WEBVTT 00:00:00.000 --> 00:00:01.916 align:middle line:90% [SQUEAKING] 00:00:01.916 --> 00:00:04.311 align:middle line:90% [RUSTLING] 00:00:04.311 --> 00:00:06.227 align:middle line:90% [CLICKING] 00:00:06.227 --> 00:00:10.370 align:middle line:90% 00:00:10.370 --> 00:00:12.710 align:middle line:84% HONG LIU: So last time, we started 00:00:12.710 --> 00:00:18.060 align:middle line:84% talking about the quantization of the Dirac theory. 00:00:18.060 --> 00:00:21.770 align:middle line:84% And so, if we start with the Lagrangian density 00:00:21.770 --> 00:00:31.880 align:middle line:84% of the Dirac theory, then we can isolate the time dependent-- 00:00:31.880 --> 00:00:35.000 align:middle line:84% the part with only time directive 00:00:35.000 --> 00:00:36.770 align:middle line:90% and has the following form. 00:00:36.770 --> 00:00:47.790 align:middle line:90% 00:00:47.790 --> 00:00:55.680 align:middle line:84% And then we interpret this as the canonical momentum for psi. 00:00:55.680 --> 00:00:58.350 align:middle line:90% 00:00:58.350 --> 00:01:01.980 align:middle line:90% And then we have pi psi. 00:01:01.980 --> 00:01:03.900 align:middle line:84% And then this one we just interpret 00:01:03.900 --> 00:01:07.740 align:middle line:84% as the Hamiltonian, Hamiltonian density. 00:01:07.740 --> 00:01:10.520 align:middle line:90% 00:01:10.520 --> 00:01:16.583 align:middle line:84% And then we can just proceed to quantize the theory. 00:01:16.583 --> 00:01:18.500 align:middle line:84% When you write down the canonical quantization 00:01:18.500 --> 00:01:21.920 align:middle line:84% condition, and then we expand psi 00:01:21.920 --> 00:01:26.130 align:middle line:84% in terms of complete sets of solutions, et cetera. 00:01:26.130 --> 00:01:30.910 align:middle line:84% And then we recognize if we do it in the standard way, 00:01:30.910 --> 00:01:33.580 align:middle line:84% say to impose the commutator relation, 00:01:33.580 --> 00:01:35.740 align:middle line:84% and then the Hamiltonian actually 00:01:35.740 --> 00:01:37.370 align:middle line:90% is unbounded from below. 00:01:37.370 --> 00:01:43.670 align:middle line:84% So, actually, the total energy can be as negative as-- 00:01:43.670 --> 00:01:48.340 align:middle line:90% yeah, can be x active-- 00:01:48.340 --> 00:01:51.190 align:middle line:84% yeah, it can be arbitrarily negative. 00:01:51.190 --> 00:01:55.040 align:middle line:84% And so, that theory does not make sense. 00:01:55.040 --> 00:01:59.440 align:middle line:84% And, also, if we use the standard way to quantize it, 00:01:59.440 --> 00:02:02.080 align:middle line:84% we would get a theory which, when 00:02:02.080 --> 00:02:05.170 align:middle line:84% you exchange two particles, which they commute, 00:02:05.170 --> 00:02:06.700 align:middle line:90% which are symmetric. 00:02:06.700 --> 00:02:11.410 align:middle line:84% And so, that looks like both a bosonic theory rather than 00:02:11.410 --> 00:02:13.270 align:middle line:90% a fermionic theory. 00:02:13.270 --> 00:02:16.120 align:middle line:90% And then, there's a simple fix. 00:02:16.120 --> 00:02:18.700 align:middle line:84% So the fix is that instead of imposing 00:02:18.700 --> 00:02:21.280 align:middle line:90% the standard quantization-- 00:02:21.280 --> 00:02:26.650 align:middle line:84% standard commutation relation, we impose the following one. 00:02:26.650 --> 00:02:32.070 align:middle line:84% So we replace everywhere the commutator by anticommutator. 00:02:32.070 --> 00:02:35.880 align:middle line:90% 00:02:35.880 --> 00:02:41.880 align:middle line:84% So what we do, say, for the commutation relation, 00:02:41.880 --> 00:02:51.780 align:middle line:84% we consider psi alpha px and then pi beta px prime. 00:02:51.780 --> 00:02:54.360 align:middle line:90% 00:02:54.360 --> 00:03:03.750 align:middle line:84% So the anticommutator is equal to i delta alpha beta 00:03:03.750 --> 00:03:04.740 align:middle line:90% and the 3. 00:03:04.740 --> 00:03:08.490 align:middle line:90% 00:03:08.490 --> 00:03:10.810 align:middle line:90% We impose this condition. 00:03:10.810 --> 00:03:15.870 align:middle line:84% So which is the same if we plug in this pi just given by i psi 00:03:15.870 --> 00:03:23.940 align:middle line:84% dagger, and then that just gives you psi alpha px times psi 00:03:23.940 --> 00:03:37.136 align:middle line:84% beta dagger px prime, gives you delta alpha beta delta 3 00:03:37.136 --> 00:03:38.460 align:middle line:90% x minus x prime. 00:03:38.460 --> 00:03:46.320 align:middle line:90% 00:03:46.320 --> 00:03:53.530 align:middle line:84% So this equation-- so one thing to notice-- 00:03:53.530 --> 00:03:58.540 align:middle line:84% so the commutator-- the anticommutator 00:03:58.540 --> 00:04:04.300 align:middle line:84% is symmetric between these two because the now 00:04:04.300 --> 00:04:06.100 align:middle line:90% is the plus rather than minus. 00:04:06.100 --> 00:04:09.660 align:middle line:90% 00:04:09.660 --> 00:04:16.829 align:middle line:84% And so, note, so this commutation relation 00:04:16.829 --> 00:04:18.959 align:middle line:90% is self-consistent. 00:04:18.959 --> 00:04:35.650 align:middle line:84% So if we call this equation one, so if one is self-consistent at 00:04:35.650 --> 00:04:39.070 align:middle line:90% x equal to x prime. 00:04:39.070 --> 00:04:41.500 align:middle line:84% So if you take these two at the same point-- sorry, 00:04:41.500 --> 00:04:43.580 align:middle line:90% so this should be prime. 00:04:43.580 --> 00:04:48.970 align:middle line:84% So if you take these two at the same point, so 00:04:48.970 --> 00:04:53.980 align:middle line:84% the left-hand side, you have psi alpha and psi beta. 00:04:53.980 --> 00:04:56.830 align:middle line:90% So you just have psi dagger. 00:04:56.830 --> 00:05:00.340 align:middle line:84% So the left-hand side is a positive definite quantity. 00:05:00.340 --> 00:05:02.200 align:middle line:90% You have psi dagger. 00:05:02.200 --> 00:05:05.770 align:middle line:84% And the right-hand side is also a positive definite quantity. 00:05:05.770 --> 00:05:09.790 align:middle line:84% And you have this delta function evaluated at 0 00:05:09.790 --> 00:05:15.280 align:middle line:90% even though it's a divergent. 00:05:15.280 --> 00:05:22.240 align:middle line:84% And so, both sides are positive definite. 00:05:22.240 --> 00:05:36.290 align:middle line:90% 00:05:36.290 --> 00:05:40.760 align:middle line:84% So the fact that this is self-consistent, 00:05:40.760 --> 00:05:46.180 align:middle line:84% if you trace it back, has to do with the psi-- 00:05:46.180 --> 00:05:50.200 align:middle line:84% this pi is actually equal to i psi beta. 00:05:50.200 --> 00:05:52.990 align:middle line:84% So if it's equal to minus i psi-- 00:05:52.990 --> 00:05:55.690 align:middle line:84% so if pi is equal to i psi dagger. 00:05:55.690 --> 00:05:59.510 align:middle line:84% So if it is equal to minus i psi dagger, and then, of course, 00:05:59.510 --> 00:06:03.168 align:middle line:84% you will have a contradiction because the left-hand side will 00:06:03.168 --> 00:06:04.585 align:middle line:84% be positive or the right side sign 00:06:04.585 --> 00:06:07.180 align:middle line:84% will be negative because you will change the sign. 00:06:07.180 --> 00:06:10.270 align:middle line:84% And that sign then trace back to this sign. 00:06:10.270 --> 00:06:15.040 align:middle line:84% So that's the reason actually we want to have this sign here. 00:06:15.040 --> 00:06:17.080 align:middle line:84% So the sign here is for a reason. 00:06:17.080 --> 00:06:19.990 align:middle line:84% Even though, classically, when you write it down, 00:06:19.990 --> 00:06:22.609 align:middle line:90% it doesn't matter what's the-- 00:06:22.609 --> 00:06:26.980 align:middle line:84% yeah-- it was not important what the sign is. 00:06:26.980 --> 00:06:31.480 align:middle line:84% But quantum mechanically it's actually important. 00:06:31.480 --> 00:06:34.610 align:middle line:90% So this comes from-- 00:06:34.610 --> 00:06:41.690 align:middle line:84% so the-- yeah, so the sign in L important. 00:06:41.690 --> 00:06:55.300 align:middle line:90% 00:06:55.300 --> 00:06:55.800 align:middle line:90% Good. 00:06:55.800 --> 00:06:56.900 align:middle line:90% Any questions on this? 00:06:56.900 --> 00:06:57.650 align:middle line:90% Yes. 00:06:57.650 --> 00:06:58.560 align:middle line:84% AUDIENCE: Is that the [INAUDIBLE] 00:06:58.560 --> 00:07:00.935 align:middle line:84% when you replace the commutator with the anti commutator? 00:07:00.935 --> 00:07:03.532 align:middle line:84% Is that the generic feature when we describe 00:07:03.532 --> 00:07:05.630 align:middle line:90% things in harmonic series? 00:07:05.630 --> 00:07:06.767 align:middle line:90% HONG LIU: Yeah. 00:07:06.767 --> 00:07:07.850 align:middle line:90% I'm going to mention that. 00:07:07.850 --> 00:07:08.270 align:middle line:90% Yeah. 00:07:08.270 --> 00:07:09.380 align:middle line:90% I'm going to mention that. 00:07:09.380 --> 00:07:10.190 align:middle line:90% Yes. 00:07:10.190 --> 00:07:12.946 align:middle line:84% AUDIENCE: So in quantum mechanics, 00:07:12.946 --> 00:07:14.700 align:middle line:84% the fundamental commutation relation 00:07:14.700 --> 00:07:17.310 align:middle line:84% tells you that x and p don't commute. 00:07:17.310 --> 00:07:20.195 align:middle line:84% But in the classical limit, h bar goes to 0, they do. 00:07:20.195 --> 00:07:20.820 align:middle line:90% HONG LIU: Yeah. 00:07:20.820 --> 00:07:24.000 align:middle line:84% AUDIENCE: And that's a very physically intuitive idea. 00:07:24.000 --> 00:07:24.770 align:middle line:90% HONG LIU: Right. 00:07:24.770 --> 00:07:27.707 align:middle line:84% AUDIENCE: But then, here, h bar goes to 0 00:07:27.707 --> 00:07:29.790 align:middle line:84% just tells you something about the anticommutator. 00:07:29.790 --> 00:07:32.220 align:middle line:84% So how do we determine the classical? 00:07:32.220 --> 00:07:33.240 align:middle line:90% HONG LIU: Yeah. 00:07:33.240 --> 00:07:36.420 align:middle line:84% So the classical limit, yeah, we will actually-- 00:07:36.420 --> 00:07:38.880 align:middle line:84% this is a very good question we will discuss later. 00:07:38.880 --> 00:07:41.940 align:middle line:84% The classical limit, they become anticommute. 00:07:41.940 --> 00:07:44.250 align:middle line:84% Yeah, in the classical limit, essentially they 00:07:44.250 --> 00:07:48.870 align:middle line:90% become a anticommute variables. 00:07:48.870 --> 00:07:51.120 align:middle line:90% Yeah. 00:07:51.120 --> 00:07:55.230 align:middle line:84% And this is crucial when we later describe path integral. 00:07:55.230 --> 00:07:58.730 align:middle line:84% We will discuss this in detail later. 00:07:58.730 --> 00:07:59.745 align:middle line:90% Other questions? 00:07:59.745 --> 00:08:00.245 align:middle line:90% Yes. 00:08:00.245 --> 00:08:02.330 align:middle line:84% AUDIENCE: So can you say anything 00:08:02.330 --> 00:08:04.810 align:middle line:84% about the commutation relation between them, 00:08:04.810 --> 00:08:06.988 align:middle line:90% or can you not impose it? 00:08:06.988 --> 00:08:07.780 align:middle line:90% HONG LIU: Oh, yeah. 00:08:07.780 --> 00:08:09.905 align:middle line:84% You can talk about the commutation relation between 00:08:09.905 --> 00:08:13.810 align:middle line:84% them, just you cannot impose them. 00:08:13.810 --> 00:08:16.786 align:middle line:84% AUDIENCE: But does it-- do you get any physical meaning from 00:08:16.786 --> 00:08:20.175 align:middle line:84% the commutation relation between two objects, or-- 00:08:20.175 --> 00:08:20.800 align:middle line:90% HONG LIU: Yeah. 00:08:20.800 --> 00:08:23.320 align:middle line:90% 00:08:23.320 --> 00:08:25.330 align:middle line:90% These are two operators. 00:08:25.330 --> 00:08:28.965 align:middle line:84% You can always consider whatever quantities about them. 00:08:28.965 --> 00:08:30.340 align:middle line:84% AUDIENCE: There isn't a general-- 00:08:30.340 --> 00:08:33.100 align:middle line:84% HONG LIU: Yeah, but there's no general rule. 00:08:33.100 --> 00:08:34.120 align:middle line:90% Yeah. 00:08:34.120 --> 00:08:38.110 align:middle line:84% So the general rule, we impose for the anticommutator. 00:08:38.110 --> 00:08:42.220 align:middle line:90% 00:08:42.220 --> 00:08:48.430 align:middle line:84% And so, and in the second step, we just 00:08:48.430 --> 00:08:56.220 align:middle line:84% expand the psi in terms of complete sets of modes. 00:08:56.220 --> 00:09:01.580 align:middle line:84% And so, in the second step, as we do before, 00:09:01.580 --> 00:09:05.735 align:middle line:84% we expand the psi x in terms of complete set of modes. 00:09:05.735 --> 00:09:09.120 align:middle line:90% 00:09:09.120 --> 00:09:23.020 align:middle line:84% So we have already worked out last time, times-- 00:09:23.020 --> 00:09:30.350 align:middle line:90% so we have ak, s for the u s. 00:09:30.350 --> 00:09:33.770 align:middle line:90% Then we have expansion ikx. 00:09:33.770 --> 00:09:37.890 align:middle line:84% And then, similarly, with the complex scalar convention-- 00:09:37.890 --> 00:09:39.890 align:middle line:90% so now let me call this c. 00:09:39.890 --> 00:09:43.220 align:middle line:84% Just last time I used b, but it's often 00:09:43.220 --> 00:09:46.070 align:middle line:90% conventional to use c. 00:09:46.070 --> 00:09:48.020 align:middle line:90% And I put the dagger here. 00:09:48.020 --> 00:09:49.260 align:middle line:90% It doesn't matter. 