WEBVTT 00:00:00.000 --> 00:00:02.982 align:middle line:90% [SQUEAKING] 00:00:02.982 --> 00:00:03.976 align:middle line:90% [RUSTLING] 00:00:03.976 --> 00:00:06.958 align:middle line:90% [CLICKING] 00:00:06.958 --> 00:00:10.440 align:middle line:90% 00:00:10.440 --> 00:00:12.120 align:middle line:90% HONG LIU: OK. 00:00:12.120 --> 00:00:13.380 align:middle line:90% So let us start. 00:00:13.380 --> 00:00:16.390 align:middle line:90% 00:00:16.390 --> 00:00:21.900 align:middle line:84% So first, I'll remind you the master formula we derived 00:00:21.900 --> 00:00:23.385 align:middle line:90% before for path integral. 00:00:23.385 --> 00:00:27.270 align:middle line:90% 00:00:27.270 --> 00:00:32.830 align:middle line:84% We said, if we consider time-ordered anything, OK, 00:00:32.830 --> 00:00:36.570 align:middle line:84% so let me just call it x will be, say, some product 00:00:36.570 --> 00:00:39.270 align:middle line:90% of operators. 00:00:39.270 --> 00:00:44.410 align:middle line:84% And then this can be written as, in the path integral form, 00:00:44.410 --> 00:00:46.320 align:middle line:90% D phi. 00:00:46.320 --> 00:00:50.460 align:middle line:84% And then so x, it just goes into here. 00:00:50.460 --> 00:00:53.190 align:middle line:90% And then you have is, OK? 00:00:53.190 --> 00:01:00.180 align:middle line:84% And then divided by D phi just without x just is, OK? 00:01:00.180 --> 00:01:04.360 align:middle line:84% So this is the master formula we derived before, 00:01:04.360 --> 00:01:09.060 align:middle line:84% which applies to any theory, OK, in principle, any theory 00:01:09.060 --> 00:01:11.790 align:middle line:90% and any x you want, OK? 00:01:11.790 --> 00:01:15.870 align:middle line:84% And so things inside this x should always 00:01:15.870 --> 00:01:17.130 align:middle line:90% be time ordered, OK? 00:01:17.130 --> 00:01:19.510 align:middle line:90% So this T means time order. 00:01:19.510 --> 00:01:22.650 align:middle line:84% So now we can apply this to interacting theory. 00:01:22.650 --> 00:01:25.580 align:middle line:90% 00:01:25.580 --> 00:01:29.260 align:middle line:84% So for interacting theory, we have 00:01:29.260 --> 00:01:34.060 align:middle line:84% S. We write S. So the action to be the free theory action 00:01:34.060 --> 00:01:37.120 align:middle line:90% plus the interaction part, OK? 00:01:37.120 --> 00:01:40.720 align:middle line:84% And the interaction part, in turn, 00:01:40.720 --> 00:01:43.690 align:middle line:84% can be written as say the integration 00:01:43.690 --> 00:01:48.250 align:middle line:84% over the interaction part of the Lagrangian density. 00:01:48.250 --> 00:01:54.490 align:middle line:84% And it's also the same as the minus 00:01:54.490 --> 00:01:57.220 align:middle line:84% the interacting part of the Hamiltonian 00:01:57.220 --> 00:01:59.710 align:middle line:90% integrated over time, OK? 00:01:59.710 --> 00:02:03.070 align:middle line:84% So the reason for this simple relation 00:02:03.070 --> 00:02:05.980 align:middle line:84% is because we assume the interacting part does not 00:02:05.980 --> 00:02:08.630 align:middle line:90% contain time derivatives. 00:02:08.630 --> 00:02:11.500 align:middle line:84% So essentially, interacting particles 00:02:11.500 --> 00:02:15.190 align:middle line:84% binding to a potential in your Lagrangian density. 00:02:15.190 --> 00:02:16.840 align:middle line:84% And then in the Hamiltonian, they just 00:02:16.840 --> 00:02:18.620 align:middle line:90% differ by a minus sign. 00:02:18.620 --> 00:02:19.120 align:middle line:90% OK. 00:02:19.120 --> 00:02:22.460 align:middle line:90% 00:02:22.460 --> 00:02:27.020 align:middle line:84% So now we can just apply this to this case, OK? 00:02:27.020 --> 00:02:27.710 align:middle line:90% So let's see. 00:02:27.710 --> 00:02:31.140 align:middle line:84% So we can just write this object. 00:02:31.140 --> 00:02:34.400 align:middle line:90% So now we can write it as-- 00:02:34.400 --> 00:02:37.580 align:middle line:90% also come here. 00:02:37.580 --> 00:02:44.420 align:middle line:84% So our goal is that we assume the Si is small compared to S0 00:02:44.420 --> 00:02:47.240 align:middle line:84% because we cannot solve this theory exactly. 00:02:47.240 --> 00:02:49.040 align:middle line:84% And so what we are going to do is 00:02:49.040 --> 00:02:54.670 align:middle line:84% we want to treat Si to be small and expand in the cautioning 00:02:54.670 --> 00:02:55.280 align:middle line:90% Si. 00:02:55.280 --> 00:02:58.880 align:middle line:84% OK, so example, of Si is this lambda phi 4, 00:02:58.880 --> 00:03:01.010 align:middle line:90% which we wrote down before. 00:03:01.010 --> 00:03:01.940 align:middle line:90% OK. 00:03:01.940 --> 00:03:07.610 align:middle line:84% So for this purpose, then we can just write this more explicitly 00:03:07.610 --> 00:03:09.680 align:middle line:90% as D phi. 00:03:09.680 --> 00:03:11.540 align:middle line:90% Then you have x. 00:03:11.540 --> 00:03:12.350 align:middle line:90% Then you have i S0. 00:03:12.350 --> 00:03:15.530 align:middle line:90% 00:03:15.530 --> 00:03:17.140 align:middle line:90% Then exponential i Si. 00:03:17.140 --> 00:03:18.820 align:middle line:90% OK? 00:03:18.820 --> 00:03:30.145 align:middle line:84% Then divide it by phi exponential i S0, i Si. 00:03:30.145 --> 00:03:30.645 align:middle line:90% OK? 00:03:30.645 --> 00:03:34.050 align:middle line:90% 00:03:34.050 --> 00:03:42.610 align:middle line:84% And now so we can now treat this as the integrand 00:03:42.610 --> 00:03:46.330 align:middle line:84% for the-- yeah, so this is the integration 00:03:46.330 --> 00:03:47.920 align:middle line:90% for the free theory. 00:03:47.920 --> 00:03:48.430 align:middle line:90% OK? 00:03:48.430 --> 00:03:50.320 align:middle line:84% So now we can integrate this just 00:03:50.320 --> 00:03:55.030 align:middle line:84% as a integrand for the free theory, as some, say, x prime. 00:03:55.030 --> 00:03:59.740 align:middle line:84% So x times the exponential i Si as another 00:03:59.740 --> 00:04:03.010 align:middle line:84% x for the free theory and similarly here, OK? 00:04:03.010 --> 00:04:06.710 align:middle line:90% So we can write this as follows. 00:04:06.710 --> 00:04:11.740 align:middle line:84% So we can write this as now in free theory. 