign, and then gamma 0 square gives you 00:04:06.450 --> 00:04:08.940 align:middle line:84% another minus sign, so you get 1. 00:04:08.940 --> 00:04:12.150 align:middle line:84% So that means that the gamma i dagger equal to gamma i. 00:04:12.150 --> 00:04:13.830 align:middle line:90% So gamma i is hermitian. 00:04:13.830 --> 00:04:14.490 align:middle line:90% Yes? 00:04:14.490 --> 00:04:16.300 align:middle line:84% STUDENT: So is there supposed to be 00:04:16.300 --> 00:04:20.190 align:middle line:84% an i in front of the derivative or do 00:04:20.190 --> 00:04:23.250 align:middle line:90% we absorb the i into the gamma? 00:04:23.250 --> 00:04:25.800 align:middle line:84% PROFESSOR: So in my convention, there is no i. 00:04:25.800 --> 00:04:33.670 align:middle line:84% So with this-- so eta mu mu here is mostly plus metric. 00:04:33.670 --> 00:04:37.710 align:middle line:84% So some people, when they use, say, mostly negative metric, 00:04:37.710 --> 00:04:41.670 align:middle line:90% then there may be an i in there. 00:04:41.670 --> 00:04:43.860 align:middle line:84% Yeah, so it depends on your convention 00:04:43.860 --> 00:04:46.110 align:middle line:90% in defining the gamma matrices. 00:04:46.110 --> 00:04:48.030 align:middle line:84% STUDENT: So for minus plus plus plus, no i-- 00:04:48.030 --> 00:04:48.900 align:middle line:90% OK, sorry. 00:04:48.900 --> 00:04:51.650 align:middle line:90% PROFESSOR: Yeah. 00:04:51.650 --> 00:04:52.160 align:middle line:90% Good. 00:04:52.160 --> 00:04:53.540 align:middle line:90% Other questions? 00:04:53.540 --> 00:04:56.420 align:middle line:84% Yeah, as I mentioned last time, different conventions 00:04:56.420 --> 00:05:00.230 align:middle line:84% are annoying, so-- but you just stick one convention-- should 00:05:00.230 --> 00:05:01.010 align:middle line:90% be fine. 00:05:01.010 --> 00:05:03.950 align:middle line:84% Stick to my convention-- would be fine. 00:05:03.950 --> 00:05:04.730 align:middle line:90% Yes? 00:05:04.730 --> 00:05:06.605 align:middle line:84% STUDENT: Sorry, how can we get the expression 00:05:06.605 --> 00:05:09.020 align:middle line:90% for gamma mu dagger? 00:05:09.020 --> 00:05:11.210 align:middle line:84% PROFESSOR: Oh, yeah, this is just a compact way 00:05:11.210 --> 00:05:13.760 align:middle line:84% to write down that gamma 0 is anti-Hermitian 00:05:13.760 --> 00:05:15.740 align:middle line:90% and gamma i is Hermitian. 00:05:15.740 --> 00:05:18.350 align:middle line:90% Yeah, just-- this just compact. 00:05:18.350 --> 00:05:21.020 align:middle line:84% Yeah, just this is useful so that you can treat all gamma 00:05:21.020 --> 00:05:23.510 align:middle line:84% mu in the same way so that you don't 00:05:23.510 --> 00:05:24.860 align:middle line:90% have to always separate them. 00:05:24.860 --> 00:05:27.440 align:middle line:90% 00:05:27.440 --> 00:05:28.910 align:middle line:90% Good. 00:05:28.910 --> 00:05:29.930 align:middle line:90% Other questions? 00:05:29.930 --> 00:05:33.280 align:middle line:90% 00:05:33.280 --> 00:05:36.400 align:middle line:84% So last time, we also said you have 00:05:36.400 --> 00:05:41.260 align:middle line:84% many, many choices for gamma mu, but they are all 00:05:41.260 --> 00:05:42.890 align:middle line:90% physically equivalent. 00:05:42.890 --> 00:05:45.490 align:middle line:84% And so different choices, they are 00:05:45.490 --> 00:05:47.740 align:middle line:84% convenient for different purposes. 00:05:47.740 --> 00:05:52.750 align:middle line:84% So often we just pick one of them, and depend on the problem 00:05:52.750 --> 00:05:55.520 align:middle line:84% we are-- yeah, so that comes with a little bit experience. 00:05:55.520 --> 00:05:57.940 align:middle line:84% So for certain problems, certain choices 00:05:57.940 --> 00:05:59.590 align:middle line:90% of gamma become more useful-- 00:05:59.590 --> 00:06:02.950 align:middle line:84% become more easier to manipulate. 00:06:02.950 --> 00:06:05.850 align:middle line:90% 00:06:05.850 --> 00:06:06.360 align:middle line:90% Good. 00:06:06.360 --> 00:06:10.190 align:middle line:84% So now, let's talk about the Lorentz covariance 00:06:10.190 --> 00:06:18.635 align:middle line:84% of the Dirac equation, which we started a little bit last time. 00:06:18.635 --> 00:06:24.690 align:middle line:90% 00:06:24.690 --> 00:06:29.170 align:middle line:84% So Lorentz covariance means that the equation looks the same 00:06:29.170 --> 00:06:32.230 align:middle line:90% in all Lorentz frames. 00:06:32.230 --> 00:06:35.270 align:middle line:84% So when you make a Lorentz transformation-- 00:06:35.270 --> 00:06:37.900 align:middle line:90% so imagine you make a-- 00:06:37.900 --> 00:06:38.920 align:middle line:90% go to another frame. 00:06:38.920 --> 00:06:41.650 align:middle line:84% You start with your frame defined by this x mu. 00:06:41.650 --> 00:06:45.900 align:middle line:84% So now, imagine you go to another frame which you 00:06:45.900 --> 00:06:47.730 align:middle line:90% make a Lorentz transformation. 00:06:47.730 --> 00:06:51.390 align:middle line:90% 00:06:51.390 --> 00:06:55.320 align:middle line:84% So the Lorentz covariance means your equation 00:06:55.320 --> 00:06:59.370 align:middle line:84% when written in this frame in terms of x prime mu 00:06:59.370 --> 00:07:01.870 align:middle line:90% have the same form. 00:07:01.870 --> 00:07:04.200 align:middle line:84% So last time, we talked about the scalar equations 00:07:04.200 --> 00:07:05.310 align:middle line:90% at the end. 00:07:05.310 --> 00:07:10.860 align:middle line:84% And another equation we have seen before is the Maxwell 00:07:10.860 --> 00:07:15.330 align:middle line:84% equation, so let me also very quickly mention 00:07:15.330 --> 00:07:17.370 align:middle line:90% for the Maxwell equation. 00:07:17.370 --> 00:07:21.880 align:middle line:84% So when you have a Maxwell equation, 00:07:21.880 --> 00:07:26.490 align:middle line:84% which is the story for the vector potential-- 00:07:26.490 --> 00:07:30.190 align:middle line:90% 00:07:30.190 --> 00:07:32.560 align:middle line:90% so now, in contrast-- 00:07:32.560 --> 00:07:34.375 align:middle line:90% remember for the scalar case-- 00:07:34.375 --> 00:07:37.020 align:middle line:90% 00:07:37.020 --> 00:07:40.410 align:middle line:84% so last time-- let me just write it down here. 00:07:40.410 --> 00:07:42.780 align:middle line:84% Last time we said for the scalar case, 00:07:42.780 --> 00:07:45.690 align:middle line:84% when you go to a different frame, 00:07:45.690 --> 00:07:48.270 align:middle line:84% the phi prime should be equal to-- 00:07:48.270 --> 00:07:51.690 align:middle line:84% evaluate the new coordinate should be equal to phi x. 00:07:51.690 --> 00:07:55.360 align:middle line:84% So this is the transformation for scalar field. 00:07:55.360 --> 00:07:59.230 align:middle line:84% But now, if you want to do Maxwell equation-- 00:07:59.230 --> 00:08:03.450 align:middle line:84% so we have to think about how this a mu transform 00:08:03.450 --> 00:08:04.920 align:middle line:90% under Lorentz transformation. 00:08:04.920 --> 00:08:07.260 align:middle line:90% And now a mu is a vector-- 00:08:07.260 --> 00:08:10.800 align:middle line:90% is a four vector in spacetime. 00:08:10.800 --> 00:08:19.090 align:middle line:84% And so that means that the a mu should transform also 00:08:19.090 --> 00:08:21.920 align:middle line:90% as the vector. 00:08:21.920 --> 00:08:28.630 align:middle line:84% So this means that when you go to a new frame, 00:08:28.630 --> 00:08:39.786 align:middle line:84% A mu transforms as lambda mu nu A nu x. 00:08:39.786 --> 00:08:42.620 align:middle line:90% So here, not only-- 00:08:42.620 --> 00:08:47.450 align:middle line:84% so when A mu evaluate at new position-- 00:08:47.450 --> 00:08:50.870 align:middle line:84% it can be written as a linear superposition of the value 00:08:50.870 --> 00:08:53.330 align:middle line:90% at your original position. 00:08:53.330 --> 00:08:56.175 align:middle line:84% And this linear superposition by itself 00:08:56.175 --> 00:08:58.550 align:middle line:84% is a Lorentz transformation because the A mu is a vector. 00:08:58.550 --> 00:09:02.670 align:middle line:90% 00:09:02.670 --> 00:09:07.160 align:middle line:84% And so that means just under Lorentz transformation, 00:09:07.160 --> 00:09:10.610 align:middle line:84% the different components of A mu should also 00:09:10.610 --> 00:09:14.240 align:middle line:90% change under the transformation. 00:09:14.240 --> 00:09:18.950 align:middle line:84% So given this-- so I will not show it now. 00:09:18.950 --> 00:09:21.980 align:middle line:84% Just will take a couple of minutes, 00:09:21.980 --> 00:09:24.200 align:middle line:84% but you should convince yourself-- 00:09:24.200 --> 00:09:26.880 align:middle line:84% try to check yourself that the Maxwell equation, 00:09:26.880 --> 00:09:31.760 align:middle line:84% which can be written-- so for example, as F mu-- 00:09:31.760 --> 00:09:33.890 align:middle line:90% mu nu equal to 0. 00:09:33.890 --> 00:09:40.320 align:middle line:84% This implies that the partial mu prime in the new frame, 00:09:40.320 --> 00:09:44.610 align:middle line:90% and then the f mu mu prime. 00:09:44.610 --> 00:09:50.010 align:middle line:84% f mu mu prime is obtained from this new A mu prime, 00:09:50.010 --> 00:09:52.800 align:middle line:84% and these two equations are the same-- 00:09:52.800 --> 00:09:53.490 align:middle line:90% are equivalent. 00:09:53.490 --> 00:09:56.820 align:middle line:84% Just in one frame, and then when you go to a different frame, 00:09:56.820 --> 00:09:58.090 align:middle line:90% then you get this equation. 00:09:58.090 --> 00:10:00.630 align:middle line:84% So the equation has exactly the same form, 00:10:00.630 --> 00:10:02.640 align:middle line:90% but now it's in the new frame. 00:10:02.640 --> 00:10:06.160 align:middle line:90% 00:10:06.160 --> 00:10:08.140 align:middle line:90% Good? 00:10:08.140 --> 00:10:11.070 align:middle line:84% So now, let's go to Dirac equation. 00:10:11.070 --> 00:10:14.730 align:middle line:90% So now, go to Dirac equation. 00:10:14.730 --> 00:10:19.960 align:middle line:84% So in the Dirac equation, again, this psi-- 00:10:19.960 --> 00:10:24.440 align:middle line:84% we need to ask how the psi should transform 00:10:24.440 --> 00:10:27.560 align:middle line:90% under Lorentz transformation. 00:10:27.560 --> 00:10:30.570 align:middle line:84% But now, this psi is a completely new object. 00:10:30.570 --> 00:10:34.110 align:middle line:84% So this is not like in the Maxwell case. 00:10:34.110 --> 00:10:36.030 align:middle line:84% We know this is a spacetime vector, 00:10:36.030 --> 00:10:39.080 align:middle line:84% so you can easily guess how you should transform. 00:10:39.080 --> 00:10:44.000 align:middle line:84% So now this psi-- this spinor space is completely new. 00:10:44.000 --> 00:10:48.530 align:middle line:84% So now, we have to figure out how psi transforms. 00:10:48.530 --> 00:10:51.500 align:middle line:90% 00:10:51.500 --> 00:11:18.270 align:middle line:84% So let's suppose under a Lorentz transformation lambda 00:11:18.270 --> 00:11:22.130 align:middle line:90% that psi transforms as follows. 00:11:22.130 --> 00:11:25.670 align:middle line:90% Psi prime alpha x prime-- 00:11:25.670 --> 00:11:28.550 align:middle line:84% because now, again, you evaluate in the new position-- 00:11:28.550 --> 00:11:30.200 align:middle line:90% new frame. 00:11:30.200 --> 00:11:32.750 align:middle line:84% Since this carries index, in principle, 00:11:32.750 --> 00:11:34.430 align:middle line:90% this can be a superposition. 00:11:34.430 --> 00:11:39.320 align:middle line:90% 00:11:39.320 --> 00:11:42.950 align:middle line:84% You can mix them in the internal space. 00:11:42.950 --> 00:11:46.130 align:middle line:90% 00:11:46.130 --> 00:11:48.860 align:middle line:84% In principle, we can have form like this. 00:11:48.860 --> 00:11:57.980 align:middle line:84% Say for some matrix S, which depends on lambda. 