WEBVTT 00:00:00.000 --> 00:00:01.948 align:middle line:90% [SQUEAKING] 00:00:01.948 --> 00:00:03.896 align:middle line:90% [RUSTLING] 00:00:03.896 --> 00:00:10.720 align:middle line:90% [CLICKING] 00:00:10.720 --> 00:00:11.950 align:middle line:90% HONG LIU: OK. 00:00:11.950 --> 00:00:12.450 align:middle line:90% Good. 00:00:12.450 --> 00:00:17.930 align:middle line:84% So last lecture, we talked about the property 00:00:17.930 --> 00:00:24.230 align:middle line:84% of this propagator, which is defined 00:00:24.230 --> 00:00:31.880 align:middle line:84% to be x, x prime which is also the same as phi x, 00:00:31.880 --> 00:00:34.910 align:middle line:90% phi x prime here. 00:00:34.910 --> 00:00:36.140 align:middle line:90% OK? 00:00:36.140 --> 00:00:40.010 align:middle line:90% So this object, we discussed. 00:00:40.010 --> 00:00:42.630 align:middle line:84% So conceptually, it can be considered-- 00:00:42.630 --> 00:00:45.330 align:middle line:90% so you have two interpretations. 00:00:45.330 --> 00:00:48.660 align:middle line:90% So one is from here. 00:00:48.660 --> 00:00:57.840 align:middle line:84% So this roughly can be heuristically interpreted 00:00:57.840 --> 00:01:16.380 align:middle line:84% as the transition amplitude for a particle from x to x prime. 00:01:16.380 --> 00:01:20.960 align:middle line:84% OK, so heuristically, you can think of it, say, 00:01:20.960 --> 00:01:22.460 align:middle line:90% if you have a point-- 00:01:22.460 --> 00:01:26.090 align:middle line:84% say, if you have a space-time point, x 00:01:26.090 --> 00:01:28.550 align:middle line:90% prime has another points x. 00:01:28.550 --> 00:01:32.510 align:middle line:84% And then you ask, what's the transition amplitude 00:01:32.510 --> 00:01:34.894 align:middle line:90% from x prime to x? 00:01:34.894 --> 00:01:40.700 align:middle line:84% OK, so one way to interpret this object is that. 00:01:40.700 --> 00:01:45.260 align:middle line:84% And as I emphasize, this is only heuristically, 00:01:45.260 --> 00:01:48.080 align:middle line:84% because this does not really describe 00:01:48.080 --> 00:01:49.880 align:middle line:90% a generally localized state. 00:01:49.880 --> 00:01:55.770 align:middle line:84% It's only a approximately localized state, OK? 00:01:55.770 --> 00:02:01.770 align:middle line:84% And similarly, the second interpretation 00:02:01.770 --> 00:02:03.315 align:middle line:90% is the correlation function. 00:02:03.315 --> 00:02:10.130 align:middle line:90% 00:02:10.130 --> 00:02:25.270 align:middle line:84% It describes correlations of phi between, say, x. 00:02:25.270 --> 00:02:30.700 align:middle line:84% Yeah, just correlations, square correlations 00:02:30.700 --> 00:02:39.190 align:middle line:84% between the value of phi at x prime and the value of phi 00:02:39.190 --> 00:02:39.755 align:middle line:90% at x. 00:02:39.755 --> 00:02:48.160 align:middle line:84% OK, so this is the interpretation most convenient, 00:02:48.160 --> 00:02:50.140 align:middle line:84% for example, from condensed matter perspective 00:02:50.140 --> 00:02:52.900 align:middle line:84% if your phi, say, describes average spins. 00:02:52.900 --> 00:02:53.620 align:middle line:90% OK. 00:02:53.620 --> 00:03:05.500 align:middle line:84% So similarly, the G minus, which is 00:03:05.500 --> 00:03:15.930 align:middle line:84% defined to be in the opposite order, 00:03:15.930 --> 00:03:20.040 align:middle line:90% also have two interpretations. 00:03:20.040 --> 00:03:26.130 align:middle line:84% So this is like a transition from x to x prime. 00:03:26.130 --> 00:03:31.170 align:middle line:84% Yeah, so if I draw x and if I still draw x that way, 00:03:31.170 --> 00:03:33.470 align:middle line:84% so it's like a transition amplitude like this, 00:03:33.470 --> 00:03:35.610 align:middle line:90% from x to x prime. 00:03:35.610 --> 00:03:37.350 align:middle line:90% OK? 00:03:37.350 --> 00:03:42.690 align:middle line:84% So you can similarly interpret the other using this language, 00:03:42.690 --> 00:03:57.530 align:middle line:84% for example, this Feynman, which is theta G plus x, x prime 00:03:57.530 --> 00:04:04.240 align:middle line:90% plus x minus. 00:04:04.240 --> 00:04:10.720 align:middle line:84% And then this means that when t is greater than t prime, 00:04:10.720 --> 00:04:13.660 align:middle line:84% say, for now, let's take this to be 00:04:13.660 --> 00:04:15.700 align:middle line:90% the direction of time going up. 00:04:15.700 --> 00:04:21.339 align:middle line:84% And now if the x has a later time than x prime, 00:04:21.339 --> 00:04:25.730 align:middle line:84% and then describe the transition from x prime to x. 00:04:25.730 --> 00:04:26.230 align:middle line:90% OK? 00:04:26.230 --> 00:04:34.930 align:middle line:84% But if the x prime has a larger time, and x has a smaller time, 00:04:34.930 --> 00:04:39.410 align:middle line:84% and then describe the opposite direction from x to x prime. 00:04:39.410 --> 00:04:42.340 align:middle line:84% But in both cases, you always go from the smaller time 00:04:42.340 --> 00:04:43.090 align:middle line:90% to bigger time. 00:04:43.090 --> 00:04:44.170 align:middle line:90% OK? 00:04:44.170 --> 00:04:47.170 align:middle line:84% And so we often call this, also GF, 00:04:47.170 --> 00:04:49.670 align:middle line:84% the time-ordered correlation function. 00:04:49.670 --> 00:04:52.270 align:middle line:90% So the GF is also called-- 00:04:52.270 --> 00:04:54.670 align:middle line:84% so it's called the Feynman-Green functions. 00:04:54.670 --> 00:05:01.480 align:middle line:84% It's also called the time-ordered correlation 00:05:01.480 --> 00:05:01.980 align:middle line:90% function. 00:05:01.980 --> 00:05:10.850 align:middle line:90% 00:05:10.850 --> 00:05:14.430 align:middle line:84% And then we wrote down the expression. 00:05:14.430 --> 00:05:18.410 align:middle line:84% So we calculate the expression explicitly for G plus. 00:05:18.410 --> 00:05:23.690 align:middle line:84% And we also almost finished the expression for-- 00:05:23.690 --> 00:05:26.780 align:middle line:84% yeah, similarly, you can do retarded et cetera. 00:05:26.780 --> 00:05:30.980 align:middle line:84% And also, so any questions on this? 00:05:30.980 --> 00:05:32.060 align:middle line:90% Yes? 00:05:32.060 --> 00:05:33.950 align:middle line:84% AUDIENCE: So on the set, we showed 00:05:33.950 --> 00:05:37.910 align:middle line:84% that for the complex scalar field, phi x phi x' is 0. 00:05:37.910 --> 00:05:38.960 align:middle line:90% HONG LIU: Yeah, yeah. 00:05:38.960 --> 00:05:40.335 align:middle line:84% AUDIENCE: So what does that mean? 00:05:40.335 --> 00:05:41.600 align:middle line:90% There's no commutator? 00:05:41.600 --> 00:05:44.570 align:middle line:84% HONG LIU: Yeah, yeah, yeah, because of the charge 00:05:44.570 --> 00:05:45.950 align:middle line:90% conservation. 00:05:45.950 --> 00:05:48.620 align:middle line:84% And you cannot propagate from one particle to the other 00:05:48.620 --> 00:05:49.790 align:middle line:90% particle. 00:05:49.790 --> 00:05:54.530 align:middle line:84% So because phi, when it acts on the 0, 00:05:54.530 --> 00:05:57.260 align:middle line:90% generates, say, a particle. 00:05:57.260 --> 00:05:59.960 align:middle line:84% But when phi acts on the left, it actually 00:05:59.960 --> 00:06:02.690 align:middle line:90% generates the antiparticle. 00:06:02.690 --> 00:06:06.100 align:middle line:84% So a particle cannot become a antiparticle, right? 00:06:06.100 --> 00:06:09.280 align:middle line:84% So only when phi and the phi dagger, and then it's 00:06:09.280 --> 00:06:10.810 align:middle line:90% from just the particle. 00:06:10.810 --> 00:06:11.310 align:middle line:90% Yeah. 00:06:11.310 --> 00:06:16.020 align:middle line:90% 00:06:16.020 --> 00:06:16.950 align:middle line:90% Other questions? 00:06:16.950 --> 00:06:20.280 align:middle line:90% 00:06:20.280 --> 00:06:22.200 align:middle line:90% OK, good. 00:06:22.200 --> 00:06:25.770 align:middle line:90% So we also discussed the-- 00:06:25.770 --> 00:06:29.670 align:middle line:84% yeah, so we wrote down this formula for R, 00:06:29.670 --> 00:06:36.290 align:middle line:90% A, F, say, in coordinate space. 00:06:36.290 --> 00:06:40.030 align:middle line:84% So x prime can be written as a Fourier transform. 00:06:40.030 --> 00:06:50.360 align:middle line:90% 00:06:50.360 --> 00:06:56.192 align:middle line:84% I think it's k squared plus m squared. 00:06:56.192 --> 00:06:57.905 align:middle line:90% And let me just check the sign. 00:06:57.905 --> 00:07:00.730 align:middle line:90% 00:07:00.730 --> 00:07:03.380 align:middle line:90% I think it's i or minus i. 00:07:03.380 --> 00:07:06.490 align:middle line:90% Yeah, minus i. 00:07:06.490 --> 00:07:11.680 align:middle line:90% So right, OK? 00:07:11.680 --> 00:07:18.490 align:middle line:84% And then we mentioned that this factor actually has a pole. 00:07:18.490 --> 00:07:22.240 align:middle line:84% In the downstairs, these have the form omega squared-- 00:07:22.240 --> 00:07:26.140 align:middle line:84% minus omega squared plus omega k squared 00:07:26.140 --> 00:07:28.510 align:middle line:90% if you write it explicitly. 00:07:28.510 --> 00:07:35.960 align:middle line:84% And then this quantity becomes singular at-- 00:07:35.960 --> 00:07:44.985 align:middle line:84% so singular at omega equal to plus-minus omega k. 00:07:44.985 --> 00:07:47.305 align:middle line:90% 00:07:47.305 --> 00:07:47.805 align:middle line:90% OK? 00:07:47.805 --> 00:07:50.370 align:middle line:90% 00:07:50.370 --> 00:07:53.850 align:middle line:84% And the singularity lies on the integration contour 00:07:53.850 --> 00:07:57.600 align:middle line:84% if we consider a complex omega plane. 00:07:57.600 --> 00:08:00.990 align:middle line:84% And so here, one of the integral is the omega. 00:08:00.990 --> 00:08:05.710 align:middle line:84% And then the omega go along the real axis. 00:08:05.710 --> 00:08:07.960 align:middle line:84% So the integration goes along the real axis. 00:08:07.960 --> 00:08:14.150 align:middle line:84% And here is minus omega k, here is omega k. 00:08:14.150 --> 00:08:18.800 align:middle line:84% So you have singularity on the integration contour. 00:08:18.800 --> 00:08:24.890 align:middle line:84% And then different ways of going around the singularity then 00:08:24.890 --> 00:08:28.160 align:middle line:84% give you different choice of retarded and advanced 00:08:28.160 --> 00:08:30.970 align:middle line:90% and the Feynman. 00:08:30.970 --> 00:08:34.600 align:middle line:84% And which one to choose based on-- which 00:08:34.600 --> 00:08:36.940 align:middle line:84% control you should choose, whether you 00:08:36.940 --> 00:08:41.200 align:middle line:84% should go around, go above, or go below the singularity. 00:08:41.200 --> 00:08:46.780 align:middle line:84% And depends on the prescription for each of them. 00:08:46.780 --> 00:08:52.360 align:middle line:84% So we discussed that for G R, which 00:08:52.360 --> 00:08:57.250 align:middle line:84% is proportional to theta, t minus t prime, then that 00:08:57.250 --> 00:09:01.330 align:middle line:84% means we should-- yeah, so let me just draw the singularity. 00:09:01.330 --> 00:09:04.220 align:middle line:90% 00:09:04.220 --> 00:09:07.690 align:middle line:84% So that means you actually need to change your contour 00:09:07.690 --> 00:09:14.200 align:middle line:84% to go above the singularity, to go above the both singularity. 00:09:14.200 --> 00:09:17.500 align:middle line:84% OK, so that will guarantee you to proportional 00:09:17.500 --> 00:09:22.480 align:middle line:84% to theta minus theta prime when you do the integral-- 00:09:22.480 --> 00:09:25.300 align:middle line:84% when you do the integral, the omega 00:09:25.300 --> 00:09:31.570 align:middle line:84% integral using the contours and using the contours. 00:09:31.570 --> 00:09:38.260 align:middle line:84% And similarly, for G A, which is proportional to theta t 00:09:38.260 --> 00:09:41.320 align:middle line:90% prime and t minus t prime-- 00:09:41.320 --> 00:09:43.540 align:middle line:84% in order to guarantee that, then you 00:09:43.540 --> 00:09:50.760 align:middle line:84% have to take your control to be below the singularity, 00:09:50.760 --> 00:09:52.210 align:middle line:90% be below the singularity. 