WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:04.410 align:middle line:90% [RUSTLING] 00:00:04.410 --> 00:00:08.330 align:middle line:90% [CLICKING] 00:00:08.330 --> 00:00:11.000 align:middle line:90% 00:00:11.000 --> 00:00:15.630 align:middle line:84% PROFESSOR: So I hope you all have had a great spring break. 00:00:15.630 --> 00:00:21.190 align:middle line:84% So before the break, we talked about Dirac equation. 00:00:21.190 --> 00:00:23.490 align:middle line:90% So we introduced the equation. 00:00:23.490 --> 00:00:30.150 align:middle line:84% We showed that it is Lorentz covariant. 00:00:30.150 --> 00:00:34.260 align:middle line:84% And then we started talking about the classical solutions 00:00:34.260 --> 00:00:40.260 align:middle line:84% of the Dirac equation, because the finding 00:00:40.260 --> 00:00:44.520 align:middle line:84% the complete set of solutions for Dirac equation 00:00:44.520 --> 00:00:50.190 align:middle line:84% is the necessary step for quantizing it. 00:00:50.190 --> 00:01:04.239 align:middle line:84% So let us consider the classical solutions of Dirac equations. 00:01:04.239 --> 00:01:09.070 align:middle line:90% 00:01:09.070 --> 00:01:12.010 align:middle line:84% And recall the Dirac equation is given by the following. 00:01:12.010 --> 00:01:17.220 align:middle line:90% 00:01:17.220 --> 00:01:19.270 align:middle line:90% So this gamma matrices-- 00:01:19.270 --> 00:01:23.130 align:middle line:84% so these gamma matrices are 4x4 matrices, 00:01:23.130 --> 00:01:27.820 align:middle line:84% and psi is a four complex vector. 00:01:27.820 --> 00:01:30.840 align:middle line:84% And we already just from setting up 00:01:30.840 --> 00:01:34.110 align:middle line:84% the Dirac equation and the properties of the gamma 00:01:34.110 --> 00:01:40.960 align:middle line:84% matrices that this should have solutions 00:01:40.960 --> 00:01:47.500 align:middle line:84% proportional to plane wave with k square equal 00:01:47.500 --> 00:01:50.170 align:middle line:90% to minus m square. 00:01:50.170 --> 00:01:58.600 align:middle line:84% So our goal is to find the pre-factors. 00:01:58.600 --> 00:02:02.180 align:middle line:84% So here, we separate into two kinds of solutions. 00:02:02.180 --> 00:02:04.060 align:middle line:90% So this just reminds you-- 00:02:04.060 --> 00:02:07.850 align:middle line:84% we discussed that at the end of last lecture-- 00:02:07.850 --> 00:02:11.560 align:middle line:84% we call uk corresponding to the positive. 00:02:11.560 --> 00:02:21.930 align:middle line:84% So we always take k to have positive time component. 00:02:21.930 --> 00:02:25.770 align:middle line:84% And so this is the positive energy solution. 00:02:25.770 --> 00:02:29.710 align:middle line:84% Then we have also so-called negative energy solution. 00:02:29.710 --> 00:02:31.440 align:middle line:90% So these are just names. 00:02:31.440 --> 00:02:34.890 align:middle line:84% Does not really means that they have positive energy 00:02:34.890 --> 00:02:36.610 align:middle line:90% or negative energy. 00:02:36.610 --> 00:02:40.800 align:middle line:84% It just means that the sign here. 00:02:40.800 --> 00:02:46.630 align:middle line:84% So now, if you plug this into the Dirac equation, 00:02:46.630 --> 00:02:49.740 align:middle line:84% and then you get the algebraic equation for u. 00:02:49.740 --> 00:02:52.020 align:middle line:84% You get the algebraic equation for u, 00:02:52.020 --> 00:03:03.332 align:middle line:84% and then you find i k slash m u equal to 0, and the i k 00:03:03.332 --> 00:03:08.160 align:middle line:90% slash plus m v equal to 0. 00:03:08.160 --> 00:03:12.150 align:middle line:84% So u and v satisfy those equations. 00:03:12.150 --> 00:03:21.120 align:middle line:84% And so I suppose you're already familiar with the rotation-- 00:03:21.120 --> 00:03:22.150 align:middle line:90% so this k slash. 00:03:22.150 --> 00:03:27.890 align:middle line:90% 00:03:27.890 --> 00:03:29.600 align:middle line:84% So those equations, in principle, 00:03:29.600 --> 00:03:31.010 align:middle line:90% you can just solve them. 00:03:31.010 --> 00:03:33.230 align:middle line:84% So these are just linear equations. 00:03:33.230 --> 00:03:39.500 align:middle line:84% And yeah, in principle, you can just solve them. 00:03:39.500 --> 00:03:44.660 align:middle line:84% But still there are tricks how we solve them so 00:03:44.660 --> 00:03:47.510 align:middle line:90% that it's mostly transparent. 00:03:47.510 --> 00:03:50.225 align:middle line:84% So there are two methods of finding the u and k. 00:03:50.225 --> 00:03:52.990 align:middle line:90% 00:03:52.990 --> 00:03:55.810 align:middle line:84% So one method is we first go to the rest frame-- 00:03:55.810 --> 00:04:00.850 align:middle line:90% 00:04:00.850 --> 00:04:08.990 align:middle line:84% say we take k mu, say, to have 0 spatial momentum. 00:04:08.990 --> 00:04:15.080 align:middle line:84% And then the k mu will be just equal to m 0. 00:04:15.080 --> 00:04:18.709 align:middle line:84% So in this case, those equations become particularly simple. 00:04:18.709 --> 00:04:21.769 align:middle line:84% And also before we solve them, actually we 00:04:21.769 --> 00:04:26.660 align:middle line:84% should also specify the gamma matrices we need to use. 00:04:26.660 --> 00:04:34.640 align:middle line:84% So the gamma matrices we will use is given by the following. 00:04:34.640 --> 00:04:36.395 align:middle line:84% Again, each are the 2 by 2 block. 00:04:36.395 --> 00:04:51.220 align:middle line:90% 00:04:51.220 --> 00:04:54.980 align:middle line:84% So this is the gamma matrices we are going to use. 00:04:54.980 --> 00:04:57.580 align:middle line:84% And as we discussed before, there 00:04:57.580 --> 00:05:01.390 align:middle line:84% are many, many different choices of gamma matrices. 00:05:01.390 --> 00:05:08.300 align:middle line:84% And some of them are convenient for certain purpose. 00:05:08.300 --> 00:05:10.310 align:middle line:84% And as we will see, this one actually 00:05:10.310 --> 00:05:14.120 align:middle line:84% makes the non-relativistic limit more manifest. 00:05:14.120 --> 00:05:20.710 align:middle line:84% So it's conveniently reduced to non-relativistic limit. 00:05:20.710 --> 00:05:25.030 align:middle line:84% And so in the rest frame, those equations simplify-- 00:05:25.030 --> 00:05:29.620 align:middle line:84% essentially, u just becomes the eigenvalue equations 00:05:29.620 --> 00:05:31.450 align:middle line:90% for gamma 0. 00:05:31.450 --> 00:05:37.240 align:middle line:84% So you find-- say, this equation for u-- 00:05:37.240 --> 00:05:40.400 align:middle line:90% 00:05:40.400 --> 00:05:46.510 align:middle line:84% so let's denote to the rest frame by 0, and then-- 00:05:46.510 --> 00:05:53.290 align:middle line:84% so the equation for u just becomes this in the rest frame, 00:05:53.290 --> 00:05:59.490 align:middle line:90% and the v given by this. 00:05:59.490 --> 00:06:05.080 align:middle line:84% So essentially, this is just corresponding-- essentially, 00:06:05.080 --> 00:06:06.610 align:middle line:90% you see the u and v-- 00:06:06.610 --> 00:06:11.710 align:middle line:84% u0 and v0 just corresponding to eigenvalues of gamma 0, 00:06:11.710 --> 00:06:14.650 align:middle line:84% because this is gamma 0 times u gives you a constant. 00:06:14.650 --> 00:06:17.525 align:middle line:84% It can be-- move that constant to the right-hand side-- it 00:06:17.525 --> 00:06:19.150 align:middle line:84% just looks like an eigenvalue equation. 00:06:19.150 --> 00:06:21.980 align:middle line:90% 00:06:21.980 --> 00:06:24.350 align:middle line:90% And so you can just easily-- 00:06:24.350 --> 00:06:27.830 align:middle line:84% you can plug this in and easily solve them, which we already 00:06:27.830 --> 00:06:30.920 align:middle line:90% discussed last time. 00:06:30.920 --> 00:06:33.050 align:middle line:84% So for example, you find that there 00:06:33.050 --> 00:06:34.800 align:middle line:90% are two independent equations. 00:06:34.800 --> 00:06:37.610 align:middle line:90% So this is a 4x4 matrix. 00:06:37.610 --> 00:06:40.160 align:middle line:90% So gamma 0 is a 4x4 matrix. 00:06:40.160 --> 00:06:45.530 align:middle line:84% And actually have two eigenvalues, say-- 00:06:45.530 --> 00:06:53.150 align:middle line:84% so each u and v have two independent solutions. 00:06:53.150 --> 00:06:55.280 align:middle line:90% So you find that the-- 00:06:55.280 --> 00:07:03.820 align:middle line:84% say, for example, one solution is just given by 1, 0, 0, 0. 00:07:03.820 --> 00:07:04.320 align:middle line:90% u2. 00:07:04.320 --> 00:07:08.000 align:middle line:90% 00:07:08.000 --> 00:07:12.600 align:middle line:84% So this-- we all just discussed at the end of last lecture 00:07:12.600 --> 00:07:13.100 align:middle line:90% here-- 00:07:13.100 --> 00:07:15.425 align:middle line:90% I'm just trying to remind you. 00:07:15.425 --> 00:07:25.480 align:middle line:90% 00:07:25.480 --> 00:07:27.910 align:middle line:84% So in this rest frame, the solutions 00:07:27.910 --> 00:07:30.010 align:middle line:90% are particularly simple. 00:07:30.010 --> 00:07:33.370 align:middle line:84% And so essentially, you have two independent solutions. 00:07:33.370 --> 00:07:36.035 align:middle line:84% And then you take and take actually arbitrary 00:07:36.035 --> 00:07:37.660 align:middle line:84% linear superposition of them because we 00:07:37.660 --> 00:07:39.400 align:middle line:90% are solving linear equations-- 00:07:39.400 --> 00:07:41.620 align:middle line:84% similarly for v. And you can take 00:07:41.620 --> 00:07:44.080 align:middle line:84% the arbitrary linear superposition of these 00:07:44.080 --> 00:07:47.145 align:middle line:90% and then you will solve them. 00:07:47.145 --> 00:07:50.450 align:middle line:90% 00:07:50.450 --> 00:07:58.330 align:middle line:84% And then for general uk and vk, you 00:07:58.330 --> 00:08:01.520 align:middle line:84% can just perform a Lorentz transformation. 00:08:01.520 --> 00:08:03.610 align:middle line:84% So you can perform Lorentz transformation. 00:08:03.610 --> 00:08:20.020 align:middle line:84% Say, the uk is obtained by S lambda uk 0 u0 and vk is 00:08:20.020 --> 00:08:22.650 align:middle line:90% given by S lambda v0. 00:08:22.650 --> 00:08:26.150 align:middle line:90% 00:08:26.150 --> 00:08:29.330 align:middle line:84% So the lambda-- so remember, our notation, S lambda, 00:08:29.330 --> 00:08:32.284 align:middle line:84% is a Lorentz transformation on the spinors. 00:08:32.284 --> 00:08:34.159 align:middle line:84% So the Lorentz transformation on the spinors. 00:08:34.159 --> 00:08:35.690 align:middle line:84% And the lambda is just determined 00:08:35.690 --> 00:08:46.790 align:middle line:84% by the Lorentz transformation to take this k0 to k. 00:08:46.790 --> 00:08:51.060 align:middle line:84% So for arbitrary-- so we take k0, 00:08:51.060 --> 00:08:53.700 align:middle line:84% so we make the Lorentz transformation to k, 00:08:53.700 --> 00:08:56.940 align:middle line:90% and then the corresponding uk. 00:08:56.940 --> 00:08:59.670 align:middle line:90% Then, it's getting by that. 00:08:59.670 --> 00:09:04.340 align:middle line:90% 00:09:04.340 --> 00:09:09.130 align:middle line:84% But in practice, if I give you an arbitrary, say-- 00:09:09.130 --> 00:09:14.590 align:middle line:84% yeah, give you arbitrary k, and you first work out the lambda, 00:09:14.590 --> 00:09:17.380 align:middle line:84% then you have to work out this S lambda. 00:09:17.380 --> 00:09:23.200 align:middle line:84% So working s lambda is possible, but again, takes some effort. 00:09:23.200 --> 00:09:26.080 align:middle line:84% But there's still a lot of shortcut. 00:09:26.080 --> 00:09:27.590 align:middle line:84% We can actually find the solution. 00:09:27.590 --> 00:09:30.535 align:middle line:84% So this is one way to do it, and there's alternative way. 00:09:30.535 --> 00:09:39.987 align:middle line:90% 00:09:39.987 --> 00:09:41.820 align:middle line:84% So before I talk about this alternative way, 00:09:41.820 --> 00:09:43.050 align:middle line:90% do you have any questions? 00:09:43.050 --> 00:09:46.020 align:middle line:84% So this is more or less where we stopped-- 00:09:46.020 --> 00:09:48.285 align:middle line:84% so at the end of-- before the break. 00:09:48.285 --> 00:09:53.070 align:middle line:90% 00:09:53.070 --> 00:09:56.240 align:middle line:90% Any questions? 