WEBVTT 00:00:00.000 --> 00:00:00.976 align:middle line:90% 00:00:00.976 --> 00:00:02.440 align:middle line:90% [SQUEAKING] 00:00:02.440 --> 00:00:03.904 align:middle line:90% [RUSTLING] 00:00:03.904 --> 00:00:10.560 align:middle line:90% [CLICKING] 00:00:10.560 --> 00:00:12.770 align:middle line:90% HONG LIU: OK, so let us start. 00:00:12.770 --> 00:00:15.420 align:middle line:90% 00:00:15.420 --> 00:00:19.110 align:middle line:84% So last time, at the end of the lecture, 00:00:19.110 --> 00:00:27.680 align:middle line:84% we discussed relativistic generalization 00:00:27.680 --> 00:00:28.685 align:middle line:90% of the cross section. 00:00:28.685 --> 00:00:32.340 align:middle line:90% 00:00:32.340 --> 00:00:37.580 align:middle line:84% So we want to derive a relativistic generalization 00:00:37.580 --> 00:00:41.030 align:middle line:84% of the cross section from some initial state 00:00:41.030 --> 00:00:42.590 align:middle line:90% to some final state. 00:00:42.590 --> 00:00:47.330 align:middle line:84% So initial state is denoted by alpha, the final state denoted 00:00:47.330 --> 00:00:49.260 align:middle line:90% by beta. 00:00:49.260 --> 00:00:54.410 align:middle line:84% So we will, for most physical process, we 00:00:54.410 --> 00:00:56.960 align:middle line:84% will consider the two particles scattering. 00:00:56.960 --> 00:00:59.300 align:middle line:90% So this is just two particles. 00:00:59.300 --> 00:01:01.680 align:middle line:90% And they scatter. 00:01:01.680 --> 00:01:06.410 align:middle line:84% So this is our initial state, always only have two particles. 00:01:06.410 --> 00:01:10.690 align:middle line:84% But the final state can have arbitrary number of particles. 00:01:10.690 --> 00:01:15.180 align:middle line:84% You can have an arbitrary number of particles. 00:01:15.180 --> 00:01:24.240 align:middle line:84% And then we define, so we require that d sigma alpha beta 00:01:24.240 --> 00:01:25.770 align:middle line:90% to be Lorentz invariant. 00:01:25.770 --> 00:01:31.586 align:middle line:90% 00:01:31.586 --> 00:01:36.050 align:middle line:84% But because the probability for-- 00:01:36.050 --> 00:01:38.630 align:middle line:84% that should not depend on say, your frame. 00:01:38.630 --> 00:01:46.010 align:middle line:84% We also want to define to be symmetric in 1 to 2. 00:01:46.010 --> 00:01:52.030 align:middle line:90% 00:01:52.030 --> 00:01:59.560 align:middle line:84% So to make a connection with the nonrelativistic story, 00:01:59.560 --> 00:02:07.290 align:middle line:84% we can consider the rest frame, say-- 00:02:07.290 --> 00:02:09.970 align:middle line:84% say as a starting point, we can consider 00:02:09.970 --> 00:02:13.250 align:middle line:90% the rest frame of particle 2. 00:02:13.250 --> 00:02:16.120 align:middle line:84% You can always go to the rest frame of particle 2. 00:02:16.120 --> 00:02:22.480 align:middle line:84% And then in that frame, then we can define this quantity 00:02:22.480 --> 00:02:31.540 align:middle line:84% to be the probability per unit time from alpha to beta dt, 00:02:31.540 --> 00:02:35.890 align:middle line:84% and then divided by the incident flux of particle 1. 00:02:35.890 --> 00:02:49.570 align:middle line:90% 00:02:49.570 --> 00:02:57.550 align:middle line:84% So this is the p alpha beta t is the probability-- 00:02:57.550 --> 00:02:59.530 align:middle line:90% let me just write it. 00:02:59.530 --> 00:03:14.390 align:middle line:84% So this is the probability per unit time for this process 00:03:14.390 --> 00:03:17.510 align:middle line:90% from alpha to beta. 00:03:17.510 --> 00:03:35.780 align:middle line:84% And then, the dN1/dt dA would be the incident flux 00:03:35.780 --> 00:03:36.740 align:middle line:90% of particle 1. 00:03:36.740 --> 00:03:40.850 align:middle line:90% 00:03:40.850 --> 00:03:44.770 align:middle line:84% So this expression is a natural generalization 00:03:44.770 --> 00:03:50.860 align:middle line:84% to relativistic context, which in the rest frame of particle 00:03:50.860 --> 00:03:54.580 align:middle line:84% 2, in the rest frame of particle 2. 00:03:54.580 --> 00:03:58.360 align:middle line:84% And so, our strategy is to first work out 00:03:58.360 --> 00:04:02.410 align:middle line:84% this quantity in the rest frame of 2. 00:04:02.410 --> 00:04:08.230 align:middle line:84% And then, we use the Lorentz, the requirement of the Lorentz 00:04:08.230 --> 00:04:10.540 align:middle line:84% invariant and the symmetric in 1 and 2 00:04:10.540 --> 00:04:13.030 align:middle line:84% to write it in the general frame. 00:04:13.030 --> 00:04:13.990 align:middle line:90% So that's our strategy. 00:04:13.990 --> 00:04:17.230 align:middle line:90% 00:04:17.230 --> 00:04:23.560 align:middle line:84% And so first, let's talk about this quantity, this probability 00:04:23.560 --> 00:04:27.230 align:middle line:90% from alpha to beta. 00:04:27.230 --> 00:04:30.670 align:middle line:84% So we have the probability from alpha to beta. 00:04:30.670 --> 00:04:35.770 align:middle line:90% 00:04:35.770 --> 00:04:37.960 align:middle line:84% So by definition, it's defined as the following. 00:04:37.960 --> 00:04:40.480 align:middle line:90% 00:04:40.480 --> 00:04:46.160 align:middle line:84% So we want to look at the transition amplitude, which 00:04:46.160 --> 00:04:51.260 align:middle line:84% gives you the beta state at time plus infinity and from alpha 00:04:51.260 --> 00:04:53.390 align:middle line:90% at time minus infinity. 00:04:53.390 --> 00:04:54.860 align:middle line:90% So this is the amplitude. 00:04:54.860 --> 00:04:58.320 align:middle line:90% And then we take the square. 00:04:58.320 --> 00:05:03.520 align:middle line:84% And then we divide by the normalization over the beta 00:05:03.520 --> 00:05:08.490 align:middle line:90% itself under the alpha itself. 00:05:08.490 --> 00:05:12.260 align:middle line:90% 00:05:12.260 --> 00:05:16.580 align:middle line:84% We divided by the normalization of the beta and alpha, 00:05:16.580 --> 00:05:17.870 align:middle line:90% the state themselves. 00:05:17.870 --> 00:05:23.870 align:middle line:84% So that is the whole thing, really defined the probability. 00:05:23.870 --> 00:05:29.400 align:middle line:84% So the upstairs, we already defined before. 00:05:29.400 --> 00:05:35.600 align:middle line:84% So previously, we defined the beta plus, 00:05:35.600 --> 00:05:37.800 align:middle line:90% infinity alpha minus infinity. 00:05:37.800 --> 00:05:44.150 align:middle line:84% So this is by our previous definition is given by 2 pi 00:05:44.150 --> 00:05:50.530 align:middle line:84% to the power 4 delta, p alpha p beta. 00:05:50.530 --> 00:05:52.840 align:middle line:84% So the p alpha p beta is the total momentum 00:05:52.840 --> 00:05:54.740 align:middle line:90% of the initial and final state. 00:05:54.740 --> 00:06:01.730 align:middle line:90% And times M alpha beta. 00:06:01.730 --> 00:06:03.770 align:middle line:84% M alpha beta is defined to be the scattering 00:06:03.770 --> 00:06:12.130 align:middle line:84% amplitude from alpha to beta, So this was essentially, 00:06:12.130 --> 00:06:15.515 align:middle line:84% our definition of the scattering amplitude. 00:06:15.515 --> 00:06:16.015 align:middle line:90% Yes? 00:06:16.015 --> 00:06:18.910 align:middle line:84% AUDIENCE: T minus-- so the plus infinity, minus infinity, 00:06:18.910 --> 00:06:21.610 align:middle line:84% is that in space time or just time? 00:06:21.610 --> 00:06:22.690 align:middle line:90% HONG LIU: Time, yeah. 00:06:22.690 --> 00:06:27.280 align:middle line:84% it means that the T equal to minus infinity. 00:06:27.280 --> 00:06:30.040 align:middle line:90% So this is the Heisenberg state. 00:06:30.040 --> 00:06:32.200 align:middle line:84% So this is the Heisenberg picture. 00:06:32.200 --> 00:06:35.890 align:middle line:84% So this means that at t equal to minus infinity, 00:06:35.890 --> 00:06:39.490 align:middle line:84% we have a free particle state described by alpha. 00:06:39.490 --> 00:06:42.010 align:middle line:84% And the t equal to plus infinity, 00:06:42.010 --> 00:06:47.790 align:middle line:84% we have free particle state, given by beta. 00:06:47.790 --> 00:06:48.810 align:middle line:90% Other questions? 00:06:48.810 --> 00:06:51.860 align:middle line:90% 00:06:51.860 --> 00:06:53.240 align:middle line:90% So now, let's take the square. 00:06:53.240 --> 00:06:56.200 align:middle line:90% 00:06:56.200 --> 00:06:59.080 align:middle line:90% Now, let's take the square. 00:06:59.080 --> 00:07:06.950 align:middle line:84% So remember, so the reason we need to put in downstairs, 00:07:06.950 --> 00:07:09.350 align:middle line:84% you say, why do we need to put the downstairs? 00:07:09.350 --> 00:07:14.090 align:middle line:84% Because are they supposed to be normalized? 00:07:14.090 --> 00:07:17.450 align:middle line:84% But in practice when we do the calculation, 00:07:17.450 --> 00:07:21.080 align:middle line:84% or in practice when we talk about the free particles, 00:07:21.080 --> 00:07:23.660 align:middle line:90% we use the plane wave basis. 00:07:23.660 --> 00:07:26.990 align:middle line:84% But plane wave, remember, is not normalizable. 00:07:26.990 --> 00:07:30.620 align:middle line:84% It's only normalizable as a delta function. 00:07:30.620 --> 00:07:34.400 align:middle line:84% So the normalization is not equal to 1, exactly. 00:07:34.400 --> 00:07:36.080 align:middle line:90% So we have to be careful. 00:07:36.080 --> 00:07:40.370 align:middle line:84% So that's why we need to include them here. 00:07:40.370 --> 00:07:43.700 align:middle line:90% So yeah. 00:07:43.700 --> 00:07:45.200 align:middle line:90% Good? 00:07:45.200 --> 00:07:48.890 align:middle line:84% So now, let's look at this quantity squared. 00:07:48.890 --> 00:07:51.630 align:middle line:90% So let's just move it down. 00:07:51.630 --> 00:07:54.480 align:middle line:84% So to save time, I just want to copy it. 00:07:54.480 --> 00:08:00.470 align:middle line:84% So this quantity squared then just equal to the right hand 00:08:00.470 --> 00:08:01.550 align:middle line:90% side square. 00:08:01.550 --> 00:08:08.020 align:middle line:84% So we have 2 pi 4 the delta function square. 00:08:08.020 --> 00:08:11.200 align:middle line:84% And then we have M alpha beta squared. 00:08:11.200 --> 00:08:15.840 align:middle line:90% 00:08:15.840 --> 00:08:19.680 align:middle line:84% And now, let's look at the delta function square. 00:08:19.680 --> 00:08:22.110 align:middle line:84% So the delta function squared will have two pieces. 00:08:22.110 --> 00:08:25.470 align:middle line:90% 00:08:25.470 --> 00:08:26.790 align:middle line:90% So the delta function square-- 00:08:26.790 --> 00:08:32.500 align:middle line:90% 00:08:32.500 --> 00:08:33.669 align:middle line:90% let me just save time. 00:08:33.669 --> 00:08:40.120 align:middle line:84% So this square, then we can take one piece, one copy of it. 00:08:40.120 --> 00:08:45.000 align:middle line:84% We just write it as p alpha minus p beta. 00:08:45.000 --> 00:08:50.970 align:middle line:84% And then the other piece will be just delta 4, 00:08:50.970 --> 00:08:54.070 align:middle line:90% we can just evaluate at the 0. 00:08:54.070 --> 00:08:57.390 align:middle line:84% Because we already set the p alpha equal 00:08:57.390 --> 00:09:00.240 align:middle line:84% to beta from the first delta function. 00:09:00.240 --> 00:09:05.370 align:middle line:84% So the second factor can be just become delta 0. 00:09:05.370 --> 00:09:07.020 align:middle line:90% So now we know how to-- 00:09:07.020 --> 00:09:10.670 align:middle line:90% so we have seen this before. 00:09:10.670 --> 00:09:12.440 align:middle line:90% So what is this object? 00:09:12.440 --> 00:09:15.250 align:middle line:90% Do you remember? 00:09:15.250 --> 00:09:15.853 align:middle line:90% Yes? 00:09:15.853 --> 00:09:17.020 align:middle line:90% AUDIENCE: Space time volume. 