00:09:49.260 --> 00:09:51.440 align:middle line:84% It's not so important whether you put dagger here. 00:09:51.440 --> 00:09:53.390 align:middle line:90% So but it's a convention. 00:09:53.390 --> 00:09:57.200 align:middle line:90% 00:09:57.200 --> 00:10:02.750 align:middle line:84% So this is our complete sets of solutions. 00:10:02.750 --> 00:10:06.500 align:middle line:84% And then we expand psi in terms of arbitrary coefficients 00:10:06.500 --> 00:10:08.630 align:middle line:90% in terms of them. 00:10:08.630 --> 00:10:12.310 align:middle line:84% And since psi is generally complex, 00:10:12.310 --> 00:10:14.920 align:middle line:84% so these two don't have to be the same. 00:10:14.920 --> 00:10:17.890 align:middle line:84% So these two, yeah, this is a complex scalar case. 00:10:17.890 --> 00:10:21.430 align:middle line:90% They don't have to be the same. 00:10:21.430 --> 00:10:24.900 align:middle line:84% And so, quantum mechanically, then a and c, 00:10:24.900 --> 00:10:26.910 align:middle line:90% they are operators. 00:10:26.910 --> 00:10:30.550 align:middle line:90% A and c, they are operators. 00:10:30.550 --> 00:10:34.470 align:middle line:84% So if I call this equation two, and then 00:10:34.470 --> 00:10:40.120 align:middle line:84% if we plug two into one, then we can get to the commutators. 00:10:40.120 --> 00:10:43.640 align:middle line:90% 00:10:43.640 --> 00:10:47.690 align:middle line:84% We can get the anticommutators between a and c. 00:10:47.690 --> 00:10:59.316 align:middle line:84% And so, so we find that, for example, ak r, 00:10:59.316 --> 00:11:21.180 align:middle line:84% ak prime s dagger equal to ckr and ck prime s dagger, 00:11:21.180 --> 00:11:28.850 align:middle line:84% now it's equal to 2 pi cube delta 3 00:11:28.850 --> 00:11:34.830 align:middle line:90% k minus k prime delta rs. 00:11:34.830 --> 00:11:51.220 align:middle line:84% And with all other anticommutator, 00:11:51.220 --> 00:11:54.100 align:middle line:84% so emphasize anticommutators zero. 00:11:54.100 --> 00:12:01.180 align:middle line:90% 00:12:01.180 --> 00:12:04.050 align:middle line:84% And, yeah, so the story is similar as before, 00:12:04.050 --> 00:12:08.240 align:middle line:84% except you just everywhere replace the commutator 00:12:08.240 --> 00:12:11.830 align:middle line:90% by anticommutator. 00:12:11.830 --> 00:12:14.410 align:middle line:84% And then, the vacuum, we can still define the vacuum. 00:12:14.410 --> 00:12:20.070 align:middle line:90% 00:12:20.070 --> 00:12:21.660 align:middle line:90% You can still divide-- 00:12:21.660 --> 00:12:24.570 align:middle line:84% later, we will see that this is justified. 00:12:24.570 --> 00:12:28.350 align:middle line:84% So we can still define the vacuum as annihilated 00:12:28.350 --> 00:12:40.850 align:middle line:90% by ak and ck for any k and s. 00:12:40.850 --> 00:12:48.490 align:middle line:90% 00:12:48.490 --> 00:12:49.600 align:middle line:90% So we define the vacuum. 00:12:49.600 --> 00:12:53.070 align:middle line:84% The reason this is the vacuum is because, now, 00:12:53.070 --> 00:12:59.240 align:middle line:84% if you find the Hamiltonian, so you 00:12:59.240 --> 00:13:05.868 align:middle line:84% can plug this in into your expression for the Hamiltonian. 00:13:05.868 --> 00:13:07.910 align:middle line:84% Because they just expressed in terms of psi, then 00:13:07.910 --> 00:13:11.330 align:middle line:84% you can just work it out, just straightforward as we 00:13:11.330 --> 00:13:13.970 align:middle line:84% did previously for the scalar field. 00:13:13.970 --> 00:13:18.695 align:middle line:84% Just now the algebra is slightly more complicated 00:13:18.695 --> 00:13:19.445 align:middle line:90% because you have-- 00:13:19.445 --> 00:13:23.130 align:middle line:90% 00:13:23.130 --> 00:13:24.380 align:middle line:90% yeah. 00:13:24.380 --> 00:13:28.030 align:middle line:90% So now you get omega k. 00:13:28.030 --> 00:13:32.410 align:middle line:90% And then you have s 1, 2. 00:13:32.410 --> 00:13:51.690 align:middle line:84% And then you have aks dagger aks plus cks dagger cks, 00:13:51.690 --> 00:13:54.270 align:middle line:90% then plus some constant. 00:13:54.270 --> 00:13:57.690 align:middle line:84% Just as in the scalar case, we always 00:13:57.690 --> 00:14:00.870 align:middle line:90% have some zero point energy. 00:14:00.870 --> 00:14:03.700 align:middle line:84% We always have some zero point energy. 00:14:03.700 --> 00:14:09.540 align:middle line:84% And so now you see, if you define-- 00:14:09.540 --> 00:14:12.300 align:middle line:84% so now you see the Hamiltonian is positive definite. 00:14:12.300 --> 00:14:14.900 align:middle line:90% 00:14:14.900 --> 00:14:17.420 align:middle line:84% The Hamiltonian is positive definite. 00:14:17.420 --> 00:14:19.550 align:middle line:84% And because we have a plus sign here 00:14:19.550 --> 00:14:23.970 align:middle line:84% rather than the minus sign if we do the commutation relation. 00:14:23.970 --> 00:14:27.120 align:middle line:84% And then, since this is positive definite, and then 00:14:27.120 --> 00:14:30.420 align:middle line:84% justify defining this 0 as the vacuum. 00:14:30.420 --> 00:14:33.160 align:middle line:84% Because this state, indeed, has the lowest energy 00:14:33.160 --> 00:14:34.580 align:middle line:90% given by this E0. 00:14:34.580 --> 00:14:37.230 align:middle line:90% And, yeah, E0 as in the bottom-- 00:14:37.230 --> 00:14:41.340 align:middle line:84% in the previous scalar case-- would be divergent. 00:14:41.340 --> 00:14:42.300 align:middle line:90% It would be divergent. 00:14:42.300 --> 00:14:49.410 align:middle line:90% 00:14:49.410 --> 00:14:50.550 align:middle line:90% Any questions on this? 00:14:50.550 --> 00:14:55.150 align:middle line:90% 00:14:55.150 --> 00:14:55.650 align:middle line:90% Yes. 00:14:55.650 --> 00:14:57.608 align:middle line:84% AUDIENCE: On that line, what is the coefficient 00:14:57.608 --> 00:14:58.561 align:middle line:90% of the exponential? 00:14:58.561 --> 00:14:59.807 align:middle line:90% It's minus i k x? 00:14:59.807 --> 00:15:00.640 align:middle line:90% HONG LIU: Oh, right. 00:15:00.640 --> 00:15:00.970 align:middle line:90% Sorry. 00:15:00.970 --> 00:15:01.593 align:middle line:90% Yeah. 00:15:01.593 --> 00:15:02.093 align:middle line:90% I did. 00:15:02.093 --> 00:15:05.550 align:middle line:90% 00:15:05.550 --> 00:15:07.020 align:middle line:90% Good. 00:15:07.020 --> 00:15:07.860 align:middle line:90% Thank you. 00:15:07.860 --> 00:15:10.740 align:middle line:90% 00:15:10.740 --> 00:15:11.980 align:middle line:90% Good. 00:15:11.980 --> 00:15:14.320 align:middle line:84% So let me make some comments on this E0. 00:15:14.320 --> 00:15:18.030 align:middle line:90% 00:15:18.030 --> 00:15:23.970 align:middle line:90% So, again, the quantization-- 00:15:23.970 --> 00:15:27.640 align:middle line:90% 00:15:27.640 --> 00:15:36.190 align:middle line:84% yeah, again, the quantized field, as in the scalar case, 00:15:36.190 --> 00:15:50.180 align:middle line:84% essentially consists of an infinite number 00:15:50.180 --> 00:15:52.920 align:middle line:90% of harmonic oscillators. 00:15:52.920 --> 00:15:59.300 align:middle line:84% So now I need to put the harmonic oscillator by codes. 00:15:59.300 --> 00:16:01.340 align:middle line:84% Because now these harmonic oscillators, 00:16:01.340 --> 00:16:04.760 align:middle line:84% they defined in terms of the anticommutators rather than 00:16:04.760 --> 00:16:05.500 align:middle line:90% commutators. 00:16:05.500 --> 00:16:12.800 align:middle line:90% 00:16:12.800 --> 00:16:17.570 align:middle line:90% And so, very important feature. 00:16:17.570 --> 00:16:19.550 align:middle line:84% And, normally, we call them fermionic 00:16:19.550 --> 00:16:23.130 align:middle line:84% for reasons will be clear very soon. 00:16:23.130 --> 00:16:26.210 align:middle line:90% So this is the first note-- 00:16:26.210 --> 00:16:29.570 align:middle line:84% for a reason will be very soon and the-- 00:16:29.570 --> 00:16:33.590 align:middle line:84% why we call them fermionic oscillator. 00:16:33.590 --> 00:16:40.480 align:middle line:84% And so, we have for each k and the value of the s, 00:16:40.480 --> 00:16:43.090 align:middle line:90% and you have an oscillator. 00:16:43.090 --> 00:16:45.640 align:middle line:84% And then you have oscillator generated by a 00:16:45.640 --> 00:16:48.055 align:middle line:84% and also have oscillator generated by c. 00:16:48.055 --> 00:16:50.970 align:middle line:90% 00:16:50.970 --> 00:16:56.700 align:middle line:90% So each fermionic oscillator-- 00:16:56.700 --> 00:17:04.030 align:middle line:90% 00:17:04.030 --> 00:17:06.480 align:middle line:84% so this is like the harmonic oscillator. 00:17:06.480 --> 00:17:20.510 align:middle line:84% They contribute to E0, to this ground state energy. 00:17:20.510 --> 00:17:24.430 align:middle line:84% So, remember, previously, for the both-- for the scalar case, 00:17:24.430 --> 00:17:28.700 align:middle line:84% each oscillator contributes 1/2 omega k. 00:17:28.700 --> 00:17:29.450 align:middle line:90% Remember. 00:17:29.450 --> 00:17:34.070 align:middle line:84% So now, in this case, they contribute 00:17:34.070 --> 00:17:35.900 align:middle line:90% by minus 1/2 omega k. 00:17:35.900 --> 00:17:39.450 align:middle line:90% 00:17:39.450 --> 00:17:43.120 align:middle line:84% It's actually opposite sign for the standard harmonic 00:17:43.120 --> 00:17:43.620 align:middle line:90% oscillator. 00:17:43.620 --> 00:17:46.440 align:middle line:84% In the standard harmonic oscillator, it's 1/2 omega, 00:17:46.440 --> 00:17:49.380 align:middle line:90% and here it's minus 1/2 omega. 00:17:49.380 --> 00:17:53.500 align:middle line:90% 00:17:53.500 --> 00:17:56.860 align:middle line:84% So let me just also saying why we call it a-- 00:17:56.860 --> 00:18:00.090 align:middle line:90% 00:18:00.090 --> 00:18:02.970 align:middle line:84% so a feature of this fermionic harmonic oscillator 00:18:02.970 --> 00:18:09.365 align:middle line:84% is that, say, you can act, you can create a state-- 00:18:09.365 --> 00:18:16.440 align:middle line:90% 00:18:16.440 --> 00:18:16.940 align:middle line:90% yeah. 00:18:16.940 --> 00:18:19.220 align:middle line:84% So let me just make some general remarks. 00:18:19.220 --> 00:18:23.330 align:middle line:84% Say if you have a harmonic oscillator satisfy-- 00:18:23.330 --> 00:18:30.540 align:middle line:84% yeah-- satisfy this kind of commutation relation. 00:18:30.540 --> 00:18:33.420 align:middle line:84% Let's just look at one set of them. 00:18:33.420 --> 00:18:37.710 align:middle line:90% 00:18:37.710 --> 00:18:42.600 align:middle line:84% So one side of them, just take any side with any choice of k 00:18:42.600 --> 00:18:46.270 align:middle line:84% and s, and then they satisfy that relation. 00:18:46.270 --> 00:18:55.530 align:middle line:84% So this defines fermionic harmonic oscillator. 00:18:55.530 --> 00:18:59.470 align:middle line:90% 00:18:59.470 --> 00:19:06.640 align:middle line:84% And so, interesting feature of this oscillator 00:19:06.640 --> 00:19:09.590 align:middle line:84% is that you can only excite them once. 00:19:09.590 --> 00:19:15.380 align:middle line:84% Because if you have a dagger acting on 0, now 00:19:15.380 --> 00:19:22.730 align:middle line:84% if you have a dagger square acting on 0, then this is 0. 00:19:22.730 --> 00:19:25.220 align:middle line:84% So the only state you have-- you only have two states. 00:19:25.220 --> 00:19:29.430 align:middle line:84% One is 0 and the one is a dagger 0. 00:19:29.430 --> 00:19:34.590 align:middle line:84% It's because the anticommutator of a dagger itself is 0. 00:19:34.590 --> 00:19:37.350 align:middle line:84% So this just tells you that a dagger squared is equal to 0. 00:19:37.350 --> 00:19:40.500 align:middle line:84% And, also, a squared is equal to 0. 00:19:40.500 --> 00:19:43.410 align:middle line:90% And so, you act twice. 00:19:43.410 --> 00:19:46.470 align:middle line:84% And this is precisely the Pauli principle, 00:19:46.470 --> 00:19:49.290 align:middle line:84% and the two particle on the same state, 00:19:49.290 --> 00:19:52.620 align:middle line:90% and you get identically 0. 00:19:52.620 --> 00:19:59.320 align:middle line:84% So if you act any of k here, a, k, and s, if they are the same, 00:19:59.320 --> 00:20:02.590 align:middle line:84% if you act them twice, you get 0. 00:20:02.590 --> 00:20:05.110 align:middle line:84% So if you have two particles in the same state, 00:20:05.110 --> 00:20:07.450 align:middle line:84% when k and s is the same, then it 00:20:07.450 --> 00:20:10.010 align:middle line:84% means that all their quantum numbers are the same. 00:20:10.010 --> 00:20:13.290 align:middle line:84% And so, this is the Pauli principle. 