WEBVTT 00:00:00.000 --> 00:00:01.479 align:middle line:90% [SQUEAKING] 00:00:01.479 --> 00:00:02.958 align:middle line:90% [RUSTLING] 00:00:02.958 --> 00:00:05.423 align:middle line:90% [CLICKING] 00:00:05.423 --> 00:00:10.670 align:middle line:90% 00:00:10.670 --> 00:00:12.880 align:middle line:90% HONG LIU: Let us start. 00:00:12.880 --> 00:00:21.120 align:middle line:84% So last lecture, we discussed the Compton scattering. 00:00:21.120 --> 00:00:23.220 align:middle line:84% So with that example essentially we 00:00:23.220 --> 00:00:29.970 align:middle line:84% covered all the tree-level diagrams in QED. 00:00:29.970 --> 00:00:34.680 align:middle line:84% So tree-level diagrams means those not involving loop. 00:00:34.680 --> 00:00:59.120 align:middle line:84% OK so far encountered diagrams, which are called the tree-level 00:00:59.120 --> 00:01:06.140 align:middle line:90% -- without loops. 00:01:06.140 --> 00:01:10.675 align:middle line:84% So these are called just tree-level diagrams. 00:01:10.675 --> 00:01:18.280 align:middle line:90% 00:01:18.280 --> 00:01:24.970 align:middle line:84% And so now you should be able to actually treat 00:01:24.970 --> 00:01:26.980 align:middle line:84% all possible tree-level diagrams. 00:01:26.980 --> 00:01:29.950 align:middle line:84% You already have all the tools to treat 00:01:29.950 --> 00:01:33.800 align:middle line:84% all possible tree-level diagrams. 00:01:33.800 --> 00:01:37.010 align:middle line:84% But tree-level diagrams, actually there 00:01:37.010 --> 00:01:40.655 align:middle line:84% are many important things missing just 00:01:40.655 --> 00:01:41.780 align:middle line:90% in the tree-level diagrams. 00:01:41.780 --> 00:01:44.450 align:middle line:90% 00:01:44.450 --> 00:01:50.550 align:middle line:84% So now today we will talk about how to go beyond that, 00:01:50.550 --> 00:01:52.820 align:middle line:84% so we talk about quantum fluctuations. 00:01:52.820 --> 00:02:11.230 align:middle line:90% 00:02:11.230 --> 00:02:14.700 align:middle line:84% So this is a very big subject and actually will 00:02:14.700 --> 00:02:20.710 align:middle line:84% consist of a big chunk of quantum field theory II. 00:02:20.710 --> 00:02:22.860 align:middle line:84% So here it's more just to give you 00:02:22.860 --> 00:02:29.145 align:middle line:84% a preview of the basic idea of the subject. 00:02:29.145 --> 00:02:32.620 align:middle line:90% 00:02:32.620 --> 00:02:35.860 align:middle line:84% So one feature of the tree-level diagram we have encountered. 00:02:35.860 --> 00:02:47.620 align:middle line:90% 00:02:47.620 --> 00:02:50.050 align:middle line:84% So if you want to summarize the say 00:02:50.050 --> 00:02:52.405 align:middle line:84% features of tree-level diagrams, what would you say? 00:02:52.405 --> 00:02:59.300 align:middle line:90% 00:02:59.300 --> 00:03:00.770 align:middle line:90% Other than the looks. 00:03:00.770 --> 00:03:02.780 align:middle line:90% Other than looks like tree. 00:03:02.780 --> 00:03:03.740 align:middle line:90% Yes? 00:03:03.740 --> 00:03:07.130 align:middle line:84% AUDIENCE: There's like no free momenta. 00:03:07.130 --> 00:03:08.060 align:middle line:90% HONG LIU: Exactly. 00:03:08.060 --> 00:03:22.380 align:middle line:84% So the key is that all intermediate momentum 00:03:22.380 --> 00:03:29.940 align:middle line:84% say in all propagators, all intermediate momenta, 00:03:29.940 --> 00:03:37.370 align:middle line:84% they are determined from external ones. 00:03:37.370 --> 00:03:45.220 align:middle line:90% 00:03:45.220 --> 00:03:48.400 align:middle line:84% So when you treat the tree-level diagram, 00:03:48.400 --> 00:03:52.780 align:middle line:90% there's no momentum integration. 00:03:52.780 --> 00:04:03.960 align:middle line:90% 00:04:03.960 --> 00:04:08.370 align:middle line:84% So this looks like a very simple mathematical feature, 00:04:08.370 --> 00:04:11.100 align:middle line:84% but actually it turns out there's 00:04:11.100 --> 00:04:15.930 align:middle line:84% a profound difference, both mathematically and physically. 00:04:15.930 --> 00:04:23.160 align:middle line:84% When you include the loops, there are profound differences. 00:04:23.160 --> 00:04:29.630 align:middle line:90% And also you can show that the-- 00:04:29.630 --> 00:04:32.420 align:middle line:90% So now let me just summarize. 00:04:32.420 --> 00:04:45.160 align:middle line:84% So now when including the loops, there are many 00:04:45.160 --> 00:04:48.370 align:middle line:90% new features arises. 00:04:48.370 --> 00:04:51.880 align:middle line:84% And so let me just first give you some simple examples 00:04:51.880 --> 00:04:53.260 align:middle line:90% including loops. 00:04:53.260 --> 00:04:58.630 align:middle line:84% So let's just consider say for example one is just consider 00:04:58.630 --> 00:05:03.370 align:middle line:84% the electron propagator and it's just a line. 00:05:03.370 --> 00:05:06.550 align:middle line:84% But now you can consider adding a photon line here. 00:05:06.550 --> 00:05:09.210 align:middle line:90% 00:05:09.210 --> 00:05:10.840 align:middle line:90% Adding a photon line. 00:05:10.840 --> 00:05:14.400 align:middle line:84% So this is the process and now you have a loop. 00:05:14.400 --> 00:05:18.850 align:middle line:84% It consists of this photon line and this electron line. 00:05:18.850 --> 00:05:22.060 align:middle line:84% And so you can consider this as a process 00:05:22.060 --> 00:05:26.170 align:middle line:84% that the electron propagates, and then can emit a photon, 00:05:26.170 --> 00:05:28.400 align:middle line:90% and then it absorbs itself. 00:05:28.400 --> 00:05:29.920 align:middle line:84% So this photon is purely virtual. 00:05:29.920 --> 00:05:33.250 align:middle line:90% 00:05:33.250 --> 00:05:35.620 align:middle line:90% It's purely a virtual photon. 00:05:35.620 --> 00:05:37.360 align:middle line:84% But such kind of process actually 00:05:37.360 --> 00:05:42.180 align:middle line:84% affects the propagation of an electron. 00:05:42.180 --> 00:05:45.690 align:middle line:84% So you can view these photons as something 00:05:45.690 --> 00:05:48.300 align:middle line:90% fluctuate from the vacuum. 00:05:48.300 --> 00:05:51.870 align:middle line:84% So you can imagine these come from the vacuum fluctuations. 00:05:51.870 --> 00:05:54.500 align:middle line:90% 00:05:54.500 --> 00:05:56.330 align:middle line:90% So this is one process. 00:05:56.330 --> 00:05:58.460 align:middle line:84% And another simple process, if you consider 00:05:58.460 --> 00:05:59.510 align:middle line:90% propagation of a photon. 00:05:59.510 --> 00:06:02.250 align:middle line:90% 00:06:02.250 --> 00:06:07.610 align:middle line:84% And then the photon can actually break up, 00:06:07.610 --> 00:06:10.550 align:middle line:84% say can compare create the electron and the positron pair. 00:06:10.550 --> 00:06:13.570 align:middle line:90% 00:06:13.570 --> 00:06:20.630 align:middle line:84% And then this pair can annihilate again and then 00:06:20.630 --> 00:06:22.100 align:middle line:90% to form another photon. 00:06:22.100 --> 00:06:26.190 align:middle line:90% 00:06:26.190 --> 00:06:31.230 align:middle line:84% Again, this is a correction to just a simple propagation 00:06:31.230 --> 00:06:32.200 align:middle line:90% of a single photon. 00:06:32.200 --> 00:06:34.080 align:middle line:84% So initial state is still one photon 00:06:34.080 --> 00:06:36.180 align:middle line:84% and the final state is still one photon, 00:06:36.180 --> 00:06:38.610 align:middle line:84% but something happens in between. 00:06:38.610 --> 00:06:42.810 align:middle line:84% And again, this something that happens in between, 00:06:42.810 --> 00:06:47.670 align:middle line:84% you can interpret it as some kind of vacuum fluctuation. 00:06:47.670 --> 00:06:51.780 align:middle line:84% So both electron and positron here are completely virtual. 00:06:51.780 --> 00:06:53.640 align:middle line:84% There's no real electrons you can detect. 00:06:53.640 --> 00:06:56.660 align:middle line:90% 00:06:56.660 --> 00:07:00.870 align:middle line:84% So the third simple process is this one, 00:07:00.870 --> 00:07:06.640 align:middle line:84% remember we have the electron, the photon interacting vertex. 00:07:06.640 --> 00:07:10.030 align:middle line:90% So this is i e gamma mu. 00:07:10.030 --> 00:07:13.090 align:middle line:84% So this is the electron phonon interaction vertex. 00:07:13.090 --> 00:07:15.400 align:middle line:84% And this vertex can also be corrected. 00:07:15.400 --> 00:07:18.700 align:middle line:90% So imagine you have a photon. 00:07:18.700 --> 00:07:23.250 align:middle line:84% So imagine the electron when it comes to and then first 00:07:23.250 --> 00:07:27.750 align:middle line:84% emit a photon and then get absorbed by the electron which 00:07:27.750 --> 00:07:29.960 align:middle line:90% comes out. 00:07:29.960 --> 00:07:34.270 align:middle line:84% And so this will correct the vertex, 00:07:34.270 --> 00:07:36.190 align:middle line:84% and so this is called the vertex corrections. 00:07:36.190 --> 00:07:39.880 align:middle line:84% So these two correct the propagator and this one 00:07:39.880 --> 00:07:41.650 align:middle line:90% is the corrected vertex. 00:07:41.650 --> 00:07:44.560 align:middle line:84% So these are the simplest loop diagrams. 00:07:44.560 --> 00:07:46.360 align:middle line:84% These are the simplest loop diagrams. 00:07:46.360 --> 00:07:51.940 align:middle line:84% In each of them here, there's a single loop 00:07:51.940 --> 00:07:55.910 align:middle line:84% and then there's a free momentum in the loop. 00:07:55.910 --> 00:07:59.200 align:middle line:84% So you have to integrate over the momentum in this loop 00:07:59.200 --> 00:08:02.670 align:middle line:84% because it's no longer determined by external momenta. 00:08:02.670 --> 00:08:06.200 align:middle line:84% It's no longer determined by external. 00:08:06.200 --> 00:08:10.580 align:middle line:84% So it turns out such kind of momentum integration, 00:08:10.580 --> 00:08:17.750 align:middle line:84% so if I call this momentum k, and then you were involving-- 00:08:17.750 --> 00:08:21.320 align:middle line:90% 00:08:21.320 --> 00:08:23.540 align:middle line:84% So let me first talk about the mathematical features. 00:08:23.540 --> 00:08:27.930 align:middle line:90% 00:08:27.930 --> 00:08:38.390 align:middle line:84% So mathematical new features for loop diagram. 00:08:38.390 --> 00:08:44.243 align:middle line:84% So mathematically now you need to integrate over 00:08:44.243 --> 00:08:45.160 align:middle line:90% the internal momentum. 00:08:45.160 --> 00:08:50.320 align:middle line:90% 00:08:50.320 --> 00:08:53.740 align:middle line:84% So now you have some free momentum to integrate. 00:08:53.740 --> 00:08:56.440 align:middle line:90% 00:08:56.440 --> 00:09:01.390 align:middle line:84% So this may look like something just not very-- 00:09:01.390 --> 00:09:03.430 align:middle line:84% you may say this is not a big deal, 00:09:03.430 --> 00:09:06.950 align:middle line:84% but actually this turns out to be a very big deal 00:09:06.950 --> 00:09:12.380 align:middle line:84% because this momentum, this integration, just 00:09:12.380 --> 00:09:16.560 align:middle line:84% by definition, is you integrate over all possible momentum. 00:09:16.560 --> 00:09:19.110 align:middle line:90% 00:09:19.110 --> 00:09:21.180 align:middle line:84% So each k is integrated from minus infinity 00:09:21.180 --> 00:09:23.520 align:middle line:90% to plus infinity. 00:09:23.520 --> 00:09:32.690 align:middle line:84% So in particular, this includes the region 00:09:32.690 --> 00:09:37.320 align:middle line:84% that k0 can go to infinity and the magnitude of k 00:09:37.320 --> 00:09:38.670 align:middle line:90% can also go to infinity. 00:09:38.670 --> 00:09:42.380 align:middle line:90% 00:09:42.380 --> 00:09:45.000 align:middle line:90% So we include those regions. 00:09:45.000 --> 00:09:48.950 align:middle line:84% So you have this infinite range of integrals, 00:09:48.950 --> 00:09:52.450 align:middle line:84% then there's a possibility you get divergence. 00:09:52.450 --> 00:09:54.200 align:middle line:84% There is a possibility you get divergence. 00:09:54.200 --> 00:09:58.300 align:middle line:84% And in fact, the divergence does happen 00:09:58.300 --> 00:10:01.390 align:middle line:90% and happened almost everywhere. 00:10:01.390 --> 00:10:03.760 align:middle line:84% Whenever you look at loop diagrams, 00:10:03.760 --> 00:10:07.200 align:middle line:84% you always, essentially always, find the divergences. 00:10:07.200 --> 00:10:12.160 align:middle line:90% 00:10:12.160 --> 00:10:18.470 align:middle line:90% So this leads to divergences. 00:10:18.470 --> 00:10:24.150 align:middle line:90% 00:10:24.150 --> 00:10:28.750 align:middle line:84% And such divergences are now more complicated. 00:10:28.750 --> 00:10:30.720 align:middle line:84% It's not like the simple divergences. 