WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:05.412 align:middle line:90% [CLICKING] 00:00:05.412 --> 00:00:10.840 align:middle line:90% 00:00:10.840 --> 00:00:12.190 align:middle line:90% HONG LIU: OK, great. 00:00:12.190 --> 00:00:13.970 align:middle line:90% So now let's talk about QED. 00:00:13.970 --> 00:00:25.320 align:middle line:90% 00:00:25.320 --> 00:00:34.700 align:middle line:90% And OK. 00:00:34.700 --> 00:00:39.080 align:middle line:90% So let's first talk about QED. 00:00:39.080 --> 00:00:42.180 align:middle line:90% 00:00:42.180 --> 00:00:46.970 align:middle line:84% And we have talked about the Maxwell action. 00:00:46.970 --> 00:00:50.270 align:middle line:84% So here is the Lagrangian density. 00:00:50.270 --> 00:01:00.920 align:middle line:84% We have F mu nu minus J mu A mu, OK? 00:01:00.920 --> 00:01:08.080 align:middle line:84% And J mu should be a conserved current 00:01:08.080 --> 00:01:12.550 align:middle line:84% because for the consistency of the Maxwell equation, OK? 00:01:12.550 --> 00:01:16.160 align:middle line:84% Remember-- J mu has to be conserved. 00:01:16.160 --> 00:01:19.810 align:middle line:84% So now, the question is what provides this J mu, OK? 00:01:19.810 --> 00:01:21.400 align:middle line:90% So now, let's imagine. 00:01:21.400 --> 00:01:24.010 align:middle line:84% Now, let's introduce some other fields. 00:01:24.010 --> 00:01:25.795 align:middle line:84% Now, imagine we have some Dirac fermions. 00:01:25.795 --> 00:01:30.560 align:middle line:90% 00:01:30.560 --> 00:01:32.420 align:middle line:90% We have some fermionic field. 00:01:32.420 --> 00:01:34.250 align:middle line:90% Same with the Lagrangian. 00:01:34.250 --> 00:01:42.590 align:middle line:90% 00:01:42.590 --> 00:01:43.090 align:middle line:90% OK. 00:01:43.090 --> 00:01:48.970 align:middle line:84% So this is the Lagrangian for the Dirac fermion. 00:01:48.970 --> 00:01:53.860 align:middle line:84% And then we discussed that for Dirac fermion-- 00:01:53.860 --> 00:01:55.690 align:middle line:84% and then there's a conserved current 00:01:55.690 --> 00:01:58.570 align:middle line:84% because there's a global symmetry. 00:01:58.570 --> 00:02:07.310 align:middle line:84% There's a U1 symmetry corresponding to psi-- 00:02:07.310 --> 00:02:12.505 align:middle line:84% goes to exponential i alpha psi with alpha to be a constant. 00:02:12.505 --> 00:02:15.590 align:middle line:90% 00:02:15.590 --> 00:02:16.280 align:middle line:90% OK? 00:02:16.280 --> 00:02:20.090 align:middle line:84% And then this leads to a Noether current-- 00:02:20.090 --> 00:02:21.980 align:middle line:90% J mu, which is conserved. 00:02:21.980 --> 00:02:30.920 align:middle line:90% 00:02:30.920 --> 00:02:32.180 align:middle line:90% OK? 00:02:32.180 --> 00:02:37.700 align:middle line:84% So now, if we want to couple the fermions to the Maxwell fields, 00:02:37.700 --> 00:02:42.710 align:middle line:84% then it's natural we identify this J mu with this J mu, OK? 00:02:42.710 --> 00:02:46.250 align:middle line:84% Because here we need the conserved current. 00:02:46.250 --> 00:02:49.340 align:middle line:84% And here we have a conserved current, OK? 00:02:49.340 --> 00:02:53.150 align:middle line:84% So it's natural to write down the theory. 00:02:53.150 --> 00:02:55.490 align:middle line:84% Then this J mu is just replaced by this J mu. 00:02:55.490 --> 00:02:57.780 align:middle line:84% That's why I use the same notation. 00:02:57.780 --> 00:03:01.910 align:middle line:84% So then that gives us the Lagrangian for QED. 00:03:01.910 --> 00:03:06.330 align:middle line:90% 00:03:06.330 --> 00:03:13.120 align:middle line:84% We have the Maxwell, which is the free photon. 00:03:13.120 --> 00:03:14.710 align:middle line:84% And then we have the free fermion. 00:03:14.710 --> 00:03:23.060 align:middle line:90% 00:03:23.060 --> 00:03:25.790 align:middle line:84% But then the fermion and the photon 00:03:25.790 --> 00:03:37.840 align:middle line:84% are coupled together through this, OK? 00:03:37.840 --> 00:03:41.770 align:middle line:84% So now, this is a cubic term, OK? 00:03:41.770 --> 00:03:45.200 align:middle line:84% So now, this is an interaction term, OK? 00:03:45.200 --> 00:03:46.970 align:middle line:84% So now, this is an interaction term. 00:03:46.970 --> 00:03:51.940 align:middle line:84% And this e can play two roles, OK? 00:03:51.940 --> 00:03:54.860 align:middle line:84% It can be considered as a coupling constant-- 00:03:54.860 --> 00:04:00.030 align:middle line:90% 00:04:00.030 --> 00:04:03.380 align:middle line:84% essentially, the e, which just determines 00:04:03.380 --> 00:04:07.020 align:middle line:84% the coupling strength between the psi and the A, OK? 00:04:07.020 --> 00:04:10.760 align:middle line:84% If e is bigger, then of course, the coupling is stronger, OK? 00:04:10.760 --> 00:04:13.590 align:middle line:84% And the e can also be considered-- 00:04:13.590 --> 00:04:15.680 align:middle line:90% so it plays a dual role. 00:04:15.680 --> 00:04:17.720 align:middle line:84% One role is you consider as a coupling. 00:04:17.720 --> 00:04:20.870 align:middle line:84% And the second role is you can consider as a unit of charge. 00:04:20.870 --> 00:04:27.520 align:middle line:90% 00:04:27.520 --> 00:04:28.330 align:middle line:90% OK? 00:04:28.330 --> 00:04:29.650 align:middle line:90% A unit of charge. 00:04:29.650 --> 00:04:34.440 align:middle line:84% Because when we talk about the Noether current, 00:04:34.440 --> 00:04:35.190 align:middle line:90% this is conserved. 00:04:35.190 --> 00:04:38.070 align:middle line:84% You can multiply this by an arbitrary number. 00:04:38.070 --> 00:04:40.680 align:middle line:90% This is still conserved, OK? 00:04:40.680 --> 00:04:41.850 align:middle line:90% This-- still conserved. 00:04:41.850 --> 00:04:47.520 align:middle line:84% And here it just means that the unit of this current 00:04:47.520 --> 00:04:49.860 align:middle line:90% is given by e, OK? 00:04:49.860 --> 00:04:55.010 align:middle line:84% So the e plays two roles here, OK? 00:04:55.010 --> 00:04:56.750 align:middle line:84% And then of course this e is just 00:04:56.750 --> 00:04:59.010 align:middle line:90% our standard electric charge. 00:04:59.010 --> 00:05:03.050 align:middle line:84% So if you take the psi to be electron, 00:05:03.050 --> 00:05:05.270 align:middle line:84% then this will be standard electric charge 00:05:05.270 --> 00:05:07.280 align:middle line:84% for the electron, and then this will 00:05:07.280 --> 00:05:11.330 align:middle line:84% be a theory which governs the interaction 00:05:11.330 --> 00:05:14.210 align:middle line:84% between the electron and the photon, OK? 00:05:14.210 --> 00:05:17.190 align:middle line:90% 00:05:17.190 --> 00:05:18.750 align:middle line:90% Any questions on this? 00:05:18.750 --> 00:05:19.887 align:middle line:90% Yes? 00:05:19.887 --> 00:05:21.720 align:middle line:84% AUDIENCE: How come different particles still 00:05:21.720 --> 00:05:23.693 align:middle line:84% have some multiple of the same charge? 00:05:23.693 --> 00:05:25.110 align:middle line:84% You might think that you could set 00:05:25.110 --> 00:05:28.970 align:middle line:84% the coupling constant to any arbitrary value. 00:05:28.970 --> 00:05:30.200 align:middle line:90% HONG LIU: Sorry? 00:05:30.200 --> 00:05:31.910 align:middle line:84% AUDIENCE: How come different particles 00:05:31.910 --> 00:05:34.555 align:middle line:84% have charges that are multiples and different multiples-- 00:05:34.555 --> 00:05:35.430 align:middle line:90% HONG LIU: Yeah, yeah. 00:05:35.430 --> 00:05:36.240 align:middle line:84% AUDIENCE: I think in this framework, 00:05:36.240 --> 00:05:38.130 align:middle line:84% you can set the coupling constant to whatever you like. 00:05:38.130 --> 00:05:38.580 align:middle line:90% HONG LIU: That's right. 00:05:38.580 --> 00:05:39.122 align:middle line:90% That's right. 00:05:39.122 --> 00:05:40.230 align:middle line:90% Yeah, yeah. 00:05:40.230 --> 00:05:45.600 align:middle line:84% So in this framework, indeed, this e can be arbitrary. 00:05:45.600 --> 00:05:48.960 align:middle line:84% And then different particles can have different charge. 00:05:48.960 --> 00:05:53.280 align:middle line:84% For example, if we couple this to electron, this will be e e. 00:05:53.280 --> 00:05:56.280 align:middle line:84% And then if we couple this to muon-- so in principle, we 00:05:56.280 --> 00:05:57.690 align:middle line:84% have some e tilde, which does not 00:05:57.690 --> 00:05:59.340 align:middle line:90% have to be the same as this e. 00:05:59.340 --> 00:06:00.090 align:middle line:90% Indeed. 00:06:00.090 --> 00:06:02.820 align:middle line:84% And it's a highly unusual feature. 00:06:02.820 --> 00:06:06.010 align:middle line:90% 00:06:06.010 --> 00:06:10.260 align:middle line:84% In nature, what we observe-- the charge seems to be quantized. 00:06:10.260 --> 00:06:15.540 align:middle line:84% There seems to be multiple of some charge. 00:06:15.540 --> 00:06:18.240 align:middle line:84% And that cannot be explained using this framework. 00:06:18.240 --> 00:06:22.838 align:middle line:84% It has to be explained using some other framework, 00:06:22.838 --> 00:06:24.630 align:middle line:84% sometimes called the grand unified theory-- 00:06:24.630 --> 00:06:26.670 align:middle line:90% can be used to explain that. 00:06:26.670 --> 00:06:28.070 align:middle line:90% Yeah. 00:06:28.070 --> 00:06:28.910 align:middle line:90% Other questions? 00:06:28.910 --> 00:06:30.860 align:middle line:90% AUDIENCE: So I'm confused. 00:06:30.860 --> 00:06:32.798 align:middle line:84% In this middle section, this J mu 00:06:32.798 --> 00:06:36.080 align:middle line:84% is the conserved current from the U1-- 00:06:36.080 --> 00:06:36.832 align:middle line:90% HONG LIU: Yeah. 00:06:36.832 --> 00:06:38.790 align:middle line:84% AUDIENCE: What was the analogy you were making? 00:06:38.790 --> 00:06:41.078 align:middle line:84% That it can do-- you could take it 00:06:41.078 --> 00:06:44.090 align:middle line:84% to do the same J mu for the Maxwell and Lagrangian? 00:06:44.090 --> 00:06:45.200 align:middle line:90% HONG LIU: Yeah. 00:06:45.200 --> 00:06:47.960 align:middle line:90% Just the natural implication. 00:06:47.960 --> 00:06:52.730 align:middle line:84% AUDIENCE: But this Lagrangian doesn't have this U1 symmetry 00:06:52.730 --> 00:06:55.365 align:middle line:84% for this fermionic field because it doesn't even exist, right? 00:06:55.365 --> 00:06:56.240 align:middle line:90% HONG LIU: Yeah, yeah. 00:06:56.240 --> 00:06:57.050 align:middle line:90% Yeah, yeah. 00:06:57.050 --> 00:06:58.968 align:middle line:90% So no, I coupled them. 00:06:58.968 --> 00:06:59.510 align:middle line:90% AUDIENCE: OK. 00:06:59.510 --> 00:07:01.052 align:middle line:84% So you're just taking inspiration now 00:07:01.052 --> 00:07:02.090 align:middle line:90% to add both of them-- 00:07:02.090 --> 00:07:02.720 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:07:02.720 --> 00:07:04.340 align:middle line:90% Really, that's what we do here. 00:07:04.340 --> 00:07:05.240 align:middle line:90% We add both of them. 00:07:05.240 --> 00:07:05.740 align:middle line:90% Yeah. 00:07:05.740 --> 00:07:10.240 align:middle line:90% 00:07:10.240 --> 00:07:14.170 align:middle line:84% Yeah, so we have a theory with a photon and fermion. 00:07:14.170 --> 00:07:16.510 align:middle line:84% And now, their coupling is through this J. 00:07:16.510 --> 00:07:18.240 align:middle line:90% And this J is given by this one. 00:07:18.240 --> 00:07:18.740 align:middle line:90% Yeah. 00:07:18.740 --> 00:07:22.660 align:middle line:90% 00:07:22.660 --> 00:07:24.160 align:middle line:90% OK? 00:07:24.160 --> 00:07:25.150 align:middle line:90% Other questions? 00:07:25.150 --> 00:07:27.910 align:middle line:90% 00:07:27.910 --> 00:07:29.205 align:middle line:90% OK, good. 00:07:29.205 --> 00:07:30.580 align:middle line:90% Good. 00:07:30.580 --> 00:07:34.960 align:middle line:84% So we can also slightly rewrite this Lagrangian 00:07:34.960 --> 00:07:36.750 align:middle line:90% in a somewhat different way. 00:07:36.750 --> 00:07:37.250 align:middle line:90% OK? 00:07:37.250 --> 00:07:45.100 align:middle line:84% So QED can also be written in the following way, OK? 00:07:45.100 --> 00:07:49.580 align:middle line:84% So we can also rewrite this as 1/4 F mu nu. 00:07:49.580 --> 00:07:52.070 align:middle line:90% So this part, we don't change. 00:07:52.070 --> 00:07:55.450 align:middle line:84% But we can actually combine these two terms together 00:07:55.450 --> 00:07:57.110 align:middle line:90% in the following way. 00:07:57.110 --> 00:08:02.770 align:middle line:84% We write it as i psi bar, and then gamma mu. 00:08:02.770 --> 00:08:08.740 align:middle line:84% I introduce a new derivative called D mu minus m psi. 00:08:08.740 --> 00:08:11.660 align:middle line:90% And this D mu-- 00:08:11.660 --> 00:08:15.290 align:middle line:84% capital D mu psi is equal to partial mu 00:08:15.290 --> 00:08:24.490 align:middle line:90% psi minus ie A mu psi, OK? 00:08:24.490 --> 00:08:29.120 align:middle line:84% So this is the same as that because the first term 00:08:29.120 --> 00:08:32.630 align:middle line:90% here just gives you this term. 00:08:32.630 --> 00:08:36.169 align:middle line:84% And the second term-- this e A mu psi term-- 00:08:36.169 --> 00:08:39.710 align:middle line:84% just gives you this e A mu psi bar. 00:08:39.710 --> 00:08:40.640 align:middle line:90% Yeah. 00:08:40.640 --> 00:08:44.500 align:middle line:90% It just gives you that, OK? 00:08:44.500 --> 00:08:47.500 align:middle line:84% And so mathematically, these two are just completely equivalent. 00:08:47.500 --> 00:08:50.740 align:middle line:90% I just slightly rewrote it. 00:08:50.740 --> 00:08:54.970 align:middle line:84% But writing this way actually makes one new property 00:08:54.970 --> 00:08:58.090 align:middle line:90% of this theory manifest, OK? 00:08:58.090 --> 00:09:06.630 align:middle line:84% So this theory-- so we know that this theory 00:09:06.630 --> 00:09:07.545 align:middle line:90% has a gauge symmetry. 00:09:07.545 --> 00:09:10.520 align:middle line:90% 00:09:10.520 --> 00:09:14.965 align:middle line:84% But this theory only has a global symmetry, OK? 00:09:14.965 --> 00:09:16.590 align:middle line:84% This theory only has a global symmetry. 00:09:16.590 --> 00:09:19.710 align:middle line:84% This alpha must be a constant, OK? 00:09:19.710 --> 00:09:23.076 align:middle line:84% So now, when we combine together, 00:09:23.076 --> 00:09:25.550 align:middle line:84% and now this new Lagrangian actually 00:09:25.550 --> 00:09:32.150 align:middle line:84% has a generalized gauge symmetry, OK? 00:09:32.150 --> 00:09:40.500 align:middle line:84% So L max-- so L QED turns out to be 00:09:40.500 --> 00:09:48.690 align:middle line:90% invariant under the following. 00:09:48.690 --> 00:09:52.800 align:middle line:90% 00:09:52.800 --> 00:09:55.340 align:middle line:90% So still, you take A mu. 00:09:55.340 --> 00:09:59.090 align:middle line:84% Goes to A mu plus that partial mu lambda x. 00:09:59.090 --> 00:10:01.940 align:middle line:90% 00:10:01.940 --> 00:10:07.220 align:middle line:84% And then now, you transform psi also by a local transformation. 