00:04:11.740 --> 00:04:15.800 align:middle line:90% 00:04:15.800 --> 00:04:19.790 align:middle line:84% So upstairs should corresponding to the 0. 00:04:19.790 --> 00:04:20.380 align:middle line:90% Yeah. 00:04:20.380 --> 00:04:24.890 align:middle line:84% Yeah, let me just not worry about it at the moment. 00:04:24.890 --> 00:04:35.450 align:middle line:84% So now you can write it as the time-ordered x exponential i Si 00:04:35.450 --> 00:04:37.130 align:middle line:90% 0. 00:04:37.130 --> 00:04:38.540 align:middle line:90% Now this is in free theory. 00:04:38.540 --> 00:04:40.610 align:middle line:90% So I put a subscript 0 here. 00:04:40.610 --> 00:04:41.585 align:middle line:90% Means in free theory. 00:04:41.585 --> 00:04:53.735 align:middle line:90% 00:04:53.735 --> 00:04:54.235 align:middle line:90% OK? 00:04:54.235 --> 00:05:01.710 align:middle line:90% 00:05:01.710 --> 00:05:05.430 align:middle line:84% So in writing from here to here, so upstairs, 00:05:05.430 --> 00:05:10.870 align:middle line:84% we interpreted as a correlation function in the free theory. 00:05:10.870 --> 00:05:14.710 align:middle line:84% So here, we should also divide by-- 00:05:14.710 --> 00:05:17.580 align:middle line:84% yeah, in order to write in terms of the expectation 00:05:17.580 --> 00:05:20.580 align:middle line:84% value in the free-- yeah, so 0 here is the free theory vacuum. 00:05:20.580 --> 00:05:30.620 align:middle line:90% 00:05:30.620 --> 00:05:33.980 align:middle line:84% So in writing the upstairs in this form, 00:05:33.980 --> 00:05:36.860 align:middle line:84% we should divide this by a path integral just 00:05:36.860 --> 00:05:38.660 align:middle line:90% with the exponential as 0. 00:05:38.660 --> 00:05:40.577 align:middle line:84% But since upstairs and downstairs 00:05:40.577 --> 00:05:42.410 align:middle line:84% we need to divide by the same path integral, 00:05:42.410 --> 00:05:43.670 align:middle line:90% they just cancel, OK? 00:05:43.670 --> 00:05:45.770 align:middle line:84% So I can just directly write down 00:05:45.770 --> 00:05:49.420 align:middle line:84% as the ratio of these two correlation functions. 00:05:49.420 --> 00:05:51.490 align:middle line:90% OK? 00:05:51.490 --> 00:05:58.530 align:middle line:84% And so now this is a very nice form, OK? 00:05:58.530 --> 00:06:04.170 align:middle line:84% Because so this is-- yeah, let me call this equation star. 00:06:04.170 --> 00:06:06.780 align:middle line:84% So this is the star is the exact expression. 00:06:06.780 --> 00:06:13.820 align:middle line:90% 00:06:13.820 --> 00:06:14.320 align:middle line:90% OK? 00:06:14.320 --> 00:06:18.070 align:middle line:90% 00:06:18.070 --> 00:06:32.540 align:middle line:84% Which writes correlation function 00:06:32.540 --> 00:06:48.520 align:middle line:84% of an interacting theory in terms of those of free theory. 00:06:48.520 --> 00:06:56.450 align:middle line:90% 00:06:56.450 --> 00:06:58.070 align:middle line:90% OK? 00:06:58.070 --> 00:07:01.310 align:middle line:84% So now these are just some correlation functions 00:07:01.310 --> 00:07:02.600 align:middle line:90% in the free theory. 00:07:02.600 --> 00:07:06.440 align:middle line:84% So both upstairs and downstairs, they can, in principle, 00:07:06.440 --> 00:07:09.680 align:middle line:84% evaluated as we discussed before in free theory. 00:07:09.680 --> 00:07:10.272 align:middle line:90% Yes? 00:07:10.272 --> 00:07:11.730 align:middle line:84% AUDIENCE: So I'm a little confused. 00:07:11.730 --> 00:07:15.763 align:middle line:84% What is there to time-order for the phase for the e to the i -- 00:07:15.763 --> 00:07:17.523 align:middle line:90% with the interaction? 00:07:17.523 --> 00:07:18.190 align:middle line:90% HONG LIU: Right. 00:07:18.190 --> 00:07:18.450 align:middle line:90% Right. 00:07:18.450 --> 00:07:19.980 align:middle line:84% As we discussed before, whenever you 00:07:19.980 --> 00:07:22.050 align:middle line:84% see time-ordered exponential, you 00:07:22.050 --> 00:07:26.130 align:middle line:84% should always expand the exponential in power series. 00:07:26.130 --> 00:07:30.510 align:middle line:84% And then you just have a bunch of factors. 00:07:30.510 --> 00:07:32.271 align:middle line:90% And you just time order them. 00:07:32.271 --> 00:07:34.920 align:middle line:90% Yeah. 00:07:34.920 --> 00:07:35.880 align:middle line:90% Other questions? 00:07:35.880 --> 00:07:40.020 align:middle line:90% 00:07:40.020 --> 00:07:42.610 align:middle line:90% OK. 00:07:42.610 --> 00:07:44.950 align:middle line:90% So yeah. 00:07:44.950 --> 00:07:47.420 align:middle line:84% And this is precisely the expression, 00:07:47.420 --> 00:07:49.560 align:middle line:90% which derived in Peskin. 00:07:49.560 --> 00:07:51.490 align:middle line:90% OK, so this is the-- 00:07:51.490 --> 00:08:10.980 align:middle line:84% so this equation is the 4.31 of Peskin and Schroeder, 00:08:10.980 --> 00:08:16.350 align:middle line:84% which they derived using the interaction picture, OK? 00:08:16.350 --> 00:08:18.930 align:middle line:84% So we see that for the path integral, 00:08:18.930 --> 00:08:21.930 align:middle line:84% to derive this equation using path integral is just trivial. 00:08:21.930 --> 00:08:27.030 align:middle line:84% Essentially, you just split the exponential for the full theory 00:08:27.030 --> 00:08:30.360 align:middle line:84% into that of the free theory and the interacting part. 00:08:30.360 --> 00:08:32.440 align:middle line:84% And then you automatically derive this equation. 00:08:32.440 --> 00:08:32.940 align:middle line:90% OK? 00:08:32.940 --> 00:08:38.850 align:middle line:84% So once you have understood how path integral works and then 00:08:38.850 --> 00:08:41.919 align:middle line:84% derive this equation becomes trivial. 00:08:41.919 --> 00:08:46.200 align:middle line:90% And so this is actually general. 00:08:46.200 --> 00:08:48.150 align:middle line:84% Later, the same expression, we will 00:08:48.150 --> 00:08:50.160 align:middle line:84% be able to apply it for, say, when 00:08:50.160 --> 00:08:53.910 align:middle line:84% we include fermions and also photons, et cetera, OK? 00:08:53.910 --> 00:08:56.640 align:middle line:90% 00:08:56.640 --> 00:09:03.460 align:middle line:84% And now we will evaluate this guy using perturbation theory, 00:09:03.460 --> 00:09:03.960 align:middle line:90% OK? 00:09:03.960 --> 00:09:23.190 align:middle line:84% So we will expand star in power series of Si. 00:09:23.190 --> 00:09:28.140 align:middle line:90% OK, we treat Si as small. 