WEBVTT 00:00:00.000 --> 00:00:02.450 align:middle line:90% [SQUEAKING] 00:00:02.450 --> 00:00:03.430 align:middle line:90% [RUSTLING] 00:00:03.430 --> 00:00:04.410 align:middle line:90% 00:00:04.410 --> 00:00:10.680 align:middle line:90% [CLICKING] 00:00:10.680 --> 00:00:15.660 align:middle line:84% HONG LIU: So again, let us first recall 00:00:15.660 --> 00:00:23.790 align:middle line:84% say if we want to calculate the n-point function 00:00:23.790 --> 00:00:25.170 align:middle line:90% in the interacting theory. 00:00:25.170 --> 00:00:33.250 align:middle line:90% 00:00:33.250 --> 00:00:35.980 align:middle line:84% And this is given by this following ratio 00:00:35.980 --> 00:00:36.850 align:middle line:90% in the free theory. 00:00:36.850 --> 00:00:41.120 align:middle line:90% 00:00:41.120 --> 00:00:44.410 align:middle line:84% You look at this T, then you look at this. 00:00:44.410 --> 00:00:47.290 align:middle line:84% Again, take this product to be X, 00:00:47.290 --> 00:00:56.840 align:middle line:84% take the T X, exponential i S_I, then in the free theory, 00:00:56.840 --> 00:01:03.891 align:middle line:84% and then divide it just by time order i 00:01:03.891 --> 00:01:06.650 align:middle line:90% S_I, in the free theory. 00:01:06.650 --> 00:01:14.030 align:middle line:90% 00:01:14.030 --> 00:01:17.390 align:middle line:84% So then what we do is we just expand these two exponentials 00:01:17.390 --> 00:01:19.880 align:middle line:90% in power series. 00:01:19.880 --> 00:01:22.950 align:middle line:84% We expand this exponential in power series. 00:01:22.950 --> 00:01:26.980 align:middle line:84% And so when you expand in power series, 00:01:26.980 --> 00:01:38.630 align:middle line:84% then, for example the nth order term in the upstairs, 00:01:38.630 --> 00:01:47.570 align:middle line:84% say upstairs, then have the following structure. 00:01:47.570 --> 00:01:52.250 align:middle line:84% You have i to the power n, n factorial. 00:01:52.250 --> 00:02:01.110 align:middle line:84% Then you have T X then S_I to the power to n S_I. 00:02:01.110 --> 00:02:07.040 align:middle line:90% 00:02:07.040 --> 00:02:11.630 align:middle line:84% And the nth order for the downstairs, 00:02:11.630 --> 00:02:14.600 align:middle line:84% you can also expand the downstairs. 00:02:14.600 --> 00:02:21.340 align:middle line:90% 00:02:21.340 --> 00:02:31.160 align:middle line:84% So in the downstairs similarly then you have almost identical 00:02:31.160 --> 00:02:36.560 align:middle line:84% structure, except you don't have this X. So you have T, 00:02:36.560 --> 00:02:38.517 align:middle line:90% then you just have n S_I's. 00:02:38.517 --> 00:02:41.810 align:middle line:90% 00:02:41.810 --> 00:02:43.430 align:middle line:90% So there's n of them. 00:02:43.430 --> 00:02:47.322 align:middle line:90% 00:02:47.322 --> 00:02:48.280 align:middle line:90% So there are n of them. 00:02:48.280 --> 00:02:51.250 align:middle line:90% 00:02:51.250 --> 00:02:57.580 align:middle line:84% And the last time we described how to compute 00:02:57.580 --> 00:03:00.030 align:middle line:90% say a typical terms in here. 00:03:00.030 --> 00:03:04.680 align:middle line:84% Let me call this equation star, equation star star. 00:03:04.680 --> 00:03:13.830 align:middle line:84% So last time we described the star and the star star 00:03:13.830 --> 00:03:20.840 align:middle line:84% can be computed using Wick's theorem. 00:03:20.840 --> 00:03:29.480 align:middle line:90% 00:03:29.480 --> 00:03:34.410 align:middle line:84% But in practice what we do is that we first 00:03:34.410 --> 00:03:46.410 align:middle line:84% draw Feynman diagrams and then we convert the Feynman diagrams 00:03:46.410 --> 00:03:49.530 align:middle line:90% into analytic expressions. 00:03:49.530 --> 00:03:52.230 align:middle line:90% 00:03:52.230 --> 00:03:58.590 align:middle line:84% Normally we go through this procedure, 00:03:58.590 --> 00:04:02.520 align:middle line:84% rather than directly do the Wick contraction. 00:04:02.520 --> 00:04:03.465 align:middle line:90% We mentioned the-- 00:04:03.465 --> 00:04:08.460 align:middle line:90% 00:04:08.460 --> 00:04:11.130 align:middle line:84% And the second line, of course, can 00:04:11.130 --> 00:04:14.970 align:middle line:84% be considered as a special case of the first line, 00:04:14.970 --> 00:04:18.310 align:middle line:90% with this x to be identity. 00:04:18.310 --> 00:04:19.589 align:middle line:90% This x to be identity. 00:04:19.589 --> 00:04:23.320 align:middle line:90% 00:04:23.320 --> 00:04:41.315 align:middle line:84% So any diagram contributing to star because the-- 00:04:41.315 --> 00:04:43.880 align:middle line:90% 00:04:43.880 --> 00:04:47.270 align:middle line:84% Sorry I should actually here I have n, let me call it m. 00:04:47.270 --> 00:04:51.785 align:middle line:90% 00:04:51.785 --> 00:04:55.610 align:middle line:84% I should call it m so that they don't-- 00:04:55.610 --> 00:04:57.980 align:middle line:90% because we already have n here. 00:04:57.980 --> 00:05:03.190 align:middle line:90% 00:05:03.190 --> 00:05:04.360 align:middle line:90% So mth order. 00:05:04.360 --> 00:05:08.020 align:middle line:90% 00:05:08.020 --> 00:05:13.935 align:middle line:84% Any diagram contributing to the star has n external legs. 00:05:13.935 --> 00:05:23.090 align:middle line:90% 00:05:23.090 --> 00:05:27.630 align:middle line:84% So that's come just from this x, from this phi x1 and the phi 00:05:27.630 --> 00:05:28.130 align:middle line:90% xn. 00:05:28.130 --> 00:05:31.040 align:middle line:90% 00:05:31.040 --> 00:05:43.570 align:middle line:84% And then also with m vertices because each vertex 00:05:43.570 --> 00:05:45.630 align:middle line:90% comes from a power of S_I. 00:05:45.630 --> 00:05:53.120 align:middle line:90% 00:05:53.120 --> 00:06:11.390 align:middle line:84% But for star star, diagrams contributing to star star 00:06:11.390 --> 00:06:17.200 align:middle line:84% have no external legs because there's no x here, 00:06:17.200 --> 00:06:19.070 align:middle line:90% there's no phi here. 00:06:19.070 --> 00:06:20.510 align:middle line:90% There's no external legs. 00:06:20.510 --> 00:06:27.400 align:middle line:90% 00:06:27.400 --> 00:06:29.935 align:middle line:84% So that's why they're all called vacuum diagrams. 00:06:29.935 --> 00:06:37.437 align:middle line:90% 00:06:37.437 --> 00:06:38.895 align:middle line:84% They're called the vacuum diagrams. 00:06:38.895 --> 00:06:42.600 align:middle line:90% 00:06:42.600 --> 00:06:47.910 align:middle line:84% The downstairs can be understood as in the case which you just 00:06:47.910 --> 00:06:51.390 align:middle line:90% essentially take X equal to 1. 00:06:51.390 --> 00:07:00.060 align:middle line:84% So essentially we are calculating some kind 00:07:00.060 --> 00:07:02.820 align:middle line:90% of unnormalized transition. 00:07:02.820 --> 00:07:06.580 align:middle line:84% This is the overlap between the vacuum itself 00:07:06.580 --> 00:07:08.940 align:middle line:84% from minus infinity to plus infinity. 00:07:08.940 --> 00:07:14.970 align:middle line:90% 00:07:14.970 --> 00:07:24.900 align:middle line:84% In Pset 6, just in the Pset we just posted last night, 00:07:24.900 --> 00:07:29.790 align:middle line:84% you can show that this quantity, which 00:07:29.790 --> 00:07:32.790 align:middle line:84% comes from summing over all possible vacuum 00:07:32.790 --> 00:07:37.860 align:middle line:84% diagrams, the diagrams without external legs-- 00:07:37.860 --> 00:07:40.870 align:middle line:84% Sorry, I should say have no external legs. 00:07:40.870 --> 00:07:44.080 align:middle line:90% This no is very important. 00:07:44.080 --> 00:07:47.410 align:middle line:84% Yeah, has no external legs and so they are called 00:07:47.410 --> 00:07:48.265 align:middle line:90% vacuum diagrams. 00:07:48.265 --> 00:07:50.830 align:middle line:90% 00:07:50.830 --> 00:07:53.410 align:middle line:84% In Pset 6 you will show yourself. 00:07:53.410 --> 00:07:56.170 align:middle line:84% It's a pretty simple calculation, 00:07:56.170 --> 00:07:57.730 align:middle line:90% but it's an instructive one. 00:07:57.730 --> 00:08:00.400 align:middle line:90% 00:08:00.400 --> 00:08:07.090 align:middle line:84% You can show that this quantity can 00:08:07.090 --> 00:08:27.080 align:middle line:84% be used to calculate the vacuum energy of the interacting 00:08:27.080 --> 00:08:27.580 align:middle line:90% theory. 00:08:27.580 --> 00:08:32.559 align:middle line:84% Of course, this is divergent, but just as the vacuum energy 00:08:32.559 --> 00:08:37.840 align:middle line:84% we calculated before in free theory, 00:08:37.840 --> 00:08:41.409 align:middle line:84% this one can be used to calculate the vacuum energy 00:08:41.409 --> 00:08:42.790 align:middle line:90% of the interacting theory. 00:08:42.790 --> 00:08:53.660 align:middle line:90% 00:08:53.660 --> 00:08:56.960 align:middle line:84% Of course, this quantity you will see its divergent, 00:08:56.960 --> 00:09:00.140 align:middle line:84% but this is the formal way to calculate it. 00:09:00.140 --> 00:09:03.020 align:middle line:90% 00:09:03.020 --> 00:09:06.470 align:middle line:84% And how to remove the divergences etc. 00:09:06.470 --> 00:09:12.680 align:middle line:84% is something we will not discuss in this class. 00:09:12.680 --> 00:09:16.260 align:middle line:90% It will be in QFT 2. 00:09:16.260 --> 00:09:17.250 align:middle line:90% Any questions on this? 00:09:17.250 --> 00:09:25.600 align:middle line:90% 00:09:25.600 --> 00:09:28.120 align:middle line:90% OK, good. 00:09:28.120 --> 00:09:29.695 align:middle line:90% Let me say a few more words. 00:09:29.695 --> 00:09:34.550 align:middle line:90% 00:09:34.550 --> 00:09:40.860 align:middle line:84% When we compute two, there we say 00:09:40.860 --> 00:09:43.480 align:middle line:84% they can be computed using the Feynman diagram, 00:09:43.480 --> 00:09:48.330 align:middle line:84% so let me be a little bit more explicit. 00:09:48.330 --> 00:09:56.970 align:middle line:84% The computation of star or star star 00:09:56.970 --> 00:09:58.380 align:middle line:90% essentially involves two steps. 00:09:58.380 --> 00:10:04.850 align:middle line:90% 00:10:04.850 --> 00:10:09.570 align:middle line:84% Just elaborate that remark in the bracket. 00:10:09.570 --> 00:10:15.735 align:middle line:84% Is that we first draw all possible -- 00:10:15.735 --> 00:10:21.490 align:middle line:84% all inequivalent Feynman diagrams. 00:10:21.490 --> 00:10:30.730 align:middle line:90% 00:10:30.730 --> 00:10:40.350 align:middle line:84% In this case, at each mth order you 00:10:40.350 --> 00:10:43.200 align:middle line:84% have an n points, n external points, 00:10:43.200 --> 00:10:44.740 align:middle line:90% and then you have m vertices. 00:10:44.740 --> 00:10:47.340 align:middle line:90% 00:10:47.340 --> 00:10:55.020 align:middle line:84% There you have say for star you just have n external points, 00:10:55.020 --> 00:11:00.120 align:middle line:90% say x1 x2 in coordinate space. 00:11:00.120 --> 00:11:03.600 align:middle line:84% You have n external points and then you have m vertices. 00:11:03.600 --> 00:11:08.050 align:middle line:90% And at each vertex you have-- 00:11:08.050 --> 00:11:13.920 align:middle line:90% let's call it y1, y2, ym-- 00:11:13.920 --> 00:11:17.790 align:middle line:84% and at each vertex you have four legs coming out. 00:11:17.790 --> 00:11:27.680 align:middle line:84% At each external point you have one line come out, 00:11:27.680 --> 00:11:31.520 align:middle line:84% and then you just need to find all possible ways to connect 00:11:31.520 --> 00:11:33.260 align:middle line:90% them. 00:11:33.260 --> 00:11:38.390 align:middle line:84% That is what we mean by joining all inequivalent diagrams. 00:11:38.390 --> 00:11:42.980 align:middle line:84% This -- you just draw it, it becomes mechanical. 