00:10:07.220 --> 00:10:12.000 align:middle line:90% 00:10:12.000 --> 00:10:14.100 align:middle line:84% You transform it by exponential i. 00:10:14.100 --> 00:10:18.420 align:middle line:90% So now, this e is different, OK? 00:10:18.420 --> 00:10:21.070 align:middle line:90% 00:10:21.070 --> 00:10:27.520 align:middle line:84% So now, you can transform psi by, actually, a local phase. 00:10:27.520 --> 00:10:29.740 align:middle line:90% Actually now, it can depend-- 00:10:29.740 --> 00:10:32.060 align:middle line:90% this arbitrary function lambda. 00:10:32.060 --> 00:10:33.890 align:middle line:84% So you can show that actually, this QED 00:10:33.890 --> 00:10:36.690 align:middle line:84% is invariant under this transformation, OK? 00:10:36.690 --> 00:10:40.840 align:middle line:90% So if I call this A mu prime-- 00:10:40.840 --> 00:10:43.550 align:middle line:90% if I call this psi prime x-- 00:10:43.550 --> 00:10:44.050 align:middle line:90% OK? 00:10:44.050 --> 00:10:46.930 align:middle line:90% So you can check yourself. 00:10:46.930 --> 00:10:50.190 align:middle line:84% With this transformation, this D mu psi-- 00:10:50.190 --> 00:10:53.440 align:middle line:90% 00:10:53.440 --> 00:10:59.180 align:middle line:90% so this D mu prime, psi prime-- 00:10:59.180 --> 00:10:59.680 align:middle line:90% OK? 00:10:59.680 --> 00:11:02.890 align:middle line:84% So D mu prime is obtained by this D mu. 00:11:02.890 --> 00:11:06.880 align:middle line:84% You replace the A mu by A mu prime. 00:11:06.880 --> 00:11:12.100 align:middle line:84% And then here you can show that this is equal to exponential 00:11:12.100 --> 00:11:20.000 align:middle line:90% i e lambda x D mu psi, OK? 00:11:20.000 --> 00:11:28.290 align:middle line:84% So it turns out, in this combination, 00:11:28.290 --> 00:11:31.700 align:middle line:84% so when you make such a local transformation 00:11:31.700 --> 00:11:36.670 align:middle line:84% on psi, when you take a derivative, 00:11:36.670 --> 00:11:39.358 align:middle line:90% then you get an extra term, OK? 00:11:39.358 --> 00:11:41.400 align:middle line:84% You get the extra term from taking the derivative 00:11:41.400 --> 00:11:43.330 align:middle line:90% on this lambda. 00:11:43.330 --> 00:11:45.490 align:middle line:84% But when you have this combination with A mu-- 00:11:45.490 --> 00:11:48.730 align:middle line:84% but A mu also transforms with additional lambda term-- 00:11:48.730 --> 00:11:49.690 align:middle line:90% partial mu lambda term. 00:11:49.690 --> 00:11:51.850 align:middle line:84% And then these two terms just cancel, OK? 00:11:51.850 --> 00:11:55.570 align:middle line:90% So in the end, this D prime mu-- 00:11:55.570 --> 00:11:57.280 align:middle line:84% D prime, mu prime-- actually transforms 00:11:57.280 --> 00:11:59.892 align:middle line:90% in a very simple way, OK? 00:11:59.892 --> 00:12:01.350 align:middle line:84% It transforms in a very simple way. 00:12:01.350 --> 00:12:01.850 align:middle line:90% Yes? 00:12:01.850 --> 00:12:04.350 align:middle line:84% AUDIENCE: Is D mu prime any different than just 00:12:04.350 --> 00:12:07.166 align:middle line:90% D mu [INAUDIBLE]? 00:12:07.166 --> 00:12:09.555 align:middle line:84% Are we changing things out [INAUDIBLE]?? 00:12:09.555 --> 00:12:10.430 align:middle line:90% HONG LIU: Yeah, yeah. 00:12:10.430 --> 00:12:12.035 align:middle line:84% We replaced the A mu by A mu prime. 00:12:12.035 --> 00:12:12.577 align:middle line:90% AUDIENCE: Oh. 00:12:12.577 --> 00:12:14.330 align:middle line:90% HONG LIU: Yeah, yeah. 00:12:14.330 --> 00:12:15.820 align:middle line:90% Yeah. 00:12:15.820 --> 00:12:19.660 align:middle line:84% So this tells you-- and then this transforms very simply. 00:12:19.660 --> 00:12:22.410 align:middle line:90% 00:12:22.410 --> 00:12:24.210 align:middle line:84% And then this transforms very simply. 00:12:24.210 --> 00:12:26.978 align:middle line:84% And psi bar-- there's no derivative here. 00:12:26.978 --> 00:12:28.770 align:middle line:84% Then this, straightforward, just transforms 00:12:28.770 --> 00:12:32.130 align:middle line:90% as the exponential minus e. 00:12:32.130 --> 00:12:35.020 align:middle line:90% And then we conclude. 00:12:35.020 --> 00:12:35.520 align:middle line:90% OK? 00:12:35.520 --> 00:12:41.100 align:middle line:84% So we conclude that L QED is invariant 00:12:41.100 --> 00:12:46.480 align:middle line:90% under this transformation, OK? 00:12:46.480 --> 00:12:47.555 align:middle line:90% Indeed, it's invariant. 00:12:47.555 --> 00:12:48.055 align:middle line:90% OK. 00:12:48.055 --> 00:12:53.370 align:middle line:90% 00:12:53.370 --> 00:12:56.860 align:middle line:90% So we call it gauge invariant. 00:12:56.860 --> 00:12:59.880 align:middle line:84% So now, this is the generalized gauge symmetry. 00:12:59.880 --> 00:13:03.092 align:middle line:84% Now, you also transform the fermions. 00:13:03.092 --> 00:13:04.425 align:middle line:90% You also transform the fermions. 00:13:04.425 --> 00:13:12.080 align:middle line:90% 00:13:12.080 --> 00:13:13.860 align:middle line:90% OK. 00:13:13.860 --> 00:13:16.260 align:middle line:90% Any questions on this? 00:13:16.260 --> 00:13:18.260 align:middle line:90% So now, we can turn it around. 00:13:18.260 --> 00:13:22.110 align:middle line:90% So let us review our logic. 00:13:22.110 --> 00:13:25.130 align:middle line:84% So we first start with the Maxwell theory. 00:13:25.130 --> 00:13:27.360 align:middle line:84% So the Maxwell theory-- there's a J here, 00:13:27.360 --> 00:13:29.660 align:middle line:90% which you need to provide. 00:13:29.660 --> 00:13:34.760 align:middle line:84% And then from the fermionic theory, you have another J. 00:13:34.760 --> 00:13:37.190 align:middle line:84% And then we just combine these two theories 00:13:37.190 --> 00:13:43.550 align:middle line:84% by replacing that J naturally from the fermions. 00:13:43.550 --> 00:13:46.850 align:middle line:84% But it turns out, when we combine them together, 00:13:46.850 --> 00:13:51.010 align:middle line:84% this theory actually has a generalized gauge symmetry. 00:13:51.010 --> 00:13:55.400 align:middle line:84% You also now can transform psi locally, OK? 00:13:55.400 --> 00:14:00.170 align:middle line:84% You can transform psi locally with this nice structure, OK? 00:14:00.170 --> 00:14:02.060 align:middle line:84% So this theory has this nice structure. 00:14:02.060 --> 00:14:08.930 align:middle line:84% And then now, you can transform psi also locally. 00:14:08.930 --> 00:14:10.580 align:middle line:90% So we can also turn this around. 00:14:10.580 --> 00:14:23.160 align:middle line:90% 00:14:23.160 --> 00:14:24.240 align:middle line:90% So we can say-- 00:14:24.240 --> 00:14:30.420 align:middle line:90% 00:14:30.420 --> 00:14:32.240 align:middle line:84% so if we call this thing to be star-- 00:14:32.240 --> 00:14:36.060 align:middle line:90% 00:14:36.060 --> 00:14:53.810 align:middle line:84% so if we require the theory of A mu and psi 00:14:53.810 --> 00:15:11.480 align:middle line:84% to be invariant under star, and then that uniquely leads to-- 00:15:11.480 --> 00:15:14.900 align:middle line:90% 00:15:14.900 --> 00:15:16.460 align:middle line:90% maybe not uniquely. 00:15:16.460 --> 00:15:21.420 align:middle line:84% That leads to the interaction of the form J mu A mu, OK? 00:15:21.420 --> 00:15:24.360 align:middle line:84% It just leads to this kind of interaction. 00:15:24.360 --> 00:15:24.900 align:middle line:90% OK? 00:15:24.900 --> 00:15:26.910 align:middle line:90% So we can turn it around. 00:15:26.910 --> 00:15:29.710 align:middle line:90% We say, OK. 00:15:29.710 --> 00:15:32.920 align:middle line:84% Using a different logic, we want to couple 00:15:32.920 --> 00:15:36.700 align:middle line:90% the photon and the psi together. 00:15:36.700 --> 00:15:39.550 align:middle line:84% We want to couple A mu and psi together. 00:15:39.550 --> 00:15:43.240 align:middle line:84% And we know that A mu previously has a gauge symmetry. 00:15:43.240 --> 00:15:45.640 align:middle line:84% And now, I want to generalize that gauge symmetry 00:15:45.640 --> 00:15:47.590 align:middle line:90% to include the fermions. 00:15:47.590 --> 00:15:51.190 align:middle line:84% And then I require the full theory 00:15:51.190 --> 00:15:54.340 align:middle line:84% to be invariant under this symmetry. 00:15:54.340 --> 00:15:56.830 align:middle line:84% And this requirement then requires 00:15:56.830 --> 00:16:02.870 align:middle line:84% that the interaction between them must have this form, OK? 00:16:02.870 --> 00:16:04.250 align:middle line:90% Yeah. 00:16:04.250 --> 00:16:07.730 align:middle line:84% Let me write more explicitly the interaction of the form. 00:16:07.730 --> 00:16:11.540 align:middle line:90% Psi bar gamma mu psi A mu, OK? 00:16:11.540 --> 00:16:14.052 align:middle line:84% So that will lead to this kind of interaction. 00:16:14.052 --> 00:16:19.340 align:middle line:90% 00:16:19.340 --> 00:16:22.820 align:middle line:84% So the reason-- even though this is 00:16:22.820 --> 00:16:28.040 align:middle line:84% just rephrase of what we just did, 00:16:28.040 --> 00:16:30.630 align:middle line:84% this rephrase is actually powerful. 00:16:30.630 --> 00:16:31.130 align:middle line:90% OK? 00:16:31.130 --> 00:16:38.860 align:middle line:84% Here it means that when we impose this gauge symmetry, 00:16:38.860 --> 00:16:42.040 align:middle line:84% we actually can deduce the interactions. 00:16:42.040 --> 00:16:45.940 align:middle line:90% 00:16:45.940 --> 00:16:48.550 align:middle line:90% Deduce the interactions, OK? 00:16:48.550 --> 00:16:52.000 align:middle line:84% We can actually deduce the interactions. 00:16:52.000 --> 00:16:54.580 align:middle line:84% We can fix the interactions by requiring 00:16:54.580 --> 00:16:56.240 align:middle line:90% certain kinds of symmetries. 00:16:56.240 --> 00:16:57.070 align:middle line:90% Yes? 00:16:57.070 --> 00:16:59.490 align:middle line:84% AUDIENCE: Is there a natural way to think about 00:16:59.490 --> 00:17:02.596 align:middle line:84% what would motivate you to assert this local gauge 00:17:02.596 --> 00:17:05.265 align:middle line:90% symmetry? 00:17:05.265 --> 00:17:05.890 align:middle line:90% HONG LIU: Yeah. 00:17:05.890 --> 00:17:10.540 align:middle line:90% That's what motivates me. 00:17:10.540 --> 00:17:13.319 align:middle line:84% Maxwell theory-- you already have a gauge symmetry. 00:17:13.319 --> 00:17:15.819 align:middle line:84% And when I couple the fermion, I just want to generalize it. 00:17:15.819 --> 00:17:19.960 align:middle line:90% 00:17:19.960 --> 00:17:21.319 align:middle line:90% OK. 00:17:21.319 --> 00:17:22.036 align:middle line:90% Yes? 00:17:22.036 --> 00:17:24.119 align:middle line:84% AUDIENCE: I guess, instead of a question, I guess, 00:17:24.119 --> 00:17:29.685 align:middle line:84% why is it that we know that it should be a scalar basis? 00:17:29.685 --> 00:17:31.590 align:middle line:84% And why couldn't it be anything else? 00:17:31.590 --> 00:17:32.460 align:middle line:90% HONG LIU: Sorry? 00:17:32.460 --> 00:17:36.318 align:middle line:84% AUDIENCE: I guess why should we know-- 00:17:36.318 --> 00:17:38.068 align:middle line:84% if you start from the gauge symmetry going 00:17:38.068 --> 00:17:40.530 align:middle line:84% to the interactions logic, then why 00:17:40.530 --> 00:17:44.180 align:middle line:84% should we know that it should be a scalar function, that it 00:17:44.180 --> 00:17:46.630 align:middle line:90% does all those [INAUDIBLE]? 00:17:46.630 --> 00:17:51.170 align:middle line:84% Why couldn't it be the [INAUDIBLE]?? 00:17:51.170 --> 00:17:53.210 align:middle line:84% HONG LIU: Because we already have this, right? 00:17:53.210 --> 00:17:56.090 align:middle line:90% We just want to generalize this. 00:17:56.090 --> 00:17:58.090 align:middle line:84% The Maxwell theory-- we already have this. 00:17:58.090 --> 00:18:00.682 align:middle line:84% We just need to generalize this to fermion. 00:18:00.682 --> 00:18:02.390 align:middle line:84% And then there's a natural generalization 00:18:02.390 --> 00:18:04.307 align:middle line:84% because the fermion is already invariant under 00:18:04.307 --> 00:18:06.410 align:middle line:90% the global phase symmetry. 00:18:06.410 --> 00:18:09.020 align:middle line:84% We just need to generalize in the way so to make it local. 00:18:09.020 --> 00:18:12.320 align:middle line:90% 00:18:12.320 --> 00:18:12.820 align:middle line:90% OK. 00:18:12.820 --> 00:18:14.010 align:middle line:90% Yes? 00:18:14.010 --> 00:18:15.870 align:middle line:84% AUDIENCE: So in quantum mechanics, 00:18:15.870 --> 00:18:19.920 align:middle line:84% when we do a single particle in an electromagnetic field, 00:18:19.920 --> 00:18:24.240 align:middle line:84% the gauge transformation on the wave function is the same. 00:18:24.240 --> 00:18:25.920 align:middle line:90% Is there a reason why that's-- 00:18:25.920 --> 00:18:27.090 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:18:27.090 --> 00:18:27.690 align:middle line:90% It's the same. 00:18:27.690 --> 00:18:30.270 align:middle line:90% 00:18:30.270 --> 00:18:31.260 align:middle line:90% Yeah, yeah. 00:18:31.260 --> 00:18:32.010 align:middle line:90% There is a reason. 00:18:32.010 --> 00:18:37.850 align:middle line:90% 00:18:37.850 --> 00:18:38.350 align:middle line:90% Yeah. 00:18:38.350 --> 00:18:49.310 align:middle line:84% The reason is that when you go to non relativistic case, 00:18:49.310 --> 00:18:51.890 align:middle line:84% this reduces just to one particle-- quantum mechanics. 00:18:51.890 --> 00:18:53.325 align:middle line:84% So one particle, quantum mechanics 00:18:53.325 --> 00:18:54.950 align:middle line:84% should also have this kind of symmetry. 00:18:54.950 --> 00:18:55.450 align:middle line:90% Yeah. 00:18:55.450 --> 00:18:58.570 align:middle line:90% 00:18:58.570 --> 00:19:00.860 align:middle line:90% Good. 00:19:00.860 --> 00:19:06.125 align:middle line:84% So this is, in fact, not just a reinterpretation 00:19:06.125 --> 00:19:07.610 align:middle line:84% of what we are looking at so far. 00:19:07.610 --> 00:19:10.730 align:middle line:84% This is actually a very deep dynamical principle, OK? 00:19:10.730 --> 00:19:18.190 align:middle line:84% So this is, in fact, a dynamical-- 00:19:18.190 --> 00:19:20.110 align:middle line:84% a deep-- let me emphasize that this 00:19:20.110 --> 00:19:23.140 align:middle line:90% is a deep, dynamical principle. 00:19:23.140 --> 00:19:31.030 align:middle line:90% 00:19:31.030 --> 00:19:36.490 align:middle line:84% So essentially, you have gauge symmetries, or in other words, 00:19:36.490 --> 00:19:37.720 align:middle line:90% local symmetries. 00:19:37.720 --> 00:19:41.210 align:middle line:90% 00:19:41.210 --> 00:19:45.620 align:middle line:84% Can be used to determine interactions, OK? 00:19:45.620 --> 00:19:50.910 align:middle line:90% 00:19:50.910 --> 00:19:51.420 align:middle line:90% OK. 00:19:51.420 --> 00:19:57.120 align:middle line:84% Or in other words, all interactions in nature-- 00:19:57.120 --> 00:20:00.100 align:middle line:84% turns out, they are related to some gauge symmetries. 