00:11:57.980 --> 00:12:00.710 align:middle line:90% 00:12:00.710 --> 00:12:05.120 align:middle line:84% So in principle, you can have a transformation like this. 00:12:05.120 --> 00:12:07.430 align:middle line:84% Say when you evaluate new position, 00:12:07.430 --> 00:12:09.500 align:middle line:84% it's given by the value at the old position, 00:12:09.500 --> 00:12:14.710 align:middle line:84% but now you can make some rotations in the internal space 00:12:14.710 --> 00:12:17.410 align:middle line:84% just as here for the Maxwell field. 00:12:17.410 --> 00:12:20.650 align:middle line:90% 00:12:20.650 --> 00:12:30.750 align:middle line:84% So except-- and so suppose this and then the Lorentz covariance 00:12:30.750 --> 00:12:37.190 align:middle line:84% would be the statement, say, for some matrix. 00:12:37.190 --> 00:12:40.000 align:middle line:90% 00:12:40.000 --> 00:12:41.920 align:middle line:84% And then the Lorentz covariance then 00:12:41.920 --> 00:12:58.350 align:middle line:84% is a statement starting from equation 1-- 00:12:58.350 --> 00:13:02.250 align:middle line:84% you should be able to find going to the new frame-- 00:13:02.250 --> 00:13:07.290 align:middle line:84% you have an equation like this-- gamma mu partial mu 00:13:07.290 --> 00:13:17.700 align:middle line:84% prime minus m psi prime x prime equal to 0. 00:13:17.700 --> 00:13:20.820 align:middle line:84% So you should get an identical equation-- not identical 00:13:20.820 --> 00:13:21.678 align:middle line:90% equation 2. 00:13:21.678 --> 00:13:23.220 align:middle line:84% You should get an identical equation, 00:13:23.220 --> 00:13:25.320 align:middle line:84% but now everything is in terms of prime. 00:13:25.320 --> 00:13:29.050 align:middle line:90% 00:13:29.050 --> 00:13:31.600 align:middle line:84% So I want to emphasize that gamma mu does not change 00:13:31.600 --> 00:13:34.180 align:middle line:90% despite a carry index mu. 00:13:34.180 --> 00:13:37.880 align:middle line:84% Mu gamma mu just some constant matrices-- 00:13:37.880 --> 00:13:42.160 align:middle line:84% just the-- it's just four matrices. 00:13:42.160 --> 00:13:45.400 align:middle line:84% It does not transform under any spacetime trans-- 00:13:45.400 --> 00:13:47.170 align:middle line:90% it's not dynamic variable. 00:13:47.170 --> 00:13:49.420 align:middle line:84% It does not transform on the spacetime transformation, 00:13:49.420 --> 00:13:52.830 align:middle line:90% so gamma mu does not change. 00:13:52.830 --> 00:13:53.790 align:middle line:90% Good. 00:13:53.790 --> 00:13:57.500 align:middle line:84% So now the question of the Lorentz covariance-- 00:13:57.500 --> 00:14:02.840 align:middle line:84% where the Dirac equation is Lorentz covariance boils down 00:14:02.840 --> 00:14:07.810 align:middle line:84% whether we can find such S. Whether-- 00:14:07.810 --> 00:14:15.670 align:middle line:84% yeah, the question is, can we find, such an S? 00:14:15.670 --> 00:14:18.290 align:middle line:90% 00:14:18.290 --> 00:14:25.120 align:middle line:84% So if we can find such an S, which this is true, 00:14:25.120 --> 00:14:28.290 align:middle line:84% then we say the Dirac equation is Lorentz covariant. 00:14:28.290 --> 00:14:33.040 align:middle line:84% And this is how the Dirac field should transform. 00:14:33.040 --> 00:14:35.460 align:middle line:84% So this psi is normally called the Dirac field, 00:14:35.460 --> 00:14:38.700 align:middle line:84% and this is how a Dirac field should transform. 00:14:38.700 --> 00:14:41.340 align:middle line:84% So psi is often called Dirac field. 00:14:41.340 --> 00:14:43.590 align:middle line:84% It's also often called spinor field. 00:14:43.590 --> 00:14:48.090 align:middle line:84% And it's also often called Dirac spinor field. 00:14:48.090 --> 00:14:51.030 align:middle line:84% And anyway, and then that should be 00:14:51.030 --> 00:14:53.040 align:middle line:90% the way how it's transformed. 00:14:53.040 --> 00:14:53.970 align:middle line:90% Yes? 00:14:53.970 --> 00:14:57.030 align:middle line:84% STUDENT: Yeah, I was just confused, because before, 00:14:57.030 --> 00:15:00.439 align:middle line:84% for the scalar fields, when we get transformations, 00:15:00.439 --> 00:15:02.387 align:middle line:90% we would either say-- 00:15:02.387 --> 00:15:03.917 align:middle line:84% like, transform the fields, like phi 00:15:03.917 --> 00:15:07.257 align:middle line:84% goes to phi prime or transform the coordinates, 00:15:07.257 --> 00:15:09.770 align:middle line:90% like x goes to x prime. 00:15:09.770 --> 00:15:11.770 align:middle line:84% So why did we transform both in this case? 00:15:11.770 --> 00:15:12.580 align:middle line:90% PROFESSOR: Yeah. 00:15:12.580 --> 00:15:14.990 align:middle line:90% No, no, no, in this case-- 00:15:14.990 --> 00:15:18.490 align:middle line:84% so when you talk about transform of the scalar field 00:15:18.490 --> 00:15:22.870 align:middle line:84% itself, of course, it's just phi equal to phi prime and-- 00:15:22.870 --> 00:15:26.260 align:middle line:90% because that function changes. 00:15:26.260 --> 00:15:30.370 align:middle line:84% But here, we are asking about the form of the equation 00:15:30.370 --> 00:15:32.150 align:middle line:90% in the new frame. 00:15:32.150 --> 00:15:34.180 align:middle line:84% And so when you go to the new frame, 00:15:34.180 --> 00:15:37.100 align:middle line:84% your function form changes, but your coordinate also changes. 00:15:37.100 --> 00:15:38.800 align:middle line:90% So that's why you need to-- 00:15:38.800 --> 00:15:40.330 align:middle line:84% when we say the covariance, you need 00:15:40.330 --> 00:15:44.690 align:middle line:84% to evaluate your new function in the new position. 00:15:44.690 --> 00:15:48.800 align:middle line:84% Yeah, that means they look the same in different frames. 00:15:48.800 --> 00:15:49.480 align:middle line:90% Yes? 00:15:49.480 --> 00:15:52.570 align:middle line:84% STUDENT: So should I think of S as the representation of lambda 00:15:52.570 --> 00:15:53.868 align:middle line:90% in the spinor space? 00:15:53.868 --> 00:15:54.910 align:middle line:90% PROFESSOR: Yeah, exactly. 00:15:54.910 --> 00:15:56.200 align:middle line:90% Good. 00:15:56.200 --> 00:16:00.220 align:middle line:84% That's exactly the right mathematical language 00:16:00.220 --> 00:16:03.790 align:middle line:84% to talk about this, which we'll mention later. 00:16:03.790 --> 00:16:06.863 align:middle line:84% At the moment, I didn't want to use mathematical language. 00:16:06.863 --> 00:16:09.280 align:middle line:84% Yeah, for those people who are familiar with group theory, 00:16:09.280 --> 00:16:10.060 align:middle line:90% indeed. 00:16:10.060 --> 00:16:14.380 align:middle line:84% So this S would be the representation of the Lorentz 00:16:14.380 --> 00:16:17.665 align:middle line:90% group in the spinor space. 00:16:17.665 --> 00:16:20.950 align:middle line:90% 00:16:20.950 --> 00:16:22.730 align:middle line:90% Good. 00:16:22.730 --> 00:16:27.220 align:middle line:84% So now, let's try to find this S. So 00:16:27.220 --> 00:16:31.000 align:middle line:84% before we do that, let's first write 00:16:31.000 --> 00:16:33.910 align:middle line:90% what is a partial prime x mu. 00:16:33.910 --> 00:16:39.227 align:middle line:84% So recall, partial prime mu, it should be partial, 00:16:39.227 --> 00:16:41.290 align:middle line:90% partial x prime mu. 00:16:41.290 --> 00:16:43.540 align:middle line:84% This transform the new coordinate. 00:16:43.540 --> 00:16:45.220 align:middle line:84% And from that transformation, you 00:16:45.220 --> 00:16:47.630 align:middle line:84% can easily figure it out-- just chain rule. 00:16:47.630 --> 00:16:53.460 align:middle line:84% So this is the same equal to partial, partial x 00:16:53.460 --> 00:17:00.060 align:middle line:90% mu and the lambda minus 1 nu mu. 00:17:00.060 --> 00:17:03.330 align:middle line:90% 00:17:03.330 --> 00:17:05.220 align:middle line:84% So this is very easy to figure it out 00:17:05.220 --> 00:17:09.210 align:middle line:90% because they have to be-- 00:17:09.210 --> 00:17:11.069 align:middle line:90% x will multiply the x. 00:17:11.069 --> 00:17:12.960 align:middle line:84% Partial x will have to be a singlet, 00:17:12.960 --> 00:17:16.060 align:middle line:84% so this transform as a inverse matrix. 00:17:16.060 --> 00:17:21.660 align:middle line:84% So this implies that the gamma mu 00:17:21.660 --> 00:17:30.480 align:middle line:84% partial mu prime should be equal to lambda minus 1 nu mu gamma 00:17:30.480 --> 00:17:32.100 align:middle line:90% mu partial nu. 00:17:32.100 --> 00:17:36.200 align:middle line:90% 00:17:36.200 --> 00:17:37.440 align:middle line:90% Let me just write it better. 00:17:37.440 --> 00:17:46.690 align:middle line:90% 00:17:46.690 --> 00:17:48.580 align:middle line:84% so with this preparation, then we 00:17:48.580 --> 00:17:54.510 align:middle line:84% can try to see whether we can find the S from this equation. 00:17:54.510 --> 00:17:57.870 align:middle line:84% No-- yeah, find this equation from 1. 00:17:57.870 --> 00:18:01.700 align:middle line:84% So what we can do is let's multiply 00:18:01.700 --> 00:18:04.970 align:middle line:84% the equation 1 from the left by S. 00:18:04.970 --> 00:18:08.300 align:middle line:90% So let's imagine we multiply S-- 00:18:08.300 --> 00:18:09.260 align:middle line:90% such a matrix. 00:18:09.260 --> 00:18:13.280 align:middle line:84% Again, I always just suppress the spinor indices 00:18:13.280 --> 00:18:16.110 align:middle line:84% and just write in the matrix form. 00:18:16.110 --> 00:18:18.740 align:middle line:90% So let's just write gamma mu. 00:18:18.740 --> 00:18:22.775 align:middle line:84% So take that equation 1, we multiply it by s. 00:18:22.775 --> 00:18:33.210 align:middle line:90% 00:18:33.210 --> 00:18:35.080 align:middle line:90% So we have two terms here. 00:18:35.080 --> 00:18:38.610 align:middle line:84% So this is a matrix in the same space as S. 00:18:38.610 --> 00:18:41.820 align:middle line:84% S act in the same space as gamma mu, 00:18:41.820 --> 00:18:44.700 align:middle line:84% so we cannot easily commute here-- 00:18:44.700 --> 00:18:47.160 align:middle line:90% normally, they don't commute. 00:18:47.160 --> 00:18:53.190 align:middle line:84% But this-- but m is a constant, so we can pass S through here. 00:18:53.190 --> 00:18:56.170 align:middle line:84% So we can rewrite this equation as follows. 00:18:56.170 --> 00:19:06.705 align:middle line:84% We can write it as S gamma mu S minus 1 partial mu minus m S 00:19:06.705 --> 00:19:09.720 align:middle line:90% psi 0. 00:19:09.720 --> 00:19:15.930 align:middle line:84% So for m, I just pass S through, but for this term, 00:19:15.930 --> 00:19:19.800 align:middle line:84% I just inserted S minus 1 and S-- and so yeah, it's the same. 00:19:19.800 --> 00:19:23.080 align:middle line:90% 00:19:23.080 --> 00:19:29.040 align:middle line:84% So now, we can use this equation in here, 00:19:29.040 --> 00:19:35.330 align:middle line:84% because S times psi just equal to psi prime x prime. 00:19:35.330 --> 00:19:45.440 align:middle line:84% So now, we get S gamma mu S minus 1 partial mu minus m psi 00:19:45.440 --> 00:19:49.340 align:middle line:90% prime x prime equal to 0. 00:19:49.340 --> 00:19:52.770 align:middle line:90% We just use that equation. 00:19:52.770 --> 00:19:53.665 align:middle line:90% Yes? 00:19:53.665 --> 00:19:55.082 align:middle line:84% STUDENT: I'm a little confused how 00:19:55.082 --> 00:19:56.583 align:middle line:84% you can pass S through partial mu 00:19:56.583 --> 00:20:00.370 align:middle line:84% because given that partial mu acts on psi 00:20:00.370 --> 00:20:02.022 align:middle line:84% and it has to have a representation 00:20:02.022 --> 00:20:05.570 align:middle line:90% in this inner space [INAUDIBLE]. 00:20:05.570 --> 00:20:08.450 align:middle line:84% PROFESSOR: Yeah, so that's the key I emphasized earlier. 00:20:08.450 --> 00:20:12.980 align:middle line:84% So x mu and psi alpha, they are in different space. 00:20:12.980 --> 00:20:16.920 align:middle line:84% And so all this S and gamma, they are all constant. 00:20:16.920 --> 00:20:18.920 align:middle line:84% They don't depend on spatial location. 00:20:18.920 --> 00:20:23.700 align:middle line:84% And so partial mu does not act on S. Yeah, 00:20:23.700 --> 00:20:25.870 align:middle line:84% for partial mu point of view, S is just a constant. 00:20:25.870 --> 00:20:27.490 align:middle line:90% It's a spacetime constant. 