00:10:30.720 --> 00:10:34.050 align:middle line:84% We already considered, we've already 00:10:34.050 --> 00:10:36.570 align:middle line:84% seen where is divergent quantities, 00:10:36.570 --> 00:10:38.640 align:middle line:84% and so far those divergent quantities 00:10:38.640 --> 00:10:41.620 align:middle line:84% can be treated in a more or less simple way. 00:10:41.620 --> 00:10:46.390 align:middle line:84% So we add or subtract the infinite constant 00:10:46.390 --> 00:10:49.480 align:middle line:84% or you multiply by the infinite constant. 00:10:49.480 --> 00:10:52.120 align:middle line:84% And in the case of when you multiply by infinite constant, 00:10:52.120 --> 00:10:54.220 align:middle line:84% you can always take the ratio and then 00:10:54.220 --> 00:10:55.720 align:middle line:84% those infinite constants go away. 00:10:55.720 --> 00:10:58.210 align:middle line:84% Or you can just subtract the infinite constant, 00:10:58.210 --> 00:11:02.730 align:middle line:84% redefine your energy, redefine your charge, et cetera. 00:11:02.730 --> 00:11:04.527 align:middle line:84% But in those diagrams, the divergences 00:11:04.527 --> 00:11:06.110 align:middle line:84% are much more complicated because they 00:11:06.110 --> 00:11:11.820 align:middle line:84% come from doing some integration of some complicated expression. 00:11:11.820 --> 00:11:15.000 align:middle line:84% And so those divergences cannot be simply removed where you 00:11:15.000 --> 00:11:16.950 align:middle line:84% just add the constant, or remove a constant, 00:11:16.950 --> 00:11:18.195 align:middle line:90% or just multiply by a factor. 00:11:18.195 --> 00:11:20.740 align:middle line:90% 00:11:20.740 --> 00:11:26.980 align:middle line:84% And so they actually cannot be removed. 00:11:26.980 --> 00:11:29.560 align:middle line:90% 00:11:29.560 --> 00:11:39.660 align:middle line:84% In general, cannot be removed simply by say subtractions. 00:11:39.660 --> 00:11:55.610 align:middle line:84% Removed by simple subtractions or divisions. 00:11:55.610 --> 00:11:59.240 align:middle line:90% 00:11:59.240 --> 00:12:00.920 align:middle line:84% So in other words, the expression 00:12:00.920 --> 00:12:02.660 align:middle line:84% you get from doing such integration just 00:12:02.660 --> 00:12:03.500 align:middle line:90% don't make sense. 00:12:03.500 --> 00:12:06.020 align:middle line:90% 00:12:06.020 --> 00:12:19.258 align:middle line:84% So the results from these diagrams are not well-defined. 00:12:19.258 --> 00:12:35.920 align:middle line:90% 00:12:35.920 --> 00:12:39.030 align:middle line:84% So the result is not well-defined. 00:12:39.030 --> 00:12:47.580 align:middle line:84% So normally when we encounter divergences, 00:12:47.580 --> 00:12:51.060 align:middle line:84% we always tell a student we encounter divergences 00:12:51.060 --> 00:12:55.980 align:middle line:84% of course, one obvious reason is that you did something wrong. 00:12:55.980 --> 00:12:59.250 align:middle line:90% You just made a mistake. 00:12:59.250 --> 00:13:01.890 align:middle line:84% But if you have done everything right, 00:13:01.890 --> 00:13:05.550 align:middle line:84% when you encounter divergences, actually that's an opportunity. 00:13:05.550 --> 00:13:07.458 align:middle line:84% Because each time you see a divergence, 00:13:07.458 --> 00:13:09.000 align:middle line:84% then that means there's something you 00:13:09.000 --> 00:13:11.600 align:middle line:90% don't understand about physics. 00:13:11.600 --> 00:13:14.240 align:middle line:84% If you understand, by definition, in physics 00:13:14.240 --> 00:13:18.020 align:middle line:84% you do not have divergences, if you understand everything well. 00:13:18.020 --> 00:13:23.990 align:middle line:84% And so divergence is mathematics telling you 00:13:23.990 --> 00:13:26.080 align:middle line:84% there's some important physics you're missing. 00:13:26.080 --> 00:13:30.960 align:middle line:90% 00:13:30.960 --> 00:13:32.460 align:middle line:84% So when you look at this divergence, 00:13:32.460 --> 00:13:36.540 align:middle line:84% you can say, oh, what physics are we missing here? 00:13:36.540 --> 00:13:40.270 align:middle line:84% So the physics missing here is very simple. 00:13:40.270 --> 00:13:43.080 align:middle line:84% The physics missing here is very simple. 00:13:43.080 --> 00:13:46.350 align:middle line:84% Because these integrals, because the divergent region 00:13:46.350 --> 00:13:49.437 align:middle line:84% corresponding to those quantities go to infinity, 00:13:49.437 --> 00:13:51.020 align:middle line:84% so those quantities go to the infinity 00:13:51.020 --> 00:13:53.010 align:middle line:84% because one is you have very high energy, 00:13:53.010 --> 00:13:54.005 align:middle line:90% very large momentum. 00:13:54.005 --> 00:13:57.800 align:middle line:90% 00:13:57.800 --> 00:14:03.580 align:middle line:84% In other words, from very short distance physics. 00:14:03.580 --> 00:14:07.030 align:middle line:84% But very short, that comes from-- 00:14:07.030 --> 00:14:10.470 align:middle line:90% 00:14:10.470 --> 00:14:22.170 align:middle line:84% So divergences come from naive extrapolation. 00:14:22.170 --> 00:14:28.100 align:middle line:90% 00:14:28.100 --> 00:14:42.240 align:middle line:84% They say consider QED to arbitrarily high energy scales. 00:14:42.240 --> 00:14:48.900 align:middle line:90% 00:14:48.900 --> 00:14:52.290 align:middle line:84% Because we used the framework of QED 00:14:52.290 --> 00:14:57.110 align:middle line:84% but when we do this integration involving very large momentum. 00:14:57.110 --> 00:15:00.380 align:middle line:84% But nobody actually tells us that a QED should be valid 00:15:00.380 --> 00:15:03.980 align:middle line:84% at not very high energies because the real experiment you 00:15:03.980 --> 00:15:09.710 align:middle line:84% cannot probe very high energy process or very short distance 00:15:09.710 --> 00:15:13.780 align:middle line:84% because it's the limit of our energy resolution. 00:15:13.780 --> 00:15:16.760 align:middle line:84% So there's no reason to trust actually QED 00:15:16.760 --> 00:15:22.280 align:middle line:84% should be valid for k 0 and k to infinity. 00:15:22.280 --> 00:15:23.060 align:middle line:90% Yes? 00:15:23.060 --> 00:15:26.780 align:middle line:84% AUDIENCE: Is the divergence the result of the problem 00:15:26.780 --> 00:15:30.730 align:middle line:90% in the perturbative expansion? 00:15:30.730 --> 00:15:35.370 align:middle line:84% Or is it a problem with the original [INAUDIBLE]?? 00:15:35.370 --> 00:15:37.500 align:middle line:84% HONG LIU: Yeah, it's not a feature 00:15:37.500 --> 00:15:39.000 align:middle line:90% of the perturbative expansion. 00:15:39.000 --> 00:15:41.550 align:middle line:84% AUDIENCE: So it's a legitimate problem. 00:15:41.550 --> 00:15:43.300 align:middle line:84% HONG LIU: Yeah, it's a legitimate problem. 00:15:43.300 --> 00:15:48.440 align:middle line:90% 00:15:48.440 --> 00:15:51.668 align:middle line:84% So this divergence just comes from we extrapolate QED 00:15:51.668 --> 00:15:53.085 align:middle line:84% to arbitrary higher energy scales. 00:15:53.085 --> 00:15:56.270 align:middle line:90% 00:15:56.270 --> 00:16:01.770 align:middle line:84% So you say, OK, then what do we do? 00:16:01.770 --> 00:16:02.880 align:middle line:90% Then what do we do? 00:16:02.880 --> 00:16:09.530 align:middle line:84% So the simplest thing you can do you say, I only can measure 00:16:09.530 --> 00:16:15.170 align:middle line:84% suppose in the 1930s when people first encountered this problem. 00:16:15.170 --> 00:16:20.000 align:middle line:84% And you say, OK, and then let's put the cutoff. 00:16:20.000 --> 00:16:24.560 align:middle line:84% Here let's only include the energy regime 00:16:24.560 --> 00:16:25.835 align:middle line:90% which we can access. 00:16:25.835 --> 00:16:29.460 align:middle line:90% 00:16:29.460 --> 00:16:34.110 align:middle line:84% But again, what cutoff do you put here? 00:16:34.110 --> 00:16:36.510 align:middle line:84% So it's not clear what kind of cutoff you put there. 00:16:36.510 --> 00:16:40.170 align:middle line:84% What is the validity of this theory? 00:16:40.170 --> 00:16:42.450 align:middle line:84% So ideally you should put a cutoff there 00:16:42.450 --> 00:16:50.820 align:middle line:84% to add the boundary of the validity of the QED. 00:16:50.820 --> 00:16:53.790 align:middle line:84% But before you actually can probe the scale 00:16:53.790 --> 00:16:56.850 align:middle line:90% which QED breaks down. 00:16:56.850 --> 00:16:58.690 align:middle line:90% Which scale should we put there? 00:16:58.690 --> 00:17:00.690 align:middle line:90% So we don't know. 00:17:00.690 --> 00:17:04.819 align:middle line:84% So for many years, people thought 00:17:04.819 --> 00:17:08.060 align:middle line:84% just doing these kinds of diagrams 00:17:08.060 --> 00:17:11.359 align:middle line:84% just doesn't make physical sense because we don't know 00:17:11.359 --> 00:17:13.800 align:middle line:90% the short distance physics. 00:17:13.800 --> 00:17:17.200 align:middle line:84% We don't know the short distance physics. 00:17:17.200 --> 00:17:23.760 align:middle line:84% And indeed, this point of view was justified. 00:17:23.760 --> 00:17:30.080 align:middle line:84% So for some years people just gave up on these ideas. 00:17:30.080 --> 00:17:32.890 align:middle line:84% They just said, maybe this does not make sense. 00:17:32.890 --> 00:17:39.520 align:middle line:84% But then after the Second World War in the late 40s, 00:17:39.520 --> 00:17:42.370 align:middle line:84% a few young physicists at the time, 00:17:42.370 --> 00:17:48.490 align:middle line:84% Schwinger, Feynman in the United States and Tomonaga in Japan, 00:17:48.490 --> 00:17:53.470 align:middle line:84% they actually say, OK, even though there 00:17:53.470 --> 00:17:56.830 align:middle line:84% are these kinds of divergences, and even though we 00:17:56.830 --> 00:17:59.110 align:middle line:84% don't understand the short distance physics, 00:17:59.110 --> 00:18:03.070 align:middle line:84% let's see whether we can just use brute force, 00:18:03.070 --> 00:18:07.150 align:middle line:84% try to design some mathematical way to get rid 00:18:07.150 --> 00:18:09.550 align:middle line:84% of those divergences, and see whether I 00:18:09.550 --> 00:18:12.540 align:middle line:90% can get sensible results. 00:18:12.540 --> 00:18:15.030 align:middle line:84% And so that was their starting point, 00:18:15.030 --> 00:18:17.430 align:middle line:84% and that was their starting point. 00:18:17.430 --> 00:18:19.740 align:middle line:84% And it was a crazy starting point 00:18:19.740 --> 00:18:22.020 align:middle line:84% because of the physical reason for those divergences 00:18:22.020 --> 00:18:24.090 align:middle line:90% are very clear. 00:18:24.090 --> 00:18:27.540 align:middle line:84% But then it turned out that they actually 00:18:27.540 --> 00:18:30.030 align:middle line:90% managed to be successful. 00:18:30.030 --> 00:18:43.570 align:middle line:84% So in the late 1940s people found the method. 00:18:43.570 --> 00:18:49.240 align:middle line:84% So Feynman, Schwinger, and Tomonaga 00:18:49.240 --> 00:19:13.750 align:middle line:84% managed to find a way to remove the divergences 00:19:13.750 --> 00:19:31.595 align:middle line:84% and create a physically sensible answer from such diagrams. 00:19:31.595 --> 00:19:39.990 align:middle line:90% 00:19:39.990 --> 00:19:44.700 align:middle line:84% And also, at the beginning, their work 00:19:44.700 --> 00:19:47.610 align:middle line:84% was very complicated and very few people actually 00:19:47.610 --> 00:19:49.110 align:middle line:90% understood them. 00:19:49.110 --> 00:19:52.620 align:middle line:84% And then Feynman used Feynman diagrams, 00:19:52.620 --> 00:19:55.710 align:middle line:84% so he invented the Feynman diagrams to do this. 00:19:55.710 --> 00:19:57.960 align:middle line:84% Because previously people do those calculations. 00:19:57.960 --> 00:20:01.380 align:middle line:84% They don't use Feynman diagrams and they use very complicated 00:20:01.380 --> 00:20:04.110 align:middle line:84% perturbation theory, as you do in 00:20:04.110 --> 00:20:06.390 align:middle line:90% the long relativistic physics. 00:20:06.390 --> 00:20:08.580 align:middle line:84% It's very complicated, and actually that's 00:20:08.580 --> 00:20:12.340 align:middle line:84% what the Schwinger and the Tomonaga they were doing. 00:20:12.340 --> 00:20:14.940 align:middle line:84% So at the beginning, this thing was very complicated, 00:20:14.940 --> 00:20:19.770 align:middle line:84% but then Freeman Dyson soon afterwards, he 00:20:19.770 --> 00:20:23.630 align:middle line:84% understood all their work and actually greatly simplified 00:20:23.630 --> 00:20:25.260 align:middle line:90% the process. 00:20:25.260 --> 00:20:28.020 align:middle line:84% And actually, the current understanding 00:20:28.020 --> 00:20:30.630 align:middle line:84% essentially came from the simplification 00:20:30.630 --> 00:20:31.950 align:middle line:90% of the Freeman Dyson. 