WEBVTT 00:00:00.000 --> 00:00:01.952 align:middle line:90% [SQUEAKING] 00:00:01.952 --> 00:00:03.904 align:middle line:90% [RUSTLING] 00:00:03.904 --> 00:00:05.856 align:middle line:90% [CLICKING] 00:00:05.856 --> 00:00:11.040 align:middle line:90% 00:00:11.040 --> 00:00:13.890 align:middle line:90% PROFESSOR: So let's start. 00:00:13.890 --> 00:00:16.680 align:middle line:84% So let me just clarify one question, 00:00:16.680 --> 00:00:18.570 align:middle line:90% which was asked last time. 00:00:18.570 --> 00:00:24.420 align:middle line:84% So when we calculate the total cross-section for, say, 00:00:24.420 --> 00:00:29.730 align:middle line:84% for this e plus e minus to mu plus, mu minus, 00:00:29.730 --> 00:00:32.369 align:middle line:84% then we find that there's a funny fact -- 00:00:32.369 --> 00:00:36.430 align:middle line:90% yeah, let me just write down-- 00:00:36.430 --> 00:00:39.820 align:middle line:84% so then there's a factor like this. 00:00:39.820 --> 00:00:48.140 align:middle line:84% m prime square E squared divided by m square e square 00:00:48.140 --> 00:00:50.120 align:middle line:90% and then times something else. 00:00:50.120 --> 00:00:54.300 align:middle line:90% OK, so times something else. 00:00:54.300 --> 00:00:57.980 align:middle line:84% And so there's funny feature that this factor seems to blow 00:00:57.980 --> 00:01:00.350 align:middle line:90% up when the E is equal to m. 00:01:00.350 --> 00:01:05.180 align:middle line:84% So if you decrease E to the value of m-- 00:01:05.180 --> 00:01:08.660 align:middle line:84% and then this seemingly blows up, OK. 00:01:08.660 --> 00:01:12.530 align:middle line:84% But of course this never happens in the real situation 00:01:12.530 --> 00:01:16.300 align:middle line:84% because this mass is much larger than this mass. 00:01:16.300 --> 00:01:19.270 align:middle line:84% So the upstairs will go to 0 before this blows up. 00:01:19.270 --> 00:01:24.110 align:middle line:84% So that's why people never cared about it, OK, including myself. 00:01:24.110 --> 00:01:26.210 align:middle line:84% But you can ask the question-- suppose 00:01:26.210 --> 00:01:29.660 align:middle line:84% the electron is more massive than the muon 00:01:29.660 --> 00:01:33.170 align:middle line:84% and then you will reach the 0s of this first 00:01:33.170 --> 00:01:34.880 align:middle line:90% before you reach the 0 of this. 00:01:34.880 --> 00:01:37.300 align:middle line:84% And then you will see something blows up. 00:01:37.300 --> 00:01:39.320 align:middle line:90% OK, then it's curious. 00:01:39.320 --> 00:01:42.860 align:middle line:84% There's a very simple mathematical reason for this. 00:01:42.860 --> 00:01:46.370 align:middle line:84% And the reason is that the cross section is defined-- 00:01:46.370 --> 00:01:50.480 align:middle line:84% remember, the cross section is defined 00:01:50.480 --> 00:01:55.160 align:middle line:84% to be defined to be the divided 1 over the flux. 00:01:55.160 --> 00:01:58.865 align:middle line:84% And the flux is proportional to 1 -- 00:01:58.865 --> 00:02:03.600 align:middle line:84% it is the density times the velocity. 00:02:03.600 --> 00:02:05.640 align:middle line:90% And the density is the same. 00:02:05.640 --> 00:02:08.370 align:middle line:84% It's just one particle per volume. 00:02:08.370 --> 00:02:11.180 align:middle line:90% And then the velocity goes to 0. 00:02:11.180 --> 00:02:15.020 align:middle line:84% This precisely comes when the velocity goes to 0 because when 00:02:15.020 --> 00:02:16.820 align:middle line:84% the velocity becomes very, very slow, 00:02:16.820 --> 00:02:19.830 align:middle line:84% then your flux become very slow and small. 00:02:19.830 --> 00:02:20.900 align:middle line:90% Then this becomes big. 00:02:20.900 --> 00:02:22.520 align:middle line:84% So mathematically, that's the reason 00:02:22.520 --> 00:02:28.520 align:middle line:84% why this becomes very big, OK, when the velocity goes to 0. 00:02:28.520 --> 00:02:34.170 align:middle line:84% But still it is a little bit funny, I should admit. 00:02:34.170 --> 00:02:36.440 align:middle line:84% So when the velocity equal to 0, of course, it's 00:02:36.440 --> 00:02:38.190 align:middle line:84% unphysical because when the velocity is 0, 00:02:38.190 --> 00:02:40.680 align:middle line:90% they just never scatter, OK. 00:02:40.680 --> 00:02:43.080 align:middle line:84% So this divergence is never an issue. 00:02:43.080 --> 00:02:48.040 align:middle line:84% But you can ask the question why physically somehow 00:02:48.040 --> 00:02:52.080 align:middle line:84% when you decrease the v and somehow the cross section 00:02:52.080 --> 00:02:53.700 align:middle line:90% should become bigger and bigger? 00:02:53.700 --> 00:02:57.690 align:middle line:84% And the mathematically it's due to the flux just due to the way 00:02:57.690 --> 00:03:01.340 align:middle line:90% the cross section is defined. 00:03:01.340 --> 00:03:03.660 align:middle line:84% But I also mentioned before that the cross section 00:03:03.660 --> 00:03:06.240 align:middle line:84% is supposed to measure the effective area 00:03:06.240 --> 00:03:07.410 align:middle line:90% of the interaction. 00:03:07.410 --> 00:03:10.170 align:middle line:84% And then why should that depend on the velocity, OK? 00:03:10.170 --> 00:03:15.120 align:middle line:84% And so this aspect that I don't have a very good explanation-- 00:03:15.120 --> 00:03:18.350 align:middle line:84% but this is the mathematical reason for it. 00:03:18.350 --> 00:03:19.280 align:middle line:90% OK, good. 00:03:19.280 --> 00:03:25.370 align:middle line:84% Yeah, so this is just clarify the question 00:03:25.370 --> 00:03:27.260 align:middle line:90% over the last time. 00:03:27.260 --> 00:03:33.440 align:middle line:84% OK, so then we talk about crossing symmetry. 00:03:33.440 --> 00:03:40.960 align:middle line:90% 00:03:40.960 --> 00:03:44.380 align:middle line:90% So let's consider two process. 00:03:44.380 --> 00:03:49.560 align:middle line:84% One is the e plus e minus goes to mu plus mu minus 00:03:49.560 --> 00:03:53.250 align:middle line:84% as we just discussed with the Feynman diagram 00:03:53.250 --> 00:03:54.030 align:middle line:90% going like this. 00:03:54.030 --> 00:03:57.380 align:middle line:90% 00:03:57.380 --> 00:03:58.510 align:middle line:90% So this is the-- 00:03:58.510 --> 00:04:02.390 align:middle line:90% 00:04:02.390 --> 00:04:06.440 align:middle line:84% OK, so this is the e minus e plus. 00:04:06.440 --> 00:04:08.795 align:middle line:90% This is mu minus mu plus, OK. 00:04:08.795 --> 00:04:12.210 align:middle line:90% 00:04:12.210 --> 00:04:14.910 align:middle line:90% So there's also another process. 00:04:14.910 --> 00:04:18.645 align:middle line:84% Let's consider another process, which is e minus-- 00:04:18.645 --> 00:04:21.230 align:middle line:90% 00:04:21.230 --> 00:04:25.650 align:middle line:84% with mu minus going to e minus plus mu minus. 00:04:25.650 --> 00:04:29.385 align:middle line:84% And the Feynman diagram for this is given by this. 00:04:29.385 --> 00:04:36.720 align:middle line:90% 00:04:36.720 --> 00:04:39.810 align:middle line:84% OK, so this is the e minus and mu minus 00:04:39.810 --> 00:04:41.280 align:middle line:90% are at the initial state now. 00:04:41.280 --> 00:04:42.750 align:middle line:90% And this is e minus. 00:04:42.750 --> 00:04:44.010 align:middle line:90% And this is mu minus. 00:04:44.010 --> 00:04:47.260 align:middle line:84% OK, again, only one diagram contributes. 00:04:47.260 --> 00:04:47.760 align:middle line:90% OK. 00:04:47.760 --> 00:04:52.420 align:middle line:90% 00:04:52.420 --> 00:04:57.620 align:middle line:84% So if you compare a and b, you can see that b-- 00:04:57.620 --> 00:05:01.070 align:middle line:84% the diagram for b is essentially the diagram for a 00:05:01.070 --> 00:05:03.500 align:middle line:90% if you view it sideways. 00:05:03.500 --> 00:05:06.260 align:middle line:84% OK, if you view it sideways-- and then you see 00:05:06.260 --> 00:05:07.590 align:middle line:90% this goes to that. 00:05:07.590 --> 00:05:09.330 align:middle line:84% And then this becomes final state. 00:05:09.330 --> 00:05:11.570 align:middle line:84% And the final state coming out e plus 00:05:11.570 --> 00:05:13.010 align:middle line:90% essentially becomes e minus. 00:05:13.010 --> 00:05:16.910 align:middle line:90% And this just becomes that, OK. 00:05:16.910 --> 00:05:21.665 align:middle line:84% So the difference between these two is that the e plus-- 00:05:21.665 --> 00:05:24.490 align:middle line:90% 00:05:24.490 --> 00:05:34.780 align:middle line:84% so e plus in initial state of a then 00:05:34.780 --> 00:05:46.860 align:middle line:84% goes to e minus in final state of b, OK, in final state of B. 00:05:46.860 --> 00:06:01.170 align:middle line:84% Similarly, the mu plus in the final state of a then 00:06:01.170 --> 00:06:16.166 align:middle line:84% become the mu minus in the initial state of b. 00:06:16.166 --> 00:06:19.290 align:middle line:90% 00:06:19.290 --> 00:06:24.040 align:middle line:90% Yeah, so essentially, mu plus-- 00:06:24.040 --> 00:06:26.680 align:middle line:84% essentially, you take the e plus going to the other side 00:06:26.680 --> 00:06:29.290 align:middle line:90% become e minus and then take-- 00:06:29.290 --> 00:06:32.680 align:middle line:84% mu plus going to this side become mu minus, OK. 00:06:32.680 --> 00:06:36.820 align:middle line:84% So essentially you just exchange that. 00:06:36.820 --> 00:06:42.400 align:middle line:84% So now let's label the quantum numbers. 00:06:42.400 --> 00:06:45.890 align:middle line:90% Let's call this p1, r1. 00:06:45.890 --> 00:06:48.410 align:middle line:90% 00:06:48.410 --> 00:06:56.640 align:middle line:90% So this is-- call it p2, r2. 00:06:56.640 --> 00:06:57.885 align:middle line:90% And call this one-- 00:06:57.885 --> 00:07:01.680 align:middle line:90% 00:07:01.680 --> 00:07:05.910 align:middle line:90% Call this k1, s1. 00:07:05.910 --> 00:07:07.800 align:middle line:90% So these are the polarization. 00:07:07.800 --> 00:07:12.060 align:middle line:90% And this is k2, s2. 00:07:12.060 --> 00:07:17.550 align:middle line:84% And now similarly I label here by p1, r1. 00:07:17.550 --> 00:07:21.280 align:middle line:90% Label this one by p2, r2 bar. 00:07:21.280 --> 00:07:24.550 align:middle line:90% So this is for the antiparticle. 00:07:24.550 --> 00:07:30.900 align:middle line:90% And so this one would be k1, s1. 00:07:30.900 --> 00:07:34.755 align:middle line:84% And then this will be k2, s2 bar, OK. 00:07:34.755 --> 00:07:39.270 align:middle line:90% 00:07:39.270 --> 00:07:51.972 align:middle line:84% So now if we look at this map, so it's like in the process A, 00:07:51.972 --> 00:07:59.860 align:middle line:84% we take this to be initial state, p2, r2. 00:07:59.860 --> 00:08:05.980 align:middle line:84% If we replace it by minus k2, s2-- 00:08:05.980 --> 00:08:08.990 align:middle line:90% OK, minus k2 means that the-- 00:08:08.990 --> 00:08:13.060 align:middle line:84% yeah, so this one e plus now becomes the final state. 00:08:13.060 --> 00:08:14.410 align:middle line:90% So this is the-- 00:08:14.410 --> 00:08:17.980 align:middle line:84% so from the momentum direction, this is the going in. 00:08:17.980 --> 00:08:19.280 align:middle line:90% And this will be going out. 00:08:19.280 --> 00:08:21.190 align:middle line:90% So we need to change the sign. 00:08:21.190 --> 00:08:25.270 align:middle line:84% OK, we to change the sign and, similarly, 00:08:25.270 --> 00:08:40.919 align:middle line:84% the k2 for s2 for k2, s2 bar and then go to minus p2, r2. 00:08:40.919 --> 00:08:44.610 align:middle line:90% 00:08:44.610 --> 00:08:50.932 align:middle line:84% If you make this placement, then we get the process b, OK. 00:08:50.932 --> 00:08:52.620 align:middle line:90% You get the process b. 00:08:52.620 --> 00:08:54.410 align:middle line:84% So you just make the replacement. 00:08:54.410 --> 00:08:58.190 align:middle line:84% And then the process in a will go through the process in b. 00:08:58.190 --> 00:09:02.370 align:middle line:90% 00:09:02.370 --> 00:09:05.250 align:middle line:84% So suppose you forget about those polarizations. 00:09:05.250 --> 00:09:10.620 align:middle line:84% If we consider how we're talking about the scalar 00:09:10.620 --> 00:09:12.510 align:middle line:84% particles-- so there's no polarization. 00:09:12.510 --> 00:09:16.210 align:middle line:90% And then this just trivially-- 00:09:16.210 --> 00:09:18.370 align:middle line:84% so for scalar, then we just have trivially-- 00:09:18.370 --> 00:09:25.130 align:middle line:90% 00:09:25.130 --> 00:09:26.900 align:middle line:90% We can just do the replacement. 