00:20:27.490 --> 00:20:33.010 align:middle line:84% It's only rotate psi at a single point-- 00:20:33.010 --> 00:20:35.530 align:middle line:84% the different component of psi at the single point. 00:20:35.530 --> 00:20:40.990 align:middle line:90% 00:20:40.990 --> 00:20:42.760 align:middle line:90% Good? 00:20:42.760 --> 00:20:45.320 align:middle line:90% So now, we have this equation. 00:20:45.320 --> 00:20:52.420 align:middle line:84% So now we can compare this object with this object. 00:20:52.420 --> 00:20:56.300 align:middle line:90% 00:20:56.300 --> 00:21:00.890 align:middle line:84% Yeah, in particular, this object is equal to that. 00:21:00.890 --> 00:21:13.900 align:middle line:84% So we conclude that 1 goes to 2 if S gamma mu S minus 1 00:21:13.900 --> 00:21:24.250 align:middle line:84% equal to this object lambda minus 1 nu mu. 00:21:24.250 --> 00:21:27.770 align:middle line:84% I think-- yeah, I think I'm messing up 00:21:27.770 --> 00:21:30.740 align:middle line:90% a little bit notation here. 00:21:30.740 --> 00:21:34.380 align:middle line:84% Gamma mu-- I think my index is a little bit wrong. 00:21:34.380 --> 00:21:36.890 align:middle line:90% Let me just make sure. 00:21:36.890 --> 00:21:38.960 align:middle line:90% Oh, yeah, here is a partial nu. 00:21:38.960 --> 00:21:43.310 align:middle line:84% Sorry, here is partial nu, so I need to exchange 00:21:43.310 --> 00:21:44.780 align:middle line:90% the mu and the nu index here. 00:21:44.780 --> 00:21:51.590 align:middle line:90% So mu and the nu, then gamma. 00:21:51.590 --> 00:21:59.070 align:middle line:90% 00:21:59.070 --> 00:22:03.520 align:middle line:90% So this has to be equal to that. 00:22:03.520 --> 00:22:05.770 align:middle line:84% So this has to be equal to that, and you 00:22:05.770 --> 00:22:08.920 align:middle line:84% need to compare partial nu with partial nu-- 00:22:08.920 --> 00:22:11.223 align:middle line:84% you have to exchange and nu and the nu here anyway. 00:22:11.223 --> 00:22:13.015 align:middle line:84% So this is the equation we have to satisfy. 00:22:13.015 --> 00:22:17.680 align:middle line:90% 00:22:17.680 --> 00:22:26.100 align:middle line:84% So we have to find a matrix S which acts on gamma mu. 00:22:26.100 --> 00:22:29.310 align:middle line:84% Gamma mu is some bunch of matrices. 00:22:29.310 --> 00:22:33.040 align:middle line:90% And gives you like this. 00:22:33.040 --> 00:22:37.680 align:middle line:84% So as if the gamma mu actually-- when you act on Sophia as 00:22:37.680 --> 00:22:41.340 align:middle line:84% if gamma mu transforms as some Lorentz transformation-- yeah, 00:22:41.340 --> 00:22:45.540 align:middle line:90% inverse Lorentz transformation. 00:22:45.540 --> 00:22:50.700 align:middle line:84% So we need to-- now let's try to find this S. So to do this, 00:22:50.700 --> 00:22:57.040 align:middle line:84% again, we use a trick which we have been using before. 00:22:57.040 --> 00:23:00.338 align:middle line:84% So how would you approach this problem? 00:23:00.338 --> 00:23:03.173 align:middle line:90% STUDENT: The identity. 00:23:03.173 --> 00:23:03.840 align:middle line:90% PROFESSOR: Good. 00:23:03.840 --> 00:23:07.650 align:middle line:84% Yes, just do infinitesimal transformations. 00:23:07.650 --> 00:23:10.200 align:middle line:84% So once you learn how to do infinitesimal transformations, 00:23:10.200 --> 00:23:12.600 align:middle line:84% then you always know how to do finite ones. 00:23:12.600 --> 00:23:17.220 align:middle line:84% And infinitesimal ones, it's much, much easier to do. 00:23:17.220 --> 00:23:30.710 align:middle line:84% So now, again, we consider Lambda mu nu close 00:23:30.710 --> 00:23:31.550 align:middle line:90% to your identity. 00:23:31.550 --> 00:23:37.840 align:middle line:90% 00:23:37.840 --> 00:23:44.170 align:middle line:84% So that means we write lambda mu nu as delta mu 00:23:44.170 --> 00:23:49.130 align:middle line:90% nu, which is the identity. 00:23:49.130 --> 00:23:54.680 align:middle line:84% Then plus omega mu nu, and take the omega to be small. 00:23:54.680 --> 00:23:57.320 align:middle line:84% And also remember previously, we discussed 00:23:57.320 --> 00:24:00.440 align:middle line:84% that omega mu nu, when you know the index, 00:24:00.440 --> 00:24:02.090 align:middle line:90% is actually antisymmetric. 00:24:02.090 --> 00:24:07.570 align:middle line:90% 00:24:07.570 --> 00:24:09.490 align:middle line:84% And this is infinitesimal, so we take 00:24:09.490 --> 00:24:11.650 align:middle line:84% this to be infinitesimal and work everything 00:24:11.650 --> 00:24:12.790 align:middle line:90% to first order in omega. 00:24:12.790 --> 00:24:17.570 align:middle line:90% 00:24:17.570 --> 00:24:22.390 align:middle line:84% So similarly, the lambda minus mu 00:24:22.390 --> 00:24:28.890 align:middle line:84% nu just equal to delta mu nu just [INAUDIBLE] 00:24:28.890 --> 00:24:32.540 align:middle line:90% minus omega mu nu. 00:24:32.540 --> 00:24:36.800 align:middle line:84% So yeah, just to leading order in omega expansion-- 00:24:36.800 --> 00:24:38.585 align:middle line:84% the inverse metric is just given by that. 00:24:38.585 --> 00:24:41.280 align:middle line:90% 00:24:41.280 --> 00:24:44.930 align:middle line:90% So now, we can try to-- 00:24:44.930 --> 00:24:48.380 align:middle line:90% so since the S-- 00:24:48.380 --> 00:24:52.990 align:middle line:84% so on the right-hand side, when lambda is close 00:24:52.990 --> 00:24:56.430 align:middle line:90% to the identity-- 00:24:56.430 --> 00:25:01.500 align:middle line:84% so the right-hand side is just the gamma mu. 00:25:01.500 --> 00:25:03.540 align:middle line:84% Just the identity does not do anything. 00:25:03.540 --> 00:25:05.790 align:middle line:84% Just gamma mu, and then plus something proportional 00:25:05.790 --> 00:25:07.320 align:middle line:90% to omega. 00:25:07.320 --> 00:25:12.570 align:middle line:84% Then, that means that S must also-- 00:25:12.570 --> 00:25:14.030 align:middle line:84% when lambda is close to identity, 00:25:14.030 --> 00:25:16.110 align:middle line:84% it means S must also have the structure 00:25:16.110 --> 00:25:20.580 align:middle line:84% to be identity and proportional to something linear in omega. 00:25:20.580 --> 00:25:23.910 align:middle line:84% So the S must also have this structure. 00:25:23.910 --> 00:25:27.390 align:middle line:90% 00:25:27.390 --> 00:25:30.420 align:middle line:84% Yeah, let me just don't write the index. 00:25:30.420 --> 00:25:32.670 align:middle line:90% Just directly write as identity. 00:25:32.670 --> 00:25:36.230 align:middle line:90% 00:25:36.230 --> 00:25:40.250 align:middle line:84% And then it should be something proportional to omega mu nu. 00:25:40.250 --> 00:25:44.150 align:middle line:84% So from convention, we write this way. 00:25:44.150 --> 00:25:48.010 align:middle line:84% So it should be linear in omega mu nu and sigma mu nu. 00:25:48.010 --> 00:25:51.580 align:middle line:84% So sigma mu nu are a bunch of matrices. 00:25:51.580 --> 00:25:57.600 align:middle line:84% So remember, S is a 4x4 matrix, and so omega mu nu is just 00:25:57.600 --> 00:25:59.590 align:middle line:90% some number. 00:25:59.590 --> 00:26:03.490 align:middle line:84% So this will be a bunch of 4x4 matrices. 00:26:03.490 --> 00:26:06.780 align:middle line:84% So for each omega can in principle 00:26:06.780 --> 00:26:10.500 align:middle line:84% independently multiply some matrix. 00:26:10.500 --> 00:26:14.700 align:middle line:84% So this is the most general way to expand this linear order 00:26:14.700 --> 00:26:16.985 align:middle line:90% in omega. 00:26:16.985 --> 00:26:19.110 align:middle line:84% So this sigma essentially just the first derivative 00:26:19.110 --> 00:26:22.920 align:middle line:84% of x with respect to each omega mu nu. 00:26:22.920 --> 00:26:26.230 align:middle line:84% And this i over 2 is just convention. 00:26:26.230 --> 00:26:31.020 align:middle line:84% And similarly, the inverse is just corresponding 00:26:31.020 --> 00:26:33.945 align:middle line:84% when you change the sign here to leading order in omega. 00:26:33.945 --> 00:26:42.060 align:middle line:90% 00:26:42.060 --> 00:26:48.220 align:middle line:84% Yeah, so emphasize each sigma mu nu is a matrix. 00:26:48.220 --> 00:26:59.110 align:middle line:84% It should be understood as some matrix in the spinor space. 00:26:59.110 --> 00:27:05.430 align:middle line:90% 00:27:05.430 --> 00:27:08.820 align:middle line:84% So now, you just need to plug in this equation. 00:27:08.820 --> 00:27:13.140 align:middle line:84% Plug in this S and S minus 1 into this equation. 00:27:13.140 --> 00:27:17.030 align:middle line:84% Yeah, just expand the both sides to the omega, 00:27:17.030 --> 00:27:19.280 align:middle line:90% and you equate the coefficients. 00:27:19.280 --> 00:27:20.810 align:middle line:90% You equate the coefficients. 00:27:20.810 --> 00:27:23.840 align:middle line:84% From that way, you determine the sigma. 00:27:23.840 --> 00:27:28.020 align:middle line:84% So let me call this equation star. 00:27:28.020 --> 00:27:31.730 align:middle line:84% Yeah, I'll just call this equation 3. 00:27:31.730 --> 00:27:40.250 align:middle line:84% So then to order omega mu nu to linear order. 00:27:40.250 --> 00:27:43.730 align:middle line:90% Then, the equation 3, if-- 00:27:43.730 --> 00:27:46.130 align:middle line:84% when you expand on both sides to omega-- 00:27:46.130 --> 00:27:48.470 align:middle line:84% to order of omega and equate both sides, 00:27:48.470 --> 00:27:50.100 align:middle line:84% then you find the following equation. 00:27:50.100 --> 00:27:51.930 align:middle line:84% This is just a couple of lines algebra, 00:27:51.930 --> 00:27:54.690 align:middle line:90% so I urge you to do it yourself. 00:27:54.690 --> 00:28:03.450 align:middle line:84% So given by i commutator lambda rho, 00:28:03.450 --> 00:28:18.215 align:middle line:84% gamma mu equal to eta lambda mu gamma rho minus rho mu gamma 00:28:18.215 --> 00:28:18.715 align:middle line:90% lambda. 00:28:18.715 --> 00:28:22.250 align:middle line:90% 00:28:22.250 --> 00:28:23.580 align:middle line:90% So you get this equation. 00:28:23.580 --> 00:28:27.320 align:middle line:84% So the left-hand side is very easy to understand. 00:28:27.320 --> 00:28:29.260 align:middle line:90% So essentially, you just-- 00:28:29.260 --> 00:28:32.160 align:middle line:90% whenever you have a commutator-- 00:28:32.160 --> 00:28:35.310 align:middle line:84% for people have done this Baker-Hausdorff et 00:28:35.310 --> 00:28:39.720 align:middle line:84% cetera, the first order, you always get the commutator. 00:28:39.720 --> 00:28:42.220 align:middle line:84% So that's where this commutator come from. 00:28:42.220 --> 00:28:45.030 align:middle line:84% So the right-hand side, when you expand this, 00:28:45.030 --> 00:28:49.020 align:middle line:84% essentially, you just get omega mu nu 00:28:49.020 --> 00:28:51.600 align:middle line:84% because we have to lower the indices. 00:28:51.600 --> 00:28:54.000 align:middle line:90% So you have some eta here. 00:28:54.000 --> 00:28:56.340 align:middle line:84% You have eta here, and yeah, so that's 00:28:56.340 --> 00:28:57.810 align:middle line:90% how you get the right-hand side. 00:28:57.810 --> 00:28:59.340 align:middle line:84% And the reason you get two terms-- 00:28:59.340 --> 00:29:03.115 align:middle line:84% because it should be antisymmetric in the-- 00:29:03.115 --> 00:29:03.615 align:middle line:90% yeah. 00:29:03.615 --> 00:29:06.810 align:middle line:90% 00:29:06.810 --> 00:29:07.890 align:middle line:90% Good. 00:29:07.890 --> 00:29:11.470 align:middle line:84% So now, it just boils down to solve this equation. 00:29:11.470 --> 00:29:14.790 align:middle line:90% So if we can find sigma-- 00:29:14.790 --> 00:29:19.670 align:middle line:84% satisfy this equation, and then we are done. 00:29:19.670 --> 00:29:23.080 align:middle line:90% 00:29:23.080 --> 00:29:27.460 align:middle line:90% So yeah, of course, this is-- 00:29:27.460 --> 00:29:30.360 align:middle line:84% now, you do it by trial and error. 00:29:30.360 --> 00:29:33.225 align:middle line:90% 00:29:33.225 --> 00:29:38.330 align:middle line:84% And the bottom line is that there's a solution. 00:29:38.330 --> 00:29:40.370 align:middle line:84% So let me just write down the solution. 