00:11:42.980 --> 00:11:44.020 align:middle line:90% Yes? 00:11:44.020 --> 00:11:46.590 align:middle line:84% AUDIENCE: Why do the vertices have to all have four legs? 00:11:46.590 --> 00:11:51.640 align:middle line:90% 00:11:51.640 --> 00:12:01.770 align:middle line:84% HONG LIU: More specifically here we're talking about the S_I 00:12:01.770 --> 00:12:07.555 align:middle line:84% equals to minus lambda 4 factorial. 00:12:07.555 --> 00:12:09.180 align:middle line:84% We can see that this particular theory. 00:12:09.180 --> 00:12:12.490 align:middle line:90% 00:12:12.490 --> 00:12:17.470 align:middle line:84% I'm going to make remarks on other more general theories, 00:12:17.470 --> 00:12:21.570 align:middle line:90% but here it's more specific. 00:12:21.570 --> 00:12:22.560 align:middle line:90% Other questions? 00:12:22.560 --> 00:12:28.510 align:middle line:90% 00:12:28.510 --> 00:12:30.490 align:middle line:90% So this is for star. 00:12:30.490 --> 00:12:35.470 align:middle line:84% So for star star, then you just don't have the x, 00:12:35.470 --> 00:12:36.490 align:middle line:90% you just have y's. 00:12:36.490 --> 00:12:41.710 align:middle line:84% You just have m y's and each y has four legs. 00:12:41.710 --> 00:12:44.770 align:middle line:84% You just connect all of them together. 00:12:44.770 --> 00:12:47.650 align:middle line:84% You connect all of them together. 00:12:47.650 --> 00:12:51.860 align:middle line:84% In the star star, in the vacuum diagrams, 00:12:51.860 --> 00:12:55.690 align:middle line:84% it will be always closed because there's no external leg. 00:12:55.690 --> 00:12:57.970 align:middle line:84% Just because all the legs will be contracted, 00:12:57.970 --> 00:13:00.790 align:middle line:84% and so it will be a closed diagram. 00:13:00.790 --> 00:13:03.040 align:middle line:84% Sometimes it's also called the vacuum bubble. 00:13:03.040 --> 00:13:05.755 align:middle line:84% Sometimes we call them vacuum diagrams or vacuum bubbles. 00:13:05.755 --> 00:13:08.940 align:middle line:90% 00:13:08.940 --> 00:13:12.555 align:middle line:84% Here just know it's always closed diagrams. 00:13:12.555 --> 00:13:24.040 align:middle line:90% 00:13:24.040 --> 00:13:30.200 align:middle line:84% And the second step is that now once you 00:13:30.200 --> 00:13:32.480 align:middle line:84% draw all the inequivalent diagrams, 00:13:32.480 --> 00:13:45.160 align:middle line:84% then you can just convert each diagram 00:13:45.160 --> 00:13:59.245 align:middle line:84% to an algebraic expression using Feynman rules. 00:13:59.245 --> 00:14:09.810 align:middle line:90% 00:14:09.810 --> 00:14:12.060 align:middle line:84% Then you just sum all of them together. 00:14:12.060 --> 00:14:14.040 align:middle line:84% Then you sum all of them together. 00:14:14.040 --> 00:14:16.710 align:middle line:84% And then we talked about the Feynman rules last time. 00:14:16.710 --> 00:14:20.273 align:middle line:84% There's coordinate space rules, there's momentum space rules. 00:14:20.273 --> 00:14:21.690 align:middle line:84% You can do it in coordinate space, 00:14:21.690 --> 00:14:23.430 align:middle line:90% you can do it in momentum space. 00:14:23.430 --> 00:14:26.895 align:middle line:84% I will not repeat them here to save time. 00:14:26.895 --> 00:14:30.870 align:middle line:90% 00:14:30.870 --> 00:14:33.450 align:middle line:84% Once we finish the second step, then 00:14:33.450 --> 00:14:35.910 align:middle line:84% we just have a bunch of analytic expressions 00:14:35.910 --> 00:14:38.910 align:middle line:84% and then you just need to do the integrals. 00:14:38.910 --> 00:14:42.840 align:middle line:84% So a bunch of integrals and you just need to do the integrals. 00:14:42.840 --> 00:14:46.130 align:middle line:90% 00:14:46.130 --> 00:14:48.340 align:middle line:90% Any questions? 00:14:48.340 --> 00:14:50.590 align:middle line:90% Good. 00:14:50.590 --> 00:14:52.195 align:middle line:90% Some further remarks on 1. 00:14:52.195 --> 00:14:56.120 align:middle line:90% 00:14:56.120 --> 00:15:06.945 align:middle line:84% Further remarks on this first step. 00:15:06.945 --> 00:15:12.020 align:middle line:90% 00:15:12.020 --> 00:15:15.780 align:middle line:84% Here I consider a very special case. 00:15:15.780 --> 00:15:17.460 align:middle line:84% Here I consider a very special case 00:15:17.460 --> 00:15:21.540 align:middle line:84% corresponding to this S_I to consist 00:15:21.540 --> 00:15:24.750 align:middle line:90% of a single term, phi 4. 00:15:24.750 --> 00:15:38.505 align:middle line:84% In principle, of course, S_I can contain multiple terms. 00:15:38.505 --> 00:15:44.710 align:middle line:90% 00:15:44.710 --> 00:15:45.985 align:middle line:90% Can contain multiple terms. 00:15:45.985 --> 00:15:53.230 align:middle line:90% 00:15:53.230 --> 00:15:55.960 align:middle line:84% It can contain multiple terms, then 00:15:55.960 --> 00:16:01.490 align:middle line:84% you will have multiple types of vertices. 00:16:01.490 --> 00:16:06.600 align:middle line:90% 00:16:06.600 --> 00:16:08.610 align:middle line:90% But the rule is similar. 00:16:08.610 --> 00:16:21.810 align:middle line:84% You just say suppose S_I contains two terms, 00:16:21.810 --> 00:16:25.380 align:middle line:90% then you just multiply them. 00:16:25.380 --> 00:16:28.167 align:middle line:90% Then you'll get many terms here. 00:16:28.167 --> 00:16:30.000 align:middle line:84% Just corresponding to different combinations 00:16:30.000 --> 00:16:35.380 align:middle line:84% of vertices for different types of vertices. 00:16:35.380 --> 00:16:37.630 align:middle line:84% But the rule is exactly the same, 00:16:37.630 --> 00:16:40.390 align:middle line:84% just now your diagrams are more complicated. 00:16:40.390 --> 00:16:43.000 align:middle line:90% 00:16:43.000 --> 00:16:47.020 align:middle line:84% But essentially otherwise it's identical. 00:16:47.020 --> 00:16:49.920 align:middle line:90% 00:16:49.920 --> 00:16:54.980 align:middle line:84% Another thing to mention is in principle we can also 00:16:54.980 --> 00:16:56.130 align:middle line:90% have multiple fields. 00:16:56.130 --> 00:16:59.630 align:middle line:84% So here we can see only a single scalar field, phi, 00:16:59.630 --> 00:17:03.200 align:middle line:84% but you can also, in principle, consider more than one field. 00:17:03.200 --> 00:17:16.339 align:middle line:84% So if there are multiple fields, the story 00:17:16.339 --> 00:17:20.880 align:middle line:84% is, again, very much parallel, and the only difference 00:17:20.880 --> 00:17:33.550 align:middle line:84% is that each field, each propagator, 00:17:33.550 --> 00:17:35.320 align:middle line:84% is represented by a different line. 00:17:35.320 --> 00:17:48.540 align:middle line:90% 00:17:48.540 --> 00:17:57.960 align:middle line:84% Other than that, these two steps again apply. 00:17:57.960 --> 00:18:01.260 align:middle line:84% Just now you actually have different type of lines. 00:18:01.260 --> 00:18:06.890 align:middle line:90% 00:18:06.890 --> 00:18:09.890 align:middle line:84% For example, if you consider a theory with two scalar fields, 00:18:09.890 --> 00:18:20.480 align:middle line:84% phi 1 phi 2 with S_I proportional to phi 1 00:18:20.480 --> 00:18:26.750 align:middle line:84% squared phi 2 squared, and then in this case 00:18:26.750 --> 00:18:30.380 align:middle line:84% you will have a propagator for phi 1, 00:18:30.380 --> 00:18:33.380 align:middle line:84% then you also have a propagator for phi 2. 00:18:33.380 --> 00:18:36.730 align:middle line:90% Now I can use a dashed line. 00:18:36.730 --> 00:18:38.672 align:middle line:84% Then the interacting vertex then will 00:18:38.672 --> 00:18:39.880 align:middle line:90% have the following structure. 00:18:39.880 --> 00:18:48.620 align:middle line:84% You have two solid lines and the two dashed lines. 00:18:48.620 --> 00:18:49.655 align:middle line:90% Now that's your vertex. 00:18:49.655 --> 00:18:57.430 align:middle line:90% 00:18:57.430 --> 00:19:02.350 align:middle line:84% Again, you just contract all possible lines 00:19:02.350 --> 00:19:04.720 align:middle line:90% according to your Feynman rules. 00:19:04.720 --> 00:19:10.140 align:middle line:90% 00:19:10.140 --> 00:19:10.640 align:middle line:90% Good? 00:19:10.640 --> 00:19:13.920 align:middle line:90% 00:19:13.920 --> 00:19:19.430 align:middle line:84% Also one remark I want to emphasize 00:19:19.430 --> 00:19:20.975 align:middle line:90% is that the momentum space-- 00:19:20.975 --> 00:19:25.670 align:middle line:90% 00:19:25.670 --> 00:19:30.025 align:middle line:84% also just another point to stress-- 00:19:30.025 --> 00:19:32.720 align:middle line:90% 00:19:32.720 --> 00:19:45.200 align:middle line:84% this is obtained by Fourier transform Gn x1 xn 00:19:45.200 --> 00:19:47.600 align:middle line:90% times the delta function. 00:19:47.600 --> 00:19:53.740 align:middle line:90% 00:19:53.740 --> 00:19:59.110 align:middle line:84% This times this gives you the Fourier transform of that. 00:19:59.110 --> 00:20:03.970 align:middle line:84% When you do the Fourier transform, p1 and pn, 00:20:03.970 --> 00:20:07.300 align:middle line:84% they just come from doing Fourier transform corresponding 00:20:07.300 --> 00:20:09.190 align:middle line:90% to momentum for x1, xn. 00:20:09.190 --> 00:20:13.990 align:middle line:84% So here the p1 and pn are just arbitrary. 00:20:13.990 --> 00:20:17.020 align:middle line:90% 00:20:17.020 --> 00:20:21.070 align:middle line:84% They don't have to satisfy any on-shell condition, 00:20:21.070 --> 00:20:23.590 align:middle line:84% they're just arbitrary momentum, and the only constraint 00:20:23.590 --> 00:20:28.630 align:middle line:84% is that this is non-vanishing only when sum of them 00:20:28.630 --> 00:20:33.790 align:middle line:84% is equal to 0, otherwise they can be arbitrary. 00:20:33.790 --> 00:20:41.040 align:middle line:90% 00:20:41.040 --> 00:20:42.270 align:middle line:90% Good? 00:20:42.270 --> 00:20:46.670 align:middle line:84% So in practice, now we have a way 00:20:46.670 --> 00:20:50.630 align:middle line:84% to calculate the diagram for both upstairs and downstairs, 00:20:50.630 --> 00:20:54.020 align:middle line:84% and now we just do the expansion. 00:20:54.020 --> 00:20:56.240 align:middle line:90% We just do the expansion. 00:20:56.240 --> 00:21:02.240 align:middle line:84% Now let me summarize the example we did last time now just only 00:21:02.240 --> 00:21:04.190 align:middle line:90% using diagrams. 00:21:04.190 --> 00:21:10.082 align:middle line:84% So for example, if we consider the two point function, 00:21:10.082 --> 00:21:11.665 align:middle line:84% let's consider the two point function. 00:21:11.665 --> 00:21:17.620 align:middle line:90% 00:21:17.620 --> 00:21:19.990 align:middle line:90% We have upstairs and downstairs. 00:21:19.990 --> 00:21:25.430 align:middle line:84% The upstairs, the lowest order you just have a single line. 00:21:25.430 --> 00:21:29.770 align:middle line:84% It just has a single line-- x1 to x2. 00:21:29.770 --> 00:21:35.450 align:middle line:84% To save effort we'll also not label x1, x2 from now on. 00:21:35.450 --> 00:21:39.150 align:middle line:90% 00:21:39.150 --> 00:21:44.230 align:middle line:84% The next order is to bring down one power of this S_I. 00:21:44.230 --> 00:21:48.940 align:middle line:84% So you bring down one power of these four indices. 00:21:48.940 --> 00:21:52.330 align:middle line:90% Now there are two possibilities. 00:21:52.330 --> 00:21:54.970 align:middle line:84% One possibility is that you still have this, 00:21:54.970 --> 00:21:59.415 align:middle line:84% but then times this vertex contracts itself. 00:21:59.415 --> 00:22:03.070 align:middle line:90% 00:22:03.070 --> 00:22:07.600 align:middle line:84% And then another possibility is that you 00:22:07.600 --> 00:22:10.740 align:middle line:90% have something like this. 00:22:10.740 --> 00:22:13.060 align:middle line:84% And you contract with the vertex, 00:22:13.060 --> 00:22:16.650 align:middle line:84% so this is the plus lambda squared. 00:22:16.650 --> 00:22:25.710 align:middle line:90% This is to lambda order. 