00:20:00.100 --> 00:20:04.595 align:middle line:84% So this applies to all interactions in nature. 00:20:04.595 --> 00:20:10.830 align:middle line:90% 00:20:10.830 --> 00:20:17.300 align:middle line:84% So this principle applies to all interactions in nature, OK? 00:20:17.300 --> 00:20:20.540 align:middle line:90% 00:20:20.540 --> 00:20:25.982 align:middle line:84% So here we see, in the electromagnetic interactions, 00:20:25.982 --> 00:20:27.440 align:middle line:84% it turns out the same thing happens 00:20:27.440 --> 00:20:30.050 align:middle line:90% for the weak interaction. 00:20:30.050 --> 00:20:32.480 align:middle line:84% Same thing happens for the strong interaction. 00:20:32.480 --> 00:20:34.850 align:middle line:84% And the same thing happens for gravity. 00:20:34.850 --> 00:20:37.520 align:middle line:84% They all can be formulated as a consequence 00:20:37.520 --> 00:20:42.230 align:middle line:90% of some local symmetries. 00:20:42.230 --> 00:20:43.977 align:middle line:90% And why is that the case? 00:20:43.977 --> 00:20:45.350 align:middle line:90% We don't really know. 00:20:45.350 --> 00:20:48.710 align:middle line:84% We don't really understand why, somehow, 00:20:48.710 --> 00:20:51.590 align:middle line:84% this dynamical principle should be there. 00:20:51.590 --> 00:20:55.550 align:middle line:84% But this is just a fact that all our fundamental interactions-- 00:20:55.550 --> 00:20:58.280 align:middle line:84% they can all be understood this way, OK? 00:20:58.280 --> 00:21:04.430 align:middle line:84% So we roughly understand, but yeah. 00:21:04.430 --> 00:21:07.430 align:middle line:84% But going into that will be a long story. 00:21:07.430 --> 00:21:09.020 align:middle line:90% Yes? 00:21:09.020 --> 00:21:11.060 align:middle line:84% AUDIENCE: So people sometimes say 00:21:11.060 --> 00:21:13.940 align:middle line:84% that gauge invariance is an ambiguity of the theories-- 00:21:13.940 --> 00:21:14.565 align:middle line:90% HONG LIU: Yeah. 00:21:14.565 --> 00:21:17.280 align:middle line:84% AUDIENCE: So you can shift your fields by whatever, 00:21:17.280 --> 00:21:18.870 align:middle line:90% and the theory's invariant. 00:21:18.870 --> 00:21:21.660 align:middle line:84% How can an ambiguity in your description 00:21:21.660 --> 00:21:24.240 align:middle line:84% lead to something very physical, which is being interacted-- 00:21:24.240 --> 00:21:25.080 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:21:25.080 --> 00:21:25.740 align:middle line:90% Exactly. 00:21:25.740 --> 00:21:29.370 align:middle line:84% So we emphasized-- so when you say ambiguity, 00:21:29.370 --> 00:21:33.180 align:middle line:84% this is what we said earlier, that the local symmetries, 00:21:33.180 --> 00:21:36.915 align:middle line:84% or gauge symmetry-- they just tell you your theory 00:21:36.915 --> 00:21:40.750 align:middle line:84% that the degrees freedom are redundant. 00:21:40.750 --> 00:21:41.250 align:middle line:90% OK? 00:21:41.250 --> 00:21:44.130 align:middle line:84% Some of the degrees freedom is not important. 00:21:44.130 --> 00:21:46.380 align:middle line:84% Then you say, this should be pretty artificial, right? 00:21:46.380 --> 00:21:48.838 align:middle line:84% They just tell you you have some redundant degrees freedom. 00:21:48.838 --> 00:21:49.930 align:middle line:90% We just get rid of them. 00:21:49.930 --> 00:21:53.290 align:middle line:84% But it turns out that they actually-- yeah. 00:21:53.290 --> 00:21:53.790 align:middle line:90% Yeah. 00:21:53.790 --> 00:21:57.900 align:middle line:84% So this is part of the mystery, OK? 00:21:57.900 --> 00:21:59.280 align:middle line:90% This is part of the mystery-- 00:21:59.280 --> 00:22:05.350 align:middle line:84% why, somehow, the gauge symmetry leads to the interactions. 00:22:05.350 --> 00:22:05.850 align:middle line:90% Yeah. 00:22:05.850 --> 00:22:08.310 align:middle line:90% So Yeah yeah. 00:22:08.310 --> 00:22:09.480 align:middle line:90% Yeah, it's a long story. 00:22:09.480 --> 00:22:14.970 align:middle line:84% Let me just briefly make some comments here, OK? 00:22:14.970 --> 00:22:19.890 align:middle line:84% So now, we more or less understand 00:22:19.890 --> 00:22:24.050 align:middle line:90% that all these interactions-- 00:22:24.050 --> 00:22:27.440 align:middle line:84% essentially, all these important interactions 00:22:27.440 --> 00:22:30.080 align:middle line:84% in nature, those fundamental interactions-- 00:22:30.080 --> 00:22:35.030 align:middle line:84% they're all governed by some massless particle, OK? 00:22:35.030 --> 00:22:37.640 align:middle line:84% They're all governed by some massless particle. 00:22:37.640 --> 00:22:42.480 align:middle line:84% But somehow, a massless particle-- 00:22:42.480 --> 00:22:46.980 align:middle line:84% you actually require to have some kind of redundancy 00:22:46.980 --> 00:22:50.520 align:middle line:84% so that you can describe them in a Lorentz covariant way. 00:22:50.520 --> 00:22:53.790 align:middle line:84% Without those redundancy, you cannot describe them 00:22:53.790 --> 00:22:55.270 align:middle line:90% in the Lorentz covariant way. 00:22:55.270 --> 00:22:58.060 align:middle line:84% So in the sense, that's where that comes from. 00:22:58.060 --> 00:22:58.620 align:middle line:90% OK? 00:22:58.620 --> 00:22:59.200 align:middle line:90% Yeah. 00:22:59.200 --> 00:23:02.640 align:middle line:84% Yeah, but you make the statement which I just said. 00:23:02.640 --> 00:23:06.430 align:middle line:84% Precisely, it's actually a long story. 00:23:06.430 --> 00:23:08.490 align:middle line:90% Do you have any questions? 00:23:08.490 --> 00:23:09.450 align:middle line:90% Yes? 00:23:09.450 --> 00:23:11.492 align:middle line:84% AUDIENCE: Would it have been possible to quantize 00:23:11.492 --> 00:23:13.914 align:middle line:84% the Maxwell theory just using the E field and the B field 00:23:13.914 --> 00:23:19.650 align:middle line:84% without involving any kind of gauge? 00:23:19.650 --> 00:23:22.770 align:middle line:90% HONG LIU: We don't know how to-- 00:23:22.770 --> 00:23:24.720 align:middle line:90% so we don't know how to-- 00:23:24.720 --> 00:23:30.360 align:middle line:84% there's no good way to quantize that to give you a nice quantum 00:23:30.360 --> 00:23:30.960 align:middle line:90% theory. 00:23:30.960 --> 00:23:36.270 align:middle line:84% And we also know that physically, A mu actually plays 00:23:36.270 --> 00:23:37.470 align:middle line:90% a role. 00:23:37.470 --> 00:23:41.555 align:middle line:84% And in situations like a Aharonov--Bohm effect, 00:23:41.555 --> 00:23:42.930 align:middle line:84% the A mu plays an important role, 00:23:42.930 --> 00:23:44.790 align:middle line:90% even when E and B is equal to 0. 00:23:44.790 --> 00:23:49.890 align:middle line:84% So A mu is a more fundamental object than E and B. 00:23:49.890 --> 00:23:54.780 align:middle line:84% So E and B should be considered as a derived object from A mu. 00:23:54.780 --> 00:23:59.950 align:middle line:90% 00:23:59.950 --> 00:24:00.670 align:middle line:90% Other questions? 00:24:00.670 --> 00:24:01.240 align:middle line:90% Yes? 00:24:01.240 --> 00:24:04.877 align:middle line:84% AUDIENCE: So here it was quite natural to extend to fermions 00:24:04.877 --> 00:24:08.782 align:middle line:84% to just have to identify-- the symmetry is already there. 00:24:08.782 --> 00:24:10.510 align:middle line:84% You didn't have to engineer it, I guess. 00:24:10.510 --> 00:24:11.135 align:middle line:90% HONG LIU: Yeah. 00:24:11.135 --> 00:24:13.880 align:middle line:84% AUDIENCE: But is there a situation where 00:24:13.880 --> 00:24:16.600 align:middle line:84% extending this symmetry to another field 00:24:16.600 --> 00:24:20.141 align:middle line:84% causes an issue when you're trying to fix your gauge like 00:24:20.141 --> 00:24:21.780 align:middle line:90% you had before-- 00:24:21.780 --> 00:24:24.550 align:middle line:84% increase your redundancy, I guess, 00:24:24.550 --> 00:24:28.550 align:middle line:84% when you're making the [INAUDIBLE]?? 00:24:28.550 --> 00:24:30.740 align:middle line:84% HONG LIU: I'm not sure I understand your question. 00:24:30.740 --> 00:24:33.010 align:middle line:84% AUDIENCE: So here we had the issue 00:24:33.010 --> 00:24:36.262 align:middle line:90% in gauge fixing was from-- 00:24:36.262 --> 00:24:36.970 align:middle line:90% HONG LIU: No, no. 00:24:36.970 --> 00:24:38.800 align:middle line:84% We're not doing any gauge fixing here. 00:24:38.800 --> 00:24:44.470 align:middle line:84% We just have a more generalized gauge symmetry. 00:24:44.470 --> 00:24:46.320 align:middle line:84% We're not doing any gauge fixing. 00:24:46.320 --> 00:24:47.195 align:middle line:90% AUDIENCE: No, no, no. 00:24:47.195 --> 00:24:50.588 align:middle line:84% I'm saying, so that came out of the local gauge symmetry 00:24:50.588 --> 00:24:52.040 align:middle line:90% of our A mu's. 00:24:52.040 --> 00:24:55.515 align:middle line:84% But I'm saying when you extend that to the psi, for example-- 00:24:55.515 --> 00:24:56.140 align:middle line:90% HONG LIU: Yeah. 00:24:56.140 --> 00:24:57.740 align:middle line:84% AUDIENCE: --you get any other issues 00:24:57.740 --> 00:24:59.910 align:middle line:90% like we had from earlier or no? 00:24:59.910 --> 00:25:01.770 align:middle line:84% HONG LIU: What do you mean by other issues? 00:25:01.770 --> 00:25:04.365 align:middle line:84% AUDIENCE: So issues when you're fixing your gauge. 00:25:04.365 --> 00:25:05.948 align:middle line:84% Or is that just a step that we already 00:25:05.948 --> 00:25:07.160 align:middle line:84% did before that we don't have to worry about? 00:25:07.160 --> 00:25:08.930 align:middle line:90% HONG LIU: Oh, oh, oh, oh, oh. 00:25:08.930 --> 00:25:09.900 align:middle line:90% You mean when we-- 00:25:09.900 --> 00:25:10.940 align:middle line:90% no, no. 00:25:10.940 --> 00:25:14.450 align:middle line:84% That won't change much because we did before 00:25:14.450 --> 00:25:16.080 align:middle line:90% for the gauge fixing already. 00:25:16.080 --> 00:25:16.670 align:middle line:90% It's enough. 00:25:16.670 --> 00:25:17.870 align:middle line:90% Yeah. 00:25:17.870 --> 00:25:18.370 align:middle line:90% Yeah. 00:25:18.370 --> 00:25:21.210 align:middle line:90% 00:25:21.210 --> 00:25:22.580 align:middle line:90% OK? 00:25:22.580 --> 00:25:26.100 align:middle line:84% So now, let's talk about the other example. 00:25:26.100 --> 00:25:34.130 align:middle line:84% So if you have a scalar field which is charged-- 00:25:34.130 --> 00:25:36.660 align:middle line:90% say, complex scalar field-- 00:25:36.660 --> 00:25:43.730 align:middle line:84% you can also couple it to the photon, OK? 00:25:43.730 --> 00:25:47.150 align:middle line:84% That's what we did for the fermions, OK? 00:25:47.150 --> 00:25:51.767 align:middle line:84% So let's consider if you have a complex scalar. 00:25:51.767 --> 00:25:53.350 align:middle line:84% So consider you have a complex scalar. 00:25:53.350 --> 00:26:03.310 align:middle line:90% 00:26:03.310 --> 00:26:03.810 align:middle line:90% Yeah. 00:26:03.810 --> 00:26:09.030 align:middle line:84% Maybe you can also have some interaction terms, OK? 00:26:09.030 --> 00:26:14.080 align:middle line:84% So let's imagine we have some complex scalar. 00:26:14.080 --> 00:26:18.180 align:middle line:84% And so this is invariant under phi. 00:26:18.180 --> 00:26:20.250 align:middle line:84% Again, there's a global symmetry. 00:26:20.250 --> 00:26:22.710 align:middle line:90% U1 global symmetry. 00:26:22.710 --> 00:26:23.730 align:middle line:90% Alpha equal to constant. 00:26:23.730 --> 00:26:26.330 align:middle line:90% 00:26:26.330 --> 00:26:27.830 align:middle line:90% OK? 00:26:27.830 --> 00:26:33.200 align:middle line:84% And then we have a conserved current here-- 00:26:33.200 --> 00:26:35.780 align:middle line:90% J mu equal to minus i. 00:26:35.780 --> 00:26:37.730 align:middle line:90% So again, your Noether current. 00:26:37.730 --> 00:26:47.700 align:middle line:90% 00:26:47.700 --> 00:26:49.830 align:middle line:90% OK. 00:26:49.830 --> 00:26:57.880 align:middle line:84% So now, you can just try to do the same trick, OK? 00:26:57.880 --> 00:27:00.250 align:middle line:84% So let's just couple this theory, 00:27:00.250 --> 00:27:03.100 align:middle line:84% combine this theory to the Maxwell theory, 00:27:03.100 --> 00:27:05.470 align:middle line:90% but identify that J with this J. 00:27:05.470 --> 00:27:10.210 align:middle line:84% So naively-- so the reason I say naively 00:27:10.210 --> 00:27:14.450 align:middle line:84% is because this procedure in this case will not work. 00:27:14.450 --> 00:27:19.810 align:middle line:84% So now, we can consider the scalar QED, say, 00:27:19.810 --> 00:27:24.280 align:middle line:84% with photon couple, and again, with this minus 1/4 00:27:24.280 --> 00:27:25.400 align:middle line:90% F square term. 00:27:25.400 --> 00:27:27.640 align:middle line:84% So let me just write the simplified version-- 00:27:27.640 --> 00:27:28.990 align:middle line:90% just write F square. 00:27:28.990 --> 00:27:34.930 align:middle line:84% And then you can add this scalar Lagrangian, OK? 00:27:34.930 --> 00:27:38.680 align:middle line:84% And now, you add the coupling E. And then 00:27:38.680 --> 00:27:42.835 align:middle line:84% you can put A mu equal to J mu, but with J mu given by this. 00:27:42.835 --> 00:27:45.640 align:middle line:90% 00:27:45.640 --> 00:27:47.470 align:middle line:90% OK? 00:27:47.470 --> 00:27:49.360 align:middle line:90% We say A mu given by this, OK? 00:27:49.360 --> 00:27:54.520 align:middle line:84% So this is the natural thing to do based on our fermion story. 00:27:54.520 --> 00:28:07.770 align:middle line:90% 00:28:07.770 --> 00:28:09.030 align:middle line:90% But now, you can check. 00:28:09.030 --> 00:28:12.060 align:middle line:90% 00:28:12.060 --> 00:28:13.440 align:middle line:90% But now, you can easily check. 00:28:13.440 --> 00:28:19.230 align:middle line:90% 00:28:19.230 --> 00:28:19.730 align:middle line:90% OK? 00:28:19.730 --> 00:28:21.752 align:middle line:84% So I will not do the exact check here. 00:28:21.752 --> 00:28:22.835 align:middle line:90% You should check yourself. 00:28:22.835 --> 00:28:26.150 align:middle line:90% 00:28:26.150 --> 00:28:28.700 align:middle line:90% So let me call this L prime, OK? 00:28:28.700 --> 00:28:31.370 align:middle line:84% L tilde because this is our naive theory. 00:28:31.370 --> 00:28:39.980 align:middle line:84% You can check that this L tilde QED, scalar QED, 00:28:39.980 --> 00:28:41.570 align:middle line:90% is not invariant-- 00:28:41.570 --> 00:28:45.480 align:middle line:90% 00:28:45.480 --> 00:28:52.160 align:middle line:90% gauge invariant, OK? 00:28:52.160 --> 00:29:00.510 align:middle line:84% Means that you cannot find the transformation on phi, OK? 00:29:00.510 --> 00:29:02.540 align:middle line:84% That this whole thing is gauge invariant. 