00:29:40.370 --> 00:29:45.500 align:middle line:84% And the nice thing about other people 00:29:45.500 --> 00:29:49.790 align:middle line:84% found the solution is that you can just check it. 00:29:49.790 --> 00:29:52.610 align:middle line:90% So you can check this quantity-- 00:29:52.610 --> 00:29:59.660 align:middle line:90% 00:29:59.660 --> 00:30:00.875 align:middle line:90% this solves the equation. 00:30:00.875 --> 00:30:08.060 align:middle line:90% 00:30:08.060 --> 00:30:11.520 align:middle line:90% So just can plug this. 00:30:11.520 --> 00:30:14.210 align:middle line:84% So this is the commutator of gamma lamda and gamma rho. 00:30:14.210 --> 00:30:16.520 align:middle line:84% You plug into that, and then you just 00:30:16.520 --> 00:30:25.100 align:middle line:84% evaluate this gamma matrices and use this kind of equation 00:30:25.100 --> 00:30:30.500 align:middle line:84% over and over, and you will find this is satisfied. 00:30:30.500 --> 00:30:32.645 align:middle line:84% Again, I will leave it as an exercise for yourself. 00:30:32.645 --> 00:30:39.860 align:middle line:90% 00:30:39.860 --> 00:30:40.957 align:middle line:90% Yes? 00:30:40.957 --> 00:30:42.790 align:middle line:84% STUDENT: I have a question about this board. 00:30:42.790 --> 00:30:47.090 align:middle line:84% So you start off with saying that partial mu is transformed 00:30:47.090 --> 00:30:50.370 align:middle line:84% and that's how you get this lambda inverse. 00:30:50.370 --> 00:30:54.300 align:middle line:84% Then, the last thing is now actually-- 00:30:54.300 --> 00:30:58.100 align:middle line:84% this gamma's transformed and it's not 00:30:58.100 --> 00:30:59.630 align:middle line:90% partial mu transformed. 00:30:59.630 --> 00:31:01.030 align:middle line:90% I'm a bit confused how it-- 00:31:01.030 --> 00:31:04.030 align:middle line:84% it seems like you're linking a transformation in partial mu 00:31:04.030 --> 00:31:06.010 align:middle line:90% with a transformation in-- 00:31:06.010 --> 00:31:07.930 align:middle line:90% PROFESSOR: No. 00:31:07.930 --> 00:31:13.450 align:middle line:84% So here, we want to match 1 and 2. 00:31:13.450 --> 00:31:19.120 align:middle line:84% So the step is we do-- well, have each equation-- 00:31:19.120 --> 00:31:21.850 align:middle line:90% from equation 1, we reach here. 00:31:21.850 --> 00:31:24.500 align:middle line:90% 00:31:24.500 --> 00:31:31.090 align:middle line:84% From equation 2, we just plug this into there. 00:31:31.090 --> 00:31:33.730 align:middle line:84% And so each equation has done one step, 00:31:33.730 --> 00:31:36.100 align:middle line:90% and then I equate them. 00:31:36.100 --> 00:31:37.600 align:middle line:84% STUDENT: Right, but I guess what I'm 00:31:37.600 --> 00:31:41.730 align:middle line:84% asking is equation 1 has partial mu transformed, 00:31:41.730 --> 00:31:45.600 align:middle line:84% and then that's how you get your lambda inverse. 00:31:45.600 --> 00:31:50.640 align:middle line:84% And then in equation 2, you are transforming your gamma. 00:31:50.640 --> 00:31:55.690 align:middle line:84% PROFESSOR: No, we want to show this-- 00:31:55.690 --> 00:31:59.770 align:middle line:84% you want to match this equation with this equation. 00:31:59.770 --> 00:32:01.720 align:middle line:90% This equation is derived from 1. 00:32:01.720 --> 00:32:03.220 align:middle line:90% So we want to derive-- 00:32:03.220 --> 00:32:06.870 align:middle line:90% we want to find the 2 from 1. 00:32:06.870 --> 00:32:07.920 align:middle line:90% This is equation 1. 00:32:07.920 --> 00:32:10.380 align:middle line:84% So I just slightly rewrite the equation 2 00:32:10.380 --> 00:32:14.640 align:middle line:84% by inserting this transformation here, then I matched them. 00:32:14.640 --> 00:32:19.600 align:middle line:90% 00:32:19.600 --> 00:32:20.470 align:middle line:90% Other questions? 00:32:20.470 --> 00:32:24.920 align:middle line:90% 00:32:24.920 --> 00:32:25.690 align:middle line:90% OK, good. 00:32:25.690 --> 00:32:40.140 align:middle line:90% 00:32:40.140 --> 00:32:43.230 align:middle line:90% So now, we can just-- 00:32:43.230 --> 00:32:48.390 align:middle line:84% now we can just immediately-- so given this equation, 00:32:48.390 --> 00:32:53.010 align:middle line:84% and now we can immediately write the final transformation. 00:32:53.010 --> 00:32:57.770 align:middle line:84% So with the finite transformation-- 00:32:57.770 --> 00:33:07.775 align:middle line:84% so for each lambda mu nu, each finite-- 00:33:07.775 --> 00:33:11.330 align:middle line:90% 00:33:11.330 --> 00:33:16.760 align:middle line:84% you can obtain the corresponding omega mu nu, 00:33:16.760 --> 00:33:18.245 align:middle line:90% and then this is also finite. 00:33:18.245 --> 00:33:21.230 align:middle line:90% 00:33:21.230 --> 00:33:27.350 align:middle line:84% And now, we can just obtain the S by exponentiating this. 00:33:27.350 --> 00:33:29.480 align:middle line:84% And then the corresponding S would 00:33:29.480 --> 00:33:39.980 align:middle line:84% be S equal to exponential minus i over 2 omega mu nu sigma. 00:33:39.980 --> 00:33:43.880 align:middle line:84% And the sigma mu nu is just given by this one. 00:33:43.880 --> 00:33:45.350 align:middle line:90% And you can check yourself. 00:33:45.350 --> 00:33:54.040 align:middle line:84% This satisfies-- you can check this satisfies equation 3. 00:33:54.040 --> 00:33:57.820 align:middle line:90% 00:33:57.820 --> 00:34:00.640 align:middle line:90% Yeah, that's finite equation. 00:34:00.640 --> 00:34:06.940 align:middle line:90% 00:34:06.940 --> 00:34:07.440 align:middle line:90% Good. 00:34:07.440 --> 00:34:08.420 align:middle line:90% Any questions on this? 00:34:08.420 --> 00:34:17.570 align:middle line:90% 00:34:17.570 --> 00:34:20.670 align:middle line:90% OK, good. 00:34:20.670 --> 00:34:22.010 align:middle line:90% So now, let me-- 00:34:22.010 --> 00:34:29.010 align:middle line:90% 00:34:29.010 --> 00:34:32.609 align:middle line:84% was there some-- so now, let's make some remarks. 00:34:32.609 --> 00:34:38.190 align:middle line:90% 00:34:38.190 --> 00:34:45.449 align:middle line:84% So then this S, it just generates 00:34:45.449 --> 00:35:01.650 align:middle line:84% Lorentz transformation in spinor space 00:35:01.650 --> 00:35:05.130 align:middle line:84% because it only act on this alpha and beta. 00:35:05.130 --> 00:35:13.620 align:middle line:90% 00:35:13.620 --> 00:35:17.190 align:middle line:90% So sigma i j-- 00:35:17.190 --> 00:35:23.120 align:middle line:84% or sigma mu nu according to our standard terminology, 00:35:23.120 --> 00:35:28.930 align:middle line:84% this is called the generator of the transformation. 00:35:28.930 --> 00:35:30.560 align:middle line:90% So these are the generators. 00:35:30.560 --> 00:35:37.280 align:middle line:90% 00:35:37.280 --> 00:35:43.520 align:middle line:90% So when omega mu nu-- 00:35:43.520 --> 00:35:47.390 align:middle line:84% when the mu nu equal to spatial directions, and then that's 00:35:47.390 --> 00:35:49.520 align:middle line:90% corresponding to a spacetime-- 00:35:49.520 --> 00:35:54.860 align:middle line:84% spatial rotation and the 0 i corresponding to a boost. 00:35:54.860 --> 00:35:56.630 align:middle line:90% So remember. 00:35:56.630 --> 00:36:04.110 align:middle line:84% So that means for sigma ij, which 00:36:04.110 --> 00:36:14.700 align:middle line:84% is i equal to 4, gamma i, gamma j-- 00:36:14.700 --> 00:36:24.910 align:middle line:84% this generates generators of rotations. 00:36:24.910 --> 00:36:29.150 align:middle line:90% 00:36:29.150 --> 00:36:33.250 align:middle line:84% So remember, previously-- so if you remember 00:36:33.250 --> 00:36:35.560 align:middle line:84% how we do the Lorentz transformation, 00:36:35.560 --> 00:36:42.490 align:middle line:84% the omega ij would correspond to the rotational angle in the ij 00:36:42.490 --> 00:36:43.930 align:middle line:90% plane. 00:36:43.930 --> 00:36:52.150 align:middle line:84% So you just rotate in the ij plane by angle omega ij. 00:36:52.150 --> 00:36:54.910 align:middle line:84% So then this just corresponding now-- 00:36:54.910 --> 00:36:59.215 align:middle line:84% corresponding to the generator of the rotations in ij plane. 00:36:59.215 --> 00:37:04.690 align:middle line:90% 00:37:04.690 --> 00:37:11.730 align:middle line:84% So you can-- because the gamma i and the gamma j are Hermitian, 00:37:11.730 --> 00:37:14.750 align:middle line:84% recall that the gamma i is Hermitian. 00:37:14.750 --> 00:37:21.380 align:middle line:84% And then the sigma ij is also Hermitian 00:37:21.380 --> 00:37:24.380 align:middle line:84% because when you take the dagger of this i 00:37:24.380 --> 00:37:26.960 align:middle line:84% gives you a minus sign, but the commutator 00:37:26.960 --> 00:37:30.350 align:middle line:90% gives you a minus sign. 00:37:30.350 --> 00:37:32.240 align:middle line:90% and so this is Hermitian. 00:37:32.240 --> 00:37:43.220 align:middle line:84% So that means that S, which corresponding to rotation, 00:37:43.220 --> 00:37:44.750 align:middle line:90% is unitary. 00:37:44.750 --> 00:37:46.550 align:middle line:90% So this is unitary. 00:37:46.550 --> 00:37:48.245 align:middle line:90% So this is a unitary matrix. 00:37:48.245 --> 00:37:56.560 align:middle line:90% 00:37:56.560 --> 00:38:02.000 align:middle line:84% So now let's consider the sigma 0 i, 00:38:02.000 --> 00:38:03.980 align:middle line:84% which corresponding to the generator for boost. 00:38:03.980 --> 00:38:07.040 align:middle line:90% 00:38:07.040 --> 00:38:10.250 align:middle line:84% So this will have the form gamma 0 gamma 00:38:10.250 --> 00:38:13.100 align:middle line:84% i, so this is the generators for boost. 00:38:13.100 --> 00:38:19.970 align:middle line:90% 00:38:19.970 --> 00:38:30.240 align:middle line:84% In i-th direction in the spinor space. 00:38:30.240 --> 00:38:32.730 align:middle line:84% And now, because the-- remember gamma 0, 00:38:32.730 --> 00:38:35.990 align:middle line:84% when you take the dagger, you get the minus sign. 00:38:35.990 --> 00:38:40.460 align:middle line:84% So the sigma 0 i, now if you take the dagger, 00:38:40.460 --> 00:38:43.310 align:middle line:84% you actually also get the minus sign because the-- 00:38:43.310 --> 00:38:46.022 align:middle line:84% yeah, so now it's anti-Hermitian. 00:38:46.022 --> 00:38:49.810 align:middle line:90% 00:38:49.810 --> 00:39:00.720 align:middle line:84% So that means the boost matrices-- 00:39:00.720 --> 00:39:02.985 align:middle line:84% so this is a boost transformation-- 00:39:02.985 --> 00:39:12.390 align:middle line:90% 00:39:12.390 --> 00:39:14.060 align:middle line:90% so this is not unitary. 00:39:14.060 --> 00:39:25.370 align:middle line:90% 00:39:25.370 --> 00:39:27.830 align:middle line:90% So this is not unitary. 00:39:27.830 --> 00:39:31.130 align:middle line:84% So in general-- so normally, as we said before, normally, 00:39:31.130 --> 00:39:33.620 align:middle line:84% when you do a symmetry transformation, 00:39:33.620 --> 00:39:35.870 align:middle line:84% the transformation is a unitary transformation. 00:39:35.870 --> 00:39:37.970 align:middle line:84% But in this case, actually-- yeah, 00:39:37.970 --> 00:39:41.180 align:middle line:84% this is a classical transformation. 00:39:41.180 --> 00:39:46.130 align:middle line:84% Here, it's actually not a unitary matrix. 00:39:46.130 --> 00:40:24.350 align:middle line:84% So this implies that S dagger, in general-- 00:40:24.350 --> 00:40:28.550 align:middle line:84% for general Lorentz transformation, say, 00:40:28.550 --> 00:40:31.685 align:middle line:84% which including both rotation boost-- 00:40:31.685 --> 00:40:38.480 align:middle line:90% 00:40:38.480 --> 00:40:41.030 align:middle line:84% say S and S dagger is not equal to 1. 00:40:41.030 --> 00:40:45.400 align:middle line:90% 00:40:45.400 --> 00:40:49.600 align:middle line:90% S dagger-- S is not equal to 1. 00:40:49.600 --> 00:40:53.800 align:middle line:84% So this has very important consequences, 00:40:53.800 --> 00:40:56.020 align:middle line:84% for example, for writing down the action 00:40:56.020 --> 00:40:57.895 align:middle line:90% for the Dirac equation. 