00:09:28.140 --> 00:09:32.630 align:middle line:84% And yeah, just expand it in power series. 00:09:32.630 --> 00:09:44.990 align:middle line:84% So more explicitly, say, we consider the Gn, so 00:09:44.990 --> 00:09:46.430 align:middle line:90% this Gn for the full theory. 00:09:46.430 --> 00:09:51.620 align:middle line:84% So let me call this Gn, just say this is some n-point function. 00:09:51.620 --> 00:09:54.650 align:middle line:90% OK, so this is the Gn. 00:09:54.650 --> 00:09:56.870 align:middle line:84% And then we can just, more explicity, 00:09:56.870 --> 00:10:00.960 align:middle line:84% we can just write Gn as the 0th order. 00:10:00.960 --> 00:10:08.780 align:middle line:84% You just have T X 0, OK, upstairs. 00:10:08.780 --> 00:10:21.860 align:middle line:84% And the next order would be i dt say 0 T X H I. 00:10:21.860 --> 00:10:26.570 align:middle line:84% So I have expanded to the first order. 00:10:26.570 --> 00:10:29.950 align:middle line:84% In Si, when I expand Si, then become 00:10:29.950 --> 00:10:31.900 align:middle line:90% dt times Hi with a minus sign. 00:10:31.900 --> 00:10:35.620 align:middle line:84% So I just write it in terms like that, OK? 00:10:35.620 --> 00:10:40.600 align:middle line:84% And so remember here, you need to time order them together. 00:10:40.600 --> 00:10:43.120 align:middle line:84% Because both of them are in the same integrand, OK? 00:10:43.120 --> 00:10:47.810 align:middle line:84% So we need to time order them together, and et cetera. 00:10:47.810 --> 00:10:49.610 align:middle line:90% OK, so this is upstairs. 00:10:49.610 --> 00:10:53.480 align:middle line:84% And downstairs, you just expand this guy. 00:10:53.480 --> 00:10:59.200 align:middle line:84% So essentially, you have 1 minus i dt. 00:10:59.200 --> 00:11:04.900 align:middle line:84% Then you can time-order the Hamiltonian, the interacting 00:11:04.900 --> 00:11:06.400 align:middle line:90% part of the Hamiltonian. 00:11:06.400 --> 00:11:10.590 align:middle line:90% And then plus higher order. 00:11:10.590 --> 00:11:12.270 align:middle line:90% OK? 00:11:12.270 --> 00:11:16.800 align:middle line:84% So you just now essentially do the Taylor series expansion. 00:11:16.800 --> 00:11:19.560 align:middle line:90% And yeah. 00:11:19.560 --> 00:11:24.080 align:middle line:84% And the first-- so you always assume 00:11:24.080 --> 00:11:25.440 align:middle line:90% the interacting part is small. 00:11:25.440 --> 00:11:28.720 align:middle line:84% So you can actually expand the downstairs also. 00:11:28.720 --> 00:11:32.270 align:middle line:84% So this will be reduced to, say, the first order. 00:11:32.270 --> 00:11:37.030 align:middle line:84% So the 0th order just gives you the free theory n-point 00:11:37.030 --> 00:11:38.080 align:middle line:90% function. 00:11:38.080 --> 00:11:41.470 align:middle line:84% OK, so I denote now with a subscript 0 00:11:41.470 --> 00:11:44.380 align:middle line:84% means the free theory correlation function. 00:11:44.380 --> 00:11:44.980 align:middle line:90% OK? 00:11:44.980 --> 00:11:51.950 align:middle line:84% And then yeah, and then you can add the rest, et cetera. 00:11:51.950 --> 00:11:53.710 align:middle line:90% OK. 00:11:53.710 --> 00:11:57.470 align:middle line:84% OK, so the higher order perturbations. 00:11:57.470 --> 00:11:59.130 align:middle line:90% OK. 00:11:59.130 --> 00:12:00.800 align:middle line:90% Good? 00:12:00.800 --> 00:12:04.030 align:middle line:90% Any questions on this? 00:12:04.030 --> 00:12:04.530 align:middle line:90% Yeah. 00:12:04.530 --> 00:12:07.560 align:middle line:84% Actually, for later purpose, let me just write one more term 00:12:07.560 --> 00:12:09.130 align:middle line:90% just explicitly. 00:12:09.130 --> 00:12:10.680 align:middle line:90% Let me just write it. 00:12:10.680 --> 00:12:16.520 align:middle line:84% So if I bring this up, so when I bring this up and then 00:12:16.520 --> 00:12:39.630 align:middle line:84% this becomes i 0 T X 0, then times dt 0 T H I. 00:12:39.630 --> 00:12:43.570 align:middle line:84% Also, so this is this term we bring up. 00:12:43.570 --> 00:12:44.880 align:middle line:90% And then multiply that term. 00:12:44.880 --> 00:12:46.030 align:middle line:90% That will give you that. 00:12:46.030 --> 00:12:50.970 align:middle line:84% And then there's another first order term is given by i 00:12:50.970 --> 00:13:02.100 align:middle line:90% dt 0 T X H I 0. 00:13:02.100 --> 00:13:06.630 align:middle line:84% And then so these are the full first-order terms. 00:13:06.630 --> 00:13:08.860 align:middle line:84% OK, so these are the full first-order order terms, 00:13:08.860 --> 00:13:13.770 align:middle line:84% which is the correction to the free theory n-point functions. 00:13:13.770 --> 00:13:15.600 align:middle line:84% And we need to evaluate those quantities. 00:13:15.600 --> 00:13:17.850 align:middle line:84% And all those correlation functions in the free theory 00:13:17.850 --> 00:13:20.550 align:middle line:84% can be just evaluated using the Wick theorem 00:13:20.550 --> 00:13:22.140 align:middle line:90% because we know how to-- 00:13:22.140 --> 00:13:24.750 align:middle line:84% because all correlation functions in free theory 00:13:24.750 --> 00:13:27.480 align:middle line:84% just can be evaluated using Wick theorem. 00:13:27.480 --> 00:13:30.780 align:middle line:84% Just use the Wick theorem over and over. 00:13:30.780 --> 00:13:31.280 align:middle line:90% OK. 00:13:31.280 --> 00:13:34.130 align:middle line:90% 00:13:34.130 --> 00:13:35.405 align:middle line:90% So any questions on this? 00:13:35.405 --> 00:13:40.410 align:middle line:90% 00:13:40.410 --> 00:13:43.390 align:middle line:84% So now we will develop techniques to evaluate 00:13:43.390 --> 00:13:46.480 align:middle line:90% such kind of series, OK? 00:13:46.480 --> 00:13:50.260 align:middle line:84% Still, even though this is not too difficult to do now, 00:13:50.260 --> 00:13:53.960 align:middle line:84% but still when you do it in practice, it's pretty tedious. 00:13:53.960 --> 00:13:58.900 align:middle line:84% So still worth making effort to actually simplify the process, 00:13:58.900 --> 00:14:01.210 align:middle line:90% OK, simplify the process. 