00:09:56.240 --> 00:09:57.980 align:middle line:90% OK, good. 00:09:57.980 --> 00:10:02.635 align:middle line:84% So alternatively, we can actually find a simpler way. 00:10:02.635 --> 00:10:06.540 align:middle line:90% 00:10:06.540 --> 00:10:12.160 align:middle line:84% So the simpler way is by observing an identity-- 00:10:12.160 --> 00:10:14.800 align:middle line:90% 00:10:14.800 --> 00:10:17.950 align:middle line:84% you say, if you look at the k slash square-- 00:10:17.950 --> 00:10:22.240 align:middle line:84% so you have done something similar in your pset-- 00:10:22.240 --> 00:10:24.940 align:middle line:84% if you look at the k slash square-- 00:10:24.940 --> 00:10:32.110 align:middle line:84% so this quantity, by definition, is k mu gamma mu k nu gamma 00:10:32.110 --> 00:10:36.780 align:middle line:84% nu And then you use the properties of gamma mu 00:10:36.780 --> 00:10:42.000 align:middle line:84% and gamma nu, and then you can easily convince yourself 00:10:42.000 --> 00:10:45.800 align:middle line:84% this is just giving you the k squared. 00:10:45.800 --> 00:10:47.570 align:middle line:84% This just gives you the k square. 00:10:47.570 --> 00:10:52.380 align:middle line:84% And then-- because we are looking at on-shell solution, 00:10:52.380 --> 00:10:54.650 align:middle line:84% so this is just equal to minus m squared. 00:10:54.650 --> 00:10:59.470 align:middle line:90% 00:10:59.470 --> 00:11:07.250 align:middle line:84% Then, this means that if I have ik slash plus m-- 00:11:07.250 --> 00:11:09.980 align:middle line:90% 00:11:09.980 --> 00:11:13.740 align:middle line:84% which-- so up here in the equation for u and v 00:11:13.740 --> 00:11:16.940 align:middle line:90% are plus or minus m. 00:11:16.940 --> 00:11:21.050 align:middle line:84% And then if I multiply by i k slash minus m-- 00:11:21.050 --> 00:11:24.160 align:middle line:90% 00:11:24.160 --> 00:11:29.560 align:middle line:84% and then essentially I have minus k slash square minus m 00:11:29.560 --> 00:11:35.630 align:middle line:84% square, and this is actually equal to 0 00:11:35.630 --> 00:11:36.785 align:middle line:90% because of this equation. 00:11:36.785 --> 00:11:44.520 align:middle line:90% 00:11:44.520 --> 00:11:46.320 align:middle line:84% So now we can immediately observe-- 00:11:46.320 --> 00:11:48.860 align:middle line:90% 00:11:48.860 --> 00:11:54.720 align:middle line:84% so now, we can immediately observe if I take a solution-- 00:11:54.720 --> 00:11:58.200 align:middle line:90% 00:11:58.200 --> 00:12:04.630 align:middle line:84% if I just-- to solve that equation, 00:12:04.630 --> 00:12:15.640 align:middle line:84% you can just write u as say some normalization constant-- 00:12:15.640 --> 00:12:18.504 align:middle line:84% N will be a normalization constant-- and i k 00:12:18.504 --> 00:12:20.080 align:middle line:90% slash plus m. 00:12:20.080 --> 00:12:23.230 align:middle line:90% 00:12:23.230 --> 00:12:29.500 align:middle line:84% And then they're acting on some arbitrary u tilde-- 00:12:29.500 --> 00:12:30.850 align:middle line:90% some arbitrary spinor. 00:12:30.850 --> 00:12:32.710 align:middle line:90% And then this equation-- 00:12:32.710 --> 00:12:34.840 align:middle line:84% that equation automatically satisfies. 00:12:34.840 --> 00:12:37.380 align:middle line:90% 00:12:37.380 --> 00:12:44.370 align:middle line:84% And similarly, v, if I take some normalization constant, 00:12:44.370 --> 00:12:49.500 align:middle line:84% and I take minus i k slash plus m-- 00:12:49.500 --> 00:12:52.590 align:middle line:90% say some v tilde-- 00:12:52.590 --> 00:12:55.740 align:middle line:84% so that equation, again, automatically-- the v equation 00:12:55.740 --> 00:13:00.010 align:middle line:84% automatically satisfied because of this identity. 00:13:00.010 --> 00:13:02.440 align:middle line:84% And now, we know that there's only two 00:13:02.440 --> 00:13:04.120 align:middle line:90% independent equations-- 00:13:04.120 --> 00:13:11.210 align:middle line:84% two independent solutions for both u and v, 00:13:11.210 --> 00:13:15.420 align:middle line:90% so we can choose a basis for u. 00:13:15.420 --> 00:13:19.035 align:middle line:84% And just by taking this to be, say, us 0-- 00:13:19.035 --> 00:13:25.730 align:middle line:90% 00:13:25.730 --> 00:13:28.340 align:middle line:90% and then this would be a basis-- 00:13:28.340 --> 00:13:30.400 align:middle line:90% a complete basis of solutions. 00:13:30.400 --> 00:13:34.000 align:middle line:90% So s would go 1 to 2. 00:13:34.000 --> 00:13:35.890 align:middle line:84% So because we have-- and for each of them, 00:13:35.890 --> 00:13:37.720 align:middle line:84% we have two independent solutions. 00:13:37.720 --> 00:13:40.910 align:middle line:90% 00:13:40.910 --> 00:13:43.850 align:middle line:84% And so the reason we can choose us here-- 00:13:43.850 --> 00:13:47.270 align:middle line:84% you can also check that when you take the m-- 00:13:47.270 --> 00:13:49.580 align:middle line:90% when you take the to be-- 00:13:49.580 --> 00:13:54.050 align:middle line:84% yeah, so when you take k to be k0-- 00:13:54.050 --> 00:13:57.350 align:middle line:84% so you can easily check yourself that this equation-- 00:13:57.350 --> 00:14:02.800 align:middle line:84% this essentially just reduced to a us 0. 00:14:02.800 --> 00:14:04.480 align:middle line:90% Yeah, this is consistent-- 00:14:04.480 --> 00:14:07.150 align:middle line:90% just say when you take that-- 00:14:07.150 --> 00:14:08.590 align:middle line:90% you just reduce to this basis. 00:14:08.590 --> 00:14:11.830 align:middle line:90% 00:14:11.830 --> 00:14:13.780 align:middle line:84% So n is a normalization constant. 00:14:13.780 --> 00:14:26.270 align:middle line:90% 00:14:26.270 --> 00:14:33.820 align:middle line:84% And so the us vs then provides a basis-- 00:14:33.820 --> 00:14:35.740 align:middle line:84% then this is a basis of solutions. 00:14:35.740 --> 00:14:45.220 align:middle line:90% 00:14:45.220 --> 00:14:47.710 align:middle line:84% And we can be sure these are the complete basis 00:14:47.710 --> 00:14:50.890 align:middle line:90% because that's the only-- 00:14:50.890 --> 00:14:56.110 align:middle line:84% yeah, essentially, we already know the solution at the rest 00:14:56.110 --> 00:14:59.710 align:middle line:84% frame, and we just have the right number 00:14:59.710 --> 00:15:02.080 align:middle line:90% of the independent solutions. 00:15:02.080 --> 00:15:03.087 align:middle line:90% Yes? 00:15:03.087 --> 00:15:05.170 align:middle line:84% STUDENT: How do we know the normalization constant 00:15:05.170 --> 00:15:07.090 align:middle line:90% is the same for uv? 00:15:07.090 --> 00:15:11.800 align:middle line:84% PROFESSOR: Oh, yeah, here, we don't know. 00:15:11.800 --> 00:15:14.390 align:middle line:84% But you can guess why it should be the same. 00:15:14.390 --> 00:15:18.820 align:middle line:84% It's just essentially-- so this u-- yeah, 00:15:18.820 --> 00:15:21.040 align:middle line:84% because the equation looks pretty symmetric 00:15:21.040 --> 00:15:24.160 align:middle line:84% and the normalization constant don't depend on the sign of m, 00:15:24.160 --> 00:15:25.060 align:middle line:90% essentially. 00:15:25.060 --> 00:15:28.340 align:middle line:90% 00:15:28.340 --> 00:15:31.910 align:middle line:84% Yeah, in principle, I should write in 1 and 2. 00:15:31.910 --> 00:15:37.460 align:middle line:84% Yeah, just-- but in the end, they are the same. 00:15:37.460 --> 00:15:38.040 align:middle line:90% Good. 00:15:38.040 --> 00:15:39.920 align:middle line:90% Any other questions? 00:15:39.920 --> 00:15:44.070 align:middle line:90% 00:15:44.070 --> 00:15:47.220 align:middle line:90% So you can also-- 00:15:47.220 --> 00:15:51.150 align:middle line:84% yeah, if you take the conjugate, then you can get the-- 00:15:51.150 --> 00:15:57.000 align:middle line:84% what's the u dagger or the u bar or v dagger v bar. 00:15:57.000 --> 00:16:00.790 align:middle line:84% So for example, you just take the conjugate, 00:16:00.790 --> 00:16:03.460 align:middle line:90% then you find the us bar. 00:16:03.460 --> 00:16:07.380 align:middle line:84% So I hope you still remember the notation-- the us bar-- 00:16:07.380 --> 00:16:12.390 align:middle line:84% which is the us dagger times gamma 0 and the us bar-- 00:16:12.390 --> 00:16:16.590 align:middle line:84% so this is for the conjugate solution-- will be an-- 00:16:16.590 --> 00:16:17.370 align:middle line:90% the same n. 00:16:17.370 --> 00:16:18.900 align:middle line:90% We can take n to be real-- 00:16:18.900 --> 00:16:25.830 align:middle line:84% say, us bar 0, then i k slash plus m. 00:16:25.830 --> 00:16:29.310 align:middle line:84% So you just use the properties of the-- 00:16:29.310 --> 00:16:43.250 align:middle line:90% 00:16:43.250 --> 00:16:44.510 align:middle line:90% so you can also find this. 00:16:44.510 --> 00:16:47.530 align:middle line:90% 00:16:47.530 --> 00:16:50.820 align:middle line:84% So now you can easily check yourself again using 00:16:50.820 --> 00:16:53.540 align:middle line:90% this equation-- 00:16:53.540 --> 00:16:57.440 align:middle line:84% you can check that the u bar and the v bar 00:16:57.440 --> 00:16:59.900 align:middle line:84% are actually-- u and v are orthogonal 00:16:59.900 --> 00:17:01.260 align:middle line:90% in the following sense. 00:17:01.260 --> 00:17:09.140 align:middle line:84% And ur bar vs is equal to v bar r us 00:17:09.140 --> 00:17:15.680 align:middle line:84% is equal to 0 for arbitrary r, s equal to 1 and 2. 00:17:15.680 --> 00:17:17.540 align:middle line:90% So u and v are orthogonal. 00:17:17.540 --> 00:17:24.630 align:middle line:90% 00:17:24.630 --> 00:17:27.200 align:middle line:90% And we can also -- 00:17:27.200 --> 00:17:32.180 align:middle line:84% and then we need to fix the normalization of u and v. 00:17:32.180 --> 00:17:36.690 align:middle line:84% So it's convenient to fix the normalization as follows. 00:17:36.690 --> 00:17:39.320 align:middle line:84% So we will use the following convention-- 00:17:39.320 --> 00:17:44.240 align:middle line:90% say us bar k-- 00:17:44.240 --> 00:17:56.510 align:middle line:84% say, ur k u bar and us k equal to i 2 m delta rs. 00:17:56.510 --> 00:18:05.965 align:middle line:84% And similarly, vr bar k and vs k - i 2m. 00:18:05.965 --> 00:18:17.790 align:middle line:90% 00:18:17.790 --> 00:18:21.150 align:middle line:84% So these 2m just for convenience. 00:18:21.150 --> 00:18:24.600 align:middle line:84% So it just convention for convenience. 00:18:24.600 --> 00:18:32.900 align:middle line:84% And so you can check yourself that actually-- 00:18:32.900 --> 00:18:37.676 align:middle line:90% yeah, so different u-- 00:18:37.676 --> 00:18:39.540 align:middle line:84% different value of r and s, they're 00:18:39.540 --> 00:18:40.790 align:middle line:90% also orthogonal to each other. 00:18:40.790 --> 00:18:45.020 align:middle line:84% You can already see already in the rest frame. 00:18:45.020 --> 00:18:47.960 align:middle line:84% You can just easily check using those expressions, 00:18:47.960 --> 00:18:51.305 align:middle line:84% but you can also check again using this identity 00:18:51.305 --> 00:18:53.510 align:middle line:84% that the different value of r and s, 00:18:53.510 --> 00:18:55.110 align:middle line:84% they are orthogonal to each other. 00:18:55.110 --> 00:18:57.410 align:middle line:84% And when they are the same, and then we 00:18:57.410 --> 00:19:02.620 align:middle line:84% normalize the constant to be 2 m. 00:19:02.620 --> 00:19:07.300 align:middle line:84% And from here, you can deduce that N 00:19:07.300 --> 00:19:13.850 align:middle line:84% is equal to 1 over square root E plus M. So from now on, 00:19:13.850 --> 00:19:17.230 align:middle line:84% I will also use this notation for E. 00:19:17.230 --> 00:19:22.630 align:middle line:84% So the E-- the energy is omega k. 00:19:22.630 --> 00:19:25.135 align:middle line:84% So now, I will also use this notation. 00:19:25.135 --> 00:19:37.970 align:middle line:90% 00:19:37.970 --> 00:19:38.540 align:middle line:90% Good. 00:19:38.540 --> 00:19:42.010 align:middle line:84% So this gives you a complete basis 00:19:42.010 --> 00:19:46.240 align:middle line:84% of solutions to the Dirac equations. 00:19:46.240 --> 00:19:48.430 align:middle line:84% So now, let me just write them down 00:19:48.430 --> 00:19:51.460 align:middle line:84% to give you a little bit intuition 00:19:51.460 --> 00:19:53.420 align:middle line:90% about the explicit expression. 00:19:53.420 --> 00:19:59.320 align:middle line:84% So with those gamma matrices, we can just plug in here. 00:19:59.320 --> 00:20:02.200 align:middle line:90% And we know what the us 0 vs 0. 00:20:02.200 --> 00:20:06.280 align:middle line:84% We know n, so in principle, we can write them down explicitly. 