00:09:26.900 --> 00:09:33.870 align:middle line:84% You can trivially see that the amplitude for the process 00:09:33.870 --> 00:09:43.820 align:middle line:84% a with p1 minus k2 and the k1 minus p2 00:09:43.820 --> 00:09:46.580 align:middle line:84% then will be just equal to the amplitude for the process 00:09:46.580 --> 00:09:52.520 align:middle line:90% b with p1, p2, k1, k2. 00:09:52.520 --> 00:09:59.300 align:middle line:84% OK, so we just rename your momentum of your process a. 00:09:59.300 --> 00:10:03.080 align:middle line:84% And then you will just get essentially the amplitude 00:10:03.080 --> 00:10:06.200 align:middle line:84% for two p because the only thing you need to do 00:10:06.200 --> 00:10:09.140 align:middle line:84% is just exchange the name of the momentum, OK. 00:10:09.140 --> 00:10:12.450 align:middle line:90% 00:10:12.450 --> 00:10:14.850 align:middle line:84% So then we'll be trivially the same. 00:10:14.850 --> 00:10:24.770 align:middle line:90% 00:10:24.770 --> 00:10:35.620 align:middle line:84% But for fermions, we should worry about-- 00:10:35.620 --> 00:10:41.130 align:middle line:90% 00:10:41.130 --> 00:10:48.915 align:middle line:84% also look at the wave function associated with external legs. 00:10:48.915 --> 00:11:01.575 align:middle line:90% 00:11:01.575 --> 00:11:04.550 align:middle line:90% OK, so in a-- 00:11:04.550 --> 00:11:07.760 align:middle line:84% let me just write down those wave function explicitly. 00:11:07.760 --> 00:11:16.400 align:middle line:84% So in a, so the p2, r2 bar are associated with the wave 00:11:16.400 --> 00:11:22.590 align:middle line:90% function is to be v bar r2, p2. 00:11:22.590 --> 00:11:24.185 align:middle line:90% OK, so this is for the e plus. 00:11:24.185 --> 00:11:28.940 align:middle line:90% 00:11:28.940 --> 00:11:38.640 align:middle line:84% And the k2, s2 bar are related to v s2, k2. 00:11:38.640 --> 00:11:40.890 align:middle line:84% OK, so this is related to the mu plus. 00:11:40.890 --> 00:11:43.880 align:middle line:84% OK, so according to our previous rule-- 00:11:43.880 --> 00:11:50.650 align:middle line:84% but in b the wave function for these two-- 00:11:50.650 --> 00:11:58.350 align:middle line:84% so the k2, s2, so the corresponding one is-- 00:11:58.350 --> 00:12:04.200 align:middle line:84% the one you want to exchange is u s2 bar, k2. 00:12:04.200 --> 00:12:10.980 align:middle line:84% This is corresponding to e minus and the p2, r2. 00:12:10.980 --> 00:12:17.070 align:middle line:84% And that's corresponding to u r2, p2, mu minus. 00:12:17.070 --> 00:12:18.510 align:middle line:90% OK. 00:12:18.510 --> 00:12:23.250 align:middle line:84% So now you see even when you make the label change-- 00:12:23.250 --> 00:12:27.460 align:middle line:84% suppose you make the label replacement from here 00:12:27.460 --> 00:12:29.050 align:middle line:90% to replace by that. 00:12:29.050 --> 00:12:33.400 align:middle line:84% You're not going to change the wave function from v to u. 00:12:33.400 --> 00:12:35.620 align:middle line:84% OK, so the wave function actually is different. 00:12:35.620 --> 00:12:41.260 align:middle line:90% 00:12:41.260 --> 00:12:43.240 align:middle line:90% So similarly this does not-- 00:12:43.240 --> 00:12:45.775 align:middle line:84% so this does not work if you have fermions, OK, 00:12:45.775 --> 00:12:48.820 align:middle line:84% if I have fermions, because the function changes. 00:12:48.820 --> 00:12:54.430 align:middle line:84% It changes from v to u, in this case here again also 00:12:54.430 --> 00:12:55.720 align:middle line:90% v to u, OK. 00:12:55.720 --> 00:12:59.300 align:middle line:90% 00:12:59.300 --> 00:13:01.080 align:middle line:84% But actually this is another problem 00:13:01.080 --> 00:13:07.070 align:middle line:84% if we consider the unpolarized spin sum actually still works, 00:13:07.070 --> 00:13:10.520 align:middle line:90% OK, for unpolarized. 00:13:10.520 --> 00:13:13.370 align:middle line:84% So if you have an unpolarized situation-- 00:13:13.370 --> 00:13:18.290 align:middle line:90% 00:13:18.290 --> 00:13:20.330 align:middle line:84% remember, we need to sum over all the spins. 00:13:20.330 --> 00:13:24.050 align:middle line:90% 00:13:24.050 --> 00:13:28.275 align:middle line:84% And consider M squared, sum of all the spins, OK. 00:13:28.275 --> 00:13:31.290 align:middle line:90% 00:13:31.290 --> 00:13:34.960 align:middle line:84% so we will involve this y sum with itself. 00:13:34.960 --> 00:13:39.460 align:middle line:84% For example, when you sum of all the r2s, 00:13:39.460 --> 00:13:41.160 align:middle line:84% then we will have something like this. 00:13:41.160 --> 00:13:43.830 align:middle line:90% 00:13:43.830 --> 00:13:46.200 align:middle line:84% So, for example, we need to sum over r2, 00:13:46.200 --> 00:13:51.835 align:middle line:84% then we will have a combination like v, r2, p2-- 00:13:51.835 --> 00:13:53.800 align:middle line:84% that's the calculation we did before-- 00:13:53.800 --> 00:13:58.390 align:middle line:90% and v, r2, bar, p2, OK. 00:13:58.390 --> 00:14:05.620 align:middle line:84% And then this one will give you just minus i p2, slash, 00:14:05.620 --> 00:14:12.090 align:middle line:90% plus m of the electron, OK. 00:14:12.090 --> 00:14:16.085 align:middle line:84% And now you can just do the replacement. 00:14:16.085 --> 00:14:17.460 align:middle line:84% Let's just do what was written -- 00:14:17.460 --> 00:14:21.390 align:middle line:90% the p2 replace it by minus k2. 00:14:21.390 --> 00:14:23.340 align:middle line:84% So if you do the replacement, then that 00:14:23.340 --> 00:14:32.590 align:middle line:84% goes to minus i, slash, k2 plus still the electron mass, OK. 00:14:32.590 --> 00:14:44.140 align:middle line:84% And then this is the same as the sum s2, u, s2, k2, u, s2, bar, 00:14:44.140 --> 00:14:46.840 align:middle line:90% k2, OK. 00:14:46.840 --> 00:14:52.720 align:middle line:84% So despite that the wave function are different, OK-- 00:14:52.720 --> 00:14:54.400 align:middle line:90% so the wave function. 00:14:54.400 --> 00:14:58.190 align:middle line:84% But after you do the spin sum, OK, they actually 00:14:58.190 --> 00:15:01.820 align:middle line:90% get the same answer, OK. 00:15:01.820 --> 00:15:05.480 align:middle line:84% So this is the same as the minus this one. 00:15:05.480 --> 00:15:10.890 align:middle line:84% OK, so in this case after you do the spin sum-- 00:15:10.890 --> 00:15:20.100 align:middle line:84% so each replacement of momentum-- 00:15:20.100 --> 00:15:27.890 align:middle line:84% so in this case just give us a minus sign, 00:15:27.890 --> 00:15:34.250 align:middle line:84% OK, when you calculate this spin sum, m squared. 00:15:34.250 --> 00:15:39.240 align:middle line:84% But now since we need to replace two of them, OK-- 00:15:39.240 --> 00:15:40.905 align:middle line:84% so 2 minus sine is a positive sign. 00:15:40.905 --> 00:15:48.200 align:middle line:90% 00:15:48.200 --> 00:16:01.240 align:middle line:84% So we conclude that the sum over spin Ma squared evaluated 00:16:01.240 --> 00:16:09.760 align:middle line:84% at p1 minus k2 and k1 minus p2 is 00:16:09.760 --> 00:16:15.820 align:middle line:84% the same as the sum over spin, the amplitude for b squared 00:16:15.820 --> 00:16:21.640 align:middle line:90% divided p1, p2, k1, k2, OK. 00:16:21.640 --> 00:16:26.290 align:middle line:84% So there's a simple relation between the amplitude square 00:16:26.290 --> 00:16:29.230 align:middle line:84% when you do the spin sum, and we just 00:16:29.230 --> 00:16:32.110 align:middle line:90% need to rename the momentum, OK. 00:16:32.110 --> 00:16:33.250 align:middle line:90% Rename the momentum. 00:16:33.250 --> 00:16:39.170 align:middle line:90% 00:16:39.170 --> 00:16:41.720 align:middle line:90% Good? 00:16:41.720 --> 00:16:45.080 align:middle line:90% Any questions on this? 00:16:45.080 --> 00:16:49.670 align:middle line:90% So for unpolarized amplitude-- 00:16:49.670 --> 00:16:52.220 align:middle line:84% so we still have this very nice relation 00:16:52.220 --> 00:16:54.800 align:middle line:90% between these two process, OK. 00:16:54.800 --> 00:16:57.268 align:middle line:84% If you calculated one of them, then you 00:16:57.268 --> 00:16:58.685 align:middle line:84% don't need to calculate the other. 00:16:58.685 --> 00:17:01.160 align:middle line:84% You can just immediately get the answer. 00:17:01.160 --> 00:17:05.510 align:middle line:84% You just change a few momenta, change a couple momenta. 00:17:05.510 --> 00:17:08.770 align:middle line:90% 00:17:08.770 --> 00:17:13.100 align:middle line:84% But if you consider the polarized amplitude-- 00:17:13.100 --> 00:17:24.640 align:middle line:90% 00:17:24.640 --> 00:17:28.180 align:middle line:84% So the story is now is much simpler. 00:17:28.180 --> 00:17:33.760 align:middle line:84% Now you do have a problem because now the wavefunction 00:17:33.760 --> 00:17:35.410 align:middle line:90% are different, OK. 00:17:35.410 --> 00:17:41.815 align:middle line:84% But nevertheless, you can choose a different basis, OK. 00:17:41.815 --> 00:17:44.710 align:middle line:90% 00:17:44.710 --> 00:18:10.542 align:middle line:84% By choosing appropriate basis, you can still directly relate 00:18:10.542 --> 00:18:12.110 align:middle line:90% -- 00:18:12.110 --> 00:18:13.505 align:middle line:90% equate the amplitudes, OK. 00:18:13.505 --> 00:18:21.690 align:middle line:90% 00:18:21.690 --> 00:18:24.540 align:middle line:84% You can still recreate the amplitude. 00:18:24.540 --> 00:18:28.250 align:middle line:84% OK, so, yeah, I will not go into there, OK. 00:18:28.250 --> 00:18:33.860 align:middle line:84% But it's possible, say, you choose a basis of v2 00:18:33.860 --> 00:18:38.780 align:middle line:84% and the u2 and u and then somehow you can relate them 00:18:38.780 --> 00:18:41.190 align:middle line:90% to each other, OK. 00:18:41.190 --> 00:18:48.861 align:middle line:84% So this relation between the amplitude of a and b-- 00:18:48.861 --> 00:18:52.170 align:middle line:84% so this is called crossing symmetry. 00:18:52.170 --> 00:18:53.850 align:middle line:84% Yeah, let me just call this star. 00:18:53.850 --> 00:18:57.930 align:middle line:84% So star is called crossing symmetry. 00:18:57.930 --> 00:19:04.200 align:middle line:90% 00:19:04.200 --> 00:19:07.440 align:middle line:84% So calling it symmetry is actually 00:19:07.440 --> 00:19:12.000 align:middle line:84% a misnomer because this is not a symmetry, OK. 00:19:12.000 --> 00:19:15.330 align:middle line:84% This is just a relation between the amplitudes 00:19:15.330 --> 00:19:16.710 align:middle line:90% of different process. 00:19:16.710 --> 00:19:19.410 align:middle line:90% OK, so this is not a symmetry. 00:19:19.410 --> 00:19:22.980 align:middle line:84% So the symmetry should be put as quotes, OK. 00:19:22.980 --> 00:19:25.740 align:middle line:84% So this just express the relation of one amplitude 00:19:25.740 --> 00:19:28.690 align:middle line:84% to the relation of another amplitude. 00:19:28.690 --> 00:19:30.930 align:middle line:84% Yeah, just relation between the amplitude of one 00:19:30.930 --> 00:19:35.690 align:middle line:84% process to the amplitude of another process. 00:19:35.690 --> 00:19:42.170 align:middle line:84% And the relation come from a fundamental fact, 00:19:42.170 --> 00:19:43.685 align:middle line:90% come from a very simple fact-- 00:19:43.685 --> 00:19:56.380 align:middle line:90% 00:19:56.380 --> 00:19:59.680 align:middle line:84% OK, so if we consider this fermionic field, psi-- 00:19:59.680 --> 00:20:05.090 align:middle line:90% 00:20:05.090 --> 00:20:07.880 align:middle line:90% so psi can play two role. 00:20:07.880 --> 00:20:14.825 align:middle line:90% It can either annihilate-- 00:20:14.825 --> 00:20:22.620 align:middle line:90% 00:20:22.620 --> 00:20:29.230 align:middle line:84% remember psi is called a plus b dagger, OK. 00:20:29.230 --> 00:20:32.840 align:middle line:90% It's called a plus b dagger. 00:20:32.840 --> 00:20:37.120 align:middle line:84% So you can either annihilate an initial particle-- 00:20:37.120 --> 00:20:43.520 align:middle line:90% 00:20:43.520 --> 00:20:44.875 align:middle line:90% so that's the a part-- 00:20:44.875 --> 00:20:47.620 align:middle line:90% 00:20:47.620 --> 00:20:56.020 align:middle line:84% or can create an initial antiparticle 00:20:56.020 --> 00:21:00.250 align:middle line:90% or create, sorry, a final-- 00:21:00.250 --> 00:21:02.050 align:middle line:90% sorry, an antiparticle. 00:21:02.050 --> 00:21:07.740 align:middle line:90% 00:21:07.740 --> 00:21:14.070 align:middle line:90% OK, so let me just elaborate. 