00:29:02.540 --> 00:29:03.040 align:middle line:90% OK? 00:29:03.040 --> 00:29:05.080 align:middle line:84% This whole thing is gauge invariant. 00:29:05.080 --> 00:29:11.540 align:middle line:84% And so for example, so it's not gauge invariant. 00:29:11.540 --> 00:29:19.020 align:middle line:84% When we can check explicitly, if you just copy what we do here, 00:29:19.020 --> 00:29:29.680 align:middle line:84% this phi x given by ie lambda x phi x-- 00:29:29.680 --> 00:29:31.780 align:middle line:90% so this won't work, OK? 00:29:31.780 --> 00:29:34.160 align:middle line:90% So it's not invariant. 00:29:34.160 --> 00:29:35.880 align:middle line:90% OK. 00:29:35.880 --> 00:29:36.630 align:middle line:90% Yes? 00:29:36.630 --> 00:29:37.297 align:middle line:90% AUDIENCE: Sorry. 00:29:37.297 --> 00:29:39.677 align:middle line:84% Can you just leave phi the same when you do-- 00:29:39.677 --> 00:29:41.510 align:middle line:84% HONG LIU: No, you cannot leave phi the same. 00:29:41.510 --> 00:29:44.750 align:middle line:90% 00:29:44.750 --> 00:29:46.890 align:middle line:84% Yeah, that's a very good question. 00:29:46.890 --> 00:29:50.870 align:middle line:84% So you cannot leave phi the same. 00:29:50.870 --> 00:29:54.590 align:middle line:90% 00:29:54.590 --> 00:29:56.360 align:middle line:84% So naively, you can ask yourself-- 00:29:56.360 --> 00:29:59.550 align:middle line:84% you say this is supposed to be conserved. 00:29:59.550 --> 00:30:01.505 align:middle line:84% And we previously said if we have 00:30:01.505 --> 00:30:05.100 align:middle line:84% A mu to a conserved current, this is gauge invariant. 00:30:05.100 --> 00:30:10.320 align:middle line:84% But this is conserved only when you use the equation of motion 00:30:10.320 --> 00:30:12.060 align:middle line:90% of this theory. 00:30:12.060 --> 00:30:14.910 align:middle line:84% But now, when you couple these two to this theory, 00:30:14.910 --> 00:30:17.610 align:middle line:84% then the equation of motion for phi changed. 00:30:17.610 --> 00:30:20.940 align:middle line:90% And then so yeah. 00:30:20.940 --> 00:30:22.770 align:middle line:90% So it no longer works, OK? 00:30:22.770 --> 00:30:26.040 align:middle line:90% And so just naively, do this. 00:30:26.040 --> 00:30:28.560 align:middle line:84% Naively, this is not guaranteed to work. 00:30:28.560 --> 00:30:30.810 align:middle line:84% It turns out in the fermion case, 00:30:30.810 --> 00:30:34.592 align:middle line:84% it just worked because of the existence of this symmetry, 00:30:34.592 --> 00:30:36.300 align:middle line:84% because of the existence of the symmetry. 00:30:36.300 --> 00:30:38.920 align:middle line:90% 00:30:38.920 --> 00:30:41.860 align:middle line:90% And it turns out, this theory-- 00:30:41.860 --> 00:30:45.050 align:middle line:84% so now, this theory will not be gauge invariant. 00:30:45.050 --> 00:30:48.090 align:middle line:90% And now, this would be bad, OK? 00:30:48.090 --> 00:30:51.210 align:middle line:84% It will be bad because you can no longer 00:30:51.210 --> 00:30:53.670 align:middle line:90% find any local symmetry. 00:30:53.670 --> 00:30:57.480 align:middle line:90% This theory is invariant, OK? 00:30:57.480 --> 00:31:00.870 align:middle line:84% And this is one where the generalization does not work. 00:31:00.870 --> 00:31:02.850 align:middle line:90% You cannot find anything. 00:31:02.850 --> 00:31:04.830 align:middle line:90% You can also not find others. 00:31:04.830 --> 00:31:08.760 align:middle line:84% But this is bad because we said we need gauge symmetry 00:31:08.760 --> 00:31:12.150 align:middle line:84% to get rid of unphysical degrees freedom in A mu 00:31:12.150 --> 00:31:14.370 align:middle line:84% because A mu should only have two polarizations, 00:31:14.370 --> 00:31:15.210 align:middle line:90% not to have four. 00:31:15.210 --> 00:31:16.950 align:middle line:84% So we need those gauge symmetries 00:31:16.950 --> 00:31:20.100 align:middle line:84% to get rid of the unphysical degrees freedom in A mu. 00:31:20.100 --> 00:31:22.410 align:middle line:84% But if this gauge symmetry is broken by coupling 00:31:22.410 --> 00:31:26.590 align:middle line:84% to the scalar field, then this theory cannot be consistent, 00:31:26.590 --> 00:31:27.090 align:middle line:90% OK? 00:31:27.090 --> 00:31:29.850 align:middle line:90% So this theory is bad. 00:31:29.850 --> 00:31:31.570 align:middle line:90% OK. 00:31:31.570 --> 00:31:33.460 align:middle line:90% This theory is bad. 00:31:33.460 --> 00:31:38.920 align:middle line:84% And it turns out, we can just generalize this principle, OK? 00:31:38.920 --> 00:31:42.850 align:middle line:84% We say let's just impose gauge symmetry, OK? 00:31:42.850 --> 00:31:46.020 align:middle line:84% We can just impose gauge symmetry. 00:31:46.020 --> 00:31:48.300 align:middle line:90% So the way out-- 00:31:48.300 --> 00:31:50.025 align:middle line:84% we can just impose the gauge symmetry. 00:31:50.025 --> 00:31:52.920 align:middle line:90% 00:31:52.920 --> 00:32:09.320 align:middle line:84% Impose the theory to be invariant under this A mu, 00:32:09.320 --> 00:32:13.370 align:middle line:84% goes to A mu plus partial mu lambda. 00:32:13.370 --> 00:32:22.150 align:middle line:84% And the phi goes to exponential minus ie lambda x phi, OK? 00:32:22.150 --> 00:32:28.830 align:middle line:84% So we require this to be under that, OK? 00:32:28.830 --> 00:32:33.870 align:middle line:84% So to write the theory invariant under this, 00:32:33.870 --> 00:32:38.870 align:middle line:84% we can just easily borrow the inside we already obtained 00:32:38.870 --> 00:32:41.940 align:middle line:90% from the fermionic case. 00:32:41.940 --> 00:32:43.380 align:middle line:84% But in the fermionic case, we were 00:32:43.380 --> 00:32:48.330 align:middle line:84% told that this derivative, this capital D, 00:32:48.330 --> 00:32:52.690 align:middle line:84% actually transforms nicely under such transformation. 00:32:52.690 --> 00:32:54.085 align:middle line:84% When we make such transformation, 00:32:54.085 --> 00:32:56.210 align:middle line:84% whether psi is a fermion or boson, and whether it's 00:32:56.210 --> 00:32:59.610 align:middle line:84% a scalar or spinor does not play any role, OK? 00:32:59.610 --> 00:33:02.040 align:middle line:84% So we can now just introduce D phi. 00:33:02.040 --> 00:33:06.290 align:middle line:90% 00:33:06.290 --> 00:33:12.430 align:middle line:84% We can just introduce D phi equal to partial mu phi 00:33:12.430 --> 00:33:15.430 align:middle line:90% minus ie A mu phi, OK? 00:33:15.430 --> 00:33:18.340 align:middle line:84% We can similarly introduce D phi like this. 00:33:18.340 --> 00:33:21.880 align:middle line:84% And now, D phi then transforms nicely 00:33:21.880 --> 00:33:23.860 align:middle line:84% under this kind of transformation, OK? 00:33:23.860 --> 00:33:34.855 align:middle line:90% So D phi-- to mu phi, OK? 00:33:34.855 --> 00:33:36.700 align:middle line:90% They transform nicely. 00:33:36.700 --> 00:33:38.680 align:middle line:84% And since this transforms nicely, 00:33:38.680 --> 00:33:40.780 align:middle line:84% then we can easily write down the Lagrangian, OK? 00:33:40.780 --> 00:33:45.700 align:middle line:84% So the L scalar QED then can be written 00:33:45.700 --> 00:33:50.370 align:middle line:90% as just minus 1/4 F squared. 00:33:50.370 --> 00:33:56.210 align:middle line:84% And now, instead of the standard derivative, 00:33:56.210 --> 00:33:59.360 align:middle line:90% I use this derivative. 00:33:59.360 --> 00:34:03.850 align:middle line:84% I use this capital D derivative, OK? 00:34:03.850 --> 00:34:07.330 align:middle line:84% And then I have m square phi star. 00:34:07.330 --> 00:34:10.900 align:middle line:84% And then I can have phi phi star, OK? 00:34:10.900 --> 00:34:13.510 align:middle line:84% And so this will be invariant because those 00:34:13.510 --> 00:34:16.300 align:middle line:84% don't matter because those don't involve derivatives, OK? 00:34:16.300 --> 00:34:20.710 align:middle line:90% And so yeah. 00:34:20.710 --> 00:34:24.011 align:middle line:90% So this is now gauge invariant. 00:34:24.011 --> 00:34:32.670 align:middle line:90% 00:34:32.670 --> 00:34:33.270 align:middle line:90% OK? 00:34:33.270 --> 00:34:40.440 align:middle line:84% So this D mu is called covariant derivative, OK? 00:34:40.440 --> 00:34:45.330 align:middle line:84% Becaues it transforms nicely under this gauge 00:34:45.330 --> 00:34:46.179 align:middle line:90% transformation. 00:34:46.179 --> 00:34:48.108 align:middle line:84% So it's called covariant derivative. 00:34:48.108 --> 00:34:56.590 align:middle line:90% 00:34:56.590 --> 00:35:02.370 align:middle line:84% So this capital D mu also has a very deep connection 00:35:02.370 --> 00:35:04.530 align:middle line:90% to mathematics. 00:35:04.530 --> 00:35:07.080 align:middle line:84% And so this is related to a subject, 00:35:07.080 --> 00:35:09.300 align:middle line:90% say, in differential geometry-- 00:35:09.300 --> 00:35:14.910 align:middle line:84% a deep connection to differential geometry, 00:35:14.910 --> 00:35:18.030 align:middle line:84% to a thing in differential geometry 00:35:18.030 --> 00:35:20.580 align:middle line:90% called the connection, OK? 00:35:20.580 --> 00:35:21.660 align:middle line:90% Called the connection. 00:35:21.660 --> 00:35:22.530 align:middle line:90% And yeah, sorry. 00:35:22.530 --> 00:35:24.428 align:middle line:90% We're not going into that. 00:35:24.428 --> 00:35:24.928 align:middle line:90% Yeah. 00:35:24.928 --> 00:35:29.380 align:middle line:90% 00:35:29.380 --> 00:35:36.520 align:middle line:84% So now, let's compare this theory with that naive theory. 00:35:36.520 --> 00:35:43.750 align:middle line:84% So you can check that L SQED is almost 00:35:43.750 --> 00:35:51.310 align:middle line:84% that L tilde SQED except this has one more term. 00:35:51.310 --> 00:35:57.520 align:middle line:84% So the difference with the fermion case-- 00:35:57.520 --> 00:36:00.680 align:middle line:84% the fermion only has one derivative, OK? 00:36:00.680 --> 00:36:04.310 align:middle line:84% And this one does not have a derivative, OK? 00:36:04.310 --> 00:36:07.850 align:middle line:84% And here you have two derivatives. 00:36:07.850 --> 00:36:13.010 align:middle line:84% And each derivative gives you additional A mu, OK? 00:36:13.010 --> 00:36:18.770 align:middle line:84% So when you expand this, so here there are four terms, OK? 00:36:18.770 --> 00:36:20.280 align:middle line:90% There are four terms. 00:36:20.280 --> 00:36:25.550 align:middle line:84% And three terms are encoded in here, OK? 00:36:25.550 --> 00:36:27.900 align:middle line:84% Three terms are encoded in here and in here. 00:36:27.900 --> 00:36:29.780 align:middle line:90% But there's one more term, OK? 00:36:29.780 --> 00:36:31.550 align:middle line:84% It turns out this leaves one more term. 00:36:31.550 --> 00:36:37.810 align:middle line:90% 00:36:37.810 --> 00:36:40.100 align:middle line:90% OK. 00:36:40.100 --> 00:36:45.780 align:middle line:84% So actually, there's a quartic inaction between A mu and phi, 00:36:45.780 --> 00:36:46.280 align:middle line:90% OK? 00:36:46.280 --> 00:36:49.590 align:middle line:84% So this only introduces cubic interaction. 00:36:49.590 --> 00:36:51.787 align:middle line:84% So if you just have this J mu A mu, 00:36:51.787 --> 00:36:53.120 align:middle line:90% you only have cubic interaction. 00:36:53.120 --> 00:36:58.040 align:middle line:84% You only have 2 phi and 1 A. But here because of this structure, 00:36:58.040 --> 00:36:59.825 align:middle line:90% you actually have 2 A. 00:36:59.825 --> 00:37:02.980 align:middle line:84% Also, a term with 2A and 2 phi, OK? 00:37:02.980 --> 00:37:06.730 align:middle line:90% 00:37:06.730 --> 00:37:09.760 align:middle line:84% And when you add this term, then the whole thing is nice, OK? 00:37:09.760 --> 00:37:15.620 align:middle line:90% 00:37:15.620 --> 00:37:16.120 align:middle line:90% Good. 00:37:16.120 --> 00:37:18.980 align:middle line:90% Any questions? 00:37:18.980 --> 00:37:19.730 align:middle line:90% Yes? 00:37:19.730 --> 00:37:21.626 align:middle line:84% AUDIENCE: Can I also add higher order terms 00:37:21.626 --> 00:37:25.643 align:middle line:84% like these quartic terms into the fermionic theory? 00:37:25.643 --> 00:37:26.310 align:middle line:90% HONG LIU: Sorry. 00:37:26.310 --> 00:37:27.000 align:middle line:90% Say it again. 00:37:27.000 --> 00:37:30.690 align:middle line:84% AUDIENCE: Can I also add quartic terms or some higher order 00:37:30.690 --> 00:37:33.895 align:middle line:84% terms into the QED -- fermionic theory? 00:37:33.895 --> 00:37:34.770 align:middle line:90% HONG LIU: No, no, no. 00:37:34.770 --> 00:37:39.420 align:middle line:84% You cannot add this term because this term by itself is not 00:37:39.420 --> 00:37:41.070 align:middle line:90% gauge invariant. 00:37:41.070 --> 00:37:43.770 align:middle line:84% Fermionic term is already gauge invariant by itself-- 00:37:43.770 --> 00:37:46.580 align:middle line:84% because there's no derivative here. 00:37:46.580 --> 00:37:50.430 align:middle line:90% And so the structure is simpler. 00:37:50.430 --> 00:37:51.000 align:middle line:90% Yes? 00:37:51.000 --> 00:37:52.875 align:middle line:84% AUDIENCE: So I guess to add to that question, 00:37:52.875 --> 00:37:55.180 align:middle line:84% can you carefully engineer higher order terms such that 00:37:55.180 --> 00:37:55.680 align:middle line:90% they're-- 00:37:55.680 --> 00:37:56.555 align:middle line:90% HONG LIU: Yeah, yeah. 00:37:56.555 --> 00:37:59.460 align:middle line:84% Yeah, you can engineer more complicated terms, but not 00:37:59.460 --> 00:38:00.930 align:middle line:90% this kind of term. 00:38:00.930 --> 00:38:01.650 align:middle line:90% Yeah. 00:38:01.650 --> 00:38:03.655 align:middle line:90% AUDIENCE: This is required for-- 00:38:03.655 --> 00:38:04.530 align:middle line:90% HONG LIU: Yeah, yeah. 00:38:04.530 --> 00:38:06.562 align:middle line:84% This is required for gauge invariance. 00:38:06.562 --> 00:38:07.990 align:middle line:90% Yeah. 00:38:07.990 --> 00:38:09.010 align:middle line:90% Yeah. 00:38:09.010 --> 00:38:11.560 align:middle line:84% For both fermion and the scalar case, 00:38:11.560 --> 00:38:14.830 align:middle line:84% you can write down more complicated terms 00:38:14.830 --> 00:38:16.330 align:middle line:90% which are gauge invariant. 00:38:16.330 --> 00:38:22.850 align:middle line:84% And the one we wrote down so far is just the simplest one. 00:38:22.850 --> 00:38:23.350 align:middle line:90% Yeah. 00:38:23.350 --> 00:38:24.010 align:middle line:90% Simplest ones. 00:38:24.010 --> 00:38:24.775 align:middle line:90% Yes? 00:38:24.775 --> 00:38:27.150 align:middle line:84% AUDIENCE: So which one should of these should I interpret 00:38:27.150 --> 00:38:28.180 align:middle line:90% as the electric current? 