00:20:13.290 --> 00:20:24.270 align:middle line:90% 00:20:24.270 --> 00:20:26.340 align:middle line:90% So this is the Pauli principle. 00:20:26.340 --> 00:20:29.535 align:middle line:84% So no two particle can be in the same states. 00:20:29.535 --> 00:20:39.090 align:middle line:90% 00:20:39.090 --> 00:20:59.080 align:middle line:84% So here, then we conclude that a dagger, aks and cks dagger, 00:20:59.080 --> 00:21:02.380 align:middle line:90% they create fermions. 00:21:02.380 --> 00:21:06.430 align:middle line:90% 00:21:06.430 --> 00:21:09.010 align:middle line:84% They create particles which obey the Pauli principle. 00:21:09.010 --> 00:21:19.295 align:middle line:90% 00:21:19.295 --> 00:21:19.795 align:middle line:90% Good? 00:21:19.795 --> 00:21:31.450 align:middle line:90% 00:21:31.450 --> 00:21:32.385 align:middle line:90% Any questions on this? 00:21:32.385 --> 00:21:35.400 align:middle line:90% 00:21:35.400 --> 00:21:36.490 align:middle line:90% Good. 00:21:36.490 --> 00:21:39.730 align:middle line:90% So now we can define-- 00:21:39.730 --> 00:21:43.300 align:middle line:84% as before, we can define single particle states. 00:21:43.300 --> 00:21:55.945 align:middle line:84% So the single particle states, say, you just act them once. 00:21:55.945 --> 00:22:00.750 align:middle line:90% 00:22:00.750 --> 00:22:03.380 align:middle line:90% So, again, we define them by-- 00:22:03.380 --> 00:22:06.980 align:middle line:84% with this normalization as in the scalar case. 00:22:06.980 --> 00:22:13.055 align:middle line:90% And then a s aks dagger on 0. 00:22:13.055 --> 00:22:15.730 align:middle line:90% 00:22:15.730 --> 00:22:23.220 align:middle line:84% And then we define ks bar to be the one created by c. 00:22:23.220 --> 00:22:35.740 align:middle line:90% 00:22:35.740 --> 00:22:38.660 align:middle line:84% So as before, in the scalar case, 00:22:38.660 --> 00:22:42.250 align:middle line:84% we can interpret this k as the momentum. 00:22:42.250 --> 00:22:45.550 align:middle line:84% You can check explicitly, as we will describe, 00:22:45.550 --> 00:22:50.080 align:middle line:84% that the k give you the momentum of the-- yeah, goes back 00:22:50.080 --> 00:22:51.910 align:middle line:90% into the momentum eigenvalue. 00:22:51.910 --> 00:22:54.010 align:middle line:84% And then, the s can be interpreted 00:22:54.010 --> 00:22:57.460 align:middle line:90% as some kind of polarization. 00:22:57.460 --> 00:23:01.100 align:middle line:90% So yeah. 00:23:01.100 --> 00:23:06.830 align:middle line:84% And yeah, so this we call particle. 00:23:06.830 --> 00:23:08.930 align:middle line:90% So we'll call this particle. 00:23:08.930 --> 00:23:10.970 align:middle line:84% And we will call this antiparticle. 00:23:10.970 --> 00:23:18.440 align:middle line:90% 00:23:18.440 --> 00:23:30.060 align:middle line:84% And so, each particle has two polarizations. 00:23:30.060 --> 00:23:35.040 align:middle line:90% 00:23:35.040 --> 00:23:36.510 align:middle line:90% Since s equal to 1, 2. 00:23:36.510 --> 00:23:39.170 align:middle line:90% 00:23:39.170 --> 00:23:42.860 align:middle line:84% And they are all satisfied, and the k square, of course, 00:23:42.860 --> 00:23:45.020 align:middle line:90% all satisfied minus m squared. 00:23:45.020 --> 00:23:52.570 align:middle line:90% 00:23:52.570 --> 00:23:53.070 align:middle line:90% good? 00:23:53.070 --> 00:23:53.987 align:middle line:90% Any questions on this? 00:23:53.987 --> 00:24:01.980 align:middle line:90% 00:24:01.980 --> 00:24:03.801 align:middle line:84% AUDIENCE: Is s and s bar different? 00:24:03.801 --> 00:24:04.384 align:middle line:90% HONG LIU: Hmm? 00:24:04.384 --> 00:24:06.672 align:middle line:84% AUDIENCE: Is s and s bar different? 00:24:06.672 --> 00:24:08.130 align:middle line:84% HONG LIU: Is s and s bar different? 00:24:08.130 --> 00:24:12.510 align:middle line:84% Yeah, because one is created by a, and one is created by c. 00:24:12.510 --> 00:24:14.580 align:middle line:84% So we just use a bar to distinguish them. 00:24:14.580 --> 00:24:17.420 align:middle line:90% 00:24:17.420 --> 00:24:21.950 align:middle line:84% Yeah, this one corresponding to a polarization 00:24:21.950 --> 00:24:22.970 align:middle line:90% of this particle. 00:24:22.970 --> 00:24:25.130 align:middle line:84% And one bar corresponding to the polarization 00:24:25.130 --> 00:24:26.210 align:middle line:90% of the antiparticle. 00:24:26.210 --> 00:24:29.400 align:middle line:90% 00:24:29.400 --> 00:24:29.900 align:middle line:90% Good? 00:24:29.900 --> 00:24:37.460 align:middle line:90% 00:24:37.460 --> 00:24:45.420 align:middle line:84% Yeah, so you can also find the Noether charge. 00:24:45.420 --> 00:24:58.450 align:middle line:84% Say you can find the Noether charge for translation. 00:24:58.450 --> 00:25:00.390 align:middle line:84% So we already talked about momentum-- 00:25:00.390 --> 00:25:02.220 align:middle line:90% Hamiltonian. 00:25:02.220 --> 00:25:07.180 align:middle line:84% You can also find the Noether charge for spatial translation. 00:25:07.180 --> 00:25:13.200 align:middle line:84% And then you just find the P, capital P, is equal to-- 00:25:13.200 --> 00:25:17.500 align:middle line:84% yeah, so this is, yeah, you can just similarly work it out. 00:25:17.500 --> 00:25:20.040 align:middle line:84% So, actually, in your pset, you will work out 00:25:20.040 --> 00:25:24.210 align:middle line:90% the stress tensor. 00:25:24.210 --> 00:25:27.000 align:middle line:84% So you find that the loss of charge 00:25:27.000 --> 00:25:29.220 align:middle line:84% for the spatial translation is given by this. 00:25:29.220 --> 00:25:33.225 align:middle line:84% And then, if you plug it in, and then you get-- 00:25:33.225 --> 00:25:43.400 align:middle line:90% 00:25:43.400 --> 00:25:46.100 align:middle line:84% and then, again, you have this a dagger. 00:25:46.100 --> 00:25:48.170 align:middle line:84% Yeah, to save the effort, I just say 00:25:48.170 --> 00:25:51.260 align:middle line:84% you have a dagger, a and c, dagger c. 00:25:51.260 --> 00:25:55.070 align:middle line:84% So you can see, immediately, these two will 00:25:55.070 --> 00:25:59.120 align:middle line:84% be a eigenstate, say, of the momentum 00:25:59.120 --> 00:26:03.620 align:middle line:90% operator with eigenvalue k. 00:26:03.620 --> 00:26:04.550 align:middle line:90% With eigenvalue k. 00:26:04.550 --> 00:26:09.380 align:middle line:84% So that justifies that these are momentum-- 00:26:09.380 --> 00:26:11.840 align:middle line:84% these are the momentum eigenstates. 00:26:11.840 --> 00:26:18.050 align:middle line:90% 00:26:18.050 --> 00:26:30.370 align:middle line:84% So we can also find we can also define normalization. 00:26:30.370 --> 00:26:33.230 align:middle line:84% We define the normalization, yeah, for those-- 00:26:33.230 --> 00:26:37.320 align:middle line:84% you can check the normalization for those state. 00:26:37.320 --> 00:26:38.510 align:middle line:90% So these are the plane wave. 00:26:38.510 --> 00:26:42.680 align:middle line:90% 00:26:42.680 --> 00:26:47.540 align:middle line:84% So you can check just by using commutation relation. 00:26:47.540 --> 00:26:51.900 align:middle line:90% And this is given by omega k. 00:26:51.900 --> 00:26:54.270 align:middle line:90% That's 2 pi cubed. 00:26:54.270 --> 00:27:00.540 align:middle line:90% 00:27:00.540 --> 00:27:15.280 align:middle line:84% And, similarly, for the antiparticle, 00:27:15.280 --> 00:27:17.330 align:middle line:90% you have the same thing. 00:27:17.330 --> 00:27:18.990 align:middle line:90% Same thing. 00:27:18.990 --> 00:27:21.140 align:middle line:84% But if you take the overlap between the particle 00:27:21.140 --> 00:27:22.640 align:middle line:84% and the antiparticle, you always get 00:27:22.640 --> 00:27:28.750 align:middle line:84% the 0, because the commutator between a and c is always 0. 00:27:28.750 --> 00:27:37.130 align:middle line:90% 00:27:37.130 --> 00:27:41.840 align:middle line:90% Take k, r, and the k prime. 00:27:41.840 --> 00:27:43.145 align:middle line:90% S bar is always 0. 00:27:43.145 --> 00:27:45.698 align:middle line:90% 00:27:45.698 --> 00:27:47.430 align:middle line:90% They are always orthogonal. 00:27:47.430 --> 00:27:57.720 align:middle line:90% 00:27:57.720 --> 00:28:00.510 align:middle line:84% So let us make some further notes here. 00:28:00.510 --> 00:28:07.870 align:middle line:90% 00:28:07.870 --> 00:28:18.970 align:middle line:84% So since we have two components, since each particle a fermion. 00:28:18.970 --> 00:28:22.870 align:middle line:84% So we already said there should be a fermion. 00:28:22.870 --> 00:28:24.580 align:middle line:90% And it has two components. 00:28:24.580 --> 00:28:31.420 align:middle line:90% 00:28:31.420 --> 00:28:34.360 align:middle line:84% So, again, we guessed they must be spin 1/2 particle. 00:28:34.360 --> 00:28:39.640 align:middle line:90% 00:28:39.640 --> 00:28:41.620 align:middle line:84% We guess they must be spin 1/2 particle. 00:28:41.620 --> 00:28:45.310 align:middle line:90% 00:28:45.310 --> 00:28:52.070 align:middle line:84% And so, you can check explicitly. 00:28:52.070 --> 00:28:55.330 align:middle line:84% Yeah, so you can guess that this would be spin 1/2 particle. 00:28:55.330 --> 00:28:56.630 align:middle line:90% But you can check explicitly. 00:28:56.630 --> 00:29:02.590 align:middle line:90% 00:29:02.590 --> 00:29:06.770 align:middle line:90% Indeed, the eigenvalues are-- 00:29:06.770 --> 00:29:14.260 align:middle line:84% so you can construct angular momentum operator 00:29:14.260 --> 00:29:17.420 align:middle line:84% because this series, Lorentz invariance, 00:29:17.420 --> 00:29:20.620 align:middle line:84% so you can construct the Noether charge associated 00:29:20.620 --> 00:29:23.000 align:middle line:90% with the Lorentz transformation. 00:29:23.000 --> 00:29:27.260 align:middle line:84% And then you can construct the angular momentum operator. 00:29:27.260 --> 00:29:30.080 align:middle line:90% And then you check. 00:29:30.080 --> 00:29:40.490 align:middle line:84% And then, indeed, so you find that indeed the ks and k bar-- 00:29:40.490 --> 00:29:52.340 align:middle line:84% ks bar has eigenvalue of spin half particles. 00:29:52.340 --> 00:30:00.430 align:middle line:90% 00:30:00.430 --> 00:30:02.470 align:middle line:84% So this is you can check explicitly. 00:30:02.470 --> 00:30:07.660 align:middle line:90% 00:30:07.660 --> 00:30:08.160 align:middle line:90% Good. 00:30:08.160 --> 00:30:10.530 align:middle line:90% Any questions on this? 00:30:10.530 --> 00:30:11.810 align:middle line:90% And this will be in your pset. 00:30:11.810 --> 00:30:14.360 align:middle line:90% 00:30:14.360 --> 00:30:16.600 align:middle line:90% This is in your pset. 00:30:16.600 --> 00:30:23.850 align:middle line:84% And this is a little bit non-trivial calculation. 00:30:23.850 --> 00:30:25.860 align:middle line:90% But it's an instructive one. 00:30:25.860 --> 00:30:28.560 align:middle line:90% So when you see from your hand-- 00:30:28.560 --> 00:30:32.520 align:middle line:84% you see it with your own eye and from your own calculation, 00:30:32.520 --> 00:30:37.030 align:middle line:84% that this is spin 1/2 is satisfying. 00:30:37.030 --> 00:30:41.200 align:middle line:90% 00:30:41.200 --> 00:30:41.700 align:middle line:90% Good. 00:30:41.700 --> 00:30:42.830 align:middle line:90% Any questions on this? 00:30:42.830 --> 00:30:46.350 align:middle line:90% 00:30:46.350 --> 00:30:49.565 align:middle line:84% So, more generally, so we will not 00:30:49.565 --> 00:30:53.570 align:middle line:84% be able to-- we will not certainly prove here. 00:30:53.570 --> 00:30:57.290 align:middle line:84% So, more generally, you can prove 00:30:57.290 --> 00:31:01.130 align:middle line:84% there exist a so-called spin statistical theorem. 00:31:01.130 --> 00:31:13.870 align:middle line:90% 00:31:13.870 --> 00:31:16.315 align:middle line:90% So a half integer spin field-- 00:31:16.315 --> 00:31:24.950 align:middle line:90% 00:31:24.950 --> 00:31:25.450 align:middle line:90% yeah. 00:31:25.450 --> 00:31:34.960 align:middle line:84% So half integer spin fields can only 00:31:34.960 --> 00:31:47.950 align:middle line:84% be quantized using anticommutation relations-- 00:31:47.950 --> 00:31:49.270 align:middle line:90% anticommutators. 00:31:49.270 --> 00:31:53.710 align:middle line:90% 00:31:53.710 --> 00:31:59.170 align:middle line:84% Not only spin half, say 3/2, 5/2, et cetera. 00:31:59.170 --> 00:32:03.580 align:middle line:84% So Dirac field is the simplest of them. 00:32:03.580 --> 00:32:10.570 align:middle line:84% And in the anticommutation relation, and so obey 00:32:10.570 --> 00:32:15.460 align:middle line:84% this so-called the Fermi-Dirac statistics. 00:32:15.460 --> 00:32:19.360 align:middle line:84% So they obey the Pauli principle. 00:32:19.360 --> 00:32:22.240 align:middle line:84% Yes, in statistical physics, which 00:32:22.240 --> 00:32:24.070 align:middle line:84% when you exchange the wave function, 00:32:24.070 --> 00:32:27.200 align:middle line:84% you get the minus sign, it's called the Fermi-Dirac 00:32:27.200 --> 00:32:29.090 align:middle line:90% statistics. 