00:22:25.710 --> 00:22:31.080 align:middle line:84% And downstairs you have 1 and then we just essentially 00:22:31.080 --> 00:22:35.790 align:middle line:90% have a single S_I. 00:22:35.790 --> 00:22:39.380 align:middle line:84% A single S_I, the only contraction 00:22:39.380 --> 00:22:40.370 align:middle line:90% is just given by this. 00:22:40.370 --> 00:22:43.680 align:middle line:90% 00:22:43.680 --> 00:22:47.490 align:middle line:84% And then plus order lambda squared. 00:22:47.490 --> 00:22:48.810 align:middle line:90% Plus order lambda squared. 00:22:48.810 --> 00:22:59.730 align:middle line:90% 00:22:59.730 --> 00:23:02.010 align:middle line:90% So now each vertex carry-- 00:23:02.010 --> 00:23:03.510 align:middle line:84% so you should remember, each vertex 00:23:03.510 --> 00:23:06.130 align:middle line:84% carries a parameter lambda, a factor of lambda, 00:23:06.130 --> 00:23:09.240 align:middle line:90% which is considered to be small. 00:23:09.240 --> 00:23:11.560 align:middle line:84% Downstairs you have 1 plus order lambda, 00:23:11.560 --> 00:23:13.530 align:middle line:84% and in the downstairs then we have 00:23:13.530 --> 00:23:16.900 align:middle line:84% 1 plus this power series of lambda, 00:23:16.900 --> 00:23:18.840 align:middle line:84% and then we can do the Taylor expansion again 00:23:18.840 --> 00:23:21.420 align:middle line:90% to bring them to the upstairs. 00:23:21.420 --> 00:23:24.270 align:middle line:90% Remember we did before. 00:23:24.270 --> 00:23:28.420 align:middle line:84% Now if you do that, then at leading order to lambda, 00:23:28.420 --> 00:23:32.310 align:middle line:84% then we still just have x1, x2 because this divided 00:23:32.310 --> 00:23:35.220 align:middle line:84% by that just x1, x2, and then you have this term. 00:23:35.220 --> 00:23:40.290 align:middle line:90% 00:23:40.290 --> 00:23:44.760 align:middle line:84% But now we can bring this term to the upstairs and then 00:23:44.760 --> 00:23:47.490 align:middle line:90% this sign changes to minus sign. 00:23:47.490 --> 00:23:49.410 align:middle line:84% This term can multiply that and then 00:23:49.410 --> 00:23:50.535 align:middle line:90% you get the term like that. 00:23:50.535 --> 00:23:53.470 align:middle line:90% 00:23:53.470 --> 00:23:57.460 align:middle line:84% Coming from bringing this term to upstairs 00:23:57.460 --> 00:23:58.870 align:middle line:90% and then you have this term. 00:23:58.870 --> 00:24:01.790 align:middle line:90% 00:24:01.790 --> 00:24:05.220 align:middle line:84% And the rest, they're all order lambda squared. 00:24:05.220 --> 00:24:09.080 align:middle line:84% So you can also have this change to minus sign, multiply that, 00:24:09.080 --> 00:24:11.720 align:middle line:84% but that will be order lambda squared. 00:24:11.720 --> 00:24:15.030 align:middle line:84% And since here we already neglected lambda squared 00:24:15.030 --> 00:24:18.260 align:middle line:84% and here neglect is lambda squared, 00:24:18.260 --> 00:24:20.570 align:middle line:90% so we don't worry about it. 00:24:20.570 --> 00:24:22.250 align:middle line:84% And if do this and multiply this, 00:24:22.250 --> 00:24:24.470 align:middle line:90% also gives you lambda squared. 00:24:24.470 --> 00:24:26.990 align:middle line:84% Up to order lambda and that you have. 00:24:26.990 --> 00:24:32.500 align:middle line:84% And then as we discussed before, these two cancel. 00:24:32.500 --> 00:24:38.980 align:middle line:84% So in the end you just have two diagrams, just one and then 00:24:38.980 --> 00:24:39.490 align:middle line:90% this one. 00:24:39.490 --> 00:24:45.020 align:middle line:90% 00:24:45.020 --> 00:24:46.250 align:middle line:90% Yes? 00:24:46.250 --> 00:24:49.760 align:middle line:84% AUDIENCE: If we have two fields with a vertex 00:24:49.760 --> 00:24:53.000 align:middle line:84% with like one solid line, one dashed line. 00:24:53.000 --> 00:24:56.660 align:middle line:84% Here, you can read it as the two different particles interacting 00:24:56.660 --> 00:24:58.880 align:middle line:84% with each other or the same species 00:24:58.880 --> 00:25:01.310 align:middle line:84% interacting with each other if you read it like that, 00:25:01.310 --> 00:25:04.078 align:middle line:90% but you lose that if you-- 00:25:04.078 --> 00:25:05.620 align:middle line:84% HONG LIU: For here it doesn't matter. 00:25:05.620 --> 00:25:08.140 align:middle line:84% In this case, it doesn't matter how you draw them. 00:25:08.140 --> 00:25:09.595 align:middle line:90% You mean if I exchange them? 00:25:09.595 --> 00:25:10.220 align:middle line:90% AUDIENCE: Yeah. 00:25:10.220 --> 00:25:11.637 align:middle line:84% HONG LIU: Yeah, it doesn't matter. 00:25:11.637 --> 00:25:14.500 align:middle line:84% AUDIENCE: But in that picture you can't read-- 00:25:14.500 --> 00:25:17.898 align:middle line:84% HONG LIU: Yeah, normally you cannot view up and up and down. 00:25:17.898 --> 00:25:19.690 align:middle line:84% Left and right here does not mean anything. 00:25:19.690 --> 00:25:22.810 align:middle line:90% 00:25:22.810 --> 00:25:26.740 align:middle line:90% The order is not important. 00:25:26.740 --> 00:25:34.030 align:middle line:84% But actually, in QFT 2, you will encounter situations 00:25:34.030 --> 00:25:36.240 align:middle line:84% which these are matrices and then 00:25:36.240 --> 00:25:37.700 align:middle line:90% the order becomes important. 00:25:37.700 --> 00:25:40.390 align:middle line:84% But here they just fields because they commute. 00:25:40.390 --> 00:25:42.800 align:middle line:84% So no matter how you order them doesn't matter. 00:25:42.800 --> 00:25:44.110 align:middle line:90% They're just ordinary fields. 00:25:44.110 --> 00:25:51.390 align:middle line:90% 00:25:51.390 --> 00:25:55.590 align:middle line:84% This order, we just have these two diagrams. 00:25:55.590 --> 00:25:58.350 align:middle line:84% This is much simpler than we did at the first time. 00:25:58.350 --> 00:26:00.900 align:middle line:84% Much simpler than we did the first time and the calculation 00:26:00.900 --> 00:26:04.010 align:middle line:90% is much easier. 00:26:04.010 --> 00:26:06.620 align:middle line:84% Let me just show you the cancellation in a slightly 00:26:06.620 --> 00:26:09.170 align:middle line:84% different way, almost identical, but I 00:26:09.170 --> 00:26:11.900 align:middle line:90% write slightly differently. 00:26:11.900 --> 00:26:13.910 align:middle line:84% Alternatively, I can also write as this. 00:26:13.910 --> 00:26:18.560 align:middle line:90% 00:26:18.560 --> 00:26:21.620 align:middle line:90% I can also write G2. 00:26:21.620 --> 00:26:25.210 align:middle line:90% We look at these two terms. 00:26:25.210 --> 00:26:28.310 align:middle line:84% We can just take a common factor out. 00:26:28.310 --> 00:26:30.250 align:middle line:84% So the upstairs, the first two terms 00:26:30.250 --> 00:26:34.570 align:middle line:84% I can also write it as this times 1 00:26:34.570 --> 00:26:42.350 align:middle line:84% plus this thing and then plus this. 00:26:42.350 --> 00:26:47.780 align:middle line:84% And downstairs I just have plus lambda squared. 00:26:47.780 --> 00:26:49.640 align:middle line:90% And downstairs I just have 1. 00:26:49.640 --> 00:27:00.930 align:middle line:90% 00:27:00.930 --> 00:27:02.620 align:middle line:84% Now you can just look at these two. 00:27:02.620 --> 00:27:05.235 align:middle line:84% You can directly see, actually, these two can be canceled. 00:27:05.235 --> 00:27:07.747 align:middle line:90% 00:27:07.747 --> 00:27:09.330 align:middle line:84% You don't have to worry about this one 00:27:09.330 --> 00:27:16.250 align:middle line:84% because when these multiplied by that, you get higher orders. 00:27:16.250 --> 00:27:18.140 align:middle line:84% Now you just directly also get from here. 00:27:18.140 --> 00:27:29.860 align:middle line:90% 00:27:29.860 --> 00:27:36.520 align:middle line:84% The reason I'm doing these in these two different ways 00:27:36.520 --> 00:27:41.110 align:middle line:84% is that this cancellation is actually not an accident. 00:27:41.110 --> 00:27:44.185 align:middle line:84% Such cancellation, this is not an accident. 00:27:44.185 --> 00:27:50.570 align:middle line:90% 00:27:50.570 --> 00:27:52.210 align:middle line:90% This is not accident. 00:27:52.210 --> 00:27:55.540 align:middle line:84% This actually happens to all orders. 00:27:55.540 --> 00:27:58.780 align:middle line:84% This happens to all orders, so you can actually show. 00:27:58.780 --> 00:28:15.550 align:middle line:90% 00:28:15.550 --> 00:28:23.920 align:middle line:84% If I call the upstairs, now let's 00:28:23.920 --> 00:28:30.265 align:middle line:84% call the upstairs to be Un and this to be U0. 00:28:30.265 --> 00:28:33.430 align:middle line:90% 00:28:33.430 --> 00:28:38.260 align:middle line:84% Un is just the upstairs U0 is the downstairs. 00:28:38.260 --> 00:28:44.110 align:middle line:84% The Un, by definition, corresponding 00:28:44.110 --> 00:28:59.145 align:middle line:84% to sum over all Feynman diagrams with n external legs. 00:28:59.145 --> 00:29:06.850 align:middle line:90% 00:29:06.850 --> 00:29:11.780 align:middle line:84% And U0, by definition, is sum of all Feynman diagrams 00:29:11.780 --> 00:29:13.460 align:middle line:90% with zero external legs. 00:29:13.460 --> 00:29:16.320 align:middle line:90% 00:29:16.320 --> 00:29:17.690 align:middle line:90% With zero external legs. 00:29:17.690 --> 00:29:26.500 align:middle line:90% 00:29:26.500 --> 00:29:31.540 align:middle line:84% So then you can show in general which 00:29:31.540 --> 00:29:38.980 align:middle line:84% the Gn is just equal to, as we wrote there, just Un over U0. 00:29:38.980 --> 00:29:44.200 align:middle line:84% You can show that this defined to be Un over U0. 00:29:44.200 --> 00:29:47.980 align:middle line:90% 00:29:47.980 --> 00:29:52.660 align:middle line:84% You find order, you can show to all orders this downstairs can 00:29:52.660 --> 00:29:53.935 align:middle line:90% always, always cancel. 00:29:53.935 --> 00:29:57.280 align:middle line:90% 00:29:57.280 --> 00:30:03.410 align:middle line:84% The statement is that this is equal to sum over 00:30:03.410 --> 00:30:23.360 align:middle line:84% the ratio of all diagrams of n external legs, 00:30:23.360 --> 00:30:34.635 align:middle line:90% but without any vacuum bubbles. 00:30:34.635 --> 00:30:48.830 align:middle line:90% 00:30:48.830 --> 00:30:50.960 align:middle line:84% Now let me just explain what this means that 00:30:50.960 --> 00:30:52.430 align:middle line:90% without any vacuum bubbles. 00:30:52.430 --> 00:31:02.580 align:middle line:84% OK, so this diagram is a diagram which 00:31:02.580 --> 00:31:07.920 align:middle line:84% contains just a straight propagator times a bubble. 00:31:07.920 --> 00:31:10.770 align:middle line:84% This diagram contains a vacuum bubble. 00:31:10.770 --> 00:31:14.640 align:middle line:84% It contains a vacuum diagram, and this diagram does not. 00:31:14.640 --> 00:31:17.370 align:middle line:84% This diagram is connected to itself. 00:31:17.370 --> 00:31:21.030 align:middle line:90% 00:31:21.030 --> 00:31:23.400 align:middle line:84% Similarly, when you go to next order, 00:31:23.400 --> 00:31:29.440 align:middle line:84% as we discussed last time, this diagram, 00:31:29.440 --> 00:31:35.310 align:middle line:84% so go to next order times, so this is also 00:31:35.310 --> 00:31:37.350 align:middle line:84% a diagram with vacuum bubble because you 00:31:37.350 --> 00:31:43.560 align:middle line:84% have this part, which does not have any external legs. 00:31:43.560 --> 00:31:46.500 align:middle line:90% This will not arise. 00:31:46.500 --> 00:31:50.820 align:middle line:84% When you calculate the ratio, then this kind of diagram 00:31:50.820 --> 00:31:53.830 align:middle line:90% we automatically don't include. 00:31:53.830 --> 00:31:57.160 align:middle line:84% Just don't need to worry about it. 00:31:57.160 --> 00:32:00.040 align:middle line:84% Now when you do the ratio you can just 00:32:00.040 --> 00:32:02.500 align:middle line:84% forget about those diagrams and directly 00:32:02.500 --> 00:32:04.930 align:middle line:84% write down those diagrams without vacuum bubbles. 