00:20:06.280 --> 00:20:10.270 align:middle line:84% So you find that in that basis, then the us k 00:20:10.270 --> 00:20:14.110 align:middle line:90% is given by the following-- 00:20:14.110 --> 00:20:22.730 align:middle line:84% given by the square root E plus M xi s, 00:20:22.730 --> 00:20:33.770 align:middle line:84% and the sigma dot k divided by square root E plus M xi s. 00:20:33.770 --> 00:20:37.090 align:middle line:90% 00:20:37.090 --> 00:20:49.750 align:middle line:84% And the vs k is given by sigma dot 00:20:49.750 --> 00:21:05.500 align:middle line:84% k divided by square root E plus M also xi s, and E plus M xi s. 00:21:05.500 --> 00:21:08.210 align:middle line:90% And xi s is a 2. 00:21:08.210 --> 00:21:13.400 align:middle line:90% So s from 1 to 2, and xi 1-- 00:21:13.400 --> 00:21:16.860 align:middle line:84% so psi s are two component vectors. 00:21:16.860 --> 00:21:21.560 align:middle line:84% One is given by 1, 0, and the other is given by 0, 1. 00:21:21.560 --> 00:21:28.200 align:middle line:90% 00:21:28.200 --> 00:21:31.250 align:middle line:90% So that you get-- 00:21:31.250 --> 00:21:37.510 align:middle line:84% so you can see in the rest frame, 00:21:37.510 --> 00:21:39.940 align:middle line:84% k equal to 0-- essentially, for u, 00:21:39.940 --> 00:21:45.440 align:middle line:84% you're just left with the upper half because this is 0. 00:21:45.440 --> 00:21:49.310 align:middle line:84% And then we have the 1, 0 and the 0, 1. 00:21:49.310 --> 00:21:51.140 align:middle line:90% And then when k equal to 0-- 00:21:51.140 --> 00:21:53.630 align:middle line:84% and for v, just left with a lower half. 00:21:53.630 --> 00:21:57.200 align:middle line:84% And again, we have the 1, 0 and the 0, 1. 00:21:57.200 --> 00:22:00.470 align:middle line:84% So this is consistent with the rest frame result 00:22:00.470 --> 00:22:03.690 align:middle line:90% we obtained earlier. 00:22:03.690 --> 00:22:04.190 align:middle line:90% Good. 00:22:04.190 --> 00:22:05.110 align:middle line:90% Any questions on this? 00:22:05.110 --> 00:22:05.710 align:middle line:90% Yes? 00:22:05.710 --> 00:22:07.460 align:middle line:84% STUDENT: What are the sig -- what's sigma? 00:22:07.460 --> 00:22:09.890 align:middle line:84% PROFESSOR: Oh, sigma just the sigma matrix. 00:22:09.890 --> 00:22:12.790 align:middle line:84% So this sigma just Pauli matrices. 00:22:12.790 --> 00:22:15.100 align:middle line:90% Sigma 1, sigma 2, sigma 3-- 00:22:15.100 --> 00:22:15.970 align:middle line:90% Pauli matrices. 00:22:15.970 --> 00:22:22.974 align:middle line:90% 00:22:22.974 --> 00:22:23.474 align:middle line:90% Good? 00:22:23.474 --> 00:22:27.910 align:middle line:90% 00:22:27.910 --> 00:22:30.580 align:middle line:90% So let me just mention a few-- 00:22:30.580 --> 00:22:33.820 align:middle line:84% make a few remarks on the solutions. 00:22:33.820 --> 00:22:43.870 align:middle line:84% And to be familiar with the properties of the solutions 00:22:43.870 --> 00:22:46.690 align:middle line:84% actually will be very useful when later you 00:22:46.690 --> 00:22:51.280 align:middle line:84% do Feynman diagram calculations, because the different-- 00:22:51.280 --> 00:22:56.020 align:middle line:84% when you involve, say, electron fermions-- and now, 00:22:56.020 --> 00:22:57.670 align:middle line:90% it's no longer plane wave. 00:22:57.670 --> 00:23:02.240 align:middle line:84% So this u and v will enter your Feynman diagram calculations. 00:23:02.240 --> 00:23:04.420 align:middle line:84% And if you are familiar with their properties-- 00:23:04.420 --> 00:23:08.830 align:middle line:84% and will help you to do such calculations. 00:23:08.830 --> 00:23:12.700 align:middle line:84% So let me just first mention some properties. 00:23:12.700 --> 00:23:17.120 align:middle line:90% 00:23:17.120 --> 00:23:19.860 align:middle line:90% Make some remarks. 00:23:19.860 --> 00:23:23.375 align:middle line:84% So first is that if we consider the non-relativistic limit-- 00:23:23.375 --> 00:23:27.310 align:middle line:90% 00:23:27.310 --> 00:23:30.250 align:middle line:84% the non-relativistic limit-- so that's corresponding to-- you 00:23:30.250 --> 00:23:37.180 align:middle line:84% take the magnitude of k divided by m goes to 0. 00:23:37.180 --> 00:23:40.720 align:middle line:84% And then in this limit, E would be approximately equal to m. 00:23:40.720 --> 00:23:46.050 align:middle line:90% 00:23:46.050 --> 00:23:47.760 align:middle line:84% And then you find the leading order 00:23:47.760 --> 00:23:52.590 align:middle line:84% when you expand, say, in-- yeah, so you can expand this in 1 00:23:52.590 --> 00:23:59.540 align:middle line:84% over M. So the leading order, you 00:23:59.540 --> 00:24:05.160 align:middle line:84% find that the us k-- so you can just easily do these two. 00:24:05.160 --> 00:24:08.670 align:middle line:84% And yeah, so this term, certainly, you 00:24:08.670 --> 00:24:10.410 align:middle line:90% can just ignore because E-- 00:24:10.410 --> 00:24:12.450 align:middle line:90% so this is M-- 00:24:12.450 --> 00:24:15.000 align:middle line:84% so this is k divided by square root of M, 00:24:15.000 --> 00:24:17.380 align:middle line:84% so this is higher order than this one. 00:24:17.380 --> 00:24:19.110 align:middle line:84% And here, it's the square root of M. 00:24:19.110 --> 00:24:28.250 align:middle line:84% So to leading order, you have square root 2m xi s 0. 00:24:28.250 --> 00:24:37.030 align:middle line:84% Similarly, vs k equal to square root of 2m leading order 00:24:37.030 --> 00:24:38.770 align:middle line:90% 0 xi s. 00:24:38.770 --> 00:24:45.490 align:middle line:90% 00:24:45.490 --> 00:24:53.540 align:middle line:84% So essentially, what you see in the non-relativistic limit-- 00:24:53.540 --> 00:24:57.150 align:middle line:84% for example, if we look at this positive energy solutions-- 00:24:57.150 --> 00:25:01.470 align:middle line:84% so this just reduced to arbitrary-- 00:25:01.470 --> 00:25:06.660 align:middle line:84% so us k-- yeah, now you can do linear superposition-- 00:25:06.660 --> 00:25:22.520 align:middle line:84% just uk just reduce to arbitrary two component complex vector. 00:25:22.520 --> 00:25:30.580 align:middle line:90% 00:25:30.580 --> 00:25:32.590 align:middle line:90% Yeah, so you just-- 00:25:32.590 --> 00:25:34.450 align:middle line:90% so this is just some constant. 00:25:34.450 --> 00:25:36.820 align:middle line:84% And as a basis, you just have this basis, 00:25:36.820 --> 00:25:39.820 align:middle line:84% and then you can just now do arbitrary-- yeah, 00:25:39.820 --> 00:25:42.920 align:middle line:84% just become the arbitrary complex vector. 00:25:42.920 --> 00:25:49.900 align:middle line:84% So the should remind you how we describe a spin-half particle 00:25:49.900 --> 00:25:51.670 align:middle line:90% in non-relativistic limit. 00:25:51.670 --> 00:25:55.510 align:middle line:84% In the non-relativistic limit, the way 00:25:55.510 --> 00:25:58.720 align:middle line:84% we describe the spin-half particle-- 00:25:58.720 --> 00:26:02.920 align:middle line:84% and it's precisely a two-component vector. 00:26:02.920 --> 00:26:18.310 align:middle line:84% So so this is exactly as the description 00:26:18.310 --> 00:26:23.705 align:middle line:90% of a spin-half particle. 00:26:23.705 --> 00:26:26.450 align:middle line:90% 00:26:26.450 --> 00:26:29.480 align:middle line:90% Yeah, the wave function-- 00:26:29.480 --> 00:26:30.950 align:middle line:84% yeah, here, you should understand 00:26:30.950 --> 00:26:33.290 align:middle line:84% this-- the wave function description 00:26:33.290 --> 00:26:37.610 align:middle line:84% of the spin-half particle in non-relativistic quantum 00:26:37.610 --> 00:26:38.240 align:middle line:90% mechanics. 00:26:38.240 --> 00:26:42.840 align:middle line:90% 00:26:42.840 --> 00:26:44.670 align:middle line:84% And as I will not go through here, 00:26:44.670 --> 00:26:51.360 align:middle line:84% but you can show in the non-relativistic limit, 00:26:51.360 --> 00:26:54.815 align:middle line:84% this solution actually decouples. 00:26:54.815 --> 00:26:56.690 align:middle line:84% In the end, when you solve the equation, only 00:26:56.690 --> 00:26:58.640 align:middle line:90% one half of the equation-- 00:26:58.640 --> 00:27:01.400 align:middle line:84% one half of the solution remains. 00:27:01.400 --> 00:27:04.474 align:middle line:84% And actually, this is no longer-- 00:27:04.474 --> 00:27:08.130 align:middle line:84% it just gets the positive energy solution. 00:27:08.130 --> 00:27:11.900 align:middle line:84% So in that limit, you essentially only 00:27:11.900 --> 00:27:13.460 align:middle line:90% reduce to two components. 00:27:13.460 --> 00:27:20.960 align:middle line:90% 00:27:20.960 --> 00:27:22.700 align:middle line:90% So this is the first remark. 00:27:22.700 --> 00:27:27.640 align:middle line:84% So this is the first indication we see that this should likely 00:27:27.640 --> 00:27:30.580 align:middle line:90% describe a spin-half particle. 00:27:30.580 --> 00:27:35.275 align:middle line:84% The Dirac theory likely describes a spin-half particle. 00:27:35.275 --> 00:27:38.820 align:middle line:90% 00:27:38.820 --> 00:27:44.880 align:middle line:84% So the second limit we can consider 00:27:44.880 --> 00:27:47.685 align:middle line:84% is considered the ultra relativistic limit. 00:27:47.685 --> 00:27:59.070 align:middle line:90% 00:27:59.070 --> 00:28:04.020 align:middle line:84% So this is a limit which E is much greater than-- 00:28:04.020 --> 00:28:08.130 align:middle line:84% energy is much greater than M. And E then 00:28:08.130 --> 00:28:13.665 align:middle line:84% will be approximate, say, to k-- to the magnitude of k. 00:28:13.665 --> 00:28:16.310 align:middle line:90% 00:28:16.310 --> 00:28:19.010 align:middle line:84% And now let's also define the direction of k 00:28:19.010 --> 00:28:26.150 align:middle line:84% by just k divided by its magnitude. 00:28:26.150 --> 00:28:30.590 align:middle line:84% So we also introduced the k hat is the unit vector 00:28:30.590 --> 00:28:32.000 align:middle line:90% along the direction of the k. 00:28:32.000 --> 00:28:37.390 align:middle line:84% So in this case, you can do a small m expansion-- 00:28:37.390 --> 00:28:41.890 align:middle line:84% just do a small expansion-- this expression for u and v. 00:28:41.890 --> 00:28:46.510 align:middle line:84% And then you find the leading order-- 00:28:46.510 --> 00:28:52.500 align:middle line:90% us k given by square root of E. 00:28:52.500 --> 00:29:02.590 align:middle line:84% And then you have xi s, and then you have sigma k hat xi s. 00:29:02.590 --> 00:29:04.850 align:middle line:90% This is for the us. 00:29:04.850 --> 00:29:06.470 align:middle line:90% And for the vs, you find-- 00:29:06.470 --> 00:29:24.090 align:middle line:90% 00:29:24.090 --> 00:29:27.790 align:middle line:84% so that's the-- so you find that solution simplify, 00:29:27.790 --> 00:29:30.480 align:middle line:84% and so that's the result you get. 00:29:30.480 --> 00:29:36.330 align:middle line:90% 00:29:36.330 --> 00:29:43.020 align:middle line:84% So it's not manifest from this expression, 00:29:43.020 --> 00:29:49.960 align:middle line:84% but actually, there's something very special in this simplified 00:29:49.960 --> 00:29:52.580 align:middle line:84% expression in the ultra relativistic limit. 00:29:52.580 --> 00:29:55.970 align:middle line:90% So if you look at-- 00:29:55.970 --> 00:29:59.120 align:middle line:90% so the expression for u and v-- 00:29:59.120 --> 00:30:03.140 align:middle line:84% these are linearly independent, in the sense that no matter how 00:30:03.140 --> 00:30:05.240 align:middle line:84% you do linear superposition of u, 00:30:05.240 --> 00:30:07.310 align:middle line:90% you will not be able to get v. 00:30:07.310 --> 00:30:09.950 align:middle line:84% And also how you do linear superposition of v, 00:30:09.950 --> 00:30:13.010 align:middle line:90% you won't get u. 00:30:13.010 --> 00:30:16.310 align:middle line:84% But in this limit, even though it's not clear-- 00:30:16.310 --> 00:30:19.110 align:middle line:84% not manifest in this form, but in fact, 00:30:19.110 --> 00:30:23.480 align:middle line:84% these two are linearly related to each other. 00:30:23.480 --> 00:30:27.390 align:middle line:90% So this can be seen as follows. 00:30:27.390 --> 00:30:46.660 align:middle line:84% So if you use the identity that's sigma dot k square-- 00:30:46.660 --> 00:30:51.610 align:middle line:84% so this is the analog of this equation for the sigma 00:30:51.610 --> 00:30:52.600 align:middle line:90% matrices-- 00:30:52.600 --> 00:30:54.460 align:middle line:90% for the Pauli matrices. 00:30:54.460 --> 00:30:56.350 align:middle line:84% And this, you can easily-- again, 00:30:56.350 --> 00:30:58.000 align:middle line:84% using the property of Pauli matrices, 00:30:58.000 --> 00:31:00.880 align:middle line:84% you can convince yourself that this is just 00:31:00.880 --> 00:31:02.410 align:middle line:90% giving you k hat squared. 