00:09:17.020 --> 00:09:20.590 align:middle line:84% HONG LIU: Yeah, it's the total space time volume. 00:09:20.590 --> 00:09:25.960 align:middle line:84% The reason is that the 2 pi delta 4 0. 00:09:25.960 --> 00:09:28.990 align:middle line:84% So this is the 0 in momentum space. 00:09:28.990 --> 00:09:35.650 align:middle line:84% So this is just equal to d4x exponential ikx evaluated 00:09:35.650 --> 00:09:39.910 align:middle line:84% at the k equal to 0, evaluated at the k equal to 0. 00:09:39.910 --> 00:09:42.250 align:middle line:84% Because this is giving you 2 two pi delta k. 00:09:42.250 --> 00:09:43.780 align:middle line:90% And so you evaluate at k = 0. 00:09:43.780 --> 00:09:45.830 align:middle line:84% When you evaluate the k = 0, then of course, 00:09:45.830 --> 00:09:48.970 align:middle line:84% this is just the space time volume. 00:09:48.970 --> 00:09:51.070 align:middle line:84% So I take the spatial volume to V. 00:09:51.070 --> 00:09:54.505 align:middle line:84% And then this T, is the time period. 00:09:54.505 --> 00:09:59.050 align:middle line:90% 00:09:59.050 --> 00:10:05.770 align:middle line:84% So physically, you can interpret the T 00:10:05.770 --> 00:10:09.300 align:middle line:84% as the duration at the time duration 00:10:09.300 --> 00:10:11.040 align:middle line:84% we actually doing the experiment. 00:10:11.040 --> 00:10:17.100 align:middle line:84% Because only that part is relevant for our experiment. 00:10:17.100 --> 00:10:19.320 align:middle line:84% So this capital T can be interpreted -- 00:10:19.320 --> 00:10:22.660 align:middle line:90% 00:10:22.660 --> 00:10:23.630 align:middle line:90% OK, good? 00:10:23.630 --> 00:10:28.330 align:middle line:84% And now, we can just replace here, 00:10:28.330 --> 00:10:37.410 align:middle line:84% so now we can just replace here by V times T. We can just 00:10:37.410 --> 00:10:48.860 align:middle line:84% replace here, V times T. So now let's look at the downstairs. 00:10:48.860 --> 00:10:50.870 align:middle line:90% Now, let's look at downstairs. 00:10:50.870 --> 00:10:55.970 align:middle line:84% The downstairs, so the alpha and the beta will be just say-- 00:10:55.970 --> 00:11:01.870 align:middle line:84% yeah, so the typical alpha, so the alpha state 00:11:01.870 --> 00:11:08.333 align:middle line:84% would be say, if your initial momentum is P1 and P2, 00:11:08.333 --> 00:11:09.250 align:middle line:90% it would be like this. 00:11:09.250 --> 00:11:13.660 align:middle line:84% For scalar particle, it just will be specified by P1 and P2. 00:11:13.660 --> 00:11:15.610 align:middle line:84% And if you have polarizations, then there 00:11:15.610 --> 00:11:17.710 align:middle line:90% will be polarizations. 00:11:17.710 --> 00:11:20.270 align:middle line:84% Here, I'll just give you an example. 00:11:20.270 --> 00:11:22.420 align:middle line:84% So now, if you look at the normalization 00:11:22.420 --> 00:11:27.940 align:middle line:84% of this kind of states, remember the P1 and P2 are defined. 00:11:27.940 --> 00:11:29.740 align:middle line:84% So remember, if you have a single particle 00:11:29.740 --> 00:11:34.210 align:middle line:90% state of momentum p. 00:11:34.210 --> 00:11:47.800 align:middle line:84% So that is normalized to be 2 pi cubed 2p0 delta 3 p minus 00:11:47.800 --> 00:11:50.925 align:middle line:84% p-prime, because they're always on-shell. 00:11:50.925 --> 00:11:52.550 align:middle line:84% Initial and final state, they on-shell. 00:11:52.550 --> 00:11:56.780 align:middle line:90% So p0 are functions of p. 00:11:56.780 --> 00:12:01.050 align:middle line:90% So that's how we normalize them. 00:12:01.050 --> 00:12:04.460 align:middle line:84% So p is normalized by square root 00:12:04.460 --> 00:12:11.570 align:middle line:84% omega k omega p, say ap acting on 0. 00:12:11.570 --> 00:12:14.060 align:middle line:84% And for example, for scalar particle, and then 00:12:14.060 --> 00:12:15.965 align:middle line:84% you look at the normalization just like this. 00:12:15.965 --> 00:12:19.470 align:middle line:90% 00:12:19.470 --> 00:12:26.550 align:middle line:84% So now if you look at the p with itself, 00:12:26.550 --> 00:12:29.430 align:middle line:84% then you have p equal to p-prime. 00:12:29.430 --> 00:12:31.830 align:middle line:84% And then you have essentially, you just 00:12:31.830 --> 00:12:38.320 align:middle line:84% have 2p0 times V. The V is the spatial volume. 00:12:38.320 --> 00:12:41.680 align:middle line:84% Because here you just have 2 pi cubed the delta 0. 00:12:41.680 --> 00:12:45.520 align:middle line:84% And delta 0 is in the spatial momentum. 00:12:45.520 --> 00:12:48.580 align:middle line:84% And spatial momentum for the same reason, 00:12:48.580 --> 00:12:50.290 align:middle line:84% just gave you the spatial volume. 00:12:50.290 --> 00:12:54.300 align:middle line:90% 00:12:54.300 --> 00:13:01.570 align:middle line:84% And so now, we can just apply these two to alpha and beta. 00:13:01.570 --> 00:13:06.180 align:middle line:84% So the alpha, So if I take the alpha to have momentum P1, 00:13:06.180 --> 00:13:10.620 align:middle line:84% P2, you can see this particle of momentum p1 and p2 00:13:10.620 --> 00:13:17.490 align:middle line:84% on beta, to consist of momentum say, k1 and kn. 00:13:17.490 --> 00:13:23.020 align:middle line:84% Suppose there are n particles, so n can be arbitrary integers. 00:13:23.020 --> 00:13:29.660 align:middle line:84% And then from here, then the alpha with itself, 00:13:29.660 --> 00:13:38.420 align:middle line:84% would be just 2 E1 times the volume and 2 E2 times volume. 00:13:38.420 --> 00:13:41.270 align:middle line:84% E1 and E2 are the energy over the p1 and p2. 00:13:41.270 --> 00:13:45.490 align:middle line:90% 00:13:45.490 --> 00:13:54.650 align:middle line:84% So E1 is equal to P 1,0 and equal to P 2,0. 00:13:54.650 --> 00:14:03.170 align:middle line:84% And similarly, the beta would be to take the product j from 1 00:14:03.170 --> 00:14:23.810 align:middle line:84% to n, 2kj, 0'th component times V. 00:14:23.810 --> 00:14:31.670 align:middle line:84% And so example, just kj0 just equal 00:14:31.670 --> 00:14:36.565 align:middle line:90% to kj squared plus mj squared. 00:14:36.565 --> 00:14:39.270 align:middle line:90% 00:14:39.270 --> 00:14:41.180 align:middle line:84% So here, we allow the different particles 00:14:41.180 --> 00:14:50.370 align:middle line:84% to have different mass and similarly with p 1,0 and p2. 00:14:50.370 --> 00:14:58.050 align:middle line:84% OK so now with those preparations, 00:14:58.050 --> 00:15:03.210 align:middle line:90% now we can write down dpt. 00:15:03.210 --> 00:15:10.740 align:middle line:84% So now, the dp alpha to beta dt then essentially 00:15:10.740 --> 00:15:15.180 align:middle line:84% just equal to p alpha beta divided by t, 00:15:15.180 --> 00:15:20.170 align:middle line:84% the total duration of the physical process. 00:15:20.170 --> 00:15:22.920 align:middle line:84% And so, the reason they just relate it. 00:15:22.920 --> 00:15:26.190 align:middle line:90% You say, how can you do this? 00:15:26.190 --> 00:15:27.510 align:middle line:90% This is a differential. 00:15:27.510 --> 00:15:30.860 align:middle line:84% And how can you just divide it by the t? 00:15:30.860 --> 00:15:33.405 align:middle line:84% Do you know a reason why we can just divide by this? 00:15:33.405 --> 00:15:43.110 align:middle line:90% 00:15:43.110 --> 00:15:44.520 align:middle line:90% Yes? 00:15:44.520 --> 00:15:47.280 align:middle line:84% AUDIENCE: Well presumably, it's the probability was small 00:15:47.280 --> 00:15:48.450 align:middle line:90% or something. 00:15:48.450 --> 00:15:51.210 align:middle line:84% You would just expect it to go up linearly in time. 00:15:51.210 --> 00:15:53.940 align:middle line:90% 00:15:53.940 --> 00:16:01.110 align:middle line:84% HONG LIU: Just because of-- yeah, that's a good statement. 00:16:01.110 --> 00:16:05.430 align:middle line:84% But the statement, I wanted you to say 00:16:05.430 --> 00:16:10.060 align:middle line:84% is that because the time translation symmetry. 00:16:10.060 --> 00:16:14.040 align:middle line:84% And so, you would expect that the probability per unit time 00:16:14.040 --> 00:16:15.880 align:middle line:90% should be independent of time. 00:16:15.880 --> 00:16:18.990 align:middle line:84% And so the probability, say for duration of time, 00:16:18.990 --> 00:16:21.060 align:middle line:84% and you can just multiply by the total time. 00:16:21.060 --> 00:16:24.550 align:middle line:84% And indeed, in order to make this equation make sense. 00:16:24.550 --> 00:16:26.580 align:middle line:84% Of course, the p has to be small. 00:16:26.580 --> 00:16:30.240 align:middle line:84% Because otherwise, if you multiply-- yeah, 00:16:30.240 --> 00:16:31.463 align:middle line:90% this thing has to be small. 00:16:31.463 --> 00:16:32.880 align:middle line:84% Otherwise, when you multiply by T, 00:16:32.880 --> 00:16:39.650 align:middle line:84% of course it will be greater than 1 But just to divide by T, 00:16:39.650 --> 00:16:41.375 align:middle line:84% it's from the time translation symmetry. 00:16:41.375 --> 00:16:44.170 align:middle line:90% 00:16:44.170 --> 00:16:49.090 align:middle line:84% So this is good because we know here, from the delta function, 00:16:49.090 --> 00:16:53.380 align:middle line:84% there's a factor of T. There's a factor of T. 00:16:53.380 --> 00:16:58.360 align:middle line:84% So if we combine everything together, 00:16:58.360 --> 00:16:59.830 align:middle line:90% and then we can just-- 00:16:59.830 --> 00:17:08.609 align:middle line:84% then the upstairs is given by, so we have the 2 pi power 00:17:08.609 --> 00:17:18.760 align:middle line:84% of 4 delta 4 p alpha p beta, the m alpha to beta 00:17:18.760 --> 00:17:22.540 align:middle line:90% squared times VT. 00:17:22.540 --> 00:17:28.089 align:middle line:84% And then divided by T, and then this T go away, 00:17:28.089 --> 00:17:33.100 align:middle line:90% so I can just erase this T. 00:17:33.100 --> 00:17:38.260 align:middle line:84% So this is the upstairs, and then divided by T, a capital T. 00:17:38.260 --> 00:17:43.360 align:middle line:84% And then the downstairs, we just copy these two expression. 00:17:43.360 --> 00:17:49.000 align:middle line:84% Downstairs, I just copy this expression and this expression. 00:17:49.000 --> 00:18:01.030 align:middle line:84% So this is 4 E1E2 times V square and then k from 1 00:18:01.030 --> 00:18:15.980 align:middle line:90% to n 2kj0 times V. Yes? 00:18:15.980 --> 00:18:18.350 align:middle line:84% AUDIENCE: So now taking T to be the total time 00:18:18.350 --> 00:18:20.090 align:middle line:84% of your experiment, are you taking 00:18:20.090 --> 00:18:22.700 align:middle line:84% V to be the volume of your experiment? 00:18:22.700 --> 00:18:24.770 align:middle line:90% HONG LIU: Yeah. 00:18:24.770 --> 00:18:26.810 align:middle line:90% AUDIENCE: Of your detector? 00:18:26.810 --> 00:18:28.310 align:middle line:84% HONG LIU: No, it's not the detector. 00:18:28.310 --> 00:18:32.480 align:middle line:84% It's just, we are computing S matrix. 00:18:32.480 --> 00:18:36.210 align:middle line:84% So x matrix, you always assume you wait for a long time 00:18:36.210 --> 00:18:37.670 align:middle line:84% so that your initial state is free, 00:18:37.670 --> 00:18:39.730 align:middle line:90% and your final state is free. 00:18:39.730 --> 00:18:43.880 align:middle line:90% So of course, it's much not. 00:18:43.880 --> 00:18:47.620 align:middle line:90% It's the range of your detector. 00:18:47.620 --> 00:18:49.930 align:middle line:84% You have to put the detector very far away in order 00:18:49.930 --> 00:18:51.400 align:middle line:90% to measure the particle. 00:18:51.400 --> 00:18:55.770 align:middle line:90% 00:18:55.770 --> 00:18:57.750 align:middle line:84% You can imagine the V just essentially 00:18:57.750 --> 00:19:07.610 align:middle line:84% is the volume, which the experiment is happening. 