00:14:01.210 --> 00:14:05.815 align:middle line:84% And the technique to simplify is called the Feynman diagrams. 00:14:05.815 --> 00:14:13.270 align:middle line:90% 00:14:13.270 --> 00:14:15.940 align:middle line:90% OK. 00:14:15.940 --> 00:14:20.150 align:middle line:84% So now we will consider, so far, those equations 00:14:20.150 --> 00:14:23.590 align:middle line:84% apply for any theory, any kind of interaction, OK? 00:14:23.590 --> 00:14:26.690 align:middle line:84% So now let's work on for a specific case. 00:14:26.690 --> 00:14:31.360 align:middle line:84% So let's consider Li equal to minus lambda 00:14:31.360 --> 00:14:36.120 align:middle line:84% divided by 4 factorial phi to the power of 4, OK? 00:14:36.120 --> 00:14:42.520 align:middle line:84% So it means that H I would be integrated over-- 00:14:42.520 --> 00:14:49.260 align:middle line:84% so H I, yeah, so yeah anyway, yeah, 00:14:49.260 --> 00:14:53.280 align:middle line:84% the others you can just directly right from here, OK? 00:14:53.280 --> 00:14:54.630 align:middle line:90% So let's consider this case. 00:14:54.630 --> 00:14:57.380 align:middle line:90% 00:14:57.380 --> 00:15:02.990 align:middle line:84% And so to be specific-- and now also, 00:15:02.990 --> 00:15:06.970 align:middle line:84% let's consider-- just in the simplest correlation 00:15:06.970 --> 00:15:10.360 align:middle line:84% function-- let's just consider the two-point function. 00:15:10.360 --> 00:15:13.940 align:middle line:84% Let's just consider the two-point function, G2. 00:15:13.940 --> 00:15:17.090 align:middle line:84% So two-point is also the Feynman function. 00:15:17.090 --> 00:15:18.710 align:middle line:84% But now this is the Feynman function 00:15:18.710 --> 00:15:19.930 align:middle line:90% in the interacting theory. 00:15:19.930 --> 00:15:23.240 align:middle line:90% 00:15:23.240 --> 00:15:25.790 align:middle line:90% So now we have two-- 00:15:25.790 --> 00:15:27.860 align:middle line:84% so imagine we have two coordinates-- 00:15:27.860 --> 00:15:30.840 align:middle line:90% and then is equal to -- now. 00:15:30.840 --> 00:15:32.040 align:middle line:90% So let's consider this. 00:15:32.040 --> 00:15:39.670 align:middle line:90% 00:15:39.670 --> 00:15:43.930 align:middle line:84% So now let's consider the Feynman function 00:15:43.930 --> 00:15:47.430 align:middle line:90% in the full theory. 00:15:47.430 --> 00:15:50.740 align:middle line:84% And now we can just now apply this equation. 00:15:50.740 --> 00:15:55.390 align:middle line:84% So let us try to calculate to the next order. 00:15:55.390 --> 00:16:02.360 align:middle line:84% So the 0-th order that's given by the free theory, two-point 00:16:02.360 --> 00:16:04.430 align:middle line:90% function. 00:16:04.430 --> 00:16:07.600 align:middle line:84% So by translation symmetry, it's always 00:16:07.600 --> 00:16:09.760 align:middle line:90% a function of x1 minus x2. 00:16:09.760 --> 00:16:12.790 align:middle line:84% So this does not depend on whether it's 00:16:12.790 --> 00:16:16.300 align:middle line:84% in the free theory or interacting theory. 00:16:16.300 --> 00:16:19.900 align:middle line:90% And now let's just plug in here. 00:16:19.900 --> 00:16:22.720 align:middle line:90% Let's just plug in here. 00:16:22.720 --> 00:16:29.340 align:middle line:84% So H I is integration of this over 00:16:29.340 --> 00:16:32.940 align:middle line:90% a three-dimensional spatial. 00:16:32.940 --> 00:16:36.030 align:middle line:84% And then combined, this become just four-dimensional spacetime 00:16:36.030 --> 00:16:37.080 align:middle line:90% integrals. 00:16:37.080 --> 00:16:41.360 align:middle line:90% So then you just get the-- 00:16:41.360 --> 00:16:49.480 align:middle line:84% so this term becomes plus i lambda divided by 4 factorial. 00:16:49.480 --> 00:16:52.890 align:middle line:84% So we will assume this lambda is small so 00:16:52.890 --> 00:16:55.155 align:middle line:90% that we can expand this lambda. 00:16:55.155 --> 00:16:58.670 align:middle line:90% 00:16:58.670 --> 00:17:02.240 align:middle line:84% And this term, just again, is the free theory correlation 00:17:02.240 --> 00:17:03.390 align:middle line:90% function. 00:17:03.390 --> 00:17:05.910 align:middle line:90% So we just have GF-- 00:17:05.910 --> 00:17:11.410 align:middle line:90% again, 0-- x1 minus x2. 00:17:11.410 --> 00:17:18.190 align:middle line:84% And then times-- my board is a little bit-- 00:17:18.190 --> 00:17:21.540 align:middle line:90% so we have to go through here-- 00:17:21.540 --> 00:17:28.520 align:middle line:84% times-- then you have d4 x then 0, T, and phi 00:17:28.520 --> 00:17:31.830 align:middle line:90% to the power 4 x here. 00:17:31.830 --> 00:17:35.770 align:middle line:90% So this is multiplied with that. 00:17:35.770 --> 00:17:37.720 align:middle line:90% So that's the second term. 00:17:37.720 --> 00:17:46.097 align:middle line:84% And the third term is minus i lambda to the 4 factorial. 00:17:46.097 --> 00:17:46.680 align:middle line:90% You have d4 x. 00:17:46.680 --> 00:17:50.200 align:middle line:90% 00:17:50.200 --> 00:17:55.740 align:middle line:84% You have 0, T. Then you have to combine them together. 00:17:55.740 --> 00:18:04.520 align:middle line:84% So this is phi x1, phi x2, and phi to the power 4 x-- 00:18:04.520 --> 00:18:07.710 align:middle line:90% and this x is integrated over-- 00:18:07.710 --> 00:18:08.700 align:middle line:90% And then 0. 00:18:08.700 --> 00:18:11.450 align:middle line:90% 00:18:11.450 --> 00:18:14.840 align:middle line:90% And so this is all time ordered. 00:18:14.840 --> 00:18:19.670 align:middle line:84% And then the rest is all times lambda squared, 00:18:19.670 --> 00:18:20.913 align:middle line:90% plus lambda squared. 00:18:20.913 --> 00:18:28.440 align:middle line:90% 00:18:28.440 --> 00:18:28.940 align:middle line:90% Good. 00:18:28.940 --> 00:18:29.857 align:middle line:90% Any questions on this? 00:18:29.857 --> 00:18:32.270 align:middle line:90% 00:18:32.270 --> 00:18:35.330 align:middle line:84% So yeah, so let me just make a quick remark. 00:18:35.330 --> 00:18:40.250 align:middle line:90% 00:18:40.250 --> 00:18:49.830 align:middle line:84% So in the end, so just note can be-- 00:18:49.830 --> 00:18:53.880 align:middle line:90% so yeah. 00:18:53.880 --> 00:18:57.930 align:middle line:84% So here, even though we only write down 00:18:57.930 --> 00:19:00.210 align:middle line:84% the first-order term, but you can continue. 