00:09:52.210 --> 00:09:54.675 align:middle line:84% So this is omega, real omega axis. 00:09:54.675 --> 00:09:57.230 align:middle line:90% 00:09:57.230 --> 00:09:59.660 align:middle line:90% And for the Feynman-- 00:09:59.660 --> 00:10:03.610 align:middle line:84% so the Feynman is half and a half. 00:10:03.610 --> 00:10:06.370 align:middle line:84% So there's a theta t minus t prime part-- there's 00:10:06.370 --> 00:10:09.280 align:middle line:90% a theta t prime minus t part. 00:10:09.280 --> 00:10:13.510 align:middle line:90% OK, if you try to check that-- 00:10:13.510 --> 00:10:15.670 align:middle line:84% so we're not going to detail it there. 00:10:15.670 --> 00:10:22.100 align:middle line:84% Here we'll give you a exercise for yourself. 00:10:22.100 --> 00:10:25.120 align:middle line:84% So if you want to satisfy that kind of condition 00:10:25.120 --> 00:10:28.870 align:middle line:84% when t greater than t prime gives you G plus 00:10:28.870 --> 00:10:30.820 align:middle line:84% and when t smaller than t prime gives you 00:10:30.820 --> 00:10:37.760 align:middle line:84% G minus, when you do the contour integration, 00:10:37.760 --> 00:10:39.570 align:middle line:84% you have to do the following way. 00:10:39.570 --> 00:10:45.240 align:middle line:84% So for this one, you go below, for this one you go from up. 00:10:45.240 --> 00:10:49.400 align:middle line:90% OK, you go up. 00:10:49.400 --> 00:10:52.010 align:middle line:84% And then there's another way, which 00:10:52.010 --> 00:10:56.780 align:middle line:84% is G F tilde, which is the opposite for G F in terms 00:10:56.780 --> 00:10:57.860 align:middle line:90% of time-ordering. 00:10:57.860 --> 00:11:00.000 align:middle line:84% And then you will do the opposite way over here. 00:11:00.000 --> 00:11:02.150 align:middle line:84% OK, so altogether, there are four different ways 00:11:02.150 --> 00:11:04.460 align:middle line:84% you can go around the singularity that give you four 00:11:04.460 --> 00:11:08.860 align:middle line:90% different kind of functions. 00:11:08.860 --> 00:11:10.830 align:middle line:84% OK, they give you four different functions. 00:11:10.830 --> 00:11:15.800 align:middle line:84% So for those of you who are not familiar with complex analysis 00:11:15.800 --> 00:11:21.230 align:middle line:84% or the contour integrals, try to refresh yourself a little bit. 00:11:21.230 --> 00:11:24.590 align:middle line:84% Because the complex analysis is definitely 00:11:24.590 --> 00:11:26.570 align:middle line:90% a very, very powerful tool. 00:11:26.570 --> 00:11:29.930 align:middle line:84% To be very familiar with it is very important. 00:11:29.930 --> 00:11:33.130 align:middle line:90% 00:11:33.130 --> 00:11:37.000 align:middle line:84% OK, so it's allowing-- each time, 00:11:37.000 --> 00:11:40.150 align:middle line:84% if you have to specifier your contour like this-- 00:11:40.150 --> 00:11:43.600 align:middle line:84% so if each time you have to specify a contour like this, 00:11:43.600 --> 00:11:44.960 align:middle line:90% it's not convenient. 00:11:44.960 --> 00:11:48.340 align:middle line:84% And the physicists developed a more convenient way 00:11:48.340 --> 00:11:50.650 align:middle line:84% to treat all of them in one shot. 00:11:50.650 --> 00:11:59.300 align:middle line:84% OK, so instead of going around the contour like this, 00:11:59.300 --> 00:12:02.680 align:middle line:84% we just try to move the singularity. 00:12:02.680 --> 00:12:05.480 align:middle line:84% OK so we keep the contour to be fixed, 00:12:05.480 --> 00:12:08.240 align:middle line:84% but we slightly move the singularity. 00:12:08.240 --> 00:12:11.400 align:middle line:84% OK, for example, for the retarded, 00:12:11.400 --> 00:12:18.500 align:middle line:84% this is the same if I just slightly move the singularity 00:12:18.500 --> 00:12:23.140 align:middle line:84% by a tiny bit below the real axis. 00:12:23.140 --> 00:12:26.380 align:middle line:84% So if my integral contour is like this, 00:12:26.380 --> 00:12:28.830 align:middle line:90% then this is equivalent to that. 00:12:28.830 --> 00:12:31.620 align:middle line:84% If this is just a tiny part, then this one 00:12:31.620 --> 00:12:33.570 align:middle line:90% won't affect your answer. 00:12:33.570 --> 00:12:37.300 align:middle line:84% OK, and similarly here, in this case, 00:12:37.300 --> 00:12:40.000 align:middle line:84% we just move both singularity above a little bit. 00:12:40.000 --> 00:12:42.360 align:middle line:84% And in this case, I move one singularity above, 00:12:42.360 --> 00:12:45.090 align:middle line:84% move one singularity down, in which case 00:12:45.090 --> 00:12:46.620 align:middle line:84% I don't have to change my contour, 00:12:46.620 --> 00:12:48.990 align:middle line:84% just slightly change my singularity. 00:12:48.990 --> 00:12:54.390 align:middle line:84% And this can be achieved by so-called epsilon prescription. 00:12:54.390 --> 00:13:04.920 align:middle line:84% So a convenient trick is in the-- for example, 00:13:04.920 --> 00:13:10.900 align:middle line:84% in the retarded case, instead of this-- 00:13:10.900 --> 00:13:17.370 align:middle line:84% so we always do the integral with the same contour. 00:13:17.370 --> 00:13:24.180 align:middle line:84% So we don't change the contour, but we slightly 00:13:24.180 --> 00:13:33.610 align:middle line:84% change the integration to be omega plus i epsilon squared 00:13:33.610 --> 00:13:37.000 align:middle line:90% plus omega k squared. 00:13:37.000 --> 00:13:40.780 align:middle line:84% OK, so the downstairs previously was minus omega 00:13:40.780 --> 00:13:44.400 align:middle line:90% squared plus omega k squared. 00:13:44.400 --> 00:13:47.070 align:middle line:90% But now, I add a tiny-- 00:13:47.070 --> 00:13:49.800 align:middle line:90% so epsilon is infinitesimal. 00:13:49.800 --> 00:13:54.027 align:middle line:90% It's positive and infinitesimal. 00:13:54.027 --> 00:13:59.940 align:middle line:90% 00:13:59.940 --> 00:14:05.700 align:middle line:84% So you see, by adding such a tiny piece to here, 00:14:05.700 --> 00:14:08.960 align:middle line:84% when you solve this singularity-- 00:14:08.960 --> 00:14:13.280 align:middle line:84% so it's now omega plus i epsilon equal to plus minus omega. 00:14:13.280 --> 00:14:17.150 align:middle line:84% And then move the epsilon to the other side, 00:14:17.150 --> 00:14:21.620 align:middle line:84% and then the solution becomes minus i epsilon. 00:14:21.620 --> 00:14:24.230 align:middle line:84% And then because epsilon is positive, 00:14:24.230 --> 00:14:27.710 align:middle line:84% then this is slightly in the lower half plane. 00:14:27.710 --> 00:14:34.950 align:middle line:84% And similarly, for the advanced, you just do a minus, OK? 00:14:34.950 --> 00:14:37.895 align:middle line:84% And then both singularity will move up. 00:14:37.895 --> 00:14:38.770 align:middle line:90% And for the Feynman-- 00:14:38.770 --> 00:14:43.930 align:middle line:90% 00:14:43.930 --> 00:14:47.035 align:middle line:84% so the trick is you can do the same thing. 00:14:47.035 --> 00:14:53.160 align:middle line:90% 00:14:53.160 --> 00:14:55.760 align:middle line:90% Yeah. 00:14:55.760 --> 00:15:00.320 align:middle line:90% Sorry, I still have i k. 00:15:00.320 --> 00:15:02.870 align:middle line:90% So this is the minus i. 00:15:02.870 --> 00:15:05.540 align:middle line:84% So I just do k squared plus m squared, 00:15:05.540 --> 00:15:09.800 align:middle line:84% the whole thing minus i epsilon, rather than change the omega, 00:15:09.800 --> 00:15:11.510 align:middle line:84% and the whole thing minus i epsilon. 00:15:11.510 --> 00:15:17.590 align:middle line:90% 00:15:17.590 --> 00:15:19.840 align:middle line:90% And you can check yourself. 00:15:19.840 --> 00:15:22.150 align:middle line:84% When you solve the omega using this, 00:15:22.150 --> 00:15:25.720 align:middle line:84% this is the whole thing minus i epsilon, in this case. 00:15:25.720 --> 00:15:27.940 align:middle line:84% So for retarded and advanced, you put it 00:15:27.940 --> 00:15:29.470 align:middle line:90% inside the omega squared. 00:15:29.470 --> 00:15:31.390 align:middle line:84% But here, you put it other side, OK? 00:15:31.390 --> 00:15:33.550 align:middle line:84% And then you can check when you solve for omega. 00:15:33.550 --> 00:15:35.740 align:middle line:84% Actually, you get this situation. 00:15:35.740 --> 00:15:38.110 align:middle line:84% One move up-- it will actually move up 00:15:38.110 --> 00:15:39.950 align:middle line:90% and the positive move down. 00:15:39.950 --> 00:15:40.450 align:middle line:90% OK? 00:15:40.450 --> 00:15:41.020 align:middle line:90% Yes. 00:15:41.020 --> 00:15:43.762 align:middle line:84% AUDIENCE: Can you outline how I would convince myself 00:15:43.762 --> 00:15:47.130 align:middle line:84% that these two pictures are equivalent? 00:15:47.130 --> 00:15:50.600 align:middle line:84% Are they going-- moving, integrating with the contour 00:15:50.600 --> 00:15:55.500 align:middle line:84% that jumps out over or under these poles for-- 00:15:55.500 --> 00:15:56.460 align:middle line:90% HONG LIU: Yeah, yeah. 00:15:56.460 --> 00:15:58.335 align:middle line:84% Yeah, because from the contour point of view, 00:15:58.335 --> 00:16:03.300 align:middle line:84% from the control integration point of view, 00:16:03.300 --> 00:16:06.210 align:middle line:84% the precise shape of the contour is not important. 00:16:06.210 --> 00:16:10.560 align:middle line:84% It only matters what singularity it encloses. 00:16:10.560 --> 00:16:13.830 align:middle line:84% So as far as it encloses these two singularity, 00:16:13.830 --> 00:16:15.750 align:middle line:90% and then the story is the same. 00:16:15.750 --> 00:16:19.050 align:middle line:84% But since I changed the singularity location only 00:16:19.050 --> 00:16:20.400 align:middle line:90% by epsilon-- 00:16:20.400 --> 00:16:23.610 align:middle line:84% and in the end, I will take this epsilon go to 0-- 00:16:23.610 --> 00:16:25.620 align:middle line:84% and then the value will be the same. 00:16:25.620 --> 00:16:28.510 align:middle line:90% 00:16:28.510 --> 00:16:29.840 align:middle line:90% Other questions? 00:16:29.840 --> 00:16:35.470 align:middle line:84% And so this is a very, very simple, but very powerful 00:16:35.470 --> 00:16:36.370 align:middle line:90% trick. 00:16:36.370 --> 00:16:38.560 align:middle line:84% This makes things much more convenient 00:16:38.560 --> 00:16:43.900 align:middle line:84% when you do many manipulations so that you don't 00:16:43.900 --> 00:16:47.770 align:middle line:84% have to constantly worry about the contour, 00:16:47.770 --> 00:16:50.005 align:middle line:90% the precise contour you take. 00:16:50.005 --> 00:16:53.800 align:middle line:90% 00:16:53.800 --> 00:16:56.080 align:middle line:90% Any other questions? 00:16:56.080 --> 00:16:56.580 align:middle line:90% Good. 00:16:56.580 --> 00:17:01.080 align:middle line:90% 00:17:01.080 --> 00:17:02.640 align:middle line:90% Good. 00:17:02.640 --> 00:17:05.470 align:middle line:84% So let me just give some final remarks. 00:17:05.470 --> 00:17:09.260 align:middle line:84% So this almost concludes our discussion 00:17:09.260 --> 00:17:10.589 align:middle line:90% of the free field theory. 00:17:10.589 --> 00:17:12.335 align:middle line:84% So let me just give some final remarks. 00:17:12.335 --> 00:17:21.319 align:middle line:90% 00:17:21.319 --> 00:17:30.520 align:middle line:84% So as I mentioned, the G plus and the G R and the retarded 00:17:30.520 --> 00:17:31.960 align:middle line:90% have many applications. 00:17:31.960 --> 00:17:40.160 align:middle line:90% 00:17:40.160 --> 00:17:42.020 align:middle line:90% Saying, in condensed matter. 00:17:42.020 --> 00:17:47.390 align:middle line:84% So you deal with them-- you deal with them all the time. 00:17:47.390 --> 00:17:54.970 align:middle line:84% And then this Feynman propagator, 00:17:54.970 --> 00:18:03.630 align:middle line:84% we will see it plays a very important role 00:18:03.630 --> 00:18:13.530 align:middle line:84% in next chapter, which we will start in the next few minutes. 