00:21:14.070 --> 00:21:20.171 align:middle line:84% So if we have a, when you act on this side-- 00:21:20.171 --> 00:21:24.276 align:middle line:84% when you act on this side, you only have-- 00:21:24.276 --> 00:21:26.310 align:middle line:84% if you have a particle in the initial state, 00:21:26.310 --> 00:21:29.560 align:middle line:84% then a can annihilate that particle. 00:21:29.560 --> 00:21:33.860 align:middle line:84% But if a acts on the left, act on the left, 00:21:33.860 --> 00:21:36.380 align:middle line:90% a becomes a dagger. 00:21:36.380 --> 00:21:38.537 align:middle line:84% And then that creates an antiparticle. 00:21:38.537 --> 00:21:45.330 align:middle line:90% 00:21:45.330 --> 00:21:54.990 align:middle line:84% Yeah, so in such a process, X plus a goes to Y. 00:21:54.990 --> 00:21:56.310 align:middle line:90% And then you can just-- 00:21:56.310 --> 00:22:00.490 align:middle line:84% so this relates to-- the X and Y are some combination, 00:22:00.490 --> 00:22:01.950 align:middle line:90% some combination of particles. 00:22:01.950 --> 00:22:03.780 align:middle line:90% And then this just relates to-- 00:22:03.780 --> 00:22:13.510 align:middle line:90% 00:22:13.510 --> 00:22:19.585 align:middle line:90% X goes to Y plus a bar, OK. 00:22:19.585 --> 00:22:24.770 align:middle line:90% 00:22:24.770 --> 00:22:29.200 align:middle line:84% So here just X and Y, some collection of particles. 00:22:29.200 --> 00:22:49.270 align:middle line:90% 00:22:49.270 --> 00:22:59.950 align:middle line:84% Yeah, a's just some particle, OK OK, so just confirm that, OK. 00:22:59.950 --> 00:23:02.850 align:middle line:90% 00:23:02.850 --> 00:23:03.350 align:middle line:90% Good. 00:23:03.350 --> 00:23:04.685 align:middle line:90% Any questions on this? 00:23:04.685 --> 00:23:05.185 align:middle line:90% Yes. 00:23:05.185 --> 00:23:10.090 align:middle line:90% 00:23:10.090 --> 00:23:12.370 align:middle line:90% e minus all mu. 00:23:12.370 --> 00:23:16.090 align:middle line:84% Yeah, just say you have one field for e. 00:23:16.090 --> 00:23:17.220 align:middle line:90% You have one field for mu. 00:23:17.220 --> 00:23:20.140 align:middle line:90% 00:23:20.140 --> 00:23:20.860 align:middle line:90% Yes? 00:23:20.860 --> 00:23:23.775 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:23:23.775 --> 00:23:37.937 align:middle line:90% 00:23:37.937 --> 00:23:38.645 align:middle line:90% PROFESSOR: Sorry? 00:23:38.645 --> 00:23:41.660 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:23:41.660 --> 00:23:43.205 align:middle line:90% PROFESSOR: Sorry, say it again. 00:23:43.205 --> 00:23:50.210 align:middle line:84% AUDIENCE: The process, like for y, can we think of it as-- 00:23:50.210 --> 00:23:54.890 align:middle line:84% first, X to Y and a bar then a bar, 00:23:54.890 --> 00:23:57.230 align:middle line:84% like annihilate or do something like that? 00:23:57.230 --> 00:23:59.360 align:middle line:84% PROFESSOR: Yeah, heuristically you may-- 00:23:59.360 --> 00:24:00.530 align:middle line:90% but it's simpler than that. 00:24:00.530 --> 00:24:04.070 align:middle line:90% Yeah, it's simpler than that. 00:24:04.070 --> 00:24:05.200 align:middle line:90% Yes? 00:24:05.200 --> 00:24:07.200 align:middle line:84% AUDIENCE: Just regarding energy conservation. 00:24:07.200 --> 00:24:09.570 align:middle line:84% So if you can keep doing that can't we just 00:24:09.570 --> 00:24:16.423 align:middle line:84% say that like out of nothing comes Y plus a bar plus X bar. 00:24:16.423 --> 00:24:17.840 align:middle line:84% PROFESSOR: Yeah, so that's why you 00:24:17.840 --> 00:24:20.540 align:middle line:90% have to change the momentum. 00:24:20.540 --> 00:24:25.700 align:middle line:84% So some momentum for this process. 00:24:25.700 --> 00:24:29.050 align:middle line:84% Corresponding to some momentum for that process. 00:24:29.050 --> 00:24:35.370 align:middle line:84% Yeah, we have to switch the initial and final momentum. 00:24:35.370 --> 00:24:38.930 align:middle line:84% Yeah, it's not to say these two processes are the same. 00:24:38.930 --> 00:24:41.860 align:middle line:84% It's just that this process for one set of momentum 00:24:41.860 --> 00:24:45.520 align:middle line:84% have the same amplitude as the other process 00:24:45.520 --> 00:24:47.637 align:middle line:90% for some other set of momentum. 00:24:47.637 --> 00:24:49.220 align:middle line:84% Yeah, there's a relation between them. 00:24:49.220 --> 00:24:55.210 align:middle line:90% 00:24:55.210 --> 00:24:58.020 align:middle line:90% Other questions? 00:24:58.020 --> 00:25:04.910 align:middle line:84% OK, so now let's consider another important process 00:25:04.910 --> 00:25:06.170 align:middle line:90% called Compton scattering. 00:25:06.170 --> 00:25:12.050 align:middle line:90% 00:25:12.050 --> 00:25:15.710 align:middle line:84% So Compton scattering played a very important role 00:25:15.710 --> 00:25:22.610 align:middle line:84% in the early days of physics and in showing that actually this 00:25:22.610 --> 00:25:25.490 align:middle line:84% is one of the early experiments to show 00:25:25.490 --> 00:25:28.100 align:middle line:84% that the macroscopic world is governed 00:25:28.100 --> 00:25:31.930 align:middle line:84% by quantum mechanics rather than classical mechanics. 00:25:31.930 --> 00:25:37.615 align:middle line:84% So now let's consider how to calculate 00:25:37.615 --> 00:25:40.510 align:middle line:90% the physics for this process. 00:25:40.510 --> 00:25:41.950 align:middle line:84% So Compton scattering is a process 00:25:41.950 --> 00:25:46.420 align:middle line:84% you have a photon hit the electron. 00:25:46.420 --> 00:25:48.790 align:middle line:90% And then you get another photon. 00:25:48.790 --> 00:25:54.030 align:middle line:84% And then you get another electron, OK. 00:25:54.030 --> 00:25:57.450 align:middle line:84% And so we can draw the Feynman diagram for it. 00:25:57.450 --> 00:25:59.160 align:middle line:84% So one Feynman diagrams is the following. 00:25:59.160 --> 00:26:03.700 align:middle line:84% Let's just imagine you have a fermionic trajectory. 00:26:03.700 --> 00:26:08.350 align:middle line:84% So you can imagine at some point there's a photon coming in. 00:26:08.350 --> 00:26:11.050 align:middle line:84% OK, so this is the electron line. 00:26:11.050 --> 00:26:14.658 align:middle line:84% So the electron line, the arrow is the charge line. 00:26:14.658 --> 00:26:16.950 align:middle line:84% So the photon line, the arrow should be understood just 00:26:16.950 --> 00:26:18.150 align:middle line:90% as a momentum, OK. 00:26:18.150 --> 00:26:24.670 align:middle line:84% And then you emit another photon as a final state, OK. 00:26:24.670 --> 00:26:31.320 align:middle line:84% So this is the simplest Feynman diagram for this process. 00:26:31.320 --> 00:26:37.230 align:middle line:84% But actually there are two diagrams because you can also-- 00:26:37.230 --> 00:26:45.330 align:middle line:84% for this electron line, you can also first emit a photon 00:26:45.330 --> 00:26:51.970 align:middle line:84% and then absorb a photon can also have that, OK. 00:26:51.970 --> 00:26:52.960 align:middle line:90% So you have two. 00:26:52.960 --> 00:26:58.855 align:middle line:84% And these two processes are not the same, OK. 00:26:58.855 --> 00:27:07.070 align:middle line:84% So now let me put some label on the diagram. 00:27:07.070 --> 00:27:10.960 align:middle line:84% So, again, the fermion is p1, r1. 00:27:10.960 --> 00:27:14.800 align:middle line:84% So electron, initial state, and let's 00:27:14.800 --> 00:27:17.150 align:middle line:84% put its final state to be k1, s1. 00:27:17.150 --> 00:27:20.060 align:middle line:90% 00:27:20.060 --> 00:27:26.760 align:middle line:84% And the photon, let's put its initial state to be p2, alpha. 00:27:26.760 --> 00:27:31.880 align:middle line:84% So alpha now is a polarization for the photon. 00:27:31.880 --> 00:27:35.910 align:middle line:90% And so call it alpha 1. 00:27:35.910 --> 00:27:42.830 align:middle line:84% And then this one is called k2, alpha 2. 00:27:42.830 --> 00:27:48.640 align:middle line:84% And then label this index to be nu and this index 00:27:48.640 --> 00:27:53.900 align:middle line:84% to be mu, OK, because the photon carry a vector 00:27:53.900 --> 00:27:56.930 align:middle line:84% index because the polarization carry a vector index. 00:27:56.930 --> 00:27:58.530 align:middle line:90% So there's mu here. 00:27:58.530 --> 00:28:02.420 align:middle line:90% 00:28:02.420 --> 00:28:04.700 align:middle line:84% So I always imagine the momentum is 00:28:04.700 --> 00:28:07.430 align:middle line:84% coming in for the initial state and the momentum come out 00:28:07.430 --> 00:28:10.130 align:middle line:90% for the final states, OK. 00:28:10.130 --> 00:28:12.480 align:middle line:84% And for this diagram, it's the same thing. 00:28:12.480 --> 00:28:13.730 align:middle line:90% So I have p1. 00:28:13.730 --> 00:28:16.430 align:middle line:90% So I label the p1, r1. 00:28:16.430 --> 00:28:20.890 align:middle line:90% So this is the p2 alpha 1. 00:28:20.890 --> 00:28:24.010 align:middle line:90% And so this is k1, s1. 00:28:24.010 --> 00:28:27.700 align:middle line:90% And this is k2, alpha 2. 00:28:27.700 --> 00:28:32.440 align:middle line:84% The only difference is the role of mu and nu is switched. 00:28:32.440 --> 00:28:34.640 align:middle line:90% So for the out leg is mu. 00:28:34.640 --> 00:28:36.780 align:middle line:90% So mu now is here. 00:28:36.780 --> 00:28:39.470 align:middle line:90% And this one is nu here, OK. 00:28:39.470 --> 00:28:43.790 align:middle line:84% So nu is associated polarization of the incoming photon. 00:28:43.790 --> 00:28:46.040 align:middle line:84% And the mu is the polarization associated 00:28:46.040 --> 00:28:48.840 align:middle line:90% with the outgoing photon. 00:28:48.840 --> 00:28:50.350 align:middle line:90% So this is my Feynman diagram. 00:28:50.350 --> 00:28:54.800 align:middle line:90% 00:28:54.800 --> 00:28:57.590 align:middle line:84% So this process compared to the one 00:28:57.590 --> 00:29:02.340 align:middle line:84% we considered, this e plus, e minus 1. 00:29:02.340 --> 00:29:05.170 align:middle line:90% So these have some new elements. 00:29:05.170 --> 00:29:14.825 align:middle line:84% So that's why this is a good example to look at. 00:29:14.825 --> 00:29:17.550 align:middle line:90% 00:29:17.550 --> 00:29:20.070 align:middle line:90% There are two new elements. 00:29:20.070 --> 00:29:23.850 align:middle line:84% First, in this example, we have-- 00:29:23.850 --> 00:29:26.220 align:middle line:84% so this propogator is a fermionic one. 00:29:26.220 --> 00:29:31.140 align:middle line:84% So now we have a intermediate fermionic propagator, OK. 00:29:31.140 --> 00:29:33.100 align:middle line:90% And the momentum for this one-- 00:29:33.100 --> 00:29:42.160 align:middle line:84% so let's call it q1 equal to k1 plus k2. 00:29:42.160 --> 00:29:43.810 align:middle line:90% So this is p1 plus p2. 00:29:43.810 --> 00:29:49.020 align:middle line:90% 00:29:49.020 --> 00:29:51.872 align:middle line:84% But this one will have a different momentum. 00:29:51.872 --> 00:29:56.390 align:middle line:90% So here let's call it q2. 00:29:56.390 --> 00:29:58.060 align:middle line:84% So what's the momentum for this one? 00:29:58.060 --> 00:30:04.110 align:middle line:90% 00:30:04.110 --> 00:30:08.870 align:middle line:84% If I draw the momentum going to the up? 00:30:08.870 --> 00:30:12.320 align:middle line:90% Yeah, so let me put it here. 00:30:12.320 --> 00:30:16.290 align:middle line:84% So what would be the momentum this one? 00:30:16.290 --> 00:30:17.220 align:middle line:90% Let's go with q2. 00:30:17.220 --> 00:30:20.270 align:middle line:90% 00:30:20.270 --> 00:30:23.477 align:middle line:84% Can you read the momentum of this intermediate line 00:30:23.477 --> 00:30:24.185 align:middle line:90% from the diagram? 00:30:24.185 --> 00:30:27.340 align:middle line:90% 00:30:27.340 --> 00:30:32.590 align:middle line:84% Yeah, it's p1 minus k2 because we have p1 coming in and then 00:30:32.590 --> 00:30:34.090 align:middle line:90% k2 come out. 00:30:34.090 --> 00:30:36.460 align:middle line:90% So that's the momentum, OK. 00:30:36.460 --> 00:30:38.290 align:middle line:90% So in this one, you have p1. 00:30:38.290 --> 00:30:40.460 align:middle line:84% And then you have the p2 coming in. 00:30:40.460 --> 00:30:43.500 align:middle line:90% And so this is p1 plus p2. 00:30:43.500 --> 00:30:47.460 align:middle line:84% OK, so the new element is that now we 00:30:47.460 --> 00:30:50.925 align:middle line:84% have a fermionic propagator as an intermediate state. 