00:32:04.930 --> 00:32:10.420 align:middle line:84% That just a greatly simplifies your life. 00:32:10.420 --> 00:32:13.280 align:middle line:90% OK One second. 00:32:13.280 --> 00:32:19.310 align:middle line:90% So this statement is simply-- 00:32:19.310 --> 00:32:22.100 align:middle line:90% 00:32:22.100 --> 00:32:24.650 align:middle line:84% If you think a little bit, then you 00:32:24.650 --> 00:32:28.370 align:middle line:84% will see that this cancellation is generic-- 00:32:28.370 --> 00:32:31.760 align:middle line:84% actually will work through all orders-- 00:32:31.760 --> 00:32:35.870 align:middle line:84% and this thinking process is actually quite instructive, 00:32:35.870 --> 00:32:38.810 align:middle line:90% so this is left as a homework. 00:32:38.810 --> 00:32:40.700 align:middle line:84% You can read the part in Peskin which 00:32:40.700 --> 00:32:44.060 align:middle line:84% tries to explain this cancellation, 00:32:44.060 --> 00:32:46.770 align:middle line:84% but that discussion is actually not great. 00:32:46.770 --> 00:32:49.850 align:middle line:84% It's not great, that discussion, but you can still 00:32:49.850 --> 00:32:52.040 align:middle line:90% read it and get the main idea. 00:32:52.040 --> 00:32:55.835 align:middle line:84% And so in homework I ask you to show this yourself. 00:32:55.835 --> 00:32:58.470 align:middle line:90% 00:32:58.470 --> 00:33:00.300 align:middle line:84% It's not difficult once you get the idea. 00:33:00.300 --> 00:33:05.750 align:middle line:84% It's actually pretty simple to see and will be illuminating. 00:33:05.750 --> 00:33:06.845 align:middle line:90% Yes? 00:33:06.845 --> 00:33:08.970 align:middle line:84% AUDIENCE: I'm pretty confused on this cancellation. 00:33:08.970 --> 00:33:12.330 align:middle line:84% So if you're pulling out an order lambda term, 00:33:12.330 --> 00:33:16.175 align:middle line:84% won't your order lambda squared would now be order lambda? 00:33:16.175 --> 00:33:18.175 align:middle line:84% HONG LIU: No, you pull the whole thing together. 00:33:18.175 --> 00:33:22.060 align:middle line:90% 00:33:22.060 --> 00:33:25.015 align:middle line:84% Those terms don't matter at this order. 00:33:25.015 --> 00:33:28.660 align:middle line:90% 00:33:28.660 --> 00:33:32.950 align:middle line:84% Whatever here, when you cancel them, 00:33:32.950 --> 00:33:35.320 align:middle line:84% you cancel the whole thing, this term doesn't matter. 00:33:35.320 --> 00:33:39.790 align:middle line:84% If you care only of all the lambda squared terms, 00:33:39.790 --> 00:33:41.860 align:middle line:84% and then you can just directly cancel these two. 00:33:41.860 --> 00:33:44.400 align:middle line:90% 00:33:44.400 --> 00:33:45.775 align:middle line:84% You can easily convince yourself. 00:33:45.775 --> 00:33:48.578 align:middle line:90% 00:33:48.578 --> 00:33:49.870 align:middle line:90% You don't have to do a diagram. 00:33:49.870 --> 00:33:55.170 align:middle line:84% Just write something-- a 1 plus lambda plus some other lambda 00:33:55.170 --> 00:33:56.617 align:middle line:90% and 1 plus-- 00:33:56.617 --> 00:33:58.200 align:middle line:84% Anyway, you can just convince yourself 00:33:58.200 --> 00:34:00.630 align:middle line:90% that this always happens. 00:34:00.630 --> 00:34:01.980 align:middle line:90% Yes. 00:34:01.980 --> 00:34:05.010 align:middle line:84% AUDIENCE: So this cancellation, does it just 00:34:05.010 --> 00:34:07.140 align:middle line:84% come out of the math or does it is there 00:34:07.140 --> 00:34:10.815 align:middle line:90% like a physical interpretation? 00:34:10.815 --> 00:34:12.940 align:middle line:84% HONG LIU: Yeah, there is a physical interpretation. 00:34:12.940 --> 00:34:16.239 align:middle line:84% Essentially you just say when you look at-- 00:34:16.239 --> 00:34:19.360 align:middle line:84% So this kind of thing essentially 00:34:19.360 --> 00:34:23.590 align:middle line:84% comes from the normalization of your state. 00:34:23.590 --> 00:34:26.500 align:middle line:84% This thing is just pure vacuum process 00:34:26.500 --> 00:34:28.792 align:middle line:84% and just comes from normalization of the states. 00:34:28.792 --> 00:34:30.250 align:middle line:84% And when you normalize your states, 00:34:30.250 --> 00:34:34.242 align:middle line:84% they just should not contribute to your correlation functions. 00:34:34.242 --> 00:34:35.409 align:middle line:90% Just roughly it's like that. 00:34:35.409 --> 00:34:39.750 align:middle line:90% 00:34:39.750 --> 00:34:40.699 align:middle line:90% Other questions? 00:34:40.699 --> 00:34:44.239 align:middle line:90% 00:34:44.239 --> 00:34:46.389 align:middle line:90% OK, good. 00:34:46.389 --> 00:34:50.110 align:middle line:84% Now let's just put this in practice. 00:34:50.110 --> 00:34:51.860 align:middle line:84% Let's put this in practice, and let 00:34:51.860 --> 00:35:00.480 align:middle line:84% me just write down the most general, 00:35:00.480 --> 00:35:03.525 align:middle line:90% all the diagrams for G2. 00:35:03.525 --> 00:35:13.020 align:middle line:90% 00:35:13.020 --> 00:35:16.470 align:middle line:90% To lambda squared order. 00:35:16.470 --> 00:35:20.190 align:middle line:84% Now let's look at G2 to lambda squared order. 00:35:20.190 --> 00:35:22.575 align:middle line:84% We have already found the lambda order. 00:35:22.575 --> 00:35:25.630 align:middle line:90% 00:35:25.630 --> 00:35:28.990 align:middle line:84% Lambda squared order, we already drew a bunch of diagrams 00:35:28.990 --> 00:35:34.270 align:middle line:84% last time, so you only need to keep those which don't 00:35:34.270 --> 00:35:35.635 align:middle line:90% include these vacuum bubbles. 00:35:35.635 --> 00:35:38.440 align:middle line:90% 00:35:38.440 --> 00:35:41.200 align:middle line:90% Let me just write them down. 00:35:41.200 --> 00:35:49.050 align:middle line:84% Including this, including that, and also including the one 00:35:49.050 --> 00:35:51.180 align:middle line:90% I forgot last time-- 00:35:51.180 --> 00:35:53.490 align:middle line:90% this one. 00:35:53.490 --> 00:35:54.720 align:middle line:90% That's it. 00:35:54.720 --> 00:35:58.180 align:middle line:84% And if I remember, last time we drew many more diagrams, 00:35:58.180 --> 00:36:01.840 align:middle line:84% but you don't need to worry about the others, 00:36:01.840 --> 00:36:03.805 align:middle line:90% such as diagrams like this. 00:36:03.805 --> 00:36:10.350 align:middle line:90% 00:36:10.350 --> 00:36:11.360 align:middle line:90% Good. 00:36:11.360 --> 00:36:16.365 align:middle line:84% Let's do another example for the 4 point function. 00:36:16.365 --> 00:36:19.750 align:middle line:90% 00:36:19.750 --> 00:36:24.430 align:middle line:84% Now it's become much easier because you have much fewer 00:36:24.430 --> 00:36:27.290 align:middle line:90% diagrams to consider. 00:36:27.290 --> 00:36:30.220 align:middle line:84% You have much fewer diagrams to consider. 00:36:30.220 --> 00:36:35.290 align:middle line:84% So at the 4 point function level, 00:36:35.290 --> 00:36:40.240 align:middle line:84% then you first have your free theory terms. 00:36:40.240 --> 00:36:42.415 align:middle line:84% So the free theory returns you just have that. 00:36:42.415 --> 00:36:45.760 align:middle line:90% 00:36:45.760 --> 00:36:48.230 align:middle line:90% It just contracted with itself. 00:36:48.230 --> 00:36:49.630 align:middle line:84% Now we have four external points. 00:36:49.630 --> 00:36:54.350 align:middle line:90% 00:36:54.350 --> 00:36:55.910 align:middle line:84% This is coming from the free theory 00:36:55.910 --> 00:36:57.950 align:middle line:90% contribution at the 0th order. 00:36:57.950 --> 00:37:01.970 align:middle line:84% And then the first order, at order lambda, 00:37:01.970 --> 00:37:06.710 align:middle line:84% you just have this, so that all the four different phi's from 00:37:06.710 --> 00:37:11.870 align:middle line:84% the vertex are contracted with each of the external phi's. 00:37:11.870 --> 00:37:14.210 align:middle line:90% This is the first order. 00:37:14.210 --> 00:37:20.765 align:middle line:84% And then you can also have a diagram like this. 00:37:20.765 --> 00:37:24.930 align:middle line:90% 00:37:24.930 --> 00:37:28.280 align:middle line:90% And you can also have-- 00:37:28.280 --> 00:37:29.655 align:middle line:84% I will not show all the diagrams. 00:37:29.655 --> 00:37:32.490 align:middle line:90% 00:37:32.490 --> 00:37:34.480 align:middle line:84% And you can draw more diagrams like this, 00:37:34.480 --> 00:37:36.660 align:middle line:84% but these are not vacuum bubbles. 00:37:36.660 --> 00:37:38.640 align:middle line:84% And you can also have diagrams like this. 00:37:38.640 --> 00:37:48.547 align:middle line:90% 00:37:48.547 --> 00:37:50.130 align:middle line:84% The reason I chose them is because you 00:37:50.130 --> 00:37:52.860 align:middle line:84% are going to do some of them in your Pset. 00:37:52.860 --> 00:38:03.860 align:middle line:90% 00:38:03.860 --> 00:38:10.190 align:middle line:84% Yeah, there's another diagram like that, et cetera. 00:38:10.190 --> 00:38:12.950 align:middle line:90% Let me just do two more. 00:38:12.950 --> 00:38:14.630 align:middle line:84% There's still some more diagrams you can 00:38:14.630 --> 00:38:20.560 align:middle line:90% draw to lambda squared order. 00:38:20.560 --> 00:38:26.520 align:middle line:90% 00:38:26.520 --> 00:38:29.190 align:middle line:90% But again, it's mechanical. 00:38:29.190 --> 00:38:35.190 align:middle line:84% With a little bit patience, there's no mystery here. 00:38:35.190 --> 00:38:36.735 align:middle line:84% Now let me make some remarks here. 00:38:36.735 --> 00:38:49.000 align:middle line:90% 00:38:49.000 --> 00:38:50.650 align:middle line:90% Let me make some remarks here. 00:38:50.650 --> 00:38:56.650 align:middle line:90% 00:38:56.650 --> 00:39:03.800 align:middle line:84% First, so the diagram, you can see separate into two types. 00:39:03.800 --> 00:39:06.240 align:middle line:84% So the full Gn separated into two types. 00:39:06.240 --> 00:39:08.495 align:middle line:84% One type is called connected diagrams. 00:39:08.495 --> 00:39:15.340 align:middle line:90% 00:39:15.340 --> 00:39:23.370 align:middle line:84% Connected diagrams, just essentially by its name, 00:39:23.370 --> 00:39:25.080 align:middle line:84% this corresponds to diagrams which 00:39:25.080 --> 00:39:28.590 align:middle line:84% all the external legs they are connected 00:39:28.590 --> 00:39:32.010 align:middle line:90% within a single diagram. 00:39:32.010 --> 00:39:37.800 align:middle line:84% This is a connected diagram or this is a connected diagram, 00:39:37.800 --> 00:39:42.780 align:middle line:84% but this is not, this is connected diagram, 00:39:42.780 --> 00:39:47.250 align:middle line:84% this is connected diagram, this connected diagram, this is not. 00:39:47.250 --> 00:39:49.185 align:middle line:90% This is also connected diagram. 00:39:49.185 --> 00:39:53.420 align:middle line:90% 00:39:53.420 --> 00:39:56.225 align:middle line:84% And you can also have disconnected diagrams. 00:39:56.225 --> 00:40:02.820 align:middle line:90% 00:40:02.820 --> 00:40:04.410 align:middle line:84% Disconnected diagrams corresponding 00:40:04.410 --> 00:40:11.470 align:middle line:84% to diagrams that's here, so your external legs 00:40:11.470 --> 00:40:13.960 align:middle line:90% separated into sub-diagrams. 00:40:13.960 --> 00:40:19.210 align:middle line:84% They belong to product of sub-diagrams. 00:40:19.210 --> 00:40:23.920 align:middle line:84% And later you will see, later we will argue, 00:40:23.920 --> 00:40:26.603 align:middle line:84% this is actually not interesting. 00:40:26.603 --> 00:40:28.270 align:middle line:84% Only connected diagrams are interesting, 00:40:28.270 --> 00:40:31.450 align:middle line:90% but in a few minutes. 00:40:31.450 --> 00:40:34.870 align:middle line:84% For the moment, let's just still define them. 00:40:34.870 --> 00:40:50.520 align:middle line:90% 00:40:50.520 --> 00:40:58.935 align:middle line:84% And emphasize again, the external pi general. 