00:31:02.410 --> 00:31:05.440 align:middle line:90% 00:31:05.440 --> 00:31:07.060 align:middle line:90% And this is actually-- 00:31:07.060 --> 00:31:10.540 align:middle line:84% because this is unit vector, this is just 1. 00:31:10.540 --> 00:31:16.440 align:middle line:84% So now, if you use this property, this is equal to 1. 00:31:16.440 --> 00:31:18.480 align:middle line:90% And then we find-- 00:31:18.480 --> 00:31:25.800 align:middle line:84% and then let eta s equal to sigma k-- 00:31:25.800 --> 00:31:30.600 align:middle line:90% sigma dot k hat xi s. 00:31:30.600 --> 00:31:35.250 align:middle line:84% And then you find that the vs k can be written 00:31:35.250 --> 00:31:37.110 align:middle line:90% in the exact form as us k. 00:31:37.110 --> 00:31:39.630 align:middle line:90% You just replace eta s by-- 00:31:39.630 --> 00:31:41.010 align:middle line:90% xi s by eta s. 00:31:41.010 --> 00:31:45.600 align:middle line:84% And then you can have it in the following form-- 00:31:45.600 --> 00:31:56.840 align:middle line:84% square root E eta s sigma dot k eta s. 00:31:56.840 --> 00:31:59.270 align:middle line:84% So obviously, this is equivalent to that 00:31:59.270 --> 00:32:03.050 align:middle line:84% once you plug in the eta s equal to sigma k psi s, 00:32:03.050 --> 00:32:05.660 align:middle line:84% because this just gives you that. 00:32:05.660 --> 00:32:08.720 align:middle line:84% And when you plug in to here, you get the sigma k square, 00:32:08.720 --> 00:32:11.390 align:middle line:84% and you get 1, and again, you reduce to that. 00:32:11.390 --> 00:32:14.380 align:middle line:90% 00:32:14.380 --> 00:32:23.530 align:middle line:84% But now, this have exactly the same form as this one, 00:32:23.530 --> 00:32:29.980 align:middle line:84% except now eta s is a linear superposition of the xi s. 00:32:29.980 --> 00:32:36.480 align:middle line:84% Then that means that the vs is a linear superposition of the us. 00:32:36.480 --> 00:32:43.870 align:middle line:84% So this means that the vs is a linear superposition of vs. 00:32:43.870 --> 00:32:59.980 align:middle line:84% So vs now is a linear superposition of vs-- of us. 00:32:59.980 --> 00:33:02.200 align:middle line:84% So they are no longer independent, 00:33:02.200 --> 00:33:10.690 align:middle line:84% so they are not linear independent. 00:33:10.690 --> 00:33:20.050 align:middle line:90% 00:33:20.050 --> 00:33:24.700 align:middle line:84% So actually, if you remember one of the ways 00:33:24.700 --> 00:33:27.970 align:middle line:84% we have discussed before, this may not appear a surprise 00:33:27.970 --> 00:33:31.150 align:middle line:84% to you, because previously, we already 00:33:31.150 --> 00:33:36.400 align:middle line:84% discussed when we derive the Dirac equation. 00:33:36.400 --> 00:33:39.900 align:middle line:84% So Dirac equation-- in order to satisfy 00:33:39.900 --> 00:33:43.590 align:middle line:84% all the properties for non-zero M, 00:33:43.590 --> 00:33:47.940 align:middle line:84% we need four dimensional matrices. 00:33:47.940 --> 00:33:50.970 align:middle line:84% So that means psi is a four vector. 00:33:50.970 --> 00:33:55.490 align:middle line:84% But we also mentioned that if you in the massless case 00:33:55.490 --> 00:33:57.670 align:middle line:90% set m equal to 0-- 00:33:57.670 --> 00:33:59.350 align:middle line:84% in the massless case, actually, you 00:33:59.350 --> 00:34:03.470 align:middle line:84% can use two components-- just use gamma matrices. 00:34:03.470 --> 00:34:06.520 align:middle line:84% So that means that if you solve a massless Dirac equation, 00:34:06.520 --> 00:34:10.150 align:middle line:84% you only need-- you only have two independent solutions. 00:34:10.150 --> 00:34:12.790 align:middle line:84% And this ultrarelativistic limit is precisely 00:34:12.790 --> 00:34:14.980 align:middle line:84% like the massless limit because the mass-- 00:34:14.980 --> 00:34:16.960 align:middle line:84% the energy is much greater than the mass, 00:34:16.960 --> 00:34:20.800 align:middle line:84% and then you can neglect the effect of mass. 00:34:20.800 --> 00:34:22.360 align:middle line:90% So this is consistent-- 00:34:22.360 --> 00:34:26.489 align:middle line:90% 00:34:26.489 --> 00:34:43.080 align:middle line:84% with that the massless Dirac equation consistent with that-- 00:34:43.080 --> 00:34:55.750 align:middle line:84% only requires two components rather than four components. 00:34:55.750 --> 00:34:59.510 align:middle line:90% 00:34:59.510 --> 00:35:00.603 align:middle line:90% Yes? 00:35:00.603 --> 00:35:02.395 align:middle line:84% STUDENT: So in the ultrarelativistic limit, 00:35:02.395 --> 00:35:07.488 align:middle line:84% theyre linearly related, but that relationship v bar times u 00:35:07.488 --> 00:35:10.300 align:middle line:90% equals 0 still holds, right? 00:35:10.300 --> 00:35:13.983 align:middle line:84% PROFESSOR: Yeah, that you have to be a little bit careful. 00:35:13.983 --> 00:35:15.400 align:middle line:84% STUDENT: Does that mean that's not 00:35:15.400 --> 00:35:17.775 align:middle line:84% to be thought of as an inner product on the spinor space? 00:35:17.775 --> 00:35:21.380 align:middle line:84% PROFESSOR: Yeah, so that's a very good question. 00:35:21.380 --> 00:35:23.890 align:middle line:84% So you have to be careful a little bit-- 00:35:23.890 --> 00:35:27.585 align:middle line:84% and when you take the limit in deriving that relation. 00:35:27.585 --> 00:35:29.825 align:middle line:90% 00:35:29.825 --> 00:35:31.450 align:middle line:84% Yeah, just you have to be a little more 00:35:31.450 --> 00:35:33.920 align:middle line:90% careful in understanding. 00:35:33.920 --> 00:35:35.890 align:middle line:90% Yeah, I forgot the exact-- 00:35:35.890 --> 00:35:37.598 align:middle line:84% what's exactly happened to that equation, 00:35:37.598 --> 00:35:39.223 align:middle line:84% but I remember there's some subtleties. 00:35:39.223 --> 00:35:40.150 align:middle line:90% Yeah, you have to be. 00:35:40.150 --> 00:35:44.200 align:middle line:90% 00:35:44.200 --> 00:35:44.700 align:middle line:90% Good. 00:35:44.700 --> 00:35:45.540 align:middle line:90% Other questions? 00:35:45.540 --> 00:35:51.650 align:middle line:90% 00:35:51.650 --> 00:35:55.520 align:middle line:84% So also there are some other useful relations 00:35:55.520 --> 00:35:58.170 align:middle line:90% which will be useful later. 00:35:58.170 --> 00:36:00.890 align:middle line:90% So one of them-- 00:36:00.890 --> 00:36:03.320 align:middle line:84% so let me also list some useful relations. 00:36:03.320 --> 00:36:17.200 align:middle line:90% 00:36:17.200 --> 00:36:24.010 align:middle line:84% So why is that if you look at ur dagger and us-- 00:36:24.010 --> 00:36:29.110 align:middle line:90% so this is k us k-- 00:36:29.110 --> 00:36:33.580 align:middle line:84% again, this is orthogonal between r and s. 00:36:33.580 --> 00:36:38.710 align:middle line:84% And then it's given by 2e dot rs. 00:36:38.710 --> 00:36:41.300 align:middle line:90% So here it's bar. 00:36:41.300 --> 00:36:43.870 align:middle line:90% So the difference here is bar. 00:36:43.870 --> 00:36:48.130 align:middle line:84% And so when you have a bar, you have additional gamma 0. 00:36:48.130 --> 00:36:52.150 align:middle line:84% And then here is 2m, and then here it becomes 2E. 00:36:52.150 --> 00:36:54.580 align:middle line:84% And here, you have additional i because you have a gamma 00:36:54.580 --> 00:36:59.110 align:middle line:84% 0 because gamma 0-- remember, gamma 0 square 2 minus 1. 00:36:59.110 --> 00:37:01.750 align:middle line:84% So gamma 0 has actually eigenvalue 00:37:01.750 --> 00:37:05.110 align:middle line:84% to be pure imaginary, so that's why there's a difference in i 00:37:05.110 --> 00:37:07.610 align:middle line:90% here. 00:37:07.610 --> 00:37:19.110 align:middle line:84% And similarly, the vr dagger k and vs k equal to 2E delta is. 00:37:19.110 --> 00:37:23.400 align:middle line:90% 00:37:23.400 --> 00:37:25.920 align:middle line:90% So those expressions to-- 00:37:25.920 --> 00:37:31.840 align:middle line:90% 00:37:31.840 --> 00:37:38.720 align:middle line:84% yeah, will be very useful later when you actually 00:37:38.720 --> 00:37:40.030 align:middle line:90% do calculations. 00:37:40.030 --> 00:37:59.320 align:middle line:90% 00:37:59.320 --> 00:38:00.940 align:middle line:84% Yeah, I forgot to mention-- yeah, 00:38:00.940 --> 00:38:04.360 align:middle line:84% the reason is that-- the reason that limit is subtle 00:38:04.360 --> 00:38:07.780 align:middle line:84% when you do that is because you remember the normalization we 00:38:07.780 --> 00:38:12.400 align:middle line:84% do here is m, and so you have to change your normalization when 00:38:12.400 --> 00:38:15.250 align:middle line:90% you go to the massless case. 00:38:15.250 --> 00:38:17.710 align:middle line:84% Yeah, so there's various things you have to be careful. 00:38:17.710 --> 00:38:22.180 align:middle line:90% 00:38:22.180 --> 00:38:25.320 align:middle line:84% So you have that, and then you also 00:38:25.320 --> 00:38:34.030 align:middle line:84% have the relation that the ur dagger k-- 00:38:34.030 --> 00:38:37.570 align:middle line:84% so because everything is on-shell so in principle, 00:38:37.570 --> 00:38:38.650 align:middle line:90% I can only-- 00:38:38.650 --> 00:38:41.640 align:middle line:84% I only need to specify the spatial momentum. 00:38:41.640 --> 00:38:45.880 align:middle line:84% So for convenience, let me just specifier the spatial momentum, 00:38:45.880 --> 00:38:48.890 align:middle line:90% which is easier in notation. 00:38:48.890 --> 00:38:57.180 align:middle line:84% So now it turns out ur and the vs-- 00:38:57.180 --> 00:38:58.860 align:middle line:90% so now it's a dagger. 00:38:58.860 --> 00:39:01.740 align:middle line:84% When you have bar, they're orthogonal. 00:39:01.740 --> 00:39:04.360 align:middle line:90% We have the expression there. 00:39:04.360 --> 00:39:07.500 align:middle line:84% But then you have dagger, it turns out 00:39:07.500 --> 00:39:10.440 align:middle line:90% it's orthogonal to minus k. 00:39:10.440 --> 00:39:14.130 align:middle line:84% You have to reverse your spatial momentum, 00:39:14.130 --> 00:39:24.210 align:middle line:84% and this is equal to vr k and us minus k. 00:39:24.210 --> 00:39:28.700 align:middle line:84% So this is equal to 0, but this is actually not equal to 0. 00:39:28.700 --> 00:39:32.170 align:middle line:90% 00:39:32.170 --> 00:39:42.570 align:middle line:84% ur dagger k us vs k is actually not equal to 0. 00:39:42.570 --> 00:39:44.460 align:middle line:90% So they're not orthogonal. 00:39:44.460 --> 00:39:46.470 align:middle line:84% Only the bar relation is orthogonal, 00:39:46.470 --> 00:39:49.320 align:middle line:84% and then this relation is orthogonal. 00:39:49.320 --> 00:39:51.910 align:middle line:90% 00:39:51.910 --> 00:39:56.410 align:middle line:84% And similarly with the conjugate of this. 00:39:56.410 --> 00:39:59.380 align:middle line:84% So those relations you can just check 00:39:59.380 --> 00:40:04.930 align:middle line:84% them using explicit form of those expressions. 00:40:04.930 --> 00:40:07.090 align:middle line:84% And we have this correction here. 00:40:07.090 --> 00:40:10.030 align:middle line:84% You can just check those expression is true. 00:40:10.030 --> 00:40:13.870 align:middle line:90% And so I will not go into that. 00:40:13.870 --> 00:40:18.080 align:middle line:84% But actually, more remarkably, those expressions-- 00:40:18.080 --> 00:40:21.370 align:middle line:84% so when we derive this expression, 00:40:21.370 --> 00:40:25.300 align:middle line:84% we actually use the explicit form of the gamma matrices. 00:40:25.300 --> 00:40:27.380 align:middle line:90% It depends on the explicit form. 00:40:27.380 --> 00:40:29.400 align:middle line:84% So those expressions depend on explicit form 00:40:29.400 --> 00:40:30.800 align:middle line:90% of these gamma matrices. 00:40:30.800 --> 00:40:33.590 align:middle line:84% So if you have a different set of gamma matrices, 00:40:33.590 --> 00:40:35.470 align:middle line:90% then you get different forms. 00:40:35.470 --> 00:40:39.080 align:middle line:84% But of course, physically, they're all equivalent. 00:40:39.080 --> 00:40:41.140 align:middle line:84% But it turns out that those relations-- 00:40:41.140 --> 00:40:44.100 align:middle line:90% 00:40:44.100 --> 00:40:47.280 align:middle line:84% so those relations actually are independent of the choice 00:40:47.280 --> 00:40:50.210 align:middle line:90% of gamma matrices. 00:40:50.210 --> 00:40:58.230 align:middle line:90% So let me just label them by-- 00:40:58.230 --> 00:41:02.220 align:middle line:84% let me call it star, and here by star star. 00:41:02.220 --> 00:41:06.150 align:middle line:90% 00:41:06.150 --> 00:41:27.810 align:middle line:84% So star and star are actually independent of choice 00:41:27.810 --> 00:41:31.235 align:middle line:84% of representations of gamma matrices. 