00:30:50.925 --> 00:30:57.060 align:middle line:90% 00:30:57.060 --> 00:30:59.820 align:middle line:84% And the second new element is now 00:30:59.820 --> 00:31:02.355 align:middle line:84% we have photon in the external state. 00:31:02.355 --> 00:31:10.040 align:middle line:90% 00:31:10.040 --> 00:31:13.213 align:middle line:84% OK, so this is a new element compared to this e plus, 00:31:13.213 --> 00:31:13.880 align:middle line:90% e minus example. 00:31:13.880 --> 00:31:18.650 align:middle line:90% 00:31:18.650 --> 00:31:21.080 align:middle line:84% But still using our previous rule, 00:31:21.080 --> 00:31:25.470 align:middle line:84% we can immediately write down the amplitude, OK. 00:31:25.470 --> 00:31:27.255 align:middle line:84% I think I may not have enough space. 00:31:27.255 --> 00:31:30.630 align:middle line:90% 00:31:30.630 --> 00:31:32.460 align:middle line:90% Yeah, let me try-- 00:31:32.460 --> 00:31:33.810 align:middle line:90% so let me start from here. 00:31:33.810 --> 00:31:36.330 align:middle line:90% 00:31:36.330 --> 00:31:42.210 align:middle line:84% So the amplitude, again, the i times the-- 00:31:42.210 --> 00:31:44.340 align:middle line:90% this i is not important. 00:31:44.340 --> 00:31:50.070 align:middle line:84% But nevertheless, let me just write it down. 00:31:50.070 --> 00:31:54.310 align:middle line:90% So the amplitude is given by-- 00:31:54.310 --> 00:31:59.410 align:middle line:84% so we follow-- you see we follow the fermionic line. 00:31:59.410 --> 00:32:02.790 align:middle line:84% So here there's one fermionic line, OK. 00:32:02.790 --> 00:32:06.390 align:middle line:84% And so we should follow that fermionic line. 00:32:06.390 --> 00:32:10.140 align:middle line:84% And then we also have the photon polarization. 00:32:10.140 --> 00:32:13.770 align:middle line:84% OK, so let's first write down the photon polarization. 00:32:13.770 --> 00:32:16.890 align:middle line:84% So for this photon final state, then we just 00:32:16.890 --> 00:32:30.870 align:middle line:84% have epsilon, mu, alpha 2, star, so k2, OK. 00:32:30.870 --> 00:32:33.420 align:middle line:84% So, remember, for the photon in the final state, 00:32:33.420 --> 00:32:35.880 align:middle line:90% we need to put the star. 00:32:35.880 --> 00:32:38.670 align:middle line:84% And also, remember, the mu is associated with the photon 00:32:38.670 --> 00:32:41.470 align:middle line:90% in the final state. 00:32:41.470 --> 00:32:44.520 align:middle line:84% And then we have epsilon nu for the initial state 00:32:44.520 --> 00:32:48.920 align:middle line:90% of the photon, alpha 1, then k1. 00:32:48.920 --> 00:32:53.370 align:middle line:84% OK, so this is the photon one, OK, photon polarization factor. 00:32:53.370 --> 00:32:56.780 align:middle line:84% And now we need to write down the-- 00:32:56.780 --> 00:33:00.500 align:middle line:84% so these two factors are the same for both processes. 00:33:00.500 --> 00:33:02.810 align:middle line:84% OK, for both processes we are not changing 00:33:02.810 --> 00:33:05.150 align:middle line:90% the external photon state, OK. 00:33:05.150 --> 00:33:07.490 align:middle line:84% So these two factors are the same, OK. 00:33:07.490 --> 00:33:11.150 align:middle line:90% 00:33:11.150 --> 00:33:14.020 align:middle line:90% And now the rest-- 00:33:14.020 --> 00:33:15.820 align:middle line:84% so now let's look at this diagram 00:33:15.820 --> 00:33:17.680 align:middle line:90% and follow this fermionic line. 00:33:17.680 --> 00:33:22.680 align:middle line:84% OK, so we start with here and then going backwards. 00:33:22.680 --> 00:33:30.000 align:middle line:84% So we just have u, bar, s1, k1 for this electron 00:33:30.000 --> 00:33:32.050 align:middle line:90% in the final state. 00:33:32.050 --> 00:33:33.810 align:middle line:90% And then we have fermionic-- 00:33:33.810 --> 00:33:40.515 align:middle line:84% and then we have this vertex should be minus ie, gamma mu. 00:33:40.515 --> 00:33:43.410 align:middle line:84% All this order is important because they are all matrices, 00:33:43.410 --> 00:33:43.920 align:middle line:90% OK. 00:33:43.920 --> 00:33:45.730 align:middle line:90% They all have spinner indices. 00:33:45.730 --> 00:33:47.310 align:middle line:90% And so this is a matrix. 00:33:47.310 --> 00:33:54.990 align:middle line:84% And then we have this fermionic propagator, which is minus 1, 00:33:54.990 --> 00:34:01.400 align:middle line:90% i q1 slash. 00:34:01.400 --> 00:34:03.470 align:middle line:84% So I just write down the fermionic propagator 00:34:03.470 --> 00:34:08.590 align:middle line:90% minus m plus i epsilon, OK. 00:34:08.590 --> 00:34:13.940 align:middle line:84% So I did the final state, intermediate propagator, 00:34:13.940 --> 00:34:19.280 align:middle line:90% and then I have another vertex. 00:34:19.280 --> 00:34:22.440 align:middle line:90% And now it's nu. 00:34:22.440 --> 00:34:23.880 align:middle line:90% And then I have the final. 00:34:23.880 --> 00:34:30.210 align:middle line:90% 00:34:30.210 --> 00:34:35.989 align:middle line:90% OK, so that's that diagram. 00:34:35.989 --> 00:34:37.260 align:middle line:90% OK, that's the diagram. 00:34:37.260 --> 00:34:39.380 align:middle line:84% And now for this diagram, we do the same thing. 00:34:39.380 --> 00:34:42.670 align:middle line:90% 00:34:42.670 --> 00:34:44.590 align:middle line:84% So the final state, again, is the same. 00:34:44.590 --> 00:34:47.270 align:middle line:90% 00:34:47.270 --> 00:34:54.170 align:middle line:90% It's the u, s1, bar, k1. 00:34:54.170 --> 00:34:56.670 align:middle line:84% But now here you have ie gamma nu. 00:34:56.670 --> 00:34:59.360 align:middle line:90% 00:34:59.360 --> 00:35:04.440 align:middle line:90% So now the order changed, OK. 00:35:04.440 --> 00:35:06.400 align:middle line:90% And then you have the photon-- 00:35:06.400 --> 00:35:09.780 align:middle line:84% then you have this fermionic propagator, 00:35:09.780 --> 00:35:14.010 align:middle line:84% which is i q2 slash plus m minus epsilon. 00:35:14.010 --> 00:35:16.640 align:middle line:90% 00:35:16.640 --> 00:35:21.790 align:middle line:84% And then you have minus i e gamma mu from here. 00:35:21.790 --> 00:35:33.780 align:middle line:84% And then you have the final state u r1, p1, OK. 00:35:33.780 --> 00:35:35.258 align:middle line:90% Yes? 00:35:35.258 --> 00:35:37.550 align:middle line:84% AUDIENCE: Where did you get these two Feynman diagrams, 00:35:37.550 --> 00:35:42.345 align:middle line:84% by doing [INAUDIBLE] which one gets absorbed or-- 00:35:42.345 --> 00:35:43.845 align:middle line:84% but if we were working with photons. 00:35:43.845 --> 00:35:45.060 align:middle line:90% It doesn't it matter. 00:35:45.060 --> 00:35:48.750 align:middle line:84% This seems like the real difference is not 00:35:48.750 --> 00:35:52.530 align:middle line:84% the kinematic difference of which one gets absorbed. 00:35:52.530 --> 00:35:55.620 align:middle line:84% It's the fact that we have this vector that 00:35:55.620 --> 00:35:58.440 align:middle line:84% describes with this polarization through the photons, right? 00:35:58.440 --> 00:36:01.260 align:middle line:84% PROFESSOR: No, these are just two inequivalent Feynman 00:36:01.260 --> 00:36:04.090 align:middle line:90% diagrams, inequivalent. 00:36:04.090 --> 00:36:09.802 align:middle line:84% AUDIENCE: But that's because of mu and nu, because they don't-- 00:36:09.802 --> 00:36:12.420 align:middle line:84% when you were writing down the Feynman diagram. 00:36:12.420 --> 00:36:17.250 align:middle line:84% PROFESSOR: No, no, no, even not for-- 00:36:17.250 --> 00:36:21.030 align:middle line:84% even they are not vectors, still they are inequivalent diagrams. 00:36:21.030 --> 00:36:24.550 align:middle line:84% Because the momentum here is different. 00:36:24.550 --> 00:36:26.530 align:middle line:90% The momentum here is different. 00:36:26.530 --> 00:36:30.540 align:middle line:84% So they're still an inequivalent diagram. 00:36:30.540 --> 00:36:32.690 align:middle line:90% OK? 00:36:32.690 --> 00:36:35.420 align:middle line:90% So these are the amplitude. 00:36:35.420 --> 00:36:38.980 align:middle line:84% So again, we look at unpolarized cross section. 00:36:38.980 --> 00:37:07.270 align:middle line:90% 00:37:07.270 --> 00:37:09.205 align:middle line:90% So unpolarized cross section. 00:37:09.205 --> 00:37:13.230 align:middle line:90% 00:37:13.230 --> 00:37:17.850 align:middle line:84% So let's look at the unpolarized situation. 00:37:17.850 --> 00:37:21.080 align:middle line:84% OK, so, again, we need to average over the initial spin 00:37:21.080 --> 00:37:24.050 align:middle line:90% and sum over the final spin. 00:37:24.050 --> 00:37:26.060 align:middle line:84% So here now the difference is now 00:37:26.060 --> 00:37:30.470 align:middle line:84% we need to average also over sum over the index for alpha 00:37:30.470 --> 00:37:33.260 align:middle line:90% and for alpha 1 and alpha 2. 00:37:33.260 --> 00:37:36.590 align:middle line:84% So alpha 1 and alpha 2, they only have two polarization. 00:37:36.590 --> 00:37:42.170 align:middle line:84% So, again, when you average over the alpha 1, 00:37:42.170 --> 00:37:44.090 align:middle line:90% you get a factor of 1/2, OK. 00:37:44.090 --> 00:37:52.080 align:middle line:84% So still we get just a factor of 1 quarter sum over all 00:37:52.080 --> 00:37:55.290 align:middle line:90% the spins of M squared. 00:37:55.290 --> 00:38:03.970 align:middle line:84% So this is just the same as sum over, say, alpha 1, alpha 2, 00:38:03.970 --> 00:38:09.160 align:middle line:90% r1, r2, OK, M squared. 00:38:09.160 --> 00:38:14.260 align:middle line:84% And each index take value 1 and 2, OK, because the photon also 00:38:14.260 --> 00:38:16.027 align:middle line:90% only have two polarizations. 00:38:16.027 --> 00:38:19.370 align:middle line:90% 00:38:19.370 --> 00:38:23.630 align:middle line:84% So now I will not do a complete calculation of this. 00:38:23.630 --> 00:38:26.530 align:middle line:90% OK, so you just square it. 00:38:26.530 --> 00:38:28.350 align:middle line:90% So you use the trick. 00:38:28.350 --> 00:38:30.900 align:middle line:90% And you just square it. 00:38:30.900 --> 00:38:33.540 align:middle line:84% And then you just try to calculate it. 00:38:33.540 --> 00:38:35.670 align:middle line:90% And now you calculate this guy. 00:38:35.670 --> 00:38:39.270 align:middle line:84% Now we need to use some more tricks, OK. 00:38:39.270 --> 00:38:44.400 align:middle line:84% So I will just explain what are the new tricks needed 00:38:44.400 --> 00:38:45.450 align:middle line:90% to compute this guy. 00:38:45.450 --> 00:38:47.615 align:middle line:84% And we will not actually do a calculation, OK. 00:38:47.615 --> 00:38:50.330 align:middle line:90% I'll do a general calculation. 00:38:50.330 --> 00:38:52.190 align:middle line:84% So the new elements we need to use-- 00:38:52.190 --> 00:39:00.160 align:middle line:90% 00:39:00.160 --> 00:39:11.620 align:middle line:84% so a few new tricks in order to do this, to do the calculation. 00:39:11.620 --> 00:39:17.600 align:middle line:90% 00:39:17.600 --> 00:39:19.600 align:middle line:84% So the first trick is how you treat 00:39:19.600 --> 00:39:21.850 align:middle line:90% this fermionic propagator. 00:39:21.850 --> 00:39:25.530 align:middle line:90% 00:39:25.530 --> 00:39:27.870 align:middle line:84% OK, so how you treat the fermionic propagator. 00:39:27.870 --> 00:39:30.650 align:middle line:90% 00:39:30.650 --> 00:39:33.530 align:middle line:90% So we can just rewrite-- 00:39:33.530 --> 00:39:37.770 align:middle line:84% yeah, so this is the inverse of a matrix. 00:39:37.770 --> 00:39:40.670 align:middle line:90% OK, so we can just do the-- 00:39:40.670 --> 00:39:49.700 align:middle line:84% we say maybe has a propagator like this, i k slash, 00:39:49.700 --> 00:39:53.120 align:middle line:90% plus m, plus epsilon. 00:39:53.120 --> 00:39:57.065 align:middle line:84% This will not worry about epsilon because the downstairs 00:39:57.065 --> 00:40:02.130 align:middle line:84% neighbor 0-- so this is the same as i k slash plus m, 00:40:02.130 --> 00:40:06.790 align:middle line:90% k squared plus m squared, OK. 00:40:06.790 --> 00:40:09.550 align:middle line:84% It's just because of the familiar relation 00:40:09.550 --> 00:40:16.360 align:middle line:84% we used before that ik slash plus m and ik slash minus m is 00:40:16.360 --> 00:40:20.510 align:middle line:84% equal to minus k squared minus m squared. 00:40:20.510 --> 00:40:22.090 align:middle line:90% OK, so you can just-- 00:40:22.090 --> 00:40:25.780 align:middle line:84% then the inverse of this matrix is just given by this matrix. 