00:40:57.895 --> 00:41:03.340 align:middle line:84% So so far, we only wrote down the Dirac equation, 00:41:03.340 --> 00:41:06.730 align:middle line:84% but we did-- remember, previously, we normally 00:41:06.730 --> 00:41:09.490 align:middle line:84% started with the action first, and then from the action, 00:41:09.490 --> 00:41:11.080 align:middle line:84% we derive the equation of motion. 00:41:11.080 --> 00:41:16.660 align:middle line:84% But in this case, since this spinor 00:41:16.660 --> 00:41:18.520 align:middle line:84% is a completely new concept, we started 00:41:18.520 --> 00:41:20.860 align:middle line:90% with actually the equation. 00:41:20.860 --> 00:41:23.500 align:middle line:84% But now, if we want to write down an action-- 00:41:23.500 --> 00:41:26.350 align:middle line:84% which is by definition, it should be Lorentz invariant-- 00:41:26.350 --> 00:41:30.370 align:middle line:84% then we should construct quantities 00:41:30.370 --> 00:41:33.310 align:middle line:84% which are invariant under Lorentz transformations. 00:41:33.310 --> 00:41:38.620 align:middle line:84% And so this property then becomes a key. 00:41:38.620 --> 00:41:39.620 align:middle line:90% Yes? 00:41:39.620 --> 00:41:41.620 align:middle line:84% STUDENT: So like, we're-- so these are operators 00:41:41.620 --> 00:41:45.770 align:middle line:84% on the spinor space, and we're talking about Hermiticity 00:41:45.770 --> 00:41:46.270 align:middle line:90% and stuff. 00:41:46.270 --> 00:41:50.130 align:middle line:84% But Hermitian is with respect to an inner product, 00:41:50.130 --> 00:41:51.210 align:middle line:90% and we haven't talked-- 00:41:51.210 --> 00:41:52.140 align:middle line:84% PROFESSOR: No, here, we are not talking 00:41:52.140 --> 00:41:53.250 align:middle line:90% about quantum mechanics. 00:41:53.250 --> 00:41:55.830 align:middle line:84% Here, we're just talking about the equations-- 00:41:55.830 --> 00:41:57.552 align:middle line:90% classical equations. 00:41:57.552 --> 00:41:59.010 align:middle line:84% We are just talking about-- they're 00:41:59.010 --> 00:42:01.050 align:middle line:90% matrix-- they're just matrix. 00:42:01.050 --> 00:42:04.530 align:middle line:84% We are talking about whether they are unitary matrix 00:42:04.530 --> 00:42:08.730 align:middle line:90% or they are not unitary matrix. 00:42:08.730 --> 00:42:10.470 align:middle line:84% They are just ordinary matrices-- 00:42:10.470 --> 00:42:11.595 align:middle line:90% 4x4 matrices. 00:42:11.595 --> 00:42:17.490 align:middle line:90% 00:42:17.490 --> 00:42:18.390 align:middle line:90% Other questions? 00:42:18.390 --> 00:42:27.350 align:middle line:90% 00:42:27.350 --> 00:42:36.820 align:middle line:90% So that means psi dagger psi-- 00:42:36.820 --> 00:42:41.560 align:middle line:84% so psi dagger is a row vector, and this is a column vector, 00:42:41.560 --> 00:42:44.350 align:middle line:90% and altogether this is a number. 00:42:44.350 --> 00:42:46.290 align:middle line:84% So this means this is not a scalar. 00:42:46.290 --> 00:42:49.440 align:middle line:90% 00:42:49.440 --> 00:42:52.230 align:middle line:84% So this transform under Lorentz transformation 00:42:52.230 --> 00:42:56.670 align:middle line:90% as psi dagger S dagger S psi. 00:42:56.670 --> 00:42:59.520 align:middle line:84% So that means this is not-- since this is not equal to 1, 00:42:59.520 --> 00:43:08.040 align:middle line:84% then this is not scalar under the Lorentz transformation. 00:43:08.040 --> 00:43:20.450 align:middle line:90% 00:43:20.450 --> 00:43:24.450 align:middle line:84% So now, we have to search a little bit harder 00:43:24.450 --> 00:43:31.230 align:middle line:90% to find the scalar. 00:43:31.230 --> 00:43:32.940 align:middle line:84% So in order to write the action, we 00:43:32.940 --> 00:43:36.315 align:middle line:84% need to find something which is invariant under the Lorentz 00:43:36.315 --> 00:43:38.750 align:middle line:90% transformation. 00:43:38.750 --> 00:43:40.810 align:middle line:84% So the easiest thing to think about 00:43:40.810 --> 00:43:43.540 align:middle line:84% is this quantity, because this automatically 00:43:43.540 --> 00:43:45.280 align:middle line:90% gives you a number. 00:43:45.280 --> 00:43:50.480 align:middle line:84% But this thing won't work, so we have to search it a bit harder. 00:43:50.480 --> 00:43:52.760 align:middle line:90% So to do that, let's-- 00:43:52.760 --> 00:43:57.140 align:middle line:84% we can get some hint from the following identity. 00:43:57.140 --> 00:44:02.490 align:middle line:84% So let's look at what this S dagger really is. 00:44:02.490 --> 00:44:04.470 align:middle line:84% So let's look at the property of S dagger. 00:44:04.470 --> 00:44:11.330 align:middle line:90% 00:44:11.330 --> 00:44:13.010 align:middle line:90% So recall that the-- 00:44:13.010 --> 00:44:17.040 align:middle line:84% I think I erased it, so let me just write it again. 00:44:17.040 --> 00:44:26.290 align:middle line:84% So gamma mu dagger is equal to gamma 0 gamma mu gamma 0. 00:44:26.290 --> 00:44:30.520 align:middle line:84% So now, let's try to find what mu nu dagger. 00:44:30.520 --> 00:44:32.950 align:middle line:84% So let's write mu nu dagger in the uniform-- 00:44:32.950 --> 00:44:36.260 align:middle line:84% so even though we wrote it separately. 00:44:36.260 --> 00:44:39.710 align:middle line:84% So you can easily check yourself because of this property. 00:44:39.710 --> 00:44:45.660 align:middle line:84% So this is just given by minus gamma 0 sigma mu nu gamma 0. 00:44:45.660 --> 00:44:48.680 align:middle line:90% 00:44:48.680 --> 00:44:50.390 align:middle line:84% Because this is just a commutator, 00:44:50.390 --> 00:44:56.960 align:middle line:84% so you can easily work it out that you just get this. 00:44:56.960 --> 00:44:59.720 align:middle line:84% So now we can find what S dagger. 00:44:59.720 --> 00:45:02.450 align:middle line:90% 00:45:02.450 --> 00:45:11.050 align:middle line:84% So S dagger is equal to the exponential 1/2 i omega 00:45:11.050 --> 00:45:14.230 align:middle line:90% mu nu, then sigma mu nu dagger. 00:45:14.230 --> 00:45:20.160 align:middle line:90% 00:45:20.160 --> 00:45:31.580 align:middle line:84% And then this is equal to i over 2 omega mu nu minus gamma 0 00:45:31.580 --> 00:45:33.920 align:middle line:90% sigma mu nu gamma 0. 00:45:33.920 --> 00:45:43.860 align:middle line:90% 00:45:43.860 --> 00:45:50.320 align:middle line:84% So now, if you remember, gamma 0 squared is equal to minus 1. 00:45:50.320 --> 00:45:52.030 align:middle line:84% So whenever you have such a situation-- 00:45:52.030 --> 00:45:56.590 align:middle line:84% and because gamma 0 squared is equal to minus 1, 00:45:56.590 --> 00:45:58.930 align:middle line:84% then you can actually take the gamma 0 outside 00:45:58.930 --> 00:46:00.190 align:middle line:90% of the exponential. 00:46:00.190 --> 00:46:09.220 align:middle line:84% So this is actually equal to minus gamma 0 exponential i 00:46:09.220 --> 00:46:13.260 align:middle line:84% over q comma 0 mu nu sigma mu nu. 00:46:13.260 --> 00:46:21.020 align:middle line:90% 00:46:21.020 --> 00:46:23.030 align:middle line:84% So if you just think-- you do a Taylor 00:46:23.030 --> 00:46:27.500 align:middle line:84% expansion of this exponential, and then 00:46:27.500 --> 00:46:34.380 align:middle line:84% when you take the power of this, then for each term, 00:46:34.380 --> 00:46:38.740 align:middle line:84% the gamma 0 at the end will pair with a gamma 0 00:46:38.740 --> 00:46:41.410 align:middle line:84% at the beginning of the other term, 00:46:41.410 --> 00:46:44.470 align:middle line:90% and then they give you minus 1. 00:46:44.470 --> 00:46:48.040 align:middle line:84% And then you have only the first gamma 0 and the last gamma 0 00:46:48.040 --> 00:46:49.180 align:middle line:90% left. 00:46:49.180 --> 00:46:50.980 align:middle line:84% And because that gives a minus 1, 00:46:50.980 --> 00:46:53.620 align:middle line:84% that changes this minus sign to a plus sign. 00:46:53.620 --> 00:46:55.840 align:middle line:84% So you just do a Taylor expansion you will find here. 00:46:55.840 --> 00:47:00.010 align:middle line:90% 00:47:00.010 --> 00:47:03.590 align:middle line:84% And now, we find the nice relation. 00:47:03.590 --> 00:47:07.060 align:middle line:84% So we find that this is just minus gamma 0, 00:47:07.060 --> 00:47:11.275 align:middle line:84% and this is just equal to S minus 1 S gamma 0. 00:47:11.275 --> 00:47:14.590 align:middle line:90% 00:47:14.590 --> 00:47:16.890 align:middle line:84% So we find that the S dagger is actually 00:47:16.890 --> 00:47:19.215 align:middle line:90% minus gamma 0 S minus 1. 00:47:19.215 --> 00:47:22.110 align:middle line:90% 00:47:22.110 --> 00:47:26.780 align:middle line:84% Then, this tells us from this property-- 00:47:26.780 --> 00:47:32.050 align:middle line:90% 00:47:32.050 --> 00:47:37.770 align:middle line:84% tells us that this quantity psi dagger 00:47:37.770 --> 00:47:44.270 align:middle line:84% gamma 0 psi should transform as a scalar. 00:47:44.270 --> 00:47:48.390 align:middle line:90% So now, let's take a look. 00:47:48.390 --> 00:47:51.540 align:middle line:84% So this, when you do a transformation-- 00:47:51.540 --> 00:47:56.180 align:middle line:84% so this gives you the psi dagger S dagger. 00:47:56.180 --> 00:47:59.390 align:middle line:84% Then, you have gamma 0, then you have S psi 00:47:59.390 --> 00:48:02.150 align:middle line:90% under Lorentz transformation. 00:48:02.150 --> 00:48:04.310 align:middle line:90% And now, you plug this into-- 00:48:04.310 --> 00:48:08.480 align:middle line:84% plug in S dagger equal to this into here. 00:48:08.480 --> 00:48:15.220 align:middle line:84% You just get minus psi dagger, so this 00:48:15.220 --> 00:48:24.040 align:middle line:84% is gamma 0 S dagger S minus 1 gamma 0, then gamma 0 S psi. 00:48:24.040 --> 00:48:27.940 align:middle line:84% So this gives you minus 1-- cancels with the 1 here. 00:48:27.940 --> 00:48:30.075 align:middle line:90% And the S one cancels with-- 00:48:30.075 --> 00:48:37.090 align:middle line:84% S minus 1 cancels with S, and then you have dagger goes here. 00:48:37.090 --> 00:48:40.133 align:middle line:84% So this actually is Lorentz invariant. 00:48:40.133 --> 00:48:44.560 align:middle line:90% 00:48:44.560 --> 00:48:48.630 align:middle line:84% So now, we find a nice Lorentz invariant quantity. 00:48:48.630 --> 00:48:54.840 align:middle line:84% So since we use this all the time, 00:48:54.840 --> 00:48:58.275 align:middle line:84% it's convenient to introduce a new notation. 00:48:58.275 --> 00:49:20.880 align:middle line:90% 00:49:20.880 --> 00:49:34.620 align:middle line:84% So now, I introduce -- convenient to introduce recall 00:49:34.620 --> 00:49:40.830 align:middle line:90% psi bar equal to psi dagger. 00:49:40.830 --> 00:49:42.810 align:middle line:84% So here, it's all just clustered. 00:49:42.810 --> 00:49:48.390 align:middle line:84% This is dagger, and it's all just matrix manipulation. 00:49:48.390 --> 00:49:52.290 align:middle line:84% Right now, we're considering a classical theory, 00:49:52.290 --> 00:49:56.190 align:middle line:84% and so it's convenient to introduce objects like this. 00:49:56.190 --> 00:49:59.880 align:middle line:90% And then we know that psi bar-- 00:49:59.880 --> 00:50:03.200 align:middle line:90% psi is a scalar. 00:50:03.200 --> 00:50:06.750 align:middle line:84% So that thing just becomes psi bar and psi. 00:50:06.750 --> 00:50:09.570 align:middle line:90% 00:50:09.570 --> 00:50:13.620 align:middle line:84% So it's convenient to work out the-- 00:50:13.620 --> 00:50:16.980 align:middle line:84% how psi bar transforms by itself. 00:50:16.980 --> 00:50:20.970 align:middle line:84% So you just use this relation so you can check yourself. 00:50:20.970 --> 00:50:24.180 align:middle line:84% So as an exercise for you to check yourself 00:50:24.180 --> 00:50:27.630 align:middle line:84% under Lorentz transformation psi bar 00:50:27.630 --> 00:50:34.990 align:middle line:84% x is equal to psi bar prime x prime. 00:50:34.990 --> 00:50:38.170 align:middle line:84% So if you go to the new psi bar prime x prime 00:50:38.170 --> 00:50:44.410 align:middle line:90% is equal to psi bar x S minus 1. 00:50:44.410 --> 00:50:47.245 align:middle line:84% So you can easily check yourself this relation. 