00:32:29.090 --> 00:32:33.200 align:middle line:84% And in contrast, if you have integer spin, say like a scalar 00:32:33.200 --> 00:32:37.760 align:middle line:84% or like a photon, they have integer spin, 00:32:37.760 --> 00:32:39.410 align:middle line:90% a photon have spin one. 00:32:39.410 --> 00:32:42.890 align:middle line:84% And then you can quantize them using commutators. 00:32:42.890 --> 00:32:55.670 align:middle line:90% 00:32:55.670 --> 00:33:10.040 align:middle line:84% So for integer spin, and then quantized-- 00:33:10.040 --> 00:33:13.980 align:middle line:84% can be consistently quantized using commutators. 00:33:13.980 --> 00:33:19.640 align:middle line:90% 00:33:19.640 --> 00:33:23.000 align:middle line:84% So, in this case, you get the Bose-Einstein statistics. 00:33:23.000 --> 00:33:29.900 align:middle line:90% 00:33:29.900 --> 00:33:32.500 align:middle line:84% Means when you exchange the wave function of the particles, 00:33:32.500 --> 00:33:34.417 align:middle line:90% the wave function remains-- 00:33:34.417 --> 00:33:36.500 align:middle line:84% when you exchange the particles, the wave function 00:33:36.500 --> 00:33:37.910 align:middle line:90% remains symmetric. 00:33:37.910 --> 00:33:43.040 align:middle line:90% 00:33:43.040 --> 00:33:45.230 align:middle line:90% And so, this is very general. 00:33:45.230 --> 00:33:46.190 align:middle line:90% This is very general. 00:33:46.190 --> 00:33:51.560 align:middle line:90% 00:33:51.560 --> 00:33:52.060 align:middle line:90% Good. 00:33:52.060 --> 00:33:53.200 align:middle line:90% Any questions on this? 00:33:53.200 --> 00:33:53.850 align:middle line:90% Yes. 00:33:53.850 --> 00:33:55.600 align:middle line:84% AUDIENCE: So are there other relationships 00:33:55.600 --> 00:33:58.227 align:middle line:90% we use to quantize a field here? 00:33:58.227 --> 00:34:00.590 align:middle line:90% 00:34:00.590 --> 00:34:03.540 align:middle line:84% HONG LIU: Yeah, that's a very good question. 00:34:03.540 --> 00:34:08.070 align:middle line:84% So in 2 plus 1 dimension, very special, 2 plus 1 dimension, 00:34:08.070 --> 00:34:13.010 align:middle line:84% so there are more general statistics than Bose-Einstein-- 00:34:13.010 --> 00:34:14.690 align:middle line:90% than bosons and fermions. 00:34:14.690 --> 00:34:18.080 align:middle line:84% There are things in between called anyons, yeah. 00:34:18.080 --> 00:34:20.060 align:middle line:84% And they play a very important role 00:34:20.060 --> 00:34:23.500 align:middle line:84% in condensed matter physics, yeah. 00:34:23.500 --> 00:34:26.360 align:middle line:84% The anyon, actually, was proposed by our colleague, 00:34:26.360 --> 00:34:28.929 align:middle line:90% Frank Wilcock, in early '80s. 00:34:28.929 --> 00:34:33.580 align:middle line:84% And, yeah, at the beginning, it sounded like a fantasy. 00:34:33.580 --> 00:34:36.988 align:middle line:84% But then later actually find many important applications 00:34:36.988 --> 00:34:38.155 align:middle line:90% in condensed matter physics. 00:34:38.155 --> 00:34:41.800 align:middle line:90% 00:34:41.800 --> 00:34:42.927 align:middle line:90% Yes. 00:34:42.927 --> 00:34:44.469 align:middle line:84% AUDIENCE: What rule we've seen so far 00:34:44.469 --> 00:34:46.511 align:middle line:84% tells us that we should interpret the a and the c 00:34:46.511 --> 00:34:49.100 align:middle line:84% operators as particles and antiparticles? 00:34:49.100 --> 00:34:53.000 align:middle line:84% HONG LIU: Yeah, it's just convention. 00:34:53.000 --> 00:34:55.730 align:middle line:90% It doesn't matter. 00:34:55.730 --> 00:34:59.720 align:middle line:84% You can-- as far as they're anti each other. 00:34:59.720 --> 00:35:01.850 align:middle line:84% AUDIENCE: I guess, what indicates they are-- 00:35:01.850 --> 00:35:03.500 align:middle line:90% HONG LIU: Good. 00:35:03.500 --> 00:35:06.643 align:middle line:84% That's what we are going to talk about. 00:35:06.643 --> 00:35:07.310 align:middle line:90% Other questions? 00:35:07.310 --> 00:35:07.880 align:middle line:90% Yes. 00:35:07.880 --> 00:35:09.530 align:middle line:84% AUDIENCE: So the spin 1/2 here comes 00:35:09.530 --> 00:35:12.290 align:middle line:84% from the possible value of s, which 00:35:12.290 --> 00:35:16.590 align:middle line:84% comes from solving the classical Dirac equation? 00:35:16.590 --> 00:35:21.840 align:middle line:84% HONG LIU: Yeah, so spin 1/2, you can guess it 00:35:21.840 --> 00:35:23.990 align:middle line:90% from the two components. 00:35:23.990 --> 00:35:27.980 align:middle line:84% But for this one, you can just work out the eigenvalues. 00:35:27.980 --> 00:35:29.330 align:middle line:90% You will see, it's 1/2. 00:35:29.330 --> 00:35:30.410 align:middle line:90% Yeah. 00:35:30.410 --> 00:35:32.887 align:middle line:84% AUDIENCE: So when we generalize it to more-- 00:35:32.887 --> 00:35:34.220 align:middle line:90% like more half spin [INAUDIBLE]. 00:35:34.220 --> 00:35:36.540 align:middle line:90% HONG LIU: Yeah. 00:35:36.540 --> 00:35:37.957 align:middle line:84% AUDIENCE: We have more components? 00:35:37.957 --> 00:35:40.123 align:middle line:84% HONG LIU: Yeah, you will have more components, yeah. 00:35:40.123 --> 00:35:41.350 align:middle line:90% AUDIENCE: And we will be-- 00:35:41.350 --> 00:35:44.910 align:middle line:84% HONG LIU: Yeah, you will have more components, yeah. 00:35:44.910 --> 00:35:48.220 align:middle line:90% Other questions? 00:35:48.220 --> 00:35:48.880 align:middle line:90% Yes. 00:35:48.880 --> 00:35:51.380 align:middle line:84% AUDIENCE: You say that the energy 00:35:51.380 --> 00:35:53.775 align:middle line:84% is positive, positive definite just because you got rid 00:35:53.775 --> 00:35:54.650 align:middle line:90% of the negative sign. 00:35:54.650 --> 00:35:57.470 align:middle line:84% But how do you know that these new operators, like c 00:35:57.470 --> 00:36:00.030 align:middle line:84% is going to behave like you want it to? 00:36:00.030 --> 00:36:02.970 align:middle line:84% I guess, because we don't know the spectrum. 00:36:02.970 --> 00:36:04.726 align:middle line:90% HONG LIU: What do you mean? 00:36:04.726 --> 00:36:06.990 align:middle line:84% AUDIENCE: The eigenvalues of c [INAUDIBLE].. 00:36:06.990 --> 00:36:08.670 align:middle line:84% HONG LIU: No, we know everything about-- 00:36:08.670 --> 00:36:11.130 align:middle line:84% we know everything about them once I 00:36:11.130 --> 00:36:15.580 align:middle line:84% specifier the anticommutator, everything is fixed. 00:36:15.580 --> 00:36:18.130 align:middle line:84% Then everything is fixed, the spectrum is fixed. 00:36:18.130 --> 00:36:21.220 align:middle line:84% The energy spectrum just follow from here. 00:36:21.220 --> 00:36:25.050 align:middle line:84% So each a or c just can create the one particle. 00:36:25.050 --> 00:36:26.160 align:middle line:90% Yeah. 00:36:26.160 --> 00:36:28.770 align:middle line:84% So the spectrum is completely fixed, the energy is fixed. 00:36:28.770 --> 00:36:31.780 align:middle line:84% Yeah, we know everything about them. 00:36:31.780 --> 00:36:32.790 align:middle line:90% Yes. 00:36:32.790 --> 00:36:34.832 align:middle line:84% AUDIENCE: Can you have a theory that is spin half 00:36:34.832 --> 00:36:37.020 align:middle line:90% particles but no antiparticles? 00:36:37.020 --> 00:36:39.790 align:middle line:84% HONG LIU: Oh, yeah, that's a very good question. 00:36:39.790 --> 00:36:44.550 align:middle line:84% And so, that's the Holy Grail of neutrino physics 00:36:44.550 --> 00:36:51.280 align:middle line:84% and also in condensed matter physics or quantum computation. 00:36:51.280 --> 00:36:53.940 align:middle line:84% So people have been looking for these Majorana fermions, which 00:36:53.940 --> 00:36:55.290 align:middle line:90% we will talk about later. 00:36:55.290 --> 00:36:59.070 align:middle line:84% And people have been looking for these Majorana fermions, which 00:36:59.070 --> 00:37:00.330 align:middle line:90% is like a real scalar. 00:37:00.330 --> 00:37:02.970 align:middle line:84% It's the counterpart of the real scalar. 00:37:02.970 --> 00:37:04.647 align:middle line:90% Here it's more like a complex-- 00:37:04.647 --> 00:37:05.355 align:middle line:90% the counterparts. 00:37:05.355 --> 00:37:07.590 align:middle line:84% Yeah, here is more like a complex scalar. 00:37:07.590 --> 00:37:09.785 align:middle line:84% But you can actually-- we will talk about it. 00:37:09.785 --> 00:37:11.160 align:middle line:84% We'll be able to define something 00:37:11.160 --> 00:37:15.420 align:middle line:84% like a real fermion and then its own antiparticle. 00:37:15.420 --> 00:37:16.660 align:middle line:90% Yeah. 00:37:16.660 --> 00:37:17.320 align:middle line:90% Yes. 00:37:17.320 --> 00:37:18.903 align:middle line:84% AUDIENCE: Can you write down something 00:37:18.903 --> 00:37:21.910 align:middle line:84% that looks analogous to the Dirac equation for spin 00:37:21.910 --> 00:37:22.757 align:middle line:90% 3/2 or 5/2? 00:37:22.757 --> 00:37:24.340 align:middle line:84% HONG LIU: Yeah, you can write it down. 00:37:24.340 --> 00:37:25.178 align:middle line:90% Yeah. 00:37:25.178 --> 00:37:26.497 align:middle line:90% AUDIENCE: You can find those? 00:37:26.497 --> 00:37:28.330 align:middle line:84% HONG LIU: Yeah, I think you can write down-- 00:37:28.330 --> 00:37:32.930 align:middle line:84% I think people are by now have written down equations for any, 00:37:32.930 --> 00:37:33.430 align:middle line:90% yeah. 00:37:33.430 --> 00:37:35.847 align:middle line:84% Yeah, in principle, you can write it down for any of them. 00:37:35.847 --> 00:37:38.810 align:middle line:90% 00:37:38.810 --> 00:37:41.480 align:middle line:84% Yeah, in the end, it boils down to group theory. 00:37:41.480 --> 00:37:42.380 align:middle line:90% Yeah. 00:37:42.380 --> 00:37:43.528 align:middle line:90% Yes. 00:37:43.528 --> 00:37:45.320 align:middle line:84% AUDIENCE: I don't know if this makes sense. 00:37:45.320 --> 00:37:48.310 align:middle line:84% But the particles that are excited, the excitations 00:37:48.310 --> 00:37:52.660 align:middle line:84% of this field, will they still obey the uncertainty relation 00:37:52.660 --> 00:37:56.352 align:middle line:84% given that we've gotten rid of position momentum 00:37:56.352 --> 00:37:58.760 align:middle line:90% and commutation relations? 00:37:58.760 --> 00:38:06.350 align:middle line:84% HONG LIU: So even in the boson case, 00:38:06.350 --> 00:38:09.020 align:middle line:84% our commutation relation is not related 00:38:09.020 --> 00:38:11.930 align:middle line:90% to position and momentum. 00:38:11.930 --> 00:38:15.986 align:middle line:84% There is the field with its conjugate. 00:38:15.986 --> 00:38:18.500 align:middle line:84% There's no position operator anymore. 00:38:18.500 --> 00:38:21.918 align:middle line:84% AUDIENCE: How do you get a single particle wave function 00:38:21.918 --> 00:38:26.400 align:middle line:84% from a [INAUDIBLE] a single particle wave equation? 00:38:26.400 --> 00:38:28.770 align:middle line:84% HONG LIU: Yeah, in the non-relativistic limit, yeah, 00:38:28.770 --> 00:38:30.860 align:middle line:84% you have to take the non-relativistic limit. 00:38:30.860 --> 00:38:31.360 align:middle line:90% Yeah. 00:38:31.360 --> 00:38:36.760 align:middle line:90% 00:38:36.760 --> 00:38:37.260 align:middle line:90% Good. 00:38:37.260 --> 00:38:38.280 align:middle line:90% Other questions? 00:38:38.280 --> 00:38:39.420 align:middle line:90% Yes. 00:38:39.420 --> 00:38:41.400 align:middle line:84% AUDIENCE: In step two, how do we know 00:38:41.400 --> 00:38:43.380 align:middle line:84% that all the solutions that we listed out 00:38:43.380 --> 00:38:45.930 align:middle line:84% form a complete set of all the solutions? 00:38:45.930 --> 00:38:48.740 align:middle line:84% HONG LIU: Yeah, because we have solved the Dirac equation. 00:38:48.740 --> 00:38:51.290 align:middle line:84% Dirac equation is a linear equation. 00:38:51.290 --> 00:38:54.000 align:middle line:84% And, yeah, you solved it, then you 00:38:54.000 --> 00:38:57.420 align:middle line:84% know you have found all the solutions. 00:38:57.420 --> 00:38:59.930 align:middle line:84% Yeah, because we do a Fourier transform, 00:38:59.930 --> 00:39:01.940 align:middle line:84% convert it into an algebraic equation. 00:39:01.940 --> 00:39:05.640 align:middle line:84% The algebraic equation we know when we find all the solutions. 00:39:05.640 --> 00:39:08.300 align:middle line:84% Yeah, so those things we can know for full confidence. 00:39:08.300 --> 00:39:11.970 align:middle line:90% 00:39:11.970 --> 00:39:13.410 align:middle line:90% Good. 00:39:13.410 --> 00:39:18.960 align:middle line:84% So now let's explain in what sense we call one of-- 00:39:18.960 --> 00:39:21.180 align:middle line:84% called the particle and antiparticle, 00:39:21.180 --> 00:39:23.160 align:middle line:84% in what sense they are anti each other. 00:39:23.160 --> 00:39:26.880 align:middle line:84% So this is very similar to in the boson case 00:39:26.880 --> 00:39:29.010 align:middle line:90% when you have a complex scalar. 