00:40:25.780 --> 00:40:30.990 align:middle line:84% Sorry, I think here should be minus sign. 00:40:30.990 --> 00:40:36.050 align:middle line:84% The inverse of this matrix is given by that matrix, OK. 00:40:36.050 --> 00:40:40.060 align:middle line:84% So then you can use this to simplify. 00:40:40.060 --> 00:40:42.770 align:middle line:84% Then you can use to simplify that expression a little bit, 00:40:42.770 --> 00:40:44.090 align:middle line:90% OK. 00:40:44.090 --> 00:40:47.660 align:middle line:90% So this is the first thing used. 00:40:47.660 --> 00:40:49.490 align:middle line:90% And the second thing used-- 00:40:49.490 --> 00:40:52.900 align:middle line:90% 00:40:52.900 --> 00:40:54.150 align:middle line:90% is when doing the spin sum. 00:40:54.150 --> 00:41:02.060 align:middle line:90% 00:41:02.060 --> 00:41:12.850 align:middle line:84% So for fermions, for spinors we use the same trick 00:41:12.850 --> 00:41:23.270 align:middle line:84% as before as the one we used for the last example. 00:41:23.270 --> 00:41:26.800 align:middle line:84% But for photons, there are some new elements, OK. 00:41:26.800 --> 00:41:28.975 align:middle line:84% So now let's discuss how we treat the photons. 00:41:28.975 --> 00:41:33.640 align:middle line:90% 00:41:33.640 --> 00:41:34.900 align:middle line:90% So now I treat photons. 00:41:34.900 --> 00:41:37.210 align:middle line:84% And the way to treat the photons actually 00:41:37.210 --> 00:41:39.310 align:middle line:90% involves some important physics. 00:41:39.310 --> 00:41:44.270 align:middle line:84% So let me explain a little bit how we do that. 00:41:44.270 --> 00:41:45.940 align:middle line:90% OK. 00:41:45.940 --> 00:41:51.030 align:middle line:84% So now if you look at the structure of this amplitude-- 00:41:51.030 --> 00:41:53.820 align:middle line:84% OK, so let's focus on one of the photons. 00:41:53.820 --> 00:41:59.460 align:middle line:84% So let's look at the final photon, OK. 00:41:59.460 --> 00:42:03.728 align:middle line:84% So the polarization has the following form. 00:42:03.728 --> 00:42:05.520 align:middle line:84% Yeah, this just look at one of the photons. 00:42:05.520 --> 00:42:07.350 align:middle line:90% It doesn't matter. 00:42:07.350 --> 00:42:10.990 align:middle line:84% Say, let's look at this one, initial photon. 00:42:10.990 --> 00:42:15.630 align:middle line:84% And then the amplitude-- then you have this. 00:42:15.630 --> 00:42:18.020 align:middle line:84% Then the whole amplitude is scalar. 00:42:18.020 --> 00:42:21.140 align:middle line:84% OK, so the amplitude then has the structure. 00:42:21.140 --> 00:42:25.090 align:middle line:90% 00:42:25.090 --> 00:42:28.350 align:middle line:90% They have epsilon nu star -- 00:42:28.350 --> 00:42:33.270 align:middle line:90% nu alpha -- times M nu. 00:42:33.270 --> 00:42:35.610 align:middle line:84% OK, just say this is the polarization. 00:42:35.610 --> 00:42:38.790 align:middle line:84% And the rest I just call it M nu because the index has 00:42:38.790 --> 00:42:43.910 align:middle line:90% to be contracted, OK. 00:42:43.910 --> 00:42:45.535 align:middle line:90% So now I do the spin sum. 00:42:45.535 --> 00:42:48.750 align:middle line:90% 00:42:48.750 --> 00:42:58.500 align:middle line:84% So if I now some of nu to the spin sum over the alpha, 00:42:58.500 --> 00:43:02.910 align:middle line:84% and then I just have alpha equal to 1, 2 to the spin 00:43:02.910 --> 00:43:04.440 align:middle line:90% sum relevant for the alpha. 00:43:04.440 --> 00:43:06.000 align:middle line:90% I just have alpha 1, 2. 00:43:06.000 --> 00:43:18.590 align:middle line:84% And then I have epsilon nu alpha star, alpha nu alpha, alpha 00:43:18.590 --> 00:43:26.930 align:middle line:84% mu star, alpha nu, alpha, and then I have M mu star 00:43:26.930 --> 00:43:28.700 align:middle line:90% and have M nu. 00:43:28.700 --> 00:43:30.590 align:middle line:84% OK, I just take the square of it. 00:43:30.590 --> 00:43:36.150 align:middle line:90% 00:43:36.150 --> 00:43:39.210 align:middle line:84% So now to do the spin sum, we looked at this object. 00:43:39.210 --> 00:43:44.050 align:middle line:90% 00:43:44.050 --> 00:43:45.520 align:middle line:84% OK, I need to look at this object. 00:43:45.520 --> 00:43:52.690 align:middle line:90% 00:43:52.690 --> 00:43:56.350 align:middle line:84% So do you remember what is the sum of this object? 00:43:56.350 --> 00:44:02.940 align:middle line:90% 00:44:02.940 --> 00:44:05.280 align:middle line:84% Yeah, exactly, the transverse projector. 00:44:05.280 --> 00:44:11.430 align:middle line:84% OK, because of the physical photon polarization 00:44:11.430 --> 00:44:16.320 align:middle line:84% is projected to the transverse space, OK. 00:44:16.320 --> 00:44:20.020 align:middle line:84% So this gives you the transverse projector. 00:44:20.020 --> 00:44:22.150 align:middle line:90% But now I claim-- 00:44:22.150 --> 00:44:34.480 align:middle line:84% OK, so if you do the alpha 1, 2, mu, alpha star, 00:44:34.480 --> 00:44:37.930 align:middle line:84% nu, alpha-- so this gives you the transverse projector. 00:44:37.930 --> 00:44:41.720 align:middle line:90% 00:44:41.720 --> 00:44:43.310 align:middle line:90% But now the claim-- 00:44:43.310 --> 00:44:46.220 align:middle line:84% but transverse projector is a little bit 00:44:46.220 --> 00:44:47.930 align:middle line:90% awkward to work with, OK. 00:44:47.930 --> 00:45:01.050 align:middle line:84% But the claim, you said actually I 00:45:01.050 --> 00:45:11.390 align:middle line:84% can replace the sum of alpha equal to 1, 2 by actually sum 00:45:11.390 --> 00:45:19.190 align:middle line:84% over alpha for all polarizations, OK. 00:45:19.190 --> 00:45:24.080 align:middle line:84% So now when I sum of all polarizations-- 00:45:24.080 --> 00:45:30.592 align:middle line:84% so if I sum of all alpha, then now do what is this? 00:45:30.592 --> 00:45:31.800 align:middle line:90% Do you remember what is this? 00:45:31.800 --> 00:45:38.660 align:middle line:90% 00:45:38.660 --> 00:45:40.970 align:middle line:90% You just get eta mu nu. 00:45:40.970 --> 00:45:44.000 align:middle line:84% OK, and this one is much simpler. 00:45:44.000 --> 00:45:47.200 align:middle line:90% So now the claim-- 00:45:47.200 --> 00:45:52.510 align:middle line:84% OK, so now the claim is that we can just simply replace 00:45:52.510 --> 00:45:55.560 align:middle line:90% this sum by eta mu nu. 00:45:55.560 --> 00:45:57.704 align:middle line:84% OK, we can replace this by eta mu nu. 00:45:57.704 --> 00:46:03.710 align:middle line:90% 00:46:03.710 --> 00:46:06.200 align:middle line:90% Yeah, so this is the claim. 00:46:06.200 --> 00:46:08.540 align:middle line:84% OK, so this amounts to the following. 00:46:08.540 --> 00:46:13.380 align:middle line:90% 00:46:13.380 --> 00:46:15.140 align:middle line:84% So this is equivalent to the following. 00:46:15.140 --> 00:46:21.640 align:middle line:90% 00:46:21.640 --> 00:46:24.820 align:middle line:84% So if you look at the difference between these two-- 00:46:24.820 --> 00:46:26.320 align:middle line:84% so now if you look at the difference 00:46:26.320 --> 00:46:29.740 align:middle line:84% between the alpha equal to sum over 1, 2 and the alpha 00:46:29.740 --> 00:46:32.250 align:middle line:90% sum over all here-- 00:46:32.250 --> 00:46:34.295 align:middle line:84% so the difference in other words-- 00:46:34.295 --> 00:46:46.160 align:middle line:90% 00:46:46.160 --> 00:47:02.880 align:middle line:84% we claim that sum alpha equal to 0 00:47:02.880 --> 00:47:16.470 align:middle line:84% and 3 epsilon alpha star mu and alpha nu 00:47:16.470 --> 00:47:21.900 align:middle line:90% alpha M mu M nu to be 0, OK. 00:47:21.900 --> 00:47:24.550 align:middle line:90% 00:47:24.550 --> 00:47:28.890 align:middle line:84% So the difference between the 2 is just from sum 0 to 3, OK. 00:47:28.890 --> 00:47:30.610 align:middle line:90% So in other words-- 00:47:30.610 --> 00:47:33.910 align:middle line:84% so if I write down this explicitly-- 00:47:33.910 --> 00:47:35.960 align:middle line:90% OK, so this corresponding to-- 00:47:35.960 --> 00:47:39.650 align:middle line:90% 00:47:39.650 --> 00:47:47.990 align:middle line:84% so this corresponding to 0, epsilon mu, 0, M mu squared 00:47:47.990 --> 00:47:57.240 align:middle line:84% is equal to epsilon mu 3 M mu squared, OK. 00:47:57.240 --> 00:48:00.270 align:middle line:90% 00:48:00.270 --> 00:48:04.420 align:middle line:84% So this claim, you see equivalent to this claim. 00:48:04.420 --> 00:48:09.240 align:middle line:84% OK, so these two will cancel each other. 00:48:09.240 --> 00:48:12.300 align:middle line:84% The equal here is because the signature, 00:48:12.300 --> 00:48:14.535 align:middle line:84% the zeroth component, there's a minus sign, OK. 00:48:14.535 --> 00:48:17.820 align:middle line:90% 00:48:17.820 --> 00:48:19.710 align:middle line:84% So it tells you that the sum of these two 00:48:19.710 --> 00:48:23.640 align:middle line:84% actually will cancel each other, OK. 00:48:23.640 --> 00:48:27.980 align:middle line:84% So now let me remind you how we choose the-- 00:48:27.980 --> 00:48:32.180 align:middle line:84% yeah, now let me try to prove this fact. 00:48:32.180 --> 00:48:37.380 align:middle line:84% OK, turns out this fact is actually very important. 00:48:37.380 --> 00:48:40.100 align:middle line:90% 00:48:40.100 --> 00:48:43.490 align:middle line:84% It contains very important physics here, OK. 00:48:43.490 --> 00:48:45.740 align:middle line:84% So do you have any questions before I do that? 00:48:45.740 --> 00:48:51.830 align:middle line:90% 00:48:51.830 --> 00:48:54.310 align:middle line:90% OK, good. 00:48:54.310 --> 00:48:58.480 align:middle line:84% So now let me try to show this is true, OK. 00:48:58.480 --> 00:49:01.420 align:middle line:84% So let me just remind you of a convention 00:49:01.420 --> 00:49:10.630 align:middle line:84% is that epsilon mu 0 is equal to just 0, just 1, 0 and mu 00:49:10.630 --> 00:49:18.250 align:middle line:84% 3 is equal to 0 and then in the direction of the k. 00:49:18.250 --> 00:49:21.040 align:middle line:84% OK, it's in the direction of the k. 00:49:21.040 --> 00:49:25.870 align:middle line:84% OK, and the epsilon mu 1, 2 will be orthogonal to both of them, 00:49:25.870 --> 00:49:28.230 align:middle line:90% OK. 00:49:28.230 --> 00:49:35.800 align:middle line:84% Be orthogonal to both of them, OK. 00:49:35.800 --> 00:49:37.810 align:middle line:84% So now to see this equation, let me 00:49:37.810 --> 00:49:39.220 align:middle line:90% call this equation star, star. 00:49:39.220 --> 00:49:49.250 align:middle line:90% 00:49:49.250 --> 00:49:52.850 align:middle line:84% So let's consider just a general physical process. 00:49:52.850 --> 00:49:55.570 align:middle line:90% 00:49:55.570 --> 00:49:58.750 align:middle line:84% We can actually make a general statement, 00:49:58.750 --> 00:50:02.240 align:middle line:84% OK, not just restricted to that particular example 00:50:02.240 --> 00:50:03.290 align:middle line:90% we have here. 00:50:03.290 --> 00:50:07.640 align:middle line:84% So let's just consider some general process. 00:50:07.640 --> 00:50:13.000 align:middle line:84% So you have a bunch of initial states and some final states, 00:50:13.000 --> 00:50:14.330 align:middle line:90% OK. 00:50:14.330 --> 00:50:18.040 align:middle line:84% But imagine one of the initial state is a photon. 00:50:18.040 --> 00:50:20.120 align:middle line:84% It's the one we are interested here, 00:50:20.120 --> 00:50:25.130 align:middle line:84% this polarization k alpha, polarization k alpha. 00:50:25.130 --> 00:50:27.650 align:middle line:90% 00:50:27.650 --> 00:50:39.240 align:middle line:84% And so now the amplitude just become mu 00:50:39.240 --> 00:50:44.000 align:middle line:84% and then this is the same as the-- you 00:50:44.000 --> 00:50:47.650 align:middle line:90% have some final state. 00:50:47.650 --> 00:50:50.100 align:middle line:84% And then you have some initial state. 00:50:50.100 --> 00:50:56.240 align:middle line:84% But within the initial state, there's a k alpha state, OK. 00:50:56.240 --> 00:50:58.570 align:middle line:84% Then the k alpha state corresponding to the photon. 00:50:58.570 --> 00:51:03.800 align:middle line:90% 00:51:03.800 --> 00:51:08.770 align:middle line:84% But now, remember, which we discuss in your homework, 00:51:08.770 --> 00:51:12.960 align:middle line:84% OK, so that's the purpose to put it in your homework. 00:51:12.960 --> 00:51:19.320 align:middle line:84% The final state for k alpha, which 00:51:19.320 --> 00:51:22.510 align:middle line:84% defined by transverse polarization, 00:51:22.510 --> 00:51:35.730 align:middle line:84% it's only a representative in the equivalence class 00:51:35.730 --> 00:51:36.230 align:middle line:90% of states. 