00:19:07.610 --> 00:19:08.570 align:middle line:90% Other questions? 00:19:08.570 --> 00:19:11.490 align:middle line:90% 00:19:11.490 --> 00:19:15.270 align:middle line:84% So, you see there's all this V flying around. 00:19:15.270 --> 00:19:18.300 align:middle line:90% So they look very unpleasant. 00:19:18.300 --> 00:19:19.080 align:middle line:90% But don't worry. 00:19:19.080 --> 00:19:22.190 align:middle line:84% Later, if you are doing the right thing, 00:19:22.190 --> 00:19:25.080 align:middle line:84% then all these unpleasant things will go away. 00:19:25.080 --> 00:19:26.900 align:middle line:90% So that's the rule of physics. 00:19:26.900 --> 00:19:28.820 align:middle line:84% And in the intermediate step, you 00:19:28.820 --> 00:19:30.720 align:middle line:84% may see a lot of unpleasant things. 00:19:30.720 --> 00:19:33.020 align:middle line:84% But if you are doing the right thing 00:19:33.020 --> 00:19:36.060 align:middle line:84% and then all this unpleasant thing will go away, or you say, 00:19:36.060 --> 00:19:37.970 align:middle line:84% well, then I must be doing the right thing. 00:19:37.970 --> 00:19:41.080 align:middle line:90% 00:19:41.080 --> 00:19:43.180 align:middle line:84% So here, actually this probability 00:19:43.180 --> 00:19:47.250 align:middle line:84% is not the probability we actually measure. 00:19:47.250 --> 00:19:51.580 align:middle line:84% Because here, I assume that the final state-- 00:19:51.580 --> 00:19:53.425 align:middle line:84% here I'll assume the final state-- 00:19:53.425 --> 00:19:56.710 align:middle line:90% 00:19:56.710 --> 00:19:58.990 align:middle line:84% yeah, here, I assume the final states 00:19:58.990 --> 00:20:03.520 align:middle line:90% have precise momentum k1, kn. 00:20:03.520 --> 00:20:05.590 align:middle line:84% So in reality, of course, we will not 00:20:05.590 --> 00:20:08.950 align:middle line:84% be able to make the precise measurements. 00:20:08.950 --> 00:20:24.910 align:middle line:84% So in real experiment, detectors have finite resolutions. 00:20:24.910 --> 00:20:32.720 align:middle line:90% 00:20:32.720 --> 00:20:38.680 align:middle line:84% So what we measure is actually when we say the particle say, 00:20:38.680 --> 00:20:50.060 align:middle line:84% have momentum k1, we actually means that the particle 1, 00:20:50.060 --> 00:20:58.460 align:middle line:90% it's within some dk 1 around k1. 00:20:58.460 --> 00:21:02.660 align:middle line:84% So there's always some finite neighborhood of k1 00:21:02.660 --> 00:21:08.600 align:middle line:84% which are allowed by our detector resolution. 00:21:08.600 --> 00:21:12.980 align:middle line:84% And similarly, in reality, it's particle 2 00:21:12.980 --> 00:21:19.010 align:middle line:90% in dk2 around k2, et cetera. 00:21:19.010 --> 00:21:25.610 align:middle line:84% So the particle n, dkn around kn. 00:21:25.610 --> 00:21:28.820 align:middle line:84% So then, we actually need to integrate over 00:21:28.820 --> 00:21:34.920 align:middle line:90% all these resolutions. 00:21:34.920 --> 00:21:39.090 align:middle line:84% So that means that we should multiply dP/T. 00:21:39.090 --> 00:21:44.730 align:middle line:84% This corresponding to sharp final momentum, 00:21:44.730 --> 00:21:48.720 align:middle line:84% and we should multiply by those uncertainties. 00:21:48.720 --> 00:21:56.880 align:middle line:84% So that means that, so if I take this equation to be star, 00:21:56.880 --> 00:22:00.570 align:middle line:84% this equation to be star, so the one actually, 00:22:00.570 --> 00:22:03.330 align:middle line:90% is measured by the experiment. 00:22:03.330 --> 00:22:08.400 align:middle line:84% When we say P alpha beta dt, from the real experiment, 00:22:08.400 --> 00:22:13.400 align:middle line:84% it's actually corresponding to the star times-- 00:22:13.400 --> 00:22:17.620 align:middle line:90% 00:22:17.620 --> 00:22:21.760 align:middle line:90% those uncertainties around the-- 00:22:21.760 --> 00:22:31.830 align:middle line:84% so j equal to 1 to n d3 kj 2 pi cubed. 00:22:31.830 --> 00:22:34.670 align:middle line:90% 00:22:34.670 --> 00:22:40.130 align:middle line:84% So we need to multiply the number of states 00:22:40.130 --> 00:22:47.450 align:middle line:84% within each dk momentum space volume, 00:22:47.450 --> 00:22:50.450 align:middle line:84% So the number of the state, and the number of states 00:22:50.450 --> 00:22:53.045 align:middle line:84% is given by this times the volume. 00:22:53.045 --> 00:22:56.010 align:middle line:90% 00:22:56.010 --> 00:22:59.280 align:middle line:84% So do you remember where this volume come from? 00:22:59.280 --> 00:23:03.540 align:middle line:90% 00:23:03.540 --> 00:23:04.230 align:middle line:90% Yes? 00:23:04.230 --> 00:23:05.610 align:middle line:90% AUDIENCE: Density of states. 00:23:05.610 --> 00:23:06.510 align:middle line:90% HONG LIU: Exactly. 00:23:06.510 --> 00:23:08.350 align:middle line:84% So this gives you the density of states. 00:23:08.350 --> 00:23:12.630 align:middle line:84% It's because the-- so if I just remind you-- 00:23:12.630 --> 00:23:16.410 align:middle line:84% remember if you-- so the way to think about it, 00:23:16.410 --> 00:23:24.550 align:middle line:84% just imagine you put your system in the box, 00:23:24.550 --> 00:23:31.600 align:middle line:84% and then your energy level will be quantized in terms of 2 pi, 00:23:31.600 --> 00:23:35.860 align:middle line:84% say 2 pi cubed divided by the volume. 00:23:35.860 --> 00:23:38.710 align:middle line:84% So this is the number of states in momentum 00:23:38.710 --> 00:23:40.690 align:middle line:90% space, the density of states. 00:23:40.690 --> 00:23:42.490 align:middle line:84% And then you multiply it by the dk. 00:23:42.490 --> 00:23:51.970 align:middle line:90% 00:23:51.970 --> 00:23:55.690 align:middle line:84% So this is actually the quantity we are interested in. 00:23:55.690 --> 00:23:59.220 align:middle line:84% This is the quantity we're interested in. 00:23:59.220 --> 00:24:00.255 align:middle line:90% Any questions on this? 00:24:00.255 --> 00:24:03.810 align:middle line:90% 00:24:03.810 --> 00:24:12.350 align:middle line:84% So now, let's just plug that star into here. 00:24:12.350 --> 00:24:18.010 align:middle line:84% So now, the nice thing, now we can count the number of Vs. 00:24:18.010 --> 00:24:22.840 align:middle line:84% If we are doing the right thing, all the V should cancel 00:24:22.840 --> 00:24:29.290 align:middle line:90% except yeah, but not quite yet. 00:24:29.290 --> 00:24:33.680 align:middle line:84% So upstairs, we have one V. Here we have V squared. 00:24:33.680 --> 00:24:37.250 align:middle line:84% So let's first cancel this V so that we 00:24:37.250 --> 00:24:39.350 align:middle line:90% don't have to worry about that. 00:24:39.350 --> 00:24:41.630 align:middle line:90% So let's first cancel that V. 00:24:41.630 --> 00:24:46.700 align:middle line:84% And now, we have downstairs, we have V for each j. 00:24:46.700 --> 00:24:49.760 align:middle line:84% And the upstairs, now we multiply this by this. 00:24:49.760 --> 00:24:52.640 align:middle line:84% And then for each j, have one V. So all this V 00:24:52.640 --> 00:24:58.310 align:middle line:84% will cancel that V. All this V will cancel that V. 00:24:58.310 --> 00:25:05.210 align:middle line:84% So we're left with only this single V, this single V. 00:25:05.210 --> 00:25:08.130 align:middle line:84% So now, we can write it as the following. 00:25:08.130 --> 00:25:11.960 align:middle line:84% So I can write it this as M alpha beta squared. 00:25:11.960 --> 00:25:14.750 align:middle line:90% So I just copy this. 00:25:14.750 --> 00:25:17.090 align:middle line:90% And then I divide it by-- 00:25:17.090 --> 00:25:25.220 align:middle line:84% let's keep this-- divided by 4 E1E2 times V. 00:25:25.220 --> 00:25:30.440 align:middle line:84% And then I group everything else into what I call d mu. 00:25:30.440 --> 00:25:32.900 align:middle line:90% And d mu is everything else. 00:25:32.900 --> 00:25:36.620 align:middle line:84% D mu is defined to be this delta function. 00:25:36.620 --> 00:25:48.680 align:middle line:90% 00:25:48.680 --> 00:26:01.490 align:middle line:84% And then, j from 1 to n, d3 kj 2 pi cubed 2kj0. 00:26:01.490 --> 00:26:05.900 align:middle line:84% So I just combine these two products, 00:26:05.900 --> 00:26:07.040 align:middle line:90% combine these two products. 00:26:07.040 --> 00:26:17.140 align:middle line:90% 00:26:17.140 --> 00:26:21.400 align:middle line:84% So the nice thing, the reason I group 00:26:21.400 --> 00:26:28.310 align:middle line:84% all this together because now d mu now is Lorenz invariant. 00:26:28.310 --> 00:26:30.830 align:middle line:90% 00:26:30.830 --> 00:26:35.330 align:middle line:84% So this is a Lorentz invariant measure. 00:26:35.330 --> 00:26:39.620 align:middle line:84% Because you remember, this Lorenz invariant. 00:26:39.620 --> 00:26:42.170 align:middle line:84% And remember, this combination is also 00:26:42.170 --> 00:26:47.420 align:middle line:84% Lorentz invariant from your first day, essentially in QFT, 00:26:47.420 --> 00:26:50.240 align:middle line:90% in your first pset. 00:26:50.240 --> 00:26:52.340 align:middle line:84% So now, this nice thing is that this now 00:26:52.340 --> 00:26:53.645 align:middle line:90% is a Lorentz invariant. 00:26:53.645 --> 00:26:59.870 align:middle line:90% 00:26:59.870 --> 00:27:03.460 align:middle line:84% So now, we have this nice expression. 00:27:03.460 --> 00:27:06.130 align:middle line:90% We have this nice expression. 00:27:06.130 --> 00:27:09.960 align:middle line:84% We have this, by definition, is Lorentz invariant. 00:27:09.960 --> 00:27:18.270 align:middle line:84% And then we have this 4 E1E2, and times V, 4 E1 times V. 00:27:18.270 --> 00:27:23.120 align:middle line:84% And now, we want to write them into a Lorentz invariant way. 00:27:23.120 --> 00:27:24.320 align:middle line:90% But we haven't done it. 00:27:24.320 --> 00:27:26.780 align:middle line:90% This is only upstairs. 00:27:26.780 --> 00:27:28.490 align:middle line:84% This by itself, is not Lorenz invariant. 00:27:28.490 --> 00:27:31.100 align:middle line:84% We have to divide it by the downstairs. 00:27:31.100 --> 00:27:35.580 align:middle line:90% So the downstairs is the flux. 00:27:35.580 --> 00:27:37.650 align:middle line:90% It's the flux. 00:27:37.650 --> 00:27:48.950 align:middle line:84% So now, in the rest frame of particle 2, 00:27:48.950 --> 00:27:52.085 align:middle line:84% so lets calculate the flux of the particle. 00:27:52.085 --> 00:27:56.360 align:middle line:90% 00:27:56.360 --> 00:27:58.250 align:middle line:90% So the flux of the particle-- 00:27:58.250 --> 00:28:02.170 align:middle line:90% 00:28:02.170 --> 00:28:04.860 align:middle line:84% so we need to calculate the flux of particle 1. 00:28:04.860 --> 00:28:09.000 align:middle line:90% 00:28:09.000 --> 00:28:11.530 align:middle line:84% So that's the thing we need to divide. 00:28:11.530 --> 00:28:15.900 align:middle line:84% So this is the flux of the particle 1. 00:28:15.900 --> 00:28:23.370 align:middle line:84% So the flux is the number of particle per unit time 00:28:23.370 --> 00:28:27.540 align:middle line:84% and per area, per unit time, per area. 00:28:27.540 --> 00:28:32.040 align:middle line:84% So this is the same as the number 00:28:32.040 --> 00:28:36.090 align:middle line:84% of the particle, number density of the particle 1 times 00:28:36.090 --> 00:28:37.770 align:middle line:90% the velocity of the particle 1. 00:28:37.770 --> 00:28:43.020 align:middle line:90% 00:28:43.020 --> 00:28:45.240 align:middle line:84% So remember, the flux is essentially 00:28:45.240 --> 00:28:49.150 align:middle line:90% the density times the velocity. 