00:39:29.010 --> 00:39:32.880 align:middle line:84% And, remember, in the complex scalar case, 00:39:32.880 --> 00:39:34.450 align:middle line:84% the reason we call one particle to be 00:39:34.450 --> 00:39:36.525 align:middle line:84% a particle, the other to be antiparticle, 00:39:36.525 --> 00:39:42.010 align:middle line:84% is they have the same mass, the same spin, spin 0 in that case, 00:39:42.010 --> 00:39:44.190 align:middle line:90% but opposite charge. 00:39:44.190 --> 00:39:46.440 align:middle line:84% So here we can also define a charge for them. 00:39:46.440 --> 00:39:49.590 align:middle line:90% 00:39:49.590 --> 00:39:50.790 align:middle line:90% Now let's talk about charge. 00:39:50.790 --> 00:39:56.580 align:middle line:90% 00:39:56.580 --> 00:40:07.030 align:middle line:84% So remember, in your pset, when we first talk 00:40:07.030 --> 00:40:10.720 align:middle line:84% about Dirac equation, in your pset you derived 00:40:10.720 --> 00:40:15.180 align:middle line:84% that if you treat the Dirac equation as a wave equation, 00:40:15.180 --> 00:40:18.180 align:middle line:84% then you can derive a probability current. 00:40:18.180 --> 00:40:20.430 align:middle line:84% And then, the zeroth component of that current 00:40:20.430 --> 00:40:22.860 align:middle line:90% is actually positive definite. 00:40:22.860 --> 00:40:23.550 align:middle line:90% Remember? 00:40:23.550 --> 00:40:33.420 align:middle line:84% So there, when we treat the Dirac equation 00:40:33.420 --> 00:40:34.290 align:middle line:90% as a wave equation. 00:40:34.290 --> 00:40:39.150 align:middle line:90% 00:40:39.150 --> 00:40:40.335 align:middle line:90% And then we find-- 00:40:40.335 --> 00:40:43.630 align:middle line:90% 00:40:43.630 --> 00:40:45.760 align:middle line:84% from what we did with Schrodinger equation, 00:40:45.760 --> 00:40:54.240 align:middle line:84% similarly from what we did with the Schrodinger equation, 00:40:54.240 --> 00:41:00.660 align:middle line:84% and we can derive a equation like partial mu, 00:41:00.660 --> 00:41:01.785 align:middle line:90% j mu equal to 0. 00:41:01.785 --> 00:41:04.770 align:middle line:90% 00:41:04.770 --> 00:41:06.120 align:middle line:90% And the j mu-- 00:41:06.120 --> 00:41:08.680 align:middle line:90% 00:41:08.680 --> 00:41:10.030 align:middle line:90% yeah, so up to a sign. 00:41:10.030 --> 00:41:13.240 align:middle line:84% So let me just choose a minus sign here. 00:41:13.240 --> 00:41:15.670 align:middle line:90% So j mu is given by this. 00:41:15.670 --> 00:41:17.770 align:middle line:90% Using our current notation. 00:41:17.770 --> 00:41:20.020 align:middle line:84% Of course, at the time, we don't know those. 00:41:20.020 --> 00:41:22.810 align:middle line:90% Yeah, do we know the bar? 00:41:22.810 --> 00:41:23.320 align:middle line:90% I forgot. 00:41:23.320 --> 00:41:25.930 align:middle line:90% Anyway, so you have this j. 00:41:25.930 --> 00:41:27.010 align:middle line:90% You can derive this j. 00:41:27.010 --> 00:41:29.140 align:middle line:84% And the zeroth component of the j, you 00:41:29.140 --> 00:41:34.090 align:middle line:84% just psi dagger psi, which is manifestly 00:41:34.090 --> 00:41:35.875 align:middle line:84% positive as a classical function. 00:41:35.875 --> 00:41:38.620 align:middle line:90% 00:41:38.620 --> 00:41:42.730 align:middle line:84% And so, I mentioned that this was the Dirac's main motivation 00:41:42.730 --> 00:41:45.340 align:middle line:84% for writing down Dirac equation is 00:41:45.340 --> 00:41:48.310 align:middle line:84% to look for probability current, which 00:41:48.310 --> 00:41:52.240 align:middle line:90% is the positive probability. 00:41:52.240 --> 00:41:55.660 align:middle line:84% And so, this is positive definite classically. 00:41:55.660 --> 00:41:59.410 align:middle line:84% Yeah, when you treat it as a wave equation. 00:41:59.410 --> 00:42:01.210 align:middle line:90% So now we are-- 00:42:01.210 --> 00:42:05.290 align:middle line:84% but now we don't think Dirac equation as a wave equation, 00:42:05.290 --> 00:42:07.940 align:middle line:90% we treat it as a field theory. 00:42:07.940 --> 00:42:13.330 align:middle line:84% So as a field theory, and then Dirac equation-- 00:42:13.330 --> 00:42:16.105 align:middle line:90% so we have the action. 00:42:16.105 --> 00:42:24.930 align:middle line:90% 00:42:24.930 --> 00:42:26.790 align:middle line:84% So this action has obvious symmetry 00:42:26.790 --> 00:42:28.800 align:middle line:84% similar to the complex boson case. 00:42:28.800 --> 00:42:37.650 align:middle line:84% So this, you can rotate psi by a phase, by a constant phase. 00:42:37.650 --> 00:42:42.870 align:middle line:84% Then psi bar goes to minus alpha psi. 00:42:42.870 --> 00:42:46.260 align:middle line:84% And so, obviously, this is the invariant if alpha 00:42:46.260 --> 00:42:48.480 align:middle line:90% is a constant. 00:42:48.480 --> 00:42:50.065 align:middle line:90% And so, this is a u1 symmetry. 00:42:50.065 --> 00:42:54.740 align:middle line:90% 00:42:54.740 --> 00:43:00.020 align:middle line:84% So when you have a symmetry by rotation by a face, 00:43:00.020 --> 00:43:02.220 align:middle line:90% and it's a U1 symmetry. 00:43:02.220 --> 00:43:04.970 align:middle line:84% And so, from the Noether theorem, 00:43:04.970 --> 00:43:09.200 align:middle line:84% then there must be a conserved current corresponding 00:43:09.200 --> 00:43:10.520 align:middle line:90% to this phase rotation. 00:43:10.520 --> 00:43:16.550 align:middle line:90% 00:43:16.550 --> 00:43:20.970 align:middle line:84% So without doing any calculation, 00:43:20.970 --> 00:43:28.440 align:middle line:84% you should already be able to guess that this is the current. 00:43:28.440 --> 00:43:30.390 align:middle line:84% So this now becomes the Noether current. 00:43:30.390 --> 00:43:33.070 align:middle line:90% 00:43:33.070 --> 00:43:38.850 align:middle line:84% So it's actually this, if you find the Noether current, you 00:43:38.850 --> 00:43:41.578 align:middle line:90% just find this one. 00:43:41.578 --> 00:43:42.370 align:middle line:90% Just find this one. 00:43:42.370 --> 00:43:48.700 align:middle line:90% 00:43:48.700 --> 00:43:56.220 align:middle line:84% So, now, instead, we interpret J0-- 00:43:56.220 --> 00:44:00.540 align:middle line:84% rather than interpret this as a probability current and this 00:44:00.540 --> 00:44:03.580 align:middle line:84% as a probability density, if you treat it as a wave equation. 00:44:03.580 --> 00:44:06.750 align:middle line:84% And here we just interpret it as the conserved current 00:44:06.750 --> 00:44:08.430 align:middle line:90% for this U1 symmetry. 00:44:08.430 --> 00:44:09.930 align:middle line:84% And you have some conserved current. 00:44:09.930 --> 00:44:14.370 align:middle line:84% And then, this is corresponding to the zeroth component 00:44:14.370 --> 00:44:16.230 align:middle line:90% of this conserved current. 00:44:16.230 --> 00:44:17.790 align:middle line:90% And so, this is-- 00:44:17.790 --> 00:44:23.580 align:middle line:84% we interpreted as some kind of charge density 00:44:23.580 --> 00:44:27.030 align:middle line:90% as we did for the scalar case. 00:44:27.030 --> 00:44:29.055 align:middle line:84% Now this is just interpreted as the-- 00:44:29.055 --> 00:44:32.720 align:middle line:90% 00:44:32.720 --> 00:44:38.660 align:middle line:84% so now, now you can just work it out, 00:44:38.660 --> 00:44:46.100 align:middle line:90% what's this, and then the Q. 00:44:46.100 --> 00:44:56.510 align:middle line:90% 00:44:56.510 --> 00:44:59.240 align:middle line:90% So the Q defined this way. 00:44:59.240 --> 00:45:02.660 align:middle line:84% So Q is the total conserved charge, 00:45:02.660 --> 00:45:06.110 align:middle line:84% which is you integrate j0 over all space. 00:45:06.110 --> 00:45:11.940 align:middle line:90% And then just equal to that. 00:45:11.940 --> 00:45:17.480 align:middle line:90% 00:45:17.480 --> 00:45:20.660 align:middle line:84% So, naively, this is a positive definite quantity. 00:45:20.660 --> 00:45:24.130 align:middle line:90% 00:45:24.130 --> 00:45:37.520 align:middle line:84% But, actually, now if you plug in the -- 00:45:37.520 --> 00:45:43.320 align:middle line:84% the expression for psi, then what you find is the following. 00:45:43.320 --> 00:45:45.485 align:middle line:84% You find the Q have the following form. 00:45:45.485 --> 00:46:14.380 align:middle line:90% 00:46:14.380 --> 00:46:24.940 align:middle line:84% Now minus cks dagger, cks, and then 00:46:24.940 --> 00:46:26.125 align:middle line:90% plus the infinite constant. 00:46:26.125 --> 00:46:31.110 align:middle line:90% 00:46:31.110 --> 00:46:32.270 align:middle line:90% Plus the infinite constant. 00:46:32.270 --> 00:46:38.100 align:middle line:90% 00:46:38.100 --> 00:46:43.760 align:middle line:84% So, yeah, so we will define the Q, the quantum operator Q, 00:46:43.760 --> 00:46:47.600 align:middle line:84% by forgetting about this infinite constant. 00:46:47.600 --> 00:46:50.400 align:middle line:90% 00:46:50.400 --> 00:46:52.805 align:middle line:84% So now you see something very interesting happens. 00:46:52.805 --> 00:46:57.940 align:middle line:90% 00:46:57.940 --> 00:47:04.000 align:middle line:84% So, yeah, so if we set this constant, yeah, 00:47:04.000 --> 00:47:05.550 align:middle line:90% they call this-- 00:47:05.550 --> 00:47:07.860 align:middle line:90% let me just call this Q0. 00:47:07.860 --> 00:47:13.950 align:middle line:84% And the Q0 is infinite, just as you 00:47:13.950 --> 00:47:18.450 align:middle line:84% have a constant energy for the-- just have E0, which 00:47:18.450 --> 00:47:21.690 align:middle line:84% is infinite, and here is also infinite. 00:47:21.690 --> 00:47:27.660 align:middle line:84% But we will define the quantum version. 00:47:27.660 --> 00:47:32.940 align:middle line:84% But we can just define the Q just to include this part. 00:47:32.940 --> 00:47:35.350 align:middle line:90% And then, by definition-- 00:47:35.350 --> 00:47:37.500 align:middle line:84% so if I define the Q just by that part, 00:47:37.500 --> 00:47:40.770 align:middle line:84% the Q acting on the vacuum just give you 0 00:47:40.770 --> 00:47:42.780 align:middle line:84% because I have c and a on the other side. 00:47:42.780 --> 00:47:44.835 align:middle line:84% So it means that the vacuum have 0 charge. 00:47:44.835 --> 00:47:49.090 align:middle line:90% 00:47:49.090 --> 00:47:49.590 align:middle line:90% Good. 00:47:49.590 --> 00:47:53.405 align:middle line:84% But now you see something interesting happens. 00:47:53.405 --> 00:47:55.280 align:middle line:84% So do you observe something interesting here? 00:47:55.280 --> 00:47:59.450 align:middle line:90% 00:47:59.450 --> 00:48:01.376 align:middle line:90% Yes. 00:48:01.376 --> 00:48:03.188 align:middle line:84% AUDIENCE: It should be [? posted ?] then 00:48:03.188 --> 00:48:05.820 align:middle line:90% and shown like this. 00:48:05.820 --> 00:48:09.460 align:middle line:84% But here, if we define [INAUDIBLE].. 00:48:09.460 --> 00:48:10.630 align:middle line:90% HONG LIU: That's right. 00:48:10.630 --> 00:48:12.190 align:middle line:90% Good. 00:48:12.190 --> 00:48:16.420 align:middle line:84% So naively, this quantity is infinitely-- 00:48:16.420 --> 00:48:20.730 align:middle line:84% naively, this quantity is positive definite. 00:48:20.730 --> 00:48:25.050 align:middle line:90% But if we define it in the way-- 00:48:25.050 --> 00:48:29.800 align:middle line:84% but this quantity by itself is divergent. 00:48:29.800 --> 00:48:32.620 align:middle line:84% When we say something is positive definite, 00:48:32.620 --> 00:48:35.980 align:middle line:84% if it's divergent, then it does not mean very much. 00:48:35.980 --> 00:48:38.200 align:middle line:84% So now, when we want to make it finite, 00:48:38.200 --> 00:48:41.320 align:middle line:84% the way we make it finite is we define this, 00:48:41.320 --> 00:48:44.080 align:middle line:84% so that when you act on the vacuum, so that it's 0, 00:48:44.080 --> 00:48:47.050 align:middle line:90% so that the vacuum has 0 charge. 00:48:47.050 --> 00:48:50.700 align:middle line:84% So when we throw away this infinite constant, 00:48:50.700 --> 00:48:54.700 align:middle line:84% and then, actually, this can be either positive or negative. 00:48:54.700 --> 00:48:57.020 align:middle line:84% This can be either positive or negative. 00:48:57.020 --> 00:49:02.050 align:middle line:84% And, yeah, so this is the magic by throwing away 00:49:02.050 --> 00:49:05.260 align:middle line:84% some constants, throwing away infinite constants. 00:49:05.260 --> 00:49:07.230 align:middle line:84% And you can make a positive quantity 00:49:07.230 --> 00:49:11.270 align:middle line:90% into an arbitrary sign. 00:49:11.270 --> 00:49:13.980 align:middle line:90% But this is a good thing. 00:49:13.980 --> 00:49:18.010 align:middle line:90% So now, if you look at-- 00:49:18.010 --> 00:49:24.190 align:middle line:84% when you act on the a and c, because 00:49:24.190 --> 00:49:30.790 align:middle line:84% of this sign difference, when you find that when 00:49:30.790 --> 00:49:39.190 align:middle line:84% you act on ks, then you get eigenvalue 1. 