00:20:31.950 --> 00:20:35.410 align:middle line:84% And the Freeman Dyson actually operated a Feynman diagram fact 00:20:35.410 --> 00:20:35.910 align:middle line:90% -- 00:20:35.910 --> 00:20:38.580 align:middle line:90% Feynman diagram training -- 00:20:38.580 --> 00:20:42.660 align:middle line:84% essentially operated the Feynman diagram training at camp, 00:20:42.660 --> 00:20:46.040 align:middle line:84% at the institute for advanced study when he was a member. 00:20:46.040 --> 00:20:48.060 align:middle line:84% So people went through there, they all 00:20:48.060 --> 00:20:51.210 align:middle line:84% learned this Feynman diagram techniques from Freeman Dyson, 00:20:51.210 --> 00:20:53.970 align:middle line:84% and then they spread it out to all other places. 00:20:53.970 --> 00:20:57.900 align:middle line:84% That's how Feynman diagram techniques got spread out 00:20:57.900 --> 00:20:59.790 align:middle line:90% in the physical community. 00:20:59.790 --> 00:21:05.790 align:middle line:84% And anyway, so Freeman Dyson, he simplified the process, 00:21:05.790 --> 00:21:09.900 align:middle line:84% and then people eventually understood. 00:21:09.900 --> 00:21:14.350 align:middle line:84% And there is a well-defined mathematical procedure 00:21:14.350 --> 00:21:16.810 align:middle line:84% you can actually get rid of those divergences 00:21:16.810 --> 00:21:21.880 align:middle line:84% and get a physically sensible answer. 00:21:21.880 --> 00:21:24.270 align:middle line:90% And so it was a funny story. 00:21:24.270 --> 00:21:27.110 align:middle line:90% 00:21:27.110 --> 00:21:29.722 align:middle line:84% This is called the program of renormalization. 00:21:29.722 --> 00:21:32.490 align:middle line:90% 00:21:32.490 --> 00:21:35.370 align:middle line:84% The process of getting rid of the divergences. 00:21:35.370 --> 00:21:41.720 align:middle line:90% 00:21:41.720 --> 00:21:44.890 align:middle line:90% And this is a funny story. 00:21:44.890 --> 00:21:51.120 align:middle line:84% It's because, as I said, there's a very physical-- 00:21:51.120 --> 00:21:54.510 align:middle line:84% This is a very funny story for the following reason. 00:21:54.510 --> 00:21:59.880 align:middle line:84% As I mentioned, we do know where the divergences come from. 00:21:59.880 --> 00:22:02.850 align:middle line:84% Divergences come from our lack understanding of short distance 00:22:02.850 --> 00:22:04.100 align:middle line:90% physics. 00:22:04.100 --> 00:22:08.690 align:middle line:84% And normally you believe you can only cure the divergence 00:22:08.690 --> 00:22:10.400 align:middle line:90% if you understand that physics. 00:22:10.400 --> 00:22:12.020 align:middle line:84% If you don't understand that physics, 00:22:12.020 --> 00:22:15.170 align:middle line:90% how you cure that divergence? 00:22:15.170 --> 00:22:17.700 align:middle line:84% So that's what most people thought at the time, 00:22:17.700 --> 00:22:20.690 align:middle line:84% including all these quantum mechanics big shots. 00:22:20.690 --> 00:22:24.740 align:middle line:84% All these old people, like Dirac, Pauli, 00:22:24.740 --> 00:22:27.020 align:middle line:84% all those giants of quantum mechanics. 00:22:27.020 --> 00:22:31.880 align:middle line:84% They all believed you have to do something radical to QED, 00:22:31.880 --> 00:22:33.755 align:middle line:84% otherwise QED cannot be made sense. 00:22:33.755 --> 00:22:37.070 align:middle line:84% You have to invent some completely new formalism 00:22:37.070 --> 00:22:41.420 align:middle line:84% to circumvent those divergences exactly. 00:22:41.420 --> 00:22:48.930 align:middle line:84% But then Dyson later commented that those young people at 00:22:48.930 --> 00:22:51.810 align:middle line:84% the time they said we were conservative. 00:22:51.810 --> 00:22:54.153 align:middle line:84% We want you just to look at those calculations 00:22:54.153 --> 00:22:56.070 align:middle line:84% just want to make sense of those calculations. 00:22:56.070 --> 00:22:58.110 align:middle line:84% We don't think of those grand physics. 00:22:58.110 --> 00:23:00.330 align:middle line:84% I just want to make sense of those calculations, 00:23:00.330 --> 00:23:02.040 align:middle line:90% and then they succeeded. 00:23:02.040 --> 00:23:04.920 align:middle line:84% So this is an example in which very old people, 00:23:04.920 --> 00:23:07.830 align:middle line:84% they want to do grand new physics, 00:23:07.830 --> 00:23:10.350 align:middle line:84% but young people wanted to do conservative physics, just 00:23:10.350 --> 00:23:15.020 align:middle line:84% two calculations, one calculation at a time 00:23:15.020 --> 00:23:18.390 align:middle line:84% without clear physical motivation. 00:23:18.390 --> 00:23:21.330 align:middle line:84% But those young people turned out to be correct, 00:23:21.330 --> 00:23:23.700 align:middle line:84% and so they came up with this program. 00:23:23.700 --> 00:23:24.360 align:middle line:90% Yes? 00:23:24.360 --> 00:23:28.860 align:middle line:84% AUDIENCE: So [INAUDIBLE] resolve is 00:23:28.860 --> 00:23:33.380 align:middle line:84% more accurate than the tree-level diagram? 00:23:33.380 --> 00:23:35.467 align:middle line:90% And then how much more accurate? 00:23:35.467 --> 00:23:37.050 align:middle line:84% HONG LIU: No, it's not about accuracy. 00:23:37.050 --> 00:23:41.880 align:middle line:84% It's about finding a sensible way to hide the divergences. 00:23:41.880 --> 00:23:44.325 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:23:44.325 --> 00:23:44.950 align:middle line:90% HONG LIU: Yeah. 00:23:44.950 --> 00:23:47.987 align:middle line:90% 00:23:47.987 --> 00:23:50.320 align:middle line:84% Because today we will not be able to go into the detail. 00:23:50.320 --> 00:23:54.070 align:middle line:84% I'm going to only tell you the result. 00:23:54.070 --> 00:23:57.970 align:middle line:84% So that's why their result was actually radical in some way. 00:23:57.970 --> 00:24:02.130 align:middle line:84% Because they realized there is some way somehow 00:24:02.130 --> 00:24:04.290 align:middle line:84% those divergences, even they are there, 00:24:04.290 --> 00:24:07.560 align:middle line:84% they do not affect those observable physics 00:24:07.560 --> 00:24:11.570 align:middle line:90% you can measure. 00:24:11.570 --> 00:24:13.410 align:middle line:90% They don't affect those things. 00:24:13.410 --> 00:24:16.160 align:middle line:84% So you can find some mathematical trick 00:24:16.160 --> 00:24:17.840 align:middle line:90% to remove them. 00:24:17.840 --> 00:24:27.550 align:middle line:90% 00:24:27.550 --> 00:24:30.970 align:middle line:84% But for many years, so after they succeeded, 00:24:30.970 --> 00:24:34.270 align:middle line:84% so this is late 1940s, and the Dyson was also 00:24:34.270 --> 00:24:39.670 align:middle line:90% late 1940s, early 1950. 00:24:39.670 --> 00:24:45.100 align:middle line:84% And then as I'm going to mention at the end of today's lecture, 00:24:45.100 --> 00:24:51.400 align:middle line:84% the first calculated quantity is this anomalous magnetic moment, 00:24:51.400 --> 00:24:56.240 align:middle line:84% which now requires doing a calculation of this one. 00:24:56.240 --> 00:24:59.440 align:middle line:84% Doing a calculation of this one and Schwinger 00:24:59.440 --> 00:25:01.240 align:middle line:90% did the first calculation. 00:25:01.240 --> 00:25:05.350 align:middle line:84% And then he calculated the correction of this 00:25:05.350 --> 00:25:09.040 align:middle line:84% to the magnetic moment of electron. 00:25:09.040 --> 00:25:11.870 align:middle line:84% And that agreed very well with experiments, 00:25:11.870 --> 00:25:15.280 align:middle line:84% so that was a triumph at the time. 00:25:15.280 --> 00:25:19.680 align:middle line:84% So the experiment verified their success. 00:25:19.680 --> 00:25:23.790 align:middle line:84% But still, many people didn't believe this actually 00:25:23.790 --> 00:25:26.948 align:middle line:90% was something sensible. 00:25:26.948 --> 00:25:28.740 align:middle line:84% For many years, people thought this is just 00:25:28.740 --> 00:25:32.890 align:middle line:84% a mathematical trick to get some answer. 00:25:32.890 --> 00:25:36.000 align:middle line:84% And so many people still, for many years 00:25:36.000 --> 00:25:40.800 align:middle line:84% people were doubting this program of renormalization. 00:25:40.800 --> 00:25:47.160 align:middle line:84% And then only much later in late 1960s or 1970s, 00:25:47.160 --> 00:25:51.720 align:middle line:84% when Ken Wilson came up with this idea 00:25:51.720 --> 00:25:54.450 align:middle line:84% called renormalization group, and then 00:25:54.450 --> 00:25:58.140 align:middle line:84% people finally understood the physical reasoning 00:25:58.140 --> 00:26:00.540 align:middle line:84% and the physical basis behind this program 00:26:00.540 --> 00:26:01.875 align:middle line:90% of renormalization. 00:26:01.875 --> 00:26:04.560 align:middle line:90% 00:26:04.560 --> 00:26:12.180 align:middle line:84% So that was a revolutionary progress made by Wilson. 00:26:12.180 --> 00:26:19.610 align:middle line:84% So all this will be covered in detail in Quantum Field Theory 00:26:19.610 --> 00:26:21.605 align:middle line:90% II. 00:26:21.605 --> 00:26:25.310 align:middle line:84% And so today let me just very quickly tell you 00:26:25.310 --> 00:26:30.470 align:middle line:84% what these three loop diagrams do. 00:26:30.470 --> 00:26:33.270 align:middle line:90% 00:26:33.270 --> 00:26:37.470 align:middle line:84% So first let me make some general remarks. 00:26:37.470 --> 00:26:40.260 align:middle line:84% So mathematically, you get divergences 00:26:40.260 --> 00:26:46.890 align:middle line:84% and then you have to find a way to get rid of the divergences. 00:26:46.890 --> 00:26:48.975 align:middle line:84% And physically, there are also some new features. 00:26:48.975 --> 00:26:53.393 align:middle line:90% 00:26:53.393 --> 00:26:54.810 align:middle line:84% New important conceptual features. 00:26:54.810 --> 00:26:57.930 align:middle line:90% 00:26:57.930 --> 00:26:59.190 align:middle line:90% So any questions on this? 00:26:59.190 --> 00:27:02.130 align:middle line:90% 00:27:02.130 --> 00:27:15.240 align:middle line:84% So the first feature, I already mentioned. 00:27:15.240 --> 00:27:18.230 align:middle line:90% We include such loops. 00:27:18.230 --> 00:27:31.740 align:middle line:84% You essentially include the quantum fluctuations 00:27:31.740 --> 00:27:32.340 align:middle line:90% of the vacuum. 00:27:32.340 --> 00:27:37.450 align:middle line:90% 00:27:37.450 --> 00:27:42.390 align:middle line:84% So all this process, so that electron pair, that photon, 00:27:42.390 --> 00:27:44.970 align:middle line:84% and that photon they all some virtual particle 00:27:44.970 --> 00:27:50.650 align:middle line:84% just come from the vacuum, just from the vacuum fluctuations. 00:27:50.650 --> 00:27:56.727 align:middle line:84% So you can actually show mathematically rigorously. 00:27:56.727 --> 00:27:57.935 align:middle line:90% That the tree-level diagrams. 00:27:57.935 --> 00:28:01.370 align:middle line:90% 00:28:01.370 --> 00:28:13.880 align:middle line:84% Any tree-level diagrams can be obtained 00:28:13.880 --> 00:28:28.827 align:middle line:84% from solving your equation of motion perturbatively. 00:28:28.827 --> 00:28:40.770 align:middle line:90% 00:28:40.770 --> 00:28:45.420 align:middle line:84% So just take your nonlinear equation of motion, 00:28:45.420 --> 00:28:48.930 align:middle line:84% take the QED, you write down the nonlinear equation of motion, 00:28:48.930 --> 00:28:50.803 align:middle line:84% and then you have this coupling, e, 00:28:50.803 --> 00:28:53.220 align:middle line:84% and then you just solve that equation perturbatively order 00:28:53.220 --> 00:28:54.150 align:middle line:90% by order. 00:28:54.150 --> 00:28:56.460 align:middle line:84% And now all tree-level diagrams can be 00:28:56.460 --> 00:28:58.650 align:middle line:90% understood from that process. 00:28:58.650 --> 00:29:00.480 align:middle line:84% Essentially just coming from solving 00:29:00.480 --> 00:29:05.900 align:middle line:90% your equation of motion. 00:29:05.900 --> 00:29:08.990 align:middle line:84% But then we know that the quantum nature comes 00:29:08.990 --> 00:29:11.270 align:middle line:90% from doing the path integral. 00:29:11.270 --> 00:29:15.830 align:middle line:84% Equation of motion captures some classical physics, 00:29:15.830 --> 00:29:17.960 align:middle line:84% but the genuine quantum physics come 00:29:17.960 --> 00:29:19.940 align:middle line:84% from doing the classic path integral, which 00:29:19.940 --> 00:29:23.040 align:middle line:84% you integrate over all possible configurations. 00:29:23.040 --> 00:29:25.290 align:middle line:90% And so that is not captured. 00:29:25.290 --> 00:29:26.930 align:middle line:84% So those features of path integral 00:29:26.930 --> 00:29:28.722 align:middle line:84% are not captured by the tree-level diagram. 