00:28:49.150 --> 00:28:53.450 align:middle line:84% So this gives you the number of particles per volume. 00:28:53.450 --> 00:28:56.440 align:middle line:84% And this is giving you the distance 00:28:56.440 --> 00:28:58.220 align:middle line:90% traveled per unit time. 00:28:58.220 --> 00:29:01.270 align:middle line:84% And so together, they give you that. 00:29:01.270 --> 00:29:05.530 align:middle line:90% 00:29:05.530 --> 00:29:13.780 align:middle line:84% So this is the velocity, so this is the density of particle 1. 00:29:13.780 --> 00:29:16.690 align:middle line:84% So this is all in the rest frame of particle 2. 00:29:16.690 --> 00:29:23.630 align:middle line:90% 00:29:23.630 --> 00:29:27.510 align:middle line:84% So now in this experiment, we have two particle scattering. 00:29:27.510 --> 00:29:31.570 align:middle line:84% So what do you think will be the density of 1. 00:29:31.570 --> 00:29:33.490 align:middle line:90% So what do you think is the n1? 00:29:33.490 --> 00:29:40.020 align:middle line:90% 00:29:40.020 --> 00:29:40.740 align:middle line:90% Yes? 00:29:40.740 --> 00:29:41.820 align:middle line:90% AUDIENCE: 1. 00:29:41.820 --> 00:29:42.570 align:middle line:90% HONG LIU: Exactly. 00:29:42.570 --> 00:29:43.820 align:middle line:90% How many particles do we have? 00:29:43.820 --> 00:29:46.050 align:middle line:90% We only have one particle. 00:29:46.050 --> 00:29:52.500 align:middle line:84% So this is just given as 1/V. So this is just given by 1/V. 00:29:52.500 --> 00:29:56.730 align:middle line:90% So now we can find the d sigma. 00:29:56.730 --> 00:30:03.200 align:middle line:84% So the d sigma alpha 2 beta, then 00:30:03.200 --> 00:30:10.690 align:middle line:84% is defined to be dp/dt now divided 00:30:10.690 --> 00:30:14.710 align:middle line:90% by the flux of 1, particle 1. 00:30:14.710 --> 00:30:17.830 align:middle line:90% So now, this is just given by-- 00:30:17.830 --> 00:30:19.660 align:middle line:90% now the volume cancel. 00:30:19.660 --> 00:30:24.260 align:middle line:90% Because n1 is 1 over volume. 00:30:24.260 --> 00:30:26.020 align:middle line:90% And then, there's a volume here. 00:30:26.020 --> 00:30:29.110 align:middle line:84% And now finally, this volume cancel. 00:30:29.110 --> 00:30:32.860 align:middle line:84% So now, if we divide that by the flux and the volume cancel, 00:30:32.860 --> 00:30:41.590 align:middle line:84% and then we get m alpha beta squared d mu. 00:30:41.590 --> 00:30:46.570 align:middle line:84% And then divide it by E1E2, and v1, 00:30:46.570 --> 00:30:50.800 align:middle line:90% the velocity of the particle 1. 00:30:50.800 --> 00:30:54.120 align:middle line:90% 00:30:54.120 --> 00:30:57.510 align:middle line:84% So v1, of course, can also be interpreted 00:30:57.510 --> 00:31:01.620 align:middle line:84% as a relative velocity between 1 and 2. 00:31:01.620 --> 00:31:06.750 align:middle line:84% Anyway, so this is the expression we find. 00:31:06.750 --> 00:31:09.990 align:middle line:84% So this is the expression in the rest frame of particle 2. 00:31:09.990 --> 00:31:13.160 align:middle line:90% 00:31:13.160 --> 00:31:23.250 align:middle line:84% So by definition, we want this thing 00:31:23.250 --> 00:31:28.200 align:middle line:84% to be Lorentz invariant and the symmetric in 1 and 2. 00:31:28.200 --> 00:31:29.130 align:middle line:90% Yes? 00:31:29.130 --> 00:31:30.755 align:middle line:84% AUDIENCE: I was thinking like regarding 00:31:30.755 --> 00:31:33.900 align:middle line:84% the cancellation of the 1 over V factor, 00:31:33.900 --> 00:31:36.000 align:middle line:84% is another way to interpret it like, you 00:31:36.000 --> 00:31:37.820 align:middle line:84% think of instead of a plain wave, 00:31:37.820 --> 00:31:42.053 align:middle line:84% like a narrow wave packet or something centered at k1. 00:31:42.053 --> 00:31:43.970 align:middle line:84% So you would get like a V from the wave packet 00:31:43.970 --> 00:31:45.595 align:middle line:84% sensitive states and that would cancel. 00:31:45.595 --> 00:31:46.460 align:middle line:90% Is that valid? 00:31:46.460 --> 00:31:50.170 align:middle line:90% 00:31:50.170 --> 00:31:55.070 align:middle line:84% HONG LIU: Yeah, I think it's the similar idea. 00:31:55.070 --> 00:31:56.780 align:middle line:90% Because let me see-- 00:31:56.780 --> 00:31:59.840 align:middle line:90% 00:31:59.840 --> 00:32:02.750 align:middle line:84% yeah, because essentially, you get rid of that V. 00:32:02.750 --> 00:32:07.160 align:middle line:84% When you consider the wave packet, 00:32:07.160 --> 00:32:10.610 align:middle line:84% then you get rid of this V. And then, you also 00:32:10.610 --> 00:32:13.640 align:middle line:84% get rid of this V. So you just get 00:32:13.640 --> 00:32:18.860 align:middle line:84% rid of the V one upstairs one downstairs, yeah, that's right. 00:32:18.860 --> 00:32:21.660 align:middle line:90% Good. 00:32:21.660 --> 00:32:24.830 align:middle line:84% So now, we want to write this in the Lorentz invariant form. 00:32:24.830 --> 00:32:27.990 align:middle line:84% So this is emphasized this is in the rest frame of 2. 00:32:27.990 --> 00:32:31.650 align:middle line:90% 00:32:31.650 --> 00:32:36.420 align:middle line:84% So now, I look for object, so now, I 00:32:36.420 --> 00:32:47.100 align:middle line:84% want to look for object, which I will call it sigma. 00:32:47.100 --> 00:32:51.370 align:middle line:90% So sigma is Lorentz invariant. 00:32:51.370 --> 00:32:57.970 align:middle line:90% 00:32:57.970 --> 00:33:01.300 align:middle line:84% And let me write here, sigma is Lorentz invariant. 00:33:01.300 --> 00:33:07.790 align:middle line:90% 00:33:07.790 --> 00:33:25.840 align:middle line:84% And the symmetric in 1 and the 2 and in the rest frame of 2, 00:33:25.840 --> 00:33:27.940 align:middle line:84% then the sigma becomes this downstairs, 00:33:27.940 --> 00:33:33.410 align:middle line:90% become this E1E2 times V1. 00:33:33.410 --> 00:33:37.310 align:middle line:84% So this is not the manifest Lorentz invariant object. 00:33:37.310 --> 00:33:40.390 align:middle line:84% It's also not symmetric in 1 or 2. 00:33:40.390 --> 00:33:42.940 align:middle line:84% But that we should be able to find the object 00:33:42.940 --> 00:33:46.210 align:middle line:84% sigma that by itself, is a Lorentz invariant, 00:33:46.210 --> 00:33:49.660 align:middle line:84% symmetric in 1 and 2 and in the rest frame of 2, 00:33:49.660 --> 00:33:52.690 align:middle line:90% reduced to this object. 00:33:52.690 --> 00:33:54.340 align:middle line:84% If I can find this object, then I'm 00:33:54.340 --> 00:33:57.690 align:middle line:84% done in finding the cross section. 00:33:57.690 --> 00:33:58.720 align:middle line:90% Yes? 00:33:58.720 --> 00:34:02.860 align:middle line:84% AUDIENCE: Should there be a factor of 4 in the denominator? 00:34:02.860 --> 00:34:04.900 align:middle line:84% HONG LIU: There should be a factor of 4-- 00:34:04.900 --> 00:34:06.350 align:middle line:90% yeah, there should be. 00:34:06.350 --> 00:34:08.420 align:middle line:90% Thank you. 00:34:08.420 --> 00:34:12.969 align:middle line:90% Yeah, yes, some other questions? 00:34:12.969 --> 00:34:13.810 align:middle line:90% Good. 00:34:13.810 --> 00:34:18.250 align:middle line:84% So I want to look for this object sigma. 00:34:18.250 --> 00:34:23.199 align:middle line:84% And so you can do a little bit trial and error 00:34:23.199 --> 00:34:24.940 align:middle line:90% to find such a sigma. 00:34:24.940 --> 00:34:27.370 align:middle line:84% So I will just write down the answer for you. 00:34:27.370 --> 00:34:34.060 align:middle line:84% And to save you the trouble, I could have put it in your pset, 00:34:34.060 --> 00:34:36.909 align:middle line:90% but I decided not to do it. 00:34:36.909 --> 00:34:39.460 align:middle line:84% So turned out the sigma is given by-- 00:34:39.460 --> 00:34:42.949 align:middle line:84% so when you write down the answer, it's very simple. 00:34:42.949 --> 00:34:46.130 align:middle line:84% You can almost guess it in a sense. 00:34:46.130 --> 00:34:54.969 align:middle line:84% So P1 dot P2 squared minus m1 squared times m2 squared. 00:34:54.969 --> 00:35:01.030 align:middle line:84% m1 and m2 are the mass of the particle 1 and 2. 00:35:01.030 --> 00:35:05.900 align:middle line:84% So this object satisfies these three conditions, 00:35:05.900 --> 00:35:08.210 align:middle line:84% this object satisfies these conditions. 00:35:08.210 --> 00:35:11.000 align:middle line:84% Also, I will leave the exercise for yourself to check it. 00:35:11.000 --> 00:35:12.450 align:middle line:90% It's very easy to check it. 00:35:12.450 --> 00:35:14.750 align:middle line:84% So you just go to the rest frame of 2. 00:35:14.750 --> 00:35:16.850 align:middle line:84% And then you can check if it's reduced to that. 00:35:16.850 --> 00:35:20.620 align:middle line:90% 00:35:20.620 --> 00:35:25.260 align:middle line:84% So now, we almost have our final answer for the cross section 00:35:25.260 --> 00:35:31.660 align:middle line:84% so the d sigma for these 2 to n scattering. 00:35:31.660 --> 00:35:48.560 align:middle line:90% 00:35:48.560 --> 00:35:52.200 align:middle line:84% So now, we just collect our final result. 00:35:52.200 --> 00:35:56.810 align:middle line:84% So we have the sigma alpha beta now, 00:35:56.810 --> 00:36:04.540 align:middle line:84% is equal to m alpha beta squared, d mu divided 00:36:04.540 --> 00:36:10.240 align:middle line:90% by 4 sigma, just d mu. 00:36:10.240 --> 00:36:13.030 align:middle line:84% So this is just our final answer. 00:36:13.030 --> 00:36:18.910 align:middle line:84% And the sigma is given by that guy, a capital sigma just given 00:36:18.910 --> 00:36:19.570 align:middle line:90% by that guy. 00:36:19.570 --> 00:36:23.020 align:middle line:90% 00:36:23.020 --> 00:36:26.290 align:middle line:84% And so, this is manifested in Lorenz invariant. 00:36:26.290 --> 00:36:30.400 align:middle line:84% And the manifest symmetric on the 1 and 2, 00:36:30.400 --> 00:36:32.965 align:middle line:84% because sigma is obviously symmetric on the 1 and 2. 00:36:32.965 --> 00:36:37.510 align:middle line:90% 00:36:37.510 --> 00:36:40.600 align:middle line:84% And this is a very beautiful formula. 00:36:40.600 --> 00:36:43.720 align:middle line:84% Even though we went through a lot of trouble, 00:36:43.720 --> 00:36:46.472 align:middle line:90% went through a lot of Vs and Ts. 00:36:46.472 --> 00:36:48.430 align:middle line:84% But in the end, we get a very beautiful answer. 00:36:48.430 --> 00:36:54.290 align:middle line:90% 00:36:54.290 --> 00:36:56.540 align:middle line:84% So now, let's can see some kinematic regimes 00:36:56.540 --> 00:36:57.245 align:middle line:90% of this formula. 00:36:57.245 --> 00:37:00.460 align:middle line:90% 00:37:00.460 --> 00:37:02.835 align:middle line:84% So now, let's talk about some kinematics of this formula. 00:37:02.835 --> 00:37:13.780 align:middle line:90% 00:37:13.780 --> 00:37:16.430 align:middle line:90% So it's convenient. 00:37:16.430 --> 00:37:18.760 align:middle line:84% So we have two particles, two initial particles. 00:37:18.760 --> 00:37:22.840 align:middle line:84% It's convenient to introduce the center of mass energy 00:37:22.840 --> 00:37:26.890 align:middle line:90% for the full system. 00:37:26.890 --> 00:37:30.100 align:middle line:84% So we can define something called s, small s. 00:37:30.100 --> 00:37:33.830 align:middle line:84% This is not big S. So big S is reserved for action. 00:37:33.830 --> 00:37:37.600 align:middle line:84% So this is small s P1 plus P2 squared. 00:37:37.600 --> 00:37:43.180 align:middle line:90% 00:37:43.180 --> 00:37:48.190 align:middle line:84% So P1 plus P2 is that of the total momentum 00:37:48.190 --> 00:37:49.390 align:middle line:90% of the initial state. 00:37:49.390 --> 00:37:52.780 align:middle line:84% Of course, it's also the total momentum of your final state, 00:37:52.780 --> 00:37:55.450 align:middle line:84% just from the momentum conservation. 