00:18:13.530 --> 00:18:15.585 align:middle line:84% So the G F will play a very important role. 00:18:15.585 --> 00:18:18.650 align:middle line:90% 00:18:18.650 --> 00:18:24.500 align:middle line:84% And so in addition to these two points-- 00:18:24.500 --> 00:18:28.780 align:middle line:84% so these are the correlation function of two points, OK? 00:18:28.780 --> 00:18:30.400 align:middle line:90% Just the x and x prime. 00:18:30.400 --> 00:18:33.640 align:middle line:84% You can, in principle, consider, say, more general correlation 00:18:33.640 --> 00:18:34.510 align:middle line:90% functions. 00:18:34.510 --> 00:18:52.460 align:middle line:90% 00:18:52.460 --> 00:18:58.710 align:middle line:84% Say, you can say, consider an arbitrary state. 00:18:58.710 --> 00:19:07.520 align:middle line:90% 00:19:07.520 --> 00:19:10.520 align:middle line:84% Say, for example, you can just consider some state psi, 00:19:10.520 --> 00:19:16.130 align:middle line:84% and then you can consider phi x1, phi x2, say, phi xn. 00:19:16.130 --> 00:19:22.860 align:middle line:90% 00:19:22.860 --> 00:19:25.530 align:middle line:84% So this is some n-point function. 00:19:25.530 --> 00:19:27.450 align:middle line:84% Yeah, what's normally called n-point function, 00:19:27.450 --> 00:19:29.967 align:middle line:90% some general state, psi. 00:19:29.967 --> 00:19:31.425 align:middle line:84% Psi does not have to be the vacuum. 00:19:31.425 --> 00:19:36.320 align:middle line:90% 00:19:36.320 --> 00:19:40.110 align:middle line:84% So general state-- remember, general state psi 00:19:40.110 --> 00:19:43.250 align:middle line:84% can all be built from a and a dagger. 00:19:43.250 --> 00:19:46.880 align:middle line:84% And the phi can also be written this way, in a dagger. 00:19:46.880 --> 00:19:48.567 align:middle line:90% So if you give me a state-- 00:19:48.567 --> 00:19:50.150 align:middle line:84% so I can, in principle, calculate this 00:19:50.150 --> 00:19:51.358 align:middle line:90% for any correlation function. 00:19:51.358 --> 00:19:54.000 align:middle line:84% OK, so this theory is fully solved. 00:19:54.000 --> 00:19:56.250 align:middle line:84% And then no matter what you want me 00:19:56.250 --> 00:20:00.780 align:middle line:84% to compute using what we have developed so far, 00:20:00.780 --> 00:20:03.600 align:middle line:90% you can now compute them. 00:20:03.600 --> 00:20:09.640 align:middle line:84% So this theory, everything is computable. 00:20:09.640 --> 00:20:13.420 align:middle line:84% And so now, let me just mention one simple fact regarding 00:20:13.420 --> 00:20:16.703 align:middle line:90% the vacuum correlation function. 00:20:16.703 --> 00:20:18.745 align:middle line:84% So let's consider the n-point function of vacuum. 00:20:18.745 --> 00:20:22.690 align:middle line:90% 00:20:22.690 --> 00:20:26.760 align:middle line:84% And then you can easily convince yourself-- 00:20:26.760 --> 00:20:28.440 align:middle line:84% OK, you can easily convince yourself 00:20:28.440 --> 00:20:48.990 align:middle line:84% this factorize into sums of product of two-point functions. 00:20:48.990 --> 00:21:06.050 align:middle line:90% 00:21:06.050 --> 00:21:08.900 align:middle line:84% So this is simple because a and a dagger, 00:21:08.900 --> 00:21:11.400 align:middle line:90% they have to be paired. 00:21:11.400 --> 00:21:16.450 align:middle line:84% Because any a dagger creates something out of 0 00:21:16.450 --> 00:21:21.270 align:middle line:84% has to be annihilated by a somewhere later. 00:21:21.270 --> 00:21:25.250 align:middle line:84% So that means all the a and a dagger has to be paired. 00:21:25.250 --> 00:21:28.320 align:middle line:84% And then that means that they have to reduce 00:21:28.320 --> 00:21:29.730 align:middle line:90% to all the two-point functions. 00:21:29.730 --> 00:21:32.140 align:middle line:90% OK, so you can check explicitly. 00:21:32.140 --> 00:21:37.360 align:middle line:84% So for example, the simplest is the four-point function. 00:21:37.360 --> 00:21:44.840 align:middle line:90% So if you consider phi x, OK? 00:21:44.840 --> 00:21:50.650 align:middle line:84% So you can easily convince yourself that the a dagger-- 00:21:50.650 --> 00:21:55.330 align:middle line:84% when a dagger of this creates on this, and then that can be, 00:21:55.330 --> 00:21:58.330 align:middle line:90% say, annihilated by any of them. 00:21:58.330 --> 00:22:01.790 align:middle line:90% And so you can have, say-- 00:22:01.790 --> 00:22:05.210 align:middle line:84% and similarly, a dagger created by this one 00:22:05.210 --> 00:22:07.060 align:middle line:84% have to be annihilated by some later. 00:22:07.060 --> 00:22:09.110 align:middle line:84% So you have to always pair them, OK? 00:22:09.110 --> 00:22:12.260 align:middle line:84% So you can show that this just reduces 00:22:12.260 --> 00:22:14.810 align:middle line:90% to all possible pairings. 00:22:14.810 --> 00:22:32.770 align:middle line:84% So you can pair 3 and 4, or pair 1 and 4 and 2 and 3, or 1 and 3 00:22:32.770 --> 00:22:34.130 align:middle line:90% and 2 and 4. 00:22:34.130 --> 00:22:37.665 align:middle line:84% OK, you just write down all possible pairings. 00:22:37.665 --> 00:22:39.415 align:middle line:84% OK, just write down all possible pairings. 00:22:39.415 --> 00:22:43.790 align:middle line:90% 00:22:43.790 --> 00:22:45.350 align:middle line:90% So in free theory-- 00:22:45.350 --> 00:22:49.300 align:middle line:84% so in this free theory, any point functions in the vacuum, 00:22:49.300 --> 00:22:52.120 align:middle line:84% they just reduce to the two-point function. 00:22:52.120 --> 00:22:54.730 align:middle line:84% If you know the two-point function, you know everything. 00:22:54.730 --> 00:22:56.270 align:middle line:90% OK, you know everything. 00:22:56.270 --> 00:23:00.870 align:middle line:90% 00:23:00.870 --> 00:23:04.385 align:middle line:84% So this, I will give you as an exercise to check yourself. 00:23:04.385 --> 00:23:10.050 align:middle line:90% 00:23:10.050 --> 00:23:14.880 align:middle line:84% And then the final remark is related to the problem 3, 00:23:14.880 --> 00:23:17.980 align:middle line:90% I think, in your Pset. 00:23:17.980 --> 00:23:27.320 align:middle line:84% So in addition to phi, we can also 00:23:27.320 --> 00:23:30.050 align:middle line:84% consider more complicated operators. 00:23:30.050 --> 00:23:37.330 align:middle line:84% We can consider, say, so-called composite operators. 00:23:37.330 --> 00:23:40.950 align:middle line:90% 00:23:40.950 --> 00:23:45.030 align:middle line:84% Means that we take the product of phi, of which is momentum. 00:23:45.030 --> 00:23:46.160 align:middle line:90% So we can have-- 00:23:46.160 --> 00:23:50.120 align:middle line:84% say, we can consider phi squared, 00:23:50.120 --> 00:23:58.730 align:middle line:84% phi cubed, say, the Hamiltonian density, the stress tensor. 00:23:58.730 --> 00:24:02.920 align:middle line:84% So these are all involving some product of phis. 00:24:02.920 --> 00:24:05.490 align:middle line:90% Or its canonical momentum. 00:24:05.490 --> 00:24:10.250 align:middle line:84% And so these are normally called the composite operators. 00:24:10.250 --> 00:24:17.680 align:middle line:84% And as you should have learned from your P 00:24:17.680 --> 00:24:22.490 align:middle line:84% set problem 3 that those operators often 00:24:22.490 --> 00:24:23.750 align:middle line:90% are not well defined. 00:24:23.750 --> 00:24:26.330 align:middle line:84% Because when you multiply them at the same point, 00:24:26.330 --> 00:24:29.210 align:middle line:90% you will get a singularity. 00:24:29.210 --> 00:24:31.128 align:middle line:90% You will get singular behavior. 00:24:31.128 --> 00:24:32.420 align:middle line:90% You will get singular behavior. 00:24:32.420 --> 00:24:34.760 align:middle line:84% And the really to renormalize them, et cetera. 00:24:34.760 --> 00:24:35.930 align:middle line:90% OK? 00:24:35.930 --> 00:24:39.680 align:middle line:84% And so such kind of divergent behavior 00:24:39.680 --> 00:24:42.410 align:middle line:84% is generic in quantum field theory. 00:24:42.410 --> 00:24:47.660 align:middle line:84% And one of the most important parts of quantum field theory 00:24:47.660 --> 00:24:51.350 align:middle line:84% is to find sensible ways to renormalize 00:24:51.350 --> 00:24:58.040 align:middle line:84% those divergent quantities in the physically sensible way. 00:24:58.040 --> 00:25:03.440 align:middle line:84% And the problem 3, it's the first example you encounter. 00:25:03.440 --> 00:25:06.500 align:middle line:84% And later, when we deal with interacting theories, 00:25:06.500 --> 00:25:10.170 align:middle line:84% there are more complicated examples, et cetera. 00:25:10.170 --> 00:25:13.040 align:middle line:84% OK, so do you have any questions? 00:25:13.040 --> 00:25:13.553 align:middle line:90% Yes? 00:25:13.553 --> 00:25:14.970 align:middle line:84% AUDIENCE: So I'm a little confused 00:25:14.970 --> 00:25:17.950 align:middle line:84% on how to interpret this n-point correlation function. 00:25:17.950 --> 00:25:21.163 align:middle line:84% Because for the two-point, we have this heuristic condition 00:25:21.163 --> 00:25:22.033 align:middle line:90% amplitude. 00:25:22.033 --> 00:25:22.700 align:middle line:90% HONG LIU: Right. 00:25:22.700 --> 00:25:24.367 align:middle line:84% AUDIENCE: But how do you interpret this, 00:25:24.367 --> 00:25:27.740 align:middle line:84% especially with theories where it's not a free scalar 00:25:27.740 --> 00:25:29.740 align:middle line:84% field where you can't decompose them like this-- 00:25:29.740 --> 00:25:30.610 align:middle line:90% HONG LIU: Right. 00:25:30.610 --> 00:25:33.110 align:middle line:84% AUDIENCE: --by being expressed in terms of ladder operators? 00:25:33.110 --> 00:25:34.570 align:middle line:90% HONG LIU: Yeah, yeah. 00:25:34.570 --> 00:25:37.180 align:middle line:84% So just heuristically, you can imagine-- say, you 00:25:37.180 --> 00:25:39.970 align:middle line:90% can imagine these correlations. 00:25:39.970 --> 00:25:44.020 align:middle line:84% So, say, you make measurements at different points. 00:25:44.020 --> 00:25:47.170 align:middle line:84% And you study some kind of joint amplitude of them. 00:25:47.170 --> 00:25:50.135 align:middle line:84% Yeah, so this is one way to interpret this quantity. 00:25:50.135 --> 00:25:51.760 align:middle line:84% AUDIENCE: Like, the value of the field? 00:25:51.760 --> 00:25:52.635 align:middle line:90% HONG LIU: Yeah, yeah. 00:25:52.635 --> 00:25:55.510 align:middle line:84% For example, if you imagine they are spin operators. 00:25:55.510 --> 00:25:58.930 align:middle line:84% And say, phi denotes some kind of spin, 00:25:58.930 --> 00:26:00.850 align:middle line:84% and then you can imagine you measure them 00:26:00.850 --> 00:26:02.650 align:middle line:90% at different space time points. 00:26:02.650 --> 00:26:05.070 align:middle line:84% And you look for their correlations. 00:26:05.070 --> 00:26:06.790 align:middle line:90% Yeah. 00:26:06.790 --> 00:26:08.860 align:middle line:84% Yeah, so this is the one example. 00:26:08.860 --> 00:26:10.390 align:middle line:90% But later, we will see-- 00:26:10.390 --> 00:26:13.300 align:middle line:84% we will actually almost see immediately-- yeah, 00:26:13.300 --> 00:26:16.212 align:middle line:84% just today, I think we will reach there. 00:26:16.212 --> 00:26:18.670 align:middle line:84% And you will see this actually is related to the scattering 00:26:18.670 --> 00:26:21.550 align:middle line:84% amplitudes when we talk about interacting series. 00:26:21.550 --> 00:26:24.220 align:middle line:90% Yeah. 00:26:24.220 --> 00:26:25.210 align:middle line:90% Other questions? 00:26:25.210 --> 00:26:29.630 align:middle line:90% 00:26:29.630 --> 00:26:30.910 align:middle line:90% OK, good. 00:26:30.910 --> 00:26:32.710 align:middle line:90% Good. 00:26:32.710 --> 00:26:38.290 align:middle line:84% So try to enjoy the problem 3 in your Pset. 00:26:38.290 --> 00:26:43.060 align:middle line:84% And after doing it, really read the problem again, OK? 00:26:43.060 --> 00:26:46.645 align:middle line:90% Read the problem again. 