00:51:36.230 --> 00:51:50.120 align:middle line:90% 00:51:50.120 --> 00:51:53.030 align:middle line:84% And within this equivalence class, 00:51:53.030 --> 00:51:56.765 align:middle line:84% they are related by these called null states, OK. 00:51:56.765 --> 00:52:04.790 align:middle line:90% 00:52:04.790 --> 00:52:11.610 align:middle line:84% So we can shift it by our null state 00:52:11.610 --> 00:52:12.985 align:middle line:84% and the physics will be the same. 00:52:12.985 --> 00:52:17.870 align:middle line:90% 00:52:17.870 --> 00:52:20.230 align:middle line:84% So, now, remember the last state-- 00:52:20.230 --> 00:52:22.980 align:middle line:84% it's like a gauge transformation. 00:52:22.980 --> 00:52:25.130 align:middle line:84% So the last state corresponding to-- 00:52:25.130 --> 00:52:29.150 align:middle line:84% and this process corresponding to you change your initial-- 00:52:29.150 --> 00:52:35.470 align:middle line:84% you change your polarization to that corresponding 00:52:35.470 --> 00:52:37.510 align:middle line:90% to a null state. 00:52:37.510 --> 00:52:42.520 align:middle line:84% And null state, the feature is that it's polarization is 00:52:42.520 --> 00:52:45.030 align:middle line:90% proportional to the momentum. 00:52:45.030 --> 00:52:47.580 align:middle line:90% OK, so that's how we are-- 00:52:47.580 --> 00:52:49.140 align:middle line:90% remember, we showed that-- 00:52:49.140 --> 00:52:51.800 align:middle line:84% because this is a gauge transformation. 00:52:51.800 --> 00:52:55.660 align:middle line:84% OK, this is like a gauge transformation. 00:52:55.660 --> 00:53:01.880 align:middle line:84% And we also discussed the reason this is a equivalence class 00:53:01.880 --> 00:53:04.640 align:middle line:84% because when you shift by a null state, 00:53:04.640 --> 00:53:09.650 align:middle line:84% the overlap between the null state to any state is 0, OK, 00:53:09.650 --> 00:53:10.910 align:middle line:90% to any state is 0. 00:53:10.910 --> 00:53:18.320 align:middle line:84% And so you can shift by a null state. 00:53:18.320 --> 00:53:20.200 align:middle line:84% So the fact that you can shift this 00:53:20.200 --> 00:53:23.330 align:middle line:84% by a null state, your physics is the same. 00:53:23.330 --> 00:53:25.315 align:middle line:84% So now you see this equation 2 here. 00:53:25.315 --> 00:54:02.270 align:middle line:90% 00:54:02.270 --> 00:54:11.090 align:middle line:84% So this implies that M mu k mu must be zero. 00:54:11.090 --> 00:54:15.610 align:middle line:90% 00:54:15.610 --> 00:54:20.530 align:middle line:90% k mu must be 0, OK. 00:54:20.530 --> 00:54:22.510 align:middle line:84% And this is a very important identity. 00:54:22.510 --> 00:54:31.330 align:middle line:84% This is called the Ward identity because this 00:54:31.330 --> 00:54:37.090 align:middle line:84% is a very important feature, which can simplify 00:54:37.090 --> 00:54:38.580 align:middle line:90% your calculation a lot, OK. 00:54:38.580 --> 00:54:42.180 align:middle line:90% 00:54:42.180 --> 00:54:45.300 align:middle line:84% So the M mu must satisfy the feature 00:54:45.300 --> 00:54:48.090 align:middle line:84% that when it contract with k mu you get 0. 00:54:48.090 --> 00:54:50.750 align:middle line:90% 00:54:50.750 --> 00:54:59.730 align:middle line:84% So now remember the k mu, you just equal to-- 00:54:59.730 --> 00:55:05.955 align:middle line:84% so this is on-shell external state, or minus. 00:55:05.955 --> 00:55:09.160 align:middle line:90% 00:55:09.160 --> 00:55:13.120 align:middle line:84% OK, this is the on-shell external state. 00:55:13.120 --> 00:55:15.660 align:middle line:90% And then this is just given by-- 00:55:15.660 --> 00:55:19.650 align:middle line:84% if I take a factor k out and this is 00:55:19.650 --> 00:55:21.870 align:middle line:90% given by minus 1 and then k-- 00:55:21.870 --> 00:55:25.110 align:middle line:90% 00:55:25.110 --> 00:55:28.410 align:middle line:84% and this is a fact I think you also used in your pset-- 00:55:28.410 --> 00:55:29.580 align:middle line:90% and this is the-- 00:55:29.580 --> 00:55:36.330 align:middle line:84% and this thing is just the same as the difference 00:55:36.330 --> 00:55:44.060 align:middle line:84% between the zeroth polarization vector and the third one 00:55:44.060 --> 00:55:46.340 align:middle line:90% from what we wrote here. 00:55:46.340 --> 00:55:50.400 align:middle line:90% I think I get the sign. 00:55:50.400 --> 00:55:52.950 align:middle line:84% Yeah, the sign does not matter very much here. 00:55:52.950 --> 00:55:54.575 align:middle line:90% I mean, just make sure. 00:55:54.575 --> 00:55:59.140 align:middle line:90% 00:55:59.140 --> 00:56:02.330 align:middle line:84% Yeah, for my sign this would be-- 00:56:02.330 --> 00:56:03.523 align:middle line:90% yeah, it's minus epsilon. 00:56:03.523 --> 00:56:04.315 align:middle line:90% Yeah, that's right. 00:56:04.315 --> 00:56:07.440 align:middle line:90% 00:56:07.440 --> 00:56:16.370 align:middle line:84% And then this equation just implies that epsilon mu 0 M mu 00:56:16.370 --> 00:56:21.020 align:middle line:90% is equal to epsilon mu 3 M mu. 00:56:21.020 --> 00:56:25.940 align:middle line:84% So that tells you that epsilon mu 0 M mu 00:56:25.940 --> 00:56:32.810 align:middle line:84% squared is equal to epsilon mu 3 M mu squared. 00:56:32.810 --> 00:56:34.835 align:middle line:84% OK, so this is the one we wanted to prove. 00:56:34.835 --> 00:56:37.437 align:middle line:90% 00:56:37.437 --> 00:56:39.020 align:middle line:84% So this is the one we wanted to prove. 00:56:39.020 --> 00:56:42.650 align:middle line:90% 00:56:42.650 --> 00:56:44.680 align:middle line:90% So this simplifies life a lot. 00:56:44.680 --> 00:56:47.980 align:middle line:90% 00:56:47.980 --> 00:56:52.520 align:middle line:84% So now we show that we can actually make this replacement 00:56:52.520 --> 00:56:54.890 align:middle line:84% because the 0 and the 3 component 00:56:54.890 --> 00:56:57.520 align:middle line:84% add together become actually a null vector. 00:56:57.520 --> 00:57:03.000 align:middle line:84% And that actually does not contribute. 00:57:03.000 --> 00:57:08.300 align:middle line:84% So now this amplitude we have two external photons. 00:57:08.300 --> 00:57:10.310 align:middle line:90% So we can write it as-- 00:57:10.310 --> 00:57:12.680 align:middle line:90% OK, so for simplicity-- 00:57:12.680 --> 00:57:22.160 align:middle line:84% so for our amplitude, so back to Compton case, 00:57:22.160 --> 00:57:24.360 align:middle line:90% to the Compton story. 00:57:24.360 --> 00:57:28.180 align:middle line:84% And then the amplitude then have the form-- 00:57:28.180 --> 00:57:30.430 align:middle line:90% so we have 2. 00:57:30.430 --> 00:57:36.270 align:middle line:84% So I have epsilon mu, alpha 2, star, and epsilon mu alpha 1. 00:57:36.270 --> 00:57:38.700 align:middle line:84% And then this will multiply some T mu nu. 00:57:38.700 --> 00:57:42.110 align:middle line:90% OK, so I call the rest T mu nu. 00:57:42.110 --> 00:57:42.610 align:middle line:90% OK. 00:57:42.610 --> 00:57:47.080 align:middle line:90% 00:57:47.080 --> 00:57:59.870 align:middle line:84% So now when I do the spin sum, M squared, 00:57:59.870 --> 00:58:06.640 align:middle line:84% then I can just use this twice for each epsilon. 00:58:06.640 --> 00:58:07.750 align:middle line:90% And then I just get-- 00:58:07.750 --> 00:58:11.620 align:middle line:90% 00:58:11.620 --> 00:58:15.070 align:middle line:90% essentially I just get-- 00:58:15.070 --> 00:58:17.830 align:middle line:90% let me just write one more step. 00:58:17.830 --> 00:58:25.080 align:middle line:84% I will get alpha 1, alpha 2 equal to 1 and 2. 00:58:25.080 --> 00:58:33.300 align:middle line:84% Then I have epsilon mu alpha 2 star, epsilon lambda alpha 2. 00:58:33.300 --> 00:58:38.060 align:middle line:90% 00:58:38.060 --> 00:58:45.280 align:middle line:84% So epsilon nu, alpha 1, epsilon rho, alpha one star. 00:58:45.280 --> 00:58:51.750 align:middle line:84% And then I have T mu nu, T lambda rho star. 00:58:51.750 --> 00:58:54.580 align:middle line:90% OK, just the square of this guy. 00:58:54.580 --> 00:58:57.760 align:middle line:84% And now I can replace this by eta mu lambda, 00:58:57.760 --> 00:59:00.100 align:middle line:90% I can replace by eta mu rho. 00:59:00.100 --> 00:59:06.910 align:middle line:84% OK, and then this just become equal to eta mu, lambda, eta 00:59:06.910 --> 00:59:14.411 align:middle line:84% mu, rho, T mu, nu, T lambda, rho, OK. 00:59:14.411 --> 00:59:27.120 align:middle line:90% 00:59:27.120 --> 00:59:32.420 align:middle line:84% So this just makes life much easier, OK. 00:59:32.420 --> 00:59:36.440 align:middle line:90% Makes life much easier. 00:59:36.440 --> 00:59:40.370 align:middle line:84% Yeah, so essentially these are the two important tricks 00:59:40.370 --> 00:59:44.570 align:middle line:84% which one need to do in the photon in this Compton case 00:59:44.570 --> 00:59:46.620 align:middle line:90% compared with this story. 00:59:46.620 --> 00:59:48.620 align:middle line:90% And once you have these-- 00:59:48.620 --> 00:59:55.070 align:middle line:84% and this part will be just the spin sum for the fermions. 00:59:55.070 --> 00:59:57.020 align:middle line:84% And then that we can just use the same trick 00:59:57.020 --> 00:59:58.190 align:middle line:90% as we did last time, OK. 00:59:58.190 --> 01:00:00.200 align:middle line:90% So we will not repeat that, OK. 01:00:00.200 --> 01:00:03.320 align:middle line:84% So now let me just write down your final answer, OK. 01:00:03.320 --> 01:00:05.760 align:middle line:84% And now we can just write down the final answer. 01:00:05.760 --> 01:00:08.540 align:middle line:84% So before writing down the final answer, 01:00:08.540 --> 01:00:11.460 align:middle line:90% let me just define the frame. 01:00:11.460 --> 01:00:13.250 align:middle line:84% So for the Compton scattering, it's 01:00:13.250 --> 01:00:18.180 align:middle line:84% often convenient to consider the reat frame of the electron. 01:00:18.180 --> 01:00:20.360 align:middle line:84% So we consider photon because there's 01:00:20.360 --> 01:00:22.220 align:middle line:90% no rest frame for photon, OK. 01:00:22.220 --> 01:00:23.810 align:middle line:84% So it's often convenient to consider 01:00:23.810 --> 01:00:26.560 align:middle line:90% the rest frame of the electron. 01:00:26.560 --> 01:00:28.485 align:middle line:84% So let's just consider the picture. 01:00:28.485 --> 01:00:54.310 align:middle line:90% 01:00:54.310 --> 01:00:56.800 align:middle line:84% When you consider the rest frame of the electron, -- 01:00:56.800 --> 01:01:01.350 align:middle line:90% 01:01:01.350 --> 01:01:11.050 align:middle line:84% so this is often the way roughly often also experiment is done. 01:01:11.050 --> 01:01:13.420 align:middle line:84% OK, so you can essentially consider the, say, 01:01:13.420 --> 01:01:17.540 align:middle line:84% electron in the matter, which they don't have much velocity. 01:01:17.540 --> 01:01:20.260 align:middle line:84% And then you can have a photon come in. 01:01:20.260 --> 01:01:24.260 align:middle line:84% So imagine you have an electron here 01:01:24.260 --> 01:01:25.690 align:middle line:90% and you have a photon come in. 01:01:25.690 --> 01:01:28.990 align:middle line:90% 01:01:28.990 --> 01:01:31.100 align:middle line:90% OK, you have a photon come in. 01:01:31.100 --> 01:01:35.950 align:middle line:84% So let's call this, the incoming axis, to be the z-axis. 01:01:35.950 --> 01:01:40.650 align:middle line:84% OK, and then after scattering the photon, 01:01:40.650 --> 01:01:44.650 align:middle line:84% say will be scattered into this direction. 01:01:44.650 --> 01:01:46.290 align:middle line:84% OK, so let's call this direction theta. 01:01:46.290 --> 01:01:49.750 align:middle line:90% 01:01:49.750 --> 01:01:50.250 align:middle line:90% OK. 01:01:50.250 --> 01:01:53.270 align:middle line:90% 01:01:53.270 --> 01:01:56.270 align:middle line:90% So in this setup-- 01:01:56.270 --> 01:02:00.950 align:middle line:84% so then the momentum of the electron 01:02:00.950 --> 01:02:06.200 align:middle line:84% is just the m, 0, OK, just the mass of the momentum. 01:02:06.200 --> 01:02:09.560 align:middle line:90% Or I labeled the momentum. 01:02:09.560 --> 01:02:14.060 align:middle line:84% And the momentum of the photon, incoming photon, p2, 01:02:14.060 --> 01:02:18.110 align:middle line:90% would be just omega 0, 0, omega. 01:02:18.110 --> 01:02:20.660 align:middle line:84% OK, so only have the momentum in the z direction. 