00:37:55.450 --> 00:37:59.500 align:middle line:84% And then if I look at P1 and P2 squared, 00:37:59.500 --> 00:38:02.380 align:middle line:84% and then this is just essentially the invariant mass 00:38:02.380 --> 00:38:06.685 align:middle line:90% for your full system. 00:38:06.685 --> 00:38:09.280 align:middle line:90% 00:38:09.280 --> 00:38:11.050 align:middle line:90% So this s is-- 00:38:11.050 --> 00:38:16.030 align:middle line:84% the square root of s is the invariant mass. 00:38:16.030 --> 00:38:26.140 align:middle line:84% It's the effective mass of the whole system. 00:38:26.140 --> 00:38:29.890 align:middle line:90% 00:38:29.890 --> 00:38:32.650 align:middle line:84% Of this though, when I say the whole system, 00:38:32.650 --> 00:38:37.390 align:middle line:84% I just mean, the whole system of particle 1, particle 2, 00:38:37.390 --> 00:38:38.500 align:middle line:90% and also the final state. 00:38:38.500 --> 00:38:46.660 align:middle line:90% 00:38:46.660 --> 00:38:52.060 align:middle line:84% So we can also write this sigma, turns out 00:38:52.060 --> 00:38:56.760 align:middle line:84% we can actually write sigma in terms of s, because the-- 00:38:56.760 --> 00:39:08.340 align:middle line:90% 00:39:08.340 --> 00:39:15.330 align:middle line:84% so because P1 dot P2 can be written as 1/2 half 00:39:15.330 --> 00:39:23.170 align:middle line:84% P1 plus P2 squared minus P1 squared minus P2 square. 00:39:23.170 --> 00:39:25.180 align:middle line:90% So this is just m squared. 00:39:25.180 --> 00:39:27.220 align:middle line:84% So P1 squared is minus m1 squared. 00:39:27.220 --> 00:39:29.510 align:middle line:90% P2 squared is minus m2 squared. 00:39:29.510 --> 00:39:37.380 align:middle line:84% So this is just equal to minus 1/2 s-- 00:39:37.380 --> 00:39:40.710 align:middle line:90% sorry, it should be totaled. 00:39:40.710 --> 00:39:45.460 align:middle line:84% S minus m1 squared minus m2 square. 00:39:45.460 --> 00:39:50.230 align:middle line:84% so P1 and P2 squared is just equal to minus s. 00:39:50.230 --> 00:39:54.280 align:middle line:84% And then, the sigma just then, just 00:39:54.280 --> 00:39:56.620 align:middle line:84% can be expressed in terms of that. 00:39:56.620 --> 00:40:00.160 align:middle line:84% So the sigma-- so let me just write down the final expression 00:40:00.160 --> 00:40:05.830 align:middle line:84% for the sigma in terms of s, just equal to square root 00:40:05.830 --> 00:40:16.690 align:middle line:84% s squared 2s m1 squared plus m2 squared plus m1 squared minus 2 00:40:16.690 --> 00:40:18.005 align:middle line:90% squared squared. 00:40:18.005 --> 00:40:20.570 align:middle line:90% 00:40:20.570 --> 00:40:23.050 align:middle line:84% So this whole kinematic factor of sigma 00:40:23.050 --> 00:40:25.045 align:middle line:84% can be just expressed in terms of s. 00:40:25.045 --> 00:40:36.660 align:middle line:90% 00:40:36.660 --> 00:40:37.625 align:middle line:90% Any questions on this? 00:40:37.625 --> 00:41:01.470 align:middle line:90% 00:41:01.470 --> 00:41:06.180 align:middle line:84% So now, another thing is that often, we-- 00:41:06.180 --> 00:41:10.930 align:middle line:90% 00:41:10.930 --> 00:41:15.370 align:middle line:84% even though this formula can be used in any frame, 00:41:15.370 --> 00:41:19.150 align:middle line:84% but sometimes, depending on your question, 00:41:19.150 --> 00:41:21.910 align:middle line:84% the expression is simpler in some frame 00:41:21.910 --> 00:41:24.400 align:middle line:90% than some other frame. 00:41:24.400 --> 00:41:26.830 align:middle line:84% So one of the very frequently used frame 00:41:26.830 --> 00:41:28.720 align:middle line:84% is so-called the center of mass frame. 00:41:28.720 --> 00:41:33.430 align:middle line:90% 00:41:33.430 --> 00:41:35.110 align:middle line:84% It's called the center of mass frame. 00:41:35.110 --> 00:41:39.330 align:middle line:84% So in the center of mass frame, so the center of mass frame, 00:41:39.330 --> 00:41:46.810 align:middle line:84% essentially, it's in this frame that the total center of mass 00:41:46.810 --> 00:41:48.820 align:middle line:90% of the system does not move. 00:41:48.820 --> 00:41:52.750 align:middle line:84% So it means that the total momentum in the center 00:41:52.750 --> 00:41:55.440 align:middle line:90% of mass frame, total momentum-- 00:41:55.440 --> 00:41:58.410 align:middle line:90% 00:41:58.410 --> 00:42:00.680 align:middle line:90% let me see. 00:42:00.680 --> 00:42:07.265 align:middle line:84% The total momentum equal to p1 p2 is taken to be 0. 00:42:07.265 --> 00:42:10.340 align:middle line:90% 00:42:10.340 --> 00:42:14.060 align:middle line:84% So the full system is not moving. 00:42:14.060 --> 00:42:16.090 align:middle line:84% And so, that means we can take -- 00:42:16.090 --> 00:42:21.170 align:middle line:84% so means that P1 equal to minus P2, 00:42:21.170 --> 00:42:26.070 align:middle line:90% so equal to, let's call it Pcm. 00:42:26.070 --> 00:42:27.820 align:middle line:84% It just means the center of mass momentum. 00:42:27.820 --> 00:42:33.100 align:middle line:90% 00:42:33.100 --> 00:42:42.610 align:middle line:84% And now, you can find the Pcm from just solving the-- 00:42:42.610 --> 00:42:46.640 align:middle line:84% you can also express the Pcm in terms of the mass, 00:42:46.640 --> 00:42:53.270 align:middle line:84% it's because E1 plus E2 equal to square root of s. 00:42:53.270 --> 00:42:55.160 align:middle line:90% So this is the total. 00:42:55.160 --> 00:43:00.110 align:middle line:84% So in the center of mass frame, the total momentum is 0. 00:43:00.110 --> 00:43:05.190 align:middle line:90% Then that means that E1E2-- 00:43:05.190 --> 00:43:08.790 align:middle line:84% so here, there's no spatial part contribution. 00:43:08.790 --> 00:43:13.020 align:middle line:84% And here, you just have E1 plus E2 squared equal to s. 00:43:13.020 --> 00:43:17.040 align:middle line:84% So E1 plus E2 equal to square root s. 00:43:17.040 --> 00:43:31.910 align:middle line:84% And then you can actually solve for p by plus m1 squared 00:43:31.910 --> 00:43:39.440 align:middle line:84% plus m2 squared equal to square root of s. 00:43:39.440 --> 00:43:43.505 align:middle line:84% So you can now, solve the center of mass in terms of s. 00:43:43.505 --> 00:43:47.130 align:middle line:90% 00:43:47.130 --> 00:43:51.540 align:middle line:84% So this is a simple equation, which you can solve. 00:43:51.540 --> 00:43:54.720 align:middle line:84% So this is a middle school equation. 00:43:54.720 --> 00:43:58.020 align:middle line:84% But turns out the result is very simple. 00:43:58.020 --> 00:44:07.030 align:middle line:84% Turns out the Pcm, the magnitude when you solve this equation, 00:44:07.030 --> 00:44:14.280 align:middle line:84% you find that this is precisely equal to this sigma divided 00:44:14.280 --> 00:44:15.540 align:middle line:90% by square root s. 00:44:15.540 --> 00:44:23.940 align:middle line:84% So this is a very beautiful simple formula, 00:44:23.940 --> 00:44:27.060 align:middle line:84% given by sigma divided by square root of s. 00:44:27.060 --> 00:44:29.820 align:middle line:84% Or in other words, the sigma can be 00:44:29.820 --> 00:44:33.930 align:middle line:84% written in terms of the center of mass, momentum and magnitude 00:44:33.930 --> 00:44:40.373 align:middle line:84% times the square root s, times square root s. 00:44:40.373 --> 00:44:44.080 align:middle line:90% 00:44:44.080 --> 00:44:48.150 align:middle line:84% So now, we can simplify that the expression 00:44:48.150 --> 00:44:49.035 align:middle line:90% center of mass frame. 00:44:49.035 --> 00:44:53.910 align:middle line:90% 00:44:53.910 --> 00:45:24.870 align:middle line:84% So in the center of mass frame, now 00:45:24.870 --> 00:45:29.880 align:middle line:84% we have in the center of mass frame, 00:45:29.880 --> 00:45:33.910 align:middle line:84% now we have the sigma is equal to-- 00:45:33.910 --> 00:45:35.750 align:middle line:90% so let me just save. 00:45:35.750 --> 00:45:38.980 align:middle line:84% Yeah, just you should assume that the subscript alpha 00:45:38.980 --> 00:45:46.990 align:middle line:84% to beta, d mu, and then divided by 4, center of mass momentum, 00:45:46.990 --> 00:45:49.720 align:middle line:84% magnitude of center of mass of momentum times 00:45:49.720 --> 00:45:51.910 align:middle line:90% square root of s. 00:45:51.910 --> 00:45:54.865 align:middle line:84% So we have a very simple formula. 00:45:54.865 --> 00:46:15.580 align:middle line:90% 00:46:15.580 --> 00:46:19.630 align:middle line:84% So the most of the time, for most questions 00:46:19.630 --> 00:46:24.037 align:middle line:84% we are interested in, as you will see in next lecture, 00:46:24.037 --> 00:46:25.870 align:middle line:84% so most question we are interesting actually 00:46:25.870 --> 00:46:27.640 align:middle line:84% corresponding to 2 to 2 scattering. 00:46:27.640 --> 00:46:31.630 align:middle line:84% So now let's specifier 2 to 2 scatterings. 00:46:31.630 --> 00:46:37.660 align:middle line:84% So the final state only also only contain two particles. 00:46:37.660 --> 00:46:41.080 align:middle line:84% So in this case, we can simplify the d mu, 00:46:41.080 --> 00:46:42.745 align:middle line:90% further simplify d mu. 00:46:42.745 --> 00:46:46.230 align:middle line:90% 00:46:46.230 --> 00:46:47.850 align:middle line:90% So let's just write down the-- 00:46:47.850 --> 00:46:50.010 align:middle line:84% so in the 2 to 2 scattering, you essentially you 00:46:50.010 --> 00:46:58.350 align:middle line:84% have some particle come in, say p1 p2, 00:46:58.350 --> 00:47:01.320 align:middle line:84% and then you have two final particle k1, k2. 00:47:01.320 --> 00:47:04.350 align:middle line:90% 00:47:04.350 --> 00:47:05.520 align:middle line:90% And then they're intact. 00:47:05.520 --> 00:47:09.890 align:middle line:90% 00:47:09.890 --> 00:47:15.560 align:middle line:84% So now let's write down this d mu explicitly for this case. 00:47:15.560 --> 00:47:18.310 align:middle line:84% Let's write down d mu explicitly for this case. 00:47:18.310 --> 00:47:22.570 align:middle line:90% So the d mu, just given by-- 00:47:22.570 --> 00:47:26.990 align:middle line:90% 00:47:26.990 --> 00:47:31.175 align:middle line:90% so P1 plus P2 minus k1 minus k2. 00:47:31.175 --> 00:47:34.240 align:middle line:90% 00:47:34.240 --> 00:47:50.130 align:middle line:84% And then times, d3 k1 divided by 2 pi cubed 2k 1,0. 00:47:50.130 --> 00:47:58.350 align:middle line:84% Let's just call it 2 1 k0 E1 prime and d3 k2 2 pi 00:47:58.350 --> 00:48:01.830 align:middle line:90% cubed, then 2 E2 prime. 00:48:01.830 --> 00:48:05.070 align:middle line:90% 00:48:05.070 --> 00:48:07.090 align:middle line:84% So the E1 prime and E2 prime, they 00:48:07.090 --> 00:48:09.810 align:middle line:90% are just energy for k1 and k2. 00:48:09.810 --> 00:48:18.910 align:middle line:84% So E1 prime equal to k 1,0 E2 prime k2. 00:48:18.910 --> 00:48:24.090 align:middle line:90% 00:48:24.090 --> 00:48:36.015 align:middle line:84% For example, yeah, so I will deload 00:48:36.015 --> 00:48:39.950 align:middle line:84% to the mass for the two final particles, 00:48:39.950 --> 00:48:43.550 align:middle line:90% m1 prime and m2 prime. 00:48:43.550 --> 00:48:46.610 align:middle line:84% So m prime 2 prime will be the mass 00:48:46.610 --> 00:48:48.110 align:middle line:90% of the two final particles. 00:48:48.110 --> 00:48:58.390 align:middle line:90% 00:48:58.390 --> 00:49:03.110 align:middle line:84% So before simplifying this a little bit further, 00:49:03.110 --> 00:49:06.670 align:middle line:84% let us first introduce some notations. 