00:26:46.645 --> 00:26:50.785 align:middle line:84% That problem contains the things which-- 00:26:50.785 --> 00:26:56.710 align:middle line:84% yeah, just the simplest situation for the divergences 00:26:56.710 --> 00:26:59.230 align:middle line:84% you will see in quantum field theory. 00:26:59.230 --> 00:27:03.610 align:middle line:84% And to be prepared for that is important. 00:27:03.610 --> 00:27:05.860 align:middle line:84% Yeah, even though we will not directly 00:27:05.860 --> 00:27:10.900 align:middle line:84% use the result of the problem 3 in the immediate future, 00:27:10.900 --> 00:27:13.660 align:middle line:84% but I think it's important as part of your education 00:27:13.660 --> 00:27:17.650 align:middle line:84% to get used to the infinities and the divergences 00:27:17.650 --> 00:27:19.030 align:middle line:90% in quantum field theory. 00:27:19.030 --> 00:27:21.580 align:middle line:90% OK, good. 00:27:21.580 --> 00:27:23.140 align:middle line:90% Other questions? 00:27:23.140 --> 00:27:24.070 align:middle line:90% Yes? 00:27:24.070 --> 00:27:28.150 align:middle line:84% AUDIENCE: So I guess over here, everything on the left hand 00:27:28.150 --> 00:27:30.790 align:middle line:84% side is, in terms of these Green's functions, 00:27:30.790 --> 00:27:32.530 align:middle line:84% which could have been solved classically 00:27:32.530 --> 00:27:34.690 align:middle line:90% without doing any quantization. 00:27:34.690 --> 00:27:38.220 align:middle line:84% So is there a reason why the correlator 00:27:38.220 --> 00:27:39.840 align:middle line:84% and the quantized field theory is 00:27:39.840 --> 00:27:42.540 align:middle line:84% the same as the correlator in the the classical? 00:27:42.540 --> 00:27:44.650 align:middle line:84% HONG LIU: Yeah, just because it's a very simple-- 00:27:44.650 --> 00:27:46.980 align:middle line:90% this is a very simple theory. 00:27:46.980 --> 00:27:47.740 align:middle line:90% Yeah. 00:27:47.740 --> 00:27:48.240 align:middle line:90% Yeah. 00:27:48.240 --> 00:27:51.810 align:middle line:84% Yeah, but even in this case, it's actually-- 00:27:51.810 --> 00:27:54.300 align:middle line:84% yeah, it depends on how you interpret the behavior 00:27:54.300 --> 00:27:56.580 align:middle line:90% of those functions. 00:27:56.580 --> 00:27:59.610 align:middle line:84% Even though this form, you might say, oh, maybe 00:27:59.610 --> 00:28:03.530 align:middle line:90% we could anticipate this. 00:28:03.530 --> 00:28:07.560 align:middle line:84% Say, if you have a, say, Gaussian statistical system, 00:28:07.560 --> 00:28:09.480 align:middle line:90% you will anticipate this. 00:28:09.480 --> 00:28:11.910 align:middle line:84% Just say you only have Gaussian fluctuations-- 00:28:11.910 --> 00:28:14.460 align:middle line:84% you only have Gaussian correlations and nothing else. 00:28:14.460 --> 00:28:16.920 align:middle line:84% But the specific form of the functions, they 00:28:16.920 --> 00:28:18.750 align:middle line:90% do encode the quantum physics. 00:28:18.750 --> 00:28:23.910 align:middle line:84% And even though this factorized form is like in-- 00:28:23.910 --> 00:28:27.196 align:middle line:84% yeah, say, like in the Gaussian statistical theory. 00:28:27.196 --> 00:28:27.696 align:middle line:90% Yeah. 00:28:27.696 --> 00:28:30.600 align:middle line:90% 00:28:30.600 --> 00:28:32.650 align:middle line:90% Good? 00:28:32.650 --> 00:28:39.220 align:middle line:84% OK, so so far, we discussed the simplest free theory, just 00:28:39.220 --> 00:28:42.430 align:middle line:84% the theory of free scalar particles. 00:28:42.430 --> 00:28:44.620 align:middle line:90% They don't have spin. 00:28:44.620 --> 00:28:49.480 align:middle line:84% They have spin 0, so they don't have any space-time component. 00:28:49.480 --> 00:28:54.040 align:middle line:84% And they have some mass, relativistic particles. 00:28:54.040 --> 00:28:58.030 align:middle line:84% And later, we will talk about electrons, which have spin 1/2, 00:28:58.030 --> 00:29:00.850 align:middle line:84% and talk about photons, which have spin 1. 00:29:00.850 --> 00:29:04.480 align:middle line:84% And before talking about them, let's talk a little bit 00:29:04.480 --> 00:29:05.620 align:middle line:90% about interactions. 00:29:05.620 --> 00:29:08.380 align:middle line:84% Because so far, this theory is absolutely boring. 00:29:08.380 --> 00:29:14.470 align:middle line:84% So even though this theory illustrates this very important 00:29:14.470 --> 00:29:18.250 align:middle line:84% conceptual point that somehow when you quantize the fields 00:29:18.250 --> 00:29:19.510 align:middle line:90% you can get particles. 00:29:19.510 --> 00:29:21.580 align:middle line:84% OK, you can get arbitrary number of particles. 00:29:21.580 --> 00:29:25.180 align:middle line:84% So it's a connection between the field and the particles. 00:29:25.180 --> 00:29:27.400 align:middle line:84% But just in terms of theory of particles, 00:29:27.400 --> 00:29:29.320 align:middle line:84% this series is absolutely boring. 00:29:29.320 --> 00:29:32.630 align:middle line:84% There's nothing-- the particles don't do anything. 00:29:32.630 --> 00:29:36.730 align:middle line:84% They just go straight, which is Newton's First Law. 00:29:36.730 --> 00:29:38.860 align:middle line:84% They just go straight like Newton's First Law. 00:29:38.860 --> 00:29:46.150 align:middle line:84% And so now, let's try to add a little bit fun to introduce 00:29:46.150 --> 00:29:47.650 align:middle line:90% a little bit interactions. 00:29:47.650 --> 00:29:49.360 align:middle line:84% And you will see that when you introduce 00:29:49.360 --> 00:29:53.840 align:middle line:84% a little bit interactions, the story becomes much richer. 00:29:53.840 --> 00:29:57.290 align:middle line:84% So now, let's talk about interactions. 00:29:57.290 --> 00:30:10.100 align:middle line:90% 00:30:10.100 --> 00:30:13.870 align:middle line:84% It turns out to describe interactions, 00:30:13.870 --> 00:30:17.810 align:middle line:84% a very powerful approach is the path integral. 00:30:17.810 --> 00:30:21.340 align:middle line:84% OK, so we will also discuss the path integral. 00:30:21.340 --> 00:30:25.970 align:middle line:84% So we will use the path integral approach to describe 00:30:25.970 --> 00:30:29.180 align:middle line:90% how to treat interactions. 00:30:29.180 --> 00:30:37.030 align:middle line:84% And yeah, so first, let's just say 00:30:37.030 --> 00:30:42.265 align:middle line:84% a few words about some general remark on interacting theories. 00:30:42.265 --> 00:30:49.450 align:middle line:90% 00:30:49.450 --> 00:30:52.805 align:middle line:84% So previously, we consider theories 00:30:52.805 --> 00:30:54.805 align:middle line:84% with a Lagrangian density of the following form. 00:30:54.805 --> 00:31:00.970 align:middle line:90% 00:31:00.970 --> 00:31:02.600 align:middle line:84% And we find it's a free particle. 00:31:02.600 --> 00:31:04.860 align:middle line:90% It's a theory of free particles. 00:31:04.860 --> 00:31:07.320 align:middle line:84% And now, we can easily make this theory 00:31:07.320 --> 00:31:11.820 align:middle line:84% to be of interacting particles if we just 00:31:11.820 --> 00:31:15.000 align:middle line:90% add some higher order powers. 00:31:15.000 --> 00:31:19.300 align:middle line:84% So for example, one of the simplest power 00:31:19.300 --> 00:31:22.352 align:middle line:90% is phi to the power of 4. 00:31:22.352 --> 00:31:26.410 align:middle line:90% 00:31:26.410 --> 00:31:31.150 align:middle line:84% So yeah, you can also add phi to the power of 3, 00:31:31.150 --> 00:31:38.620 align:middle line:84% but phi to the power of 3 is a little bit sick. 00:31:38.620 --> 00:31:45.860 align:middle line:84% It's because the phi cubed doesn't have a definite sign. 00:31:45.860 --> 00:31:50.405 align:middle line:84% I don't have a definite sign, and so phi cubed, say, 00:31:50.405 --> 00:31:52.460 align:middle line:84% does not have a well defined energy. 00:31:52.460 --> 00:31:54.440 align:middle line:84% So your energy can go arbitrarily negative. 00:31:54.440 --> 00:31:59.120 align:middle line:84% And so this is the simplest interacting theory 00:31:59.120 --> 00:32:00.620 align:middle line:90% with a well defined energy. 00:32:00.620 --> 00:32:03.650 align:middle line:90% 00:32:03.650 --> 00:32:11.200 align:middle line:84% So we will write this as L0, which is the free theory 00:32:11.200 --> 00:32:14.690 align:middle line:84% part, which we can see there, and then the interacting part, 00:32:14.690 --> 00:32:15.190 align:middle line:90% OK? 00:32:15.190 --> 00:32:20.480 align:middle line:84% This lambda factorial, phi fourth. 00:32:20.480 --> 00:32:24.890 align:middle line:84% And so this lambda, essentially heuristic, this lambda 00:32:24.890 --> 00:32:28.170 align:middle line:84% should characterize the importance of this term. 00:32:28.170 --> 00:32:29.810 align:middle line:84% So if lambda goes to 0, of course, 00:32:29.810 --> 00:32:31.670 align:middle line:84% then this term goes away, and then this term 00:32:31.670 --> 00:32:32.810 align:middle line:90% is not important. 00:32:32.810 --> 00:32:34.820 align:middle line:84% And when lambda becomes bigger, then this term 00:32:34.820 --> 00:32:36.480 align:middle line:90% becomes more and more important. 00:32:36.480 --> 00:32:40.550 align:middle line:84% And so we call the lambda to be the coupling, which 00:32:40.550 --> 00:32:42.440 align:middle line:84% characterizes the importance of this term. 00:32:42.440 --> 00:32:49.740 align:middle line:90% 00:32:49.740 --> 00:32:52.980 align:middle line:84% So now, first, let's follow what we 00:32:52.980 --> 00:32:56.790 align:middle line:84% were doing before for the general quantization procedure. 00:32:56.790 --> 00:32:59.950 align:middle line:84% We tried to write down the classical equation of motion, 00:32:59.950 --> 00:33:01.800 align:middle line:84% and then we quantized-- and then we 00:33:01.800 --> 00:33:04.890 align:middle line:84% solved the most general classical equations, 00:33:04.890 --> 00:33:07.740 align:middle line:84% the most general solutions for the classical equations. 00:33:07.740 --> 00:33:11.850 align:middle line:84% And then we promote those solutions to operators, right? 00:33:11.850 --> 00:33:14.880 align:middle line:84% Yeah, so let's try to do this one. 00:33:14.880 --> 00:33:22.498 align:middle line:90% 00:33:22.498 --> 00:33:24.665 align:middle line:84% So now, when you write down the equation of motion-- 00:33:24.665 --> 00:33:30.237 align:middle line:90% 00:33:30.237 --> 00:33:31.945 align:middle line:84% so the equation of motion is very simple. 00:33:31.945 --> 00:33:34.640 align:middle line:90% 00:33:34.640 --> 00:33:39.280 align:middle line:84% You just get partial square phi, m square phi. 00:33:39.280 --> 00:33:41.500 align:middle line:90% So this is the free theory part. 00:33:41.500 --> 00:33:47.260 align:middle line:84% And now, you have a nonlinear part, the cube, OK? 00:33:47.260 --> 00:33:51.337 align:middle line:90% 00:33:51.337 --> 00:33:52.920 align:middle line:84% So now, you have a nonlinear equation. 00:33:52.920 --> 00:33:59.840 align:middle line:90% 00:33:59.840 --> 00:34:01.265 align:middle line:90% So who can solve this equation? 00:34:01.265 --> 00:34:06.130 align:middle line:90% 00:34:06.130 --> 00:34:07.270 align:middle line:90% Any volunteers? 00:34:07.270 --> 00:34:11.469 align:middle line:90% 00:34:11.469 --> 00:34:13.270 align:middle line:84% It turns out actually this equation, 00:34:13.270 --> 00:34:16.250 align:middle line:84% nobody has been able to solve it so far. 00:34:16.250 --> 00:34:18.940 align:middle line:90% OK? 00:34:18.940 --> 00:34:21.520 align:middle line:84% So we don't know how to solve it. 00:34:21.520 --> 00:34:24.340 align:middle line:84% And so our previous strategy, even though 00:34:24.340 --> 00:34:27.770 align:middle line:84% sounds very powerful and immediately it breaks down, OK? 00:34:27.770 --> 00:34:31.449 align:middle line:84% Because rely on you can actually find the solutions 00:34:31.449 --> 00:34:34.000 align:middle line:90% to this equation of motion. 