00:29:28.722 --> 00:29:30.850 align:middle line:90% 00:29:30.850 --> 00:29:33.760 align:middle line:84% But you may ask, does tree-level diagram actually 00:29:33.760 --> 00:29:35.170 align:middle line:90% capture any quantum physics? 00:29:35.170 --> 00:29:36.250 align:middle line:90% Of course it does. 00:29:36.250 --> 00:29:40.330 align:middle line:84% It does capture quantum physics because all this notion 00:29:40.330 --> 00:29:42.490 align:middle line:84% of the particles, even the notion 00:29:42.490 --> 00:29:46.530 align:middle line:84% of quantizing the particles, that's already 00:29:46.530 --> 00:29:49.480 align:middle line:90% included in the quantum effect. 00:29:49.480 --> 00:29:53.118 align:middle line:84% Just the tree-level diagram does not include the fluctuations 00:29:53.118 --> 00:29:53.785 align:middle line:90% from the vacuum. 00:29:53.785 --> 00:29:57.650 align:middle line:90% 00:29:57.650 --> 00:29:59.450 align:middle line:90% Good? 00:29:59.450 --> 00:30:02.330 align:middle line:84% And so this is the first feature. 00:30:02.330 --> 00:30:05.660 align:middle line:84% So when you include the loop, you actually now 00:30:05.660 --> 00:30:08.090 align:middle line:84% finally include the quantum fluctuations. 00:30:08.090 --> 00:30:13.260 align:middle line:84% And the second feature, it's very important, 00:30:13.260 --> 00:30:29.490 align:middle line:84% is that the physical mass charge and the physical fields 00:30:29.490 --> 00:30:46.285 align:middle line:84% are not the same as that appearing in the Lagrangian. 00:30:46.285 --> 00:30:54.660 align:middle line:90% 00:30:54.660 --> 00:30:56.800 align:middle line:84% So let me elaborate what this means. 00:30:56.800 --> 00:31:00.510 align:middle line:84% So remember, when we write down the Lagrangian for QED, 00:31:00.510 --> 00:31:07.230 align:middle line:84% you have i psi bar gamma mu partial mu minus i 00:31:07.230 --> 00:31:20.040 align:middle line:84% e A mu psi minus m psi minus one half one quarter F squared. 00:31:20.040 --> 00:31:21.790 align:middle line:90% So that's Lagrangian. 00:31:21.790 --> 00:31:27.960 align:middle line:84% And then we say that this is the charge of the electron, 00:31:27.960 --> 00:31:31.150 align:middle line:84% and this is the mass of the electron, 00:31:31.150 --> 00:31:35.650 align:middle line:84% and this psi denotes the electron field. 00:31:35.650 --> 00:31:38.680 align:middle line:84% Means that when you acted on the vacuum, 00:31:38.680 --> 00:31:42.250 align:middle line:84% you can create the electron or anti electron. 00:31:42.250 --> 00:31:47.680 align:middle line:84% And similarly, A mu because A mu is massless, 00:31:47.680 --> 00:31:49.690 align:middle line:84% and when A mu acting on the vacuum, 00:31:49.690 --> 00:31:51.550 align:middle line:90% it can create the photon. 00:31:51.550 --> 00:31:54.700 align:middle line:90% 00:31:54.700 --> 00:31:58.340 align:middle line:84% Turns out when you include the loop effect, 00:31:58.340 --> 00:32:01.410 align:middle line:90% the mass we actually measure. 00:32:01.410 --> 00:32:03.950 align:middle line:84% So by physical mass, means the mass 00:32:03.950 --> 00:32:06.650 align:middle line:84% of the electron we actually measure in the lab, 00:32:06.650 --> 00:32:09.620 align:middle line:84% and the charge we actually measure in the lab. 00:32:09.620 --> 00:32:13.390 align:middle line:84% They are no longer the same as the one 00:32:13.390 --> 00:32:15.420 align:middle line:90% appearing in the Lagrangian. 00:32:15.420 --> 00:32:17.550 align:middle line:84% So the y in the Lagrangian should be just viewed 00:32:17.550 --> 00:32:19.610 align:middle line:90% as some kind of parameter. 00:32:19.610 --> 00:32:23.830 align:middle line:84% And so and the geophysical mass and the charge 00:32:23.830 --> 00:32:28.190 align:middle line:84% are functions of those parameters. 00:32:28.190 --> 00:32:30.200 align:middle line:84% Now there's a more complicated relation 00:32:30.200 --> 00:32:32.755 align:middle line:84% between those parameters and your physical mass 00:32:32.755 --> 00:32:33.380 align:middle line:90% and the charge. 00:32:33.380 --> 00:32:36.120 align:middle line:90% 00:32:36.120 --> 00:32:39.450 align:middle line:84% Also, the physical field, which actually creates the electron, 00:32:39.450 --> 00:32:43.520 align:middle line:84% is no longer directly given by this psi. 00:32:43.520 --> 00:32:45.325 align:middle line:84% It's related by psi by some relation. 00:32:45.325 --> 00:32:48.880 align:middle line:90% 00:32:48.880 --> 00:32:53.640 align:middle line:84% So this difference is also called renormalization. 00:32:53.640 --> 00:32:57.870 align:middle line:90% 00:32:57.870 --> 00:33:02.790 align:middle line:84% So here this true sense of renormalization, 00:33:02.790 --> 00:33:04.650 align:middle line:84% which is actually used for completely 00:33:04.650 --> 00:33:08.800 align:middle line:90% two different contexts. 00:33:08.800 --> 00:33:11.460 align:middle line:84% So this renormalization just means the removal 00:33:11.460 --> 00:33:14.520 align:middle line:84% of the divergences, and this renormalization 00:33:14.520 --> 00:33:20.250 align:middle line:84% means the parameters in your Lagrangian 00:33:20.250 --> 00:33:23.160 align:middle line:84% may not be the same as the physical mass and the charge 00:33:23.160 --> 00:33:25.350 align:middle line:90% you measure. 00:33:25.350 --> 00:33:27.240 align:middle line:84% So the thing you measure actually 00:33:27.240 --> 00:33:29.250 align:middle line:84% it's related to the parameters in the Lagrangian 00:33:29.250 --> 00:33:31.270 align:middle line:90% in a non-trivial way. 00:33:31.270 --> 00:33:35.710 align:middle line:84% And this renormalization is in principle 00:33:35.710 --> 00:33:36.910 align:middle line:90% not due to the divergence. 00:33:36.910 --> 00:33:38.470 align:middle line:84% It's just due to the interactions. 00:33:38.470 --> 00:33:43.520 align:middle line:90% 00:33:43.520 --> 00:33:46.660 align:middle line:84% So heuristically the physical is as follows-- 00:33:46.660 --> 00:33:50.060 align:middle line:84% just imagine you have a free electron. 00:33:50.060 --> 00:33:55.130 align:middle line:84% So if you have a free electron, and then 00:33:55.130 --> 00:33:59.450 align:middle line:84% the mass of the electron, then it is what you measure. 00:33:59.450 --> 00:34:03.050 align:middle line:84% But now if you include the interactions, and now electron 00:34:03.050 --> 00:34:05.570 align:middle line:84% can interact with the photon in a complicated way, 00:34:05.570 --> 00:34:08.293 align:middle line:84% and that interaction then can change 00:34:08.293 --> 00:34:09.335 align:middle line:90% the mass of the electron. 00:34:09.335 --> 00:34:12.260 align:middle line:90% 00:34:12.260 --> 00:34:17.989 align:middle line:84% So that's why, in general for interacting theory, what 00:34:17.989 --> 00:34:21.620 align:middle line:84% appears in your Lagrangian, they don't 00:34:21.620 --> 00:34:25.949 align:middle line:84% have to be the quantity you actually measure. 00:34:25.949 --> 00:34:29.429 align:middle line:84% Because the interactions can change 00:34:29.429 --> 00:34:31.260 align:middle line:84% how you translate those quantities 00:34:31.260 --> 00:34:33.929 align:middle line:84% into the physical quantities you measure. 00:34:33.929 --> 00:34:39.423 align:middle line:90% 00:34:39.423 --> 00:34:40.840 align:middle line:84% So this renormalization just means 00:34:40.840 --> 00:34:46.030 align:middle line:84% the renormalization changes the value of your parameters. 00:34:46.030 --> 00:34:51.310 align:middle line:90% 00:34:51.310 --> 00:34:53.882 align:middle line:84% So now let me just make this a little bit more explicit. 00:34:53.882 --> 00:34:55.840 align:middle line:84% So before I do that, do you have any questions? 00:34:55.840 --> 00:34:56.590 align:middle line:90% Yes. 00:34:56.590 --> 00:35:02.350 align:middle line:84% AUDIENCE: [INAUDIBLE] looking at QED by itself [INAUDIBLE] 00:35:02.350 --> 00:35:04.950 align:middle line:84% any other theories, wouldn't we have 00:35:04.950 --> 00:35:10.030 align:middle line:84% to assume that this parameter m corresponds to a physical mass? 00:35:10.030 --> 00:35:16.310 align:middle line:84% [INAUDIBLE] how we say [INAUDIBLE].. 00:35:16.310 --> 00:35:18.020 align:middle line:84% But you're saying that QED on its own 00:35:18.020 --> 00:35:21.260 align:middle line:84% has its way of explaining how this parameter m relates 00:35:21.260 --> 00:35:22.150 align:middle line:90% to the physical m? 00:35:22.150 --> 00:35:26.295 align:middle line:90% 00:35:26.295 --> 00:35:30.480 align:middle line:84% HONG LIU: Yeah, so when I write down this Lagrangian and then 00:35:30.480 --> 00:35:34.940 align:middle line:84% I have parameters here for the e and m, 00:35:34.940 --> 00:35:40.480 align:middle line:84% so if I ignore the interaction, then the m 00:35:40.480 --> 00:35:44.630 align:middle line:84% will be the mass of the particle you actually measure. 00:35:44.630 --> 00:35:46.730 align:middle line:84% But now when you include the interaction, 00:35:46.730 --> 00:35:50.850 align:middle line:84% then the interaction can change that mass. 00:35:50.850 --> 00:35:54.580 align:middle line:84% It's not related to the Higgs story. 00:35:54.580 --> 00:35:58.510 align:middle line:84% The Higgs explains where this m comes from. 00:35:58.510 --> 00:36:03.760 align:middle line:84% Hicks explains where this parameter comes from. 00:36:03.760 --> 00:36:05.890 align:middle line:84% So this is a renormalization on top of that. 00:36:05.890 --> 00:36:09.040 align:middle line:90% 00:36:09.040 --> 00:36:09.830 align:middle line:90% Good. 00:36:09.830 --> 00:36:12.560 align:middle line:84% So now let me just elaborate what this means. 00:36:12.560 --> 00:36:16.220 align:middle line:90% 00:36:16.220 --> 00:36:20.420 align:middle line:84% So for this purpose now let me put a subscript B 00:36:20.420 --> 00:36:25.403 align:middle line:84% here on everything just to emphasize 00:36:25.403 --> 00:36:26.445 align:middle line:90% that they are no longer-- 00:36:26.445 --> 00:36:29.958 align:middle line:90% 00:36:29.958 --> 00:36:31.250 align:middle line:90% AUDIENCE: Can I ask a question? 00:36:31.250 --> 00:36:31.875 align:middle line:90% HONG LIU: Yeah. 00:36:31.875 --> 00:36:34.610 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:36:34.610 --> 00:36:52.490 align:middle line:90% 00:36:52.490 --> 00:36:54.608 align:middle line:84% HONG LIU: Yeah, it changes the energy you 00:36:54.608 --> 00:36:55.775 align:middle line:90% need to create the particle. 00:36:55.775 --> 00:37:01.520 align:middle line:90% 00:37:01.520 --> 00:37:05.430 align:middle line:84% So now I denoted everything with B, 00:37:05.430 --> 00:37:07.580 align:middle line:84% so those are called the bare parameters. 00:37:07.580 --> 00:37:19.670 align:middle line:84% So eB is called the bare charge, and mB called the bare mass, 00:37:19.670 --> 00:37:26.300 align:middle line:84% and this psi B and A mu B is called the bare fields. 00:37:26.300 --> 00:37:31.560 align:middle line:90% 00:37:31.560 --> 00:37:34.830 align:middle line:84% And when I write this F square means that the F square 00:37:34.830 --> 00:37:42.550 align:middle line:90% constructed out of A mu B. 00:37:42.550 --> 00:37:46.840 align:middle line:84% So now let's just illustrate this thing using these three 00:37:46.840 --> 00:37:49.510 align:middle line:90% simple diagrams. 00:37:49.510 --> 00:37:55.278 align:middle line:84% So let's look at first that diagram, the propagation 00:37:55.278 --> 00:37:55.945 align:middle line:90% of the electron. 00:37:55.945 --> 00:38:00.987 align:middle line:90% 00:38:00.987 --> 00:38:02.320 align:middle line:90% The propagation of the electron. 00:38:02.320 --> 00:38:07.070 align:middle line:90% 00:38:07.070 --> 00:38:15.010 align:middle line:84% So what you need to do is you calculate this diagram 00:38:15.010 --> 00:38:18.520 align:middle line:84% and then you find the correction to the propagator. 00:38:18.520 --> 00:38:21.580 align:middle line:84% So this diagram will induce a correction 00:38:21.580 --> 00:38:23.200 align:middle line:84% to the propagator of the electron. 00:38:23.200 --> 00:38:25.840 align:middle line:90% 00:38:25.840 --> 00:38:29.290 align:middle line:84% And from this correction, then you find that the-- 00:38:29.290 --> 00:38:32.050 align:middle line:90% 00:38:32.050 --> 00:38:37.740 align:middle line:84% So let's first write down the tree-level propagator. 00:38:37.740 --> 00:38:48.690 align:middle line:90% 00:38:48.690 --> 00:38:57.090 align:middle line:84% So this is given by minus 1 i k slash minus m plus i epsilon. 00:38:57.090 --> 00:38:58.100 align:middle line:90% So this is just mB. 00:38:58.100 --> 00:39:05.540 align:middle line:90% 00:39:05.540 --> 00:39:08.780 align:middle line:84% So you can interpret the mass of the electron 00:39:08.780 --> 00:39:10.670 align:middle line:90% as the pole of this k. 00:39:10.670 --> 00:39:14.810 align:middle line:84% So when k is equal to mB and then that corresponding 00:39:14.810 --> 00:39:16.790 align:middle line:90% to the mass of the electron. 