00:49:06.670 --> 00:49:09.970 align:middle line:84% So it's often convenient so many of you 00:49:09.970 --> 00:49:13.000 align:middle line:84% may have already seen this before, 00:49:13.000 --> 00:49:17.680 align:middle line:84% often convenient to introduce the following quantities-- 00:49:17.680 --> 00:49:20.890 align:middle line:90% 00:49:20.890 --> 00:49:23.160 align:middle line:84% so essentially, they characterize all the Lorentz 00:49:23.160 --> 00:49:33.280 align:middle line:84% invariant quantities we can build up from k P1 P2, k1 k2. 00:49:33.280 --> 00:49:37.030 align:middle line:84% So you can have so-called t is defined 00:49:37.030 --> 00:49:42.130 align:middle line:84% to be minus p1 minus k1 square, which is also 00:49:42.130 --> 00:49:45.520 align:middle line:84% the same from the momentum conservation, p2 minus k2 00:49:45.520 --> 00:49:47.900 align:middle line:90% square. 00:49:47.900 --> 00:49:58.670 align:middle line:84% So u minus p1 minus k2 square, the same, minus p2 00:49:58.670 --> 00:49:59.750 align:middle line:90% minus k1 square. 00:49:59.750 --> 00:50:03.290 align:middle line:90% 00:50:03.290 --> 00:50:10.040 align:middle line:84% So let me just copy the s is p1 plus p2 squared. 00:50:10.040 --> 00:50:18.430 align:middle line:90% 00:50:18.430 --> 00:50:21.760 align:middle line:84% So these are the quantities, which are obviously 00:50:21.760 --> 00:50:23.910 align:middle line:90% Lorentz invariant. 00:50:23.910 --> 00:50:25.710 align:middle line:84% And they are the Lorentz invariant. 00:50:25.710 --> 00:50:27.990 align:middle line:90% You can build up-- 00:50:27.990 --> 00:50:29.670 align:middle line:84% Lorentz invariant quantities, you 00:50:29.670 --> 00:50:34.650 align:middle line:84% can build up from say, p1 p2, k1 and k2. 00:50:34.650 --> 00:50:38.100 align:middle line:84% So any Lorentz invariant quantities, 00:50:38.100 --> 00:50:42.780 align:middle line:84% which you can build from those four momentum, 00:50:42.780 --> 00:50:48.210 align:middle line:84% can be expressed in terms of some combinations of s t u. 00:50:48.210 --> 00:50:51.870 align:middle line:84% In fact, s to u themselves, are not independent of each other. 00:50:51.870 --> 00:50:55.140 align:middle line:84% There's only two independent Lorentz invariants. 00:50:55.140 --> 00:50:56.550 align:middle line:84% With momentum conservation, there 00:50:56.550 --> 00:51:00.840 align:middle line:84% are only two independent Lorentz invariants you can introduce. 00:51:00.840 --> 00:51:05.307 align:middle line:84% So s plus t and plus u is actually not 00:51:05.307 --> 00:51:06.390 align:middle line:90% independent of each other. 00:51:06.390 --> 00:51:08.010 align:middle line:90% You can show. 00:51:08.010 --> 00:51:10.545 align:middle line:84% So this again, I give you the exercise for yourself. 00:51:10.545 --> 00:51:18.210 align:middle line:90% 00:51:18.210 --> 00:51:20.400 align:middle line:90% You can show that. 00:51:20.400 --> 00:51:22.920 align:middle line:84% So if you know any of the two, then you know the s, 00:51:22.920 --> 00:51:24.270 align:middle line:90% you know the other one. 00:51:24.270 --> 00:51:32.120 align:middle line:84% And anything can be expressed in terms of these variables 00:51:32.120 --> 00:51:33.050 align:middle line:90% Any questions? 00:51:33.050 --> 00:51:36.790 align:middle line:90% 00:51:36.790 --> 00:51:37.978 align:middle line:90% Yes? 00:51:37.978 --> 00:51:41.950 align:middle line:84% AUDIENCE: Written down, definitions are useful, 00:51:41.950 --> 00:51:44.020 align:middle line:90% for instance, these. 00:51:44.020 --> 00:51:47.613 align:middle line:84% How do they correspond to different kinematics? 00:51:47.613 --> 00:51:49.280 align:middle line:84% HONG LIU: What do you mean, how do they? 00:51:49.280 --> 00:51:50.522 align:middle line:90% AUDIENCE: Feynman diagrams. 00:51:50.522 --> 00:51:51.980 align:middle line:84% HONG LIU: Oh yeah, then you Feynman 00:51:51.980 --> 00:51:56.060 align:middle line:84% diagrams can be conveniently expressed in terms of them, 00:51:56.060 --> 00:51:58.535 align:middle line:84% as we will see in next lecture and in homework. 00:51:58.535 --> 00:52:01.810 align:middle line:90% 00:52:01.810 --> 00:52:06.260 align:middle line:84% So this quantity this quantity is Lorentz invariant. 00:52:06.260 --> 00:52:08.740 align:middle line:84% So that can be conveniently expressed 00:52:08.740 --> 00:52:10.150 align:middle line:90% in terms of those quantities. 00:52:10.150 --> 00:52:19.080 align:middle line:90% 00:52:19.080 --> 00:52:26.520 align:middle line:84% So now, including the d mu, so now, let's further 00:52:26.520 --> 00:52:29.700 align:middle line:84% simplify d mu in the center of mass frame. 00:52:29.700 --> 00:52:33.160 align:middle line:84% So this quantity, we can try to simplify further. 00:52:33.160 --> 00:53:14.820 align:middle line:84% So we can-- so now, let's consider 00:53:14.820 --> 00:53:17.820 align:middle line:84% to do this in the center of mass frame. 00:53:17.820 --> 00:53:21.090 align:middle line:84% So in the center of mass frame, for 2 to 2 00:53:21.090 --> 00:53:23.760 align:middle line:84% scattering is particularly simple, 00:53:23.760 --> 00:53:28.380 align:middle line:84% because your incoming particle have the opposite momentum. 00:53:28.380 --> 00:53:30.750 align:middle line:90% So one is Pcm. 00:53:30.750 --> 00:53:39.180 align:middle line:84% And the y will be minus Pcm and the spatial momentum. 00:53:39.180 --> 00:53:42.360 align:middle line:84% And then the final particle, they 00:53:42.360 --> 00:53:45.990 align:middle line:84% must also have opposite momentum. 00:53:45.990 --> 00:53:48.990 align:middle line:84% Because the total momentum, total spatial momentum 00:53:48.990 --> 00:53:52.170 align:middle line:84% in the center of mass frame is 0. 00:53:52.170 --> 00:53:57.240 align:middle line:84% So let's can call this k cm then this 00:53:57.240 --> 00:54:05.960 align:middle line:84% must be minus k cm OK, must be minus cm. 00:54:05.960 --> 00:54:08.780 align:middle line:84% So for simplicity, let's just call this k cm just k. 00:54:08.780 --> 00:54:13.100 align:middle line:90% 00:54:13.100 --> 00:54:16.820 align:middle line:84% So now, we can just -- now let's just with this in mind, 00:54:16.820 --> 00:54:20.420 align:middle line:84% and let's just look at this d mu. 00:54:20.420 --> 00:54:21.920 align:middle line:90% Look at this d mu. 00:54:21.920 --> 00:54:26.950 align:middle line:90% 00:54:26.950 --> 00:54:30.290 align:middle line:90% So there are a few steps. 00:54:30.290 --> 00:54:34.360 align:middle line:84% So maybe we don't want to go through all 00:54:34.360 --> 00:54:35.965 align:middle line:90% the details of these steps. 00:54:35.965 --> 00:54:41.240 align:middle line:90% 00:54:41.240 --> 00:54:41.740 align:middle line:90% Let me. 00:54:41.740 --> 00:54:48.880 align:middle line:84% See I think, yeah, let me just outline some here. 00:54:48.880 --> 00:54:53.500 align:middle line:84% So we'll try to only go through some of the steps, 00:54:53.500 --> 00:54:56.770 align:middle line:90% not to doing all the steps. 00:54:56.770 --> 00:55:01.480 align:middle line:84% So now in the center of mass frame, then 00:55:01.480 --> 00:55:05.720 align:middle line:84% you find this d mu can be written as-- 00:55:05.720 --> 00:55:08.600 align:middle line:90% so first, these two pi forth. 00:55:08.600 --> 00:55:11.120 align:middle line:84% And then you have two factor of 2 pi cubed. 00:55:11.120 --> 00:55:12.980 align:middle line:90% And then you cancel. 00:55:12.980 --> 00:55:14.645 align:middle line:90% So you have this 2 pi squared. 00:55:14.645 --> 00:55:17.390 align:middle line:90% 00:55:17.390 --> 00:55:23.870 align:middle line:84% Then you have 4 E1 prime E2 prime. 00:55:23.870 --> 00:55:25.880 align:middle line:84% Then you have one set of functions. 00:55:25.880 --> 00:55:32.450 align:middle line:84% So you have k1, you have E1 prime plus E2 00:55:32.450 --> 00:55:41.000 align:middle line:84% prime minus square root s times delta 3 k1 plus 00:55:41.000 --> 00:55:48.950 align:middle line:90% k2 and the d3 k1 and d3 k2. 00:55:48.950 --> 00:55:53.210 align:middle line:84% So as we said, the k3, they have to be equal and opposite 00:55:53.210 --> 00:55:53.795 align:middle line:90% to each other. 00:55:53.795 --> 00:55:56.340 align:middle line:90% 00:55:56.340 --> 00:55:58.880 align:middle line:84% And then you can just evaluate one delta function. 00:55:58.880 --> 00:56:01.550 align:middle line:84% And then the other dk will remain. 00:56:01.550 --> 00:56:04.960 align:middle line:90% The other dk will remain. 00:56:04.960 --> 00:56:08.590 align:middle line:90% So essentially, you just have-- 00:56:08.590 --> 00:56:11.220 align:middle line:84% essentially, we can just forget about this. 00:56:11.220 --> 00:56:17.770 align:middle line:84% And then you just have dk, just have dk, d3k. 00:56:17.770 --> 00:56:20.800 align:middle line:90% 00:56:20.800 --> 00:56:27.480 align:middle line:84% And this d3k, you can write it as dk, 00:56:27.480 --> 00:56:33.000 align:middle line:84% the magnitude of k and k square, and then the center of mass -- 00:56:33.000 --> 00:56:36.210 align:middle line:84% solid angle the angle in the center of mass frame. 00:56:36.210 --> 00:56:40.170 align:middle line:84% So this k vector can be decomposed into the angular 00:56:40.170 --> 00:56:43.290 align:middle line:90% directions and the magnitude. 00:56:43.290 --> 00:56:48.370 align:middle line:84% And then, now both E1 and E2 are expressed 00:56:48.370 --> 00:56:53.280 align:middle line:84% in terms of k squared plus m1 prime squared 00:56:53.280 --> 00:57:00.810 align:middle line:84% and the E2 prime is equal to k squared plus m2 prime squared. 00:57:00.810 --> 00:57:10.370 align:middle line:90% 00:57:10.370 --> 00:57:14.105 align:middle line:84% So, now you can further evaluate this delta function. 00:57:14.105 --> 00:57:18.470 align:middle line:90% 00:57:18.470 --> 00:57:22.190 align:middle line:84% Now you can further evaluate this function. 00:57:22.190 --> 00:57:24.950 align:middle line:84% Because the E1 prime and E2 prime, 00:57:24.950 --> 00:57:28.450 align:middle line:84% they're just the functions of magnitude of k squared. 00:57:28.450 --> 00:57:31.180 align:middle line:84% So you can evaluate this delta function. 00:57:31.180 --> 00:57:36.130 align:middle line:84% You have dk over the magnitude of k here. 00:57:36.130 --> 00:57:39.700 align:middle line:84% So you can do that, so I will not go into detail. 00:57:39.700 --> 00:57:42.140 align:middle line:90% I will not going to detail. 00:57:42.140 --> 00:57:46.430 align:middle line:84% And so, when you do that, when you solve that delta function 00:57:46.430 --> 00:57:50.850 align:middle line:84% and then you find from here, then you 00:57:50.850 --> 00:57:54.850 align:middle line:84% find that the final answer is given by-- 00:57:54.850 --> 00:57:57.800 align:middle line:84% you find the final answer is given by very simple. 00:57:57.800 --> 00:58:07.730 align:middle line:84% You find given by k cm divided by 14 pi square square root s 00:58:07.730 --> 00:58:09.715 align:middle line:90% d omega center of mass. 00:58:09.715 --> 00:58:13.390 align:middle line:90% 00:58:13.390 --> 00:58:18.640 align:middle line:84% And this k cm is defined to be the solution of this equation. 00:58:18.640 --> 00:58:29.590 align:middle line:84% So k cm is the solution to E1 prime plus E2 00:58:29.590 --> 00:58:32.680 align:middle line:90% prime equal to square root s. 00:58:32.680 --> 00:58:35.755 align:middle line:84% So again, this is expressed in terms of the s. 00:58:35.755 --> 00:58:43.180 align:middle line:90% 00:58:43.180 --> 00:58:46.450 align:middle line:84% So through some, just technically, 00:58:46.450 --> 00:58:47.905 align:middle line:84% just evaluate this delta function. 00:58:47.905 --> 00:58:51.370 align:middle line:90% 00:58:51.370 --> 00:58:52.000 align:middle line:90% Yes? 00:58:52.000 --> 00:58:54.587 align:middle line:84% AUDIENCE: So when we calculate this, 00:58:54.587 --> 00:58:56.170 align:middle line:84% are we integrating over all the cases? 