00:34:34.000 --> 00:34:38.020 align:middle line:84% And yeah, so we don't know how to solve this. 00:34:38.020 --> 00:34:53.159 align:middle line:90% 00:34:53.159 --> 00:34:56.420 align:middle line:84% But heuristically, you can imagine 00:34:56.420 --> 00:34:58.970 align:middle line:84% that this, indeed, should be a theory of interacting 00:34:58.970 --> 00:35:00.910 align:middle line:90% particles. 00:35:00.910 --> 00:35:03.388 align:middle line:84% Because previously, when you don't have this term, 00:35:03.388 --> 00:35:04.930 align:middle line:84% you just have a linear equation, then 00:35:04.930 --> 00:35:06.980 align:middle line:90% the phi is not doing anything. 00:35:06.980 --> 00:35:08.780 align:middle line:84% OK, now, you have a phi cubed term. 00:35:08.780 --> 00:35:11.800 align:middle line:84% So that means that the different phis can now doing something 00:35:11.800 --> 00:35:12.580 align:middle line:90% together, OK? 00:35:12.580 --> 00:35:15.340 align:middle line:84% So when different phi's doing something together, 00:35:15.340 --> 00:35:17.350 align:middle line:84% by definition, that's called interactions. 00:35:17.350 --> 00:35:19.510 align:middle line:84% OK, and that's called interactions. 00:35:19.510 --> 00:35:22.660 align:middle line:84% And so just heuristically, this should already-- 00:35:22.660 --> 00:35:26.405 align:middle line:84% this should give you a theory of interacting particles. 00:35:26.405 --> 00:35:26.905 align:middle line:90% OK. 00:35:26.905 --> 00:35:35.400 align:middle line:90% 00:35:35.400 --> 00:35:38.850 align:middle line:84% So we will talk about how to treat such a theory. 00:35:38.850 --> 00:35:40.800 align:middle line:84% Since this cannot be solved exactly, 00:35:40.800 --> 00:35:44.730 align:middle line:84% we can only try to solve it approximately. 00:35:44.730 --> 00:35:48.630 align:middle line:84% So the only way we know how to solve it approximately so far 00:35:48.630 --> 00:35:51.680 align:middle line:84% is to imagine the lambda is small. 00:35:51.680 --> 00:35:54.198 align:middle line:84% And then you just expand the lambda perturbatively. 00:35:54.198 --> 00:35:55.990 align:middle line:84% Just treat the lambda as a small parameter, 00:35:55.990 --> 00:35:57.700 align:middle line:90% you just expand the lambda. 00:35:57.700 --> 00:36:01.840 align:middle line:84% And so I will outline this procedure in a little bit. 00:36:01.840 --> 00:36:04.820 align:middle line:84% But before that, let me just make some general remarks, 00:36:04.820 --> 00:36:08.675 align:middle line:84% which apply to general interacting theories, which 00:36:08.675 --> 00:36:10.050 align:middle line:84% this is just one of the examples. 00:36:10.050 --> 00:36:14.260 align:middle line:90% 00:36:14.260 --> 00:36:20.470 align:middle line:84% So actually figuring out interactions from experiments-- 00:36:20.470 --> 00:36:26.120 align:middle line:84% it's actually one of the main tasks of the particle physics 00:36:26.120 --> 00:36:29.910 align:middle line:84% and many areas of condensed matter physics. 00:36:29.910 --> 00:36:36.500 align:middle line:84% OK, and so for example, we build LHC at CERN 00:36:36.500 --> 00:36:41.420 align:middle line:84% and you collide these protons together, billions of dollars. 00:36:41.420 --> 00:36:44.390 align:middle line:84% The whole purpose is to figure out the interactions 00:36:44.390 --> 00:36:46.520 align:middle line:90% between the particles. 00:36:46.520 --> 00:36:52.790 align:middle line:84% Between the elementary particles and to verify them, et cetera. 00:36:52.790 --> 00:36:57.290 align:middle line:84% And similarly, with many condensed matter experiments. 00:36:57.290 --> 00:36:59.940 align:middle line:84% So now, I'll ask you a simple question. 00:36:59.940 --> 00:37:02.930 align:middle line:84% So what's the most powerful way, experimentally, 00:37:02.930 --> 00:37:06.920 align:middle line:84% to probe the interactions between particles? 00:37:06.920 --> 00:37:10.130 align:middle line:84% What is the word to describe that kind of experiment? 00:37:10.130 --> 00:37:14.660 align:middle line:90% 00:37:14.660 --> 00:37:16.035 align:middle line:90% What? 00:37:16.035 --> 00:37:16.910 align:middle line:90% AUDIENCE: Scattering. 00:37:16.910 --> 00:37:19.963 align:middle line:90% HONG LIU: Yeah, scattering. 00:37:19.963 --> 00:37:21.380 align:middle line:84% So you should already have learned 00:37:21.380 --> 00:37:23.540 align:middle line:90% this in quantum mechanics. 00:37:23.540 --> 00:37:28.280 align:middle line:84% So scattering is the key thing, OK? 00:37:28.280 --> 00:37:37.980 align:middle line:84% So that's essentially the universal approach 00:37:37.980 --> 00:37:41.770 align:middle line:84% we have been using for more than 100 years, starting 00:37:41.770 --> 00:37:48.520 align:middle line:84% from Rutherford, who first shoot the helium atom at the-- 00:37:48.520 --> 00:37:53.600 align:middle line:84% who first shooting electrons at the helium atom. 00:37:53.600 --> 00:37:54.240 align:middle line:90% OK? 00:37:54.240 --> 00:37:56.020 align:middle line:90% No, no, no, no, no. 00:37:56.020 --> 00:37:58.330 align:middle line:84% Shoot the alpha particles at the-- 00:37:58.330 --> 00:38:00.350 align:middle line:90% alpha particles at atoms. 00:38:00.350 --> 00:38:09.600 align:middle line:84% And so the scattering-- and then, so you have Rutherford, 00:38:09.600 --> 00:38:15.790 align:middle line:84% and then you have the so-called deep inelastic scattering, DIS, 00:38:15.790 --> 00:38:18.280 align:middle line:84% which you figured out that the proton actually 00:38:18.280 --> 00:38:22.900 align:middle line:84% has substructure, which leads to the discovery of quarks. 00:38:22.900 --> 00:38:27.940 align:middle line:84% And actually, MIT played a key role, Friedman and Kendall, 00:38:27.940 --> 00:38:32.710 align:middle line:84% in the DIS experiment, which figured out the structure-- 00:38:32.710 --> 00:38:37.900 align:middle line:84% which discovered that the quarks were-- 00:38:37.900 --> 00:38:42.980 align:middle line:84% yeah, they played the key role in that. 00:38:42.980 --> 00:38:49.540 align:middle line:84% So and still, many, many interactions and the particles 00:38:49.540 --> 00:38:51.880 align:middle line:90% were discovered this way. 00:38:51.880 --> 00:39:00.500 align:middle line:84% And so the basic idea of scattering-- 00:39:00.500 --> 00:39:06.170 align:middle line:90% 00:39:06.170 --> 00:39:07.735 align:middle line:84% so essentially, it's very simple. 00:39:07.735 --> 00:39:10.280 align:middle line:90% 00:39:10.280 --> 00:39:23.220 align:middle line:84% You just collide a bunch of particles, OK? 00:39:23.220 --> 00:39:24.660 align:middle line:90% And just examine the outcome. 00:39:24.660 --> 00:39:34.520 align:middle line:90% 00:39:34.520 --> 00:39:36.860 align:middle line:84% OK, so from the outcome, you try to deduce 00:39:36.860 --> 00:39:39.560 align:middle line:84% what are the interactions between those particles? 00:39:39.560 --> 00:39:42.050 align:middle line:84% OK, so that's what the experiment-- 00:39:42.050 --> 00:39:44.570 align:middle line:90% scattering experiment about. 00:39:44.570 --> 00:39:50.930 align:middle line:84% OK, so from here, you deduce the interaction, which has always 00:39:50.930 --> 00:39:53.040 align:middle line:90% been very successful. 00:39:53.040 --> 00:39:57.350 align:middle line:84% So in condensed matter, you have neutron scattering, et cetera. 00:39:57.350 --> 00:40:00.110 align:middle line:84% X-ray, you can use X-ray and the neutrons 00:40:00.110 --> 00:40:02.135 align:middle line:90% to shine on your samples. 00:40:02.135 --> 00:40:04.830 align:middle line:90% 00:40:04.830 --> 00:40:08.960 align:middle line:84% And one of the most important observables 00:40:08.960 --> 00:40:14.200 align:middle line:84% for the scattering experiment is what? 00:40:14.200 --> 00:40:16.943 align:middle line:84% Again, this is a simple question of quantum mechanics. 00:40:16.943 --> 00:40:17.735 align:middle line:90% AUDIENCE: Momentum? 00:40:17.735 --> 00:40:18.398 align:middle line:90% HONG LIU: Hmm? 00:40:18.398 --> 00:40:20.240 align:middle line:90% AUDIENCE: Momentum? 00:40:20.240 --> 00:40:21.267 align:middle line:90% HONG LIU: Sorry? 00:40:21.267 --> 00:40:22.338 align:middle line:90% AUDIENCE: Momentum? 00:40:22.338 --> 00:40:23.838 align:middle line:84% HONG LIU: No, momentum is not that-- 00:40:23.838 --> 00:40:25.152 align:middle line:90% AUDIENCE: And cross section. 00:40:25.152 --> 00:40:26.030 align:middle line:90% Cross section. 00:40:26.030 --> 00:40:26.120 align:middle line:90% 00:40:26.120 --> 00:40:27.495 align:middle line:84% HONG LIU: Cross section is close, 00:40:27.495 --> 00:40:30.573 align:middle line:90% but there's a more fancy word. 00:40:30.573 --> 00:40:31.448 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:40:31.448 --> 00:40:33.840 align:middle line:90% HONG LIU: Hmm? 00:40:33.840 --> 00:40:36.680 align:middle line:84% AUDIENCE: Differential cross section? 00:40:36.680 --> 00:40:40.890 align:middle line:84% HONG LIU: That's fancier, but not fancy enough. 00:40:40.890 --> 00:40:41.765 align:middle line:90% It's called S-matrix. 00:40:41.765 --> 00:40:46.310 align:middle line:90% 00:40:46.310 --> 00:40:47.810 align:middle line:84% AUDIENCE: No, I said S-matrix, but I 00:40:47.810 --> 00:40:48.740 align:middle line:90% didn't say it loud enough. 00:40:48.740 --> 00:40:49.448 align:middle line:90% HONG LIU: Oh, OK. 00:40:49.448 --> 00:40:50.900 align:middle line:90% Yeah, just be brave. 00:40:50.900 --> 00:40:53.090 align:middle line:90% Be braver next time. 00:40:53.090 --> 00:40:58.440 align:middle line:90% 00:40:58.440 --> 00:41:01.520 align:middle line:90% So one of the key elements-- 00:41:01.520 --> 00:41:07.910 align:middle line:84% yeah, so one of the key observables for scattering 00:41:07.910 --> 00:41:09.680 align:middle line:90% is a so-called the S-matrix. 00:41:09.680 --> 00:41:18.190 align:middle line:84% OK, to define S-matrix, we need a little bit 00:41:18.190 --> 00:41:23.980 align:middle line:84% of mathematical idealization, just as always in physics. 00:41:23.980 --> 00:41:26.160 align:middle line:84% So the mathematical idealization, 00:41:26.160 --> 00:41:28.980 align:middle line:84% in order to define S-matrix, is the following. 00:41:28.980 --> 00:41:36.770 align:middle line:90% 00:41:36.770 --> 00:41:42.780 align:middle line:84% So at t equal to infinity, t equal to minus infinity, 00:41:42.780 --> 00:41:47.775 align:middle line:90% prepare initial state-- 00:41:47.775 --> 00:41:56.750 align:middle line:90% 00:41:56.750 --> 00:42:06.200 align:middle line:84% yeah, initial state may be the initial state as localized, 00:42:06.200 --> 00:42:13.105 align:middle line:84% say, wave packet infinitely far apart. 00:42:13.105 --> 00:42:20.290 align:middle line:90% 00:42:20.290 --> 00:42:23.520 align:middle line:84% So this is the mathematical idealization part, OK? 00:42:23.520 --> 00:42:25.245 align:middle line:84% We want them to be infinitely far apart. 00:42:25.245 --> 00:42:32.260 align:middle line:90% 00:42:32.260 --> 00:42:34.120 align:middle line:90% But you aim them, OK? 00:42:34.120 --> 00:42:38.050 align:middle line:84% So you prepare them spatially infinitely far apart, 00:42:38.050 --> 00:42:41.470 align:middle line:84% but you aim of them-- you prepare their initial momentum 00:42:41.470 --> 00:42:43.780 align:middle line:84% so that they can come together at some point. 00:42:43.780 --> 00:42:46.020 align:middle line:90% You aim them. 00:42:46.020 --> 00:42:48.690 align:middle line:84% And then they will come together at some point, 00:42:48.690 --> 00:42:50.610 align:middle line:90% and then they will scatter. 00:42:50.610 --> 00:42:53.190 align:middle line:84% They will interact with each other. 00:42:53.190 --> 00:42:57.930 align:middle line:84% And then they will form some final product, 00:42:57.930 --> 00:43:01.290 align:middle line:84% and then the final product will then run away from each other, 00:43:01.290 --> 00:43:01.950 align:middle line:90% OK? 00:43:01.950 --> 00:43:03.820 align:middle line:84% They will have velocity momentum. 