00:39:16.790 --> 00:39:19.160 align:middle line:84% And the mass of the electron corresponding 00:39:19.160 --> 00:39:22.250 align:middle line:90% to the pole of this propagator. 00:39:22.250 --> 00:39:23.000 align:middle line:90% Yes? 00:39:23.000 --> 00:39:25.760 align:middle line:84% AUDIENCE: What does it mean [INAUDIBLE]?? 00:39:25.760 --> 00:39:30.710 align:middle line:84% HONG LIU: Yeah, what I mean is just when you invert it, 00:39:30.710 --> 00:39:36.260 align:middle line:84% this becomes k squared plus m squared minus i epsilon. 00:39:36.260 --> 00:39:38.510 align:middle line:84% It just means k squared equals this. 00:39:38.510 --> 00:39:42.530 align:middle line:84% So upstairs we have k slash plus m. 00:39:42.530 --> 00:39:45.650 align:middle line:84% It just means k squared plus m square. 00:39:45.650 --> 00:39:48.590 align:middle line:84% Just essentially the mass electron 00:39:48.590 --> 00:39:52.870 align:middle line:84% corresponding to the pole of the propagator. 00:39:52.870 --> 00:39:55.880 align:middle line:84% So now when you include such corrections, 00:39:55.880 --> 00:39:59.000 align:middle line:84% then you find your propagator modified 00:39:59.000 --> 00:40:00.100 align:middle line:90% into the following thing-- 00:40:00.100 --> 00:40:08.780 align:middle line:84% Z divided by i k slash minus some other mass plus epsilon. 00:40:08.780 --> 00:40:13.080 align:middle line:84% And then now this m mass is no longer the same. 00:40:13.080 --> 00:40:23.690 align:middle line:84% So this physical mass, it's now just mB plus some correction. 00:40:23.690 --> 00:40:24.935 align:middle line:90% So this is a physical mass. 00:40:24.935 --> 00:40:38.980 align:middle line:90% 00:40:38.980 --> 00:40:44.340 align:middle line:84% So this tells you that the mass of the electron when 00:40:44.340 --> 00:40:45.840 align:middle line:84% included such diagram actually gets 00:40:45.840 --> 00:40:49.790 align:middle line:90% shifted from your bare mass. 00:40:49.790 --> 00:40:53.260 align:middle line:90% 00:40:53.260 --> 00:40:56.290 align:middle line:84% So if we calculate it and then you 00:40:56.290 --> 00:41:00.160 align:middle line:84% find the contribution of this diagram 00:41:00.160 --> 00:41:02.530 align:middle line:84% is infinite and so that's why you need to include 00:41:02.530 --> 00:41:03.880 align:middle line:90% this renormalization. 00:41:03.880 --> 00:41:07.330 align:middle line:84% And then to calculate the finite quantity of this delta m. 00:41:07.330 --> 00:41:12.440 align:middle line:90% 00:41:12.440 --> 00:41:17.520 align:middle line:84% So you find that the pole of the propagator changed. 00:41:17.520 --> 00:41:20.850 align:middle line:84% You also find actually the residue also changed. 00:41:20.850 --> 00:41:23.810 align:middle line:90% So here the residue just 1. 00:41:23.810 --> 00:41:30.100 align:middle line:84% Here the residue becomes some other constant Z. So 00:41:30.100 --> 00:41:38.480 align:middle line:84% that means that when you act psi B on the vacuum, 00:41:38.480 --> 00:41:42.410 align:middle line:84% you no longer create just the k-- 00:41:42.410 --> 00:41:44.840 align:middle line:84% here I'm just writing schematically-- 00:41:44.840 --> 00:41:49.290 align:middle line:84% you're actually creating square root of Z times k 00:41:49.290 --> 00:41:51.840 align:middle line:90% OK because of the prefactor. 00:41:51.840 --> 00:41:55.770 align:middle line:84% So remember, our previous psi B is equal to a dagger etc., 00:41:55.770 --> 00:41:58.410 align:middle line:84% when you act on the 0 you will create a particle 00:41:58.410 --> 00:41:59.820 align:middle line:90% normalized in a certain way. 00:41:59.820 --> 00:42:02.910 align:middle line:84% But now with this Z factor, and now 00:42:02.910 --> 00:42:11.220 align:middle line:84% you create a lot of the factor now with a prefactor Z. 00:42:11.220 --> 00:42:19.620 align:middle line:84% So we normally define the physical field 00:42:19.620 --> 00:42:23.760 align:middle line:84% as the one which psi, when you act on the vacuum, which just 00:42:23.760 --> 00:42:25.170 align:middle line:90% precisely created the state. 00:42:25.170 --> 00:42:29.620 align:middle line:90% 00:42:29.620 --> 00:42:31.660 align:middle line:84% And so now there's a relation between this bear 00:42:31.660 --> 00:42:34.015 align:middle line:90% field and the physical field. 00:42:34.015 --> 00:42:36.880 align:middle line:90% 00:42:36.880 --> 00:42:39.070 align:middle line:84% And now there's a relation between the bear field 00:42:39.070 --> 00:42:40.800 align:middle line:84% and physical field through this factor. 00:42:40.800 --> 00:42:46.850 align:middle line:90% 00:42:46.850 --> 00:42:54.710 align:middle line:84% So this tells you that the magnitude of the field itself 00:42:54.710 --> 00:42:56.245 align:middle line:90% also get renormalized. 00:42:56.245 --> 00:42:59.890 align:middle line:90% 00:42:59.890 --> 00:43:02.120 align:middle line:90% Any questions on this? 00:43:02.120 --> 00:43:06.450 align:middle line:84% So the similar thing is happening for the photon case. 00:43:06.450 --> 00:43:10.620 align:middle line:84% So if you consider this process, so this 00:43:10.620 --> 00:43:13.560 align:middle line:84% corresponded to a modification to the photon propagator. 00:43:13.560 --> 00:43:24.640 align:middle line:90% 00:43:24.640 --> 00:43:27.940 align:middle line:84% So this created a propagation to the photon propagator. 00:43:27.940 --> 00:43:33.330 align:middle line:84% So now you find that actually the photon mass is not changed. 00:43:33.330 --> 00:43:35.150 align:middle line:84% So even when you include such corrections, 00:43:35.150 --> 00:43:37.720 align:middle line:90% the photon is still massless. 00:43:37.720 --> 00:43:42.120 align:middle line:84% So remember, previously we say why you cannot add a mass 00:43:42.120 --> 00:43:45.050 align:middle line:90% to a photon? 00:43:45.050 --> 00:43:51.960 align:middle line:84% So remember why we cannot add a mass term to the Maxwell field? 00:43:51.960 --> 00:43:54.180 align:middle line:90% The guage invariance. 00:43:54.180 --> 00:43:58.710 align:middle line:84% So the fact that the mass cannot be generated by such correction 00:43:58.710 --> 00:44:02.910 align:middle line:84% just means that the gauge symmetry is maintained, 00:44:02.910 --> 00:44:04.515 align:middle line:90% so the photon remains massless. 00:44:04.515 --> 00:44:13.180 align:middle line:90% 00:44:13.180 --> 00:44:16.960 align:middle line:84% But you do have similar renormalization of the strength 00:44:16.960 --> 00:44:19.450 align:middle line:90% of the Maxwell field. 00:44:19.450 --> 00:44:28.500 align:middle line:84% So now the bare field is related to your physical field 00:44:28.500 --> 00:44:32.635 align:middle line:84% through some factor normally called Z3. 00:44:32.635 --> 00:44:36.050 align:middle line:90% 00:44:36.050 --> 00:44:36.740 align:middle line:90% Yes? 00:44:36.740 --> 00:44:39.620 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:44:39.620 --> 00:44:42.500 align:middle line:90% 00:44:42.500 --> 00:44:44.090 align:middle line:84% HONG LIU: Yeah, this just a constant. 00:44:44.090 --> 00:44:48.410 align:middle line:90% Yeah, this is just a constant. 00:44:48.410 --> 00:44:56.440 align:middle line:90% 00:44:56.440 --> 00:44:59.010 align:middle line:84% So this is defined to be the residue 00:44:59.010 --> 00:45:02.151 align:middle line:90% of the pole of the propagator. 00:45:02.151 --> 00:45:06.000 align:middle line:90% 00:45:06.000 --> 00:45:07.290 align:middle line:90% Good? 00:45:07.290 --> 00:45:11.210 align:middle line:84% So finally, let me mention a little bit the last diagram. 00:45:11.210 --> 00:45:31.420 align:middle line:90% 00:45:31.420 --> 00:45:34.530 align:middle line:84% So this last diagram, which I will redraw it here. 00:45:34.530 --> 00:45:41.760 align:middle line:90% 00:45:41.760 --> 00:45:48.240 align:middle line:84% So remember, if you just have the original one, 00:45:48.240 --> 00:45:51.195 align:middle line:90% then here is just i eB gamma mu. 00:45:51.195 --> 00:45:53.920 align:middle line:90% 00:45:53.920 --> 00:45:56.860 align:middle line:84% And now when you add such a correction, 00:45:56.860 --> 00:45:59.980 align:middle line:84% so this is the tree-level one and then at one loop level 00:45:59.980 --> 00:46:01.580 align:middle line:90% you can add such a thing. 00:46:01.580 --> 00:46:04.450 align:middle line:84% And when you add such a thing, then you 00:46:04.450 --> 00:46:07.060 align:middle line:90% renormalize the charge. 00:46:07.060 --> 00:46:09.910 align:middle line:90% Then the charge gets changed. 00:46:09.910 --> 00:46:18.550 align:middle line:84% So now you find interestingly that the new electric charge is 00:46:18.550 --> 00:46:19.670 align:middle line:90% related. 00:46:19.670 --> 00:46:23.170 align:middle line:84% So this is the measured charge, physical charge. 00:46:23.170 --> 00:46:28.215 align:middle line:90% 00:46:28.215 --> 00:46:29.590 align:middle line:84% Through a complicated calculation 00:46:29.590 --> 00:46:36.320 align:middle line:84% you find this physical charge is related to the bare charge 00:46:36.320 --> 00:46:40.310 align:middle line:84% by the same factor, Z3, as in the case 00:46:40.310 --> 00:46:45.350 align:middle line:84% of the relation between the bare field 00:46:45.350 --> 00:46:48.050 align:middle line:90% and the renormalized field. 00:46:48.050 --> 00:46:50.420 align:middle line:84% You find that the same number that 00:46:50.420 --> 00:46:52.933 align:middle line:84% appears in the relation between the physical electron charge 00:46:52.933 --> 00:46:53.975 align:middle line:90% and then the bare charge. 00:46:53.975 --> 00:46:59.770 align:middle line:90% 00:46:59.770 --> 00:47:04.550 align:middle line:84% So this is important for the following reason. 00:47:04.550 --> 00:47:13.590 align:middle line:84% So now you observe because of these two relation, 00:47:13.590 --> 00:47:24.490 align:middle line:84% that eB A mu B is actually equal to e A mu. 00:47:24.490 --> 00:47:29.930 align:middle line:84% So the product here is the same because the square root 3 00:47:29.930 --> 00:47:30.635 align:middle line:90% factor cancels. 00:47:30.635 --> 00:47:38.950 align:middle line:90% 00:47:38.950 --> 00:47:40.090 align:middle line:90% And this is important. 00:47:40.090 --> 00:47:48.560 align:middle line:84% Because of this, that means your covariant derivative does not 00:47:48.560 --> 00:47:49.060 align:middle line:90% change. 00:47:49.060 --> 00:48:05.310 align:middle line:90% 00:48:05.310 --> 00:48:09.830 align:middle line:84% So remember, the structure of this covariant derivative 00:48:09.830 --> 00:48:13.400 align:middle line:84% is needed for the gauge invariance. 00:48:13.400 --> 00:48:17.150 align:middle line:84% It's needed for the gauge invariance. 00:48:17.150 --> 00:48:20.510 align:middle line:84% So that this is maintained, again, 00:48:20.510 --> 00:48:26.550 align:middle line:84% is an implication that the gauge symmetry is maintained. 00:48:26.550 --> 00:48:29.670 align:middle line:84% In other words, you want to say that if we 00:48:29.670 --> 00:48:34.350 align:middle line:84% want to maintain the gauge symmetry, then eB and A mu B, 00:48:34.350 --> 00:48:36.640 align:middle line:84% they have to renormalize in the opposite way 00:48:36.640 --> 00:48:37.860 align:middle line:90% so that this is the same. 00:48:37.860 --> 00:48:43.990 align:middle line:90% 00:48:43.990 --> 00:48:46.900 align:middle line:90% So this diagram has two effects. 00:48:46.900 --> 00:48:50.500 align:middle line:90% 00:48:50.500 --> 00:48:54.400 align:middle line:84% One effect is correct the electric charge. 00:48:54.400 --> 00:48:57.280 align:middle line:84% And another effect is generate the magnetic moment. 00:48:57.280 --> 00:49:00.710 align:middle line:84% Generate an anomalous magnetic moment. 00:49:00.710 --> 00:49:02.410 align:middle line:90% So this has two effects. 00:49:02.410 --> 00:49:05.560 align:middle line:84% The first effect is to renormalize, 00:49:05.560 --> 00:49:09.910 align:middle line:84% to change your charge, and the second effect 00:49:09.910 --> 00:49:15.555 align:middle line:84% is generate an anomalous magnetic moment. 00:49:15.555 --> 00:49:33.790 align:middle line:90% 00:49:33.790 --> 00:49:41.695 align:middle line:84% So you may have learned in your quantum mechanics 00:49:41.695 --> 00:49:55.210 align:middle line:84% that e minus has a magnetic moment, which is normally 00:49:55.210 --> 00:50:00.220 align:middle line:84% written as following-- mu e equal to minus e 00:50:00.220 --> 00:50:05.960 align:middle line:84% divided by 2m g and the spin operator. 00:50:05.960 --> 00:50:08.660 align:middle line:84% Just because the electron has spin, spin half, 00:50:08.660 --> 00:50:10.700 align:middle line:90% and so it has a magnetic moment. 00:50:10.700 --> 00:50:13.880 align:middle line:84% And this spin half of course is normally 00:50:13.880 --> 00:50:17.540 align:middle line:90% written this spin operator. 