00:50:47.245 --> 00:50:53.070 align:middle line:90% 00:50:53.070 --> 00:50:56.190 align:middle line:84% Similarly, using this transformation 00:50:56.190 --> 00:50:58.800 align:middle line:84% of gamma partial mu, you can also 00:50:58.800 --> 00:51:07.410 align:middle line:84% check that the gamma mu partial mu so prime 00:51:07.410 --> 00:51:16.310 align:middle line:84% psi prime x prime equal to S gamma mu partial mu psi. 00:51:16.310 --> 00:51:19.480 align:middle line:90% 00:51:19.480 --> 00:51:21.812 align:middle line:90% So you can also check yourself-- 00:51:21.812 --> 00:51:22.645 align:middle line:90% check this equation. 00:51:22.645 --> 00:51:29.580 align:middle line:90% 00:51:29.580 --> 00:51:34.710 align:middle line:84% So yeah, just by using these properties of S dagger. 00:51:34.710 --> 00:51:37.740 align:middle line:84% Yeah, just-- yeah, this, you don't use the-- 00:51:37.740 --> 00:51:39.690 align:middle line:84% you only use for above the S dagger. 00:51:39.690 --> 00:51:42.630 align:middle line:84% Just use how partial mu prime transform 00:51:42.630 --> 00:51:46.080 align:middle line:84% and how psi transform, and then you can show this is true. 00:51:46.080 --> 00:51:51.010 align:middle line:90% 00:51:51.010 --> 00:51:53.000 align:middle line:90% Good. 00:51:53.000 --> 00:51:57.190 align:middle line:84% So this-- also this gamma mu partial mu 00:51:57.190 --> 00:52:01.130 align:middle line:84% will appear a lot because this appears in the Dirac equation. 00:52:01.130 --> 00:52:05.620 align:middle line:84% So it's convenient to introduce a new notation. 00:52:05.620 --> 00:52:14.170 align:middle line:84% Say gamma mu partial mu we define to be partial slash. 00:52:14.170 --> 00:52:19.600 align:middle line:84% So essentially, anything slash corresponding 00:52:19.600 --> 00:52:21.400 align:middle line:84% to that thing contract with gamma mu. 00:52:21.400 --> 00:52:24.750 align:middle line:90% 00:52:24.750 --> 00:52:26.340 align:middle line:84% And then this equation then can be 00:52:26.340 --> 00:52:32.460 align:middle line:84% written as the partial prime slash psi 00:52:32.460 --> 00:52:38.582 align:middle line:84% prime x prime equal to S partial slash psi x. 00:52:38.582 --> 00:52:47.970 align:middle line:90% 00:52:47.970 --> 00:52:48.470 align:middle line:90% Good? 00:52:48.470 --> 00:52:55.320 align:middle line:90% 00:52:55.320 --> 00:52:57.420 align:middle line:84% So let me just mention one more thing 00:52:57.420 --> 00:53:00.030 align:middle line:90% you can work out yourself. 00:53:00.030 --> 00:53:03.030 align:middle line:90% So you can check-- 00:53:03.030 --> 00:53:07.500 align:middle line:84% so these are all things you can check yourself once you 00:53:07.500 --> 00:53:09.810 align:middle line:90% equip those transformations. 00:53:09.810 --> 00:53:19.290 align:middle line:84% You can also check that gamma psi bar gamma mu 00:53:19.290 --> 00:53:21.705 align:middle line:90% psi transforms as a vector. 00:53:21.705 --> 00:53:36.400 align:middle line:90% 00:53:36.400 --> 00:53:44.590 align:middle line:84% That means psi bar prime x prime gamma mu psi prime 00:53:44.590 --> 00:54:00.700 align:middle line:84% x prime is equal to lambda mu nu psi bar x gamma nu psi x. 00:54:00.700 --> 00:54:03.970 align:middle line:84% So if you view this whole thing as a vector, 00:54:03.970 --> 00:54:07.740 align:middle line:84% and then you see the prime-- the quantity 00:54:07.740 --> 00:54:11.770 align:middle line:84% is equal to just a Lorentz transformation 00:54:11.770 --> 00:54:13.770 align:middle line:90% lambda on itself. 00:54:13.770 --> 00:54:17.317 align:middle line:90% 00:54:17.317 --> 00:54:18.900 align:middle line:84% So again, this is something just based 00:54:18.900 --> 00:54:21.300 align:middle line:84% on the transformations of S and the relations 00:54:21.300 --> 00:54:25.980 align:middle line:84% between different gamma matrices you can just show that. 00:54:25.980 --> 00:54:30.030 align:middle line:90% 00:54:30.030 --> 00:54:30.530 align:middle line:90% Good. 00:54:30.530 --> 00:54:31.447 align:middle line:90% Any questions on this? 00:54:31.447 --> 00:54:39.010 align:middle line:90% 00:54:39.010 --> 00:54:40.900 align:middle line:84% So now, with these preparations, we 00:54:40.900 --> 00:54:45.040 align:middle line:84% can now write down the actions which gives rise 00:54:45.040 --> 00:54:45.985 align:middle line:90% to the Dirac equation. 00:54:45.985 --> 00:54:50.650 align:middle line:90% 00:54:50.650 --> 00:54:52.500 align:middle line:84% So now, we can write down the Dirac action. 00:54:52.500 --> 00:54:57.900 align:middle line:90% 00:54:57.900 --> 00:55:00.040 align:middle line:84% The action which gives rise to the Dirac equation. 00:55:00.040 --> 00:55:01.890 align:middle line:84% So I will just write down the answer. 00:55:01.890 --> 00:55:04.390 align:middle line:90% It's very intuitive. 00:55:04.390 --> 00:55:15.040 align:middle line:84% So you have just essentially S equal to minus i d4 x 00:55:15.040 --> 00:55:23.110 align:middle line:84% psi bar gamma mu partial mu minus m psi. 00:55:23.110 --> 00:55:24.235 align:middle line:90% So this is the answer. 00:55:24.235 --> 00:55:32.400 align:middle line:90% 00:55:32.400 --> 00:55:38.820 align:middle line:84% Yeah, so there are various things to uncompact here. 00:55:38.820 --> 00:55:43.380 align:middle line:84% So first, just based on those relations, based on this 00:55:43.380 --> 00:55:52.640 align:middle line:84% is a scalar, this is a scalar, and the transformation of this, 00:55:52.640 --> 00:55:55.400 align:middle line:84% you can immediately see that this is a Lorentz scalar 00:55:55.400 --> 00:55:58.610 align:middle line:84% because it only involves two variables-- one quantities-- 00:55:58.610 --> 00:56:01.820 align:middle line:84% psi bar, psi with this m term, and then psi bar 00:56:01.820 --> 00:56:03.440 align:middle line:90% gamma mu partial mu psi. 00:56:03.440 --> 00:56:05.990 align:middle line:90% And that we already show here-- 00:56:05.990 --> 00:56:09.620 align:middle line:90% that the transform as S. 00:56:09.620 --> 00:56:11.750 align:middle line:84% So yeah, you can easily check just 00:56:11.750 --> 00:56:14.480 align:middle line:84% based on those equations that this is a scalar. 00:56:14.480 --> 00:56:18.630 align:middle line:90% And this is Lorentz invariant. 00:56:18.630 --> 00:56:25.440 align:middle line:84% So the second thing is that this i here is for-- 00:56:25.440 --> 00:56:30.450 align:middle line:84% to make the action is real, because you 00:56:30.450 --> 00:56:32.830 align:middle line:84% can show if you take the complex conjugate of this, 00:56:32.830 --> 00:56:36.240 align:middle line:84% you actually get the minus sign, and so 00:56:36.240 --> 00:56:38.880 align:middle line:90% you need the i to make it real. 00:56:38.880 --> 00:56:42.930 align:middle line:84% And this minus sign, which cannot be explained now, 00:56:42.930 --> 00:56:47.100 align:middle line:84% which we will talk about it later when we quantize 00:56:47.100 --> 00:56:50.430 align:middle line:84% the theory and we see actually, we need to put the minus sign 00:56:50.430 --> 00:56:51.390 align:middle line:90% here. 00:56:51.390 --> 00:56:52.470 align:middle line:90% Yeah? 00:56:52.470 --> 00:56:57.630 align:middle line:84% STUDENT: And the definition of psi bar has psi dagger gamma 0. 00:56:57.630 --> 00:56:59.850 align:middle line:84% It feels like we're singling out the time 00:56:59.850 --> 00:57:03.930 align:middle line:84% dimension and the definition, because gamma 0-- 00:57:03.930 --> 00:57:06.780 align:middle line:84% is that true that we're singling out the time dimension? 00:57:06.780 --> 00:57:09.120 align:middle line:84% PROFESSOR: We are not really singling out 00:57:09.120 --> 00:57:11.790 align:middle line:90% the time direction. 00:57:11.790 --> 00:57:15.450 align:middle line:84% just due to the property that gamma 0 is not a Hermitian. 00:57:15.450 --> 00:57:21.060 align:middle line:84% Yeah, just because if you look at all this complicate-- 00:57:21.060 --> 00:57:24.000 align:middle line:84% just related to that this S dagger is not-- 00:57:24.000 --> 00:57:25.860 align:middle line:90% yeah, S is not unitary. 00:57:25.860 --> 00:57:29.170 align:middle line:84% And the reason S is not unitary is because gamma 0 is not 00:57:29.170 --> 00:57:30.010 align:middle line:90% a hermitian. 00:57:30.010 --> 00:57:35.110 align:middle line:84% Yeah, so you have to put in the gamma 0 in various places 00:57:35.110 --> 00:57:36.205 align:middle line:90% to compensate for that. 00:57:36.205 --> 00:57:41.933 align:middle line:90% 00:57:41.933 --> 00:57:42.600 align:middle line:90% Other questions. 00:57:42.600 --> 00:57:43.365 align:middle line:90% Yes? 00:57:43.365 --> 00:57:46.650 align:middle line:84% STUDENT: So I guess the action, if you wanted a Lorentz 00:57:46.650 --> 00:57:48.510 align:middle line:84% invariant-- a Lorentz scalar, you 00:57:48.510 --> 00:57:51.390 align:middle line:84% could have constructed some dot product of that object that 00:57:51.390 --> 00:57:54.840 align:middle line:90% transforms as a vector, right? 00:57:54.840 --> 00:57:57.990 align:middle line:84% So the psi bar gamma-- so that would be allowed in principle. 00:57:57.990 --> 00:57:59.490 align:middle line:84% I guess this gives the right answer, 00:57:59.490 --> 00:58:02.458 align:middle line:84% but that could've appeared in the action. 00:58:02.458 --> 00:58:04.500 align:middle line:84% PROFESSOR: Which-- yeah, this appeared in action. 00:58:04.500 --> 00:58:08.560 align:middle line:84% Yeah, so this is the-- so because this is a vector-- 00:58:08.560 --> 00:58:10.470 align:middle line:84% so when this contracts with partial mu, 00:58:10.470 --> 00:58:14.320 align:middle line:90% that gives you a scalar. 00:58:14.320 --> 00:58:16.890 align:middle line:84% Yeah, so that gives you a scalar. 00:58:16.890 --> 00:58:19.390 align:middle line:84% So yeah, you can understand that this is-- 00:58:19.390 --> 00:58:23.640 align:middle line:84% in both ways, this transform-- the fact this transform vector 00:58:23.640 --> 00:58:25.740 align:middle line:90% is also related to here. 00:58:25.740 --> 00:58:26.675 align:middle line:90% I mean, just-- yeah. 00:58:26.675 --> 00:58:30.770 align:middle line:90% 00:58:30.770 --> 00:58:32.490 align:middle line:90% Other questions? 00:58:32.490 --> 00:58:33.852 align:middle line:90% So-- yes? 00:58:33.852 --> 00:58:36.435 align:middle line:84% STUDENT: To follow up on that, if I contract that with itself, 00:58:36.435 --> 00:58:38.102 align:middle line:84% would it give like another [INAUDIBLE].. 00:58:38.102 --> 00:58:43.293 align:middle line:90% 00:58:43.293 --> 00:58:43.960 align:middle line:90% PROFESSOR: What? 00:58:43.960 --> 00:58:46.930 align:middle line:84% STUDENT: if I contract that [INAUDIBLE] with itself. 00:58:46.930 --> 00:58:53.800 align:middle line:84% PROFESSOR: Yeah, but then you will have four psi's. 00:58:53.800 --> 00:58:57.478 align:middle line:84% So four psi's will give you rise to an interacting theory. 00:58:57.478 --> 00:58:58.270 align:middle line:90% Yeah, that's right. 00:58:58.270 --> 00:59:01.530 align:middle line:84% Here, I'm writing down a free theory right now. 00:59:01.530 --> 00:59:05.110 align:middle line:90% 00:59:05.110 --> 00:59:10.130 align:middle line:84% So you can also see that this equation gives rise 00:59:10.130 --> 00:59:11.480 align:middle line:90% to the Dirac equation. 00:59:11.480 --> 00:59:14.030 align:middle line:84% Just imagine psi bar is independent of psi, 00:59:14.030 --> 00:59:15.860 align:middle line:84% because this corresponding to psi dagger. 00:59:15.860 --> 00:59:17.690 align:middle line:84% So if you do the variation of this one, 00:59:17.690 --> 00:59:20.510 align:middle line:84% you just automatically get the Dirac equation. 00:59:20.510 --> 00:59:23.730 align:middle line:84% Then you can also easily check by integration by part. 00:59:23.730 --> 00:59:28.070 align:middle line:84% And if you were psi, you get the complex conjugate version 00:59:28.070 --> 00:59:32.780 align:middle line:84% of the Dirac equation acting on psi bar. 00:59:32.780 --> 00:59:35.870 align:middle line:84% So now let me just say a few words 00:59:35.870 --> 00:59:37.760 align:middle line:90% on why you need the i here. 00:59:37.760 --> 00:59:39.590 align:middle line:90% So you can check. 