00:40:58.935 --> 00:41:01.830 align:middle line:90% 00:41:01.830 --> 00:41:06.780 align:middle line:84% Each external line, which you can assign a momentum 00:41:06.780 --> 00:41:10.140 align:middle line:90% pi from the Fourier transform. 00:41:10.140 --> 00:41:12.780 align:middle line:90% 00:41:12.780 --> 00:41:23.320 align:middle line:84% For example here, you have p1, p2, p3, p4 in momentum space. 00:41:23.320 --> 00:41:26.290 align:middle line:84% Each one you can assign a momentum and note momentum 00:41:26.290 --> 00:41:26.990 align:middle line:90% general. 00:41:26.990 --> 00:41:28.782 align:middle line:84% It can be in principle, it can be anything. 00:41:28.782 --> 00:41:34.260 align:middle line:90% 00:41:34.260 --> 00:41:34.760 align:middle line:90% Good. 00:41:34.760 --> 00:41:37.330 align:middle line:90% Any questions on this? 00:41:37.330 --> 00:41:38.703 align:middle line:90% Yes. 00:41:38.703 --> 00:41:40.120 align:middle line:84% AUDIENCE: Is there a trick to know 00:41:40.120 --> 00:41:44.750 align:middle line:84% that you've got all of them, like do you count them? 00:41:44.750 --> 00:41:46.640 align:middle line:84% HONG LIU: No, there's no trick to make 00:41:46.640 --> 00:41:48.630 align:middle line:90% sure you counted all of them. 00:41:48.630 --> 00:41:49.865 align:middle line:90% The only trick is patience. 00:41:49.865 --> 00:41:55.680 align:middle line:90% 00:41:55.680 --> 00:41:59.220 align:middle line:84% Just try to eliminate all possibilities. 00:41:59.220 --> 00:42:01.470 align:middle line:84% It's a finite number of them, you can always do it. 00:42:01.470 --> 00:42:01.970 align:middle line:90% Yes. 00:42:01.970 --> 00:42:04.500 align:middle line:84% AUDIENCE: Can you use metrics, like 00:42:04.500 --> 00:42:06.360 align:middle line:90% count the number of diagrams. 00:42:06.360 --> 00:42:07.230 align:middle line:90% HONG LIU: Sorry? 00:42:07.230 --> 00:42:09.180 align:middle line:84% AUDIENCE: Can you count the number of diagrams 00:42:09.180 --> 00:42:12.330 align:middle line:84% with a certain number of objects and external points? 00:42:12.330 --> 00:42:15.060 align:middle line:84% HONG LIU: No, I think there's no general formula 00:42:15.060 --> 00:42:17.910 align:middle line:84% to tell you at each order how many diagrams you should have. 00:42:17.910 --> 00:42:19.635 align:middle line:84% There's no such kind of magic formula. 00:42:19.635 --> 00:42:23.890 align:middle line:90% 00:42:23.890 --> 00:42:28.510 align:middle line:90% There's only one formula. 00:42:28.510 --> 00:42:29.590 align:middle line:90% It's not the formula. 00:42:29.590 --> 00:42:33.130 align:middle line:84% There's only one trend which you can show, 00:42:33.130 --> 00:42:42.040 align:middle line:84% that when you go to mth order, there are how many roughly 00:42:42.040 --> 00:42:45.870 align:middle line:90% how many diagrams at mth order? 00:42:45.870 --> 00:42:46.370 align:middle line:90% Yes. 00:42:46.370 --> 00:42:47.660 align:middle line:90% AUDIENCE: m factorial. 00:42:47.660 --> 00:42:49.350 align:middle line:84% HONG LIU: That's right, m factorial, 00:42:49.350 --> 00:42:51.500 align:middle line:90% so it's quite a lot of them. 00:42:51.500 --> 00:42:55.430 align:middle line:84% Once you go to say fourth or fifth order, it becomes a lot. 00:42:55.430 --> 00:43:02.270 align:middle line:90% 00:43:02.270 --> 00:43:06.740 align:middle line:84% You can be sure that we will not ask you to eliminate 00:43:06.740 --> 00:43:08.750 align:middle line:90% to fourth order in your Pset. 00:43:08.750 --> 00:43:16.580 align:middle line:90% 00:43:16.580 --> 00:43:21.170 align:middle line:84% Actually, the growth of the number of Feynman diagrams 00:43:21.170 --> 00:43:22.655 align:middle line:90% implies something deep. 00:43:22.655 --> 00:43:26.370 align:middle line:90% 00:43:26.370 --> 00:43:28.310 align:middle line:90% It's a side remark. 00:43:28.310 --> 00:43:33.160 align:middle line:84% It implies that this perturbation theory in lambda, 00:43:33.160 --> 00:43:37.000 align:middle line:84% it's actually not convergent because the number of diagrams 00:43:37.000 --> 00:43:38.300 align:middle line:90% grow too fast. 00:43:38.300 --> 00:43:42.550 align:middle line:84% And so if at mth order becomes m factorial, 00:43:42.550 --> 00:43:46.780 align:middle line:84% then that means the mth order contribution also 00:43:46.780 --> 00:43:49.210 align:middle line:84% grows as m factorial, and then that 00:43:49.210 --> 00:43:51.700 align:middle line:84% means this power series is actually not convergent. 00:43:51.700 --> 00:43:54.160 align:middle line:90% It's only asymptotic series. 00:43:54.160 --> 00:43:56.935 align:middle line:84% But still, for many physical purposes, it's enough. 00:43:56.935 --> 00:44:00.830 align:middle line:90% 00:44:00.830 --> 00:44:01.330 align:middle line:90% Good. 00:44:01.330 --> 00:44:03.310 align:middle line:90% Any other questions? 00:44:03.310 --> 00:44:07.630 align:middle line:84% Because this concludes our discussion 00:44:07.630 --> 00:44:14.230 align:middle line:84% of calculating such kind of Feynman correlation functions 00:44:14.230 --> 00:44:17.620 align:middle line:90% for interacting theory. 00:44:17.620 --> 00:44:18.308 align:middle line:90% Yes. 00:44:18.308 --> 00:44:20.350 align:middle line:84% AUDIENCE: You just said that it's not convergent, 00:44:20.350 --> 00:44:22.270 align:middle line:90% but it is asymptotic. 00:44:22.270 --> 00:44:26.230 align:middle line:84% HONG LIU: Yeah, it's an asymptotic series. 00:44:26.230 --> 00:44:29.415 align:middle line:84% AUDIENCE: Sorry, what's the distinction between those? 00:44:29.415 --> 00:44:30.790 align:middle line:84% HONG LIU: Yeah, asymptotic series 00:44:30.790 --> 00:44:32.470 align:middle line:84% is the kind of series which you-- 00:44:32.470 --> 00:44:37.060 align:middle line:90% 00:44:37.060 --> 00:44:39.520 align:middle line:84% Just the higher order terms will be small 00:44:39.520 --> 00:44:42.550 align:middle line:84% or will be smaller and smaller for a while 00:44:42.550 --> 00:44:45.680 align:middle line:90% and then grow again. 00:44:45.680 --> 00:44:49.587 align:middle line:84% And so if you don't calculate to high of an order, 00:44:49.587 --> 00:44:50.920 align:middle line:90% they're actually quite reliable. 00:44:50.920 --> 00:44:53.625 align:middle line:90% You can bound the errors. 00:44:53.625 --> 00:44:56.170 align:middle line:90% 00:44:56.170 --> 00:44:58.390 align:middle line:84% This is the heuristic way to say it. 00:44:58.390 --> 00:45:01.320 align:middle line:90% 00:45:01.320 --> 00:45:06.700 align:middle line:84% It's just when you look at the first few order terms, 00:45:06.700 --> 00:45:08.970 align:middle line:84% it's a reliable approximation to your true answer. 00:45:08.970 --> 00:45:12.420 align:middle line:90% 00:45:12.420 --> 00:45:15.120 align:middle line:84% AUDIENCE: I guess we started out by setting the interaction 00:45:15.120 --> 00:45:19.980 align:middle line:84% term to be phi to the 4th, but the action side. 00:45:19.980 --> 00:45:24.725 align:middle line:84% But in general, you're going to have some series of terms 00:45:24.725 --> 00:45:25.600 align:middle line:90% added to your action. 00:45:25.600 --> 00:45:29.333 align:middle line:84% Is that how you would do a general interaction? 00:45:29.333 --> 00:45:31.250 align:middle line:84% HONG LIU: No, depends on your specific theory. 00:45:31.250 --> 00:45:34.410 align:middle line:90% 00:45:34.410 --> 00:45:38.820 align:middle line:90% Maybe that's just your theory. 00:45:38.820 --> 00:45:41.430 align:middle line:84% AUDIENCE: How do you apply this to a physical problem? 00:45:41.430 --> 00:45:42.408 align:middle line:90% How do you know what-- 00:45:42.408 --> 00:45:43.950 align:middle line:84% HONG LIU: Yeah, in physical problem-- 00:45:43.950 --> 00:45:49.110 align:middle line:90% 00:45:49.110 --> 00:45:51.510 align:middle line:84% In one second we will tell you how 00:45:51.510 --> 00:45:54.780 align:middle line:84% you come from here to calculate the scattering amplitude. 00:45:54.780 --> 00:45:58.140 align:middle line:84% And when you find the scattering amplitude, 00:45:58.140 --> 00:46:00.600 align:middle line:84% then you can measure the experiment, 00:46:00.600 --> 00:46:04.120 align:middle line:84% then you try to deduce back the action. 00:46:04.120 --> 00:46:06.620 align:middle line:84% AUDIENCE: And in general what changes is just how many edges 00:46:06.620 --> 00:46:07.742 align:middle line:90% your vertex has? 00:46:07.742 --> 00:46:09.358 align:middle line:90% Is that the phi to the n? 00:46:09.358 --> 00:46:11.150 align:middle line:84% HONG LIU: Yeah, that's right, that's right. 00:46:11.150 --> 00:46:12.692 align:middle line:84% AUDIENCE: What if it's, I don't know, 00:46:12.692 --> 00:46:15.480 align:middle line:84% 1 over like something [INAUDIBLE],, is that-- 00:46:15.480 --> 00:46:16.920 align:middle line:90% HONG LIU: It can happen. 00:46:16.920 --> 00:46:22.230 align:middle line:84% As a toy model, you can always write whatever theory you want, 00:46:22.230 --> 00:46:26.220 align:middle line:84% but in nature somehow the series we discovered so 00:46:26.220 --> 00:46:27.420 align:middle line:90% is all polynomial. 00:46:27.420 --> 00:46:33.220 align:middle line:90% 00:46:33.220 --> 00:46:34.150 align:middle line:90% OK, good? 00:46:34.150 --> 00:46:36.670 align:middle line:90% 00:46:36.670 --> 00:46:41.860 align:middle line:84% Earlier we mentioned that there is this LSZ theorem, which 00:46:41.860 --> 00:46:48.700 align:middle line:84% tells you that the scattering amplitude can 00:46:48.700 --> 00:46:50.950 align:middle line:84% be obtained from this time-ordered correlation 00:46:50.950 --> 00:46:52.840 align:middle line:90% function. 00:46:52.840 --> 00:46:56.230 align:middle line:84% Now let's go back to this LSZ story 00:46:56.230 --> 00:47:00.490 align:middle line:84% again to tell you actually how to obtain scattering amplitude 00:47:00.490 --> 00:47:03.880 align:middle line:90% from correlation functions. 00:47:03.880 --> 00:47:12.830 align:middle line:90% 00:47:12.830 --> 00:47:17.030 align:middle line:90% The basic idea is the following. 00:47:17.030 --> 00:47:18.890 align:middle line:84% The basic idea is following-- is that you 00:47:18.890 --> 00:47:26.390 align:middle line:84% take your n-point function with n external momentum p1 to pn. 00:47:26.390 --> 00:47:30.340 align:middle line:90% 00:47:30.340 --> 00:47:35.450 align:middle line:84% So as I emphasized here, each pi is general. 00:47:35.450 --> 00:47:44.390 align:middle line:84% But now suppose you take all momentum, all pi, 00:47:44.390 --> 00:47:45.350 align:middle line:90% to be on shell. 00:47:45.350 --> 00:47:50.910 align:middle line:90% 00:47:50.910 --> 00:47:55.800 align:middle line:84% Now you consider them to satisfy pi squared 00:47:55.800 --> 00:47:57.195 align:middle line:90% equal to minus m squared. 00:47:57.195 --> 00:48:05.447 align:middle line:90% 00:48:05.447 --> 00:48:07.780 align:middle line:84% In order to get the amplitude, you need to do this step. 00:48:07.780 --> 00:48:09.500 align:middle line:84% You take all the external momentum. 00:48:09.500 --> 00:48:11.800 align:middle line:84% This is very reasonable to expect 00:48:11.800 --> 00:48:15.350 align:middle line:84% because when we're scattering particles, 00:48:15.350 --> 00:48:17.680 align:middle line:84% remember we say the initial particle can be considered 00:48:17.680 --> 00:48:19.510 align:middle line:84% as a free particle, final particle 00:48:19.510 --> 00:48:21.670 align:middle line:84% can also be considered as a free particle. 00:48:21.670 --> 00:48:25.060 align:middle line:84% If they are free particles, then their momentum 00:48:25.060 --> 00:48:28.210 align:middle line:84% satisfies this on-shell condition. 