00:49:39.190 --> 00:49:45.670 align:middle line:84% When you act on ks bar, you get eigenvalue minus 1. 00:49:45.670 --> 00:49:51.070 align:middle line:90% 00:49:51.070 --> 00:49:52.900 align:middle line:90% So this has charge 1. 00:49:52.900 --> 00:49:56.350 align:middle line:90% So this has charge minus 1. 00:49:56.350 --> 00:49:59.272 align:middle line:84% So you see that this particle and antiparticle, everything 00:49:59.272 --> 00:49:59.980 align:middle line:90% else is the same. 00:49:59.980 --> 00:50:01.540 align:middle line:90% They have the same mass. 00:50:01.540 --> 00:50:03.880 align:middle line:84% They have the same spin, spin half, 00:50:03.880 --> 00:50:07.910 align:middle line:84% but they only differ charge by opposite charge. 00:50:07.910 --> 00:50:11.110 align:middle line:84% So that's why we call one of them is particle 00:50:11.110 --> 00:50:14.250 align:middle line:90% and one of them antiparticle. 00:50:14.250 --> 00:50:14.750 align:middle line:90% Yes. 00:50:14.750 --> 00:50:16.250 align:middle line:84% AUDIENCE: Can we interpret this just 00:50:16.250 --> 00:50:17.775 align:middle line:90% like the probability charge? 00:50:17.775 --> 00:50:18.650 align:middle line:90% HONG LIU: No, no, no. 00:50:18.650 --> 00:50:20.990 align:middle line:84% This is not probability, this is just charge. 00:50:20.990 --> 00:50:23.450 align:middle line:84% There's no probability interpretation anymore. 00:50:23.450 --> 00:50:27.980 align:middle line:84% Probability is only when you treat it as a wave function-- 00:50:27.980 --> 00:50:29.662 align:middle line:90% a wave equation. 00:50:29.662 --> 00:50:31.370 align:middle line:84% But we don't treat it as a wave equation. 00:50:31.370 --> 00:50:33.000 align:middle line:90% We treat it as a field theory. 00:50:33.000 --> 00:50:38.210 align:middle line:84% So in the field theory, and this is just some charge 00:50:38.210 --> 00:50:39.600 align:middle line:90% particles can carry. 00:50:39.600 --> 00:50:42.440 align:middle line:84% Yeah, you have nothing to do with probability. 00:50:42.440 --> 00:50:48.700 align:middle line:90% 00:50:48.700 --> 00:50:49.390 align:middle line:90% Yes. 00:50:49.390 --> 00:50:51.580 align:middle line:84% AUDIENCE: Is the fact that Q not is positive 00:50:51.580 --> 00:50:53.955 align:middle line:84% break any symmetry between particle and antiparticle? 00:50:53.955 --> 00:50:56.070 align:middle line:90% 00:50:56.070 --> 00:50:57.585 align:middle line:90% HONG LIU: What do you mean by-- 00:50:57.585 --> 00:51:00.630 align:middle line:84% AUDIENCE: So just right there, Q not 00:51:00.630 --> 00:51:02.255 align:middle line:84% goes to just a positive infinity. 00:51:02.255 --> 00:51:02.880 align:middle line:90% HONG LIU: Yeah. 00:51:02.880 --> 00:51:03.922 align:middle line:90% AUDIENCE: Does it break-- 00:51:03.922 --> 00:51:07.050 align:middle line:84% I don't know-- particles and antiparticles are not the same? 00:51:07.050 --> 00:51:10.440 align:middle line:84% HONG LIU: Yeah, you always have to-- 00:51:10.440 --> 00:51:12.990 align:middle line:84% because you have to choose a reference. 00:51:12.990 --> 00:51:15.240 align:middle line:84% Because the particle and antiparticle, 00:51:15.240 --> 00:51:17.760 align:middle line:84% they create something out of the vacuum. 00:51:17.760 --> 00:51:22.710 align:middle line:84% Still, even if you keep this Q0, so that 00:51:22.710 --> 00:51:25.980 align:middle line:84% means that your vacuum has a charge, which is Q0. 00:51:25.980 --> 00:51:30.120 align:middle line:84% And, again, then the particle will increase the charge by 1 00:51:30.120 --> 00:51:33.000 align:middle line:84% and the antiparticle will decrease the charge by 1. 00:51:33.000 --> 00:51:34.970 align:middle line:90% So this aspect is the same. 00:51:34.970 --> 00:51:35.910 align:middle line:90% Yeah. 00:51:35.910 --> 00:51:38.168 align:middle line:84% So when you act this on the vacuum, 00:51:38.168 --> 00:51:39.210 align:middle line:90% you just increase charge. 00:51:39.210 --> 00:51:40.680 align:middle line:84% Yeah, just here, for convenience, 00:51:40.680 --> 00:51:44.685 align:middle line:84% we choose the vacuum to have 0 charge. 00:51:44.685 --> 00:51:47.570 align:middle line:90% 00:51:47.570 --> 00:51:49.130 align:middle line:90% Good. 00:51:49.130 --> 00:51:52.215 align:middle line:90% Any other questions? 00:51:52.215 --> 00:51:52.715 align:middle line:90% Yes. 00:51:52.715 --> 00:51:54.923 align:middle line:84% AUDIENCE: Does charge have a physical interpretation, 00:51:54.923 --> 00:51:58.940 align:middle line:84% or how do I interpret the charge in this instance? 00:51:58.940 --> 00:52:03.120 align:middle line:84% HONG LIU: Yeah, so 1 is the electron, 1 is the positron. 00:52:03.120 --> 00:52:07.300 align:middle line:84% Yeah, so these are the electric charge we observe. 00:52:07.300 --> 00:52:07.800 align:middle line:90% Yeah. 00:52:07.800 --> 00:52:10.230 align:middle line:84% So this is the electron-- so when 00:52:10.230 --> 00:52:12.240 align:middle line:84% you apply this theory to the electron, 00:52:12.240 --> 00:52:15.405 align:middle line:84% then this is just the charge of the electron. 00:52:15.405 --> 00:52:24.040 align:middle line:90% 00:52:24.040 --> 00:52:27.610 align:middle line:84% So, yeah, so this actually my next remark. 00:52:27.610 --> 00:52:32.880 align:middle line:90% So applied to electron. 00:52:32.880 --> 00:52:36.740 align:middle line:90% 00:52:36.740 --> 00:52:49.490 align:middle line:84% So Q can be interpreted up to a sign as the charge-- 00:52:49.490 --> 00:52:50.660 align:middle line:90% as the electric charge. 00:52:50.660 --> 00:52:53.210 align:middle line:90% 00:52:53.210 --> 00:52:55.468 align:middle line:90% Yeah, up to assign and a unit. 00:52:55.468 --> 00:52:57.260 align:middle line:84% So this is essentially the electric charge. 00:52:57.260 --> 00:52:59.930 align:middle line:90% 00:52:59.930 --> 00:53:03.480 align:middle line:84% So we will see later, this is the charge, 00:53:03.480 --> 00:53:06.370 align:middle line:84% which couples to the electromagnetism. 00:53:06.370 --> 00:53:06.870 align:middle line:90% Yes. 00:53:06.870 --> 00:53:08.453 align:middle line:84% AUDIENCE: So I'm a little bit confused 00:53:08.453 --> 00:53:10.156 align:middle line:84% how you can define the notion of charge 00:53:10.156 --> 00:53:13.003 align:middle line:84% without even defining the notion of force or force 00:53:13.003 --> 00:53:14.170 align:middle line:90% carrier, anything like that. 00:53:14.170 --> 00:53:17.200 align:middle line:84% HONG LIU: Yeah, the concept of charge 00:53:17.200 --> 00:53:19.720 align:middle line:90% is independent of the force. 00:53:19.720 --> 00:53:20.220 align:middle line:90% Yeah. 00:53:20.220 --> 00:53:21.912 align:middle line:84% Even though sometimes the force, you're 00:53:21.912 --> 00:53:23.370 align:middle line:84% used to this concept with a charge. 00:53:23.370 --> 00:53:26.650 align:middle line:84% But charge by itself is an independent concept. 00:53:26.650 --> 00:53:31.980 align:middle line:84% Yeah, when we talk about the Maxwell theory and then 00:53:31.980 --> 00:53:35.310 align:middle line:84% couple this thing to the Maxwell theory and then 00:53:35.310 --> 00:53:37.800 align:middle line:90% will become clearer. 00:53:37.800 --> 00:53:41.220 align:middle line:84% Yeah, but the bottom line is that the charge by-- 00:53:41.220 --> 00:53:45.930 align:middle line:84% yeah, you can define independent of the force. 00:53:45.930 --> 00:53:48.520 align:middle line:90% It's just some quantum number. 00:53:48.520 --> 00:53:49.318 align:middle line:90% Yes. 00:53:49.318 --> 00:53:50.860 align:middle line:84% AUDIENCE: Yeah, so how do you get out 00:53:50.860 --> 00:53:53.210 align:middle line:90% a value for what that term is? 00:53:53.210 --> 00:53:55.163 align:middle line:84% Right now, it's just one or minus one. 00:53:55.163 --> 00:53:55.830 align:middle line:90% HONG LIU: Right. 00:53:55.830 --> 00:53:59.880 align:middle line:84% Yeah, so indeed, so here, there's a unit. 00:53:59.880 --> 00:54:05.220 align:middle line:84% Because you can multiply that j by arbitrary constant. 00:54:05.220 --> 00:54:08.760 align:middle line:84% And then, that defines your unit for your charge. 00:54:08.760 --> 00:54:12.270 align:middle line:84% And so, in principle, yeah, so that unit 00:54:12.270 --> 00:54:13.980 align:middle line:84% have to be determined by experiments. 00:54:13.980 --> 00:54:23.350 align:middle line:90% 00:54:23.350 --> 00:54:23.850 align:middle line:90% Good? 00:54:23.850 --> 00:54:26.530 align:middle line:90% 00:54:26.530 --> 00:54:29.110 align:middle line:90% So the Dirac equation-- 00:54:29.110 --> 00:54:32.900 align:middle line:90% 00:54:32.900 --> 00:54:45.520 align:middle line:84% so it's very important, is that Dirac theory predict 00:54:45.520 --> 00:54:49.110 align:middle line:84% if you say, this is the theory of electron, 00:54:49.110 --> 00:54:53.370 align:middle line:84% so predict that electron has an antiparticle. 00:54:53.370 --> 00:55:03.150 align:middle line:90% 00:55:03.150 --> 00:55:03.900 align:middle line:90% We call it e-plus. 00:55:03.900 --> 00:55:06.870 align:middle line:90% 00:55:06.870 --> 00:55:11.660 align:middle line:84% So the e-plus have the same mass spin but just opposite charge 00:55:11.660 --> 00:55:12.410 align:middle line:90% from the electron. 00:55:12.410 --> 00:55:16.180 align:middle line:90% 00:55:16.180 --> 00:55:21.780 align:middle line:84% And, of course, when Dirac wrote down this theory, 00:55:21.780 --> 00:55:23.630 align:middle line:90% there was no e-plus yet. 00:55:23.630 --> 00:55:26.940 align:middle line:90% People only know electron. 00:55:26.940 --> 00:55:30.450 align:middle line:84% Even though Dirac had all the wrong motivation to write down 00:55:30.450 --> 00:55:34.800 align:middle line:84% this theory, try to treat it as a wave equation et cetera, 00:55:34.800 --> 00:55:41.620 align:middle line:84% but he correctly predicted somehow electron has-- 00:55:41.620 --> 00:55:44.980 align:middle line:84% this theory predict another positive charged particle. 00:55:44.980 --> 00:55:47.890 align:middle line:90% 00:55:47.890 --> 00:55:51.340 align:middle line:90% And so, that was in 1930-- 00:55:51.340 --> 00:55:53.140 align:middle line:90% that was in 19-- 00:55:53.140 --> 00:55:56.993 align:middle line:84% I think-- when he did that there was a-- 00:55:56.993 --> 00:55:57.910 align:middle line:90% first wrote this down. 00:55:57.910 --> 00:56:00.940 align:middle line:90% Maybe it was 1929 or 1930. 00:56:00.940 --> 00:56:06.290 align:middle line:84% So, at that time, you predict a new particle, that 00:56:06.290 --> 00:56:08.690 align:middle line:90% was considered to be crazy. 00:56:08.690 --> 00:56:12.620 align:middle line:84% And 20 years later, people try to predict one particle a day. 00:56:12.620 --> 00:56:13.400 align:middle line:90% [LAUGHTER] 00:56:13.400 --> 00:56:14.780 align:middle line:90% It become a fashion. 00:56:14.780 --> 00:56:22.010 align:middle line:84% But in 1929, 1930, to predict existence of new particles, 00:56:22.010 --> 00:56:23.810 align:middle line:90% people just absolutely crazy. 00:56:23.810 --> 00:56:29.680 align:middle line:90% 00:56:29.680 --> 00:56:32.830 align:middle line:84% So, yeah, I will not explain how he predicts this particle. 00:56:32.830 --> 00:56:37.060 align:middle line:84% Anyway, he tried to understand this wave equations 00:56:37.060 --> 00:56:41.480 align:middle line:84% and then predict there must be some antiparticle. 00:56:41.480 --> 00:56:45.770 align:middle line:84% But at the beginning, he was a little bit afraid. 00:56:45.770 --> 00:56:50.150 align:middle line:84% He was worried people will just call him crazy. 00:56:50.150 --> 00:56:52.430 align:middle line:84% So he said, this particle, maybe it's a proton. 00:56:52.430 --> 00:56:54.440 align:middle line:90% [LAUGHTER] 00:56:54.440 --> 00:56:58.010 align:middle line:84% Say, oh, because people knew proton at the time. 00:56:58.010 --> 00:57:00.650 align:middle line:84% He said, maybe this particle is proton. 00:57:00.650 --> 00:57:02.840 align:middle line:84% But, of course, he should know immediately 00:57:02.840 --> 00:57:05.440 align:middle line:84% himself that the proton does not have 00:57:05.440 --> 00:57:06.690 align:middle line:90% the same mass as the electron. 00:57:06.690 --> 00:57:07.400 align:middle line:90% [LAUGHTER] 00:57:07.400 --> 00:57:09.650 align:middle line:90% So this cannot be proton. 00:57:09.650 --> 00:57:13.490 align:middle line:84% So and then he quickly gave up that idea. 00:57:13.490 --> 00:57:16.610 align:middle line:84% And he said, oh, maybe this is a new particle. 00:57:16.610 --> 00:57:18.440 align:middle line:90% Maybe this is a new particle. 00:57:18.440 --> 00:57:21.410 align:middle line:90% And so, yeah. 00:57:21.410 --> 00:57:26.700 align:middle line:84% But, luckily, so in 1931 he changed his mind. 00:57:26.700 --> 00:57:28.670 align:middle line:90% He said, this is a new particle. 