00:38:28.180 --> 00:38:32.650 align:middle line:84% J mu there or I guess J mu plus the A mu psi psi psi? 00:38:32.650 --> 00:38:33.850 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:38:33.850 --> 00:38:39.160 align:middle line:84% So still, so when you interpret the electric current, 00:38:39.160 --> 00:38:40.070 align:middle line:90% it's still this J mu. 00:38:40.070 --> 00:38:40.570 align:middle line:90% Yeah. 00:38:40.570 --> 00:38:47.700 align:middle line:90% 00:38:47.700 --> 00:38:50.190 align:middle line:90% Other questions? 00:38:50.190 --> 00:38:52.130 align:middle line:90% OK, great. 00:38:52.130 --> 00:38:56.110 align:middle line:84% So now, so let me just make a comment. 00:38:56.110 --> 00:39:04.795 align:middle line:84% So in this story, now the global charge before now becomes the-- 00:39:04.795 --> 00:39:06.670 align:middle line:84% just to emphasize again-- in the scalar case, 00:39:06.670 --> 00:39:10.370 align:middle line:84% again, the global charge for this global current 00:39:10.370 --> 00:39:13.180 align:middle line:84% has now become the electric charge 00:39:13.180 --> 00:39:16.270 align:middle line:84% coupled to the electro- magnetic fields, OK? 00:39:16.270 --> 00:39:19.150 align:middle line:84% Become electric charge coupled to electro- magnetic field, OK? 00:39:19.150 --> 00:39:21.900 align:middle line:90% 00:39:21.900 --> 00:39:22.630 align:middle line:90% OK, good. 00:39:22.630 --> 00:39:26.430 align:middle line:84% So let's draw the Feynman rules, OK? 00:39:26.430 --> 00:39:29.535 align:middle line:84% So from here we can just talk about the Feynman rules. 00:39:29.535 --> 00:39:39.830 align:middle line:90% 00:39:39.830 --> 00:39:43.210 align:middle line:84% So let's first do the fermionic case. 00:39:43.210 --> 00:39:45.680 align:middle line:84% So the rules for fermions is the same as before. 00:39:45.680 --> 00:39:48.054 align:middle line:84% The only thing we just need to write down-- 00:39:48.054 --> 00:39:56.960 align:middle line:84% the propagators are the same as previously for the Maxwell 00:39:56.960 --> 00:39:57.460 align:middle line:90% theory. 00:39:57.460 --> 00:40:00.070 align:middle line:90% 00:40:00.070 --> 00:40:02.770 align:middle line:90% Yeah, let's first do the QED. 00:40:02.770 --> 00:40:12.300 align:middle line:84% And the propagator for A mu and phi-- 00:40:12.300 --> 00:40:14.730 align:middle line:90% it's the same as before, OK? 00:40:14.730 --> 00:40:16.320 align:middle line:84% Because the propagator only cares 00:40:16.320 --> 00:40:17.580 align:middle line:90% about the quadratic theory. 00:40:17.580 --> 00:40:19.560 align:middle line:84% It doesn't care about interactions. 00:40:19.560 --> 00:40:23.100 align:middle line:90% Same as before. 00:40:23.100 --> 00:40:28.160 align:middle line:84% So we can draw a wavy line corresponding to A mu, OK? 00:40:28.160 --> 00:40:34.490 align:middle line:84% And then the solid line corresponding to, say, psi, OK? 00:40:34.490 --> 00:40:35.960 align:middle line:90% Solid line equals 1 and 2 psi. 00:40:35.960 --> 00:40:38.630 align:middle line:84% So this is related to A mu propagator. 00:40:38.630 --> 00:40:42.920 align:middle line:84% This is related to psi propagator, OK? 00:40:42.920 --> 00:40:44.820 align:middle line:84% And now, the interaction between them-- 00:40:44.820 --> 00:40:49.850 align:middle line:84% this interaction between them is given by this term. 00:40:49.850 --> 00:40:51.680 align:middle line:84% And then these have the structure. 00:40:51.680 --> 00:40:54.440 align:middle line:90% 00:40:54.440 --> 00:40:57.780 align:middle line:90% You have one fermion come in. 00:40:57.780 --> 00:41:00.750 align:middle line:84% You have two fermions and then coupled to a photon, OK? 00:41:00.750 --> 00:41:03.660 align:middle line:84% So this is the same, very similar to this Yukawa 00:41:03.660 --> 00:41:04.560 align:middle line:90% coupling. 00:41:04.560 --> 00:41:06.300 align:middle line:90% We can that before-- 00:41:06.300 --> 00:41:08.850 align:middle line:84% just now, this one becomes a photon, OK? 00:41:08.850 --> 00:41:10.920 align:middle line:90% This one becomes a photon. 00:41:10.920 --> 00:41:12.780 align:middle line:84% And this effective vertex is given 00:41:12.780 --> 00:41:19.610 align:middle line:90% by minus ie A mu gamma mu, OK? 00:41:19.610 --> 00:41:26.020 align:middle line:84% So the minus comes from minus here. 00:41:26.020 --> 00:41:28.540 align:middle line:84% The i comes from the i in the action, OK? 00:41:28.540 --> 00:41:30.160 align:middle line:84% When you do the passing equal action. 00:41:30.160 --> 00:41:31.810 align:middle line:90% And you have minus ie. 00:41:31.810 --> 00:41:37.040 align:middle line:84% And then phi, psi, and A mu are taken care of by those lines. 00:41:37.040 --> 00:41:39.700 align:middle line:84% And then you only have a gamma mu, OK? 00:41:39.700 --> 00:41:43.780 align:middle line:84% So the vertex here is the gamma mu. 00:41:43.780 --> 00:41:44.830 align:middle line:90% But pay attention. 00:41:44.830 --> 00:41:48.140 align:middle line:84% This gamma mu is actually a matrix in the spinor space. 00:41:48.140 --> 00:41:51.310 align:middle line:84% And so you have to be careful when you contract to the spinor 00:41:51.310 --> 00:41:54.850 align:middle line:84% indices because now, these have spinor indices. 00:41:54.850 --> 00:41:58.220 align:middle line:90% And it's a matrix, OK? 00:41:58.220 --> 00:41:58.720 align:middle line:90% Good? 00:41:58.720 --> 00:42:04.300 align:middle line:90% So this is for the fermion. 00:42:04.300 --> 00:42:05.640 align:middle line:90% And oh, yeah. 00:42:05.640 --> 00:42:09.250 align:middle line:90% So I should also mention-- 00:42:09.250 --> 00:42:21.980 align:middle line:84% for the external legs, so this is for the vertices 00:42:21.980 --> 00:42:23.120 align:middle line:90% and the propagator. 00:42:23.120 --> 00:42:25.750 align:middle line:90% 00:42:25.750 --> 00:42:28.240 align:middle line:84% And if we consider the scattering amplitude-- 00:42:28.240 --> 00:42:36.450 align:middle line:90% 00:42:36.450 --> 00:42:39.470 align:middle line:84% so for scattering amplitude, we mentioned it before. 00:42:39.470 --> 00:42:44.190 align:middle line:84% For fermion, we need to include the polarization vector. 00:42:44.190 --> 00:42:46.370 align:middle line:84% Similarly, for the photon, we also 00:42:46.370 --> 00:42:54.180 align:middle line:84% need to include the polarization vector for the external legs, 00:42:54.180 --> 00:42:54.680 align:middle line:90% OK? 00:42:54.680 --> 00:43:01.140 align:middle line:84% So for external legs, for scattering amplitude-- 00:43:01.140 --> 00:43:11.020 align:middle line:90% 00:43:11.020 --> 00:43:14.560 align:middle line:84% yeah, if you calculate the green function, it doesn't matter. 00:43:14.560 --> 00:43:16.870 align:middle line:84% You just include the external propagator. 00:43:16.870 --> 00:43:18.940 align:middle line:84% But for the scattering amplitude, 00:43:18.940 --> 00:43:22.720 align:middle line:84% then we need to include the polarization vector 00:43:22.720 --> 00:43:23.900 align:middle line:90% for photons. 00:43:23.900 --> 00:43:29.170 align:middle line:84% So for the initial state, supposing the initial state, 00:43:29.170 --> 00:43:33.100 align:middle line:84% we have, say, some vector k and alpha. 00:43:33.100 --> 00:43:36.560 align:middle line:84% So alpha gives its polarization vector. 00:43:36.560 --> 00:43:38.410 align:middle line:90% And k is the momentum. 00:43:38.410 --> 00:43:41.450 align:middle line:90% And then we can denote it as-- 00:43:41.450 --> 00:43:46.070 align:middle line:84% so alpha denotes the polarization for the-- 00:43:46.070 --> 00:43:47.870 align:middle line:90% and then say we can have k. 00:43:47.870 --> 00:43:50.490 align:middle line:90% So this is the momentum index. 00:43:50.490 --> 00:43:52.760 align:middle line:84% So this is the momentum arrow, OK? 00:43:52.760 --> 00:43:55.730 align:middle line:84% And so this [INAUDIBLE] corresponding to-- 00:43:55.730 --> 00:43:59.660 align:middle line:90% 00:43:59.660 --> 00:44:00.290 align:middle line:90% OK. 00:44:00.290 --> 00:44:01.610 align:middle line:90% OK? 00:44:01.610 --> 00:44:03.710 align:middle line:84% And so this is for the initial state. 00:44:03.710 --> 00:44:10.660 align:middle line:84% Then for the final state, if you have a photon 00:44:10.660 --> 00:44:14.530 align:middle line:90% in the final state, like this-- 00:44:14.530 --> 00:44:18.850 align:middle line:84% and then again, we just have an alpha. 00:44:18.850 --> 00:44:21.700 align:middle line:84% And so normally, we throw the momentum arrow to come out 00:44:21.700 --> 00:44:23.330 align:middle line:90% if you have a final state. 00:44:23.330 --> 00:44:31.230 align:middle line:84% And then the polarization vector will be just this star, OK? 00:44:31.230 --> 00:44:32.720 align:middle line:90% Just the complex conjugate. 00:44:32.720 --> 00:44:33.720 align:middle line:90% OK, it's very easy. 00:44:33.720 --> 00:44:34.220 align:middle line:90% OK? 00:44:34.220 --> 00:44:37.410 align:middle line:90% 00:44:37.410 --> 00:44:38.040 align:middle line:90% Yeah. 00:44:38.040 --> 00:44:40.500 align:middle line:84% So when you write down the amplitude, 00:44:40.500 --> 00:44:42.810 align:middle line:90% you just have to be careful. 00:44:42.810 --> 00:44:45.390 align:middle line:84% For external photon legs, you have 00:44:45.390 --> 00:44:48.970 align:middle line:84% to include the polarization vector. 00:44:48.970 --> 00:44:51.060 align:middle line:84% So for scalar QED, it's very similar. 00:44:51.060 --> 00:44:59.390 align:middle line:90% 00:44:59.390 --> 00:45:05.480 align:middle line:84% So here we can introduce a dashed line 00:45:05.480 --> 00:45:09.510 align:middle line:84% that's corresponding to a scalar propagator. 00:45:09.510 --> 00:45:10.010 align:middle line:90% OK? 00:45:10.010 --> 00:45:13.790 align:middle line:84% So corresponding to the scalar propagator, OK? 00:45:13.790 --> 00:45:18.410 align:middle line:84% And now, this arrow now has a meaning 00:45:18.410 --> 00:45:20.330 align:middle line:84% because the scalar has a charge, OK? 00:45:20.330 --> 00:45:21.990 align:middle line:90% A scalar has a charge. 00:45:21.990 --> 00:45:23.990 align:middle line:84% And so this arrow is not the momentum arrow 00:45:23.990 --> 00:45:26.490 align:middle line:90% you can do arbitrarily. 00:45:26.490 --> 00:45:31.140 align:middle line:90% And so there are various-- 00:45:31.140 --> 00:45:35.130 align:middle line:90% and yeah. 00:45:35.130 --> 00:45:39.960 align:middle line:84% And so for QED, let's look at the interaction 00:45:39.960 --> 00:45:40.950 align:middle line:90% with the photon. 00:45:40.950 --> 00:45:44.550 align:middle line:90% And then we have this vertex. 00:45:44.550 --> 00:45:52.500 align:middle line:84% And we also have this vertex with J mu coupled to that, OK? 00:45:52.500 --> 00:45:56.530 align:middle line:84% So then that kind of interaction will have the following form. 00:45:56.530 --> 00:45:59.220 align:middle line:90% So this J mu coupling is easy. 00:45:59.220 --> 00:46:03.360 align:middle line:84% We can schematically write it down. 00:46:03.360 --> 00:46:08.745 align:middle line:84% So again, you have 2 phi and then coupled to a photon, OK? 00:46:08.745 --> 00:46:11.460 align:middle line:90% 00:46:11.460 --> 00:46:14.410 align:middle line:90% Coupled to a photon, OK? 00:46:14.410 --> 00:46:19.840 align:middle line:90% And so A mu will have index. 00:46:19.840 --> 00:46:23.880 align:middle line:84% And so we can put the mu here, OK? 00:46:23.880 --> 00:46:33.060 align:middle line:84% And this vertex is given by minus i e k plus k prime mu, 00:46:33.060 --> 00:46:33.560 align:middle line:90% OK? 00:46:33.560 --> 00:46:35.710 align:middle line:90% Suppose the momentum here is k. 00:46:35.710 --> 00:46:37.990 align:middle line:84% And the momentum here is k prime, OK? 00:46:37.990 --> 00:46:42.640 align:middle line:84% So yeah, the reason you have this k plus k prime mu 00:46:42.640 --> 00:46:48.990 align:middle line:84% is because here there's a derivative acting on phi. 00:46:48.990 --> 00:46:54.120 align:middle line:84% And so this derivative acting on this phi gives you a k1, a k. 00:46:54.120 --> 00:46:57.450 align:middle line:84% And this derivative on phi gives you the k prime, OK? 00:46:57.450 --> 00:47:00.280 align:middle line:84% And you have the sum of these two terms. 00:47:00.280 --> 00:47:04.140 align:middle line:84% So that's why you have this kind of term for the interaction. 00:47:04.140 --> 00:47:04.943 align:middle line:90% Yes? 00:47:04.943 --> 00:47:05.610 align:middle line:90% AUDIENCE: Sorry. 00:47:05.610 --> 00:47:08.325 align:middle line:84% Going back to the fermion QED for a second-- 00:47:08.325 --> 00:47:08.950 align:middle line:90% HONG LIU: Yeah. 00:47:08.950 --> 00:47:12.640 align:middle line:84% AUDIENCE: Do you still also have, for the external legs, 00:47:12.640 --> 00:47:14.600 align:middle line:90% for fermions that are usually-- 00:47:14.600 --> 00:47:16.100 align:middle line:84% HONG LIU: Yeah, the same [AUDIO OUT] 00:47:16.100 --> 00:47:16.460 align:middle line:90% AUDIENCE: OK. 00:47:16.460 --> 00:47:17.085 align:middle line:90% HONG LIU: Yeah. 00:47:17.085 --> 00:47:20.748 align:middle line:84% So fermion-- external legs, exactly the same as before. 00:47:20.748 --> 00:47:21.290 align:middle line:90% AUDIENCE: OK. 00:47:21.290 --> 00:47:21.915 align:middle line:90% HONG LIU: Yeah. 00:47:21.915 --> 00:47:23.090 align:middle line:90% And yeah. 00:47:23.090 --> 00:47:25.950 align:middle line:84% So we only need to introduce new rules for photons. 00:47:25.950 --> 00:47:29.600 align:middle line:90% 00:47:29.600 --> 00:47:30.230 align:middle line:90% Yes? 00:47:30.230 --> 00:47:33.320 align:middle line:84% AUDIENCE: Still follow the same rules for starting from one-- 00:47:33.320 --> 00:47:34.460 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:47:34.460 --> 00:47:36.050 align:middle line:90% That rule is exactly the same. 00:47:36.050 --> 00:47:37.280 align:middle line:84% AUDIENCE: If you got a [INAUDIBLE] interaction, 00:47:37.280 --> 00:47:38.450 align:middle line:90% [INAUDIBLE] in the right order? 00:47:38.450 --> 00:47:39.020 align:middle line:90% HONG LIU: That's right. 00:47:39.020 --> 00:47:39.500 align:middle line:90% That's right. 00:47:39.500 --> 00:47:40.000 align:middle line:90% Yeah. 00:47:40.000 --> 00:47:41.880 align:middle line:84% And now, you have to just be careful. 00:47:41.880 --> 00:47:45.440 align:middle line:84% Some of the vertices are now including matrices. 00:47:45.440 --> 00:47:47.205 align:middle line:90% Yeah. 00:47:47.205 --> 00:47:47.705 align:middle line:90% Yes? 00:47:47.705 --> 00:47:49.588 align:middle line:84% AUDIENCE: Doesn't photon polarization 00:47:49.588 --> 00:47:51.912 align:middle line:84% have to transverse the momentum-- 00:47:51.912 --> 00:47:52.495 align:middle line:90% HONG LIU: Hmm? 00:47:52.495 --> 00:47:53.953 align:middle line:84% AUDIENCE: Isn't photon polarization 00:47:53.953 --> 00:47:55.575 align:middle line:90% have to transverse the momentum? 