00:41:31.235 --> 00:41:34.790 align:middle line:90% 00:41:34.790 --> 00:41:40.290 align:middle line:84% So they're actually universal for any choice of matrices. 00:41:40.290 --> 00:41:44.240 align:middle line:84% So this is very useful because when you do calculations, 00:41:44.240 --> 00:41:48.770 align:middle line:84% then you don't have to worry about which choice of gamma 00:41:48.770 --> 00:41:53.572 align:middle line:84% matrices you make, and those expressions are true. 00:41:53.572 --> 00:41:57.620 align:middle line:90% 00:41:57.620 --> 00:42:06.320 align:middle line:84% So you can actually derive those expressions 00:42:06.320 --> 00:42:10.310 align:middle line:84% without using gamma matrices-- just abstractly 00:42:10.310 --> 00:42:15.650 align:middle line:84% only using the properties of the Dirac equation itself 00:42:15.650 --> 00:42:18.620 align:middle line:84% and the general properties of gamma matrices 00:42:18.620 --> 00:42:23.120 align:middle line:84% without using explicit representations. 00:42:23.120 --> 00:42:23.980 align:middle line:90% Yes? 00:42:23.980 --> 00:42:25.730 align:middle line:84% STUDENT: So is the other normalization one 00:42:25.730 --> 00:42:28.885 align:middle line:90% dependent on representation? 00:42:28.885 --> 00:42:29.760 align:middle line:90% PROFESSOR: Which one? 00:42:29.760 --> 00:42:31.880 align:middle line:90% STUDENT: The one with ur bar v. 00:42:31.880 --> 00:42:34.580 align:middle line:84% PROFESSOR: No, that also does not depend on representation. 00:42:34.580 --> 00:42:36.020 align:middle line:90% Yeah, that also does not-- 00:42:36.020 --> 00:42:39.890 align:middle line:84% or this normalization does not depend on the representation. 00:42:39.890 --> 00:42:42.710 align:middle line:90% But this expression somehow is-- 00:42:42.710 --> 00:42:44.280 align:middle line:84% yeah, that's just our definition. 00:42:44.280 --> 00:42:46.990 align:middle line:90% We just define them this way. 00:42:46.990 --> 00:42:48.320 align:middle line:90% We define them in this way. 00:42:48.320 --> 00:42:51.610 align:middle line:84% So this just follows from those general expressions 00:42:51.610 --> 00:42:53.740 align:middle line:90% just from this equation. 00:42:53.740 --> 00:42:58.330 align:middle line:84% But this equation is less obvious because without gamma 00:42:58.330 --> 00:43:01.690 align:middle line:84% 0, then it's less directly related 00:43:01.690 --> 00:43:04.370 align:middle line:90% to the Dirac equation itself. 00:43:04.370 --> 00:43:07.330 align:middle line:84% But it turns out this expression also independent 00:43:07.330 --> 00:43:09.220 align:middle line:90% of the choice of gamma matrices. 00:43:09.220 --> 00:43:14.160 align:middle line:90% 00:43:14.160 --> 00:43:15.230 align:middle line:90% Any other questions? 00:43:15.230 --> 00:43:17.770 align:middle line:90% 00:43:17.770 --> 00:43:21.220 align:middle line:84% I can give you a derivation of this 00:43:21.220 --> 00:43:24.310 align:middle line:90% without using gamma matrices. 00:43:24.310 --> 00:43:27.460 align:middle line:84% but that take a few minutes, So I will skip that. 00:43:27.460 --> 00:43:29.140 align:middle line:90% You can read it in Peskin. 00:43:29.140 --> 00:43:32.905 align:middle line:84% So Peskin has a derivation of this. 00:43:32.905 --> 00:43:35.680 align:middle line:90% 00:43:35.680 --> 00:43:36.180 align:middle line:90% Good? 00:43:36.180 --> 00:43:41.290 align:middle line:90% 00:43:41.290 --> 00:43:44.470 align:middle line:84% But you should remember this yourself-- just remember it. 00:43:44.470 --> 00:43:49.680 align:middle line:90% 00:43:49.680 --> 00:43:53.610 align:middle line:84% So the first remark is that often we actually need 00:43:53.610 --> 00:43:57.740 align:middle line:90% to use the projection-- 00:43:57.740 --> 00:44:01.820 align:middle line:84% so you can also construct projections-- 00:44:01.820 --> 00:44:13.415 align:middle line:84% projectors to the positive and the negative solution space-- 00:44:13.415 --> 00:44:19.900 align:middle line:90% 00:44:19.900 --> 00:44:20.400 align:middle line:90% spaces. 00:44:20.400 --> 00:44:23.960 align:middle line:90% 00:44:23.960 --> 00:44:54.020 align:middle line:84% So consider-- so the projector to the space spanned by us 00:44:54.020 --> 00:44:55.920 align:middle line:90% can be defined as the follows. 00:44:55.920 --> 00:44:59.720 align:middle line:90% So let me call it p plus. 00:44:59.720 --> 00:45:03.020 align:middle line:84% So this is just from 2i m, so that's 00:45:03.020 --> 00:45:06.040 align:middle line:90% coming from the normalization. 00:45:06.040 --> 00:45:16.507 align:middle line:84% Then, just sum s equal to 1 and 2 us k and us bar k. 00:45:16.507 --> 00:45:19.860 align:middle line:90% 00:45:19.860 --> 00:45:23.160 align:middle line:84% So you can easily check yourself because 00:45:23.160 --> 00:45:26.050 align:middle line:90% of orthogonal relations-- 00:45:26.050 --> 00:45:29.740 align:middle line:84% or because u and v are orthogonal. 00:45:29.740 --> 00:45:33.280 align:middle line:84% When you act this on the general solution, 00:45:33.280 --> 00:45:36.460 align:middle line:84% we'll just project into solutions proportional 00:45:36.460 --> 00:45:39.220 align:middle line:84% that only involve linear superpositions of u. 00:45:39.220 --> 00:45:41.870 align:middle line:90% 00:45:41.870 --> 00:45:50.710 align:middle line:84% So now, you can check yourself explicitly with this. 00:45:50.710 --> 00:45:52.480 align:middle line:90% So just use various-- 00:45:52.480 --> 00:45:53.785 align:middle line:90% use property of this u. 00:45:53.785 --> 00:45:55.810 align:middle line:84% You can check yourself explicitly with your-- 00:45:55.810 --> 00:45:57.160 align:middle line:90% I will not do there. 00:45:57.160 --> 00:46:04.170 align:middle line:84% And you can check that this is indeed a projector. 00:46:04.170 --> 00:46:07.140 align:middle line:84% If you look at the p squared, it gives back to itself. 00:46:07.140 --> 00:46:12.570 align:middle line:90% 00:46:12.570 --> 00:46:16.020 align:middle line:84% And with a little bit more effort, 00:46:16.020 --> 00:46:21.270 align:middle line:84% you can further show that the p plus can be 00:46:21.270 --> 00:46:23.910 align:middle line:90% written in the following form. 00:46:23.910 --> 00:46:27.330 align:middle line:90% ik slash plus m divided by 2m. 00:46:27.330 --> 00:46:31.940 align:middle line:90% 00:46:31.940 --> 00:46:33.900 align:middle line:84% So this requires a little bit more effort 00:46:33.900 --> 00:46:38.570 align:middle line:84% so I will leave as an exercise for yourself, 00:46:38.570 --> 00:46:42.810 align:middle line:90% or you can get it from the book. 00:46:42.810 --> 00:46:48.250 align:middle line:90% 00:46:48.250 --> 00:46:52.070 align:middle line:84% And similarly, you can define the projector 00:46:52.070 --> 00:46:57.710 align:middle line:84% to the space of v. So the p minus-- 00:46:57.710 --> 00:47:02.930 align:middle line:84% you can also define p minus to be minus 2 im 00:47:02.930 --> 00:47:05.460 align:middle line:90% because of the minus sign here. 00:47:05.460 --> 00:47:15.030 align:middle line:84% So 1 over minus 2 im some s vs k vs bar k. 00:47:15.030 --> 00:47:19.110 align:middle line:84% And you can show that this also have a-- 00:47:19.110 --> 00:47:21.120 align:middle line:84% can be written as something like this-- 00:47:21.120 --> 00:47:27.490 align:middle line:84% minus ik slash plus m divided by 2m. 00:47:27.490 --> 00:47:31.580 align:middle line:84% So again, this I will leave as an exercise to yourself. 00:47:31.580 --> 00:47:37.340 align:middle line:84% So this is the projector to the space of v's. 00:47:37.340 --> 00:47:54.010 align:middle line:84% And then from these two expression, 00:47:54.010 --> 00:47:59.140 align:middle line:84% p plus p minus-- you easily see that p plus plus p minus 00:47:59.140 --> 00:47:59.785 align:middle line:90% equal to 1. 00:47:59.785 --> 00:48:03.430 align:middle line:90% 00:48:03.430 --> 00:48:06.490 align:middle line:84% And you see the expression I just erased-- 00:48:06.490 --> 00:48:11.320 align:middle line:84% so from the-- taking the product and the p plus p 00:48:11.320 --> 00:48:15.520 align:middle line:84% minus is equal to p minus p plus equal to 0. 00:48:15.520 --> 00:48:18.430 align:middle line:84% So indeed, they are just projectors-- orthogonal 00:48:18.430 --> 00:48:20.455 align:middle line:90% projectors into different space. 00:48:20.455 --> 00:48:23.970 align:middle line:90% 00:48:23.970 --> 00:48:28.020 align:middle line:84% So this concludes our discussion of the classical solutions 00:48:28.020 --> 00:48:31.330 align:middle line:84% of Dirac equations and their properties. 00:48:31.330 --> 00:48:35.410 align:middle line:90% So do you have other questions? 00:48:35.410 --> 00:48:36.070 align:middle line:90% Yes? 00:48:36.070 --> 00:48:38.153 align:middle line:84% STUDENT: So we've seen hints that this corresponds 00:48:38.153 --> 00:48:41.440 align:middle line:84% to spin 1/2, but wouldn't it make more sense 00:48:41.440 --> 00:48:43.990 align:middle line:84% to just define the angular momentum operator 00:48:43.990 --> 00:48:46.550 align:middle line:84% and see whether it has eigenvalues or not? 00:48:46.550 --> 00:48:49.780 align:middle line:84% PROFESSOR: Yeah, that you would need to quantize the theory, 00:48:49.780 --> 00:48:52.840 align:middle line:84% but we haven't quantized the theory. 00:48:52.840 --> 00:48:55.190 align:middle line:84% Yeah, that would be our next step. 00:48:55.190 --> 00:48:58.940 align:middle line:84% So once we have found the full set of classical solutions 00:48:58.940 --> 00:49:00.590 align:middle line:90% and now we can quantize it. 00:49:00.590 --> 00:49:04.538 align:middle line:84% And once quantize it, then we can actually really find 00:49:04.538 --> 00:49:05.455 align:middle line:90% this angular momentum. 00:49:05.455 --> 00:49:08.080 align:middle line:90% 00:49:08.080 --> 00:49:09.190 align:middle line:90% Other questions? 00:49:09.190 --> 00:49:10.930 align:middle line:84% And indeed, actually, I think that would 00:49:10.930 --> 00:49:16.450 align:middle line:84% be one of your pset problems to show yourself 00:49:16.450 --> 00:49:17.590 align:middle line:90% that this is spin-half. 00:49:17.590 --> 00:49:20.410 align:middle line:90% 00:49:20.410 --> 00:49:23.080 align:middle line:90% Yeah, but yeah. 00:49:23.080 --> 00:49:23.950 align:middle line:90% Other questions? 00:49:23.950 --> 00:49:27.190 align:middle line:90% 00:49:27.190 --> 00:49:27.900 align:middle line:90% OK, good. 00:49:27.900 --> 00:49:31.160 align:middle line:90% 00:49:31.160 --> 00:49:33.800 align:middle line:84% So now we conclude the discussion. 00:49:33.800 --> 00:49:35.225 align:middle line:90% Now we can quantize the theory. 00:49:35.225 --> 00:49:44.680 align:middle line:90% 00:49:44.680 --> 00:49:46.885 align:middle line:84% Now we can go to the quantum system. 00:49:46.885 --> 00:49:58.730 align:middle line:90% 00:49:58.730 --> 00:50:06.270 align:middle line:84% So the-- remember, the action for the Dirac 00:50:06.270 --> 00:50:18.710 align:middle line:84% is d4x psi bar gamma mu partial mu psi partial mu minus m psi. 00:50:18.710 --> 00:50:25.160 align:middle line:84% So this is the action which gives us the Dirac equation. 00:50:25.160 --> 00:50:28.810 align:middle line:90% 00:50:28.810 --> 00:50:31.570 align:middle line:84% So in order to quantize this, we first-- 00:50:31.570 --> 00:50:36.450 align:middle line:84% remember our procedure to quantize a theory. 00:50:36.450 --> 00:50:41.540 align:middle line:84% So we need to first find the canonical momentum, 00:50:41.540 --> 00:50:45.710 align:middle line:84% and we also need to impose the canonical commutation relation, 00:50:45.710 --> 00:50:50.000 align:middle line:84% and then we expand the operators in terms 00:50:50.000 --> 00:50:53.807 align:middle line:84% of complete sets of classical solutions, 00:50:53.807 --> 00:50:56.390 align:middle line:84% and then that's corresponding to the solutions to the operator 00:50:56.390 --> 00:50:58.490 align:middle line:90% equations. 00:50:58.490 --> 00:51:02.630 align:middle line:84% So let's try to find out what's the canonical momentum for psi. 00:51:02.630 --> 00:51:05.780 align:middle line:90% 00:51:05.780 --> 00:51:08.490 align:middle line:84% So as in the complex scalar case, 00:51:08.490 --> 00:51:11.570 align:middle line:84% we can treat the psi and psi dagger 00:51:11.570 --> 00:51:15.410 align:middle line:84% as independent variables because of the-- 00:51:15.410 --> 00:51:16.610 align:middle line:90% yeah. 00:51:16.610 --> 00:51:19.970 align:middle line:84% So then the canonical momentum for psi-- 00:51:19.970 --> 00:51:22.500 align:middle line:90% if you look at this expression. 