00:50:17.540 --> 00:50:20.810 align:middle line:84% It's in the spin half space, then just a sigma matrix. 00:50:20.810 --> 00:50:23.630 align:middle line:90% 00:50:23.630 --> 00:50:27.590 align:middle line:84% And this g factor, so this is a g factor which 00:50:27.590 --> 00:50:31.340 align:middle line:90% relates the g, just a number. 00:50:31.340 --> 00:50:32.700 align:middle line:90% g is just a number. 00:50:32.700 --> 00:50:33.990 align:middle line:90% So normally called g factor-- 00:50:33.990 --> 00:50:34.490 align:middle line:90% Yes. 00:50:34.490 --> 00:50:41.180 align:middle line:84% AUDIENCE: [INAUDIBLE] diagrammatic expansion 00:50:41.180 --> 00:50:44.630 align:middle line:84% of the perturbation theory in powers of lambda. 00:50:44.630 --> 00:50:49.400 align:middle line:84% So you're saying you add a term and you change e with the -- 00:50:49.400 --> 00:50:53.480 align:middle line:84% HONG LIU: Yeah, the e can change itself Yeah. 00:50:53.480 --> 00:50:58.610 align:middle line:84% So you just shift the value of e right. 00:50:58.610 --> 00:51:02.140 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:51:02.140 --> 00:51:05.590 align:middle line:84% HONG LIU: Yeah, change is also small. 00:51:05.590 --> 00:51:06.870 align:middle line:90% The change is a higher order. 00:51:06.870 --> 00:51:09.930 align:middle line:90% 00:51:09.930 --> 00:51:13.150 align:middle line:84% But this is a very good question. 00:51:13.150 --> 00:51:15.510 align:middle line:84% This is a very good question because, as I mentioned, 00:51:15.510 --> 00:51:18.190 align:middle line:84% naively if you look at those corrections, 00:51:18.190 --> 00:51:25.240 align:middle line:84% say this delta m and this change between e and eB, 00:51:25.240 --> 00:51:27.360 align:middle line:90% they are all divergent. 00:51:27.360 --> 00:51:31.463 align:middle line:84% So that's why they're all very complicated. 00:51:31.463 --> 00:51:32.880 align:middle line:84% These are all divergent, so that's 00:51:32.880 --> 00:51:37.110 align:middle line:84% why it's non-trivial to find a scheme to make 00:51:37.110 --> 00:51:38.990 align:middle line:90% sense of those equations. 00:51:38.990 --> 00:51:43.573 align:middle line:84% But we'll have all those divergences flying around. 00:51:43.573 --> 00:51:44.240 align:middle line:90% Other questions? 00:51:44.240 --> 00:51:45.100 align:middle line:90% Yes? 00:51:45.100 --> 00:51:47.475 align:middle line:84% AUDIENCE: What is the difference between square root of Z 00:51:47.475 --> 00:51:48.753 align:middle line:90% and the square root of Z3? 00:51:48.753 --> 00:51:49.420 align:middle line:90% HONG LIU: Sorry? 00:51:49.420 --> 00:51:51.311 align:middle line:84% AUDIENCE: What is the difference between square root of Z 00:51:51.311 --> 00:51:52.060 align:middle line:90% and the square root of Z3? 00:51:52.060 --> 00:51:54.220 align:middle line:84% HONG LIU: Oh, they're just different numbers. 00:51:54.220 --> 00:51:57.040 align:middle line:84% So this is one number, this is some other number. 00:51:57.040 --> 00:52:00.850 align:middle line:84% Just the electron field renormalized in a different way 00:52:00.850 --> 00:52:04.000 align:middle line:84% from the photon, but somehow the charge 00:52:04.000 --> 00:52:06.310 align:middle line:84% has to be renormalized in the same way. 00:52:06.310 --> 00:52:09.740 align:middle line:90% 00:52:09.740 --> 00:52:11.210 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:52:11.210 --> 00:52:30.210 align:middle line:90% 00:52:30.210 --> 00:52:36.540 align:middle line:84% HONG LIU: Yeah, but that come out from-- 00:52:36.540 --> 00:52:42.213 align:middle line:84% It comes out from the gauge symmetry, 00:52:42.213 --> 00:52:43.880 align:middle line:84% from the requirements of gauge symmetry. 00:52:43.880 --> 00:52:47.740 align:middle line:90% 00:52:47.740 --> 00:52:49.540 align:middle line:90% Good? 00:52:49.540 --> 00:53:00.520 align:middle line:84% So this g factor, so one of the triumphs of the Dirac equation, 00:53:00.520 --> 00:53:02.830 align:middle line:84% when Dirac first wrote it down, it 00:53:02.830 --> 00:53:09.100 align:middle line:84% said the Dirac equation predicts that g equals to 2. 00:53:09.100 --> 00:53:15.040 align:middle line:90% 00:53:15.040 --> 00:53:21.238 align:middle line:84% But in reality, in the late 30s and early 40s, 00:53:21.238 --> 00:53:22.780 align:middle line:84% the experiments were accurate enough. 00:53:22.780 --> 00:53:25.210 align:middle line:84% At the beginning, people thought that g equals to 2, 00:53:25.210 --> 00:53:28.420 align:middle line:84% so that was considered to be a triumph of the Dirac equation. 00:53:28.420 --> 00:53:30.010 align:middle line:84% But then later, when the experiment 00:53:30.010 --> 00:53:33.550 align:middle line:84% became more accurate, they find that g is actually 00:53:33.550 --> 00:53:34.960 align:middle line:90% slightly greater than 2. 00:53:34.960 --> 00:53:41.200 align:middle line:90% 00:53:41.200 --> 00:53:45.310 align:middle line:84% Of course, this accuracy is later. 00:53:45.310 --> 00:53:47.920 align:middle line:84% At the beginning, in the 40s, they 00:53:47.920 --> 00:53:50.990 align:middle line:84% can already see that the g is no longer 2. 00:53:50.990 --> 00:53:52.720 align:middle line:90% There's some small corrections. 00:53:52.720 --> 00:53:56.660 align:middle line:84% So there was a big question at the time-- 00:53:56.660 --> 00:53:58.933 align:middle line:84% how do you calculate this correction? 00:53:58.933 --> 00:54:04.420 align:middle line:90% 00:54:04.420 --> 00:54:07.160 align:middle line:84% So that amounts to calculate this diagram. 00:54:07.160 --> 00:54:08.920 align:middle line:90% So that's what Schwinger did. 00:54:08.920 --> 00:54:11.330 align:middle line:84% And so he calculated this diagram, 00:54:11.330 --> 00:54:13.330 align:middle line:84% he managed to get rid of the divergence, 00:54:13.330 --> 00:54:16.260 align:middle line:84% and he managed to reproduce these small corrections. 00:54:16.260 --> 00:54:20.210 align:middle line:90% 00:54:20.210 --> 00:54:25.210 align:middle line:84% So now let me just very quickly tell you tell you 00:54:25.210 --> 00:54:26.590 align:middle line:90% where this result comes from. 00:54:26.590 --> 00:54:29.530 align:middle line:90% 00:54:29.530 --> 00:54:33.130 align:middle line:84% Why Dirac equation predicts the g equal to 2. 00:54:33.130 --> 00:54:36.270 align:middle line:84% So this actually I had at some point 00:54:36.270 --> 00:54:40.950 align:middle line:84% was thinking as a homework exercise, but never got 00:54:40.950 --> 00:54:45.060 align:middle line:84% to put it in the homework, but it's an instructive exercise. 00:54:45.060 --> 00:54:53.100 align:middle line:84% And so now let me try to explain where it comes from. 00:54:53.100 --> 00:55:03.300 align:middle line:90% 00:55:03.300 --> 00:55:04.952 align:middle line:90% So we have the Dirac equation. 00:55:04.952 --> 00:55:06.660 align:middle line:84% So now let's consider the Dirac equation. 00:55:06.660 --> 00:55:15.100 align:middle line:90% 00:55:15.100 --> 00:55:23.490 align:middle line:84% Of electron coupled to the electromagnetic field. 00:55:23.490 --> 00:55:24.855 align:middle line:90% So let's consider this equation. 00:55:24.855 --> 00:55:29.830 align:middle line:90% 00:55:29.830 --> 00:55:36.550 align:middle line:84% So the claim-- from this equation, of course, 00:55:36.550 --> 00:55:39.540 align:middle line:84% we don't see anything about the magnetic moment. 00:55:39.540 --> 00:55:41.770 align:middle line:84% We don't see anything about the magnetic moment. 00:55:41.770 --> 00:55:45.180 align:middle line:84% So the claim is that if you take this equation and go 00:55:45.180 --> 00:55:51.250 align:middle line:84% to nonrelativistic limit, if you go to nonrelativistic limit, 00:55:51.250 --> 00:55:54.180 align:middle line:84% you will get the Schrodinger equation, 00:55:54.180 --> 00:55:57.070 align:middle line:84% you get the so-called Pauli equation. 00:55:57.070 --> 00:55:59.560 align:middle line:84% So you get the Schrodinger equation, 00:55:59.560 --> 00:56:05.280 align:middle line:84% and with a spin half particle and with a magnetic moment 00:56:05.280 --> 00:56:07.140 align:middle line:90% precisely given by g equal to 2. 00:56:07.140 --> 00:56:10.840 align:middle line:90% 00:56:10.840 --> 00:56:13.590 align:middle line:90% So now let's try to see this. 00:56:13.590 --> 00:56:17.880 align:middle line:90% 00:56:17.880 --> 00:56:20.970 align:middle line:90% Do you have any questions? 00:56:20.970 --> 00:56:23.460 align:middle line:84% So this is just the Dirac equation itself, 00:56:23.460 --> 00:56:27.070 align:middle line:84% but now if you include that vertex correction, 00:56:27.070 --> 00:56:30.370 align:middle line:84% and then that will generate this additional term. 00:56:30.370 --> 00:56:33.820 align:middle line:90% 00:56:33.820 --> 00:56:35.910 align:middle line:84% So now let's look at this nonrelativistic limit. 00:56:35.910 --> 00:56:43.710 align:middle line:90% 00:56:43.710 --> 00:56:55.290 align:middle line:84% So remember, the Dirac equation, as it was originally conceived, 00:56:55.290 --> 00:56:59.820 align:middle line:84% was already an equation of the Schrodinger type. 00:56:59.820 --> 00:57:07.140 align:middle line:84% So you have the form of the i H t psi equal to H psi. 00:57:07.140 --> 00:57:12.810 align:middle line:84% So you just multiply the i gamma 0 on both sides 00:57:12.810 --> 00:57:16.110 align:middle line:90% and then remove everything away. 00:57:16.110 --> 00:57:19.350 align:middle line:84% So just this side keeps only i partial t. 00:57:19.350 --> 00:57:21.300 align:middle line:84% So through this procedure you have 00:57:21.300 --> 00:57:35.865 align:middle line:84% H is equal to m times beta plus alpha p plus eA then plus eA0. 00:57:35.865 --> 00:57:42.270 align:middle line:84% eA0 is 0 component of A and eA is the vector component of A. 00:57:42.270 --> 00:57:47.670 align:middle line:84% So the beta is just the minus gamma i gamma 0, 00:57:47.670 --> 00:57:57.970 align:middle line:84% and alpha i is equal to minus i gamma 0 gamma i, and the p 00:57:57.970 --> 00:58:02.770 align:middle line:84% goes to minus i, the standard gradient operator. 00:58:02.770 --> 00:58:05.600 align:middle line:90% 00:58:05.600 --> 00:58:07.370 align:middle line:84% So now the claim is that if I want 00:58:07.370 --> 00:58:10.370 align:middle line:84% to take the nonrelativistic limit, 00:58:10.370 --> 00:58:14.210 align:middle line:84% this equation becomes an ordinary Schrodinger equation, 00:58:14.210 --> 00:58:15.560 align:middle line:90% including the magnetic moment. 00:58:15.560 --> 00:58:21.210 align:middle line:90% 00:58:21.210 --> 00:58:24.370 align:middle line:84% Is this clear what we want to do? 00:58:24.370 --> 00:58:28.310 align:middle line:84% So now let's choose the basis for the gamma matrices. 00:58:28.310 --> 00:58:32.290 align:middle line:84% So for this purpose, we go to the nonrelativistic limit, 00:58:32.290 --> 00:58:36.040 align:middle line:84% it's convenient to use this basis, which actually I 00:58:36.040 --> 00:58:38.260 align:middle line:84% wrote down at the beginning when we talked 00:58:38.260 --> 00:58:39.880 align:middle line:90% about the Dirac equations. 00:58:39.880 --> 00:58:48.190 align:middle line:90% 00:58:48.190 --> 00:58:53.300 align:middle line:84% Again, 1 is always 2 by 2 identity matrix. 00:58:53.300 --> 00:58:59.600 align:middle line:84% So from here, you find that the beta just equals to 1 0 00:58:59.600 --> 00:59:07.150 align:middle line:84% 0 minus 1 and the alpha just becomes 0 sigma sigma 0. 00:59:07.150 --> 00:59:17.890 align:middle line:90% 00:59:17.890 --> 00:59:19.480 align:middle line:90% So to reduce-- 00:59:19.480 --> 00:59:22.240 align:middle line:84% Right now we just choose then we also 00:59:22.240 --> 00:59:27.140 align:middle line:84% need to write psi in terms of the two components. 00:59:27.140 --> 00:59:29.260 align:middle line:84% So each of phi and the chi has two components, 00:59:29.260 --> 00:59:31.030 align:middle line:90% altogether four components. 00:59:31.030 --> 00:59:34.120 align:middle line:84% So let's write psi in terms of two upper components, two lower 00:59:34.120 --> 00:59:37.180 align:middle line:90% components. 00:59:37.180 --> 00:59:41.560 align:middle line:84% And now let's plug all this into that equation. 00:59:41.560 --> 00:59:43.690 align:middle line:90% All this into this equation. 00:59:43.690 --> 00:59:51.060 align:middle line:84% And then we get partial t phi equal to m 00:59:51.060 --> 01:00:02.880 align:middle line:84% phi plus sigma dot p plus eA chi then plus eA0 psi. 01:00:02.880 --> 01:00:05.660 align:middle line:90% 01:00:05.660 --> 01:00:08.390 align:middle line:84% And then for the equation for chi, 01:00:08.390 --> 01:00:12.290 align:middle line:84% you get a similar equation, but here it's 01:00:12.290 --> 01:00:23.390 align:middle line:84% minus m chi plus sigma dot p plus eA phi plus eA0 chi. 01:00:23.390 --> 01:00:27.730 align:middle line:90% 01:00:27.730 --> 01:00:29.155 align:middle line:90% So you get these two equations. 