00:43:03.820 --> 00:43:04.920 align:middle line:90% They will run away. 00:43:04.920 --> 00:43:06.720 align:middle line:90% And then you just wait. 00:43:06.720 --> 00:43:09.030 align:middle line:84% Wait until the products are far apart, 00:43:09.030 --> 00:43:11.160 align:middle line:90% then you match with them. 00:43:11.160 --> 00:43:15.300 align:middle line:84% So mathematically, this is called t equals infinity-- 00:43:15.300 --> 00:43:17.760 align:middle line:84% yeah, of course, t equal to plus-minus infinity 00:43:17.760 --> 00:43:20.550 align:middle line:84% is also a mathematical idealization. 00:43:20.550 --> 00:43:28.860 align:middle line:84% And the final particles are also far apart. 00:43:28.860 --> 00:43:32.250 align:middle line:90% Again, infinitely far apart. 00:43:32.250 --> 00:43:38.520 align:middle line:84% Because they are all relativistic particles and they 00:43:38.520 --> 00:43:41.450 align:middle line:90% all move. 00:43:41.450 --> 00:43:43.430 align:middle line:84% And even with slightly momentum difference, 00:43:43.430 --> 00:43:44.810 align:middle line:90% eventually, they will be-- 00:43:44.810 --> 00:43:47.570 align:middle line:84% if you wait long enough time, then they will be-- 00:43:47.570 --> 00:43:51.120 align:middle line:90% 00:43:51.120 --> 00:43:53.740 align:middle line:84% so this final path are very important, 00:43:53.740 --> 00:44:11.050 align:middle line:84% these two, because this means we can neglect the interactions. 00:44:11.050 --> 00:44:17.770 align:middle line:90% 00:44:17.770 --> 00:44:22.856 align:middle line:84% OK, as t goes to plus-minus infinity. 00:44:22.856 --> 00:44:27.600 align:middle line:84% OK, so with this set up, and then t equals 00:44:27.600 --> 00:44:31.200 align:middle line:84% plus-minus infinity, they are very far away. 00:44:31.200 --> 00:44:32.950 align:middle line:84% And then you can neglect the interactions. 00:44:32.950 --> 00:44:33.630 align:middle line:90% This is important. 00:44:33.630 --> 00:44:35.100 align:middle line:84% Then we can identify each particle, 00:44:35.100 --> 00:44:36.720 align:middle line:84% because when they are together, they 00:44:36.720 --> 00:44:39.420 align:middle line:84% all interacting with each other, it's 00:44:39.420 --> 00:44:41.730 align:middle line:84% not easy to cleanly identify each particles 00:44:41.730 --> 00:44:43.710 align:middle line:90% and discuss their properties. 00:44:43.710 --> 00:44:47.610 align:middle line:84% And so now, we can neglect the interactions. 00:44:47.610 --> 00:44:56.620 align:middle line:90% 00:44:56.620 --> 00:45:15.990 align:middle line:84% So the initial and final states, and then a collection 00:45:15.990 --> 00:45:17.160 align:middle line:90% of free particles. 00:45:17.160 --> 00:45:24.890 align:middle line:90% 00:45:24.890 --> 00:45:26.615 align:middle line:84% And they can be described by free theory. 00:45:26.615 --> 00:45:35.050 align:middle line:90% 00:45:35.050 --> 00:45:38.950 align:middle line:84% So this is very important because remember, 00:45:38.950 --> 00:45:41.290 align:middle line:84% we cannot solve this kind of interacting theory exactly. 00:45:41.290 --> 00:45:44.040 align:middle line:90% 00:45:44.040 --> 00:45:49.470 align:middle line:84% So now, but we can reliably almost to infinitely accuracy 00:45:49.470 --> 00:45:53.640 align:middle line:84% if you talk about the initial state and the final state. 00:45:53.640 --> 00:45:55.830 align:middle line:84% And we can talk about them as precisely 00:45:55.830 --> 00:45:59.820 align:middle line:84% as we can, as far as we go to t going to plus-minus infinity. 00:45:59.820 --> 00:46:00.600 align:middle line:90% Yes? 00:46:00.600 --> 00:46:04.503 align:middle line:84% AUDIENCE: So how do you know adding a phi to the fourth term 00:46:04.503 --> 00:46:05.420 align:middle line:90% leads to interactions? 00:46:05.420 --> 00:46:08.035 align:middle line:84% Just because its the simplest term that [INAUDIBLE].. 00:46:08.035 --> 00:46:09.520 align:middle line:84% HONG LIU: Yeah, just in principle, 00:46:09.520 --> 00:46:14.100 align:middle line:84% any nonlinear term will lead to interactions. 00:46:14.100 --> 00:46:16.410 align:middle line:84% Yeah, the intuition is that anything 00:46:16.410 --> 00:46:19.680 align:middle line:84% beyond the linear level, you will have different phi's 00:46:19.680 --> 00:46:20.620 align:middle line:90% coming together. 00:46:20.620 --> 00:46:24.870 align:middle line:84% So whether it's phi cubed, or phi four, so higher powers 00:46:24.870 --> 00:46:26.940 align:middle line:90% or exponentials. 00:46:26.940 --> 00:46:30.108 align:middle line:84% Yeah, just anything which is not linear phi's 00:46:30.108 --> 00:46:31.650 align:middle line:84% have to do something with each other, 00:46:31.650 --> 00:46:33.690 align:middle line:90% and that just means interaction. 00:46:33.690 --> 00:46:34.290 align:middle line:90% Yeah. 00:46:34.290 --> 00:46:34.635 align:middle line:90% Yes? 00:46:34.635 --> 00:46:36.427 align:middle line:84% AUDIENCE: Why are you doing the interaction 00:46:36.427 --> 00:46:40.950 align:middle line:84% from the minus one half m squared phi squared term? 00:46:40.950 --> 00:46:43.383 align:middle line:90% HONG LIU: Sorry, say it again. 00:46:43.383 --> 00:46:44.300 align:middle line:90% AUDIENCE: So we have-- 00:46:44.300 --> 00:46:45.550 align:middle line:90% HONG LIU: Oh yeah, yeah, yeah. 00:46:45.550 --> 00:46:47.420 align:middle line:84% Yeah, this does not give you interactions 00:46:47.420 --> 00:46:48.530 align:middle line:90% because this just-- 00:46:48.530 --> 00:46:52.130 align:middle line:84% remember, when you do equation of motion, that still gives you 00:46:52.130 --> 00:46:52.880 align:middle line:90% a linear term. 00:46:52.880 --> 00:46:53.300 align:middle line:90% AUDIENCE: Oh, OK. 00:46:53.300 --> 00:46:54.510 align:middle line:84% HONG LIU: Linear term just give you a mass. 00:46:54.510 --> 00:46:54.790 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:46:54.790 --> 00:46:55.873 align:middle line:90% HONG LIU: Yeah yeah, yeah. 00:46:55.873 --> 00:46:56.840 align:middle line:90% Right, right. 00:46:56.840 --> 00:46:58.610 align:middle line:90% Yeah, yeah, yeah, yeah, yeah. 00:46:58.610 --> 00:47:03.320 align:middle line:84% Yeah, in the Lagrangian, the quadratic term, a free term. 00:47:03.320 --> 00:47:05.210 align:middle line:84% So since above the quadratic term 00:47:05.210 --> 00:47:06.380 align:middle line:90% gives you the interactions. 00:47:06.380 --> 00:47:07.270 align:middle line:90% Yeah. 00:47:07.270 --> 00:47:09.210 align:middle line:90% Yes. 00:47:09.210 --> 00:47:09.710 align:middle line:90% Yes? 00:47:09.710 --> 00:47:11.607 align:middle line:84% AUDIENCE: If I know what the interaction potential is 00:47:11.607 --> 00:47:14.240 align:middle line:84% to the particles, like gluon or something like that, how 00:47:14.240 --> 00:47:15.425 align:middle line:90% can I translate that into-- 00:47:15.425 --> 00:47:16.550 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:47:16.550 --> 00:47:19.250 align:middle line:90% Here you can not translate. 00:47:19.250 --> 00:47:22.070 align:middle line:84% Here, I'm just giving you the intuition that somehow when 00:47:22.070 --> 00:47:23.810 align:middle line:84% you add such kind of nonlinear term 00:47:23.810 --> 00:47:26.510 align:middle line:84% that should lead to some kind of interactions. 00:47:26.510 --> 00:47:29.720 align:middle line:84% But to understand what precisely kind of interactions, 00:47:29.720 --> 00:47:31.643 align:middle line:84% you have to look at physical observables. 00:47:31.643 --> 00:47:33.560 align:middle line:84% So that's why we need to look at the S-matrix. 00:47:33.560 --> 00:47:37.250 align:middle line:84% Even for this theory, we want to say, suppose 00:47:37.250 --> 00:47:41.000 align:middle line:84% we have this theory, but in real life, 00:47:41.000 --> 00:47:46.550 align:middle line:84% how do we see in the real experiment? 00:47:46.550 --> 00:47:49.460 align:middle line:84% Yeah, if this theory can be realized 00:47:49.460 --> 00:47:51.500 align:middle line:84% in the real experiment, what it looks like? 00:47:51.500 --> 00:47:54.410 align:middle line:84% And the only way to see it is through this S-matrix. 00:47:54.410 --> 00:47:55.340 align:middle line:90% Yeah, yeah. 00:47:55.340 --> 00:48:00.020 align:middle line:84% And so this S-matrix, and then yeah, we can deduce back. 00:48:00.020 --> 00:48:00.520 align:middle line:90% Good? 00:48:00.520 --> 00:48:05.470 align:middle line:90% 00:48:05.470 --> 00:48:06.510 align:middle line:90% OK, good? 00:48:06.510 --> 00:48:09.810 align:middle line:84% So this is important because this makes sure 00:48:09.810 --> 00:48:14.790 align:middle line:84% that even when we're unable to solve the theory exactly, 00:48:14.790 --> 00:48:18.510 align:middle line:84% we can still talk about initial and final state exactly. 00:48:18.510 --> 00:48:19.780 align:middle line:90% OK? 00:48:19.780 --> 00:48:23.770 align:middle line:84% At least we can define our problem precisely. 00:48:23.770 --> 00:48:26.530 align:middle line:84% So let's denote the collection of initial particle 00:48:26.530 --> 00:48:31.010 align:middle line:84% to be alpha, the momentum of initial particles. 00:48:31.010 --> 00:48:32.540 align:middle line:84% So essentially, this characterizes 00:48:32.540 --> 00:48:35.780 align:middle line:90% your initial state. 00:48:35.780 --> 00:48:38.472 align:middle line:84% And because here we're talking about scalar particles. 00:48:38.472 --> 00:48:40.055 align:middle line:84% They don't have spin, so the momentum, 00:48:40.055 --> 00:48:43.100 align:middle line:90% they are only quantum numbers. 00:48:43.100 --> 00:48:48.020 align:middle line:84% And say I use beta to describe their final momentum. 00:48:48.020 --> 00:48:51.530 align:middle line:90% I call them p 1 prime, say p n. 00:48:51.530 --> 00:48:53.780 align:middle line:84% So the initial number of particles 00:48:53.780 --> 00:48:57.980 align:middle line:84% and final number of particles don't have to be the same. 00:48:57.980 --> 00:49:01.010 align:middle line:84% And then the scattering process is just 00:49:01.010 --> 00:49:03.350 align:middle line:90% to go from alpha to beta. 00:49:03.350 --> 00:49:06.200 align:middle line:84% OK, you start a bunch of particles at t equal to 00:49:06.200 --> 00:49:08.120 align:middle line:84% minus infinity with those momentum, 00:49:08.120 --> 00:49:11.780 align:middle line:84% and then at t plus infinity, you observe some other particles 00:49:11.780 --> 00:49:13.400 align:middle line:90% with those momentum, OK? 00:49:13.400 --> 00:49:15.740 align:middle line:84% And then we want to understand the transition 00:49:15.740 --> 00:49:16.860 align:middle line:90% amplitude between them. 00:49:16.860 --> 00:49:19.490 align:middle line:84% So that's why the experiment matches. 00:49:19.490 --> 00:49:21.950 align:middle line:84% And from here, of course, you get the cross section, 00:49:21.950 --> 00:49:23.682 align:middle line:90% et cetera. 00:49:23.682 --> 00:49:25.265 align:middle line:84% But this is a more fundamental object. 00:49:25.265 --> 00:49:33.520 align:middle line:90% 00:49:33.520 --> 00:49:36.680 align:middle line:90% So to write this precisely-- 00:49:36.680 --> 00:49:48.810 align:middle line:90% 00:49:48.810 --> 00:49:50.850 align:middle line:84% so essentially, we are looking at, say, 00:49:50.850 --> 00:49:56.800 align:middle line:84% in the Heisenberg picture, this is my final state. 00:49:56.800 --> 00:50:01.830 align:middle line:84% And I look at the evolution from minus infinity 00:50:01.830 --> 00:50:05.200 align:middle line:84% to plus infinity, start from initial state alpha. 00:50:05.200 --> 00:50:10.990 align:middle line:84% So this is the evolution operator 00:50:10.990 --> 00:50:14.310 align:middle line:90% of your quantum system. 00:50:14.310 --> 00:50:18.180 align:middle line:84% And so sometimes, we also write this as the-- so this 00:50:18.180 --> 00:50:20.190 align:middle line:90% is in the Schrodinger picture. 