00:51:22.500 --> 00:51:26.720 align:middle line:84% So you look at the partial 0 psi. 00:51:26.720 --> 00:51:31.370 align:middle line:84% Then you find-- so the Lagrangian density-- 00:51:31.370 --> 00:51:34.290 align:middle line:84% if you look at that, if I write it bit-- 00:51:34.290 --> 00:51:37.760 align:middle line:90% I have the form psi dagger. 00:51:37.760 --> 00:51:47.790 align:middle line:84% So this psi dagger gamma 0, and then gamma 0 psi 0 psi, then 00:51:47.790 --> 00:51:49.290 align:middle line:90% plus the rest of the terms. 00:51:49.290 --> 00:51:51.360 align:middle line:84% So in order to find the canonical momentum, 00:51:51.360 --> 00:51:56.200 align:middle line:84% we are only interested in the part of the Lagrangian density 00:51:56.200 --> 00:51:58.800 align:middle line:84% which are related to the derivative of psi-- or the time 00:51:58.800 --> 00:52:00.000 align:middle line:90% derivative. 00:52:00.000 --> 00:52:02.550 align:middle line:84% And so that's the only term involving 00:52:02.550 --> 00:52:06.750 align:middle line:84% time derivative of either psi or psi bar or psi dagger. 00:52:06.750 --> 00:52:09.330 align:middle line:90% 00:52:09.330 --> 00:52:12.750 align:middle line:84% So this gamma 0 squared equal to minus 1-- 00:52:12.750 --> 00:52:19.400 align:middle line:84% so this is just i psi dagger partial 0 psi. 00:52:19.400 --> 00:52:23.300 align:middle line:84% So now, we can immediately find out 00:52:23.300 --> 00:52:31.650 align:middle line:84% that the conjugate momentum for psi is given by i psi dagger, 00:52:31.650 --> 00:52:34.695 align:middle line:84% because you take a derivative on psi 0-- 00:52:34.695 --> 00:52:37.290 align:middle line:84% partial 0 psi, you just get i psi dagger. 00:52:37.290 --> 00:52:43.680 align:middle line:84% But the conjugate momentum for psi dagger 00:52:43.680 --> 00:52:44.775 align:middle line:90% is actually equal to what? 00:52:44.775 --> 00:52:47.710 align:middle line:90% 00:52:47.710 --> 00:52:51.406 align:middle line:90% So what is this? 00:52:51.406 --> 00:52:52.804 align:middle line:90% STUDENT: 0. 00:52:52.804 --> 00:52:55.850 align:middle line:84% PROFESSOR: Thi is equal to 0 because there is no time 00:52:55.850 --> 00:53:00.190 align:middle line:90% derivative on psi dagger. 00:53:00.190 --> 00:53:01.650 align:middle line:90% So this tells you-- 00:53:01.650 --> 00:53:05.440 align:middle line:90% 00:53:05.440 --> 00:53:09.350 align:middle line:84% so this-- so the fact that this-- 00:53:09.350 --> 00:53:15.080 align:middle line:84% equal to 0 tells you actually the Dirac system-- 00:53:15.080 --> 00:53:17.900 align:middle line:84% the action-- this Dirac system is actually 00:53:17.900 --> 00:53:18.845 align:middle line:90% a constrained system. 00:53:18.845 --> 00:53:23.310 align:middle line:90% 00:53:23.310 --> 00:53:26.780 align:middle line:84% So this is considered to be a constraint, because you 00:53:26.780 --> 00:53:29.320 align:middle line:90% can no longer normally-- 00:53:29.320 --> 00:53:36.790 align:middle line:84% so remember, when we say from the Lagrangian, 00:53:36.790 --> 00:53:39.670 align:middle line:84% you find the canonical momentum, and then 00:53:39.670 --> 00:53:42.160 align:middle line:84% you invert the canonical momentum 00:53:42.160 --> 00:53:43.480 align:middle line:90% to find the Hamiltonian. 00:53:43.480 --> 00:53:46.310 align:middle line:90% 00:53:46.310 --> 00:53:50.300 align:middle line:84% But when you have this equal to 0, you cannot invert it. 00:53:50.300 --> 00:53:54.530 align:middle line:84% You cannot express your, say, psi or psi dagger in terms 00:53:54.530 --> 00:53:56.330 align:middle line:90% of pi or pi dagger. 00:53:56.330 --> 00:53:58.980 align:middle line:90% And so this is a constrained. 00:53:58.980 --> 00:54:03.695 align:middle line:84% So that means you actually have a constrained system. 00:54:03.695 --> 00:54:07.510 align:middle line:84% It means you have a constrained system. 00:54:07.510 --> 00:54:11.380 align:middle line:84% And frankly, constrained systems are very annoying. 00:54:11.380 --> 00:54:15.830 align:middle line:90% 00:54:15.830 --> 00:54:19.740 align:middle line:90% So they have involved lots of-- 00:54:19.740 --> 00:54:21.420 align:middle line:84% you have a lot of formalism in order 00:54:21.420 --> 00:54:23.400 align:middle line:84% to, say, treat constrained system-- 00:54:23.400 --> 00:54:25.860 align:middle line:90% to quantize them, et cetera. 00:54:25.860 --> 00:54:27.900 align:middle line:84% But fortunately, in this case, there's 00:54:27.900 --> 00:54:31.710 align:middle line:84% actually a simple trick to go around it. 00:54:31.710 --> 00:54:37.200 align:middle line:84% And so now, I will not use this full-fledged constrained 00:54:37.200 --> 00:54:38.460 align:middle line:90% quantization to do the job. 00:54:38.460 --> 00:54:40.470 align:middle line:84% I will just use a simple trick which 00:54:40.470 --> 00:54:42.150 align:middle line:90% will reach the same answer. 00:54:42.150 --> 00:54:42.812 align:middle line:90% Yes? 00:54:42.812 --> 00:54:44.520 align:middle line:84% STUDENT: Is it like the canonical momenta 00:54:44.520 --> 00:54:46.614 align:middle line:84% not well-defined, because I could integrate 00:54:46.614 --> 00:54:50.091 align:middle line:84% by parts to move around the time derivative 00:54:50.091 --> 00:54:52.760 align:middle line:84% because I can move the time derivative from psi 00:54:52.760 --> 00:54:54.080 align:middle line:90% to psi dagger, right? 00:54:54.080 --> 00:54:55.740 align:middle line:84% PROFESSOR: Yeah, so there's ambiguity. 00:54:55.740 --> 00:54:58.040 align:middle line:84% So that's part of the reflection that this 00:54:58.040 --> 00:55:00.360 align:middle line:90% is a constrained system. 00:55:00.360 --> 00:55:05.120 align:middle line:84% So this is actually typical to have these kind of constraints. 00:55:05.120 --> 00:55:07.920 align:middle line:84% It's typical whenever you have a first-order system. 00:55:07.920 --> 00:55:11.000 align:middle line:84% So here, it's different from the Klein-Gordon action is 00:55:11.000 --> 00:55:15.860 align:middle line:84% because here, we only involving the first derivative-- 00:55:15.860 --> 00:55:18.350 align:middle line:84% one derivative rather than two derivatives. 00:55:18.350 --> 00:55:22.950 align:middle line:84% And so this is not our standard system. 00:55:22.950 --> 00:55:29.620 align:middle line:84% So the way to go around this is the following. 00:55:29.620 --> 00:55:33.230 align:middle line:84% The way to go around it is the following. 00:55:33.230 --> 00:55:38.500 align:middle line:84% It's now-- so this equation-- if we look at this equation-- 00:55:38.500 --> 00:55:42.490 align:middle line:90% 00:55:42.490 --> 00:55:44.850 align:middle line:84% so this equation tells you that the canonical momentum 00:55:44.850 --> 00:55:50.220 align:middle line:84% conjugate to psi is actually psi dagger. 00:55:50.220 --> 00:55:53.670 align:middle line:84% So also here it's a little bit funny, 00:55:53.670 --> 00:55:57.240 align:middle line:84% because you remember in the standard story, 00:55:57.240 --> 00:56:01.630 align:middle line:84% the canonical momentum is related to the time derivative. 00:56:01.630 --> 00:56:03.130 align:middle line:84% And here, there's no time derivative 00:56:03.130 --> 00:56:05.650 align:middle line:90% here-- just psi dagger itself. 00:56:05.650 --> 00:56:08.350 align:middle line:84% And this is another indication that this 00:56:08.350 --> 00:56:11.440 align:middle line:90% is a constraint rather than-- 00:56:11.440 --> 00:56:12.130 align:middle line:90% yeah. 00:56:12.130 --> 00:56:20.030 align:middle line:84% So anyway, here, it tells you that the dagger is, in fact, 00:56:20.030 --> 00:56:22.790 align:middle line:90% the canonical moment of psi. 00:56:22.790 --> 00:56:24.670 align:middle line:90% And now, we can just-- 00:56:24.670 --> 00:56:26.530 align:middle line:84% now in this equation, we can just 00:56:26.530 --> 00:56:30.310 align:middle line:84% try to interpret the psi dagger as the momentum 00:56:30.310 --> 00:56:33.040 align:middle line:84% rather than as the configuration variable. 00:56:33.040 --> 00:56:35.620 align:middle line:84% So now, let's rewrite the Lagrangian. 00:56:35.620 --> 00:56:38.620 align:middle line:84% So the Lagrangian has the following form-- 00:56:38.620 --> 00:56:45.910 align:middle line:84% minus i psi bar gamma mu partial mu minus m psi. 00:56:45.910 --> 00:56:48.170 align:middle line:84% So this is the Lagrangian density. 00:56:48.170 --> 00:56:51.260 align:middle line:84% So we'll rewrite it as the following. 00:56:51.260 --> 00:56:54.800 align:middle line:84% So let's first separate the time derivative. 00:56:54.800 --> 00:56:56.440 align:middle line:90% So this is psi dagger. 00:56:56.440 --> 00:57:01.420 align:middle line:84% So just this term involving the time derivative. 00:57:01.420 --> 00:57:06.570 align:middle line:84% And then the rest, I have psi bar gamma i partial 00:57:06.570 --> 00:57:08.160 align:middle line:90% i minus m psi. 00:57:08.160 --> 00:57:12.570 align:middle line:84% But here, they are no longer involving the time derivative. 00:57:12.570 --> 00:57:17.540 align:middle line:84% And now, we just interpret this as the canonical momentum 00:57:17.540 --> 00:57:18.180 align:middle line:90% for psi. 00:57:18.180 --> 00:57:24.660 align:middle line:90% 00:57:24.660 --> 00:57:27.240 align:middle line:84% And then this is just something involving 00:57:27.240 --> 00:57:30.450 align:middle line:90% spatial derivative of psi. 00:57:30.450 --> 00:57:32.940 align:middle line:84% And I just interpret this as the Hamiltonian for psi. 00:57:32.940 --> 00:57:45.440 align:middle line:90% 00:57:45.440 --> 00:57:47.120 align:middle line:84% So this expression just involving 00:57:47.120 --> 00:57:49.190 align:middle line:90% psi dagger and the psi-- 00:57:49.190 --> 00:57:50.990 align:middle line:84% involving some spatial derivative. 00:57:50.990 --> 00:57:55.160 align:middle line:84% And so this is essentially just some functions of psi-- 00:57:55.160 --> 00:57:59.720 align:middle line:84% pi psi and psi and maybe with some spatial derivatives. 00:57:59.720 --> 00:58:04.330 align:middle line:84% So now, you see that this Lagrangian density-- 00:58:04.330 --> 00:58:07.210 align:middle line:84% actually, if I interpret the diagram 00:58:07.210 --> 00:58:08.770 align:middle line:84% as the canonical momentum, then it 00:58:08.770 --> 00:58:11.170 align:middle line:84% has the form of this canonical transformation 00:58:11.170 --> 00:58:14.080 align:middle line:84% to go from the Lagrangian density to Hamiltonian. 00:58:14.080 --> 00:58:16.510 align:middle line:84% And then I can just interpret this as the Hamiltonian. 00:58:16.510 --> 00:58:21.180 align:middle line:90% 00:58:21.180 --> 00:58:24.856 align:middle line:84% So now, I can just interpret this as my Hamiltonian. 00:58:24.856 --> 00:58:28.800 align:middle line:90% 00:58:28.800 --> 00:58:32.635 align:middle line:90% So we just treat-- 00:58:32.635 --> 00:59:03.010 align:middle line:90% 00:59:03.010 --> 00:59:08.900 align:middle line:90% so now, we just-- 00:59:08.900 --> 00:59:14.540 align:middle line:84% so this is-- so psi dagger and psi. 00:59:14.540 --> 00:59:18.230 align:middle line:84% And now, this is the full phase space. 00:59:18.230 --> 00:59:21.210 align:middle line:84% So this is the momentum and this is the coordinate, 00:59:21.210 --> 00:59:26.318 align:middle line:84% so this is the full space, rather than previously, we 00:59:26.318 --> 00:59:28.235 align:middle line:84% would interpret them as a configuration space. 00:59:28.235 --> 00:59:31.970 align:middle line:90% 00:59:31.970 --> 00:59:37.690 align:middle line:84% And then the Hamiltonian just equal 00:59:37.690 --> 00:59:45.290 align:middle line:84% to i psi bar gamma i partial i minus m psi. 00:59:45.290 --> 00:59:50.290 align:middle line:90% 00:59:50.290 --> 00:59:52.000 align:middle line:84% So this is the Hamiltonian density. 00:59:52.000 --> 01:00:01.350 align:middle line:90% 01:00:01.350 --> 01:00:04.770 align:middle line:84% So we now then have-- now, this just goes back 01:00:04.770 --> 01:00:06.000 align:middle line:90% to the standard formalism. 01:00:06.000 --> 01:00:08.660 align:middle line:90% 01:00:08.660 --> 01:00:10.340 align:middle line:90% Any questions on this? 01:00:10.340 --> 01:00:11.000 align:middle line:90% Yes? 01:00:11.000 --> 01:00:13.248 align:middle line:84% STUDENT: So can we still interpret psi dagger 01:00:13.248 --> 01:00:16.570 align:middle line:90% as a different field than psi? 01:00:16.570 --> 01:00:19.216 align:middle line:84% PROFESSOR: it's canonical momentum. 