00:48:28.210 --> 00:48:31.180 align:middle line:84% In order to relate this to scattering amplitude, 00:48:31.180 --> 00:48:33.910 align:middle line:84% then we need to take external momentum to be on-shell. 00:48:33.910 --> 00:48:37.177 align:middle line:90% 00:48:37.177 --> 00:48:38.760 align:middle line:84% Now if you look at this, you say there 00:48:38.760 --> 00:48:43.230 align:middle line:84% are two possible choices for this equation 00:48:43.230 --> 00:48:45.750 align:middle line:84% because this equation has two solutions. 00:48:45.750 --> 00:48:54.480 align:middle line:90% 00:48:54.480 --> 00:49:06.240 align:middle line:84% So for each pi, pi0 can either be plus minus omega pi. 00:49:06.240 --> 00:49:09.390 align:middle line:84% We are using the same notation we used before, 00:49:09.390 --> 00:49:12.840 align:middle line:84% so this Pi vector is the spatial part, 00:49:12.840 --> 00:49:15.780 align:middle line:90% and then we have omega Pi. 00:49:15.780 --> 00:49:18.120 align:middle line:90% This omega pi is just the--ok? 00:49:18.120 --> 00:49:28.220 align:middle line:90% 00:49:28.220 --> 00:49:30.800 align:middle line:84% So you have two possible solutions. 00:49:30.800 --> 00:49:36.690 align:middle line:90% 00:49:36.690 --> 00:49:38.400 align:middle line:90% Now the rule is the following-- 00:49:38.400 --> 00:49:45.620 align:middle line:90% 00:49:45.620 --> 00:49:51.350 align:middle line:90% let's take p1, take m of them. 00:49:51.350 --> 00:49:56.570 align:middle line:90% 00:49:56.570 --> 00:49:58.805 align:middle line:90% We take the negative root. 00:49:58.805 --> 00:50:04.850 align:middle line:90% 00:50:04.850 --> 00:50:10.290 align:middle line:90% So alpha would be 1 over 2m. 00:50:10.290 --> 00:50:15.530 align:middle line:84% Then for the rest, say Pm plus 1, 00:50:15.530 --> 00:50:18.530 align:middle line:84% let's consider the following situation. 00:50:18.530 --> 00:50:21.035 align:middle line:84% And then you take the positive root. 00:50:21.035 --> 00:50:34.540 align:middle line:90% 00:50:34.540 --> 00:50:35.040 align:middle line:90% Ok? 00:50:35.040 --> 00:50:36.790 align:middle line:84% So let's consider the following situation. 00:50:36.790 --> 00:50:44.810 align:middle line:90% 00:50:44.810 --> 00:50:52.060 align:middle line:84% Let's imagine we take such an n-point function in momentum 00:50:52.060 --> 00:50:57.700 align:middle line:84% space, and then we take all the momentum on-shell, 00:50:57.700 --> 00:51:04.120 align:middle line:84% but for some of them we take the negative root and some of them 00:51:04.120 --> 00:51:08.890 align:middle line:84% we take the positive root, and then there's a theorem. 00:51:08.890 --> 00:51:12.745 align:middle line:84% So this is called LSZ theorem, which is 3 person-- 00:51:12.745 --> 00:51:16.150 align:middle line:90% 00:51:16.150 --> 00:51:20.420 align:middle line:84% Lehmann, Symanzik, and Zimmerman. 00:51:20.420 --> 00:51:22.835 align:middle line:84% This theorem is says, under this limit. 00:51:22.835 --> 00:51:29.190 align:middle line:90% 00:51:29.190 --> 00:51:31.740 align:middle line:90% Under this limit. 00:51:31.740 --> 00:51:32.970 align:middle line:90% Under this on-shell limit. 00:51:32.970 --> 00:51:38.800 align:middle line:90% 00:51:38.800 --> 00:51:45.090 align:middle line:84% So you will see why we emphasize that we call it on-shell limit. 00:51:45.090 --> 00:51:47.730 align:middle line:84% So imagine you don't actually put 00:51:47.730 --> 00:51:52.220 align:middle line:90% them to be exactly a momentum. 00:51:52.220 --> 00:51:56.280 align:middle line:84% You take the momentum to approach that. 00:51:56.280 --> 00:52:00.040 align:middle line:84% We approach that, we just emphasize. 00:52:00.040 --> 00:52:05.955 align:middle line:84% So in this limit then you find your momentum space correlation 00:52:05.955 --> 00:52:06.455 align:middle line:90% function. 00:52:06.455 --> 00:52:18.550 align:middle line:90% 00:52:18.550 --> 00:52:21.160 align:middle line:84% Can you find your momentum space correlation function approach 00:52:21.160 --> 00:52:26.765 align:middle line:84% to the following quantity in this limit. 00:52:26.765 --> 00:52:45.880 align:middle line:90% 00:52:45.880 --> 00:52:49.540 align:middle line:84% First, we write down the expression 00:52:49.540 --> 00:52:51.505 align:middle line:84% and then we will explain what this means. 00:52:51.505 --> 00:53:02.960 align:middle line:90% 00:53:02.960 --> 00:53:04.980 align:middle line:84% And then we just do the downstairs. 00:53:04.980 --> 00:53:10.900 align:middle line:90% 00:53:10.900 --> 00:53:12.010 align:middle line:90% Do the downstairs. 00:53:12.010 --> 00:53:20.960 align:middle line:90% 00:53:20.960 --> 00:53:22.040 align:middle line:90% Plus infinity. 00:53:22.040 --> 00:53:32.030 align:middle line:90% 00:53:32.030 --> 00:53:34.790 align:middle line:84% So the statement is the following, 00:53:34.790 --> 00:53:36.665 align:middle line:84% and here Z is just some constant. 00:53:36.665 --> 00:53:40.470 align:middle line:90% 00:53:40.470 --> 00:53:42.870 align:middle line:90% We don't need to worry about it. 00:53:42.870 --> 00:53:46.010 align:middle line:90% Z is just some constant. 00:53:46.010 --> 00:53:48.650 align:middle line:84% This is square root Z divided by this factor 00:53:48.650 --> 00:53:54.980 align:middle line:84% for each external momentum, and this 00:53:54.980 --> 00:54:00.080 align:middle line:84% is the scattering amplitude for m initial particles. 00:54:00.080 --> 00:54:02.020 align:middle line:84% Oh, sorry, I forgot the minus sign. 00:54:02.020 --> 00:54:07.270 align:middle line:84% It should be minus p1, minus p2, to minus pm. 00:54:07.270 --> 00:54:11.380 align:middle line:84% This is the scattering amplitude for particles 00:54:11.380 --> 00:54:17.770 align:middle line:84% with initial momentum minus p1 to minus pm into particles 00:54:17.770 --> 00:54:23.590 align:middle line:84% with final momentum m plus 1 to pn. 00:54:23.590 --> 00:54:27.730 align:middle line:84% Essentially, whether you choose the negative root 00:54:27.730 --> 00:54:31.660 align:middle line:84% or the positive root decides whether you 00:54:31.660 --> 00:54:35.763 align:middle line:84% are in the initial state or in the final state. 00:54:35.763 --> 00:54:37.930 align:middle line:84% Whether you are in the final state in the scattering 00:54:37.930 --> 00:54:38.620 align:middle line:90% amplitude. 00:54:38.620 --> 00:54:49.620 align:middle line:90% 00:54:49.620 --> 00:54:50.625 align:middle line:90% Let me just write here. 00:54:50.625 --> 00:54:55.340 align:middle line:90% 00:54:55.340 --> 00:54:58.880 align:middle line:90% Let me just copy this again. 00:54:58.880 --> 00:55:02.480 align:middle line:90% This scattering amplitude-- 00:55:02.480 --> 00:55:12.410 align:middle line:84% Yeah, let me just write here this pm, pm plus 1, 00:55:12.410 --> 00:55:23.230 align:middle line:84% pm plus infinity minus p1 minus pm minus infinity. 00:55:23.230 --> 00:55:27.410 align:middle line:84% This is the scattering amplitude. 00:55:27.410 --> 00:55:31.240 align:middle line:84% This is the scattering amplitude. 00:55:31.240 --> 00:55:40.360 align:middle line:84% This is an amplitude for the particle 00:55:40.360 --> 00:55:47.980 align:middle line:84% with momentum p1 pm at t equal to minus 00:55:47.980 --> 00:56:02.710 align:middle line:84% infinity transitioning to the rest particles at this momentum 00:56:02.710 --> 00:56:04.000 align:middle line:90% at t plus infinity. 00:56:04.000 --> 00:56:24.980 align:middle line:90% 00:56:24.980 --> 00:56:32.300 align:middle line:84% To save the location, again, remember previously 00:56:32.300 --> 00:56:32.930 align:middle line:90% we defined-- 00:56:32.930 --> 00:56:49.870 align:middle line:90% 00:56:49.870 --> 00:56:54.730 align:middle line:84% If you remember before, we defined this object. 00:56:54.730 --> 00:56:56.530 align:middle line:84% This object should also preserve, 00:56:56.530 --> 00:57:00.400 align:middle line:84% because of time translation symmetry, 00:57:00.400 --> 00:57:04.090 align:middle line:84% this object will also preserve the energy and momentum. 00:57:04.090 --> 00:57:07.660 align:middle line:84% This object is also proportional, 00:57:07.660 --> 00:57:18.860 align:middle line:84% so you can extract out the delta function and then times 00:57:18.860 --> 00:57:21.680 align:middle line:84% a quantity, which we called the scattering amplitude for, 00:57:21.680 --> 00:57:23.660 align:middle line:90% say alpha to beta. 00:57:23.660 --> 00:57:27.560 align:middle line:84% Alpha denotes the initial state, so this is the initial state 00:57:27.560 --> 00:57:30.950 align:middle line:84% alpha, and this is the final state beta. 00:57:30.950 --> 00:57:36.940 align:middle line:90% 00:57:36.940 --> 00:57:43.438 align:middle line:84% Now if you plug this expression into here, 00:57:43.438 --> 00:57:45.730 align:middle line:84% then the delta function will just cancel on both sides. 00:57:45.730 --> 00:57:49.650 align:middle line:90% 00:57:49.650 --> 00:57:55.170 align:middle line:84% And then you find Gn p1 pn is just 00:57:55.170 --> 00:58:00.750 align:middle line:84% equal to, approach this product of the delta functions, 00:58:00.750 --> 00:58:08.760 align:middle line:84% approach this factor, and then M alpha to beta. 00:58:08.760 --> 00:58:12.780 align:middle line:90% 00:58:12.780 --> 00:58:19.120 align:middle line:84% This is a statement of the LSZ theorem. 00:58:19.120 --> 00:58:29.710 align:middle line:84% This essentially tells you that such a scattering amplitude 00:58:29.710 --> 00:58:35.980 align:middle line:84% just can be obtained from this, and you multiply those things, 00:58:35.980 --> 00:58:38.260 align:middle line:90% and then just equal to that. 00:58:38.260 --> 00:58:43.870 align:middle line:84% Now you observe, up to this factor of Z, 00:58:43.870 --> 00:58:45.910 align:middle line:84% these are precisely just propagators 00:58:45.910 --> 00:58:46.975 align:middle line:90% for the external legs. 00:58:46.975 --> 00:58:49.720 align:middle line:90% 00:58:49.720 --> 00:58:57.860 align:middle line:84% Now you conclude, in other words, that 00:58:57.860 --> 00:59:00.560 align:middle line:84% means that M alpha beta then corresponding to you 00:59:00.560 --> 00:59:04.490 align:middle line:84% just take the G, then you get rid of all 00:59:04.490 --> 00:59:07.400 align:middle line:84% the external propagators because they're 00:59:07.400 --> 00:59:10.250 align:middle line:84% just corresponding to the external propagators 00:59:10.250 --> 00:59:11.930 align:middle line:90% for each external leg. 00:59:11.930 --> 00:59:14.660 align:middle line:84% Just get rid of the external legs. 00:59:14.660 --> 00:59:19.650 align:middle line:84% Here I'm just telling you the theorem and we will use it, 00:59:19.650 --> 00:59:22.130 align:middle line:84% but of course, the derivation is a little bit complicated. 00:59:22.130 --> 00:59:29.850 align:middle line:90% 00:59:29.850 --> 00:59:31.485 align:middle line:90% Let me just write it in words. 00:59:31.485 --> 00:59:36.550 align:middle line:90% 00:59:36.550 --> 00:59:46.830 align:middle line:84% To obtain the scattering amplitude M to alpha beta, 00:59:46.830 --> 00:59:48.690 align:middle line:90% you first calculate. 00:59:48.690 --> 00:59:56.990 align:middle line:90% 00:59:56.990 --> 01:00:00.200 align:middle line:84% Suppose the total number of initial particles 01:00:00.200 --> 01:00:02.450 align:middle line:84% and the final particles are n and then 01:00:02.450 --> 01:00:11.090 align:middle line:84% you just take Gn for n particles. 01:00:11.090 --> 01:00:14.060 align:middle line:90% 01:00:14.060 --> 01:00:37.080 align:middle line:84% And then take pi on-shell, and the initial momentum 01:00:37.080 --> 01:00:48.870 align:middle line:84% take the negative root, and in the final momentum 01:00:48.870 --> 01:00:53.835 align:middle line:84% you take a positive root of that equation. 01:00:53.835 --> 01:00:56.590 align:middle line:90% 01:00:56.590 --> 01:01:04.520 align:middle line:84% And then the last step you just truncate 01:01:04.520 --> 01:01:06.320 align:middle line:90% all the external propagators. 