00:19:00.210 --> 00:19:23.110 align:middle line:84% So all terms in the expansion can be evaluated by 00:19:23.110 --> 00:19:34.040 align:middle line:90% repeated apply the Wick theorem. 00:19:34.040 --> 00:19:37.190 align:middle line:84% So any such kind of functions, they all 00:19:37.190 --> 00:19:41.570 align:middle line:84% factorize into two-point functions. 00:19:41.570 --> 00:19:43.430 align:middle line:84% And the two-point functions are essentially 00:19:43.430 --> 00:19:48.800 align:middle line:84% just given by GF 0, just the GF 0, free theory two-point 00:19:48.800 --> 00:19:50.540 align:middle line:90% functions. 00:19:50.540 --> 00:19:58.565 align:middle line:84% So the final result always have the form-- 00:19:58.565 --> 00:20:04.720 align:middle line:90% 00:20:04.720 --> 00:20:16.065 align:middle line:84% final result always have the form sum over products of GF 00:20:16.065 --> 00:20:16.565 align:middle line:90% 0's-- 00:20:16.565 --> 00:20:20.530 align:middle line:90% 00:20:20.530 --> 00:20:23.350 align:middle line:84% and plus some integration, so sum of the-- 00:20:23.350 --> 00:20:30.600 align:middle line:90% 00:20:30.600 --> 00:20:33.270 align:middle line:90% plus possible integrations. 00:20:33.270 --> 00:20:36.480 align:middle line:84% So, in general, the structure is pretty simple. 00:20:36.480 --> 00:20:40.190 align:middle line:84% They all just reduce to the product 00:20:40.190 --> 00:20:42.005 align:middle line:90% of those two-point functions. 00:20:42.005 --> 00:20:44.980 align:middle line:90% 00:20:44.980 --> 00:20:50.970 align:middle line:84% So now, as an example, let's just evaluate these two pieces, 00:20:50.970 --> 00:20:53.050 align:middle line:90% evaluate these two pieces. 00:20:53.050 --> 00:20:55.090 align:middle line:90% So let's continue. 00:20:55.090 --> 00:21:00.475 align:middle line:84% So we find that the GF x1 minus x2-- 00:21:00.475 --> 00:21:02.920 align:middle line:90% yeah, so GF is the same as G2. 00:21:02.920 --> 00:21:06.730 align:middle line:84% So for two-point function, this is just the Feynman function 00:21:06.730 --> 00:21:07.855 align:middle line:90% for the interacting theory. 00:21:07.855 --> 00:21:10.360 align:middle line:90% 00:21:10.360 --> 00:21:13.870 align:middle line:84% So the leading term is just GF 0. 00:21:13.870 --> 00:21:16.940 align:middle line:90% 00:21:16.940 --> 00:21:20.820 align:middle line:84% And then now let's evaluate this term. 00:21:20.820 --> 00:21:23.030 align:middle line:90% Now evaluate this. 00:21:23.030 --> 00:21:26.990 align:middle line:84% So this is all phi contracted on the same point. 00:21:26.990 --> 00:21:28.670 align:middle line:90% So there are four phi's. 00:21:28.670 --> 00:21:32.280 align:middle line:84% We could just contract them within themselves. 00:21:32.280 --> 00:21:36.130 align:middle line:90% So we essentially get to the-- 00:21:36.130 --> 00:21:38.800 align:middle line:90% so you have GF. 00:21:38.800 --> 00:21:49.090 align:middle line:84% And then the next term is i, 4 factorial-- 00:21:49.090 --> 00:21:50.950 align:middle line:90% i lambda 4 factorial. 00:21:50.950 --> 00:21:54.370 align:middle line:90% And then you have GF 0. 00:21:54.370 --> 00:21:59.680 align:middle line:84% So x1,2-- so let me use x1,2 to denote x1 minus x2 just to save 00:21:59.680 --> 00:22:02.890 align:middle line:90% a little bit effort. 00:22:02.890 --> 00:22:07.150 align:middle line:84% And then we have this integration over d4 x. 00:22:07.150 --> 00:22:09.050 align:middle line:90% And then let's evaluate this. 00:22:09.050 --> 00:22:13.720 align:middle line:84% So this just reduce to two Feynman propagators. 00:22:13.720 --> 00:22:15.760 align:middle line:90% But both of them have 0. 00:22:15.760 --> 00:22:23.230 align:middle line:84% So essentially, you just have GF 0, 0 is x minus x. 00:22:23.230 --> 00:22:26.830 align:middle line:90% 00:22:26.830 --> 00:22:30.300 align:middle line:90% So we just get that, x minus x. 00:22:30.300 --> 00:22:32.410 align:middle line:84% So let me just write one of them. 00:22:32.410 --> 00:22:38.520 align:middle line:90% 00:22:38.520 --> 00:22:42.570 align:middle line:84% But we are actually missing some factors because we have four 00:22:42.570 --> 00:22:44.690 align:middle line:90% phi's. 00:22:44.690 --> 00:22:48.220 align:middle line:84% So when you contract the four phi's, there's 00:22:48.220 --> 00:22:50.720 align:middle line:84% three different ways to group them. 00:22:50.720 --> 00:22:55.150 align:middle line:90% so We should have a factor of 3. 00:22:55.150 --> 00:22:56.710 align:middle line:84% There's three different ways to group 00:22:56.710 --> 00:22:58.520 align:middle line:90% four phi's into two groups. 00:22:58.520 --> 00:23:00.230 align:middle line:90% Then we have a factor of 3. 00:23:00.230 --> 00:23:08.080 align:middle line:90% 00:23:08.080 --> 00:23:12.038 align:middle line:84% OK, so now let's look at this term. 00:23:12.038 --> 00:23:13.205 align:middle line:90% Now let's look at this term. 00:23:13.205 --> 00:23:16.960 align:middle line:90% 00:23:16.960 --> 00:23:18.990 align:middle line:84% So this term, now have now we have 00:23:18.990 --> 00:23:23.820 align:middle line:84% to contract six phi's, so phi x1, phi x2, 00:23:23.820 --> 00:23:25.590 align:middle line:84% and then there are four phi's here. 00:23:25.590 --> 00:23:28.580 align:middle line:84% So we need to contract six phi's. 00:23:28.580 --> 00:23:32.130 align:middle line:84% So we can have two possible patterns. 00:23:32.130 --> 00:23:36.460 align:middle line:84% Inside this, two phi's contract with themselves. 00:23:36.460 --> 00:23:40.120 align:middle line:84% These two phi's-- this phi x1, phi x2 contract with 00:23:40.120 --> 00:23:40.840 align:middle line:90% themselves. 00:23:40.840 --> 00:23:46.010 align:middle line:84% And then four phi's will have to contract among themselves. 00:23:46.010 --> 00:23:50.660 align:middle line:84% Or phi x1, if phi x1 contract with one of the phi here, 00:23:50.660 --> 00:23:53.090 align:middle line:84% then phi x2 also have to contract with one of the phi 00:23:53.090 --> 00:23:53.660 align:middle line:90% here. 00:23:53.660 --> 00:23:56.990 align:middle line:84% So you have two possible patterns for contraction. 