01:00:19.216 --> 01:00:21.482 align:middle line:84% STUDENT: So it's no longer two different fields? 01:00:21.482 --> 01:00:23.690 align:middle line:84% PROFESSOR: Yeah, it's no longer two different fields. 01:00:23.690 --> 01:00:26.230 align:middle line:84% For example, if there are two different fields, then 01:00:26.230 --> 01:00:29.110 align:middle line:84% psi and psi dagger, they should commute. 01:00:29.110 --> 01:00:31.300 align:middle line:84% But now, since they become canonical momentum, 01:00:31.300 --> 01:00:34.010 align:middle line:90% they no longer commute. 01:00:34.010 --> 01:00:35.020 align:middle line:90% yes? 01:00:35.020 --> 01:00:37.930 align:middle line:84% STUDENT: If you take psi bar and then you bring the, you pull 01:00:37.930 --> 01:00:42.760 align:middle line:84% out the gamma 0, isn't this just like the right-hand side 01:00:42.760 --> 01:00:46.090 align:middle line:84% of the-- like our earlier version of the Dirac equation-- 01:00:46.090 --> 01:00:50.944 align:middle line:84% the one where you like [INAUDIBLE]?? 01:00:50.944 --> 01:00:52.563 align:middle line:84% Like, the one with the alpha and beta, 01:00:52.563 --> 01:00:54.730 align:middle line:84% like the other Dirac equation we wrote down earlier? 01:00:54.730 --> 01:00:56.650 align:middle line:90% PROFESSOR: Yeah, that's right. 01:00:56.650 --> 01:00:59.800 align:middle line:84% Yeah, it's very-- essentially just that Hamiltonian. 01:00:59.800 --> 01:01:01.090 align:middle line:90% Yeah, that's right. 01:01:01.090 --> 01:01:02.710 align:middle line:90% Yes? 01:01:02.710 --> 01:01:05.300 align:middle line:84% STUDENT: Is there not a problem with this 01:01:05.300 --> 01:01:07.725 align:middle line:90% treating the time differently? 01:01:07.725 --> 01:01:10.620 align:middle line:84% Is it no longer Lorentz invariant? 01:01:10.620 --> 01:01:12.060 align:middle line:90% PROFESSOR: Yeah, so when you-- 01:01:12.060 --> 01:01:17.790 align:middle line:84% yeah, when we do the client encoding equation in the scalar 01:01:17.790 --> 01:01:20.250 align:middle line:84% theory, when you quantize, you always 01:01:20.250 --> 01:01:22.230 align:middle line:90% have to treat time differently. 01:01:22.230 --> 01:01:28.590 align:middle line:84% And then the Lorentz symmetry come out 01:01:28.590 --> 01:01:31.830 align:middle line:90% in the property of the state. 01:01:31.830 --> 01:01:34.020 align:middle line:90% Yeah, but you always-- 01:01:34.020 --> 01:01:39.520 align:middle line:84% when you quantize, you always have to treat time separately. 01:01:39.520 --> 01:01:40.210 align:middle line:90% Yes? 01:01:40.210 --> 01:01:40.791 align:middle line:84% STUDENT: So now we have the momentum operator, 01:01:40.791 --> 01:01:42.374 align:middle line:84% its just the complex conjugate of phi? 01:01:42.374 --> 01:01:47.070 align:middle line:90% 01:01:47.070 --> 01:01:48.434 align:middle line:90% PROFESSOR: Yeah, exactly. 01:01:48.434 --> 01:01:52.174 align:middle line:84% STUDENT: So this like, its still like constrained. 01:01:52.174 --> 01:01:54.665 align:middle line:90% PROFESSOR: Yeah, it's a-- 01:01:54.665 --> 01:02:03.240 align:middle line:90% 01:02:03.240 --> 01:02:04.650 align:middle line:90% yeah, this is just-- 01:02:04.650 --> 01:02:06.970 align:middle line:84% yeah, so that's why we call this a constrained system. 01:02:06.970 --> 01:02:09.430 align:middle line:84% But now, I'm just giving you a shortcut 01:02:09.430 --> 01:02:13.620 align:middle line:84% so that we have to don't have to go through this-- 01:02:13.620 --> 01:02:18.638 align:middle line:84% yeah, the intricacy of constrained quantization. 01:02:18.638 --> 01:02:21.740 align:middle line:90% 01:02:21.740 --> 01:02:24.980 align:middle line:84% So now, the canonical quantization becomes easy. 01:02:24.980 --> 01:02:28.920 align:middle line:90% So now, this is my momentum. 01:02:28.920 --> 01:02:33.670 align:middle line:84% So now, we just have the canonical quantization. 01:02:33.670 --> 01:02:42.640 align:middle line:90% 01:02:42.640 --> 01:02:46.360 align:middle line:84% I think today, we may have unfortunate timing. 01:02:46.360 --> 01:02:47.980 align:middle line:90% So now, become psi-- 01:02:47.980 --> 01:02:49.300 align:middle line:90% is coordinate. 01:02:49.300 --> 01:02:52.540 align:middle line:90% 01:02:52.540 --> 01:02:54.700 align:middle line:90% And now, we have psi dagger. 01:02:54.700 --> 01:03:00.670 align:middle line:84% We have pi psi, which is the momentum. 01:03:00.670 --> 01:03:09.340 align:middle line:84% So this, by definition, should give you the delta function, so 01:03:09.340 --> 01:03:11.180 align:middle line:90% equal time correlation function. 01:03:11.180 --> 01:03:12.820 align:middle line:90% So here, we should-- 01:03:12.820 --> 01:03:15.380 align:middle line:84% because remember, this is a four vector, 01:03:15.380 --> 01:03:17.660 align:middle line:84% so this also have four components. 01:03:17.660 --> 01:03:22.930 align:middle line:84% So let me just now suppress this subscript psi 01:03:22.930 --> 01:03:26.440 align:middle line:84% because there's only one variable here now. 01:03:26.440 --> 01:03:28.430 align:middle line:84% So here, there's also coefficient-- 01:03:28.430 --> 01:03:30.850 align:middle line:84% so this is also four vector beta, 01:03:30.850 --> 01:03:33.520 align:middle line:84% and then here should be delta alpha beta. 01:03:33.520 --> 01:03:35.290 align:middle line:84% So alpha beta are spinor indices. 01:03:35.290 --> 01:03:43.060 align:middle line:90% 01:03:43.060 --> 01:03:44.340 align:middle line:90% So if you impose-- 01:03:44.340 --> 01:03:51.190 align:middle line:84% and this is equivalent just to psi alpha t, 01:03:51.190 --> 01:04:06.830 align:middle line:84% x psi dagger beta t, x prime equal to i delta alpha beta 01:04:06.830 --> 01:04:08.020 align:middle line:90% x x minus prime. 01:04:08.020 --> 01:04:15.480 align:middle line:90% 01:04:15.480 --> 01:04:22.560 align:middle line:84% And so we can now expand psi x in terms 01:04:22.560 --> 01:04:24.910 align:middle line:90% of complete set of solutions. 01:04:24.910 --> 01:04:29.380 align:middle line:84% So this is the first step of the quantization. 01:04:29.380 --> 01:04:32.940 align:middle line:84% So first is-- this is the commutation relation. 01:04:32.940 --> 01:04:34.950 align:middle line:84% So the second step-- remember, we 01:04:34.950 --> 01:04:39.270 align:middle line:84% can expand the psi in terms of complete basis 01:04:39.270 --> 01:04:41.460 align:middle line:90% of classical solutions. 01:04:41.460 --> 01:04:45.540 align:middle line:90% So psi x just equal to-- 01:04:45.540 --> 01:04:48.465 align:middle line:84% again, we integrate over all spatial momentum. 01:04:48.465 --> 01:04:51.430 align:middle line:90% 01:04:51.430 --> 01:04:55.870 align:middle line:84% Again, we use this factor for convenience 01:04:55.870 --> 01:04:57.820 align:middle line:90% as in the scalar case. 01:04:57.820 --> 01:05:01.800 align:middle line:84% And now, we just need to sum over all possible solutions. 01:05:01.800 --> 01:05:09.960 align:middle line:84% So here, we have a sum over s equal to 1 and 2. 01:05:09.960 --> 01:05:21.170 align:middle line:84% So we have-- then we have ak s us exponential ik x. 01:05:21.170 --> 01:05:25.115 align:middle line:90% And then plus bk-- 01:05:25.115 --> 01:05:25.615 align:middle line:90% sorry. 01:05:25.615 --> 01:05:28.140 align:middle line:90% 01:05:28.140 --> 01:05:29.850 align:middle line:90% k should be here-- sorry. 01:05:29.850 --> 01:05:38.840 align:middle line:84% bk s and vs exponential minus ik x. 01:05:38.840 --> 01:05:43.940 align:middle line:90% 01:05:43.940 --> 01:05:46.890 align:middle line:90% So these are the complete-- 01:05:46.890 --> 01:05:50.040 align:middle line:84% because complete set of solution are u and v. 01:05:50.040 --> 01:05:52.935 align:middle line:84% And we call the coefficients of a and b, 01:05:52.935 --> 01:05:56.430 align:middle line:84% so let me call it b dagger as we did for the complex scalar case 01:05:56.430 --> 01:05:59.040 align:middle line:90% before. 01:05:59.040 --> 01:06:02.430 align:middle line:84% So we-- so the coefficient, we call them a and b. 01:06:02.430 --> 01:06:05.250 align:middle line:90% Actually, it doesn't matter. 01:06:05.250 --> 01:06:06.930 align:middle line:90% We call it b. 01:06:06.930 --> 01:06:08.880 align:middle line:90% Actually, let me just call it b. 01:06:08.880 --> 01:06:12.180 align:middle line:84% Yeah, so you can just-- you have u and v, 01:06:12.180 --> 01:06:14.630 align:middle line:84% and then the coefficient will be a and b. 01:06:14.630 --> 01:06:19.575 align:middle line:90% 01:06:19.575 --> 01:06:30.650 align:middle line:84% And so now, you can just plug this. 01:06:30.650 --> 01:06:36.160 align:middle line:84% So if this equation is 1 and this equation is 2, 01:06:36.160 --> 01:06:38.770 align:middle line:84% and you take the complex conjugate, psi dagger, 01:06:38.770 --> 01:06:41.350 align:middle line:84% you can plug it in here to find what's the commutation 01:06:41.350 --> 01:06:42.748 align:middle line:90% relation between a and b. 01:06:42.748 --> 01:06:46.960 align:middle line:90% 01:06:46.960 --> 01:06:51.895 align:middle line:90% So you find that the-- 01:06:51.895 --> 01:06:54.980 align:middle line:90% 01:06:54.980 --> 01:07:03.030 align:middle line:90% yeah, so plug 2 into 1. 01:07:03.030 --> 01:07:19.750 align:middle line:84% So you find the commutation relation ak s ak prime p, then 01:07:19.750 --> 01:07:29.630 align:middle line:90% equal to bk s and bk prime t. 01:07:29.630 --> 01:07:39.350 align:middle line:84% Then, equal to 2 pi cubed delta st, delta 3 k minus k prime. 01:07:39.350 --> 01:07:45.250 align:middle line:90% 01:07:45.250 --> 01:07:49.990 align:middle line:84% So now-- and you can also define the vacuum. 01:07:49.990 --> 01:08:15.770 align:middle line:90% 01:08:15.770 --> 01:08:19.340 align:middle line:84% And all other commutators are 0-- 01:08:19.340 --> 01:08:20.494 align:middle line:90% with all others 0. 01:08:20.494 --> 01:08:26.840 align:middle line:90% 01:08:26.840 --> 01:08:31.310 align:middle line:84% And you can also define the vacuum, 01:08:31.310 --> 01:08:40.569 align:middle line:84% as we did for the scalar case, to be annihilated by ak and bk. 01:08:40.569 --> 01:08:51.620 align:middle line:90% 01:08:51.620 --> 01:08:56.240 align:middle line:84% And now, you can, in principle, build your Hilbert space. 01:08:56.240 --> 01:08:56.960 align:middle line:90% Yes? 01:08:56.960 --> 01:08:58.943 align:middle line:84% STUDENT: For equation one up there, 01:08:58.943 --> 01:09:04.620 align:middle line:84% are you supposed to not have factor pi just because momentum 01:09:04.620 --> 01:09:07.080 align:middle line:90% is i psi dagger? 01:09:07.080 --> 01:09:07.878 align:middle line:90% PROFESSOR: Yeah. 01:09:07.878 --> 01:09:08.670 align:middle line:90% STUDENT: And then-- 01:09:08.670 --> 01:09:09.545 align:middle line:90% PROFESSOR: Oh, right. 01:09:09.545 --> 01:09:12.220 align:middle line:84% Sorry, there's no-- yeah, thank you. 01:09:12.220 --> 01:09:13.109 align:middle line:90% There's no i here. 01:09:13.109 --> 01:09:15.399 align:middle line:84% Yeah, because the momentum is i psi dagger. 01:09:15.399 --> 01:09:22.859 align:middle line:84% Yeah, there's no i anymore-- just the delta alpha beta. 01:09:22.859 --> 01:09:28.130 align:middle line:84% So all look pretty straightforward, 01:09:28.130 --> 01:09:31.269 align:middle line:90% but you actually have problems. 01:09:31.269 --> 01:09:34.660 align:middle line:90% 01:09:34.660 --> 01:09:37.160 align:middle line:90% You actually have problems. 01:09:37.160 --> 01:09:49.170 align:middle line:84% So the first problem some of you may be already asking 01:09:49.170 --> 01:09:55.300 align:middle line:84% in your head, is that, are we supposed 01:09:55.300 --> 01:09:58.820 align:middle line:90% to describe spin-half particles? 01:09:58.820 --> 01:10:03.400 align:middle line:84% But if we are suppose to describe spin-half particles, 01:10:03.400 --> 01:10:05.200 align:middle line:90% then why do we find the bosons? 01:10:05.200 --> 01:10:09.030 align:middle line:84% So these are commutation relations for bosons. 01:10:09.030 --> 01:10:11.760 align:middle line:90% It's because they commute. 01:10:11.760 --> 01:10:16.960 align:middle line:84% So different k-- sorry, I think I'm missing dagger. 01:10:16.960 --> 01:10:17.460 align:middle line:90% Sorry. 01:10:17.460 --> 01:10:20.550 align:middle line:90% 01:10:20.550 --> 01:10:25.310 align:middle line:84% So different ak, they commute with each other. 