00:57:28.670 --> 00:57:33.440 align:middle line:84% So luckily, just in 1932, and Anderson 00:57:33.440 --> 00:57:35.740 align:middle line:90% discovered it in cosmic string-- 00:57:35.740 --> 00:57:37.730 align:middle line:90% oh, no, no, in cosmic rays. 00:57:37.730 --> 00:57:40.790 align:middle line:84% And then they found this new particle 00:57:40.790 --> 00:57:43.520 align:middle line:84% which have exactly the same mass as the electron 00:57:43.520 --> 00:57:46.860 align:middle line:84% but opposite charge in cosmic ray. 00:57:46.860 --> 00:57:49.720 align:middle line:90% So that became very happy story. 00:57:49.720 --> 00:57:53.208 align:middle line:90% It became very happy story. 00:57:53.208 --> 00:57:54.250 align:middle line:90% So any questions on this? 00:57:54.250 --> 00:57:54.750 align:middle line:90% Yes. 00:57:54.750 --> 00:57:56.816 align:middle line:84% AUDIENCE: So wouldn't any [INAUDIBLE]?? 00:57:56.816 --> 00:58:04.440 align:middle line:90% 00:58:04.440 --> 00:58:09.880 align:middle line:84% HONG LIU: Yeah, so indeed, when he first wrote it down, 00:58:09.880 --> 00:58:13.540 align:middle line:90% he treated as a wave equation. 00:58:13.540 --> 00:58:16.410 align:middle line:84% So wave equation-- wave function is complex. 00:58:16.410 --> 00:58:19.020 align:middle line:84% So he likely just treat it as a complex. 00:58:19.020 --> 00:58:22.110 align:middle line:84% He didn't even try to make it to be real. 00:58:22.110 --> 00:58:24.960 align:middle line:84% Yeah, it is very natural for it to be complex. 00:58:24.960 --> 00:58:27.090 align:middle line:90% Yeah, and then, yeah. 00:58:27.090 --> 00:58:32.870 align:middle line:90% 00:58:32.870 --> 00:58:36.710 align:middle line:84% It turns out, if you want to write the real equation, 00:58:36.710 --> 00:58:39.330 align:middle line:84% it requires a little bit more effort, 00:58:39.330 --> 00:58:41.630 align:middle line:84% which we will describe, I think, maybe next lecture. 00:58:41.630 --> 00:58:45.530 align:middle line:84% Yeah, and you can get to the real version of this. 00:58:45.530 --> 00:58:48.410 align:middle line:90% 00:58:48.410 --> 00:58:49.080 align:middle line:90% Other questions? 00:58:49.080 --> 00:58:50.890 align:middle line:84% But that takes a little bit more effort. 00:58:50.890 --> 00:58:51.390 align:middle line:90% Yeah. 00:58:51.390 --> 00:58:57.090 align:middle line:84% And that will wait until Majorana, who discovered it. 00:58:57.090 --> 00:58:58.350 align:middle line:90% Yes. 00:58:58.350 --> 00:59:00.120 align:middle line:84% AUDIENCE: So is this whole framework 00:59:00.120 --> 00:59:04.770 align:middle line:84% specific to charging spin one half particle? 00:59:04.770 --> 00:59:06.440 align:middle line:84% Because it's saying it has to be charged 00:59:06.440 --> 00:59:07.990 align:middle line:90% and it has to be spin 1/2 right? 00:59:07.990 --> 00:59:09.970 align:middle line:90% HONG LIU: Yeah, exactly. 00:59:09.970 --> 00:59:10.570 align:middle line:90% Yeah. 00:59:10.570 --> 00:59:11.945 align:middle line:84% AUDIENCE: Is there a version that 00:59:11.945 --> 00:59:14.770 align:middle line:84% could accommodate a chargeless than 1/2, 00:59:14.770 --> 00:59:17.040 align:middle line:84% or is that just a different formalism? 00:59:17.040 --> 00:59:17.790 align:middle line:90% HONG LIU: Yeah. 00:59:17.790 --> 00:59:20.820 align:middle line:84% Chargeless spin 1/2, that is the Majorana fermion, which we will 00:59:20.820 --> 00:59:23.050 align:middle line:90% talk about maybe next lecture. 00:59:23.050 --> 00:59:23.592 align:middle line:90% Yeah. 00:59:23.592 --> 00:59:24.550 align:middle line:90% AUDIENCE: But I guess-- 00:59:24.550 --> 00:59:25.383 align:middle line:90% [INTERPOSING VOICES] 00:59:25.383 --> 00:59:27.390 align:middle line:84% --build out of this framework, or is it-- 00:59:27.390 --> 00:59:29.473 align:middle line:84% HONG LIU: Yeah, it's built from this breakthrough, 00:59:29.473 --> 00:59:31.938 align:middle line:84% but with a little bit more elaboration. 00:59:31.938 --> 00:59:38.360 align:middle line:90% 00:59:38.360 --> 00:59:40.780 align:middle line:90% Other questions? 00:59:40.780 --> 00:59:42.100 align:middle line:90% Good? 00:59:42.100 --> 00:59:46.960 align:middle line:84% So now lets-- so after talk about this quantization, 00:59:46.960 --> 00:59:49.900 align:middle line:84% now we can talk about correlation functions 00:59:49.900 --> 00:59:52.405 align:middle line:90% of these Dirac fields. 00:59:52.405 --> 01:00:34.690 align:middle line:90% 01:00:34.690 --> 01:00:37.690 align:middle line:84% So let's first look at the, say, the Wightman function, 01:00:37.690 --> 01:00:40.285 align:middle line:84% which you just don't do any time ordering. 01:00:40.285 --> 01:00:43.930 align:middle line:90% 01:00:43.930 --> 01:00:45.760 align:middle line:90% So let's look at this object. 01:00:45.760 --> 01:00:51.160 align:middle line:84% I call d plus alpha beta, which is defined, again, 01:00:51.160 --> 01:00:52.780 align:middle line:84% due to the time translation symmetry, 01:00:52.780 --> 01:00:54.190 align:middle line:90% we only depend on the-- 01:00:54.190 --> 01:01:03.060 align:middle line:90% 01:01:03.060 --> 01:01:03.560 align:middle line:90% OK. 01:01:03.560 --> 01:01:04.643 align:middle line:90% Let's look at this object. 01:01:04.643 --> 01:01:08.630 align:middle line:90% 01:01:08.630 --> 01:01:10.480 align:middle line:84% Yeah, you can also exchange them. 01:01:10.480 --> 01:01:13.580 align:middle line:84% That's called d minus, so it doesn't matter. 01:01:13.580 --> 01:01:17.150 align:middle line:84% And so, yeah, so let's look at this object. 01:01:17.150 --> 01:01:20.870 align:middle line:84% So this is a 2-point correlation function between these two 01:01:20.870 --> 01:01:22.830 align:middle line:90% fermionic fields. 01:01:22.830 --> 01:01:29.150 align:middle line:84% And then, you can just plug in the expansion for each of them. 01:01:29.150 --> 01:01:34.010 align:middle line:84% Then you just work it out as we did before for the boson. 01:01:34.010 --> 01:01:36.530 align:middle line:84% And so, let me just outline one step-- 01:01:36.530 --> 01:01:38.670 align:middle line:90% one intermediate step. 01:01:38.670 --> 01:01:44.600 align:middle line:84% So when you plug them in, then you find something like this. 01:01:44.600 --> 01:01:49.980 align:middle line:84% And then you find this sum over s equal to 1, 2, 01:01:49.980 --> 01:01:54.265 align:middle line:84% and then you find u s alpha, and the u 01:01:54.265 --> 01:02:02.540 align:middle line:84% s beta bar, and exponential i k x minus y. 01:02:02.540 --> 01:02:06.490 align:middle line:84% So now, do you recognize this object? 01:02:06.490 --> 01:02:18.610 align:middle line:90% 01:02:18.610 --> 01:02:21.970 align:middle line:90% Anybody recognize it? 01:02:21.970 --> 01:02:25.330 align:middle line:84% So this is the object we discussed last lecture. 01:02:25.330 --> 01:02:26.935 align:middle line:90% But even though, yeah, but-- 01:02:26.935 --> 01:02:30.160 align:middle line:90% 01:02:30.160 --> 01:02:34.630 align:middle line:84% so this is the projector to the positive-- to the space 01:02:34.630 --> 01:02:36.880 align:middle line:84% of the positive energy solutions. 01:02:36.880 --> 01:02:40.750 align:middle line:84% So this is the projector to the space of the U solutions. 01:02:40.750 --> 01:02:42.410 align:middle line:90% So it appears here. 01:02:42.410 --> 01:02:49.030 align:middle line:84% So and I mentioned that this actually-- 01:02:49.030 --> 01:02:51.550 align:middle line:90% you can work it out. 01:02:51.550 --> 01:02:53.050 align:middle line:90% So this is given by this form. 01:02:53.050 --> 01:02:56.840 align:middle line:90% 01:02:56.840 --> 01:03:04.600 align:middle line:84% And then, yeah, so try to check in your notes of last week-- 01:03:04.600 --> 01:03:06.260 align:middle line:90% on Monday. 01:03:06.260 --> 01:03:13.250 align:middle line:84% Yeah, so now, if you plug this in, and then you get d3x, 01:03:13.250 --> 01:03:24.960 align:middle line:90% d3k, i, ik slash plus m. 01:03:24.960 --> 01:03:28.050 align:middle line:84% So this is a matrix, and you take alpha beta component. 01:03:28.050 --> 01:03:30.660 align:middle line:90% 01:03:30.660 --> 01:03:34.500 align:middle line:84% And then exponential ik x minus y. 01:03:34.500 --> 01:03:44.079 align:middle line:90% 01:03:44.079 --> 01:03:54.660 align:middle line:84% So now, so remember, in Fourier transform, any factor k, 01:03:54.660 --> 01:03:56.910 align:middle line:84% in the integrand, you can take it out 01:03:56.910 --> 01:03:58.810 align:middle line:84% in terms of a differential operator. 01:03:58.810 --> 01:04:01.440 align:middle line:84% So we can actually rewrite this as i, 01:04:01.440 --> 01:04:07.945 align:middle line:84% take this i out, and replace this by partial slash, 01:04:07.945 --> 01:04:18.460 align:middle line:84% but with derivative on x plus m alpha beta d3k 2 pi. 01:04:18.460 --> 01:04:27.790 align:middle line:90% 01:04:27.790 --> 01:04:30.190 align:middle line:84% So you can just take this outside of this Fourier 01:04:30.190 --> 01:04:33.137 align:middle line:84% transform and then replace k slash 01:04:33.137 --> 01:04:36.123 align:middle line:90% by partial derivative on x. 01:04:36.123 --> 01:04:37.540 align:middle line:84% When you take the derivative on x, 01:04:37.540 --> 01:04:39.610 align:middle line:90% you bring down a factor of k. 01:04:39.610 --> 01:04:40.990 align:middle line:90% Yeah, bring down a factor of ik. 01:04:40.990 --> 01:04:45.490 align:middle line:84% So ik can be replaced by partial x. 01:04:45.490 --> 01:04:48.190 align:middle line:84% But now, if you recognize this guy-- 01:04:48.190 --> 01:04:51.580 align:middle line:90% so do you recognize this guy? 01:04:51.580 --> 01:04:53.300 align:middle line:90% Yes? 01:04:53.300 --> 01:04:54.170 align:middle line:90% What is this? 01:04:54.170 --> 01:04:59.600 align:middle line:84% AUDIENCE: [INAUDIBLE] do a function? 01:04:59.600 --> 01:05:00.600 align:middle line:90% HONG LIU: Yeah, exactly. 01:05:00.600 --> 01:05:05.760 align:middle line:84% This is the Wightman function for a scalar field. 01:05:05.760 --> 01:05:12.200 align:middle line:84% So now we can just write it as i partial x slash 01:05:12.200 --> 01:05:17.750 align:middle line:84% plus m alpha beta and D plus x minus y. 01:05:17.750 --> 01:05:19.145 align:middle line:90% And so, this is the-- 01:05:19.145 --> 01:05:22.580 align:middle line:90% 01:05:22.580 --> 01:05:31.095 align:middle line:84% so D plus x minus y, say, is the 0 by x by dagger 01:05:31.095 --> 01:05:38.030 align:middle line:84% y of a scalar field, say, of a complex scalar. 01:05:38.030 --> 01:05:43.780 align:middle line:90% 01:05:43.780 --> 01:05:47.920 align:middle line:84% So we see that, actually, there's 01:05:47.920 --> 01:05:55.990 align:middle line:84% a very nice relation between the complex scalar Wightman 01:05:55.990 --> 01:06:01.460 align:middle line:90% function and the fermionic ones. 01:06:01.460 --> 01:06:04.410 align:middle line:84% There's a very nice relation between them. 01:06:04.410 --> 01:06:06.245 align:middle line:84% So they just differ by this factor. 01:06:06.245 --> 01:06:09.880 align:middle line:90% 01:06:09.880 --> 01:06:13.650 align:middle line:84% So, similarly, you can work out other kind 01:06:13.650 --> 01:06:16.330 align:middle line:90% of correlation function. 01:06:16.330 --> 01:06:20.560 align:middle line:84% So let me just write down the result for the others. 01:06:20.560 --> 01:06:27.600 align:middle line:84% So you can also define D minus alpha beta x minus y 01:06:27.600 --> 01:06:31.047 align:middle line:84% to be-- so you just exchange the order between them. 01:06:31.047 --> 01:06:32.505 align:middle line:84% So, in general, they don't commute. 01:06:32.505 --> 01:06:35.520 align:middle line:90% 01:06:35.520 --> 01:06:44.100 align:middle line:90% So beta y psi alpha x 0. 01:06:44.100 --> 01:06:45.600 align:middle line:90% And then, you find-- 01:06:45.600 --> 01:06:49.320 align:middle line:84% in this case, you find that given by minus i 01:06:49.320 --> 01:06:54.810 align:middle line:84% partial x slash plus m alpha beta. 01:06:54.810 --> 01:06:57.240 align:middle line:90% And the D minus-- 01:06:57.240 --> 01:06:58.290 align:middle line:90% x minus y. 01:06:58.290 --> 01:07:01.470 align:middle line:84% The D minus, again, is defined as for the scalar field, 01:07:01.470 --> 01:07:02.250 align:middle line:90% you put-- 01:07:02.250 --> 01:07:04.980 align:middle line:90% you exchange these two. 01:07:04.980 --> 01:07:17.180 align:middle line:84% And for the retarded, you can define the Dr alpha plus beta 01:07:17.180 --> 01:07:19.320 align:middle line:90% to be theta. 01:07:19.320 --> 01:07:22.760 align:middle line:90% So x minus 0 minus y 0. 01:07:22.760 --> 01:07:27.170 align:middle line:84% And now you define retarded using the anticommutator rather 01:07:27.170 --> 01:07:30.390 align:middle line:84% than the commutator we used before. 01:07:30.390 --> 01:07:34.970 align:middle line:84% So for when we define a retarded for scalar, 01:07:34.970 --> 01:07:36.680 align:middle line:90% we use the commutator. 