00:23:56.990 --> 00:24:02.000 align:middle line:84% So the first one is important, is just first contract phi x1 00:24:02.000 --> 00:24:04.550 align:middle line:90% and phi x2. 00:24:04.550 --> 00:24:15.140 align:middle line:84% So again, so we have minus i lambda divided by 4 factorial. 00:24:15.140 --> 00:24:24.370 align:middle line:84% So when we contract x1, x2, then we just get GF 0 x1,2. 00:24:24.370 --> 00:24:27.700 align:middle line:84% And then we contract phi 4 themselves. 00:24:27.700 --> 00:24:29.360 align:middle line:90% So that's what we did before. 00:24:29.360 --> 00:24:30.680 align:middle line:90% It's the same thing as above. 00:24:30.680 --> 00:24:37.960 align:middle line:84% So we just get a factor of 3 d4 x and then GF 0. 00:24:37.960 --> 00:24:41.930 align:middle line:90% We evaluate at 0 squared-- 00:24:41.930 --> 00:24:44.810 align:middle line:90% just the same as this. 00:24:44.810 --> 00:24:49.080 align:middle line:84% But then the other term, phi contract with this-- 00:24:49.080 --> 00:24:54.260 align:middle line:84% so each of these, x1, x2 contract with here. 00:24:54.260 --> 00:24:55.950 align:middle line:90% So in this case-- 00:24:55.950 --> 00:24:58.910 align:middle line:84% so let's imagine this contract with them. 00:24:58.910 --> 00:25:01.420 align:middle line:84% Then there are four possibilities. 00:25:01.420 --> 00:25:03.630 align:middle line:84% So after this contract with one of them, 00:25:03.630 --> 00:25:06.870 align:middle line:84% this contract with that, then I have three possibilities. 00:25:06.870 --> 00:25:11.910 align:middle line:84% So you have 4 times three combinatorial factors. 00:25:11.910 --> 00:25:15.720 align:middle line:84% So you have i lambda 4 factorial. 00:25:15.720 --> 00:25:19.070 align:middle line:90% Then you have 4 times 3. 00:25:19.070 --> 00:25:20.030 align:middle line:90% Then you have d4 x. 00:25:20.030 --> 00:25:23.020 align:middle line:90% 00:25:23.020 --> 00:25:29.740 align:middle line:84% Then here, we have GF 0 x minus x1-- 00:25:29.740 --> 00:25:31.180 align:middle line:90% so x1 minus x. 00:25:31.180 --> 00:25:37.000 align:middle line:84% It doesn't matter-- and GF 0 x2 minus x. 00:25:37.000 --> 00:25:39.040 align:middle line:84% And then the last one just contract 00:25:39.040 --> 00:25:41.500 align:middle line:90% with self, just again, GF 0, 0-- 00:25:41.500 --> 00:25:44.760 align:middle line:90% 00:25:44.760 --> 00:25:48.880 align:middle line:84% times-- and order lambda squared. 00:25:48.880 --> 00:25:51.750 align:middle line:90% 00:25:51.750 --> 00:25:54.340 align:middle line:84% So these are the explicit expressions. 00:25:54.340 --> 00:25:56.010 align:middle line:84% So these are the explicit expressions. 00:25:56.010 --> 00:25:59.600 align:middle line:90% 00:25:59.600 --> 00:26:03.610 align:middle line:84% So now you can actually-- notice that these two terms actually 00:26:03.610 --> 00:26:04.480 align:middle line:90% are the same, right? 00:26:04.480 --> 00:26:07.610 align:middle line:90% 00:26:07.610 --> 00:26:10.490 align:middle line:84% So these two terms actually are identical. 00:26:10.490 --> 00:26:14.480 align:middle line:84% And they just differ by a sign, so actually, they cancel. 00:26:14.480 --> 00:26:15.620 align:middle line:90% So these cancel. 00:26:15.620 --> 00:26:19.870 align:middle line:84% So we just left with that and then this term. 00:26:19.870 --> 00:26:21.974 align:middle line:90% We just with this term. 00:26:21.974 --> 00:26:26.260 align:middle line:90% 00:26:26.260 --> 00:26:30.250 align:middle line:84% So this procedure can actually be 00:26:30.250 --> 00:26:37.650 align:middle line:84% represented diagrammatically, be represented diagrammatically. 00:26:37.650 --> 00:26:43.260 align:middle line:84% And I will just introduce a single line, 00:26:43.260 --> 00:26:46.530 align:middle line:84% just introduce a single line for GF-- 00:26:46.530 --> 00:26:48.500 align:middle line:90% say, x and y-- 00:26:48.500 --> 00:26:54.930 align:middle line:90% for GF 0 x minus y. 00:26:54.930 --> 00:26:58.410 align:middle line:84% So we just represent the line like that. 00:26:58.410 --> 00:27:00.360 align:middle line:90% So if I use that-- 00:27:00.360 --> 00:27:04.680 align:middle line:84% and then we can write them in the diagrammatic form, 00:27:04.680 --> 00:27:09.480 align:middle line:84% then the first term is just x1, x2. 00:27:09.480 --> 00:27:13.390 align:middle line:90% 00:27:13.390 --> 00:27:18.890 align:middle line:84% Another second term would be 3 multiply 4 factorial. 00:27:18.890 --> 00:27:23.610 align:middle line:84% So this is i lambda divided by 8. 00:27:23.610 --> 00:27:30.330 align:middle line:84% And then you have x1 times x1, x2 then times-- 00:27:30.330 --> 00:27:33.000 align:middle line:90% this contracted with itself. 00:27:33.000 --> 00:27:34.400 align:middle line:90% And so you have two propagator. 00:27:34.400 --> 00:27:39.130 align:middle line:84% One is contracted like that, and one is contracted like that. 00:27:39.130 --> 00:27:41.760 align:middle line:90% And this point is x. 00:27:41.760 --> 00:27:48.500 align:middle line:84% So you have x, but they all contract with itself. 00:27:48.500 --> 00:27:52.630 align:middle line:84% So it's x minus x for this one and x minus x for that one. 00:27:52.630 --> 00:27:55.450 align:middle line:90% 00:27:55.450 --> 00:27:57.070 align:middle line:84% And then similarly, of course, this 00:27:57.070 --> 00:28:02.230 align:middle line:84% is identical to that, is i lambda divided by 8. 00:28:02.230 --> 00:28:11.460 align:middle line:90% So this is times, so x1, x2 x. 00:28:11.460 --> 00:28:15.540 align:middle line:90% And then the last one-- 00:28:15.540 --> 00:28:18.480 align:middle line:90% so this one corresponding to-- 00:28:18.480 --> 00:28:21.360 align:middle line:90% so now this is 12 divided by 24. 00:28:21.360 --> 00:28:26.130 align:middle line:90% This is minus i 2 lambda-- 00:28:26.130 --> 00:28:29.630 align:middle line:90% i lambda divided by 2. 00:28:29.630 --> 00:28:33.920 align:middle line:90% So x1 is to x, x and x2 to x. 00:28:33.920 --> 00:28:38.840 align:middle line:84% And then you have something like this, x1, x2. 