01:00:29.155 --> 01:00:32.560 align:middle line:90% 01:00:32.560 --> 01:00:34.990 align:middle line:84% So now so far we haven't done anything. 01:00:34.990 --> 01:00:39.460 align:middle line:84% We just wrote this equation in a different form. 01:00:39.460 --> 01:00:41.505 align:middle line:84% So now let's consider the nonrelativistic limit. 01:00:41.505 --> 01:00:49.670 align:middle line:90% 01:00:49.670 --> 01:00:53.420 align:middle line:84% Now consider the nonrelativistic limit. 01:00:53.420 --> 01:00:58.870 align:middle line:84% So in the nonrelativistic limit, this p 01:00:58.870 --> 01:01:02.480 align:middle line:84% is essentially the spatial momentum 01:01:02.480 --> 01:01:04.520 align:middle line:90% if you act on the wave. 01:01:04.520 --> 01:01:06.770 align:middle line:90% And so we take it to be mv. 01:01:06.770 --> 01:01:09.610 align:middle line:90% 01:01:09.610 --> 01:01:12.085 align:middle line:84% And of course, v to be much, much smaller than 1. 01:01:12.085 --> 01:01:15.940 align:middle line:90% 01:01:15.940 --> 01:01:22.430 align:middle line:84% And then this e A0, which has a unit of energy, 01:01:22.430 --> 01:01:24.310 align:middle line:84% so it should be of order mv squared. 01:01:24.310 --> 01:01:28.520 align:middle line:90% 01:01:28.520 --> 01:01:31.910 align:middle line:84% And this e A, which should be the same order as p, 01:01:31.910 --> 01:01:38.030 align:middle line:84% and then e A should also be of the same order as mv. 01:01:38.030 --> 01:01:41.800 align:middle line:90% 01:01:41.800 --> 01:01:44.080 align:middle line:84% And now let's look at this nonrelativistic limit. 01:01:44.080 --> 01:01:49.520 align:middle line:90% 01:01:49.520 --> 01:01:50.930 align:middle line:90% So far I have not done anything. 01:01:50.930 --> 01:01:53.660 align:middle line:90% I just specified this. 01:01:53.660 --> 01:01:58.080 align:middle line:84% And now let's consider a transformation. 01:01:58.080 --> 01:02:01.460 align:middle line:84% So in the nonrelativistic limit we 01:02:01.460 --> 01:02:02.675 align:middle line:90% know that your total energy-- 01:02:02.675 --> 01:02:26.860 align:middle line:90% 01:02:26.860 --> 01:02:29.910 align:middle line:84% So remember, a typical wave function 01:02:29.910 --> 01:02:33.540 align:middle line:84% is always the time dependence, so you always 01:02:33.540 --> 01:02:39.420 align:middle line:84% have something like iEt, and in the nonrelativistic limit, 01:02:39.420 --> 01:02:44.370 align:middle line:84% E would be m plus one half mv squared, et cetera. 01:02:44.370 --> 01:02:46.590 align:middle line:84% So this is much, much larger than this term. 01:02:46.590 --> 01:02:53.940 align:middle line:90% 01:02:53.940 --> 01:02:57.390 align:middle line:84% So we can just isolate this bigger piece. 01:02:57.390 --> 01:03:00.000 align:middle line:84% So we can try to isolate this big piece, 01:03:00.000 --> 01:03:03.970 align:middle line:84% so we will make a slight transformation. 01:03:03.970 --> 01:03:12.700 align:middle line:84% So we write phi equal to exponential minus i m t capital 01:03:12.700 --> 01:03:13.730 align:middle line:90% phi. 01:03:13.730 --> 01:03:23.480 align:middle line:84% And the chi equal to i m t x or capital chi. 01:03:23.480 --> 01:03:29.150 align:middle line:84% So now this will only contain this low energy part, 01:03:29.150 --> 01:03:31.250 align:middle line:84% so those field will only contain the low energy 01:03:31.250 --> 01:03:35.600 align:middle line:84% part that already isolated the big things in here. 01:03:35.600 --> 01:03:38.330 align:middle line:84% So the time dependence here will only include the low energy. 01:03:38.330 --> 01:03:41.130 align:middle line:90% 01:03:41.130 --> 01:03:46.570 align:middle line:84% So we assume that as a t phi would be mv squared, 01:03:46.570 --> 01:04:04.560 align:middle line:84% so phi similarly this X. Similarly this X. So this is 01:04:04.560 --> 01:04:07.110 align:middle line:84% the fast oscillator part in time, 01:04:07.110 --> 01:04:09.720 align:middle line:90% this is the slow part in time. 01:04:09.720 --> 01:04:15.770 align:middle line:84% So now we plug this in to these two equations. 01:04:15.770 --> 01:04:19.860 align:middle line:90% Plugged into those equations. 01:04:19.860 --> 01:04:24.750 align:middle line:90% Then we find i partial t. 01:04:24.750 --> 01:04:29.080 align:middle line:90% 01:04:29.080 --> 01:04:34.510 align:middle line:84% i partial t phi equal to sigma dot 01:04:34.510 --> 01:04:49.390 align:middle line:84% pi x plus eA0 phi and i partial tx equal to 2 minus mx. 01:04:49.390 --> 01:04:53.040 align:middle line:90% 01:04:53.040 --> 01:04:55.980 align:middle line:84% Let me just call it x for simplicity. 01:04:55.980 --> 01:05:05.280 align:middle line:90% 01:05:05.280 --> 01:05:07.890 align:middle line:84% So I also have introduced the notation 01:05:07.890 --> 01:05:13.680 align:middle line:84% that the pi is equal to p plus eA. 01:05:13.680 --> 01:05:24.040 align:middle line:90% 01:05:24.040 --> 01:05:26.650 align:middle line:84% So now you see the difference between these two equations. 01:05:26.650 --> 01:05:29.540 align:middle line:90% 01:05:29.540 --> 01:05:35.260 align:middle line:84% So the mass terms here have opposite sign. 01:05:35.260 --> 01:05:39.990 align:middle line:84% When I isolate that piece, and then 01:05:39.990 --> 01:05:43.770 align:middle line:84% the mass term for this equation canceled, but in this equation 01:05:43.770 --> 01:05:45.930 align:middle line:84% they add together, this minus 2m. 01:05:45.930 --> 01:05:48.660 align:middle line:90% 01:05:48.660 --> 01:05:51.850 align:middle line:90% We get this minus 2m. 01:05:51.850 --> 01:05:53.320 align:middle line:84% So now let's look at the magnitude 01:05:53.320 --> 01:05:55.070 align:middle line:90% of each term in this equation. 01:05:55.070 --> 01:06:04.350 align:middle line:84% So this is like mv squared X. So this is like m times X. 01:06:04.350 --> 01:06:08.700 align:middle line:90% So this is just something phi. 01:06:08.700 --> 01:06:10.230 align:middle line:84% We don't know the relative magnitude 01:06:10.230 --> 01:06:12.520 align:middle line:84% between X and the phi, and so this is also 01:06:12.520 --> 01:06:17.760 align:middle line:84% mv squared X. We said eA0 should be of order mv squared. 01:06:17.760 --> 01:06:20.400 align:middle line:90% 01:06:20.400 --> 01:06:25.020 align:middle line:84% So these three terms, they're all related to X, 01:06:25.020 --> 01:06:29.780 align:middle line:84% and this term dominates over these two. 01:06:29.780 --> 01:06:32.540 align:middle line:84% So you can forget about these two terms 01:06:32.540 --> 01:06:33.800 align:middle line:90% in the nonrelativistic limit. 01:06:33.800 --> 01:06:39.840 align:middle line:90% 01:06:39.840 --> 01:06:46.560 align:middle line:84% So from here, if I call this equation 1 and this equation 2, 01:06:46.560 --> 01:06:53.550 align:middle line:84% and from 2 we get in the nonrelativistic limit, 01:06:53.550 --> 01:07:02.190 align:middle line:84% we just get 2mX plus sigma pi acting on phi equal to 0. 01:07:02.190 --> 01:07:09.530 align:middle line:90% 01:07:09.530 --> 01:07:17.510 align:middle line:84% So now we find that X, we can now solve X in terms of phi-- 01:07:17.510 --> 01:07:21.890 align:middle line:90% 1 over 2m sigma dot pi phi. 01:07:21.890 --> 01:07:32.970 align:middle line:90% 01:07:32.970 --> 01:07:37.540 align:middle line:84% So now we can plug this back into this equation. 01:07:37.540 --> 01:07:40.800 align:middle line:90% So this is equation 3. 01:07:40.800 --> 01:07:47.430 align:middle line:84% If we plug 3 into 1 again, then we 01:07:47.430 --> 01:07:52.410 align:middle line:84% get Schrodinger equation for phi. 01:07:52.410 --> 01:07:54.660 align:middle line:84% So X is no longer independent field, 01:07:54.660 --> 01:07:58.040 align:middle line:84% it's just expressed in terms of phi. 01:07:58.040 --> 01:07:59.665 align:middle line:84% So now we get the equation for phi. 01:07:59.665 --> 01:08:17.470 align:middle line:90% 01:08:17.470 --> 01:08:19.270 align:middle line:84% So now you see each term, they are 01:08:19.270 --> 01:08:20.990 align:middle line:90% of the same order of magnitude. 01:08:20.990 --> 01:08:26.160 align:middle line:84% So this is mv squared and this is mv. 01:08:26.160 --> 01:08:28.170 align:middle line:90% Upstairs pi of order mv. 01:08:28.170 --> 01:08:30.689 align:middle line:84% So this is m squared v squared divided by m, 01:08:30.689 --> 01:08:34.479 align:middle line:84% so this again is mv squared and eA0 is mv squared. 01:08:34.479 --> 01:08:37.229 align:middle line:84% So now everything is nonrelativistic 01:08:37.229 --> 01:08:39.803 align:middle line:90% and of the same order. 01:08:39.803 --> 01:08:41.720 align:middle line:84% So now let's look at the meaning of this term. 01:08:41.720 --> 01:08:48.058 align:middle line:90% 01:08:48.058 --> 01:08:49.850 align:middle line:84% Now let's look at the meaning of that term. 01:08:49.850 --> 01:09:08.670 align:middle line:90% 01:09:08.670 --> 01:09:10.718 align:middle line:84% So that term you can write explicitly. 01:09:10.718 --> 01:09:19.290 align:middle line:90% 01:09:19.290 --> 01:09:23.370 align:middle line:84% Sigma i pi i, sigma j pi j, but remember pi i and pi j, 01:09:23.370 --> 01:09:28.710 align:middle line:84% they are operators, so you need to keep the ordering there. 01:09:28.710 --> 01:09:32.069 align:middle line:90% 01:09:32.069 --> 01:09:34.380 align:middle line:84% So we can just write down, so we just 01:09:34.380 --> 01:09:43.660 align:middle line:84% have sigma i sigma j just gives me delta ij plus i epsilon ijk 01:09:43.660 --> 01:09:46.899 align:middle line:90% sigma k and then pi i pi j. 01:09:46.899 --> 01:09:51.410 align:middle line:90% 01:09:51.410 --> 01:09:58.300 align:middle line:84% And so the first term gives me the pi squared, 01:09:58.300 --> 01:10:04.840 align:middle line:84% and the second term I can just, using the antisymmetric in ij, 01:10:04.840 --> 01:10:11.720 align:middle line:84% I can write it as epsilon ijk sigma k and the pi i and the pi 01:10:11.720 --> 01:10:12.220 align:middle line:90% j. 01:10:12.220 --> 01:10:15.150 align:middle line:90% 01:10:15.150 --> 01:10:17.430 align:middle line:84% So I just write the commutator in terms of them. 01:10:17.430 --> 01:10:20.840 align:middle line:90% 01:10:20.840 --> 01:10:28.750 align:middle line:84% So now it's easy to calculate pi i pi j. 01:10:28.750 --> 01:10:41.235 align:middle line:84% So this is just minus partial i plus eAi minus partial j -- 01:10:41.235 --> 01:10:42.810 align:middle line:90% minus i partial j -- 01:10:42.810 --> 01:10:45.952 align:middle line:90% plus eAj. 01:10:45.952 --> 01:10:47.410 align:middle line:84% So if you calculate the commutator, 01:10:47.410 --> 01:10:49.000 align:middle line:90% you just get minus e Fij. 01:10:49.000 --> 01:10:56.470 align:middle line:90% 01:10:56.470 --> 01:11:01.330 align:middle line:84% And now if you plug this Fij into here, so epsilon ijk 01:11:01.330 --> 01:11:02.980 align:middle line:90% times Fij, what do you get? 01:11:02.980 --> 01:11:08.050 align:middle line:90% 01:11:08.050 --> 01:11:10.600 align:middle line:84% Do you remember when epsilon ijk multiplied 01:11:10.600 --> 01:11:11.920 align:middle line:90% by Fij, what do you get? 01:11:11.920 --> 01:11:15.020 align:middle line:90% 01:11:15.020 --> 01:11:17.000 align:middle line:90% You get B. 01:11:17.000 --> 01:11:20.330 align:middle line:84% So now this equation, when you plug this in here, 01:11:20.330 --> 01:11:26.770 align:middle line:84% and then this just becomes pi squared plus-- 01:11:26.770 --> 01:11:27.670 align:middle line:90% just make sure. 01:11:27.670 --> 01:11:34.490 align:middle line:90% 01:11:34.490 --> 01:11:49.310 align:middle line:84% e sigma dot B. Because Fij multiplied by the Fijk 01:11:49.310 --> 01:11:53.944 align:middle line:84% just gets Bk and Bk sigma k, just sigma dot B. 01:11:53.944 --> 01:11:57.353 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:11:57.353 --> 01:11:58.630 align:middle line:90% 01:11:58.630 --> 01:12:02.420 align:middle line:84% HONG LIU: Yeah, it's one half epsilon ijk times Fij give you. 01:12:02.420 --> 01:12:06.520 align:middle line:90% 01:12:06.520 --> 01:12:11.370 align:middle line:84% So if you use one half epsilon ijk Fij. 01:12:11.370 --> 01:12:18.600 align:middle line:90% 01:12:18.600 --> 01:12:19.890 align:middle line:90% OK, good. 01:12:19.890 --> 01:12:22.770 align:middle line:90% We are there. 01:12:22.770 --> 01:12:26.340 align:middle line:90% So now we are done, essentially. 01:12:26.340 --> 01:12:29.160 align:middle line:84% So now the Schrodinger equation, now we 01:12:29.160 --> 01:12:33.360 align:middle line:84% get the partial t phi equal to H phi. 01:12:33.360 --> 01:12:37.730 align:middle line:90% Now H is equal to p squared. 01:12:37.730 --> 01:12:40.940 align:middle line:90% So now you have pi squared. 