01:01:06.320 --> 01:01:29.310 align:middle line:90% 01:01:29.310 --> 01:01:32.640 align:middle line:84% And then you truncated all the external legs. 01:01:32.640 --> 01:01:37.920 align:middle line:84% So that just comes from you need to obtain 01:01:37.920 --> 01:01:42.630 align:middle line:84% M. You need to divide this side by this factor 01:01:42.630 --> 01:01:45.480 align:middle line:84% then up to this factor of Z. This factor Z will only 01:01:45.480 --> 01:01:53.620 align:middle line:84% give you a constant, which let's not worry about it for now. 01:01:53.620 --> 01:01:54.340 align:middle line:90% Yes? 01:01:54.340 --> 01:01:55.825 align:middle line:90% AUDIENCE: [INAUDIBLE]. 01:01:55.825 --> 01:02:01.140 align:middle line:90% 01:02:01.140 --> 01:02:04.040 align:middle line:84% it seems like you just put it in ad hoc. 01:02:04.040 --> 01:02:06.812 align:middle line:84% HONG LIU: Which is putting ad hoc? 01:02:06.812 --> 01:02:08.770 align:middle line:84% AUDIENCE: Why are you just setting some of them 01:02:08.770 --> 01:02:09.880 align:middle line:90% to be negative? 01:02:09.880 --> 01:02:11.230 align:middle line:90% HONG LIU: Oh, yeah. 01:02:11.230 --> 01:02:12.400 align:middle line:90% That I will explain. 01:02:12.400 --> 01:02:13.315 align:middle line:90% That I will explain. 01:02:13.315 --> 01:02:16.300 align:middle line:90% 01:02:16.300 --> 01:02:18.550 align:middle line:84% First, let me just state the rule. 01:02:18.550 --> 01:02:20.650 align:middle line:84% First, you understand the rule, then 01:02:20.650 --> 01:02:24.340 align:middle line:84% I try to make you more comfortable with the rule. 01:02:24.340 --> 01:02:26.550 align:middle line:90% Yes? 01:02:26.550 --> 01:02:27.120 align:middle line:90% OK. 01:02:27.120 --> 01:02:29.520 align:middle line:90% Yeah, here is just the rule. 01:02:29.520 --> 01:02:34.140 align:middle line:84% Here I'm just explaining here is the rule. 01:02:34.140 --> 01:02:38.230 align:middle line:84% This is a statement you can prove mathematically, 01:02:38.230 --> 01:02:40.485 align:middle line:84% so this is a mathematical statement you can prove. 01:02:40.485 --> 01:02:43.100 align:middle line:90% 01:02:43.100 --> 01:02:46.700 align:middle line:84% And then following from this statement, 01:02:46.700 --> 01:02:57.900 align:middle line:84% here is the rule to obtain the M alpha beta from the G. 01:02:57.900 --> 01:03:00.120 align:middle line:84% If I want to write it in one sentence-- 01:03:00.120 --> 01:03:03.020 align:middle line:90% 01:03:03.020 --> 01:03:18.430 align:middle line:84% scattering amplitude equals to sum over 01:03:18.430 --> 01:03:37.944 align:middle line:84% all truncated diagrams with external momentum on-shell. 01:03:37.944 --> 01:03:45.500 align:middle line:90% 01:03:45.500 --> 01:03:48.380 align:middle line:84% If I summarize it into one sentence, it's just this. 01:03:48.380 --> 01:03:58.530 align:middle line:90% 01:03:58.530 --> 01:04:04.260 align:middle line:84% I would say the only thing unintuitive here 01:04:04.260 --> 01:04:05.520 align:middle line:90% is this sign choice. 01:04:05.520 --> 01:04:08.100 align:middle line:90% 01:04:08.100 --> 01:04:10.890 align:middle line:84% Everything else I think is intuitive. 01:04:10.890 --> 01:04:12.300 align:middle line:90% Everything else is intuitive. 01:04:12.300 --> 01:04:15.630 align:middle line:84% You take this Green function, which contains 01:04:15.630 --> 01:04:18.210 align:middle line:90% essentially any external legs. 01:04:18.210 --> 01:04:21.870 align:middle line:84% And now you just imagine this internal legs corresponding 01:04:21.870 --> 01:04:24.570 align:middle line:84% to initial and final particles, and then it's 01:04:24.570 --> 01:04:28.390 align:middle line:84% very reasonable we make them on-shell because they 01:04:28.390 --> 01:04:30.710 align:middle line:90% are physical particles. 01:04:30.710 --> 01:04:32.930 align:middle line:84% Then the only thing which may be unintuitive 01:04:32.930 --> 01:04:34.490 align:middle line:84% is that why for the initial momentum 01:04:34.490 --> 01:04:37.100 align:middle line:84% we need to take the negative root, but for the final one 01:04:37.100 --> 01:04:40.340 align:middle line:90% we need to take a positive root. 01:04:40.340 --> 01:04:43.610 align:middle line:84% Then the final thing here is just 01:04:43.610 --> 01:04:47.510 align:middle line:84% if you imagine when you scatter particles 01:04:47.510 --> 01:04:52.670 align:middle line:84% and how they propagate to the scattering does not matter. 01:04:52.670 --> 01:04:54.380 align:middle line:84% The propagator just essentially tells you 01:04:54.380 --> 01:04:58.820 align:middle line:84% this particle how they propagate to the scattering point. 01:04:58.820 --> 01:05:01.640 align:middle line:90% 01:05:01.640 --> 01:05:03.890 align:middle line:84% This should not matter for the scattering process. 01:05:03.890 --> 01:05:08.990 align:middle line:90% 01:05:08.990 --> 01:05:14.820 align:middle line:84% We will explain this at the end, but before we do that, 01:05:14.820 --> 01:05:18.320 align:middle line:90% let's just look at one example. 01:05:18.320 --> 01:05:20.280 align:middle line:90% Do you have any questions? 01:05:20.280 --> 01:05:20.780 align:middle line:90% Yes. 01:05:20.780 --> 01:05:22.738 align:middle line:84% AUDIENCE: I just to make sure I understand when 01:05:22.738 --> 01:05:25.480 align:middle line:84% we say truncate we mean divide n and divide Gn 01:05:25.480 --> 01:05:26.620 align:middle line:90% and by this product. 01:05:26.620 --> 01:05:31.600 align:middle line:90% 01:05:31.600 --> 01:05:33.700 align:middle line:84% HONG LIU: Yeah, you will see explicitly we just 01:05:33.700 --> 01:05:39.210 align:middle line:84% don't include all the external propagators. 01:05:39.210 --> 01:05:40.335 align:middle line:90% Let me give you an example. 01:05:40.335 --> 01:05:42.597 align:middle line:90% It will be clear. 01:05:42.597 --> 01:05:43.930 align:middle line:90% Let me just give you an example. 01:05:43.930 --> 01:05:48.825 align:middle line:84% Let's consider four particles scattering. 01:05:48.825 --> 01:05:49.450 align:middle line:90% Let's consider. 01:05:49.450 --> 01:05:58.370 align:middle line:90% 01:05:58.370 --> 01:06:02.330 align:middle line:84% Following the convention here is the minus p1 p2, 01:06:02.330 --> 01:06:11.830 align:middle line:84% so you have minus p1 and minus p2, 01:06:11.830 --> 01:06:16.330 align:middle line:90% and then you have p3 and p4. 01:06:16.330 --> 01:06:17.800 align:middle line:84% Let's suppose we want to calculate 01:06:17.800 --> 01:06:22.150 align:middle line:84% such kind of scattering amplitude to initial particles 01:06:22.150 --> 01:06:25.630 align:middle line:84% with momentum minus p1 and minus p2. 01:06:25.630 --> 01:06:28.090 align:middle line:84% When you take this to be the negative root, 01:06:28.090 --> 01:06:31.960 align:middle line:84% then the minus p1 will actually have positive energy. 01:06:31.960 --> 01:06:34.490 align:middle line:84% The minus p1 will actually have a positive energy, 01:06:34.490 --> 01:06:38.190 align:middle line:84% so all of them have positive energy. 01:06:38.190 --> 01:06:39.880 align:middle line:84% All of them have positive energy. 01:06:39.880 --> 01:06:42.560 align:middle line:90% 01:06:42.560 --> 01:06:45.530 align:middle line:84% If you want to calculate in this case, 01:06:45.530 --> 01:06:55.030 align:middle line:84% then you just take the G4 p1 p4, and then 01:06:55.030 --> 01:07:01.180 align:middle line:84% you just take p1 and p2 to take to be negative root. 01:07:01.180 --> 01:07:05.320 align:middle line:90% 01:07:05.320 --> 01:07:10.660 align:middle line:84% Just to save the time to take to the negative root. 01:07:10.660 --> 01:07:16.180 align:middle line:84% And then p3 p4 to be the positive root. 01:07:16.180 --> 01:07:21.760 align:middle line:90% 01:07:21.760 --> 01:07:24.250 align:middle line:84% And then you put the t here, then 01:07:24.250 --> 01:07:30.610 align:middle line:84% you truncate your external propagator. 01:07:30.610 --> 01:07:33.580 align:middle line:84% Then you truncate your external propagator. 01:07:33.580 --> 01:07:37.520 align:middle line:84% What this means, let's look at the leading order contribution 01:07:37.520 --> 01:07:38.020 align:middle line:90% to this. 01:07:38.020 --> 01:07:41.925 align:middle line:90% 01:07:41.925 --> 01:07:43.550 align:middle line:84% The leading order contribution to that, 01:07:43.550 --> 01:07:44.967 align:middle line:84% we already have the diagram there. 01:07:44.967 --> 01:07:52.450 align:middle line:90% 01:07:52.450 --> 01:07:56.960 align:middle line:84% The leading order, in the free theory, 01:07:56.960 --> 01:07:59.870 align:middle line:84% those processes don't contribute because those things, 01:07:59.870 --> 01:08:01.370 align:middle line:90% there's no interaction here. 01:08:01.370 --> 01:08:03.600 align:middle line:90% Particle just propagates. 01:08:03.600 --> 01:08:07.070 align:middle line:84% So the lowest diagram is this one. 01:08:07.070 --> 01:08:09.560 align:middle line:84% The lowest diagram is this one, so we just 01:08:09.560 --> 01:08:12.350 align:middle line:84% consider that diagram, so that's just leading order 01:08:12.350 --> 01:08:13.415 align:middle line:90% given by that diagram. 01:08:13.415 --> 01:08:18.990 align:middle line:90% 01:08:18.990 --> 01:08:28.939 align:middle line:84% Then you have minus p1 minus p2 and then p3 and p4. 01:08:28.939 --> 01:08:32.609 align:middle line:90% 01:08:32.609 --> 01:08:38.279 align:middle line:84% Now according to our Feynman rule, this vertex just 01:08:38.279 --> 01:08:42.390 align:middle line:84% gives you a factor of minus i lambda. 01:08:42.390 --> 01:08:45.870 align:middle line:84% So if we want to calculate this Green function, 01:08:45.870 --> 01:08:50.490 align:middle line:84% then we need to include all these four propagators. 01:08:50.490 --> 01:08:53.640 align:middle line:84% But here it tells us to truncate, 01:08:53.640 --> 01:08:56.810 align:middle line:84% but all the propagators here are external, 01:08:56.810 --> 01:08:58.439 align:middle line:90% so we don't include them. 01:08:58.439 --> 01:09:00.770 align:middle line:84% So the answer here is simple-- just minus i lambda. 01:09:00.770 --> 01:09:03.460 align:middle line:90% 01:09:03.460 --> 01:09:06.040 align:middle line:84% The leading order contribution to these 4 particles 01:09:06.040 --> 01:09:07.540 align:middle line:84% scattering is just minus i lambda. 01:09:07.540 --> 01:09:10.979 align:middle line:90% 01:09:10.979 --> 01:09:17.200 align:middle line:84% And then you have all the lambda squared contributions. 01:09:17.200 --> 01:09:17.830 align:middle line:90% Yes? 01:09:17.830 --> 01:09:19.622 align:middle line:84% AUDIENCE: Why didn't we consider the other, 01:09:19.622 --> 01:09:22.000 align:middle line:84% the one where one is just propagating 01:09:22.000 --> 01:09:23.314 align:middle line:90% and the other has a loop? 01:09:23.314 --> 01:09:28.460 align:middle line:90% 01:09:28.460 --> 01:09:31.550 align:middle line:84% HONG LIU: I'm going to mention that good. 01:09:31.550 --> 01:09:32.810 align:middle line:90% That's a good question. 01:09:32.810 --> 01:09:35.000 align:middle line:90% I'm going to mention that. 01:09:35.000 --> 01:09:37.532 align:middle line:84% Again, that one does not correspond to the scattering. 01:09:37.532 --> 01:09:38.990 align:middle line:84% Essentially it's just corresponding 01:09:38.990 --> 01:09:42.200 align:middle line:84% to one particle propagated because there's 01:09:42.200 --> 01:09:45.680 align:middle line:84% no interaction between the particles. 01:09:45.680 --> 01:09:46.819 align:middle line:90% They factorize. 01:09:46.819 --> 01:09:50.510 align:middle line:90% 01:09:50.510 --> 01:09:53.903 align:middle line:84% This one involves the two particles then, there's 01:09:53.903 --> 01:09:55.820 align:middle line:84% an interaction, you go to two other particles. 