01:02:20.660 --> 01:02:24.390 align:middle line:90% 01:02:24.390 --> 01:02:28.860 align:middle line:84% Then the final momentum of the electron is k mu, k1. 01:02:28.860 --> 01:02:30.660 align:middle line:90% So this can be something-- 01:02:30.660 --> 01:02:32.760 align:middle line:90% OK, so let's not worry about it. 01:02:32.760 --> 01:02:36.540 align:middle line:90% Then for the photon-- 01:02:36.540 --> 01:02:39.430 align:middle line:84% so photon momentum, final photon momentum-- 01:02:39.430 --> 01:02:43.300 align:middle line:84% so let's call it-- yeah, so this is k2. 01:02:43.300 --> 01:02:49.230 align:middle line:84% OK, so k2 we can parameterize it by omega prime. 01:02:49.230 --> 01:02:57.660 align:middle line:84% And then omega prime then going into the spatial direction, 01:02:57.660 --> 01:03:00.030 align:middle line:84% the unit vector in the spatial direction for k2-- 01:03:00.030 --> 01:03:01.050 align:middle line:90% let's call it n. 01:03:01.050 --> 01:03:04.150 align:middle line:84% OK, so this is the direction of n, 01:03:04.150 --> 01:03:07.920 align:middle line:84% which is the theta angle of respect to the z direction, OK. 01:03:07.920 --> 01:03:10.910 align:middle line:90% 01:03:10.910 --> 01:03:12.920 align:middle line:84% And then the k1 just can be obtained 01:03:12.920 --> 01:03:16.730 align:middle line:84% by momentum conservation from these quantities, OK. 01:03:16.730 --> 01:03:19.720 align:middle line:90% 01:03:19.720 --> 01:03:26.440 align:middle line:84% So k1 would be just p1 plus p2 minus k2. 01:03:26.440 --> 01:03:29.850 align:middle line:90% 01:03:29.850 --> 01:03:32.880 align:middle line:84% So from the fact that the electron 01:03:32.880 --> 01:03:38.295 align:middle line:84% have a static mass minus m squared, OK. 01:03:38.295 --> 01:03:42.230 align:middle line:90% 01:03:42.230 --> 01:03:44.610 align:middle line:90% And then this relation-- 01:03:44.610 --> 01:03:49.670 align:middle line:84% then the p1 plus p2 minus k2 should satisfy the constraint 01:03:49.670 --> 01:03:55.818 align:middle line:84% that this should be equal to minus m squared, OK. 01:03:55.818 --> 01:03:58.730 align:middle line:84% It should satisfy the constraint -- 01:03:58.730 --> 01:04:03.590 align:middle line:84% from this equation, you can actually solve omega prime 01:04:03.590 --> 01:04:05.210 align:middle line:90% in terms of omega. 01:04:05.210 --> 01:04:08.060 align:middle line:84% OK, so we will not write down this equation explicitly. 01:04:08.060 --> 01:04:11.420 align:middle line:90% You can easily check yourself. 01:04:11.420 --> 01:04:13.160 align:middle line:90% You can easily check yourself. 01:04:13.160 --> 01:04:16.670 align:middle line:84% So you find the omega prime is equal to omega 01:04:16.670 --> 01:04:23.370 align:middle line:84% divided by 1 plus omega m and the 1 minus cosine theta. 01:04:23.370 --> 01:04:26.640 align:middle line:90% 01:04:26.640 --> 01:04:34.940 align:middle line:84% OK, so this is the final frequency of the photon, 01:04:34.940 --> 01:04:39.270 align:middle line:84% expressed in terms of initial frequency of the photon 01:04:39.270 --> 01:04:42.660 align:middle line:84% and the mass of the electron and then 01:04:42.660 --> 01:04:44.160 align:middle line:90% this scattering angle, theta. 01:04:44.160 --> 01:04:48.640 align:middle line:90% 01:04:48.640 --> 01:04:52.220 align:middle line:84% So now let me just write down the final answer, 01:04:52.220 --> 01:04:53.720 align:middle line:90% with this set up. 01:04:53.720 --> 01:04:57.940 align:middle line:84% And then you find in the rest frame of the electron, 01:04:57.940 --> 01:05:01.780 align:middle line:84% the differential cross section can be written. 01:05:01.780 --> 01:05:06.970 align:middle line:84% So again the phi direction would be symmetric, OK. 01:05:06.970 --> 01:05:11.600 align:middle line:84% And so this only worry about the theta direction. 01:05:11.600 --> 01:05:16.310 align:middle line:84% So this can be written as pi alpha square, so 01:05:16.310 --> 01:05:19.190 align:middle line:84% alpha against the fine structure constant divided 01:05:19.190 --> 01:05:25.930 align:middle line:84% by m squared and then given by omega prime, the final momentum 01:05:25.930 --> 01:05:29.650 align:middle line:84% frequency for the photon divided by omega 01:05:29.650 --> 01:05:35.260 align:middle line:84% squared and omega prime divided by omega and omega, omega 01:05:35.260 --> 01:05:39.950 align:middle line:90% prime minus sine squared. 01:05:39.950 --> 01:05:41.375 align:middle line:90% OK, so this is the final answer. 01:05:41.375 --> 01:05:45.500 align:middle line:90% 01:05:45.500 --> 01:05:48.213 align:middle line:84% OK, so this is the final answer for the Compton scattering 01:05:48.213 --> 01:05:49.130 align:middle line:90% and the cross section. 01:05:49.130 --> 01:05:51.710 align:middle line:90% 01:05:51.710 --> 01:05:56.100 align:middle line:84% OK, so this formula actually contains a lot of physics, 01:05:56.100 --> 01:05:58.090 align:middle line:90% contains a lot of physics. 01:05:58.090 --> 01:06:00.990 align:middle line:84% So let me just describe some of the physics here. 01:06:00.990 --> 01:06:11.090 align:middle line:90% 01:06:11.090 --> 01:06:12.970 align:middle line:84% So let's first consider the regime 01:06:12.970 --> 01:06:15.700 align:middle line:84% that the photons have very low energy. 01:06:15.700 --> 01:06:19.450 align:middle line:84% OK, just low energy photons, shining some light, 01:06:19.450 --> 01:06:20.590 align:middle line:90% shining on the-- 01:06:20.590 --> 01:06:23.410 align:middle line:90% 01:06:23.410 --> 01:06:27.100 align:middle line:84% so suppose the photon have very low frequency. 01:06:27.100 --> 01:06:31.030 align:middle line:84% They're much smaller than the electron mass. 01:06:31.030 --> 01:06:34.100 align:middle line:84% So now if you look at this formula-- 01:06:34.100 --> 01:06:36.320 align:middle line:84% so if the initial photon is much, 01:06:36.320 --> 01:06:39.370 align:middle line:84% much smaller than the mass and then this factor 01:06:39.370 --> 01:06:44.080 align:middle line:84% is approximately 0, then you find in this regime 01:06:44.080 --> 01:06:47.170 align:middle line:84% that omega prime is actually approximately 01:06:47.170 --> 01:06:51.520 align:middle line:90% equal to the omega. 01:06:51.520 --> 01:06:55.030 align:middle line:84% OK, so you find in this regime the frequency actually 01:06:55.030 --> 01:07:02.430 align:middle line:84% does not change when the photon scatter away from the electron, 01:07:02.430 --> 01:07:05.040 align:middle line:90% OK. 01:07:05.040 --> 01:07:07.560 align:middle line:84% And then now omega prime equal to omega. 01:07:07.560 --> 01:07:08.970 align:middle line:90% So this factor becomes 1. 01:07:08.970 --> 01:07:11.550 align:middle line:84% This factor become 1 and 1 minus sine theta 01:07:11.550 --> 01:07:13.020 align:middle line:90% equal to cosine square theta. 01:07:13.020 --> 01:07:14.700 align:middle line:90% And this becomes one. 01:07:14.700 --> 01:07:17.580 align:middle line:84% And there you find then d sigma, d 01:07:17.580 --> 01:07:23.150 align:middle line:84% cosine theta just equal to pi alpha 01:07:23.150 --> 01:07:30.390 align:middle line:84% squared divided by m squared then plus 1 cosine squared 01:07:30.390 --> 01:07:32.520 align:middle line:90% theta. 01:07:32.520 --> 01:07:47.330 align:middle line:84% OK, so this is a famous formula, more than 100 years ago 01:07:47.330 --> 01:07:50.060 align:middle line:90% or even more. 01:07:50.060 --> 01:07:54.110 align:middle line:90% Because this is the classical-- 01:07:54.110 --> 01:07:56.480 align:middle line:84% so this is called the Thompson cross-section. 01:07:56.480 --> 01:08:06.640 align:middle line:90% 01:08:06.640 --> 01:08:08.230 align:middle line:84% OK, so these are Thomson cross-section 01:08:08.230 --> 01:08:13.390 align:middle line:84% derived from the classical electrodynamics. 01:08:13.390 --> 01:08:16.779 align:middle line:84% OK, and so this is a result already known 01:08:16.779 --> 01:08:22.580 align:middle line:84% in the 19th century from the classical electrodynamics. 01:08:22.580 --> 01:08:25.580 align:middle line:84% So how people derived this formula 01:08:25.580 --> 01:08:29.550 align:middle line:84% in the classical electrodynamics-- 01:08:29.550 --> 01:08:33.134 align:middle line:84% so in classical electrodynamics, you do it this way. 01:08:33.134 --> 01:09:01.020 align:middle line:90% 01:09:01.020 --> 01:09:03.020 align:middle line:84% So in classical electrodynamics-- 01:09:03.020 --> 01:09:08.029 align:middle line:90% 01:09:08.029 --> 01:09:13.149 align:middle line:84% so light is believed to be a wave, electromagnetic wave, OK. 01:09:13.149 --> 01:09:15.500 align:middle line:84% So essentially, it's an electromagnetic wave. 01:09:15.500 --> 01:09:18.984 align:middle line:84% So this electron, or this electric field, 01:09:18.984 --> 01:09:23.660 align:middle line:84% so with a polarization, and then the magnitude. 01:09:23.660 --> 01:09:25.729 align:middle line:90% And then it's a wave, OK. 01:09:25.729 --> 01:09:26.510 align:middle line:90% It's a plane wave. 01:09:26.510 --> 01:09:31.529 align:middle line:90% 01:09:31.529 --> 01:09:32.700 align:middle line:90% OK. 01:09:32.700 --> 01:09:35.250 align:middle line:84% So from the classical electrodynamics, 01:09:35.250 --> 01:09:37.170 align:middle line:90% the light is just a wave, OK. 01:09:37.170 --> 01:09:39.540 align:middle line:84% When you shine the light on the material, 01:09:39.540 --> 01:09:42.069 align:middle line:90% the light will scatter. 01:09:42.069 --> 01:09:44.890 align:middle line:84% Under the physical description of the light 01:09:44.890 --> 01:09:46.420 align:middle line:90% scatter is the following. 01:09:46.420 --> 01:09:54.410 align:middle line:90% 01:09:54.410 --> 01:09:57.780 align:middle line:84% So this is an oscillating electric field. 01:09:57.780 --> 01:10:02.940 align:middle line:84% So if you have a charged particle like electron, 01:10:02.940 --> 01:10:05.840 align:middle line:84% then it will oscillate because it 01:10:05.840 --> 01:10:22.990 align:middle line:84% has a charge under such an E. So you 01:10:22.990 --> 01:10:25.030 align:middle line:84% can approximate the electron in the matter 01:10:25.030 --> 01:10:27.580 align:middle line:84% just by a forced harmonic oscillator. 01:10:27.580 --> 01:10:31.070 align:middle line:90% OK, it's the forced oscillation. 01:10:31.070 --> 01:10:32.711 align:middle line:90% It's a forced oscillation. 01:10:32.711 --> 01:10:35.870 align:middle line:90% 01:10:35.870 --> 01:10:41.510 align:middle line:84% So you should have learned in 8.03 01:10:41.510 --> 01:10:47.150 align:middle line:84% that under a forced oscillation, essentially 01:10:47.150 --> 01:10:52.190 align:middle line:84% the electron velocity electron will have acceleration 01:10:52.190 --> 01:10:55.655 align:middle line:84% essentially given by the given by this. 01:10:55.655 --> 01:10:56.750 align:middle line:90% Yeah, let me just-- 01:10:56.750 --> 01:11:02.290 align:middle line:84% proportional through the exponential i omega T plus i 01:11:02.290 --> 01:11:04.980 align:middle line:90% kx. 01:11:04.980 --> 01:11:09.770 align:middle line:84% OK, essentially just the force acceleration were like this. 01:11:09.770 --> 01:11:13.040 align:middle line:84% And then from classical electrodynamics because this 01:11:13.040 --> 01:11:14.390 align:middle line:90% is accelerated motion. 01:11:14.390 --> 01:11:16.370 align:middle line:90% This is accelerated charge. 01:11:16.370 --> 01:11:20.540 align:middle line:84% And accelerated oscillators in the classical electrodynamics 01:11:20.540 --> 01:11:22.910 align:middle line:90% will emit waves. 01:11:22.910 --> 01:11:26.400 align:middle line:84% OK, we emit electromagnetic wave. 01:11:26.400 --> 01:11:28.170 align:middle line:90% So this will radiate. 01:11:28.170 --> 01:11:33.575 align:middle line:84% So this will lead to radiation with frequency omega. 01:11:33.575 --> 01:11:39.320 align:middle line:90% 01:11:39.320 --> 01:11:43.820 align:middle line:84% So essentially controlled by this acceleration, OK. 01:11:43.820 --> 01:11:45.440 align:middle line:84% So this is the classical result in 01:11:45.440 --> 01:11:48.260 align:middle line:84% the classical electrodynamics that such a oscillator 01:11:48.260 --> 01:11:51.810 align:middle line:84% will emit the light with frequency omega. 