01:10:25.310 --> 01:10:27.140 align:middle line:84% As we discussed in the scalar case, 01:10:27.140 --> 01:10:32.660 align:middle line:84% when the a mu all commute with each other, this is boson. 01:10:32.660 --> 01:10:34.190 align:middle line:84% It means the particle you created, 01:10:34.190 --> 01:10:38.750 align:middle line:84% you can exchange them and don't-- 01:10:38.750 --> 01:10:41.000 align:middle line:84% yeah, you can arbitrarily exchange them. 01:10:41.000 --> 01:10:42.800 align:middle line:90% It's symmetric. 01:10:42.800 --> 01:10:48.542 align:middle line:84% But fermions, they are supposed to be Pauli principle. 01:10:48.542 --> 01:10:50.000 align:middle line:84% So you're supposed to get the minus 01:10:50.000 --> 01:10:52.160 align:middle line:90% sign when you exchange them. 01:10:52.160 --> 01:10:55.870 align:middle line:84% So this cannot-- this doesn't look to be right. 01:10:55.870 --> 01:10:59.260 align:middle line:84% So this is-- the first thing it seems 01:10:59.260 --> 01:11:05.720 align:middle line:84% to be we were finding bosons rather than fermions. 01:11:05.720 --> 01:11:08.750 align:middle line:84% And the second is that now if you find that the Hamiltonian-- 01:11:08.750 --> 01:11:12.430 align:middle line:90% 01:11:12.430 --> 01:11:14.620 align:middle line:84% so now, if you try to find the Hamiltonian-- 01:11:14.620 --> 01:11:18.360 align:middle line:90% 01:11:18.360 --> 01:11:25.020 align:middle line:84% so H is just the integration of this Hamiltonian density. 01:11:25.020 --> 01:11:32.390 align:middle line:90% 01:11:32.390 --> 01:11:34.580 align:middle line:90% And you can just plug it in-- 01:11:34.580 --> 01:11:38.210 align:middle line:84% plug that expression for Hamiltonian density in. 01:11:38.210 --> 01:11:40.370 align:middle line:90% Where is my Hamiltonian density? 01:11:40.370 --> 01:11:46.930 align:middle line:84% Yeah, plug that in, and then you will find the-- 01:11:46.930 --> 01:11:49.970 align:middle line:90% 01:11:49.970 --> 01:11:56.300 align:middle line:84% yeah so you find just this just given by i d3 01:11:56.300 --> 01:12:02.420 align:middle line:84% x psi bar gamma i partial i minus m psi. 01:12:02.420 --> 01:12:11.790 align:middle line:90% 01:12:11.790 --> 01:12:15.990 align:middle line:84% So now, if you plug them in, you just plug them in-- plug this 01:12:15.990 --> 01:12:17.430 align:middle line:90% mode expansion in-- 01:12:17.430 --> 01:12:19.410 align:middle line:90% and then you find actually-- 01:12:19.410 --> 01:12:22.990 align:middle line:90% 01:12:22.990 --> 01:12:24.880 align:middle line:84% then you find the following expression-- 01:12:24.880 --> 01:12:34.580 align:middle line:90% 01:12:34.580 --> 01:12:37.190 align:middle line:90% omega k. 01:12:37.190 --> 01:12:41.940 align:middle line:90% And you sum over s from 1 to 2. 01:12:41.940 --> 01:12:47.220 align:middle line:84% And then you find it's given by ak 01:12:47.220 --> 01:13:05.160 align:middle line:84% s dagger ak s minus bk s dagger bk 01:13:05.160 --> 01:13:11.290 align:middle line:84% s, and then up to some infinite constants. 01:13:11.290 --> 01:13:15.450 align:middle line:84% So now, you see something problematic here. 01:13:15.450 --> 01:13:17.250 align:middle line:90% So what's problematic you see? 01:13:17.250 --> 01:13:22.000 align:middle line:90% 01:13:22.000 --> 01:13:22.500 align:middle line:90% Yes? 01:13:22.500 --> 01:13:24.780 align:middle line:84% STUDENT: The energy can be arbitrarily small. 01:13:24.780 --> 01:13:27.840 align:middle line:84% PROFESSOR: Yeah, energy is not positive definite 01:13:27.840 --> 01:13:29.070 align:middle line:90% because of this minus sign. 01:13:29.070 --> 01:13:35.600 align:middle line:90% 01:13:35.600 --> 01:13:40.550 align:middle line:84% So this minus sign tells you that if you excite 01:13:40.550 --> 01:13:45.140 align:middle line:84% a lot of these b particles, you can make the energy 01:13:45.140 --> 01:13:48.200 align:middle line:84% as negative as you want, and then the theory 01:13:48.200 --> 01:13:49.850 align:middle line:90% don't make sense. 01:13:49.850 --> 01:13:54.510 align:middle line:84% And then this is a theory with energy unbounded from below. 01:13:54.510 --> 01:13:57.990 align:middle line:84% It's a theory with energy unbounded from below. 01:13:57.990 --> 01:14:02.105 align:middle line:84% So that means this vacuum is unstable. 01:14:02.105 --> 01:14:16.970 align:middle line:90% 01:14:16.970 --> 01:14:19.760 align:middle line:84% So there must be something wrong with this procedure. 01:14:19.760 --> 01:14:24.080 align:middle line:90% 01:14:24.080 --> 01:14:28.170 align:middle line:84% So it turns out that there's a very quick fix to this problem, 01:14:28.170 --> 01:14:34.160 align:middle line:84% and this quick fix was invented by Jordan. 01:14:34.160 --> 01:14:38.660 align:middle line:84% So he recognized that if you don't 01:14:38.660 --> 01:14:41.240 align:middle line:84% do the standard commutation relation-- 01:14:41.240 --> 01:14:43.910 align:middle line:90% 01:14:43.910 --> 01:14:48.910 align:middle line:84% if you do instead of commutator-- 01:14:48.910 --> 01:14:52.060 align:middle line:84% here, you change it to anti-commutator. 01:14:52.060 --> 01:14:59.800 align:middle line:90% 01:14:59.800 --> 01:15:02.350 align:middle line:84% Remember, this bracket-- this curly bracket 01:15:02.350 --> 01:15:04.270 align:middle line:90% means anticommutator. 01:15:04.270 --> 01:15:05.980 align:middle line:84% And that means that here, you change it 01:15:05.980 --> 01:15:15.790 align:middle line:84% to anticommutator, and then still with the same thing-- 01:15:15.790 --> 01:15:30.490 align:middle line:90% 01:15:30.490 --> 01:15:35.040 align:middle line:84% and now this becomes anti-commutator. 01:15:35.040 --> 01:15:48.050 align:middle line:90% 01:15:48.050 --> 01:15:49.100 align:middle line:90% So let me see. 01:15:49.100 --> 01:15:54.380 align:middle line:90% 01:15:54.380 --> 01:15:56.520 align:middle line:84% I just want to make sure I get the-- 01:15:56.520 --> 01:15:58.925 align:middle line:90% 01:15:58.925 --> 01:16:00.550 align:middle line:84% yeah, if we change anti-commutator also 01:16:00.550 --> 01:16:04.540 align:middle line:84% from the convention, you change this to dagger. 01:16:04.540 --> 01:16:07.390 align:middle line:84% Now you change this to dagger as we did for the complex scalar 01:16:07.390 --> 01:16:08.080 align:middle line:90% case. 01:16:08.080 --> 01:16:10.400 align:middle line:90% And then you just get this. 01:16:10.400 --> 01:16:13.180 align:middle line:84% And now, if you do that, and then this becomes plus sign. 01:16:13.180 --> 01:16:18.980 align:middle line:90% 01:16:18.980 --> 01:16:22.550 align:middle line:84% And now, not only this becomes plus sign-- 01:16:22.550 --> 01:16:30.740 align:middle line:84% and also all different ak for different s and t-- 01:16:30.740 --> 01:16:32.630 align:middle line:84% they anti-commute with each other. 01:16:32.630 --> 01:16:42.840 align:middle line:90% 01:16:42.840 --> 01:16:45.590 align:middle line:90% They anticommute each other. 01:16:45.590 --> 01:16:51.070 align:middle line:84% So anticommute each other means that if you create a particle-- 01:16:51.070 --> 01:16:53.720 align:middle line:90% for example, ak dagger-- 01:16:53.720 --> 01:16:58.400 align:middle line:84% and you create another particle, ak prime dagger on 0, 01:16:58.400 --> 01:17:00.635 align:middle line:90% and now these two anticommute. 01:17:00.635 --> 01:17:04.300 align:middle line:84% It means when you exchange all, you get the minus sign. 01:17:04.300 --> 01:17:06.820 align:middle line:84% That precisely gives you the minus sign 01:17:06.820 --> 01:17:08.530 align:middle line:90% in the Pauli principle-- 01:17:08.530 --> 01:17:14.230 align:middle line:90% in the statistics for fermions. 01:17:14.230 --> 01:17:19.360 align:middle line:84% And so changing this thing to the anti-commutator-- 01:17:19.360 --> 01:17:22.660 align:middle line:90% and so solves all your problem. 01:17:22.660 --> 01:17:25.420 align:middle line:84% And your energy now is bounded from below, 01:17:25.420 --> 01:17:28.150 align:middle line:84% and then you actually now actually describe 01:17:28.150 --> 01:17:32.290 align:middle line:84% some particles which obey the Pauli principle. 01:17:32.290 --> 01:17:38.500 align:middle line:90% 01:17:38.500 --> 01:17:48.130 align:middle line:84% So yeah, I think we still have 1 minute left, but we-- 01:17:48.130 --> 01:17:52.540 align:middle line:84% but it's a good time to stop, because otherwise, other things 01:17:52.540 --> 01:17:53.900 align:middle line:90% will take more time. 01:17:53.900 --> 01:17:57.220 align:middle line:84% So let me just make a couple remark 01:17:57.220 --> 01:18:02.470 align:middle line:84% why the commutator-- why here, you need to do something 01:18:02.470 --> 01:18:03.580 align:middle line:90% unconventional. 01:18:03.580 --> 01:18:10.990 align:middle line:90% 01:18:10.990 --> 01:18:12.910 align:middle line:84% So if you look at this Hamiltonian density-- 01:18:12.910 --> 01:18:19.290 align:middle line:90% 01:18:19.290 --> 01:18:25.240 align:middle line:84% so even without doing this calculation, 01:18:25.240 --> 01:18:30.790 align:middle line:84% you can already conclude from here that this H cannot be 01:18:30.790 --> 01:18:32.920 align:middle line:84% bounded from below for the following reason. 01:18:32.920 --> 01:18:36.320 align:middle line:90% 01:18:36.320 --> 01:18:38.890 align:middle line:90% So from the Dirac equation-- 01:18:38.890 --> 01:18:40.120 align:middle line:90% let me just erase here. 01:18:40.120 --> 01:18:43.390 align:middle line:90% 01:18:43.390 --> 01:18:47.230 align:middle line:84% So from the Dirac equation, you have 01:18:47.230 --> 01:18:53.620 align:middle line:84% gamma mu partial mu minus m psi equal to 0. 01:18:53.620 --> 01:18:59.230 align:middle line:84% So that means that the gamma i partial i minus m psi 01:18:59.230 --> 01:19:02.710 align:middle line:84% is equal to minus gamma 0 partial 0 psi. 01:19:02.710 --> 01:19:05.380 align:middle line:90% 01:19:05.380 --> 01:19:13.360 align:middle line:84% So now, if you plug this equation into here, 01:19:13.360 --> 01:19:15.980 align:middle line:90% and then what you get-- 01:19:15.980 --> 01:19:21.670 align:middle line:84% you should find that H is equal to i d3 01:19:21.670 --> 01:19:26.020 align:middle line:90% x psi dagger partial 0 psi. 01:19:26.020 --> 01:19:29.340 align:middle line:90% 01:19:29.340 --> 01:19:33.240 align:middle line:90% So because the psi-- 01:19:33.240 --> 01:19:35.010 align:middle line:84% now this becomes a operate equation, 01:19:35.010 --> 01:19:36.960 align:middle line:90% and psi satisfies this equation. 01:19:36.960 --> 01:19:41.200 align:middle line:84% And now, you find H becomes this form, which 01:19:41.200 --> 01:19:44.770 align:middle line:84% is psi dagger times the time derivative of psi. 01:19:44.770 --> 01:19:47.110 align:middle line:84% And time derivative of psi-- remember, 01:19:47.110 --> 01:19:51.340 align:middle line:84% psi expanded both in terms of the positive energy 01:19:51.340 --> 01:19:55.390 align:middle line:84% solutions and the negative energy solutions. 01:19:55.390 --> 01:19:57.750 align:middle line:84% So depends on which omega you have. 01:19:57.750 --> 01:20:01.200 align:middle line:84% You can have either positive omega negative omega, 01:20:01.200 --> 01:20:03.750 align:middle line:84% because this is the first derivative-- a first time 01:20:03.750 --> 01:20:04.920 align:middle line:90% derivative of psi. 01:20:04.920 --> 01:20:09.270 align:middle line:84% And so this cannot be a positive definite. 01:20:09.270 --> 01:20:11.610 align:middle line:84% So this quantity cannot be positive definite. 01:20:11.610 --> 01:20:18.960 align:middle line:84% Anyway-- so you have to do something more radical. 01:20:18.960 --> 01:20:21.750 align:middle line:84% If these are conventional functions, 01:20:21.750 --> 01:20:24.720 align:middle line:84% and then this won't be positive definite. 01:20:24.720 --> 01:20:27.810 align:middle line:84% But now, what we did is that we make these into functions 01:20:27.810 --> 01:20:29.700 align:middle line:90% which anticommute. 01:20:29.700 --> 01:20:31.590 align:middle line:84% When they anticommute, turns out this 01:20:31.590 --> 01:20:35.080 align:middle line:84% to be a positive definite quantity. 01:20:35.080 --> 01:20:37.840 align:middle line:84% As the ordinary function, this cannot be positive definite. 01:20:37.840 --> 01:20:41.800 align:middle line:84% But if they anticommute, then become a positive definite. 01:20:41.800 --> 01:20:43.980 align:middle line:90% So let's stop here. 01:20:43.980 --> 01:20:47.000 align:middle line:90%