01:09:55.820 --> 01:10:00.000 align:middle line:84% Here just one particle straight goes to the other particle, 01:10:00.000 --> 01:10:02.870 align:middle line:84% so this particle by definition has to be the same as this one, 01:10:02.870 --> 01:10:07.220 align:middle line:84% this has to be the same, so there's no scattering. 01:10:07.220 --> 01:10:12.700 align:middle line:84% I will mention this point separately a little bit later. 01:10:12.700 --> 01:10:17.850 align:middle line:84% Another possibility if we want to consider 01:10:17.850 --> 01:10:28.580 align:middle line:84% the decay of a particle into three other particles. 01:10:28.580 --> 01:10:32.510 align:middle line:90% 01:10:32.510 --> 01:10:36.320 align:middle line:84% The decay of a particle into three other particles. 01:10:36.320 --> 01:10:41.090 align:middle line:84% In this case then, again, we take G4, 01:10:41.090 --> 01:10:46.670 align:middle line:84% so now we take minus p1, and then p2, p3, p4, 01:10:46.670 --> 01:10:49.670 align:middle line:90% and now we take p1, p4. 01:10:49.670 --> 01:10:56.400 align:middle line:84% Now we take p1 goes to minus root, p1 0 is the minus omega, 01:10:56.400 --> 01:11:03.620 align:middle line:84% and then p3, 2, 3, 4 take it to be the positive root. 01:11:03.620 --> 01:11:06.110 align:middle line:84% And then that's corresponding to this process. 01:11:06.110 --> 01:11:12.120 align:middle line:84% And again, the leading term, the nontrivial term is just this. 01:11:12.120 --> 01:11:16.650 align:middle line:84% The leading nontrivial term is this, so now this is minus p1, 01:11:16.650 --> 01:11:22.320 align:middle line:90% so now this is p2, p3, p4. 01:11:22.320 --> 01:11:25.988 align:middle line:84% Again, the leading order, this is just minus i lambda 01:11:25.988 --> 01:11:28.155 align:middle line:84% because you don't need to include the external legs. 01:11:28.155 --> 01:11:31.890 align:middle line:90% 01:11:31.890 --> 01:11:34.110 align:middle line:84% This goes on to a single particle decay 01:11:34.110 --> 01:11:35.550 align:middle line:90% into three other particles. 01:11:35.550 --> 01:11:51.150 align:middle line:90% 01:11:51.150 --> 01:11:52.830 align:middle line:84% AUDIENCE: In the phi 4 theory, there's 01:11:52.830 --> 01:11:58.050 align:middle line:84% no conservation law or anything for preventing this, right? 01:11:58.050 --> 01:11:58.830 align:middle line:90% HONG LIU: No, no. 01:11:58.830 --> 01:12:02.280 align:middle line:90% 01:12:02.280 --> 01:12:03.900 align:middle line:90% So now let me make some remarks. 01:12:03.900 --> 01:12:33.290 align:middle line:90% 01:12:33.290 --> 01:12:33.800 align:middle line:90% Oh, yeah. 01:12:33.800 --> 01:12:36.050 align:middle line:84% I erased this, but yeah, I'm fine. 01:12:36.050 --> 01:12:48.610 align:middle line:90% 01:12:48.610 --> 01:12:49.510 align:middle line:90% So the remarks. 01:12:49.510 --> 01:12:59.460 align:middle line:90% 01:12:59.460 --> 01:13:01.860 align:middle line:84% You already asked, when we calculate 01:13:01.860 --> 01:13:03.930 align:middle line:84% this to the leading order, we only 01:13:03.930 --> 01:13:05.595 align:middle line:90% included the connected diagrams. 01:13:05.595 --> 01:13:09.230 align:middle line:90% 01:13:09.230 --> 01:13:20.620 align:middle line:84% The disconnected diagrams in general factorize. 01:13:20.620 --> 01:13:25.030 align:middle line:84% It means that when you have disconnected diagram, 01:13:25.030 --> 01:13:33.920 align:middle line:84% it means that your process factorizes into subscattering 01:13:33.920 --> 01:13:43.385 align:middle line:90% process of lower orders. 01:13:43.385 --> 01:13:53.990 align:middle line:90% 01:13:53.990 --> 01:13:58.220 align:middle line:84% As we were doing here, there's one other diagram which 01:13:58.220 --> 01:14:01.310 align:middle line:90% corresponds to just this one. 01:14:01.310 --> 01:14:04.520 align:middle line:90% 01:14:04.520 --> 01:14:06.020 align:middle line:84% Just corresponding to this, you have 01:14:06.020 --> 01:14:14.750 align:middle line:90% p1 minus p1, minus p2, p3, p4. 01:14:14.750 --> 01:14:17.900 align:middle line:84% You can also have a diagram like this. 01:14:17.900 --> 01:14:19.400 align:middle line:84% But in this case, just corresponding 01:14:19.400 --> 01:14:23.690 align:middle line:84% to p1 separately go to p3 and p2 separately goes to p4, 01:14:23.690 --> 01:14:26.690 align:middle line:90% so nothing really happens. 01:14:26.690 --> 01:14:32.440 align:middle line:84% So in general, you have m particle scattering. 01:14:32.440 --> 01:14:35.260 align:middle line:90% You have m particle scattering. 01:14:35.260 --> 01:14:39.310 align:middle line:84% So with some number of initial particles 01:14:39.310 --> 01:14:43.960 align:middle line:84% and some number of final particles. 01:14:43.960 --> 01:14:47.920 align:middle line:84% When you have a disconnected diagram, 01:14:47.920 --> 01:14:51.170 align:middle line:84% it means that you have a situation like this. 01:14:51.170 --> 01:14:55.120 align:middle line:84% You have one diagram with some subset of particles 01:14:55.120 --> 01:14:58.780 align:middle line:84% go to some subset of particles, and with a lot of diagrams 01:14:58.780 --> 01:15:01.225 align:middle line:84% some set of particles, some set of particles. 01:15:01.225 --> 01:15:03.940 align:middle line:90% 01:15:03.940 --> 01:15:05.890 align:middle line:84% This is a three to three scattering, 01:15:05.890 --> 01:15:08.560 align:middle line:84% and then essentially this factorizes into two to two 01:15:08.560 --> 01:15:10.360 align:middle line:84% scattering, and then one particle just 01:15:10.360 --> 01:15:12.670 align:middle line:90% goes to another particle. 01:15:12.670 --> 01:15:14.620 align:middle line:84% This can already be understood in terms 01:15:14.620 --> 01:15:15.670 align:middle line:90% of lower order process. 01:15:15.670 --> 01:15:20.590 align:middle line:90% 01:15:20.590 --> 01:15:21.685 align:middle line:90% We don't include them. 01:15:21.685 --> 01:15:24.162 align:middle line:90% 01:15:24.162 --> 01:15:25.870 align:middle line:84% Essentially, they are already understood. 01:15:25.870 --> 01:15:28.970 align:middle line:90% 01:15:28.970 --> 01:15:31.280 align:middle line:84% When we define the scattering amplitude, 01:15:31.280 --> 01:15:32.330 align:middle line:90% we don't include them. 01:15:32.330 --> 01:15:37.060 align:middle line:90% 01:15:37.060 --> 01:15:56.280 align:middle line:84% We are only interested in process involving 01:15:56.280 --> 01:15:57.960 align:middle line:90% all participating particles. 01:15:57.960 --> 01:16:09.800 align:middle line:90% 01:16:09.800 --> 01:16:11.525 align:middle line:84% We are actually running out of time. 01:16:11.525 --> 01:16:19.150 align:middle line:90% 01:16:19.150 --> 01:16:20.170 align:middle line:90% This is first remark. 01:16:20.170 --> 01:16:23.630 align:middle line:90% 01:16:23.630 --> 01:16:24.380 align:middle line:90% Second remark. 01:16:24.380 --> 01:16:29.120 align:middle line:90% 01:16:29.120 --> 01:16:33.360 align:middle line:84% Since we truncate all the external legs, 01:16:33.360 --> 01:16:36.690 align:middle line:84% we truncate all the external propagators. 01:16:36.690 --> 01:16:43.750 align:middle line:84% This kind of diagram, you have a loop on one 01:16:43.750 --> 01:16:47.350 align:middle line:84% of the external propagators, so this diagram 01:16:47.350 --> 01:16:48.260 align:middle line:90% doesn't contribute. 01:16:48.260 --> 01:16:54.340 align:middle line:90% 01:16:54.340 --> 01:16:59.020 align:middle line:84% Because this one, after you have truncated the external legs, 01:16:59.020 --> 01:17:02.320 align:middle line:84% it's no different from this one, it's 01:17:02.320 --> 01:17:05.620 align:middle line:84% no different from that one, so you don't have 01:17:05.620 --> 01:17:08.380 align:middle line:90% to count this one separately. 01:17:08.380 --> 01:17:09.820 align:middle line:84% So this one is just corresponding 01:17:09.820 --> 01:17:15.160 align:middle line:84% to when the particle propagates to infinity somehow there's 01:17:15.160 --> 01:17:18.910 align:middle line:84% some other process to happen to this particular particle. 01:17:18.910 --> 01:17:22.360 align:middle line:84% When you truncate the external legs, they don't include them. 01:17:22.360 --> 01:17:28.920 align:middle line:90% 01:17:28.920 --> 01:17:32.580 align:middle line:84% So that means, with these two, it 01:17:32.580 --> 01:17:34.650 align:middle line:84% means that the scattering amplitude-- 01:17:34.650 --> 01:17:42.570 align:middle line:90% 01:17:42.570 --> 01:17:48.460 align:middle line:84% Essentially can forget anything happening to the external legs. 01:17:48.460 --> 01:17:54.590 align:middle line:90% 01:17:54.590 --> 01:17:57.230 align:middle line:84% You can have complicated things happen external legs, 01:17:57.230 --> 01:17:58.210 align:middle line:90% but they don't matter. 01:17:58.210 --> 01:18:02.700 align:middle line:90% 01:18:02.700 --> 01:18:09.000 align:middle line:84% To summarize, the scattering amplitude, 01:18:09.000 --> 01:18:13.740 align:middle line:84% when you compute the scattering amplitude, you just sum over-- 01:18:13.740 --> 01:18:17.030 align:middle line:90% 01:18:17.030 --> 01:18:20.990 align:middle line:84% following this definition-- you sum over all truncated. 01:18:20.990 --> 01:18:24.395 align:middle line:84% It means that you truncate all the external legs. 01:18:24.395 --> 01:18:27.580 align:middle line:90% 01:18:27.580 --> 01:18:34.555 align:middle line:84% Connected because we want all particles to participate. 01:18:34.555 --> 01:18:36.640 align:middle line:84% And then the external particle on-shell. 01:18:36.640 --> 01:18:44.240 align:middle line:90% 01:18:44.240 --> 01:18:46.820 align:middle line:84% This is a simplification compared 01:18:46.820 --> 01:18:49.687 align:middle line:84% to calculating the general correlation functions. 01:18:49.687 --> 01:18:52.220 align:middle line:90% 01:18:52.220 --> 01:18:54.980 align:middle line:84% I have two other remarks to make, but we don't have time. 01:18:54.980 --> 01:18:58.970 align:middle line:84% One remark contains the choice of this sign-- 01:18:58.970 --> 01:19:01.880 align:middle line:84% why initial momentum corresponding to negative root 01:19:01.880 --> 01:19:04.790 align:middle line:84% and why this corresponding to positive root, 01:19:04.790 --> 01:19:12.980 align:middle line:84% and a lot of remarks related to this factor of Z. 01:19:12.980 --> 01:19:17.150 align:middle line:84% We will talk about them at the beginning of next lecture. 01:19:17.150 --> 01:19:20.060 align:middle line:84% And then after that, then we are done 01:19:20.060 --> 01:19:23.700 align:middle line:84% with the general formulation of interacting theory. 01:19:23.700 --> 01:19:28.310 align:middle line:84% And now we will be equipped with the power. 01:19:28.310 --> 01:19:31.520 align:middle line:84% Essentially, you can calculate any interacting theory, 01:19:31.520 --> 01:19:36.080 align:middle line:84% any amplitude, using perturbation theory. 01:19:36.080 --> 01:19:39.380 align:middle line:84% But in order to calculate anything useful, 01:19:39.380 --> 01:19:42.560 align:middle line:84% we still have to first learn how to treat fermions, 01:19:42.560 --> 01:19:46.950 align:middle line:84% how to treat photons, and then that's what we will do next. 01:19:46.950 --> 01:19:51.030 align:middle line:84% So we will, starting next lecture, 01:19:51.030 --> 01:19:54.810 align:middle line:84% talk about how to deal with fermions, 01:19:54.810 --> 01:19:57.780 align:middle line:84% how to introduce fermions, and after that we'll talk about how 01:19:57.780 --> 01:20:00.930 align:middle line:84% to introduce photons, and then we can talk about QED, 01:20:00.930 --> 01:20:04.700 align:middle line:84% how photons interact with electrons, et cetera. 01:20:04.700 --> 01:20:06.850 align:middle line:90% OK, good. 01:20:06.850 --> 01:20:10.000 align:middle line:90%