00:59:39.590 --> 00:59:41.810 align:middle line:84% So here, I'm listing some relations. 00:59:41.810 --> 00:59:45.290 align:middle line:84% Again, each of them, you need to really write down the paper 00:59:45.290 --> 00:59:49.340 align:middle line:84% yourself, stare at it, maybe do a derivation to get intuition 00:59:49.340 --> 00:59:50.210 align:middle line:90% about yourself. 00:59:50.210 --> 00:59:53.210 align:middle line:84% Right now, I'm just writing down those relations. 00:59:53.210 --> 01:00:00.380 align:middle line:84% So you can also check yourself that when 01:00:00.380 --> 01:00:07.520 align:middle line:84% you take the psi bar psi dagger, again, 01:00:07.520 --> 01:00:10.070 align:middle line:84% just go through all these things. 01:00:10.070 --> 01:00:13.670 align:middle line:84% You find it's equal to minus psi bar psi. 01:00:13.670 --> 01:00:17.100 align:middle line:90% So this is just two lines here. 01:00:17.100 --> 01:00:20.640 align:middle line:84% I will not write them explicitly for you. 01:00:20.640 --> 01:00:25.700 align:middle line:84% And similarly, you can check related to this-- 01:00:25.700 --> 01:00:31.230 align:middle line:84% the other term that the gamma psi bar 01:00:31.230 --> 01:00:35.620 align:middle line:90% gamma mu partial mu psi-- 01:00:35.620 --> 01:00:37.480 align:middle line:90% if you take the dagger-- 01:00:37.480 --> 01:00:41.920 align:middle line:84% so here-- so this one is slightly more complicated, 01:00:41.920 --> 01:00:43.150 align:middle line:90% but you can still-- 01:00:43.150 --> 01:00:44.830 align:middle line:90% you can just walk it through. 01:00:44.830 --> 01:00:54.090 align:middle line:84% You find it's just equal to negative of itself, 01:00:54.090 --> 01:00:57.130 align:middle line:90% then plus total derivatives. 01:00:57.130 --> 01:00:59.010 align:middle line:84% So this one is not exactly minus sign, 01:00:59.010 --> 01:01:01.140 align:middle line:84% but you have to throw away some total derivatives. 01:01:01.140 --> 01:01:03.960 align:middle line:84% So total derivatives which give you-- 01:01:03.960 --> 01:01:07.590 align:middle line:84% when you plug into the action, it 01:01:07.590 --> 01:01:10.710 align:middle line:84% gives rise to boundary terms, but it will always assume 01:01:10.710 --> 01:01:13.800 align:middle line:90% the field vanish at infinity. 01:01:13.800 --> 01:01:17.460 align:middle line:84% So that explains why you need this i, because when 01:01:17.460 --> 01:01:21.870 align:middle line:84% you take the complex conjugate of the quantity here, 01:01:21.870 --> 01:01:23.880 align:middle line:90% you always get the minus sign. 01:01:23.880 --> 01:01:26.750 align:middle line:90% 01:01:26.750 --> 01:01:27.250 align:middle line:90% Good. 01:01:27.250 --> 01:01:28.210 align:middle line:90% Any questions? 01:01:28.210 --> 01:01:39.570 align:middle line:90% 01:01:39.570 --> 01:01:40.770 align:middle line:90% Good. 01:01:40.770 --> 01:01:43.110 align:middle line:84% So now, it's getting a little bit awkward 01:01:43.110 --> 01:01:45.840 align:middle line:90% because our printer today broke. 01:01:45.840 --> 01:01:49.930 align:middle line:84% So I didn't bring enough of my notes, 01:01:49.930 --> 01:01:54.960 align:middle line:84% so now I have to look at my computer to remember my notes. 01:01:54.960 --> 01:02:00.605 align:middle line:90% 01:02:00.605 --> 01:02:01.105 align:middle line:90% One second. 01:02:01.105 --> 01:02:06.820 align:middle line:90% 01:02:06.820 --> 01:02:08.970 align:middle line:90% Good. 01:02:08.970 --> 01:02:09.470 align:middle line:90% Ugh. 01:02:09.470 --> 01:02:24.760 align:middle line:90% 01:02:24.760 --> 01:02:25.930 align:middle line:90% I have to find the location. 01:02:25.930 --> 01:02:41.570 align:middle line:90% 01:02:41.570 --> 01:02:43.080 align:middle line:90% OK, good. 01:02:43.080 --> 01:02:46.740 align:middle line:90% So any questions on this? 01:02:46.740 --> 01:02:47.240 align:middle line:90% No? 01:02:47.240 --> 01:02:50.670 align:middle line:84% OK, so now let's move to the next topic. 01:02:50.670 --> 01:02:53.240 align:middle line:84% So now, this concludes our discussion 01:02:53.240 --> 01:02:55.020 align:middle line:90% of the Dirac equation. 01:02:55.020 --> 01:02:59.660 align:middle line:84% So we have derived the Dirac equation by-- 01:02:59.660 --> 01:03:03.230 align:middle line:84% and also we have discussed how various quantities 01:03:03.230 --> 01:03:07.550 align:middle line:84% in Dirac equation should transform, and also 01:03:07.550 --> 01:03:08.840 align:middle line:90% finally the Dirac action. 01:03:08.840 --> 01:03:11.540 align:middle line:90% 01:03:11.540 --> 01:03:14.000 align:middle line:90% So the next step-- 01:03:14.000 --> 01:03:17.150 align:middle line:84% the logical next step is to quantize 01:03:17.150 --> 01:03:18.650 align:middle line:84% to go to the quantum field theory. 01:03:18.650 --> 01:03:21.920 align:middle line:84% So right now, so far, everything is classical. 01:03:21.920 --> 01:03:27.170 align:middle line:84% So now, the next step is that we want to quantize this theory. 01:03:27.170 --> 01:03:29.570 align:middle line:84% And when we quantize the theory, we 01:03:29.570 --> 01:03:33.860 align:middle line:90% will see remarkably fermions. 01:03:33.860 --> 01:03:35.300 align:middle line:90% So we will see fermions. 01:03:35.300 --> 01:03:38.240 align:middle line:90% We will see Pauli principle. 01:03:38.240 --> 01:03:42.890 align:middle line:84% But before-- but as we said before, 01:03:42.890 --> 01:03:47.550 align:middle line:84% that when we quantize the theory, we first need to-- 01:03:47.550 --> 01:03:50.100 align:middle line:84% the simplest way is to first find 01:03:50.100 --> 01:03:53.730 align:middle line:84% this all is classical solutions, and then 01:03:53.730 --> 01:03:55.650 align:middle line:84% the classical solutions then become solutions 01:03:55.650 --> 01:03:57.840 align:middle line:84% to the operator equations, and then we 01:03:57.840 --> 01:03:59.910 align:middle line:90% can automatically quantize. 01:03:59.910 --> 01:04:03.810 align:middle line:84% So then before, we actually do the quantization. 01:04:03.810 --> 01:04:08.250 align:middle line:84% It's better we try to find all the classical solutions 01:04:08.250 --> 01:04:10.380 align:middle line:90% of Dirac equation. 01:04:10.380 --> 01:04:12.740 align:middle line:84% And so it's not like a Klein-Gordon equation-- 01:04:12.740 --> 01:04:15.140 align:middle line:84% we can just immediately write down the solutions. 01:04:15.140 --> 01:04:18.360 align:middle line:84% And the Dirac equation is a little bit more intricate, 01:04:18.360 --> 01:04:23.960 align:middle line:84% so we need to spend a little bit of effort to write down the-- 01:04:23.960 --> 01:04:27.620 align:middle line:84% to find all the solutions of the Dirac equations. 01:04:27.620 --> 01:04:30.110 align:middle line:90% So that's what we will do. 01:04:30.110 --> 01:04:34.920 align:middle line:84% Unfortunately, we are not going to finish today, 01:04:34.920 --> 01:04:38.670 align:middle line:84% and then we will have a long spring break. 01:04:38.670 --> 01:04:44.370 align:middle line:84% So I hope you still remember what we talked about today when 01:04:44.370 --> 01:04:45.045 align:middle line:90% you come back. 01:04:45.045 --> 01:04:54.510 align:middle line:90% 01:04:54.510 --> 01:05:06.190 align:middle line:90% So this is [INAUDIBLE]. 01:05:06.190 --> 01:05:09.640 align:middle line:84% So now, we have classical solutions. 01:05:09.640 --> 01:05:19.300 align:middle line:90% 01:05:19.300 --> 01:05:26.440 align:middle line:84% So by construction, the Dirac equation has solutions. 01:05:26.440 --> 01:05:29.620 align:middle line:90% 01:05:29.620 --> 01:05:31.660 align:middle line:84% We know must be proportional to e to the i k 01:05:31.660 --> 01:05:38.470 align:middle line:84% x with k squared equal to minus m squared. 01:05:38.470 --> 01:05:39.670 align:middle line:90% So this is by construction. 01:05:39.670 --> 01:05:45.680 align:middle line:84% Because we square it, we get the Klein-Gordon equation. 01:05:45.680 --> 01:05:52.060 align:middle line:84% So this k mu would be omega k and k. 01:05:52.060 --> 01:05:56.870 align:middle line:90% 01:05:56.870 --> 01:06:01.760 align:middle line:84% But this is not enough because the Dirac equation-- 01:06:01.760 --> 01:06:04.940 align:middle line:84% because psi have four components. 01:06:04.940 --> 01:06:08.220 align:middle line:84% So this just determines one factor of it. 01:06:08.220 --> 01:06:10.550 align:middle line:84% And we also have to determine it's-- 01:06:10.550 --> 01:06:14.180 align:middle line:84% the behavior of all its four components. 01:06:14.180 --> 01:06:18.990 align:middle line:84% And so now, that's we are going to do now. 01:06:18.990 --> 01:06:22.770 align:middle line:84% So we will separate the solutions into two types. 01:06:22.770 --> 01:06:27.870 align:middle line:84% So one type is we call psi plus x, which corresponding 01:06:27.870 --> 01:06:33.760 align:middle line:90% to uk x, the expression ik x. 01:06:33.760 --> 01:06:37.850 align:middle line:84% And another we call the psi minus x, 01:06:37.850 --> 01:06:40.900 align:middle line:90% which corresponding to vk x. 01:06:40.900 --> 01:06:42.400 align:middle line:90% vk-- yeah, sorry-- 01:06:42.400 --> 01:06:45.680 align:middle line:90% no x-- because it's minus ik x. 01:06:45.680 --> 01:06:49.950 align:middle line:90% 01:06:49.950 --> 01:06:53.550 align:middle line:84% Yeah, so uk vk, they just-- they're 01:06:53.550 --> 01:06:59.280 align:middle line:84% all four-component spinors which have the same thing as psi, 01:06:59.280 --> 01:07:01.240 align:middle line:84% because this is just some number. 01:07:01.240 --> 01:07:03.720 align:middle line:90% So because this is-- 01:07:03.720 --> 01:07:07.528 align:middle line:84% because k have the positive frequency, 01:07:07.528 --> 01:07:09.945 align:middle line:84% so this is some kind of-- called positive energy solution. 01:07:09.945 --> 01:07:14.820 align:middle line:90% 01:07:14.820 --> 01:07:18.960 align:middle line:84% And this is called the negative energy solution. 01:07:18.960 --> 01:07:22.410 align:middle line:84% But it's the same thing as in the scalar case. 01:07:22.410 --> 01:07:25.568 align:middle line:84% They don't really-- there's not really-- 01:07:25.568 --> 01:07:27.360 align:middle line:84% when we quantize the theory, there's really 01:07:27.360 --> 01:07:31.770 align:middle line:90% no negative energy excitations. 01:07:31.770 --> 01:07:33.360 align:middle line:90% It's just a name. 01:07:33.360 --> 01:07:36.330 align:middle line:84% So we call the positive energy and negative energy 01:07:36.330 --> 01:07:37.350 align:middle line:90% just a name. 01:07:37.350 --> 01:07:39.120 align:middle line:84% So the actual physical excitations 01:07:39.120 --> 01:07:42.240 align:middle line:90% always have positive energy. 01:07:42.240 --> 01:07:52.890 align:middle line:84% And this uk and vk, they are all four-component complex vectors. 01:07:52.890 --> 01:08:01.420 align:middle line:90% 01:08:01.420 --> 01:08:02.080 align:middle line:90% Yes? 01:08:02.080 --> 01:08:08.140 align:middle line:84% STUDENT: So it seems like the psi should be labeled by k. 01:08:08.140 --> 01:08:13.990 align:middle line:84% PROFESSOR: Yeah, psi-- so this is just the basis of solutions. 01:08:13.990 --> 01:08:16.165 align:middle line:84% Indeed, we should label them by k. 01:08:16.165 --> 01:08:19.649 align:middle line:90% 01:08:19.649 --> 01:08:20.149 align:middle line:90% Good. 01:08:20.149 --> 01:08:24.050 align:middle line:84% So this-- so essentially, we just-- 01:08:24.050 --> 01:08:26.029 align:middle line:84% in the scalar case, you just expand it 01:08:26.029 --> 01:08:29.640 align:middle line:84% in terms of plane wave, and then you just get some constant. 01:08:29.640 --> 01:08:33.390 align:middle line:84% So here, its a little bit indicate we have a vector. 01:08:33.390 --> 01:08:35.016 align:middle line:84% Now, we need to solve these vectors. 01:08:35.016 --> 01:08:36.599 align:middle line:84% So our goal is to solve these vectors. 