01:11:51.810 --> 01:11:54.950 align:middle line:84% So that's the scattering process from the classical point 01:11:54.950 --> 01:11:56.630 align:middle line:90% of view. 01:11:56.630 --> 01:11:57.530 align:middle line:90% OK. 01:11:57.530 --> 01:11:59.750 align:middle line:84% From a classical point of view, drive the oscillator, 01:11:59.750 --> 01:12:01.880 align:middle line:90% the oscillator will emit, OK. 01:12:01.880 --> 01:12:07.000 align:middle line:84% And if you go from this process, then you precisely 01:12:07.000 --> 01:12:09.130 align:middle line:90% find this formula, OK. 01:12:09.130 --> 01:12:11.650 align:middle line:84% You precisely find this formula, OK. 01:12:11.650 --> 01:12:14.220 align:middle line:90% 01:12:14.220 --> 01:12:15.540 align:middle line:90% So in this case-- 01:12:15.540 --> 01:12:21.980 align:middle line:90% 01:12:21.980 --> 01:12:39.400 align:middle line:84% so regarding this, the value of omega, 01:12:39.400 --> 01:12:43.450 align:middle line:84% the omega prime, the emitted photon always-- 01:12:43.450 --> 01:12:45.370 align:middle line:84% so in this story, omega prime always 01:12:45.370 --> 01:12:49.450 align:middle line:84% equal to omega, OK, because it's a driven oscillation, OK. 01:12:49.450 --> 01:12:57.470 align:middle line:90% 01:12:57.470 --> 01:13:00.360 align:middle line:84% So classically, you always have this. 01:13:00.360 --> 01:13:04.890 align:middle line:84% So you always have elastic scattering. 01:13:04.890 --> 01:13:06.920 align:middle line:84% So the photon frequency does not change. 01:13:06.920 --> 01:13:09.860 align:middle line:90% 01:13:09.860 --> 01:13:13.800 align:middle line:90% And then the equation-- 01:13:13.800 --> 01:13:20.540 align:middle line:84% so let me call this equation 1 and the equation 1 holds. 01:13:20.540 --> 01:13:24.980 align:middle line:84% OK, yeah, and then equal this a and b. 01:13:24.980 --> 01:13:26.550 align:middle line:90% And equation 1 holds. 01:13:26.550 --> 01:13:31.690 align:middle line:84% So the classical, very robust prediction 01:13:31.690 --> 01:13:34.780 align:middle line:84% is that you always have elastic scattering when you 01:13:34.780 --> 01:13:38.020 align:middle line:90% shine light on this Compton. 01:13:38.020 --> 01:13:42.310 align:middle line:84% And then the equation, you have this Thomson cross-section. 01:13:42.310 --> 01:13:46.660 align:middle line:84% But then now in the early part of the 20th century, 01:13:46.660 --> 01:13:50.030 align:middle line:84% when Compton did this experiment-- 01:13:50.030 --> 01:13:51.770 align:middle line:90% and then he said-- 01:13:51.770 --> 01:13:56.600 align:middle line:84% then he observed actually the frequency can be smaller, OK. 01:13:56.600 --> 01:13:59.090 align:middle line:84% So in this formula, generically omega prime 01:13:59.090 --> 01:14:01.700 align:middle line:90% is smaller than omega. 01:14:01.700 --> 01:14:03.290 align:middle line:90% OK, but in quantum mechanics-- 01:14:03.290 --> 01:14:09.040 align:middle line:90% 01:14:09.040 --> 01:14:14.530 align:middle line:84% But in quantum mechanics, we see that omega prime is generically 01:14:14.530 --> 01:14:19.900 align:middle line:84% smaller than omega big because this is 1 plus a factor, which 01:14:19.900 --> 01:14:22.325 align:middle line:90% is greater than 1, OK. 01:14:22.325 --> 01:14:25.200 align:middle line:90% 01:14:25.200 --> 01:14:26.640 align:middle line:84% And in particular, this deviation 01:14:26.640 --> 01:14:28.500 align:middle line:84% will become more obvious when omega 01:14:28.500 --> 01:14:35.130 align:middle line:84% become comparable to m when omega become comparable to m. 01:14:35.130 --> 01:14:39.390 align:middle line:84% Yeah, so this is the quantum mechanics. 01:14:39.390 --> 01:14:43.020 align:middle line:84% So you will always have inelastic scattering. 01:14:43.020 --> 01:14:48.300 align:middle line:90% 01:14:48.300 --> 01:14:56.420 align:middle line:84% So this makes sense because just for momentum conservation 01:14:56.420 --> 01:14:58.760 align:middle line:90% in actual-- 01:14:58.760 --> 01:15:02.240 align:middle line:84% if some photon hit the electron, electron needs to move. 01:15:02.240 --> 01:15:03.920 align:middle line:90% The electron need to move. 01:15:03.920 --> 01:15:06.350 align:middle line:84% Then, of course, the energy will increase. 01:15:06.350 --> 01:15:08.113 align:middle line:84% And then the energy will increase, 01:15:08.113 --> 01:15:09.530 align:middle line:84% in the omega prime of course, have 01:15:09.530 --> 01:15:11.480 align:middle line:90% to be smaller than omega, OK. 01:15:11.480 --> 01:15:17.190 align:middle line:84% So in the rest frame, an electron always gain energy. 01:15:17.190 --> 01:15:19.070 align:middle line:84% So that's why in quantum mechanics 01:15:19.070 --> 01:15:23.680 align:middle line:90% always inelastic scattering, OK. 01:15:23.680 --> 01:15:33.940 align:middle line:84% And then the star under this equation 1 no longer holds OK. 01:15:33.940 --> 01:15:36.470 align:middle line:90% 01:15:36.470 --> 01:15:40.190 align:middle line:90% So in the early days-- 01:15:40.190 --> 01:15:45.860 align:middle line:84% so in the early days, this just cannot be explained using 01:15:45.860 --> 01:15:47.220 align:middle line:90% classical electrodynamics. 01:15:47.220 --> 01:15:54.490 align:middle line:84% OK, so this is decisive evidence that actually photon-- 01:15:54.490 --> 01:15:58.600 align:middle line:84% the light behaves very different quantum mechanically 01:15:58.600 --> 01:16:01.510 align:middle line:90% from classical wave, OK. 01:16:01.510 --> 01:16:05.380 align:middle line:84% In fact, this simple fact can be just explained 01:16:05.380 --> 01:16:08.050 align:middle line:84% by treating a photon as a particle. 01:16:08.050 --> 01:16:09.910 align:middle line:84% OK, so this is the decisive support 01:16:09.910 --> 01:16:12.220 align:middle line:84% that the photon actually behave like a particle, 01:16:12.220 --> 01:16:13.885 align:middle line:90% OK, behave like a particle. 01:16:13.885 --> 01:16:17.110 align:middle line:90% 01:16:17.110 --> 01:16:20.930 align:middle line:84% So leads to the particle picture of photon, OK. 01:16:20.930 --> 01:16:33.360 align:middle line:90% 01:16:33.360 --> 01:16:37.660 align:middle line:90% OK, so we are out of time soon. 01:16:37.660 --> 01:16:42.600 align:middle line:84% So let me just mention one simple application of this. 01:16:42.600 --> 01:16:49.040 align:middle line:90% 01:16:49.040 --> 01:16:54.430 align:middle line:84% So in particular, let me just mention two small things. 01:16:54.430 --> 01:17:00.250 align:middle line:84% So one thing is that if omega is much, much greater than m, 01:17:00.250 --> 01:17:04.600 align:middle line:84% then omega prime divided by n becomes very small. 01:17:04.600 --> 01:17:13.480 align:middle line:90% 01:17:13.480 --> 01:17:23.000 align:middle line:84% OK, so this actually becomes much, much smaller than 1, 01:17:23.000 --> 01:17:25.310 align:middle line:84% except for theta equal to zero, OK. 01:17:25.310 --> 01:17:33.060 align:middle line:90% 01:17:33.060 --> 01:17:37.200 align:middle line:84% So if the initial photon momentum energy is very big, 01:17:37.200 --> 01:17:41.220 align:middle line:84% then actually most of the energy will go to the electron, OK. 01:17:41.220 --> 01:17:42.990 align:middle line:84% And then the final frequency is actually 01:17:42.990 --> 01:17:46.480 align:middle line:84% become much, much smaller than initial frequency. 01:17:46.480 --> 01:17:48.760 align:middle line:90% This is one remark. 01:17:48.760 --> 01:17:58.820 align:middle line:84% And another remark-- so this is in the electron rest frame. 01:17:58.820 --> 01:18:02.380 align:middle line:84% But in a different frame, you will actually 01:18:02.380 --> 01:18:04.610 align:middle line:90% get a very different answer. 01:18:04.610 --> 01:18:08.590 align:middle line:84% So now let's imagine we have a frame-- 01:18:08.590 --> 01:18:12.080 align:middle line:84% So in this frame the electron just sits here. 01:18:12.080 --> 01:18:15.170 align:middle line:84% The photon comes in the z direction. 01:18:15.170 --> 01:18:20.120 align:middle line:84% Now, imagine we boost the system in the negative z direction, 01:18:20.120 --> 01:18:21.560 align:middle line:90% the negative z direction. 01:18:21.560 --> 01:18:24.160 align:middle line:84% So the result of this boost is that the photon energy 01:18:24.160 --> 01:18:25.810 align:middle line:90% becomes very small. 01:18:25.810 --> 01:18:27.670 align:middle line:84% And the electron energy will become 01:18:27.670 --> 01:18:29.620 align:middle line:84% bigger and bigger when you increase the boost, 01:18:29.620 --> 01:18:32.350 align:middle line:84% OK, because the boost is opposite to the direction 01:18:32.350 --> 01:18:35.830 align:middle line:84% of the photon so that you boost more. 01:18:35.830 --> 01:18:38.440 align:middle line:84% The photon energy will become smaller and smaller, OK. 01:18:38.440 --> 01:18:40.360 align:middle line:84% So now imagine you go to a frame, which 01:18:40.360 --> 01:18:45.410 align:middle line:84% photons have a very small energy compared to electron. 01:18:45.410 --> 01:18:46.880 align:middle line:90% And then this is a very-- 01:18:46.880 --> 01:18:50.165 align:middle line:84% then in that frame will be a very fast electron heat 01:18:50.165 --> 01:18:52.550 align:middle line:90% of a very low energy photon. 01:18:52.550 --> 01:18:55.640 align:middle line:84% OK, so this is called the inverse Compton scattering. 01:18:55.640 --> 01:19:07.530 align:middle line:90% 01:19:07.530 --> 01:19:12.080 align:middle line:84% OK, so in this case, then the energy 01:19:12.080 --> 01:19:15.560 align:middle line:84% in-- then the energy in the-- initial energy in the electron 01:19:15.560 --> 01:19:19.490 align:middle line:84% then can be transferred into the photon. 01:19:19.490 --> 01:19:23.030 align:middle line:84% Yeah, you have a very fast, very high energy electron 01:19:23.030 --> 01:19:24.830 align:middle line:90% hit a a low energy photon. 01:19:24.830 --> 01:19:27.200 align:middle line:84% Then you just give a big kick to the photon. 01:19:27.200 --> 01:19:29.830 align:middle line:84% And the photon can have very high energy. 01:19:29.830 --> 01:19:32.220 align:middle line:90% So this is a very simple effect. 01:19:32.220 --> 01:19:39.990 align:middle line:84% But it can have very important astrophysical applications. 01:19:39.990 --> 01:19:42.720 align:middle line:84% So this is called the important application in this called 01:19:42.720 --> 01:19:45.630 align:middle line:90% the Sunyaev--Zeldovich effect. 01:19:45.630 --> 01:19:54.540 align:middle line:90% 01:19:54.540 --> 01:19:59.960 align:middle line:84% So essentially they have the simple observation like this. 01:19:59.960 --> 01:20:01.985 align:middle line:84% So in the universe, we have the-- 01:20:01.985 --> 01:20:04.970 align:middle line:90% 01:20:04.970 --> 01:20:08.120 align:middle line:84% we have microwave background radiation. 01:20:08.120 --> 01:20:10.448 align:middle line:84% OK, the microwave background radiation, the photon 01:20:10.448 --> 01:20:12.740 align:middle line:84% is very low energy because the temperature is very low. 01:20:12.740 --> 01:20:17.700 align:middle line:90% 01:20:17.700 --> 01:20:20.100 align:middle line:90% But around a galaxy-- 01:20:20.100 --> 01:20:23.160 align:middle line:84% so if you're near the galaxy center, 01:20:23.160 --> 01:20:25.440 align:middle line:90% so near the galaxy cluster-- 01:20:25.440 --> 01:20:27.870 align:middle line:84% and the galaxy cluster, some of the electrons, 01:20:27.870 --> 01:20:29.830 align:middle line:90% they can have very high energy. 01:20:29.830 --> 01:20:33.710 align:middle line:84% And when they scatter and the microwave background 01:20:33.710 --> 01:20:37.310 align:middle line:84% photon and then they can give those microwave photon 01:20:37.310 --> 01:20:41.240 align:middle line:90% a big kick, and then they will-- 01:20:41.240 --> 01:20:44.220 align:middle line:84% then the photon will get a very high energy. 01:20:44.220 --> 01:20:48.050 align:middle line:84% So then by looking at the photon spectrum in the sky, 01:20:48.050 --> 01:20:50.810 align:middle line:90% look for this kind of hotspot. 01:20:50.810 --> 01:20:55.400 align:middle line:84% And then this is a way to detect the galaxy cluster. 01:20:55.400 --> 01:20:58.730 align:middle line:84% OK, then you can just use this inverse Compton scattering 01:20:58.730 --> 01:21:03.830 align:middle line:84% that detects the location of the galaxy cluster. 01:21:03.830 --> 01:21:07.220 align:middle line:90% So it's a very cool-- 01:21:07.220 --> 01:21:10.210 align:middle line:90% it's a very cool application. 01:21:10.210 --> 01:21:12.000 align:middle line:90%