00:47:55.575 --> 00:47:56.700 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:47:56.700 --> 00:47:57.090 align:middle line:90% They do. 00:47:57.090 --> 00:47:57.590 align:middle line:90% Yeah, yeah. 00:47:57.590 --> 00:47:59.010 align:middle line:90% For the physical photon, yeah. 00:47:59.010 --> 00:48:03.585 align:middle line:84% Yeah, but for the physical photon, indeed, yeah. 00:48:03.585 --> 00:48:06.548 align:middle line:84% AUDIENCE: Also, why is it complex? 00:48:06.548 --> 00:48:07.090 align:middle line:90% HONG LIU: Oh. 00:48:07.090 --> 00:48:08.960 align:middle line:90% You can choose it to be complex. 00:48:08.960 --> 00:48:10.130 align:middle line:90% Yeah. 00:48:10.130 --> 00:48:14.440 align:middle line:84% For example, if you choose a spherically-- 00:48:14.440 --> 00:48:17.050 align:middle line:84% polarization, then it's a complex vector. 00:48:17.050 --> 00:48:18.310 align:middle line:90% Yeah. 00:48:18.310 --> 00:48:20.620 align:middle line:84% For the one we wrote down, it's real. 00:48:20.620 --> 00:48:22.360 align:middle line:84% But then this complex is the same. 00:48:22.360 --> 00:48:23.140 align:middle line:90% Yeah. 00:48:23.140 --> 00:48:23.800 align:middle line:90% Yes? 00:48:23.800 --> 00:48:25.300 align:middle line:84% AUDIENCE: [INAUDIBLE] this, but when 00:48:25.300 --> 00:48:28.030 align:middle line:84% you have an internal photon like in a diagram, 00:48:28.030 --> 00:48:32.877 align:middle line:84% do you sum over all four polarizations or just two? 00:48:32.877 --> 00:48:34.460 align:middle line:84% HONG LIU: Internal photon-- you always 00:48:34.460 --> 00:48:38.440 align:middle line:90% just use the photon propagator. 00:48:38.440 --> 00:48:41.570 align:middle line:84% And the photon propagator will include everything. 00:48:41.570 --> 00:48:42.297 align:middle line:90% Yeah. 00:48:42.297 --> 00:48:43.130 align:middle line:90% AUDIENCE: Thank you. 00:48:43.130 --> 00:48:45.517 align:middle line:90% 00:48:45.517 --> 00:48:46.600 align:middle line:90% HONG LIU: Other questions? 00:48:46.600 --> 00:48:47.440 align:middle line:90% Yes? 00:48:47.440 --> 00:48:50.080 align:middle line:84% AUDIENCE: For external vertices having matrices-- now, 00:48:50.080 --> 00:48:52.570 align:middle line:84% that's just a byproduct of the number of degrees of freedom 00:48:52.570 --> 00:48:54.290 align:middle line:90% you have per particle? 00:48:54.290 --> 00:48:55.070 align:middle line:90% HONG LIU: Yeah. 00:48:55.070 --> 00:48:56.400 align:middle line:90% Yeah. 00:48:56.400 --> 00:48:57.515 align:middle line:90% You mean the gamma mu? 00:48:57.515 --> 00:48:58.140 align:middle line:90% AUDIENCE: Yeah. 00:48:58.140 --> 00:48:58.765 align:middle line:90% HONG LIU: Yeah. 00:48:58.765 --> 00:49:02.610 align:middle line:84% It's just because of spinor nature, right? 00:49:02.610 --> 00:49:03.940 align:middle line:90% OK, good. 00:49:03.940 --> 00:49:08.100 align:middle line:84% So this vertex-- so A mu J mu gave you this vertex, 00:49:08.100 --> 00:49:10.620 align:middle line:90% but we also have this vertex. 00:49:10.620 --> 00:49:13.590 align:middle line:90% And this vertex is easy. 00:49:13.590 --> 00:49:20.920 align:middle line:84% We essentially just have two fermions or two scalars, 00:49:20.920 --> 00:49:26.590 align:middle line:84% but this one with two photons, OK? 00:49:26.590 --> 00:49:28.190 align:middle line:90% With two photons. 00:49:28.190 --> 00:49:33.640 align:middle line:84% And so these have A mu A mu contracted. 00:49:33.640 --> 00:49:38.500 align:middle line:84% So if we write mu here and nu here, 00:49:38.500 --> 00:49:47.610 align:middle line:84% then we will have minus ie squared eta mu nu, OK? 00:49:47.610 --> 00:49:53.670 align:middle line:90% 00:49:53.670 --> 00:49:54.170 align:middle line:90% OK. 00:49:54.170 --> 00:49:55.790 align:middle line:90% Minus ie squared. 00:49:55.790 --> 00:49:56.720 align:middle line:90% OK? 00:49:56.720 --> 00:49:59.160 align:middle line:84% So minus i for the same reason, and the e 00:49:59.160 --> 00:50:00.540 align:middle line:90% squared for the same reason. 00:50:00.540 --> 00:50:05.880 align:middle line:84% And this A mu contracted, so you have eta mu nu there, OK? 00:50:05.880 --> 00:50:06.380 align:middle line:90% Good? 00:50:06.380 --> 00:50:10.700 align:middle line:90% 00:50:10.700 --> 00:50:17.640 align:middle line:84% And depending on what is V here, phi 00:50:17.640 --> 00:50:20.300 align:middle line:84% may have some additional interaction. 00:50:20.300 --> 00:50:22.890 align:middle line:90% But suppose V is given by phi 4. 00:50:22.890 --> 00:50:23.390 align:middle line:90% OK? 00:50:23.390 --> 00:50:33.040 align:middle line:84% Suppose V, say, is equal to lambda 4 and phi phi star 00:50:33.040 --> 00:50:35.770 align:middle line:90% squared, OK? 00:50:35.770 --> 00:50:44.630 align:middle line:84% And then you will also have interactions like these-- 00:50:44.630 --> 00:50:47.030 align:middle line:90% four scalar interactions. 00:50:47.030 --> 00:50:50.760 align:middle line:90% 00:50:50.760 --> 00:50:51.420 align:middle line:90% OK. 00:50:51.420 --> 00:50:53.172 align:middle line:90% Yeah. 00:50:53.172 --> 00:50:54.630 align:middle line:84% You have two come in, two come out. 00:50:54.630 --> 00:50:55.130 align:middle line:90% OK? 00:50:55.130 --> 00:50:58.050 align:middle line:90% 00:50:58.050 --> 00:51:03.190 align:middle line:84% And this would be just minus i lambda, OK? 00:51:03.190 --> 00:51:03.880 align:middle line:90% Yes? 00:51:03.880 --> 00:51:06.520 align:middle line:84% AUDIENCE: Could you clarify-- the vertex with two scalars 00:51:06.520 --> 00:51:08.592 align:middle line:90% and one photon comes from? 00:51:08.592 --> 00:51:09.175 align:middle line:90% HONG LIU: Hmm? 00:51:09.175 --> 00:51:11.330 align:middle line:84% AUDIENCE: Can you explain where the V 00:51:11.330 --> 00:51:13.270 align:middle line:90% is a vertex with two scalars? 00:51:13.270 --> 00:51:14.260 align:middle line:90% HONG LIU: Oh, right. 00:51:14.260 --> 00:51:14.760 align:middle line:90% Right. 00:51:14.760 --> 00:51:17.865 align:middle line:90% It comes from this term. 00:51:17.865 --> 00:51:20.620 align:middle line:90% It comes from this term. 00:51:20.620 --> 00:51:27.160 align:middle line:84% So this is A mu times J. And the J has two phis here. 00:51:27.160 --> 00:51:28.078 align:middle line:90% AUDIENCE: Oh. 00:51:28.078 --> 00:51:29.286 align:middle line:90% HONG LIU: Right, right, yeah. 00:51:29.286 --> 00:51:35.240 align:middle line:90% 00:51:35.240 --> 00:51:36.200 align:middle line:90% Other questions? 00:51:36.200 --> 00:51:39.640 align:middle line:90% 00:51:39.640 --> 00:51:40.480 align:middle line:90% Yes? 00:51:40.480 --> 00:51:42.730 align:middle line:84% AUDIENCE: Would it be possible just to hypothetically, 00:51:42.730 --> 00:51:45.070 align:middle line:84% if I wanted to, give the photon mass? 00:51:45.070 --> 00:51:47.000 align:middle line:84% I'd just add a mass term to the Lagrangian? 00:51:47.000 --> 00:51:48.205 align:middle line:90% HONG LIU: Yeah, yeah. 00:51:48.205 --> 00:51:50.540 align:middle line:90% 00:51:50.540 --> 00:51:51.040 align:middle line:90% OK. 00:51:51.040 --> 00:51:52.760 align:middle line:90% What's the question? 00:51:52.760 --> 00:51:53.260 align:middle line:90% [LAUGHTER] 00:51:53.260 --> 00:51:54.130 align:middle line:90% AUDIENCE: I'm sorry. 00:51:54.130 --> 00:51:56.620 align:middle line:84% Is it possible to add a mass term 00:51:56.620 --> 00:51:58.420 align:middle line:84% for the photon in the Lagrangian? 00:51:58.420 --> 00:51:59.992 align:middle line:90% HONG LIU: No, you cannot. 00:51:59.992 --> 00:52:01.825 align:middle line:84% No because that violates the gauge symmetry. 00:52:01.825 --> 00:52:02.170 align:middle line:90% AUDIENCE: OK. 00:52:02.170 --> 00:52:02.810 align:middle line:90% HONG LIU: Yeah. 00:52:02.810 --> 00:52:03.310 align:middle line:90% Yeah. 00:52:03.310 --> 00:52:07.970 align:middle line:84% So the gauge symmetry is what ensures the photon is massless. 00:52:07.970 --> 00:52:11.110 align:middle line:84% Yeah, so that's why this is something 00:52:11.110 --> 00:52:12.250 align:middle line:90% we don't want to break. 00:52:12.250 --> 00:52:14.400 align:middle line:90% Yeah. 00:52:14.400 --> 00:52:15.533 align:middle line:90% Other questions? 00:52:15.533 --> 00:52:16.200 align:middle line:90% AUDIENCE: Sorry. 00:52:16.200 --> 00:52:17.492 align:middle line:90% This might be a silly question. 00:52:17.492 --> 00:52:22.140 align:middle line:84% But if you have the scalar QED and then the fermion QED, 00:52:22.140 --> 00:52:22.925 align:middle line:90% can you add them? 00:52:22.925 --> 00:52:23.550 align:middle line:90% HONG LIU: Yeah. 00:52:23.550 --> 00:52:24.280 align:middle line:90% You can add them. 00:52:24.280 --> 00:52:27.370 align:middle line:84% AUDIENCE: So then it's the one theory and [INAUDIBLE]?? 00:52:27.370 --> 00:52:28.110 align:middle line:90% It's combined? 00:52:28.110 --> 00:52:28.360 align:middle line:90% HONG LIU: Yeah. 00:52:28.360 --> 00:52:29.010 align:middle line:90% Yeah, yeah. 00:52:29.010 --> 00:52:33.400 align:middle line:84% You can combine them just like you have a theory-- 00:52:33.400 --> 00:52:35.800 align:middle line:84% the electromagnetic field coupled both to 00:52:35.800 --> 00:52:38.850 align:middle line:84% charged scalar and the charged fermions. 00:52:38.850 --> 00:52:41.000 align:middle line:84% Yeah, you can certainly combine them. 00:52:41.000 --> 00:52:41.500 align:middle line:90% Yeah. 00:52:41.500 --> 00:52:45.815 align:middle line:84% The reason I'm separating them is just for convenience. 00:52:45.815 --> 00:52:47.440 align:middle line:84% AUDIENCE: So can we have an interaction 00:52:47.440 --> 00:52:50.750 align:middle line:84% term directly between the fermion term and the scalar 00:52:50.750 --> 00:52:51.250 align:middle line:90% term? 00:52:51.250 --> 00:52:52.042 align:middle line:90% HONG LIU: Oh, yeah. 00:52:52.042 --> 00:52:53.770 align:middle line:90% You can add whatever. 00:52:53.770 --> 00:52:57.910 align:middle line:84% You only have to make sure things are gauge invariant. 00:52:57.910 --> 00:53:00.820 align:middle line:84% And you can add arbitrary interactions you want. 00:53:00.820 --> 00:53:02.410 align:middle line:84% Just, you have to make sure they're 00:53:02.410 --> 00:53:05.680 align:middle line:84% invariant under those gauge transformations. 00:53:05.680 --> 00:53:07.140 align:middle line:84% AUDIENCE: So on the other question, 00:53:07.140 --> 00:53:14.520 align:middle line:84% can we make another theory that has U1 gauge theory, 00:53:14.520 --> 00:53:20.240 align:middle line:90% but with massive particles? 00:53:20.240 --> 00:53:21.650 align:middle line:84% HONG LIU: With massive particles? 00:53:21.650 --> 00:53:22.550 align:middle line:90% Massive photons? 00:53:22.550 --> 00:53:24.230 align:middle line:90% AUDIENCE: With the photons. 00:53:24.230 --> 00:53:28.080 align:middle line:84% We have something that is massive 00:53:28.080 --> 00:53:32.550 align:middle line:90% and coupled to the [INAUDIBLE]. 00:53:32.550 --> 00:53:33.330 align:middle line:90% Can we do that? 00:53:33.330 --> 00:53:33.630 align:middle line:90% HONG LIU: No. 00:53:33.630 --> 00:53:34.110 align:middle line:90% No. 00:53:34.110 --> 00:53:34.610 align:middle line:90% No. 00:53:34.610 --> 00:53:39.450 align:middle line:84% You're asking whether the massive photon coupled 00:53:39.450 --> 00:53:42.060 align:middle line:90% to fermionic field exists? 00:53:42.060 --> 00:53:43.050 align:middle line:90% AUDIENCE: Yeah. 00:53:43.050 --> 00:53:43.890 align:middle line:90% HONG LIU: OK. 00:53:43.890 --> 00:53:44.490 align:middle line:90% Yeah. 00:53:44.490 --> 00:53:51.610 align:middle line:84% So this question can be answered in several-- 00:53:51.610 --> 00:53:55.070 align:middle line:84% so if you just directly add a mass term-- 00:53:55.070 --> 00:53:59.110 align:middle line:84% so let's add a mass term for the photon. 00:53:59.110 --> 00:54:01.840 align:middle line:90% 00:54:01.840 --> 00:54:02.890 align:middle line:90% OK? 00:54:02.890 --> 00:54:06.220 align:middle line:90% So you can immediately check. 00:54:06.220 --> 00:54:07.910 align:middle line:90% This is not gauge invariant. 00:54:07.910 --> 00:54:10.900 align:middle line:84% So that's the reason we didn't add such a term to the Maxwell 00:54:10.900 --> 00:54:13.900 align:middle line:84% theory because in the Maxwell theory, if you add this term, 00:54:13.900 --> 00:54:15.760 align:middle line:84% it will violate the gauge symmetry. 00:54:15.760 --> 00:54:17.120 align:middle line:90% The same reason here. 00:54:17.120 --> 00:54:20.620 align:middle line:84% So you cannot add such a term, OK? 00:54:20.620 --> 00:54:23.710 align:middle line:90% So you cannot add such a term. 00:54:23.710 --> 00:54:26.170 align:middle line:90% So this is ruled out, OK? 00:54:26.170 --> 00:54:29.070 align:middle line:90% You cannot add this. 00:54:29.070 --> 00:54:32.580 align:middle line:84% But now, you say is it possible somehow, 00:54:32.580 --> 00:54:38.080 align:middle line:84% through some other way, that the photon actually gains mass, OK? 00:54:38.080 --> 00:54:40.450 align:middle line:90% That's possible. 00:54:40.450 --> 00:54:41.920 align:middle line:84% That's called the Higgs mechanism. 00:54:41.920 --> 00:54:44.680 align:middle line:84% That's how Higgs got a Nobel Prize. 00:54:44.680 --> 00:54:47.510 align:middle line:90% And that comes from this term. 00:54:47.510 --> 00:54:51.930 align:middle line:84% So now, you imagine somehow, this phi now 00:54:51.930 --> 00:54:54.680 align:middle line:90% has a vacuum expectation value. 00:54:54.680 --> 00:54:58.260 align:middle line:84% And now, phi has a constant part. 00:54:58.260 --> 00:55:02.970 align:middle line:84% If phi has a constant path, then this becomes a mass term 00:55:02.970 --> 00:55:04.320 align:middle line:90% for the photon, OK? 00:55:04.320 --> 00:55:06.688 align:middle line:84% And then the photon becomes massive. 00:55:06.688 --> 00:55:08.230 align:middle line:84% And actually, that's precisely what's 00:55:08.230 --> 00:55:10.810 align:middle line:84% happening inside the superconductor. 00:55:10.810 --> 00:55:14.760 align:middle line:84% So in every superconductor, something like this happens. 00:55:14.760 --> 00:55:15.330 align:middle line:90% OK? 00:55:15.330 --> 00:55:21.060 align:middle line:84% And the photon inside the superconductor is massive, OK? 00:55:21.060 --> 00:55:27.190 align:middle line:84% That's what responsible for this London effect, Meisner effect, 00:55:27.190 --> 00:55:28.680 align:middle line:90% et cetera, OK? 00:55:28.680 --> 00:55:31.440 align:middle line:84% And in the superconductor, that's 00:55:31.440 --> 00:55:35.190 align:middle line:90% understood by Phillip Anderson. 00:55:35.190 --> 00:55:40.020 align:middle line:84% He was a little bit unhappy because he 00:55:40.020 --> 00:55:43.860 align:middle line:84% thought he should have claimed for the Higgs effect 00:55:43.860 --> 00:55:45.900 align:middle line:84% because he understood it earlier. 00:55:45.900 --> 00:55:47.280 align:middle line:90% And yeah. 00:55:47.280 --> 00:55:50.040 align:middle line:84% But he did understand the mechanism earlier. 