01:08:36.599 --> 01:08:39.609 align:middle line:90% 01:08:39.609 --> 01:08:42.640 align:middle line:84% And then you just plug them into the Dirac equation. 01:08:42.640 --> 01:08:45.080 align:middle line:84% Just plug these two into the Dirac equation, 01:08:45.080 --> 01:08:52.149 align:middle line:84% and then you get the equations for the uk and the vk. 01:08:52.149 --> 01:08:55.450 align:middle line:84% And I will also suppress the k in the uv 01:08:55.450 --> 01:08:59.470 align:middle line:84% and v just for notational simplicity. 01:08:59.470 --> 01:09:02.920 align:middle line:90% So when you have the gamma mu-- 01:09:02.920 --> 01:09:06.430 align:middle line:84% partial mu minus m, psi equal to 0, 01:09:06.430 --> 01:09:09.490 align:middle line:84% you just plug the psi plus minus m, 01:09:09.490 --> 01:09:14.500 align:middle line:84% and then you find that the following equations for u 01:09:14.500 --> 01:09:19.450 align:middle line:84% if you get i k slash minus m u equal to 0, 01:09:19.450 --> 01:09:24.890 align:middle line:84% and the i k slash and the minus-- 01:09:24.890 --> 01:09:27.560 align:middle line:84% yeah, I think-- let me just double-check the sign. 01:09:27.560 --> 01:09:28.970 align:middle line:90% Yeah, indeed. 01:09:28.970 --> 01:09:34.490 align:middle line:84% So you get i k slash plus m V equal to 0. 01:09:34.490 --> 01:09:36.510 align:middle line:90% So you get these two equations. 01:09:36.510 --> 01:09:39.500 align:middle line:84% So our goal is just to solve those equations. 01:09:39.500 --> 01:09:46.680 align:middle line:84% And the k slash, just as defined before, it just 01:09:46.680 --> 01:09:52.356 align:middle line:84% defines as in the partial slash k slash equal to mu gamma. 01:09:52.356 --> 01:09:58.080 align:middle line:90% 01:09:58.080 --> 01:09:59.610 align:middle line:90% Good. 01:09:59.610 --> 01:10:02.120 align:middle line:84% So you can also work out the complex conjugate. 01:10:02.120 --> 01:10:06.230 align:middle line:84% So let me just write down the equations, 01:10:06.230 --> 01:10:08.960 align:middle line:84% because sometimes they will be used later. 01:10:08.960 --> 01:10:14.750 align:middle line:84% So the u bar i k slash minus m equal to 0, 01:10:14.750 --> 01:10:15.980 align:middle line:90% the complex conjugate. 01:10:15.980 --> 01:10:20.000 align:middle line:90% And v bar i k slash plus m. 01:10:20.000 --> 01:10:33.480 align:middle line:90% 01:10:33.480 --> 01:10:41.770 align:middle line:84% So now, let's try to work out the uk and the vk 01:10:41.770 --> 01:10:45.450 align:middle line:90% by solving those equations. 01:10:45.450 --> 01:10:50.830 align:middle line:90% So we do this by-- 01:10:50.830 --> 01:10:54.890 align:middle line:84% you can, in principle, do it by brute force. 01:10:54.890 --> 01:10:57.360 align:middle line:90% So after all, you just-- 01:10:57.360 --> 01:10:59.760 align:middle line:90% these are just 4x4 equations-- 01:10:59.760 --> 01:11:02.940 align:middle line:90% 4x4 matrix equations. 01:11:02.940 --> 01:11:04.200 align:middle line:90% Just linear algebra. 01:11:04.200 --> 01:11:08.470 align:middle line:84% You can-- in principle, we can solve it. 01:11:08.470 --> 01:11:11.580 align:middle line:84% But the physicists are often lazy, so we often-- 01:11:11.580 --> 01:11:14.610 align:middle line:84% still even for the problem, you can solve-- 01:11:14.610 --> 01:11:16.650 align:middle line:90% we still look for shortcuts. 01:11:16.650 --> 01:11:19.570 align:middle line:84% And so in this case, there are two possible shortcuts. 01:11:19.570 --> 01:11:23.310 align:middle line:84% There are two possible ways we can do, and so 01:11:23.310 --> 01:11:26.110 align:middle line:90% let me describe both ways. 01:11:26.110 --> 01:11:30.250 align:middle line:84% Actually, I think we only have time to describe one way. 01:11:30.250 --> 01:11:39.820 align:middle line:84% So to find the explicit form of u and v-- 01:11:39.820 --> 01:11:43.710 align:middle line:90% 01:11:43.710 --> 01:11:45.630 align:middle line:84% so one simple thing to do is let's 01:11:45.630 --> 01:11:47.850 align:middle line:84% just consider in the simple case-- 01:11:47.850 --> 01:11:49.830 align:middle line:84% say, consider the particle is at rest. 01:11:49.830 --> 01:11:53.550 align:middle line:90% 01:11:53.550 --> 01:11:56.250 align:middle line:84% When the particle is at rest, then the omega 01:11:56.250 --> 01:12:01.740 align:middle line:84% is just equal to m, and then k equal to 0. 01:12:01.740 --> 01:12:03.750 align:middle line:84% And then that equation just becomes 01:12:03.750 --> 01:12:12.638 align:middle line:84% i m gamma 0 minus m u equal to 0 and i 01:12:12.638 --> 01:12:18.740 align:middle line:90% m gamma 0 plus m v equal to 0. 01:12:18.740 --> 01:12:21.700 align:middle line:90% 01:12:21.700 --> 01:12:23.350 align:middle line:90% I think-- do I-- 01:12:23.350 --> 01:12:28.590 align:middle line:90% 01:12:28.590 --> 01:12:31.770 align:middle line:90% so I think the mu-- 01:12:31.770 --> 01:12:32.520 align:middle line:90% oh, yeah. 01:12:32.520 --> 01:12:38.561 align:middle line:84% Sorry, I should have a minus sign here because of the-- 01:12:38.561 --> 01:12:41.840 align:middle line:90% it's the k upper index. 01:12:41.840 --> 01:12:44.750 align:middle line:84% So it's the k upper index equal to omega. 01:12:44.750 --> 01:12:47.900 align:middle line:84% Then k lower index equal to minus omega. 01:12:47.900 --> 01:12:51.310 align:middle line:90% 01:12:51.310 --> 01:12:53.440 align:middle line:84% And here, when we contract with gamma 0, 01:12:53.440 --> 01:12:56.030 align:middle line:84% we use the lower index, so we have a minus sign here. 01:12:56.030 --> 01:13:00.560 align:middle line:90% 01:13:00.560 --> 01:13:02.240 align:middle line:84% So essentially, this just becomes-- 01:13:02.240 --> 01:13:04.490 align:middle line:84% so m can be canceled on both sides. 01:13:04.490 --> 01:13:08.180 align:middle line:84% So essentially, it just becomes the i gamma 01:13:08.180 --> 01:13:16.775 align:middle line:84% 0 u equal to minus u and i gamma 0 v equal to-- 01:13:16.775 --> 01:13:17.275 align:middle line:90% OK. 01:13:17.275 --> 01:13:27.990 align:middle line:90% 01:13:27.990 --> 01:13:30.890 align:middle line:84% So this just tells you essentially gamma u-- 01:13:30.890 --> 01:13:35.230 align:middle line:84% u-- it's an eigenvector of u and v. 01:13:35.230 --> 01:13:37.600 align:middle line:84% They are just essentially the eigenvectors of gamma 0. 01:13:37.600 --> 01:13:41.500 align:middle line:90% 01:13:41.500 --> 01:13:45.400 align:middle line:84% And so let's just now-- to write them explicitly. 01:13:45.400 --> 01:13:47.800 align:middle line:84% So now, let's use the explicit representation 01:13:47.800 --> 01:13:49.780 align:middle line:90% of gamma matrices. 01:13:49.780 --> 01:13:51.940 align:middle line:90% Let me just copy this. 01:13:51.940 --> 01:13:58.210 align:middle line:84% So now, let's use this representation with the gamma 0 01:13:58.210 --> 01:14:02.920 align:middle line:90% is equal to i0 0i minus i. 01:14:02.920 --> 01:14:09.385 align:middle line:84% And then gamma i is equal to 0i sigma i minus i sigma i0. 01:14:09.385 --> 01:14:13.460 align:middle line:90% 01:14:13.460 --> 01:14:16.160 align:middle line:84% And then now, if you plug in this gamma 0 01:14:16.160 --> 01:14:20.820 align:middle line:84% into those equations, and then you 01:14:20.820 --> 01:14:23.790 align:middle line:84% find the equation for u and v becomes very simple. 01:14:23.790 --> 01:14:31.280 align:middle line:84% For the u, it just becomes 0, 0, 0, 1 u equal to 0. 01:14:31.280 --> 01:14:34.760 align:middle line:84% So again, this is 2 by 2 blocks because this here 01:14:34.760 --> 01:14:36.410 align:middle line:90% is 2 by 2 blocks. 01:14:36.410 --> 01:14:47.720 align:middle line:84% And for v, just the 1, 0, 0 v equal to 0. 01:14:47.720 --> 01:14:53.410 align:middle line:84% So that means u-- you can just take it to be the upper-- 01:14:53.410 --> 01:14:55.780 align:middle line:84% so that means the solution of this equation-- 01:14:55.780 --> 01:15:05.820 align:middle line:90% 01:15:05.820 --> 01:15:09.090 align:middle line:84% so that means that the solution of this equation for u-- 01:15:09.090 --> 01:15:11.570 align:middle line:84% so now let me write the upper index 01:15:11.570 --> 01:15:15.750 align:middle line:84% 0 means that this is for the 0 momentum. 01:15:15.750 --> 01:15:19.280 align:middle line:84% So here, we can just choose to pick psi 0. 01:15:19.280 --> 01:15:23.610 align:middle line:84% So psi is some arbitrary two vector. 01:15:23.610 --> 01:15:31.320 align:middle line:84% For v, we can just choose the lower 0 eta. 01:15:31.320 --> 01:15:40.280 align:middle line:84% So psi eta are arbitrary two vectors-- 01:15:40.280 --> 01:15:43.285 align:middle line:84% two complex vectors-- two vector-- 01:15:43.285 --> 01:15:46.310 align:middle line:90% 01:15:46.310 --> 01:15:52.070 align:middle line:84% two component vectors-- two component complex vectors. 01:15:52.070 --> 01:15:57.650 align:middle line:90% 01:15:57.650 --> 01:16:02.180 align:middle line:84% So once you have u0 and v0, and we can choose a basis. 01:16:02.180 --> 01:16:08.351 align:middle line:84% For example, u0 1 equal to 0, 1, -- 01:16:08.351 --> 01:16:16.130 align:middle line:84% 1, 0, 0, 0, and u0 2 equal to 0, 1, 0, 0. 01:16:16.130 --> 01:16:22.370 align:middle line:84% And similarly, for v0 1 and v0 2. 01:16:22.370 --> 01:16:26.760 align:middle line:84% So you can choose to be here 1, and-- 01:16:26.760 --> 01:16:32.580 align:middle line:90% 01:16:32.580 --> 01:16:34.580 align:middle line:84% so you can just choose as a basis-- you can just 01:16:34.580 --> 01:16:38.870 align:middle line:90% choose [INAUDIBLE]. 01:16:38.870 --> 01:16:40.100 align:middle line:90% And now, once you have-- 01:16:40.100 --> 01:16:45.230 align:middle line:90% 01:16:45.230 --> 01:16:55.970 align:middle line:84% so once you have the vector at k equal to 0, for general k, 01:16:55.970 --> 01:16:58.170 align:middle line:90% what do you do? 01:16:58.170 --> 01:16:58.670 align:middle line:90% Yes? 01:16:58.670 --> 01:16:59.180 align:middle line:90% STUDENT: Lorentz boost. 01:16:59.180 --> 01:17:00.847 align:middle line:84% PROFESSOR: Yes, just do a Lorentz boost. 01:17:00.847 --> 01:17:04.370 align:middle line:84% So we know the matrix S. And then you just-- 01:17:04.370 --> 01:17:15.560 align:middle line:84% for the general k, you just S u0 and vk Sv 0. 01:17:15.560 --> 01:17:20.440 align:middle line:84% And then you can find the behavior general k. 01:17:20.440 --> 01:17:25.296 align:middle line:84% But this is-- but this is easily said than do. 01:17:25.296 --> 01:17:28.720 align:middle line:84% To do this actually is not quite easy. 01:17:28.720 --> 01:17:31.060 align:middle line:84% Even though this sounds like a great idea-- 01:17:31.060 --> 01:17:36.820 align:middle line:84% OK, let's find the 0 momentum and then let's just do a boost. 01:17:36.820 --> 01:17:39.340 align:middle line:84% But this step is still a little bit tedious. 01:17:39.340 --> 01:17:43.540 align:middle line:84% But still it's doable and a little bit simpler 01:17:43.540 --> 01:17:47.980 align:middle line:84% than solve the original equation by brute force. 01:17:47.980 --> 01:17:52.960 align:middle line:84% But there's, again, still another simpler methods, 01:17:52.960 --> 01:17:55.000 align:middle line:84% which you can actually just guess the answer. 01:17:55.000 --> 01:17:57.040 align:middle line:84% You don't have to do any calculations. 01:17:57.040 --> 01:17:59.680 align:middle line:90% You can just guess the answer-- 01:17:59.680 --> 01:18:02.980 align:middle line:84% guess the solution for the full equation. 01:18:02.980 --> 01:18:06.320 align:middle line:84% And I think we don't have time to talk about it today, 01:18:06.320 --> 01:18:08.890 align:middle line:84% and so we will talk about it next time. 01:18:08.890 --> 01:18:13.210 align:middle line:84% And yeah, so hopefully, you still 01:18:13.210 --> 01:18:16.750 align:middle line:84% remember what we talked about today when we come back. 01:18:16.750 --> 01:18:20.360 align:middle line:84% And hope you have a good spring break. 01:18:20.360 --> 01:18:21.910 align:middle line:90% Yeah. 01:18:21.910 --> 01:18:26.000 align:middle line:90%