00:58:56.170 --> 00:59:00.790 align:middle line:84% Is that why we can evaluate all these type of functions? 00:59:00.790 --> 00:59:02.440 align:middle line:84% HONG LIU: So you only evaluate the k 00:59:02.440 --> 00:59:05.615 align:middle line:90% around the momentum shell. 00:59:05.615 --> 00:59:07.240 align:middle line:84% So you can evaluate the delta function, 00:59:07.240 --> 00:59:10.330 align:middle line:84% because we always have a finite range of k to integrate over. 00:59:10.330 --> 00:59:14.380 align:middle line:84% So it doesn't matter how wide the range of this. 00:59:14.380 --> 00:59:18.805 align:middle line:84% Because the non-zero value of the mu is always around that 00:59:18.805 --> 00:59:21.070 align:middle line:84% will satisfy the momentum conservation. 00:59:21.070 --> 00:59:23.898 align:middle line:90% 00:59:23.898 --> 00:59:25.190 align:middle line:90% Does this answer your question? 00:59:25.190 --> 00:59:26.800 align:middle line:90% AUDIENCE: Thanks. 00:59:26.800 --> 00:59:28.340 align:middle line:90% HONG LIU: Yeah, good. 00:59:28.340 --> 00:59:30.170 align:middle line:90% Other questions? 00:59:30.170 --> 00:59:31.850 align:middle line:90% Here, I didn't write-- 00:59:31.850 --> 00:59:36.860 align:middle line:84% I think, you may wonder here, I didn't write the integral sign. 00:59:36.860 --> 00:59:40.950 align:middle line:84% How can I just evaluate the delta function? 00:59:40.950 --> 00:59:43.010 align:middle line:84% The reason you can evaluate delta function 00:59:43.010 --> 00:59:48.350 align:middle line:84% is just no matter what is the range of dk you integrate over, 00:59:48.350 --> 00:59:52.670 align:middle line:84% you will always evaluate around those delta functions. 00:59:52.670 --> 00:59:55.660 align:middle line:84% And so you can always take care of them. 00:59:55.660 --> 00:59:59.310 align:middle line:90% 00:59:59.310 --> 01:00:09.270 align:middle line:84% So now, if you combine this result and this result, 01:00:09.270 --> 01:00:14.030 align:middle line:84% and now, you can write a simple expression for d sigma 01:00:14.030 --> 01:00:17.810 align:middle line:90% in the center of mass frame. 01:00:17.810 --> 01:00:19.550 align:middle line:84% So if you combine them together, then we 01:00:19.550 --> 01:00:27.080 align:middle line:84% find the sigma, the omega center of mass frame, 01:00:27.080 --> 01:00:36.605 align:middle line:84% is equal to M alpha beta squared divided by 16, 01:00:36.605 --> 01:00:45.100 align:middle line:84% 64 pi square s and k cm divided by pcm. 01:00:45.100 --> 01:00:49.240 align:middle line:84% And the k cm is the solution to this equation. 01:00:49.240 --> 01:00:53.340 align:middle line:84% And the pcm is the solution to this equation. 01:00:53.340 --> 01:00:57.770 align:middle line:84% So the pcm is the equation to this equation. 01:00:57.770 --> 01:00:59.780 align:middle line:84% So they are given by the same equation, 01:00:59.780 --> 01:01:05.420 align:middle line:84% just replace the m1 m2 by m1 prime and m2 prime. 01:01:05.420 --> 01:01:09.800 align:middle line:84% So this is the final answer for the differential cross section 01:01:09.800 --> 01:01:15.560 align:middle line:84% for 2 to 2 scattering in the center of mass frame, 01:01:15.560 --> 01:01:19.310 align:middle line:90% in the center of mass frame. 01:01:19.310 --> 01:01:21.570 align:middle line:84% And so this is the expression, which 01:01:21.570 --> 01:01:25.540 align:middle line:90% we will use later for certain-- 01:01:25.540 --> 01:01:30.290 align:middle line:90% 01:01:30.290 --> 01:01:31.460 align:middle line:90% do you have any questions? 01:01:31.460 --> 01:01:35.490 align:middle line:90% 01:01:35.490 --> 01:01:37.200 align:middle line:90% OK, good. 01:01:37.200 --> 01:01:43.660 align:middle line:84% So this concludes the discussion of the cross section. 01:01:43.660 --> 01:01:45.670 align:middle line:90% So it's finally over. 01:01:45.670 --> 01:01:48.340 align:middle line:90% It's finally over. 01:01:48.340 --> 01:01:51.190 align:middle line:84% But we have to go through this, because this 01:01:51.190 --> 01:01:57.450 align:middle line:84% is the kind of thing which we compare with experiments. 01:01:57.450 --> 01:01:59.580 align:middle line:84% And in particular, if you calculate the total cross 01:01:59.580 --> 01:02:01.500 align:middle line:84% section, and then you can just integrate 01:02:01.500 --> 01:02:03.060 align:middle line:90% this over all the solid angle. 01:02:03.060 --> 01:02:08.260 align:middle line:90% 01:02:08.260 --> 01:02:12.950 align:middle line:84% So before now, talking about the physical process, 01:02:12.950 --> 01:02:15.560 align:middle line:84% there's one more thing we need to consider. 01:02:15.560 --> 01:02:22.820 align:middle line:84% So here, we consider the initial state to be two particles. 01:02:22.820 --> 01:02:25.850 align:middle line:84% Because we say we don't normally do scattering 01:02:25.850 --> 01:02:29.900 align:middle line:90% with more than two particles. 01:02:29.900 --> 01:02:35.825 align:middle line:84% But there's another situation, which still often could happen. 01:02:35.825 --> 01:02:38.630 align:middle line:90% 01:02:38.630 --> 01:02:42.410 align:middle line:84% So this is the situation, which initial state only 01:02:42.410 --> 01:02:43.175 align:middle line:90% have one particle. 01:02:43.175 --> 01:02:45.810 align:middle line:90% 01:02:45.810 --> 01:02:49.050 align:middle line:84% So when your initial state only have one particle, 01:02:49.050 --> 01:02:50.040 align:middle line:90% what do you have? 01:02:50.040 --> 01:02:50.753 align:middle line:90% AUDIENCE: Decay. 01:02:50.753 --> 01:02:52.920 align:middle line:84% HONG LIU: Yeah, then that is corresponding to decay. 01:02:52.920 --> 01:02:57.270 align:middle line:84% Yeah, so we can still have the situation. 01:02:57.270 --> 01:03:00.930 align:middle line:84% If you have an unstable particle, then we can decay. 01:03:00.930 --> 01:03:06.390 align:middle line:84% And then it's very important for many, many physical situations, 01:03:06.390 --> 01:03:07.560 align:middle line:90% to calculate the decay rate. 01:03:07.560 --> 01:03:13.880 align:middle line:90% 01:03:13.880 --> 01:03:18.560 align:middle line:84% So now we have p1, say, suppose this is the initial momentum. 01:03:18.560 --> 01:03:25.460 align:middle line:84% And decay into k1 plus kn, say final n particles. 01:03:25.460 --> 01:03:28.250 align:middle line:84% So now the initial state, only one particle. 01:03:28.250 --> 01:03:31.190 align:middle line:90% And the final state is k1 kn. 01:03:31.190 --> 01:03:35.870 align:middle line:84% So beta remain the same, but alpha only one particle. 01:03:35.870 --> 01:03:42.895 align:middle line:84% And the decay rate is much simpler to define, 01:03:42.895 --> 01:03:49.770 align:middle line:90% it's just the dP alpha beta dt. 01:03:49.770 --> 01:03:53.000 align:middle line:84% And we don't need to divide it by flux, all those things. 01:03:53.000 --> 01:03:57.290 align:middle line:90% 01:03:57.290 --> 01:04:00.690 align:middle line:84% So now let me explain a little bit. 01:04:00.690 --> 01:04:03.815 align:middle line:84% Again, this dP alpha beta, we always mean-- 01:04:03.815 --> 01:04:09.400 align:middle line:90% 01:04:09.400 --> 01:04:15.280 align:middle line:84% we mean that the probability of P1 01:04:15.280 --> 01:04:21.820 align:middle line:90% decaying into n final particles. 01:04:21.820 --> 01:04:28.350 align:middle line:90% 01:04:28.350 --> 01:04:44.330 align:middle line:84% with again, particle 1 with range, 01:04:44.330 --> 01:04:58.020 align:middle line:84% with the decay 1 around k1 and the decay around k2, et cetera, 01:04:58.020 --> 01:05:01.580 align:middle line:90% and decay n around kn. 01:05:01.580 --> 01:05:06.050 align:middle line:84% So when we write d alpha beta, you should imagine this. 01:05:06.050 --> 01:05:07.220 align:middle line:90% We already include that. 01:05:07.220 --> 01:05:11.120 align:middle line:90% 01:05:11.120 --> 01:05:16.380 align:middle line:84% So now, we can just repeat what we did before. 01:05:16.380 --> 01:05:26.670 align:middle line:84% So then, this dP alpha beta dt, dP alpha beta, then just given 01:05:26.670 --> 01:05:34.510 align:middle line:84% by this thing squared, beta plus infinity square and alpha minus 01:05:34.510 --> 01:05:43.390 align:middle line:84% infinity squared, divided by beta and alpha. 01:05:43.390 --> 01:05:58.550 align:middle line:84% And times j from 1 to n, d3 kj 2 pi cubed times V. 01:05:58.550 --> 01:06:00.890 align:middle line:90% So now, it's the same thing. 01:06:00.890 --> 01:06:02.870 align:middle line:90% Now, it's same thing. 01:06:02.870 --> 01:06:07.100 align:middle line:90% So the only thing you need to -- 01:06:07.100 --> 01:06:12.440 align:middle line:84% and the alpha one, everything else is the same. 01:06:12.440 --> 01:06:14.640 align:middle line:90% So everything else is the same. 01:06:14.640 --> 01:06:17.400 align:middle line:84% So you just repeat our previous analysis, 01:06:17.400 --> 01:06:19.530 align:middle line:90% which I will not repeat. 01:06:19.530 --> 01:06:22.130 align:middle line:84% So the only thing different is just now the initial state 01:06:22.130 --> 01:06:25.520 align:middle line:90% is just to a single energy. 01:06:25.520 --> 01:06:27.510 align:middle line:84% So you just repeat the whole thing. 01:06:27.510 --> 01:06:31.100 align:middle line:90% And then you find-- 01:06:31.100 --> 01:06:35.420 align:middle line:84% then you find d Gamma alpha beta equal 01:06:35.420 --> 01:06:43.330 align:middle line:90% to dP alpha beta divided by t. 01:06:43.330 --> 01:06:45.220 align:middle line:84% And then you find that this is just given 01:06:45.220 --> 01:06:55.180 align:middle line:90% by 2E1 alpha beta square d mu. 01:06:55.180 --> 01:06:58.060 align:middle line:90% D mu is defined in the same way. 01:06:58.060 --> 01:06:59.500 align:middle line:90% D mu is defined in the same way. 01:06:59.500 --> 01:07:03.130 align:middle line:90% 01:07:03.130 --> 01:07:08.110 align:middle line:84% So this is the final answer for the decay case. 01:07:08.110 --> 01:07:12.600 align:middle line:84% And the total decay rate would be you integrate just 01:07:12.600 --> 01:07:13.530 align:middle line:90% over all momentum. 01:07:13.530 --> 01:07:19.790 align:middle line:90% 01:07:19.790 --> 01:07:21.095 align:middle line:90% So the total decay rate-- 01:07:21.095 --> 01:07:50.940 align:middle line:90% 01:07:50.940 --> 01:08:00.350 align:middle line:84% so the total decay rate is just gamma total rate. 01:08:00.350 --> 01:08:04.760 align:middle line:84% Then gamma is summing over all possible choices of beta. 01:08:04.760 --> 01:08:06.785 align:middle line:84% And then you integrate over all momentum. 01:08:06.785 --> 01:08:11.090 align:middle line:90% 01:08:11.090 --> 01:08:18.859 align:middle line:84% And the lifetime of the particle, tau 01:08:18.859 --> 01:08:20.180 align:middle line:90% is just equal to 1 over gamma. 01:08:20.180 --> 01:08:26.800 align:middle line:90% 01:08:26.800 --> 01:08:32.600 align:middle line:84% So one thing, one difference with the cross section case-- 01:08:32.600 --> 01:08:35.540 align:middle line:84% so the cross action we mentioned before, 01:08:35.540 --> 01:08:38.850 align:middle line:90% say it's Lorentz invariant. 01:08:38.850 --> 01:08:41.130 align:middle line:90% But decay rate is not. 01:08:41.130 --> 01:08:45.160 align:middle line:84% And decay rate does depend on the frame of the particle, 01:08:45.160 --> 01:08:49.450 align:middle line:84% does depend on the frame of the particle. 01:08:49.450 --> 01:08:54.970 align:middle line:84% So decay rate-- so this lifetime does 01:08:54.970 --> 01:08:57.910 align:middle line:84% depend on the frame of the particle 01:08:57.910 --> 01:09:09.569 align:middle line:84% and the rest frame of the particle, 01:09:09.569 --> 01:09:12.660 align:middle line:84% corresponding to just E equal to m1, 01:09:12.660 --> 01:09:16.109 align:middle line:84% just equal to mass of the particle. 