00:55:50.040 --> 00:55:50.940 align:middle line:90% Yeah, yeah. 00:55:50.940 --> 00:55:53.340 align:middle line:84% But Higgs got the Nobel Prize because he 00:55:53.340 --> 00:55:54.990 align:middle line:90% predicted the particle. 00:55:54.990 --> 00:56:00.210 align:middle line:84% Anyway, that's a long story I'm not going to go into. 00:56:00.210 --> 00:56:00.737 align:middle line:90% Yeah. 00:56:00.737 --> 00:56:02.070 align:middle line:90% AUDIENCE: Just a quick question. 00:56:02.070 --> 00:56:05.910 align:middle line:84% For the photon propagator, do we give the same factor? 00:56:05.910 --> 00:56:08.430 align:middle line:84% Because I don't think we said in the previous lecture 00:56:08.430 --> 00:56:13.433 align:middle line:84% that we give the same factor as for the scalar field. 00:56:13.433 --> 00:56:14.100 align:middle line:90% HONG LIU: Sorry. 00:56:14.100 --> 00:56:14.580 align:middle line:90% Say it again. 00:56:14.580 --> 00:56:15.990 align:middle line:84% AUDIENCE: For photon propagator-- 00:56:15.990 --> 00:56:16.615 align:middle line:90% HONG LIU: Yeah. 00:56:16.615 --> 00:56:18.480 align:middle line:84% AUDIENCE: --which factor do we give? 00:56:18.480 --> 00:56:19.500 align:middle line:90% HONG LIU: Which factor? 00:56:19.500 --> 00:56:21.660 align:middle line:90% AUDIENCE: Yeah. 00:56:21.660 --> 00:56:23.400 align:middle line:90% HONG LIU: You mean-- 00:56:23.400 --> 00:56:24.935 align:middle line:84% we just use the photon propagator. 00:56:24.935 --> 00:56:25.560 align:middle line:90% AUDIENCE: Yeah. 00:56:25.560 --> 00:56:28.422 align:middle line:84% Because I don't think we did it in this lecture. 00:56:28.422 --> 00:56:29.130 align:middle line:90% HONG LIU: We did. 00:56:29.130 --> 00:56:29.630 align:middle line:90% Yeah. 00:56:29.630 --> 00:56:31.020 align:middle line:84% Yeah, at the end of last lecture. 00:56:31.020 --> 00:56:32.353 align:middle line:90% Yeah, let me just write it down. 00:56:32.353 --> 00:56:42.090 align:middle line:84% So the photon propagator depends on this parameter xi. 00:56:42.090 --> 00:56:46.470 align:middle line:84% For xi equal to 1, the photon propagator-- 00:56:46.470 --> 00:56:48.630 align:middle line:90% if I call it D mu nu, yeah. 00:56:48.630 --> 00:56:50.760 align:middle line:84% Anyway, let me just write it down. 00:56:50.760 --> 00:56:54.015 align:middle line:84% For xi equal to 1, this is just given by eta mu nu 00:56:54.015 --> 00:56:56.530 align:middle line:90% k squared minus i epsilon. 00:56:56.530 --> 00:56:59.310 align:middle line:90% So this is for xi equal to 1. 00:56:59.310 --> 00:57:02.760 align:middle line:84% So for xi not equal to 1, so yeah. 00:57:02.760 --> 00:57:04.800 align:middle line:90% So this is for xi equal to 1. 00:57:04.800 --> 00:57:06.390 align:middle line:84% For xi not equal to 1, then there 00:57:06.390 --> 00:57:09.530 align:middle line:90% are some additional terms. 00:57:09.530 --> 00:57:10.190 align:middle line:90% Yes? 00:57:10.190 --> 00:57:12.398 align:middle line:84% AUDIENCE: I might be leaving the scope of this class, 00:57:12.398 --> 00:57:14.905 align:middle line:84% but the weak force has a carrier that has mass, right? 00:57:14.905 --> 00:57:15.530 align:middle line:90% HONG LIU: Yeah. 00:57:15.530 --> 00:57:17.030 align:middle line:84% AUDIENCE: And it's called a fermion. 00:57:17.030 --> 00:57:17.710 align:middle line:90% HONG LIU: Right. 00:57:17.710 --> 00:57:19.885 align:middle line:84% AUDIENCE: So that is possible to construct. 00:57:19.885 --> 00:57:21.010 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:57:21.010 --> 00:57:21.510 align:middle line:90% Yeah. 00:57:21.510 --> 00:57:24.450 align:middle line:84% Again, this mechanism is what's responsible 00:57:24.450 --> 00:57:26.903 align:middle line:84% for the weak interaction in short ranged. 00:57:26.903 --> 00:57:27.570 align:middle line:90% AUDIENCE: I see. 00:57:27.570 --> 00:57:27.990 align:middle line:90% Oh. 00:57:27.990 --> 00:57:29.365 align:middle line:84% HONG LIU: So the weak interaction 00:57:29.365 --> 00:57:31.212 align:middle line:90% started as massless particle-- 00:57:31.212 --> 00:57:31.920 align:middle line:90% AUDIENCE: Got it. 00:57:31.920 --> 00:57:34.320 align:middle line:84% HONG LIU: --but then through the Higgs mechanism 00:57:34.320 --> 00:57:38.850 align:middle line:84% and then the analog of A mu for the weak interaction, 00:57:38.850 --> 00:57:40.950 align:middle line:84% becomes massive, and then becomes short ranged. 00:57:40.950 --> 00:57:41.410 align:middle line:90% AUDIENCE: Got it. 00:57:41.410 --> 00:57:41.630 align:middle line:90% That makes sense. 00:57:41.630 --> 00:57:42.780 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:57:42.780 --> 00:57:43.890 align:middle line:90% And yeah, yeah. 00:57:43.890 --> 00:57:45.190 align:middle line:90% And yeah. 00:57:45.190 --> 00:57:45.690 align:middle line:90% Good. 00:57:45.690 --> 00:57:46.860 align:middle line:90% Other questions? 00:57:46.860 --> 00:57:47.580 align:middle line:90% Yes? 00:57:47.580 --> 00:57:50.580 align:middle line:84% AUDIENCE: In the standard model is the Higgs the only scalar 00:57:50.580 --> 00:57:54.080 align:middle line:84% that couples to the photon [INAUDIBLE]?? 00:57:54.080 --> 00:57:58.430 align:middle line:84% HONG LIU: So in the standard model, 00:57:58.430 --> 00:58:00.864 align:middle line:90% Higgs is the only scalar field. 00:58:00.864 --> 00:58:01.364 align:middle line:90% Yeah. 00:58:01.364 --> 00:58:06.780 align:middle line:90% 00:58:06.780 --> 00:58:10.410 align:middle line:84% But you can have some other composite scalars, like pions. 00:58:10.410 --> 00:58:15.030 align:middle line:84% Fundamental scalars-- it's just the Higgs field. 00:58:15.030 --> 00:58:16.450 align:middle line:90% Good. 00:58:16.450 --> 00:58:17.550 align:middle line:90% OK, good. 00:58:17.550 --> 00:58:22.380 align:middle line:84% So this concludes the discussion of the QED. 00:58:22.380 --> 00:58:28.380 align:middle line:84% And now, ready just to study the physical processes in QED, OK? 00:58:28.380 --> 00:58:31.170 align:middle line:84% And before doing that, we need to develop 00:58:31.170 --> 00:58:35.340 align:middle line:84% a little bit of formalism because when 00:58:35.340 --> 00:58:36.870 align:middle line:84% we talk about physical process, then 00:58:36.870 --> 00:58:38.910 align:middle line:90% we need to make connection. 00:58:38.910 --> 00:58:39.540 align:middle line:90% Yeah. 00:58:39.540 --> 00:58:41.430 align:middle line:90% QED is real life, OK? 00:58:41.430 --> 00:58:46.180 align:middle line:84% And then we can make connection with the real experiments. 00:58:46.180 --> 00:58:46.887 align:middle line:90% OK? 00:58:46.887 --> 00:58:48.720 align:middle line:84% To make connection with the real experiment, 00:58:48.720 --> 00:58:51.930 align:middle line:84% we need to develop one more thing. 00:58:51.930 --> 00:58:55.240 align:middle line:84% We need to develop one more thing. 00:58:55.240 --> 00:59:01.080 align:middle line:84% So we need to learn how to calculate 00:59:01.080 --> 00:59:03.130 align:middle line:90% the cross-section, OK? 00:59:03.130 --> 00:59:23.250 align:middle line:90% 00:59:23.250 --> 00:59:26.410 align:middle line:84% So let's talk about how to calculate the cross-section. 00:59:26.410 --> 00:59:33.030 align:middle line:90% So this is a digression, OK? 00:59:33.030 --> 00:59:37.860 align:middle line:84% So first, let's remind you how we 00:59:37.860 --> 00:59:41.820 align:middle line:84% define a cross-section in nonrelativistic quantum 00:59:41.820 --> 00:59:42.630 align:middle line:90% mechanics, OK? 00:59:42.630 --> 00:59:45.180 align:middle line:84% In the scattering theory of nonrelativistic quantum 00:59:45.180 --> 00:59:46.020 align:middle line:90% mechanics. 00:59:46.020 --> 00:59:53.740 align:middle line:84% So in nonrelativistic quantum mechanics, 00:59:53.740 --> 00:59:57.220 align:middle line:90% you consider there is a target. 00:59:57.220 --> 00:59:59.860 align:middle line:84% The target is normally considered to be a point, OK? 00:59:59.860 --> 01:00:04.070 align:middle line:84% Here I just make it big so that we can see it. 01:00:04.070 --> 01:00:08.210 align:middle line:90% So here is the target. 01:00:08.210 --> 01:00:11.170 align:middle line:90% And then we define our axis-- 01:00:11.170 --> 01:00:14.750 align:middle line:84% the scattering axis, which the particle come in. 01:00:14.750 --> 01:00:17.425 align:middle line:84% So this is some incident beam of particle. 01:00:17.425 --> 01:00:24.960 align:middle line:90% 01:00:24.960 --> 01:00:26.650 align:middle line:90% OK, coming into this direction. 01:00:26.650 --> 01:00:29.940 align:middle line:84% So say this is a Z-direction, OK? 01:00:29.940 --> 01:00:33.720 align:middle line:84% And moving in this direction to-- 01:00:33.720 --> 01:00:36.990 align:middle line:84% and then it will interact with this particle. 01:00:36.990 --> 01:00:39.690 align:middle line:90% And then it will scatter, OK? 01:00:39.690 --> 01:00:41.250 align:middle line:90% So the particle will scatter. 01:00:41.250 --> 01:00:48.300 align:middle line:84% And then we put the detector around this target particle 01:00:48.300 --> 01:00:51.190 align:middle line:84% to detect the scattered particle. 01:00:51.190 --> 01:00:51.690 align:middle line:90% OK? 01:00:51.690 --> 01:00:53.520 align:middle line:90% So for example, yeah. 01:00:53.520 --> 01:00:59.440 align:middle line:84% So you should view this solid angle here. 01:00:59.440 --> 01:00:59.940 align:middle line:90% OK? 01:00:59.940 --> 01:01:03.290 align:middle line:84% And so this is the phi direction. 01:01:03.290 --> 01:01:06.850 align:middle line:84% And so this is the theta direction, OK? 01:01:06.850 --> 01:01:13.410 align:middle line:90% So here is the theta direction. 01:01:13.410 --> 01:01:20.780 align:middle line:84% And this is some small solid angle, which the detector-- 01:01:20.780 --> 01:01:22.850 align:middle line:84% suppose there's a detector here, OK? 01:01:22.850 --> 01:01:25.800 align:middle line:90% 01:01:25.800 --> 01:01:27.570 align:middle line:90% Suppose there's a detector here. 01:01:27.570 --> 01:01:32.970 align:middle line:84% And then this detector spread the solid angle, D omega, 01:01:32.970 --> 01:01:35.910 align:middle line:90% with respect to the target, OK? 01:01:35.910 --> 01:01:41.490 align:middle line:84% So this is the standard setup for nonrelativistic-- 01:01:41.490 --> 01:01:45.540 align:middle line:84% scattering process in the nonrelativistic quantum 01:01:45.540 --> 01:01:47.940 align:middle line:90% mechanics. 01:01:47.940 --> 01:02:04.630 align:middle line:84% So what we measure is dN dt outgoing particle, OK? 01:02:04.630 --> 01:02:08.920 align:middle line:84% Scattered particle in the theta and phi direction, OK? 01:02:08.920 --> 01:02:11.750 align:middle line:90% So this is theta and phi. 01:02:11.750 --> 01:02:14.770 align:middle line:84% This is where the detector locates in the theta phi 01:02:14.770 --> 01:02:16.210 align:middle line:90% direction. 01:02:16.210 --> 01:02:19.620 align:middle line:90% So this is the number. 01:02:19.620 --> 01:02:24.740 align:middle line:84% So this dN dt is the number of particles-- 01:02:24.740 --> 01:02:36.850 align:middle line:84% number of scattered particles, number of scattered particles 01:02:36.850 --> 01:02:37.510 align:middle line:90% in -- 01:02:37.510 --> 01:02:45.945 align:middle line:90% per unit time in the detector. 01:02:45.945 --> 01:02:49.430 align:middle line:90% 01:02:49.430 --> 01:02:52.880 align:middle line:84% It means registered by the detector, detected 01:02:52.880 --> 01:02:58.910 align:middle line:84% by the detector at the location theta phi, OK? 01:02:58.910 --> 01:03:02.585 align:middle line:90% 01:03:02.585 --> 01:03:03.085 align:middle line:90% OK? 01:03:03.085 --> 01:03:06.650 align:middle line:90% 01:03:06.650 --> 01:03:12.618 align:middle line:84% So the quantity we measure is this quantity. 01:03:12.618 --> 01:03:13.910 align:middle line:90% We just put the detector there. 01:03:13.910 --> 01:03:17.270 align:middle line:90% We just measure particles, OK? 01:03:17.270 --> 01:03:22.810 align:middle line:84% And clearly, so this scene depends on many things, OK? 01:03:22.810 --> 01:03:24.400 align:middle line:84% This quantity depends on many things, 01:03:24.400 --> 01:03:30.980 align:middle line:84% depends on the interaction between the incident particle 01:03:30.980 --> 01:03:32.900 align:middle line:90% and the target, et cetera. 01:03:32.900 --> 01:03:36.710 align:middle line:84% But they also depend on many other kinematic factors, OK? 01:03:36.710 --> 01:03:43.630 align:middle line:84% So for example, it will depend on -- this out-- 01:03:43.630 --> 01:03:48.940 align:middle line:84% clearly, it will depend on the number of incident particles. 01:03:48.940 --> 01:03:53.480 align:middle line:90% 01:03:53.480 --> 01:03:53.980 align:middle line:90% OK? 01:03:53.980 --> 01:03:55.810 align:middle line:84% So this is the number of incident particles 01:03:55.810 --> 01:03:56.605 align:middle line:90% per unit time. 01:03:56.605 --> 01:04:08.820 align:middle line:90% 01:04:08.820 --> 01:04:11.020 align:middle line:84% If you have more incident particles, of course, 01:04:11.020 --> 01:04:14.850 align:middle line:84% you will detect more outgoing particles, OK? 01:04:14.850 --> 01:04:17.760 align:middle line:84% And of course, this will also be proportional to D omega, 01:04:17.760 --> 01:04:23.800 align:middle line:84% the solid angle extended by the detector. 01:04:23.800 --> 01:04:25.810 align:middle line:84% So if you have a larger detector, of course, 01:04:25.810 --> 01:04:28.260 align:middle line:84% you will detect more particles, OK? 01:04:28.260 --> 01:04:34.250 align:middle line:90% 01:04:34.250 --> 01:04:41.990 align:middle line:84% And then if we divide it by those obvious kinematic factors 01:04:41.990 --> 01:04:44.930 align:middle line:84% away, then we will get something more 01:04:44.930 --> 01:04:48.050 align:middle line:84% intrinsic to the interactions, OK? 01:04:48.050 --> 01:05:11.450 align:middle line:90% So we can consider-- 01:05:11.450 --> 01:05:14.170 align:middle line:90% 01:05:14.170 --> 01:05:16.404 align:middle line:84% we can divide by D omega and divide it by this thing. 01:05:16.404 --> 01:05:21.640 align:middle line:90% So this object, dN dt. 01:05:21.640 --> 01:05:26.650 align:middle line:84% Then we divide it by D omega per solid angle out 01:05:26.650 --> 01:05:29.830 align:middle line:84% and divide it by the incident number 01:05:29.830 --> 01:05:32.750 align:middle line:90% of particles per unit time. 01:05:32.750 --> 01:05:44.810 align:middle line:84% So this should give us the probability 01:05:44.810 --> 01:06:00.860 align:middle line:84% of a particle to be scattered to theta phi direction, OK? 01:06:00.860 --> 01:06:05.900 align:middle line:84% So this is the quantity which captures the effect 01:06:05.900 --> 01:06:07.760 align:middle line:90% of the interaction, OK? 01:06:07.760 --> 01:06:09.920 align:middle line:84% So this is a more intrinsic quantity. 