00:50:20.190 --> 00:50:23.520 align:middle line:84% And in the Heisenberg picture, we write it as this. 00:50:23.520 --> 00:50:26.220 align:middle line:90% 00:50:26.220 --> 00:50:29.790 align:middle line:84% So now, this means that this is the state defined at the minus 00:50:29.790 --> 00:50:31.920 align:middle line:84% infinity, this is the state that's 00:50:31.920 --> 00:50:33.557 align:middle line:90% defined as plus infinity. 00:50:33.557 --> 00:50:35.640 align:middle line:84% So this is the notation in the Heisenberg picture. 00:50:35.640 --> 00:50:37.860 align:middle line:84% And this is the notation in the Schrodinger picture. 00:50:37.860 --> 00:50:45.855 align:middle line:84% OK, so this object we often denote. 00:50:45.855 --> 00:50:48.420 align:middle line:90% 00:50:48.420 --> 00:50:50.070 align:middle line:90% So you can worry-- 00:50:50.070 --> 00:50:51.930 align:middle line:84% you can run alpha and the beta over all 00:50:51.930 --> 00:50:55.320 align:middle line:84% possible initial states and all possible final states. 00:50:55.320 --> 00:50:58.470 align:middle line:84% And then that forms a big matrix, 00:50:58.470 --> 00:51:00.930 align:middle line:84% in fact infinite times infinite matrix. 00:51:00.930 --> 00:51:02.790 align:middle line:90% And so this is our S-matrix. 00:51:02.790 --> 00:51:06.160 align:middle line:90% 00:51:06.160 --> 00:51:09.850 align:middle line:84% So essentially, all your secret of interactions 00:51:09.850 --> 00:51:12.350 align:middle line:90% are encoded in here. 00:51:12.350 --> 00:51:12.890 align:middle line:90% Yes? 00:51:12.890 --> 00:51:15.040 align:middle line:84% AUDIENCE: Why is it an infinite dimensional matrix? 00:51:15.040 --> 00:51:16.540 align:middle line:84% HONG LIU: Yeah, because you can-- 00:51:16.540 --> 00:51:21.640 align:middle line:84% yeah, so you have a value for any choice of alpha 00:51:21.640 --> 00:51:22.990 align:middle line:90% and the beta, right? 00:51:22.990 --> 00:51:27.430 align:middle line:84% And so any choice of alpha and the beta is one matrix element. 00:51:27.430 --> 00:51:30.463 align:middle line:84% But you have infinite possible choice of initial momentum. 00:51:30.463 --> 00:51:32.630 align:middle line:84% Yeah, you can have an arbitrary number of particles, 00:51:32.630 --> 00:51:34.460 align:middle line:84% you can have a arbitrary momentum. 00:51:34.460 --> 00:51:37.585 align:middle line:84% And similarly, in principle, for the final state. 00:51:37.585 --> 00:51:38.085 align:middle line:90% Yeah. 00:51:38.085 --> 00:51:41.058 align:middle line:84% AUDIENCE: And all this is one kind of particle? 00:51:41.058 --> 00:51:41.850 align:middle line:90% Or how do you know? 00:51:41.850 --> 00:51:42.475 align:middle line:90% HONG LIU: Yeah. 00:51:42.475 --> 00:51:44.280 align:middle line:84% Yeah, here, for this particular problem, 00:51:44.280 --> 00:51:45.570 align:middle line:90% it's one kind of particle. 00:51:45.570 --> 00:51:47.072 align:middle line:84% But this definition is very general. 00:51:47.072 --> 00:51:48.280 align:middle line:90% This is a very good question. 00:51:48.280 --> 00:51:51.550 align:middle line:84% It doesn't matter whether you have one kind of particle, 00:51:51.550 --> 00:51:54.780 align:middle line:84% two kinds of particle, even an infinite number of particles. 00:51:54.780 --> 00:51:55.620 align:middle line:90% Yeah. 00:51:55.620 --> 00:51:57.520 align:middle line:84% An infinite species of particles. 00:51:57.520 --> 00:52:00.540 align:middle line:84% So here, even for one kind the particle, 00:52:00.540 --> 00:52:02.970 align:middle line:84% I have infinite possible choices because I 00:52:02.970 --> 00:52:05.580 align:middle line:84% have infinite possible choice of all those momentums. 00:52:05.580 --> 00:52:09.240 align:middle line:84% I can choose k equal to 1, 2, 3, to infinity. 00:52:09.240 --> 00:52:11.430 align:middle line:84% And also, the value of those k can all change. 00:52:11.430 --> 00:52:13.010 align:middle line:90% P can change. 00:52:13.010 --> 00:52:13.670 align:middle line:90% So yeah. 00:52:13.670 --> 00:52:15.587 align:middle line:84% So this is the infinite times infinite matrix. 00:52:15.587 --> 00:52:20.440 align:middle line:90% 00:52:20.440 --> 00:52:25.212 align:middle line:90% OK, so essentially, this S-- 00:52:25.212 --> 00:52:26.920 align:middle line:84% this is essentially just a matrix element 00:52:26.920 --> 00:52:29.598 align:middle line:90% of this evolution operator. 00:52:29.598 --> 00:52:31.390 align:middle line:84% OK, essentially, this is the matrix element 00:52:31.390 --> 00:52:32.473 align:middle line:90% of the evolution operator. 00:52:32.473 --> 00:52:36.010 align:middle line:84% You can see it from here explicitly from minus infinity 00:52:36.010 --> 00:52:36.910 align:middle line:90% to plus infinity. 00:52:36.910 --> 00:52:42.190 align:middle line:90% OK, good? 00:52:42.190 --> 00:52:45.450 align:middle line:90% 00:52:45.450 --> 00:52:48.770 align:middle line:84% So it's convenient-- so let me just 00:52:48.770 --> 00:52:55.020 align:middle line:84% say a few things about the S-matrix. 00:52:55.020 --> 00:52:58.290 align:middle line:84% So it's convenient-- so when the interaction is weak-- 00:52:58.290 --> 00:53:06.940 align:middle line:90% 00:53:06.940 --> 00:53:10.708 align:middle line:84% so when the interaction is weak, you can imagine, 00:53:10.708 --> 00:53:12.500 align:middle line:84% most of the time, the particles, they don't 00:53:12.500 --> 00:53:15.740 align:middle line:90% interact with each other, OK? 00:53:15.740 --> 00:53:18.590 align:middle line:84% So when you do the scattering, you 00:53:18.590 --> 00:53:20.400 align:middle line:84% put a bunch of particles scattering-- 00:53:20.400 --> 00:53:22.160 align:middle line:84% if the interaction is very weak, then 00:53:22.160 --> 00:53:26.660 align:middle line:84% they don't actually interact very much. 00:53:26.660 --> 00:53:32.590 align:middle line:84% So we expect-- so when the S equals to identity, 00:53:32.590 --> 00:53:35.340 align:middle line:84% then that means there's no interaction at all. 00:53:35.340 --> 00:53:39.100 align:middle line:84% OK, for free theory, it will be just the identity. 00:53:39.100 --> 00:53:41.350 align:middle line:84% So now, if you're seeing that the interaction is weak, 00:53:41.350 --> 00:53:43.720 align:middle line:84% and then it's convenient, we write the S-matrix 00:53:43.720 --> 00:53:48.940 align:middle line:84% in the following form, i T, and this includes the interaction 00:53:48.940 --> 00:53:49.540 align:middle line:90% part. 00:53:49.540 --> 00:53:54.170 align:middle line:90% OK, captures interacting effect. 00:53:54.170 --> 00:53:59.475 align:middle line:90% 00:53:59.475 --> 00:53:59.975 align:middle line:90% OK? 00:53:59.975 --> 00:54:06.790 align:middle line:90% 00:54:06.790 --> 00:54:08.270 align:middle line:90% Good? 00:54:08.270 --> 00:54:13.860 align:middle line:84% And by definition-- yeah, any questions on this? 00:54:13.860 --> 00:54:14.370 align:middle line:90% Yes? 00:54:14.370 --> 00:54:17.194 align:middle line:84% AUDIENCE: So I was interpreting the elements of the S-matrix 00:54:17.194 --> 00:54:19.294 align:middle line:84% as sort of the probability to end up 00:54:19.294 --> 00:54:21.270 align:middle line:84% in a certain state from the initial state. 00:54:21.270 --> 00:54:21.895 align:middle line:90% HONG LIU: Yeah. 00:54:21.895 --> 00:54:25.590 align:middle line:84% Yeah, S-matrix is the amplitude, the probability 00:54:25.590 --> 00:54:26.580 align:middle line:90% you have to square it. 00:54:26.580 --> 00:54:27.090 align:middle line:90% AUDIENCE: Oh, yes. 00:54:27.090 --> 00:54:27.715 align:middle line:90% HONG LIU: Yeah. 00:54:27.715 --> 00:54:30.030 align:middle line:84% AUDIENCE: But then why does it make sense 00:54:30.030 --> 00:54:33.132 align:middle line:84% that if it's a weak interaction, that it would-- 00:54:33.132 --> 00:54:35.510 align:middle line:84% like, as the limit goes to no interaction, 00:54:35.510 --> 00:54:38.140 align:middle line:84% the S-matrix would be normal at the identity? 00:54:38.140 --> 00:54:39.390 align:middle line:90% HONG LIU: Sorry, say it again. 00:54:39.390 --> 00:54:41.640 align:middle line:84% AUDIENCE: So why do we see that for weak interactions, 00:54:41.640 --> 00:54:42.410 align:middle line:90% it's the identity? 00:54:42.410 --> 00:54:44.160 align:middle line:84% HONG LIU: Yeah, if there's no interaction, 00:54:44.160 --> 00:54:45.150 align:middle line:90% it will be identity. 00:54:45.150 --> 00:54:46.775 align:middle line:84% AUDIENCE: But why does that make sense? 00:54:46.775 --> 00:54:48.780 align:middle line:84% Because that's just saying the amplitude 00:54:48.780 --> 00:54:52.140 align:middle line:84% is 1 to end up in a certain state? 00:54:52.140 --> 00:54:54.420 align:middle line:84% HONG LIU: Yeah, just you always go back to-- 00:54:54.420 --> 00:54:57.390 align:middle line:90% the state does not change. 00:54:57.390 --> 00:54:59.310 align:middle line:90% What it means is that-- 00:54:59.310 --> 00:55:00.910 align:middle line:90% it means delta alpha beta. 00:55:00.910 --> 00:55:02.317 align:middle line:84% Because alpha must equal to beta. 00:55:02.317 --> 00:55:03.150 align:middle line:90% AUDIENCE: Oh, I see. 00:55:03.150 --> 00:55:03.650 align:middle line:90% I see. 00:55:03.650 --> 00:55:07.530 align:middle line:84% HONG LIU: Yeah, what it means is just delta alpha beta, right? 00:55:07.530 --> 00:55:09.255 align:middle line:84% So it means the alpha equals to beta. 00:55:09.255 --> 00:55:09.963 align:middle line:90% AUDIENCE: Thanks. 00:55:09.963 --> 00:55:11.760 align:middle line:90% HONG LIU: Yeah, yeah. 00:55:11.760 --> 00:55:12.480 align:middle line:90% Yes? 00:55:12.480 --> 00:55:14.840 align:middle line:84% AUDIENCE: Is S a Hermitian Matrix? 00:55:14.840 --> 00:55:16.760 align:middle line:84% HONG LIU: Yeah, we will talk about that. 00:55:16.760 --> 00:55:18.530 align:middle line:90% It's not Hermitian it's unitary. 00:55:18.530 --> 00:55:19.070 align:middle line:90% Yeah. 00:55:19.070 --> 00:55:19.817 align:middle line:90% Yes? 00:55:19.817 --> 00:55:21.650 align:middle line:84% AUDIENCE: Does the phase of the entries in S 00:55:21.650 --> 00:55:23.320 align:middle line:90% carry any physical consequences? 00:55:23.320 --> 00:55:25.210 align:middle line:84% HONG LIU: Yeah, that's a good question. 00:55:25.210 --> 00:55:28.120 align:middle line:90% 00:55:28.120 --> 00:55:30.880 align:middle line:84% So if you can see the particular amplitude when you square it, 00:55:30.880 --> 00:55:32.560 align:middle line:90% of course, the phase don't. 00:55:32.560 --> 00:55:35.200 align:middle line:84% But in principle, you can design the different-- 00:55:35.200 --> 00:55:38.487 align:middle line:84% you can design experiments in the different things they 00:55:38.487 --> 00:55:39.070 align:middle line:90% can interfere. 00:55:39.070 --> 00:55:39.570 align:middle line:90% Yeah. 00:55:39.570 --> 00:55:42.010 align:middle line:84% Yeah, so in principle, they contain physical information. 00:55:42.010 --> 00:55:43.990 align:middle line:90% Yeah. 00:55:43.990 --> 00:55:44.900 align:middle line:90% Yes? 00:55:44.900 --> 00:55:48.010 align:middle line:84% AUDIENCE: Why is this entry complex? 00:55:48.010 --> 00:55:49.260 align:middle line:90% HONG LIU: Or just convenience. 00:55:49.260 --> 00:55:49.818 align:middle line:90% AUDIENCE: OK. 00:55:49.818 --> 00:55:51.360 align:middle line:84% HONG LIU: Yeah, just for convenience. 00:55:51.360 --> 00:55:51.860 align:middle line:90% Yeah. 00:55:51.860 --> 00:55:54.160 align:middle line:90% 00:55:54.160 --> 00:55:58.300 align:middle line:84% AUDIENCE: Yeah, if alpha was a particle at minus x infinity, 00:55:58.300 --> 00:56:04.790 align:middle line:84% beta was the same particle at plus x infinity? 00:56:04.790 --> 00:56:06.580 align:middle line:90% HONG LIU: Sorry, say it again. 00:56:06.580 --> 00:56:09.360 align:middle line:84% AUDIENCE: If my alpha to beta was just particle here 00:56:09.360 --> 00:56:11.825 align:middle line:84% goes to particle there, is that captured by S equals 1? 