00:28:38.840 --> 00:28:41.430 align:middle line:90% 00:28:41.430 --> 00:28:42.570 align:middle line:90% So this here is x. 00:28:42.570 --> 00:28:51.780 align:middle line:84% So x1 x, x2 x, and then x to itself. 00:28:51.780 --> 00:28:54.240 align:middle line:84% And then you have all the lambda squared. 00:28:54.240 --> 00:28:59.670 align:middle line:90% 00:28:59.670 --> 00:29:03.860 align:middle line:84% So this-- again, these two terms-- 00:29:03.860 --> 00:29:06.020 align:middle line:84% as we already said, these two terms cancel. 00:29:06.020 --> 00:29:10.140 align:middle line:84% And then you just get these two terms, just 00:29:10.140 --> 00:29:11.183 align:middle line:90% get these two terms. 00:29:11.183 --> 00:29:14.500 align:middle line:90% 00:29:14.500 --> 00:29:20.690 align:middle line:84% And we see that each factor of H I or each factor of S I you 00:29:20.690 --> 00:29:22.070 align:middle line:90% bring down-- 00:29:22.070 --> 00:29:25.550 align:middle line:84% so each factor of H I or each factor of-- 00:29:25.550 --> 00:29:29.963 align:middle line:84% you bring down corresponding to a interacting vertex like this, 00:29:29.963 --> 00:29:31.880 align:middle line:84% corresponding to interaction vertex like this. 00:29:31.880 --> 00:29:35.300 align:middle line:90% You have four legs or phi's. 00:29:35.300 --> 00:29:37.880 align:middle line:84% And so we call this interaction vertex. 00:29:37.880 --> 00:29:55.170 align:middle line:90% 00:29:55.170 --> 00:30:01.930 align:middle line:84% So in practice, it's actually much easier 00:30:01.930 --> 00:30:06.040 align:middle line:84% to just draw the diagram first, and then from the diagram, 00:30:06.040 --> 00:30:08.960 align:middle line:84% to write down the analytic expression. 00:30:08.960 --> 00:30:12.740 align:middle line:84% Because the diagram is much more intuitive. 00:30:12.740 --> 00:30:16.880 align:middle line:84% Essentially, what's happening here, whatever terms here, 00:30:16.880 --> 00:30:19.880 align:middle line:84% you just find all possible way to contract them. 00:30:19.880 --> 00:30:23.260 align:middle line:84% And then diagrammatically, we can just do it diagrammatically 00:30:23.260 --> 00:30:25.010 align:middle line:84% through all possible way to contract them. 00:30:25.010 --> 00:30:28.580 align:middle line:84% That's much easier to do by drawing a diagram 00:30:28.580 --> 00:30:32.960 align:middle line:90% than by doing the counting. 00:30:32.960 --> 00:30:51.415 align:middle line:84% So in practice, much easier to draw the diagram first. 00:30:51.415 --> 00:30:55.430 align:middle line:90% 00:30:55.430 --> 00:30:57.155 align:middle line:84% You just draw all possible diagrams. 00:30:57.155 --> 00:31:01.360 align:middle line:90% 00:31:01.360 --> 00:31:16.320 align:middle line:84% And then from diagram, write down the expressions. 00:31:16.320 --> 00:31:24.970 align:middle line:90% 00:31:24.970 --> 00:31:29.770 align:middle line:84% So those diagrams are just called the Feynman diagrams. 00:31:29.770 --> 00:31:31.780 align:middle line:84% So we just draw possible diagrams. 00:31:31.780 --> 00:31:34.420 align:middle line:84% And then from the diagrams, we write down the expressions. 00:31:34.420 --> 00:31:36.010 align:middle line:90% That's called the Feynman rules. 00:31:36.010 --> 00:31:36.970 align:middle line:90% I gave you a diagram. 00:31:36.970 --> 00:31:38.553 align:middle line:84% Then there's a rule which convert them 00:31:38.553 --> 00:31:40.660 align:middle line:90% into such kind of expression. 00:31:40.660 --> 00:31:44.590 align:middle line:84% We can convert them into such kind of expression. 00:31:44.590 --> 00:31:47.580 align:middle line:84% So now let's give you an example. 00:31:47.580 --> 00:31:49.490 align:middle line:84% So now let's give you an example. 00:31:49.490 --> 00:31:52.858 align:middle line:84% So now let's look at the example in the second order. 00:31:52.858 --> 00:31:54.775 align:middle line:84% Let's look at the example in the second order. 00:31:54.775 --> 00:31:58.180 align:middle line:90% 00:31:58.180 --> 00:32:07.250 align:middle line:84% So let's consider order lambda squared example. 00:32:07.250 --> 00:32:10.610 align:middle line:90% 00:32:10.610 --> 00:32:14.020 align:middle line:84% So the lambda squared, there are many terms. 00:32:14.020 --> 00:32:18.730 align:middle line:84% So one of the term is just to say 00:32:18.730 --> 00:32:21.720 align:middle line:90% you now have H I squared here. 00:32:21.720 --> 00:32:23.220 align:middle line:84% You also have downstairs, et cetera. 00:32:23.220 --> 00:32:24.095 align:middle line:90% There are many terms. 00:32:24.095 --> 00:32:25.335 align:middle line:90% But one of the term-- 00:32:25.335 --> 00:32:28.050 align:middle line:90% 00:32:28.050 --> 00:32:29.290 align:middle line:90% yeah, let me just say here. 00:32:29.290 --> 00:32:32.750 align:middle line:84% So one of the term just x and then you have H I squared here. 00:32:32.750 --> 00:32:33.750 align:middle line:90% This is one of the term. 00:32:33.750 --> 00:32:35.110 align:middle line:90% They also have other terms. 00:32:35.110 --> 00:32:40.030 align:middle line:84% So let's now look at just that term as example. 00:32:40.030 --> 00:32:42.940 align:middle line:84% So at the lambda squared example, 00:32:42.940 --> 00:32:46.030 align:middle line:90% there's a term like this. 00:32:46.030 --> 00:32:49.750 align:middle line:84% 1 factorial-- 1 factorial, when you expand this to the second 00:32:49.750 --> 00:32:51.820 align:middle line:90% order, then this-- 00:32:51.820 --> 00:32:55.480 align:middle line:90% yeah, a 1/2. 00:32:55.480 --> 00:33:00.280 align:middle line:84% And then you have minus i lambda 4 factorial squared 00:33:00.280 --> 00:33:03.560 align:middle line:84% because now this coefficient also expand to second order. 00:33:03.560 --> 00:33:19.700 align:middle line:84% And then you have d4 x d4 y x1 x2 phi 4 x phi 4 y here. 00:33:19.700 --> 00:33:21.905 align:middle line:84% So now one of the term is like this. 00:33:21.905 --> 00:33:28.950 align:middle line:90% 00:33:28.950 --> 00:33:31.770 align:middle line:84% So later, we don't even need to write down such a term. 00:33:31.770 --> 00:33:34.590 align:middle line:84% We just immediately write down all the diagram. 