01:06:09.920 --> 01:06:14.100 align:middle line:90% 01:06:14.100 --> 01:06:16.290 align:middle line:84% But experimentally, this is actually 01:06:16.290 --> 01:06:21.470 align:middle line:84% not the thing we directly measure in the experiment 01:06:21.470 --> 01:06:22.940 align:middle line:84% because in the experiment, it's not 01:06:22.940 --> 01:06:26.240 align:middle line:84% easy to count the number of the incident particles, OK? 01:06:26.240 --> 01:06:30.600 align:middle line:84% We always consider a beam of particles. 01:06:30.600 --> 01:06:33.810 align:middle line:84% We always consider a large number of particles, 01:06:33.810 --> 01:06:36.760 align:middle line:90% a beam of particles. 01:06:36.760 --> 01:06:41.460 align:middle line:84% So experimentally, it's more convenient. 01:06:41.460 --> 01:06:49.400 align:middle line:90% 01:06:49.400 --> 01:06:52.150 align:middle line:90% We often use the incident flux. 01:06:52.150 --> 01:06:57.580 align:middle line:90% 01:06:57.580 --> 01:07:09.740 align:middle line:84% It's the number of particles per unit time and per unit area, 01:07:09.740 --> 01:07:10.340 align:middle line:90% OK? 01:07:10.340 --> 01:07:11.810 align:middle line:90% So we just send the beam. 01:07:11.810 --> 01:07:14.300 align:middle line:84% And we just need to know the density of the beam, OK? 01:07:14.300 --> 01:07:18.980 align:middle line:84% So that will give us the flux, the number 01:07:18.980 --> 01:07:22.350 align:middle line:84% of particles ingoing per unit time and per unit area. 01:07:22.350 --> 01:07:28.470 align:middle line:84% So A is the cross-section of the incident beam, OK? 01:07:28.470 --> 01:07:30.290 align:middle line:90% So now, this is the quantity. 01:07:30.290 --> 01:07:32.320 align:middle line:90% And now, we have to divide-- 01:07:32.320 --> 01:07:35.140 align:middle line:84% so instead of dividing by this one, we divide by this one. 01:07:35.140 --> 01:07:35.640 align:middle line:90% OK? 01:07:35.640 --> 01:07:46.790 align:middle line:84% So now, this D sigma D omega, which is defined to be dN 01:07:46.790 --> 01:07:51.790 align:middle line:90% dt D omega out-- 01:07:51.790 --> 01:07:54.940 align:middle line:84% and then you divide it by incident flux, OK? 01:07:54.940 --> 01:07:58.890 align:middle line:90% 01:07:58.890 --> 01:08:02.860 align:middle line:90% Incident flux per unit time, OK? 01:08:02.860 --> 01:08:08.310 align:middle line:84% So that's the object we actually measure, OK? 01:08:08.310 --> 01:08:14.590 align:middle line:90% So some comments on this. 01:08:14.590 --> 01:08:18.269 align:middle line:84% So the first comment is that this has dimension of area. 01:08:18.269 --> 01:08:26.890 align:middle line:90% 01:08:26.890 --> 01:08:30.250 align:middle line:84% So because this thing is dimensionless-- 01:08:30.250 --> 01:08:32.800 align:middle line:84% this thing is dimensionless because T canceled, 01:08:32.800 --> 01:08:35.109 align:middle line:90% but now, we have divided by dA. 01:08:35.109 --> 01:08:36.649 align:middle line:90% So A is going up. 01:08:36.649 --> 01:08:38.649 align:middle line:84% And then this has dimension of area. 01:08:38.649 --> 01:08:41.899 align:middle line:90% 01:08:41.899 --> 01:08:45.470 align:middle line:84% So the physical meaning of this, which 01:08:45.470 --> 01:08:50.029 align:middle line:84% I'm sure you learned in your nonrelativistic quantum 01:08:50.029 --> 01:09:00.250 align:middle line:84% mechanics class, is that this gives you the physical meaning 01:09:00.250 --> 01:09:02.750 align:middle line:90% of this object. 01:09:02.750 --> 01:09:18.279 align:middle line:84% So this gives you the effective area of interaction, OK? 01:09:18.279 --> 01:09:20.939 align:middle line:90% 01:09:20.939 --> 01:09:27.010 align:middle line:84% So heuristically, you can imagine the following. 01:09:27.010 --> 01:09:30.942 align:middle line:84% So if you imagine the incident particle, 01:09:30.942 --> 01:09:33.359 align:middle line:84% this target particle-- they actually have some short range 01:09:33.359 --> 01:09:34.710 align:middle line:90% interaction, OK? 01:09:34.710 --> 01:09:38.910 align:middle line:84% And then they no longer interact, say, 01:09:38.910 --> 01:09:40.770 align:middle line:90% outside some distance. 01:09:40.770 --> 01:09:43.800 align:middle line:84% And then heuristically, then this quantity then 01:09:43.800 --> 01:09:51.810 align:middle line:84% can be considered as-- give you the area of interaction 01:09:51.810 --> 01:09:56.770 align:middle line:90% around this target particle, OK? 01:09:56.770 --> 01:09:58.330 align:middle line:90% Around the target particle. 01:09:58.330 --> 01:10:03.220 align:middle line:84% So that's why this D sigma D omega is called 01:10:03.220 --> 01:10:04.780 align:middle line:90% the differential cross-section. 01:10:04.780 --> 01:10:14.420 align:middle line:90% 01:10:14.420 --> 01:10:19.220 align:middle line:84% So the area which is going into the direction of theta phi-- 01:10:19.220 --> 01:10:23.120 align:middle line:84% and the sigma total, which is defined 01:10:23.120 --> 01:10:27.510 align:middle line:84% to be D sigma, D omega-- you integrate them over all angles, 01:10:27.510 --> 01:10:28.010 align:middle line:90% OK? 01:10:28.010 --> 01:10:30.530 align:middle line:84% So this is a function of theta and phi. 01:10:30.530 --> 01:10:33.270 align:middle line:84% And now, you integrate over all solid angles. 01:10:33.270 --> 01:10:35.420 align:middle line:84% And then that gives you the total cross-section. 01:10:35.420 --> 01:10:44.060 align:middle line:90% 01:10:44.060 --> 01:10:44.600 align:middle line:90% OK? 01:10:44.600 --> 01:10:51.840 align:middle line:84% And to see this effective area of interaction, 01:10:51.840 --> 01:10:56.100 align:middle line:84% can be seen in the very simple classical example. 01:10:56.100 --> 01:10:58.350 align:middle line:84% So this definition actually is not quantum mechanical. 01:10:58.350 --> 01:11:00.000 align:middle line:84% You can also do it classically, OK? 01:11:00.000 --> 01:11:03.060 align:middle line:84% It's not restricted to quantum mechanics. 01:11:03.060 --> 01:11:06.160 align:middle line:84% This can be defined for any scattering event, 01:11:06.160 --> 01:11:08.970 align:middle line:84% including in classical mechanics. 01:11:08.970 --> 01:11:20.130 align:middle line:90% 01:11:20.130 --> 01:11:29.570 align:middle line:84% So for example, if you have classical scattering, 01:11:29.570 --> 01:11:38.970 align:middle line:84% say, of some bullets, off a billiard ball, 01:11:38.970 --> 01:11:42.330 align:middle line:84% and then you find the sigma total-- 01:11:42.330 --> 01:11:45.960 align:middle line:84% so suppose you have classical scattering, OK? 01:11:45.960 --> 01:11:48.660 align:middle line:84% And then you find the sigma total. 01:11:48.660 --> 01:11:50.460 align:middle line:90% It's just the cross-section. 01:11:50.460 --> 01:11:56.130 align:middle line:84% It's just the total area, total surface area. 01:11:56.130 --> 01:11:56.670 align:middle line:90% Yeah. 01:11:56.670 --> 01:11:57.360 align:middle line:90% No. 01:11:57.360 --> 01:12:05.180 align:middle line:84% It's the cross-section area over the billiards, OK? 01:12:05.180 --> 01:12:11.910 align:middle line:90% 01:12:11.910 --> 01:12:14.982 align:middle line:84% Because classically, when you hit the billiard ball, 01:12:14.982 --> 01:12:16.440 align:middle line:84% then you hit the billiard ball, OK? 01:12:16.440 --> 01:12:18.240 align:middle line:84% If you don't hit the billiard ball, 01:12:18.240 --> 01:12:20.280 align:middle line:90% then you don't change direction. 01:12:20.280 --> 01:12:23.130 align:middle line:84% And then the total cross-section just 01:12:23.130 --> 01:12:27.180 align:middle line:84% is the cross-section of the billiard ball, OK? 01:12:27.180 --> 01:12:33.390 align:middle line:84% And so when you go to quantum theory, 01:12:33.390 --> 01:12:35.940 align:middle line:84% then this is the heuristic way you understand 01:12:35.940 --> 01:12:40.830 align:middle line:84% there's some kind of effective cross-section for this target, 01:12:40.830 --> 01:12:41.430 align:middle line:90% OK? 01:12:41.430 --> 01:12:46.710 align:middle line:84% And outside that area then you no longer have-- 01:12:46.710 --> 01:12:47.610 align:middle line:90% yeah. 01:12:47.610 --> 01:12:50.820 align:middle line:84% Heuristically, you no longer have interactions. 01:12:50.820 --> 01:12:51.510 align:middle line:90% OK. 01:12:51.510 --> 01:12:56.610 align:middle line:84% So now, let me just make two quick remarks 01:12:56.610 --> 01:12:58.065 align:middle line:90% in a relativistic theory. 01:12:58.065 --> 01:13:02.850 align:middle line:90% 01:13:02.850 --> 01:13:05.590 align:middle line:90% So now, we can generalize. 01:13:05.590 --> 01:13:10.550 align:middle line:84% So this discussion is in the nonrelativistic context. 01:13:10.550 --> 01:13:12.940 align:middle line:84% So in the relativistic context, we essentially just 01:13:12.940 --> 01:13:20.630 align:middle line:84% have D sigma cross-section from initial state to final state, 01:13:20.630 --> 01:13:21.130 align:middle line:90% OK? 01:13:21.130 --> 01:13:23.620 align:middle line:84% We no longer have-- so this is very specific. 01:13:23.620 --> 01:13:27.000 align:middle line:84% We have one particle scattering the other particle. 01:13:27.000 --> 01:13:29.880 align:middle line:84% And this particle scatters away, OK? 01:13:29.880 --> 01:13:33.870 align:middle line:84% But in relativistic, you can have two particles. 01:13:33.870 --> 01:13:36.840 align:middle line:84% You can create 10,000 particles, OK? 01:13:36.840 --> 01:13:40.320 align:middle line:84% And so this language no longer works, OK? 01:13:40.320 --> 01:13:43.230 align:middle line:84% So but still, you can define some kind of cross-section 01:13:43.230 --> 01:13:47.890 align:middle line:84% of initial state alpha to some final state beta, OK? 01:13:47.890 --> 01:13:52.900 align:middle line:84% And so we will be interested in the situation in which we 01:13:52.900 --> 01:13:55.030 align:middle line:90% have two initial particles, OK? 01:13:55.030 --> 01:14:01.360 align:middle line:84% Because experimentally, it's convenient just 01:14:01.360 --> 01:14:03.040 align:middle line:84% to have two particles scattering. 01:14:03.040 --> 01:14:04.660 align:middle line:84% So you have two initial particles. 01:14:04.660 --> 01:14:13.390 align:middle line:84% But final state, in principle, can be arbitrary, OK? 01:14:13.390 --> 01:14:18.960 align:middle line:90% And so this object-- 01:14:18.960 --> 01:14:20.890 align:middle line:84% since, in the relativistic theory, 01:14:20.890 --> 01:14:27.212 align:middle line:84% this should be physically observable, so this should be-- 01:14:27.212 --> 01:14:29.670 align:middle line:84% so because this essentially is a measure of the probability 01:14:29.670 --> 01:14:32.930 align:middle line:84% of the scattering, so we require this 01:14:32.930 --> 01:14:38.961 align:middle line:90% object to be Lorentz invariant. 01:14:38.961 --> 01:14:46.040 align:middle line:90% 01:14:46.040 --> 01:14:47.450 align:middle line:90% OK? 01:14:47.450 --> 01:14:52.760 align:middle line:84% And also, in the relativistic, we just scatter two particles. 01:14:52.760 --> 01:14:54.500 align:middle line:84% There's no notion of what is the target 01:14:54.500 --> 01:14:56.990 align:middle line:84% or what is the incident particle. 01:14:56.990 --> 01:14:59.240 align:middle line:84% So these two particles should have equal rho. 01:14:59.240 --> 01:15:08.040 align:middle line:84% So this should be symmetric in 1 2, OK? 01:15:08.040 --> 01:15:10.260 align:middle line:84% So if you exchange 1 2, this cross-section 01:15:10.260 --> 01:15:12.020 align:middle line:90% should be the same. 01:15:12.020 --> 01:15:12.520 align:middle line:90% OK? 01:15:12.520 --> 01:15:15.470 align:middle line:90% It should be the same. 01:15:15.470 --> 01:15:18.820 align:middle line:84% So now, if you consider-- but you can always 01:15:18.820 --> 01:15:22.180 align:middle line:84% consider in the rest frame, say, of particle 2, OK? 01:15:22.180 --> 01:15:27.265 align:middle line:84% So suppose we consider the rest frame. 01:15:27.265 --> 01:15:30.390 align:middle line:90% 01:15:30.390 --> 01:15:35.120 align:middle line:84% Suppose we go into the rest frame of particle 2. 01:15:35.120 --> 01:15:40.010 align:middle line:84% And then we roughly have this situation, OK? 01:15:40.010 --> 01:15:41.630 align:middle line:90% So this is particle 2. 01:15:41.630 --> 01:15:44.820 align:middle line:84% This is the particle 1, which comes to particle 2. 01:15:44.820 --> 01:15:53.520 align:middle line:84% And then in this situation, we can define the alpha to beta 01:15:53.520 --> 01:15:56.350 align:middle line:84% based on our nonrelativistic considerations. 01:15:56.350 --> 01:15:58.450 align:middle line:90% And then we consider-- 01:15:58.450 --> 01:16:07.990 align:middle line:84% so here I use the probability from alpha to beta dP, OK? 01:16:07.990 --> 01:16:12.970 align:middle line:84% So this essentially replaces that. 01:16:12.970 --> 01:16:13.600 align:middle line:90% OK? 01:16:13.600 --> 01:16:16.510 align:middle line:84% Here the final particle only has one particle. 01:16:16.510 --> 01:16:19.490 align:middle line:90% So you can define a solid angle. 01:16:19.490 --> 01:16:24.370 align:middle line:84% But if we have 10,000 particles here in the beta, of course, 01:16:24.370 --> 01:16:25.960 align:middle line:84% I cannot define a solid angle, OK? 01:16:25.960 --> 01:16:30.370 align:middle line:84% So I can just talk about probability from alpha to beta, 01:16:30.370 --> 01:16:31.720 align:middle line:90% OK? 01:16:31.720 --> 01:16:34.390 align:middle line:84% From alpha to beta, but per unit time. 01:16:34.390 --> 01:16:40.120 align:middle line:84% And then again, we divide it by the incident flux of 1. 01:16:40.120 --> 01:16:43.550 align:middle line:90% 01:16:43.550 --> 01:16:44.690 align:middle line:90% OK? 01:16:44.690 --> 01:16:47.940 align:middle line:90% The incident flux of 1, OK? 01:16:47.940 --> 01:16:48.510 align:middle line:90% Oh. 01:16:48.510 --> 01:16:51.710 align:middle line:90% And let me just call it in. 01:16:51.710 --> 01:16:56.690 align:middle line:84% And 1 means the number of particles of 1, OK? 01:16:56.690 --> 01:16:59.390 align:middle line:90% So yeah. 01:16:59.390 --> 01:17:04.160 align:middle line:84% So what we will do is that we will first 01:17:04.160 --> 01:17:08.540 align:middle line:84% calculate this object in the rest frame of 2. 01:17:08.540 --> 01:17:13.460 align:middle line:84% And then we construct the Lorentz invariant version 01:17:13.460 --> 01:17:15.140 align:middle line:90% of this, OK? 01:17:15.140 --> 01:17:19.490 align:middle line:84% So that's our strategy to define what 01:17:19.490 --> 01:17:21.540 align:middle line:84% will be the relativistic generalization of this. 01:17:21.540 --> 01:17:22.040 align:middle line:90% OK. 01:17:22.040 --> 01:17:25.960 align:middle line:84% So we will discuss this next time. 01:17:25.960 --> 01:17:30.000 align:middle line:90%