01:09:16.109 --> 01:09:23.729 align:middle line:84% And so the rest frame decay rate is the smallest 01:09:23.729 --> 01:09:28.470 align:middle line:84% among all possible decay rate because of the time dilation. 01:09:28.470 --> 01:09:33.010 align:middle line:90% 01:09:33.010 --> 01:09:37.500 align:middle line:84% In all other moving frame, because the time dilation, 01:09:37.500 --> 01:09:39.779 align:middle line:90% the decay rate become faster. 01:09:39.779 --> 01:09:45.090 align:middle line:84% So that's what you meant, that the particles are moving 01:09:45.090 --> 01:09:47.910 align:middle line:90% and they have a longer lifetime. 01:09:47.910 --> 01:09:50.990 align:middle line:90% They have a longer lifetime. 01:09:50.990 --> 01:09:55.390 align:middle line:90% So this makes perfect sense. 01:09:55.390 --> 01:09:58.890 align:middle line:90% So any questions on this? 01:09:58.890 --> 01:09:59.390 align:middle line:90% Yes? 01:09:59.390 --> 01:10:02.040 align:middle line:90% 01:10:02.040 --> 01:10:04.160 align:middle line:84% AUDIENCE: Do you need a finite set? 01:10:04.160 --> 01:10:08.120 align:middle line:84% HONG LIU: Yeah, in general, in general, for the real-- 01:10:08.120 --> 01:10:15.990 align:middle line:84% you never know, but beta is what we discovered. 01:10:15.990 --> 01:10:18.960 align:middle line:84% We observe what are the decay final product. 01:10:18.960 --> 01:10:21.550 align:middle line:84% But you can also predict, from your theory, 01:10:21.550 --> 01:10:23.790 align:middle line:90% what are the possible decay? 01:10:23.790 --> 01:10:25.770 align:middle line:84% But in real experiment, there always 01:10:25.770 --> 01:10:29.160 align:middle line:84% may be some particles we don't know. 01:10:29.160 --> 01:10:32.790 align:middle line:84% There may be some hidden interactions. 01:10:32.790 --> 01:10:35.920 align:middle line:90% 01:10:35.920 --> 01:10:36.880 align:middle line:90% Other questions? 01:10:36.880 --> 01:10:39.420 align:middle line:90% 01:10:39.420 --> 01:10:41.870 align:middle line:90% OK, good. 01:10:41.870 --> 01:10:45.260 align:middle line:84% So that finally concludes this discussion of the cross section 01:10:45.260 --> 01:10:47.480 align:middle line:90% and the decay rate. 01:10:47.480 --> 01:10:51.330 align:middle line:84% And now, we can study some process. 01:10:51.330 --> 01:10:53.870 align:middle line:84% So we only have 10 minutes to do it. 01:10:53.870 --> 01:10:58.530 align:middle line:90% 01:10:58.530 --> 01:11:05.630 align:middle line:84% So we will not really be able to do it, just maybe to start it. 01:11:05.630 --> 01:11:18.150 align:middle line:90% 01:11:18.150 --> 01:11:22.630 align:middle line:84% So in general, we will consider 2 to 2 scatterings. 01:11:22.630 --> 01:11:33.550 align:middle line:84% OK In general, we consider 2 to 2 scatterings. 01:11:33.550 --> 01:11:39.220 align:middle line:90% And so one remark to make-- 01:11:39.220 --> 01:11:41.380 align:middle line:84% and in that case, we have this formula. 01:11:41.380 --> 01:11:44.920 align:middle line:90% We have this very nice formula. 01:11:44.920 --> 01:12:03.220 align:middle line:84% So for particles, with spin, say whether spin 1/2 or spin 1. 01:12:03.220 --> 01:12:07.480 align:middle line:84% Spin 1/2 would be electrons, protons, et cetera. 01:12:07.480 --> 01:12:09.700 align:middle line:90% And spin 1 would be photon. 01:12:09.700 --> 01:12:15.370 align:middle line:84% And the scattering amplitude then 01:12:15.370 --> 01:12:30.610 align:middle line:84% will depend on the polarization of the initial and final 01:12:30.610 --> 01:12:31.420 align:middle line:90% particles. 01:12:31.420 --> 01:12:42.960 align:middle line:90% 01:12:42.960 --> 01:12:44.880 align:middle line:90% So in most experiments -- 01:12:44.880 --> 01:12:48.330 align:middle line:90% 01:12:48.330 --> 01:12:57.520 align:middle line:84% in most experiments, so the initial beam, 01:12:57.520 --> 01:13:00.490 align:middle line:84% they are unpolarized, it's not easy, 01:13:00.490 --> 01:13:02.500 align:middle line:84% sometimes to do the polarized beam. 01:13:02.500 --> 01:13:10.110 align:middle line:84% So initial state, initial beams are unpolarized-- 01:13:10.110 --> 01:13:14.140 align:middle line:90% 01:13:14.140 --> 01:13:16.210 align:middle line:84% unpolarized means they're just corresponding 01:13:16.210 --> 01:13:19.810 align:middle line:84% to a superposition of all possible kinds 01:13:19.810 --> 01:13:25.930 align:middle line:84% of polarizations, incoherent superposition. 01:13:25.930 --> 01:13:39.840 align:middle line:84% And then the final state and the spin polarization, 01:13:39.840 --> 01:13:50.110 align:middle line:84% a final state normally, it cannot be detected. 01:13:50.110 --> 01:13:53.500 align:middle line:90% 01:13:53.500 --> 01:13:56.650 align:middle line:84% It's a difficult question to detect 01:13:56.650 --> 01:13:58.650 align:middle line:90% the polarization of a particle. 01:13:58.650 --> 01:14:01.450 align:middle line:90% 01:14:01.450 --> 01:14:04.030 align:middle line:90% And actually, it's not even-- 01:14:04.030 --> 01:14:06.550 align:middle line:84% say if you say, observe a new particle, 01:14:06.550 --> 01:14:10.420 align:middle line:84% it's not even easy to tell whether this particle is 01:14:10.420 --> 01:14:12.010 align:middle line:90% a boson or fermion. 01:14:12.010 --> 01:14:16.600 align:middle line:84% So to measure the specific polarization is even harder. 01:14:16.600 --> 01:14:21.340 align:middle line:84% Anyway, so the polarization of final state 01:14:21.340 --> 01:14:24.260 align:middle line:90% are normally not detected. 01:14:24.260 --> 01:14:27.620 align:middle line:84% So in this case, when we compare with experiment, 01:14:27.620 --> 01:14:30.940 align:middle line:84% so when we calculate this kind of cross section 01:14:30.940 --> 01:14:38.035 align:middle line:84% compare with experiment, we should-- 01:14:38.035 --> 01:14:52.160 align:middle line:90% 01:14:52.160 --> 01:15:01.130 align:middle line:84% that means that we should to compare with experiment. 01:15:01.130 --> 01:15:09.890 align:middle line:84% We should sum over, should sum over 01:15:09.890 --> 01:15:16.895 align:middle line:90% polarizations of final state. 01:15:16.895 --> 01:15:24.720 align:middle line:90% 01:15:24.720 --> 01:15:38.535 align:middle line:84% and average over polarizations of initial state. 01:15:38.535 --> 01:15:43.440 align:middle line:90% 01:15:43.440 --> 01:15:46.500 align:middle line:84% So for the final state, we need to sum over them. 01:15:46.500 --> 01:15:49.880 align:middle line:84% Because we need to sum over all possibilities. 01:15:49.880 --> 01:15:54.350 align:middle line:84% But for the initial state all different polarizations 01:15:54.350 --> 01:15:55.070 align:middle line:90% contribute. 01:15:55.070 --> 01:15:57.470 align:middle line:90% So we need to average over them. 01:15:57.470 --> 01:15:58.730 align:middle line:90% We need to average over them. 01:15:58.730 --> 01:16:01.470 align:middle line:90% 01:16:01.470 --> 01:16:21.080 align:middle line:84% So for example, if we consider such a process, 01:16:21.080 --> 01:16:26.500 align:middle line:84% it's a very important process, say 01:16:26.500 --> 01:16:29.360 align:middle line:84% in QED is your annhilation of particles. 01:16:29.360 --> 01:16:31.540 align:middle line:84% So if you have a particle and antiparticle, 01:16:31.540 --> 01:16:33.990 align:middle line:90% then they can annihilate. 01:16:33.990 --> 01:16:38.250 align:middle line:84% When they annihilate, then they annihilate into photon. 01:16:38.250 --> 01:16:41.040 align:middle line:84% And then the photon can split into some other particle 01:16:41.040 --> 01:16:42.570 align:middle line:90% and antiparticle. 01:16:42.570 --> 01:16:45.960 align:middle line:84% So this is the process of particle a come in. 01:16:45.960 --> 01:16:54.370 align:middle line:90% 01:16:54.370 --> 01:16:59.770 align:middle line:84% So this is a, a-bar, this is b and b-bar. 01:16:59.770 --> 01:17:02.920 align:middle line:84% So this is a production, the pair creation 01:17:02.920 --> 01:17:07.270 align:middle line:84% of b and b-bar from colliding and its antiparticle. 01:17:07.270 --> 01:17:11.700 align:middle line:84% So in real life, in real life, by colliding, 01:17:11.700 --> 01:17:17.010 align:middle line:84% say, for example, electron plus positron, then 01:17:17.010 --> 01:17:19.120 align:middle line:90% you can create many particles. 01:17:19.120 --> 01:17:25.200 align:middle line:84% So this is one of the most important way 01:17:25.200 --> 01:17:27.450 align:middle line:90% to discover new particles. 01:17:27.450 --> 01:17:29.790 align:middle line:84% Many new particles are discovered this way. 01:17:29.790 --> 01:17:32.370 align:middle line:84% You just collide the electron and positron. 01:17:32.370 --> 01:17:35.820 align:middle line:84% And then you will be able to create new particles. 01:17:35.820 --> 01:17:41.190 align:middle line:84% For example, you can create muon and anti-muon. 01:17:41.190 --> 01:17:49.150 align:middle line:84% And then you can create quarks and antiquark, et cetera. 01:17:49.150 --> 01:17:57.980 align:middle line:90% 01:17:57.980 --> 01:18:01.790 align:middle line:84% So in all these cases, both the initial and final particles, 01:18:01.790 --> 01:18:03.710 align:middle line:90% they are fermions. 01:18:03.710 --> 01:18:05.880 align:middle line:84% So you need to specify their polarization. 01:18:05.880 --> 01:18:09.290 align:middle line:84% So this one -- supposed we have p1 r1. 01:18:09.290 --> 01:18:11.150 align:middle line:90% So here we have p2. 01:18:11.150 --> 01:18:18.240 align:middle line:84% So since this is antiparticle, let's call it r2-bar. 01:18:18.240 --> 01:18:20.910 align:middle line:90% And the b would be called-- 01:18:20.910 --> 01:18:23.460 align:middle line:90% say this is k1 s1. 01:18:23.460 --> 01:18:25.200 align:middle line:90% Say this is k2 s2-bar. 01:18:25.200 --> 01:18:29.940 align:middle line:90% 01:18:29.940 --> 01:18:33.130 align:middle line:84% And then for the unpolarized process, 01:18:33.130 --> 01:18:35.445 align:middle line:84% let me just write down one last formula. 01:18:35.445 --> 01:18:47.780 align:middle line:90% 01:18:47.780 --> 01:18:51.670 align:middle line:84% So you need to say, suppose the scattering amplitude is 01:18:51.670 --> 01:18:59.190 align:middle line:84% M squared, so M. And then we need to, 01:18:59.190 --> 01:19:09.300 align:middle line:84% for unpolarized process, so we need 01:19:09.300 --> 01:19:14.730 align:middle line:84% to consider-- we need to average initial spin, r1, and average 01:19:14.730 --> 01:19:17.835 align:middle line:90% over the final, the r2. 01:19:17.835 --> 01:19:20.730 align:middle line:90% 01:19:20.730 --> 01:19:26.730 align:middle line:84% And then we need to sum over s1 and the sum over s2. 01:19:26.730 --> 01:19:30.420 align:middle line:90% And then, I have Musk squared. 01:19:30.420 --> 01:19:33.990 align:middle line:84% So essentially, this becomes 1 over 4. 01:19:33.990 --> 01:19:37.740 align:middle line:84% And you sum over all possible spin polarizations 01:19:37.740 --> 01:19:40.050 align:middle line:90% of the M squared. 01:19:40.050 --> 01:19:45.330 align:middle line:84% So this is the one we can say compare with experiments. 01:19:45.330 --> 01:19:50.400 align:middle line:84% So next time, we will write this down explicitly 01:19:50.400 --> 01:19:51.460 align:middle line:90% for this kind of process. 01:19:51.460 --> 01:19:54.000 align:middle line:90% 01:19:54.000 --> 01:19:56.940 align:middle line:90% Yeah, so let's stop here. 01:19:56.940 --> 01:20:01.000 align:middle line:90%