01:07:36.680 --> 01:07:38.630 align:middle line:90% But now you use anticommutator. 01:07:38.630 --> 01:07:43.850 align:middle line:90% 01:07:43.850 --> 01:07:47.360 align:middle line:84% And then, now you find that this is given by, again, 01:07:47.360 --> 01:07:57.560 align:middle line:84% i partial x plus m, the corresponding retarded scalar 01:07:57.560 --> 01:07:58.295 align:middle line:90% 2-point function. 01:07:58.295 --> 01:08:02.090 align:middle line:90% 01:08:02.090 --> 01:08:07.640 align:middle line:84% So, finally, we can also define the time-ordered. 01:08:07.640 --> 01:08:23.520 align:middle line:90% 01:08:23.520 --> 01:08:25.229 align:middle line:84% So, finally, we can also define the time 01:08:25.229 --> 01:08:26.271 align:middle line:90% ordered Feynman function. 01:08:26.271 --> 01:08:34.000 align:middle line:90% 01:08:34.000 --> 01:08:37.485 align:middle line:90% So define to be DF alpha beta. 01:08:37.485 --> 01:08:40.029 align:middle line:90% 01:08:40.029 --> 01:08:44.260 align:middle line:84% So this is defined to be time-ordered, 01:08:44.260 --> 01:08:52.120 align:middle line:90% psi alpha x, psi beta bar y 0. 01:08:52.120 --> 01:08:57.460 align:middle line:84% So now, this time-ordering is defined as following. 01:08:57.460 --> 01:08:59.819 align:middle line:84% So, again, everything for fermionic 01:08:59.819 --> 01:09:05.370 align:middle line:84% fields you replace commutator by anticommutator. 01:09:05.370 --> 01:09:09.450 align:middle line:84% You replace commutation by anticommute. 01:09:09.450 --> 01:09:15.350 align:middle line:84% So remember, previously, when you change the order, 01:09:15.350 --> 01:09:20.529 align:middle line:84% so if the x0 is greater than y0, and then you 01:09:20.529 --> 01:09:22.370 align:middle line:90% just maintain this order. 01:09:22.370 --> 01:09:29.904 align:middle line:84% So you just have 0 psi alpha x psi bar beta y0. 01:09:29.904 --> 01:09:32.910 align:middle line:90% 01:09:32.910 --> 01:09:38.050 align:middle line:84% But now when y0 become greater than x0, 01:09:38.050 --> 01:09:40.104 align:middle line:84% you exchange the order between them. 01:09:40.104 --> 01:09:47.870 align:middle line:90% 01:09:47.870 --> 01:09:51.399 align:middle line:84% And now you add the extra minus sign for fermion. 01:09:51.399 --> 01:09:54.250 align:middle line:84% So for boson, you would just have this. 01:09:54.250 --> 01:09:58.780 align:middle line:84% But now with fermion, you add an additional minus sign. 01:09:58.780 --> 01:10:08.490 align:middle line:84% Then you can show that this is the same as, again, just 01:10:08.490 --> 01:10:11.700 align:middle line:84% plus m alpha beta, the scalar Gf. 01:10:11.700 --> 01:10:27.480 align:middle line:90% 01:10:27.480 --> 01:10:31.080 align:middle line:90% So now we can go to-- 01:10:31.080 --> 01:10:32.880 align:middle line:84% so if you want to do Feynman diagrams, 01:10:32.880 --> 01:10:37.170 align:middle line:84% exactly as we discussed before, we often 01:10:37.170 --> 01:10:40.000 align:middle line:84% need the momentum space expression. 01:10:40.000 --> 01:10:48.420 align:middle line:84% So if we go to momentum space, essentially just 01:10:48.420 --> 01:10:52.200 align:middle line:84% do a Fourier transform of x minus y. 01:10:52.200 --> 01:10:55.380 align:middle line:84% So this is a function of x minus y. 01:10:55.380 --> 01:10:58.870 align:middle line:84% Do a Fourier transform, x minus y. 01:10:58.870 --> 01:11:03.730 align:middle line:90% And then you will get dfk. 01:11:03.730 --> 01:11:06.850 align:middle line:84% So now I will suppress this alpha beta indices. 01:11:06.850 --> 01:11:09.550 align:middle line:84% And now you treat this as a matrix-- 01:11:09.550 --> 01:11:12.650 align:middle line:84% treat this as a matrix in the spinor space. 01:11:12.650 --> 01:11:17.650 align:middle line:84% And so, you can just Fourier transform this guy. 01:11:17.650 --> 01:11:18.770 align:middle line:90% So this is easy. 01:11:18.770 --> 01:11:21.040 align:middle line:84% We know how to do this Fourier transform. 01:11:21.040 --> 01:11:23.940 align:middle line:84% And this just gives us some factor of k. 01:11:23.940 --> 01:11:32.070 align:middle line:84% So essentially, you just get i, ik slash plus m. 01:11:32.070 --> 01:11:36.960 align:middle line:84% So i partial x slash just become ik slash. 01:11:36.960 --> 01:11:45.060 align:middle line:84% And then, we just plug in the expression for scalar 01:11:45.060 --> 01:11:46.755 align:middle line:90% propagator, just given by this. 01:11:46.755 --> 01:11:53.840 align:middle line:90% 01:11:53.840 --> 01:11:56.750 align:middle line:90% So this is a c number. 01:11:56.750 --> 01:11:59.400 align:middle line:90% This is a matrix. 01:11:59.400 --> 01:12:01.210 align:middle line:90% This is a matrix. 01:12:01.210 --> 01:12:05.340 align:middle line:84% And now, if you remember, again, this formula, 01:12:05.340 --> 01:12:09.780 align:middle line:84% that i k that we discussed last time, ik slash 01:12:09.780 --> 01:12:21.154 align:middle line:84% m, ik slash minus m, equal to minus k squared, 01:12:21.154 --> 01:12:25.280 align:middle line:84% slash minus m squared equal to, say, minus 01:12:25.280 --> 01:12:29.010 align:middle line:90% k squared minus m square. 01:12:29.010 --> 01:12:31.770 align:middle line:84% And so, essentially, you see that the k 01:12:31.770 --> 01:12:35.200 align:middle line:84% squared plus m square is essentially 01:12:35.200 --> 01:12:36.800 align:middle line:90% the product of these two. 01:12:36.800 --> 01:12:39.040 align:middle line:90% So this is two matrix product. 01:12:39.040 --> 01:12:43.210 align:middle line:84% And the right-hand side is the identity matrix. 01:12:43.210 --> 01:12:45.622 align:middle line:84% It is this number times the identity matrix. 01:12:45.622 --> 01:12:47.080 align:middle line:84% And so, this becomes-- essentially, 01:12:47.080 --> 01:12:49.330 align:middle line:84% you should view this as times identity matrix. 01:12:49.330 --> 01:12:53.280 align:middle line:90% 01:12:53.280 --> 01:12:55.680 align:middle line:84% And you see that, essentially, these two 01:12:55.680 --> 01:12:59.640 align:middle line:84% up to a constant, these two are inverse matrix of each other. 01:12:59.640 --> 01:13:01.620 align:middle line:84% So we can actually rewrite this as just 01:13:01.620 --> 01:13:10.590 align:middle line:84% inverse of this matrix minus ik slash plus m minus epsilon. 01:13:10.590 --> 01:13:20.770 align:middle line:90% 01:13:20.770 --> 01:13:25.580 align:middle line:84% So this ik slash minus m is the inverse matrix of this. 01:13:25.580 --> 01:13:30.710 align:middle line:84% And so, yeah, use this equation, you find them. 01:13:30.710 --> 01:13:34.040 align:middle line:90% 01:13:34.040 --> 01:13:34.540 align:middle line:90% Good? 01:13:34.540 --> 01:13:38.420 align:middle line:90% 01:13:38.420 --> 01:13:44.990 align:middle line:84% So now we conclude with the momentum space, 01:13:44.990 --> 01:13:48.230 align:middle line:90% a key expression. 01:13:48.230 --> 01:13:49.640 align:middle line:90% Is that, this one. 01:13:49.640 --> 01:13:56.240 align:middle line:84% So let me just write this in very prominent position. 01:13:56.240 --> 01:13:57.950 align:middle line:84% So this will be used over and over. 01:13:57.950 --> 01:14:01.070 align:middle line:90% 01:14:01.070 --> 01:14:03.050 align:middle line:90% So this is a matrix. 01:14:03.050 --> 01:14:05.090 align:middle line:90% And given by that. 01:14:05.090 --> 01:14:10.870 align:middle line:90% 01:14:10.870 --> 01:14:13.300 align:middle line:90% Any questions? 01:14:13.300 --> 01:14:14.410 align:middle line:90% Yes. 01:14:14.410 --> 01:14:16.780 align:middle line:84% AUDIENCE: Is it still true that the primary function 01:14:16.780 --> 01:14:18.910 align:middle line:84% is the Green's function of some wave operator 01:14:18.910 --> 01:14:20.114 align:middle line:90% for the Dirac equation? 01:14:20.114 --> 01:14:22.680 align:middle line:90% 01:14:22.680 --> 01:14:26.490 align:middle line:84% HONG LIU: Oh, you mean when the Dirac equation acting on it 01:14:26.490 --> 01:14:27.780 align:middle line:90% you get some delta function? 01:14:27.780 --> 01:14:28.140 align:middle line:90% AUDIENCE: Yeah. 01:14:28.140 --> 01:14:28.410 align:middle line:90% HONG LIU: Yeah. 01:14:28.410 --> 01:14:29.243 align:middle line:90% AUDIENCE: All right. 01:14:29.243 --> 01:14:33.830 align:middle line:90% 01:14:33.830 --> 01:14:35.365 align:middle line:90% HONG LIU: Other questions? 01:14:35.365 --> 01:14:37.240 align:middle line:84% AUDIENCE: That's not the same epsilon, right? 01:14:37.240 --> 01:14:37.823 align:middle line:90% HONG LIU: Hmm? 01:14:37.823 --> 01:14:40.870 align:middle line:84% AUDIENCE: The epsilon is really different physically? 01:14:40.870 --> 01:14:43.030 align:middle line:84% HONG LIU: No, the epsilon is the same. 01:14:43.030 --> 01:14:44.120 align:middle line:90% It's similar. 01:14:44.120 --> 01:14:47.870 align:middle line:84% It's just it's the m minus i epsilon. 01:14:47.870 --> 01:14:48.890 align:middle line:90% Yeah. 01:14:48.890 --> 01:14:51.140 align:middle line:84% Yeah, it is always, whether it's m 01:14:51.140 --> 01:14:55.370 align:middle line:84% squared minus epsilon or m minus epsilon, it's the same. 01:14:55.370 --> 01:14:57.490 align:middle line:90% Yeah. 01:14:57.490 --> 01:14:59.020 align:middle line:90% Just become-- yeah. 01:14:59.020 --> 01:15:01.250 align:middle line:84% Here, you only have one factor of m. 01:15:01.250 --> 01:15:06.380 align:middle line:90% 01:15:06.380 --> 01:15:08.720 align:middle line:90% Good? 01:15:08.720 --> 01:15:14.810 align:middle line:84% So let's conclude our discussion over the quantization 01:15:14.810 --> 01:15:22.670 align:middle line:90% of the Dirac theory. 01:15:22.670 --> 01:15:26.980 align:middle line:84% And now we go to the next topic, which you have already 01:15:26.980 --> 01:15:29.530 align:middle line:90% asked several times. 01:15:29.530 --> 01:15:34.690 align:middle line:84% But we only have a couple of minutes to motivate it 01:15:34.690 --> 01:15:35.620 align:middle line:90% and to discuss it. 01:15:35.620 --> 01:15:42.820 align:middle line:90% 01:15:42.820 --> 01:15:49.435 align:middle line:90% So the goal is chiral fermions. 01:15:49.435 --> 01:15:53.320 align:middle line:90% 01:15:53.320 --> 01:15:54.728 align:middle line:90% And [INAUDIBLE]. 01:15:54.728 --> 01:16:10.550 align:middle line:90% 01:16:10.550 --> 01:16:13.280 align:middle line:84% So, yes, we will not not have time 01:16:13.280 --> 01:16:15.140 align:middle line:90% to really start discussing. 01:16:15.140 --> 01:16:19.510 align:middle line:84% And let me just say a few remarks. 01:16:19.510 --> 01:16:25.060 align:middle line:84% So we derived before that the Dirac spinor-- 01:16:25.060 --> 01:16:27.070 align:middle line:84% so, so far, what we discussed normally 01:16:27.070 --> 01:16:28.630 align:middle line:90% is called Dirac spinor. 01:16:28.630 --> 01:16:33.760 align:middle line:84% So they transform under Lorentz transformation in the way 01:16:33.760 --> 01:16:40.600 align:middle line:84% so that the Dirac equation is covariant. 01:16:40.600 --> 01:16:43.130 align:middle line:90% So it's a natural question-- 01:16:43.130 --> 01:16:45.890 align:middle line:84% and the Dirac spinor have eight components. 01:16:45.890 --> 01:16:48.070 align:middle line:90% So I have eight real component. 01:16:48.070 --> 01:16:49.840 align:middle line:90% You have four complex component. 01:16:49.840 --> 01:16:54.100 align:middle line:84% So altogether, there are eight variables. 01:16:54.100 --> 01:17:01.300 align:middle line:84% So it's natural to ask whether you can reduce the Dirac spinor 01:17:01.300 --> 01:17:04.330 align:middle line:90% into a smaller set. 01:17:04.330 --> 01:17:07.630 align:middle line:84% So you have-- instead you have eight independent variables 01:17:07.630 --> 01:17:09.910 align:middle line:90% into a smaller set. 01:17:09.910 --> 01:17:16.340 align:middle line:84% That still can give rise to a Dirac equation, 01:17:16.340 --> 01:17:24.670 align:middle line:84% which transform covariantly under Lorentz transformation. 01:17:24.670 --> 01:17:27.250 align:middle line:90% And so, the answer is, yes. 01:17:27.250 --> 01:17:29.600 align:middle line:84% So there are two ways of doing it. 01:17:29.600 --> 01:17:32.410 align:middle line:84% And one is called the chiral fermions. 01:17:32.410 --> 01:17:34.900 align:middle line:84% And the other is called the Majorana fermions. 01:17:34.900 --> 01:17:44.050 align:middle line:84% And the chiral fermions is to have two component complex 01:17:44.050 --> 01:17:44.960 align:middle line:90% vector. 01:17:44.960 --> 01:17:48.160 align:middle line:84% So altogether you have four real components. 01:17:48.160 --> 01:17:50.860 align:middle line:84% Or you have so-called Majorana fermions, which 01:17:50.860 --> 01:17:55.670 align:middle line:90% you have four real components. 01:17:55.670 --> 01:18:00.670 align:middle line:84% So we will discuss that next time. 01:18:00.670 --> 01:18:02.880 align:middle line:90% good, that's all for today. 01:18:02.880 --> 01:18:06.000 align:middle line:90%