00:56:11.825 --> 00:56:12.450 align:middle line:90% HONG LIU: Yeah. 00:56:12.450 --> 00:56:13.230 align:middle line:90% Yeah, yeah. 00:56:13.230 --> 00:56:13.772 align:middle line:90% AUDIENCE: OK. 00:56:13.772 --> 00:56:16.380 align:middle line:84% HONG LIU: Just say in the free theory, 00:56:16.380 --> 00:56:19.380 align:middle line:84% if you start with a bunch of particles at p 1 and p k 00:56:19.380 --> 00:56:23.230 align:middle line:84% then you end with the same p 1 p k Yeah. 00:56:23.230 --> 00:56:23.730 align:middle line:90% Yeah. 00:56:23.730 --> 00:56:26.190 align:middle line:90% Yeah. 00:56:26.190 --> 00:56:28.960 align:middle line:90% OK? 00:56:28.960 --> 00:56:33.030 align:middle line:90% So due to translation symmetry-- 00:56:33.030 --> 00:56:37.250 align:middle line:84% so we expect energy momentum conservation. 00:56:37.250 --> 00:56:50.090 align:middle line:90% 00:56:50.090 --> 00:56:52.340 align:middle line:84% OK, so yeah, so this, if you write it 00:56:52.340 --> 00:56:57.570 align:middle line:84% in terms of the component rotation, 00:56:57.570 --> 00:57:02.690 align:middle line:84% so this is delta beta alpha plus i T beta alpha. 00:57:02.690 --> 00:57:05.630 align:middle line:84% OK, and i just purely convention. 00:57:05.630 --> 00:57:07.820 align:middle line:84% i is purely convention, just for convenience. 00:57:07.820 --> 00:57:12.930 align:middle line:90% 00:57:12.930 --> 00:57:14.970 align:middle line:84% So due to the momentum conservation-- 00:57:14.970 --> 00:57:17.830 align:middle line:90% 00:57:17.830 --> 00:57:20.530 align:middle line:84% so this corresponding to the interaction part. 00:57:20.530 --> 00:57:23.310 align:middle line:84% So in this part, we expect alpha not equal to beta 00:57:23.310 --> 00:57:26.230 align:middle line:90% because you have interactions. 00:57:26.230 --> 00:57:28.270 align:middle line:84% But the total momentum must be conserved. 00:57:28.270 --> 00:57:33.550 align:middle line:84% So the sum of here must be the sum of here. 00:57:33.550 --> 00:57:37.320 align:middle line:84% Just the total momentum must be conserved. 00:57:37.320 --> 00:57:41.720 align:middle line:84% So that means this scene must contain a delta function 00:57:41.720 --> 00:57:43.670 align:middle line:84% between the total initial momentum 00:57:43.670 --> 00:57:46.970 align:middle line:90% and the total final momentum. 00:57:46.970 --> 00:57:50.345 align:middle line:84% And so we can just convenient to extract-- 00:57:50.345 --> 00:57:53.047 align:middle line:90% 00:57:53.047 --> 00:57:54.380 align:middle line:90% to separate that delta function. 00:57:54.380 --> 00:58:08.460 align:middle line:84% OK, so convenient to write i T beta alpha equal to i, 00:58:08.460 --> 00:58:10.340 align:middle line:84% you separate out this delta function. 00:58:10.340 --> 00:58:13.640 align:middle line:84% Because we know the delta function must be there, 00:58:13.640 --> 00:58:15.500 align:middle line:90% so p alpha minus p beta. 00:58:15.500 --> 00:58:18.950 align:middle line:84% So p alpha and p beta means total momentum 00:58:18.950 --> 00:58:20.060 align:middle line:90% for alpha and the beta. 00:58:20.060 --> 00:58:23.630 align:middle line:90% And then, you have M alpha beta. 00:58:23.630 --> 00:58:25.310 align:middle line:90% So this is the nontrivial part. 00:58:25.310 --> 00:58:26.850 align:middle line:90% So this part is trivial. 00:58:26.850 --> 00:58:28.700 align:middle line:90% You expect from kinematics. 00:58:28.700 --> 00:58:30.770 align:middle line:84% And so this M alpha beta is called 00:58:30.770 --> 00:58:32.720 align:middle line:90% the scattering amplitude. 00:58:32.720 --> 00:58:42.770 align:middle line:90% 00:58:42.770 --> 00:58:51.795 align:middle line:84% So now some properties of the S-matrix because you see that 00:58:51.795 --> 00:58:56.070 align:middle line:84% S-matrix is essentially just a matrix element-- 00:58:56.070 --> 00:59:00.160 align:middle line:84% essentially just the matrix of the U. And the U 00:59:00.160 --> 00:59:04.250 align:middle line:84% is a unitary matrix, because it's a evolution operator. 00:59:04.250 --> 00:59:18.370 align:middle line:84% OK, so that means that the S-matrix is unitary since U-- 00:59:18.370 --> 00:59:24.010 align:middle line:84% this is a unitary operator, so that means 00:59:24.010 --> 00:59:29.060 align:middle line:90% that S is a unitary matrix. 00:59:29.060 --> 00:59:35.820 align:middle line:90% 00:59:35.820 --> 00:59:38.430 align:middle line:90% Also, you can show-- 00:59:38.430 --> 00:59:41.520 align:middle line:84% somebody be running out of time because today, 00:59:41.520 --> 00:59:43.740 align:middle line:90% we have to end at 5:40. 00:59:43.740 --> 00:59:48.090 align:middle line:84% And so for any symmetry of the Hamiltonian-- 00:59:48.090 --> 00:59:57.172 align:middle line:90% 00:59:57.172 --> 01:00:02.670 align:middle line:84% for any symmetry of the Hamiltonian, 01:00:02.670 --> 01:00:06.360 align:middle line:84% that means that if you have some Q-- 01:00:06.360 --> 01:00:10.140 align:middle line:84% yeah, if you have some, say, unitary operator, which 01:00:10.140 --> 01:00:11.625 align:middle line:90% commutes with the Hamiltonian. 01:00:11.625 --> 01:00:14.940 align:middle line:90% 01:00:14.940 --> 01:00:19.160 align:middle line:84% OK, so remember all the symmetries, 01:00:19.160 --> 01:00:25.190 align:middle line:84% the interaction are generated by unitary operators. 01:00:25.190 --> 01:00:28.060 align:middle line:84% OK, so suppose we have some unitary, which 01:00:28.060 --> 01:00:29.500 align:middle line:90% commutes with the Hamiltonian. 01:00:29.500 --> 01:00:36.040 align:middle line:90% 01:00:36.040 --> 01:00:38.420 align:middle line:84% Then, you can easily show yourself-- 01:00:38.420 --> 01:00:43.650 align:middle line:84% so I'll leave this exercise to yourself. 01:00:43.650 --> 01:00:52.200 align:middle line:84% So lambda is a unitary operator generating some symmetries. 01:00:52.200 --> 01:01:01.457 align:middle line:90% 01:01:01.457 --> 01:01:03.540 align:middle line:84% OK, so for example, for the Lorentz transformation 01:01:03.540 --> 01:01:05.340 align:middle line:84% would be like this e to the i omega 01:01:05.340 --> 01:01:09.540 align:middle line:84% M, which you have done in your homework. 01:01:09.540 --> 01:01:12.465 align:middle line:90% And then it will satisfy-- 01:01:12.465 --> 01:01:17.070 align:middle line:90% 01:01:17.070 --> 01:01:19.830 align:middle line:84% then when you act initial state and the final state 01:01:19.830 --> 01:01:27.380 align:middle line:84% by this lambda, and then the matrix should be the same. 01:01:27.380 --> 01:01:30.560 align:middle line:84% So you can easily prove this by using the commute. 01:01:30.560 --> 01:01:32.290 align:middle line:90% OK? 01:01:32.290 --> 01:01:32.950 align:middle line:90% Yes? 01:01:32.950 --> 01:01:34.408 align:middle line:84% AUDIENCE: Is another equivalent way 01:01:34.408 --> 01:01:37.390 align:middle line:84% of saying that like S and lambda also commute? 01:01:37.390 --> 01:01:40.848 align:middle line:90% 01:01:40.848 --> 01:01:41.390 align:middle line:90% HONG LIU: No. 01:01:41.390 --> 01:01:41.950 align:middle line:90% No, no, no. 01:01:41.950 --> 01:01:47.230 align:middle line:90% No, they don't. 01:01:47.230 --> 01:01:48.730 align:middle line:90% Yeah. 01:01:48.730 --> 01:01:51.430 align:middle line:84% Yeah, that's the statement, right? 01:01:51.430 --> 01:01:52.730 align:middle line:90% This is a statement. 01:01:52.730 --> 01:01:54.290 align:middle line:90% So as S-matrix element. 01:01:54.290 --> 01:01:55.090 align:middle line:90% Yeah. 01:01:55.090 --> 01:01:55.750 align:middle line:90% Yeah. 01:01:55.750 --> 01:01:59.502 align:middle line:84% Yes, it's a statement that the lambda commute with U. 01:01:59.502 --> 01:02:00.210 align:middle line:90% AUDIENCE: OK, OK. 01:02:00.210 --> 01:02:02.020 align:middle line:84% HONG LIU: Yeah, lambda commute with U and S 01:02:02.020 --> 01:02:03.760 align:middle line:90% is the matrix element of U. 01:02:03.760 --> 01:02:04.480 align:middle line:90% AUDIENCE: OK, OK. 01:02:04.480 --> 01:02:04.670 align:middle line:90% Thank you. 01:02:04.670 --> 01:02:05.795 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 01:02:05.795 --> 01:02:08.290 align:middle line:90% 01:02:08.290 --> 01:02:12.280 align:middle line:84% OK, so this is the key object we want to calculate 01:02:12.280 --> 01:02:13.690 align:middle line:90% in the interacting theory. 01:02:13.690 --> 01:02:16.220 align:middle line:90% 01:02:16.220 --> 01:02:20.000 align:middle line:90% So now, without proof-- 01:02:20.000 --> 01:02:24.890 align:middle line:84% because the proof requires something going beyond what 01:02:24.890 --> 01:02:26.820 align:middle line:90% we discussed so far. 01:02:26.820 --> 01:02:34.640 align:middle line:84% So let me just code this LSZ theorem, which 01:02:34.640 --> 01:02:40.300 align:middle line:84% said that the M alpha beta, this scattering amplitude, 01:02:40.300 --> 01:02:40.975 align:middle line:90% can be obtained. 01:02:40.975 --> 01:02:46.740 align:middle line:90% 01:02:46.740 --> 01:02:49.920 align:middle line:84% So we will discuss how you obtain in nature 01:02:49.920 --> 01:02:57.470 align:middle line:84% from such correlation functions, time-ordered correlation 01:02:57.470 --> 01:02:58.220 align:middle line:90% functions. 01:02:58.220 --> 01:03:02.670 align:middle line:90% 01:03:02.670 --> 01:03:08.270 align:middle line:84% So this omega is the vacuum for the full interaction theory. 01:03:08.270 --> 01:03:10.770 align:middle line:90% OK, so this is the vacuum. 01:03:10.770 --> 01:03:13.070 align:middle line:84% So I write omega to distinguish it 01:03:13.070 --> 01:03:17.130 align:middle line:84% from 0, which is the free theory we discussed earlier. 01:03:17.130 --> 01:03:19.800 align:middle line:84% And this time-ordering means that you order the operator 01:03:19.800 --> 01:03:21.180 align:middle line:90% according to their time. 01:03:21.180 --> 01:03:27.720 align:middle line:84% So whichever time is bigger sits earlier, OK? 01:03:27.720 --> 01:03:31.050 align:middle line:84% So this is the object, key object, 01:03:31.050 --> 01:03:33.450 align:middle line:84% if you want to calculate the M alpha beta. 01:03:33.450 --> 01:03:36.900 align:middle line:84% And you have to calculate this object first. 01:03:36.900 --> 01:03:40.510 align:middle line:90% OK, and so I will stop here. 01:03:40.510 --> 01:03:45.180 align:middle line:84% And so next time, we will say a few more words 01:03:45.180 --> 01:03:49.620 align:middle line:84% about how the general approach to treat this kind of theory, 01:03:49.620 --> 01:03:52.890 align:middle line:84% and then we will start talking about the path integral. 01:03:52.890 --> 01:03:57.100 align:middle line:84% So have people studied the path integral before? 01:03:57.100 --> 01:03:59.453 align:middle line:84% So you are a expert of path integral or not quite? 01:03:59.453 --> 01:04:00.870 align:middle line:84% AUDIENCE: Expert is a strong word. 01:04:00.870 --> 01:04:02.070 align:middle line:84% AUDIENCE: I know we've heard of it. 01:04:02.070 --> 01:04:02.640 align:middle line:90% HONG LIU: OK. 01:04:02.640 --> 01:04:07.440 align:middle line:84% Yeah, so I will review the path integral of-- 01:04:07.440 --> 01:04:10.770 align:middle line:90% yeah, I will actually-- 01:04:10.770 --> 01:04:13.830 align:middle line:84% so I will actually introduce the path integral from scratch, 01:04:13.830 --> 01:04:18.400 align:middle line:84% but I will do it a little bit faster than ordinary quantum 01:04:18.400 --> 01:04:19.390 align:middle line:90% mechanical class. 01:04:19.390 --> 01:04:21.670 align:middle line:84% And even for people who have not seen it before, 01:04:21.670 --> 01:04:24.340 align:middle line:84% I think you should be able to follow 01:04:24.340 --> 01:04:26.290 align:middle line:84% with a little bit of effort after the class. 01:04:26.290 --> 01:04:26.790 align:middle line:90% OK. 01:04:26.790 --> 01:04:28.140 align:middle line:90% AUDIENCE: Good point. 01:04:28.140 --> 01:04:32.000 align:middle line:90%