01:12:40.940 --> 01:12:43.370 align:middle line:84% So pi squared, if I write it explicitly, 01:12:43.370 --> 01:12:49.400 align:middle line:84% is p plus eA squared divided by 2m, then 01:12:49.400 --> 01:12:56.360 align:middle line:84% plus e divided by 2m sigma dot B, come from this term. 01:12:56.360 --> 01:12:59.520 align:middle line:90% 01:12:59.520 --> 01:13:02.310 align:middle line:90% And then the last term is eA0. 01:13:02.310 --> 01:13:06.110 align:middle line:90% 01:13:06.110 --> 01:13:12.340 align:middle line:84% So now if you recognize this part of the Hamiltonian 01:13:12.340 --> 01:13:17.050 align:middle line:84% is precisely what you get when you couple Schrodinger equation 01:13:17.050 --> 01:13:20.170 align:middle line:90% to your electromagnetic fields. 01:13:20.170 --> 01:13:24.020 align:middle line:84% You just add the p plus eA for the vector potential 01:13:24.020 --> 01:13:26.740 align:middle line:90% and here you get eA0. 01:13:26.740 --> 01:13:28.820 align:middle line:84% But now here you get the additional term, 01:13:28.820 --> 01:13:31.980 align:middle line:84% so this is called the Pauli term. 01:13:31.980 --> 01:13:34.990 align:middle line:84% When the particle has spin, has magnetic moment, 01:13:34.990 --> 01:13:36.460 align:middle line:84% then you have an additional term. 01:13:36.460 --> 01:13:38.350 align:middle line:90% So this term can be interpreted. 01:13:38.350 --> 01:13:42.865 align:middle line:84% So remember, when a particle has a magnetic moment. 01:13:42.865 --> 01:13:54.640 align:middle line:90% 01:13:54.640 --> 01:13:59.790 align:middle line:84% So when a particle has a magnetic moment, 01:13:59.790 --> 01:14:05.640 align:middle line:84% then you have a piece corresponding to mu dot B. 01:14:05.640 --> 01:14:07.980 align:middle line:84% If you compare this with that term, 01:14:07.980 --> 01:14:09.930 align:middle line:90% then we conclude that the mu-- 01:14:09.930 --> 01:14:12.360 align:middle line:84% So in this case, compare that term. 01:14:12.360 --> 01:14:22.380 align:middle line:84% That term will be equal to e 2m sigma dot B. 01:14:22.380 --> 01:14:27.390 align:middle line:84% So now we conclude, from here you can read what's the mu. 01:14:27.390 --> 01:14:35.050 align:middle line:84% mu just minus e divided by 2m times 2 times sigma 01:14:35.050 --> 01:14:36.730 align:middle line:90% divided by 2. 01:14:36.730 --> 01:14:41.770 align:middle line:84% So if you compare with this, you find that the g 01:14:41.770 --> 01:14:44.916 align:middle line:90% is exactly equal to 2. 01:14:44.916 --> 01:14:49.010 align:middle line:90% You find g exactly equal to 2. 01:14:49.010 --> 01:14:52.100 align:middle line:84% And so this way, you see actually the Dirac equation 01:14:52.100 --> 01:14:57.140 align:middle line:84% contains the information the electron has magnetic moment. 01:14:57.140 --> 01:14:58.940 align:middle line:90% It has a magnetic moment. 01:14:58.940 --> 01:15:00.020 align:middle line:90% Yes. 01:15:00.020 --> 01:15:00.468 align:middle line:84% AUDIENCE: [INAUDIBLE] would we expect 01:15:00.468 --> 01:15:01.926 align:middle line:84% g equal to 2 for all fermions then? 01:15:01.926 --> 01:15:07.860 align:middle line:90% 01:15:07.860 --> 01:15:10.200 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 01:15:10.200 --> 01:15:13.080 align:middle line:84% AUDIENCE: Even if we look at these perturbative calculations 01:15:13.080 --> 01:15:15.247 align:middle line:84% like the 1 plus 0.001, etc., that should apply also? 01:15:15.247 --> 01:15:19.650 align:middle line:90% 01:15:19.650 --> 01:15:21.535 align:middle line:90% HONG LIU: No, no, no. 01:15:21.535 --> 01:15:23.160 align:middle line:84% Because the correction can be different 01:15:23.160 --> 01:15:26.060 align:middle line:84% for different formulas because different formulas, 01:15:26.060 --> 01:15:28.080 align:middle line:84% they have different interactions. 01:15:28.080 --> 01:15:31.440 align:middle line:84% And so that will change this vertex differently. 01:15:31.440 --> 01:15:34.310 align:middle line:84% For example, muons and electrons will be different. 01:15:34.310 --> 01:15:38.030 align:middle line:90% 01:15:38.030 --> 01:15:38.540 align:middle line:90% Yes. 01:15:38.540 --> 01:15:39.606 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:15:39.606 --> 01:15:49.700 align:middle line:90% 01:15:49.700 --> 01:15:52.640 align:middle line:84% HONG LIU: Well, in the high speed you can no longer-- 01:15:52.640 --> 01:15:58.290 align:middle line:84% So when you have high speeds or this effect, they mix together. 01:15:58.290 --> 01:16:01.180 align:middle line:84% So relativistically, you can no longer 01:16:01.180 --> 01:16:03.610 align:middle line:84% isolate the contribution from the angular momentum 01:16:03.610 --> 01:16:05.153 align:middle line:84% from your orbital angular momentum, 01:16:05.153 --> 01:16:06.070 align:middle line:90% spin angular momentum. 01:16:06.070 --> 01:16:07.320 align:middle line:90% They all come together. 01:16:07.320 --> 01:16:09.070 align:middle line:84% So it's only in the long relativistic case 01:16:09.070 --> 01:16:11.410 align:middle line:90% you can isolate like this. 01:16:11.410 --> 01:16:13.605 align:middle line:84% So normally when we talk about magnetic moments, 01:16:13.605 --> 01:16:14.980 align:middle line:84% we always think it about in terms 01:16:14.980 --> 01:16:16.510 align:middle line:90% of nonrelativistic situation. 01:16:16.510 --> 01:16:20.350 align:middle line:90% 01:16:20.350 --> 01:16:24.610 align:middle line:90% So now just some final words. 01:16:24.610 --> 01:16:27.320 align:middle line:90% Just erased that number. 01:16:27.320 --> 01:16:31.250 align:middle line:84% So Schwinger then did this calculation. 01:16:31.250 --> 01:16:38.410 align:middle line:84% So including this diagram, Schwinger did this calculation, 01:16:38.410 --> 01:16:42.850 align:middle line:84% and then he found that there's a small correction. 01:16:42.850 --> 01:16:45.370 align:middle line:84% So normally, its called -- say if I call it -- 01:16:45.370 --> 01:16:51.890 align:middle line:84% 1 plus F. And so this is the correction. 01:16:51.890 --> 01:16:56.570 align:middle line:84% And then the correction, which Schwinger first calculated, 01:16:56.570 --> 01:17:00.800 align:middle line:84% is given by alpha just divided by 2pi. 01:17:00.800 --> 01:17:03.740 align:middle line:84% The reason it's alpha is because here there's a e here, 01:17:03.740 --> 01:17:07.700 align:middle line:84% here there's an e, so e squared is alpha. 01:17:07.700 --> 01:17:09.980 align:middle line:84% So all this complicated calculation 01:17:09.980 --> 01:17:13.703 align:middle line:84% you do, essentially you find this 1 over 2pi. 01:17:13.703 --> 01:17:15.620 align:middle line:84% Because this correction has to be proportional 01:17:15.620 --> 01:17:20.450 align:middle line:84% to alpha, and this number you do a very hard calculation, 01:17:20.450 --> 01:17:22.850 align:middle line:90% you find it's 1 over 2pi. 01:17:22.850 --> 01:17:26.270 align:middle line:84% And then this actually agreed with experiment very well. 01:17:26.270 --> 01:17:31.310 align:middle line:84% So this is first calculated by Schwinger in 1948. 01:17:31.310 --> 01:17:36.200 align:middle line:84% So this was really a theoretical triumph at that time 01:17:36.200 --> 01:17:40.910 align:middle line:84% and agreed very well with experiments. 01:17:40.910 --> 01:17:45.710 align:middle line:84% So then people then calculate to higher orders. 01:17:45.710 --> 01:17:50.950 align:middle line:84% So the next order must be proportional to alpha squared, 01:17:50.950 --> 01:17:54.932 align:middle line:84% so we can parameterize by alpha divided by 2pi squared, 01:17:54.932 --> 01:17:56.140 align:middle line:90% and then with some number a2. 01:17:56.140 --> 01:17:59.340 align:middle line:90% 01:17:59.340 --> 01:18:01.840 align:middle line:84% You do an even more complicated calculation, much, much more 01:18:01.840 --> 01:18:04.420 align:middle line:84% complicated calculation, you find that this a2, again, it's 01:18:04.420 --> 01:18:05.910 align:middle line:90% a single number. 01:18:05.910 --> 01:18:08.700 align:middle line:84% And this contains seven diagrams. 01:18:08.700 --> 01:18:12.210 align:middle line:84% So this is one diagram, but when you go to the next order, 01:18:12.210 --> 01:18:14.810 align:middle line:84% actually involving seven diagrams. 01:18:14.810 --> 01:18:18.290 align:middle line:90% And that was calculated in 1957. 01:18:18.290 --> 01:18:23.780 align:middle line:84% Again, it was a very important thing to do at that time, 01:18:23.780 --> 01:18:26.420 align:middle line:84% because just to check whether QED, 01:18:26.420 --> 01:18:30.530 align:middle line:84% because the experimental accuracy was at the level 01:18:30.530 --> 01:18:32.315 align:middle line:84% actually you can compare this number. 01:18:32.315 --> 01:18:37.160 align:middle line:90% 01:18:37.160 --> 01:18:39.920 align:middle line:84% Actually, also there's some kind of story 01:18:39.920 --> 01:18:42.830 align:middle line:84% behind this number, which at the beginning 01:18:42.830 --> 01:18:45.500 align:middle line:84% they calculated wrong, et cetera. 01:18:45.500 --> 01:18:48.224 align:middle line:84% So now you can also calculate the next one, a3. 01:18:48.224 --> 01:18:50.990 align:middle line:90% 01:18:50.990 --> 01:18:53.720 align:middle line:84% You can parameterize again by cube. 01:18:53.720 --> 01:18:57.395 align:middle line:84% And now this now includes 72 diagrams. 01:18:57.395 --> 01:19:00.780 align:middle line:90% 01:19:00.780 --> 01:19:03.700 align:middle line:90% And it took two physicists-- 01:19:03.700 --> 01:19:05.730 align:middle line:90% I won't write down their names-- 01:19:05.730 --> 01:19:09.370 align:middle line:84% it took them 25 years to calculate it. 01:19:09.370 --> 01:19:10.530 align:middle line:90% So actually they used it. 01:19:10.530 --> 01:19:14.320 align:middle line:90% 01:19:14.320 --> 01:19:18.600 align:middle line:84% So in 1996 they finally calculated this. 01:19:18.600 --> 01:19:26.090 align:middle line:84% But now if you want to say, oh, let's try to calculate a4, 01:19:26.090 --> 01:19:29.290 align:middle line:84% this includes 891 diagrams just in QED. 01:19:29.290 --> 01:19:32.350 align:middle line:90% 01:19:32.350 --> 01:19:35.180 align:middle line:84% And actually, this is also calculated using computer. 01:19:35.180 --> 01:19:39.170 align:middle line:90% 01:19:39.170 --> 01:19:40.920 align:middle line:84% And also, you can calculate this for muon. 01:19:40.920 --> 01:19:45.540 align:middle line:90% 01:19:45.540 --> 01:19:47.220 align:middle line:84% At a certain level, you also need 01:19:47.220 --> 01:19:49.810 align:middle line:84% to include the contribution from other interactions, 01:19:49.810 --> 01:19:53.800 align:middle line:84% not just QED, weak interactions, et cetera. 01:19:53.800 --> 01:19:59.160 align:middle line:84% Anyway, so the reason you want to calculate such a number 01:19:59.160 --> 01:20:03.420 align:middle line:84% to very high accuracy is because this F 01:20:03.420 --> 01:20:08.585 align:middle line:84% can be measured to very, very high accuracy experimentally. 01:20:08.585 --> 01:20:09.960 align:middle line:84% So actually, experimentally, this 01:20:09.960 --> 01:20:15.010 align:middle line:84% is one of the most accurately measured number in physics. 01:20:15.010 --> 01:20:18.755 align:middle line:84% They can measure this F to order of 10 to the -12. 01:20:18.755 --> 01:20:29.790 align:middle line:90% 01:20:29.790 --> 01:20:32.550 align:middle line:84% I think at the theoretical level, 01:20:32.550 --> 01:20:34.950 align:middle line:84% you actually cannot calculate to this level accuracy, 01:20:34.950 --> 01:20:42.790 align:middle line:84% but still you can compare to very, very good agreement. 01:20:42.790 --> 01:20:48.975 align:middle line:84% So QED actually describes nature very, very well. 01:20:48.975 --> 01:20:52.020 align:middle line:90% 01:20:52.020 --> 01:20:55.785 align:middle line:90% So I think that's all. 01:20:55.785 --> 01:21:01.170 align:middle line:90% 01:21:01.170 --> 01:21:04.020 align:middle line:84% I'm honored and very happy to have this opportunity 01:21:04.020 --> 01:21:09.270 align:middle line:84% to take you to this first part of the journey to quantum field 01:21:09.270 --> 01:21:14.580 align:middle line:84% theory, and I want to thank you all 01:21:14.580 --> 01:21:20.430 align:middle line:84% for making this journey very enjoyable 01:21:20.430 --> 01:21:24.900 align:middle line:84% and rewarding for myself, and I hope you have learned something 01:21:24.900 --> 01:21:31.770 align:middle line:84% from this class, and whatever your career path takes you, 01:21:31.770 --> 01:21:36.140 align:middle line:84% I hope you will later find this course useful. 01:21:36.140 --> 01:21:37.270 align:middle line:90% Thank you. 01:21:37.270 --> 01:21:40.320 align:middle line:90% [APPLAUSE] 01:21:40.320 --> 01:21:45.000 align:middle line:90%