00:33:34.590 --> 00:33:39.960 align:middle line:84% Just here, to help you to understand the process, 00:33:39.960 --> 00:33:43.240 align:middle line:84% let's just say let's look at this particular term. 00:33:43.240 --> 00:33:47.610 align:middle line:84% So now let's try to write down all possible contractions 00:33:47.610 --> 00:33:52.900 align:middle line:84% by just using diagrams, just by using diagrams. 00:33:52.900 --> 00:33:59.875 align:middle line:84% So now we have x1, x2, and the four phi-- four x and four y's. 00:33:59.875 --> 00:34:01.750 align:middle line:84% So essentially, you have something like this. 00:34:01.750 --> 00:34:04.780 align:middle line:90% You have x1. 00:34:04.780 --> 00:34:06.100 align:middle line:90% You have x2. 00:34:06.100 --> 00:34:09.159 align:middle line:84% So these are the two external points. 00:34:09.159 --> 00:34:13.345 align:middle line:84% And then now you have four phi's coming from x 00:34:13.345 --> 00:34:16.489 align:middle line:90% and four phi's come from y. 00:34:16.489 --> 00:34:20.850 align:middle line:84% So we need to connect, find all possible ways to connect. 00:34:20.850 --> 00:34:22.830 align:middle line:84% We find all possible ways to connect. 00:34:22.830 --> 00:34:28.800 align:middle line:90% So that's the goal. 00:34:28.800 --> 00:34:31.580 align:middle line:90% So yeah. 00:34:31.580 --> 00:34:33.754 align:middle line:84% So here are the all possible ways-- 00:34:33.754 --> 00:34:36.540 align:middle line:90% 00:34:36.540 --> 00:34:38.870 align:middle line:84% so here are the all possible ways. 00:34:38.870 --> 00:34:46.159 align:middle line:84% So first-- there are actually five in inequivalent way. 00:34:46.159 --> 00:34:48.000 align:middle line:90% So let me just write them down. 00:34:48.000 --> 00:34:51.139 align:middle line:84% So the first one is you just connect one of them 00:34:51.139 --> 00:34:53.870 align:middle line:90% to x and one of them to x-- 00:34:53.870 --> 00:34:56.810 align:middle line:90% yeah, one of the x-- 00:34:56.810 --> 00:34:58.200 align:middle line:84% yeah, let me just draw them down. 00:34:58.200 --> 00:34:59.480 align:middle line:90% OK. 00:34:59.480 --> 00:35:01.140 align:middle line:90% So one of them is the following. 00:35:01.140 --> 00:35:02.495 align:middle line:90% So you take one of the x-- 00:35:02.495 --> 00:35:06.090 align:middle line:90% 00:35:06.090 --> 00:35:08.100 align:middle line:90% so there are five of them. 00:35:08.100 --> 00:35:10.330 align:middle line:90% Let me write equal sign here. 00:35:10.330 --> 00:35:15.280 align:middle line:84% So each one of them, let me give a label for later use. 00:35:15.280 --> 00:35:20.070 align:middle line:84% So here, you can have, say, x1 and x2. 00:35:20.070 --> 00:35:24.540 align:middle line:90% Then they can connect to x. 00:35:24.540 --> 00:35:27.400 align:middle line:90% And then x can connect to y. 00:35:27.400 --> 00:35:29.890 align:middle line:90% So this is one of them. 00:35:29.890 --> 00:35:33.600 align:middle line:90% 00:35:33.600 --> 00:35:37.140 align:middle line:84% And the second one possibility is you 00:35:37.140 --> 00:35:46.970 align:middle line:84% just x1 connect to x and x2 to connect to y. 00:35:46.970 --> 00:35:48.170 align:middle line:90% And now there are three-- 00:35:48.170 --> 00:35:51.800 align:middle line:90% 00:35:51.800 --> 00:35:53.750 align:middle line:84% between x and the y, we have to connect. 00:35:53.750 --> 00:35:56.750 align:middle line:90% So we have something like that. 00:35:56.750 --> 00:36:00.060 align:middle line:90% So this is the second. 00:36:00.060 --> 00:36:37.160 align:middle line:84% So the third-- and the third way, 00:36:37.160 --> 00:36:45.140 align:middle line:84% if you connect x1 to x2 itself and then times-- 00:36:45.140 --> 00:36:47.330 align:middle line:84% and then you have to connect x and y. 00:36:47.330 --> 00:36:49.910 align:middle line:90% So there are four of them. 00:36:49.910 --> 00:36:51.860 align:middle line:90% Just connect x and y like this. 00:36:51.860 --> 00:36:54.260 align:middle line:90% You can connect them like this. 00:36:54.260 --> 00:36:59.030 align:middle line:90% Or you connect x1, x2 itself. 00:36:59.030 --> 00:37:03.260 align:middle line:84% And then you, again, have to connect with x and y. 00:37:03.260 --> 00:37:04.130 align:middle line:90% Or you can connect. 00:37:04.130 --> 00:37:07.222 align:middle line:84% Then the y-- and that's just between x and y. 00:37:07.222 --> 00:37:08.180 align:middle line:90% You can also have that. 00:37:08.180 --> 00:37:12.240 align:middle line:90% 00:37:12.240 --> 00:37:15.500 align:middle line:84% So some of x is contracted y, and some of x 00:37:15.500 --> 00:37:16.635 align:middle line:90% is contracted with itself. 00:37:16.635 --> 00:37:19.200 align:middle line:90% 00:37:19.200 --> 00:37:22.200 align:middle line:84% Or you have the last possibility. 00:37:22.200 --> 00:37:25.920 align:middle line:90% So you have x1, x2. 00:37:25.920 --> 00:37:29.720 align:middle line:84% But then the x just contract with itself. 00:37:29.720 --> 00:37:31.650 align:middle line:84% And the y also contract with itself. 00:37:31.650 --> 00:37:38.240 align:middle line:90% 00:37:38.240 --> 00:37:40.950 align:middle line:90% So any questions on this? 00:37:40.950 --> 00:37:43.970 align:middle line:84% So these are essentially all possible contractions 00:37:43.970 --> 00:37:45.800 align:middle line:90% you can have between them. 00:37:45.800 --> 00:37:48.730 align:middle line:90% We just draw the diagrams. 00:37:48.730 --> 00:37:52.090 align:middle line:84% But now we also have to calculate-- 00:37:52.090 --> 00:37:57.200 align:middle line:90% 00:37:57.200 --> 00:37:59.990 align:middle line:84% so we also have to find the combinatorial factors 00:37:59.990 --> 00:38:02.680 align:middle line:90% between them. 00:38:02.680 --> 00:38:05.350 align:middle line:84% We also have to find the combinatorial factors 00:38:05.350 --> 00:38:06.910 align:middle line:90% between them. 00:38:06.910 --> 00:38:12.010 align:middle line:84% So now let's try to work out the combinatorial factors. 00:38:12.010 --> 00:38:16.623 align:middle line:84% So let me just give you some examples. 00:38:16.623 --> 00:38:18.790 align:middle line:84% And then the rest, I will just write down the answer 00:38:18.790 --> 00:38:22.150 align:middle line:84% because it just take too much time to do all the examples one 00:38:22.150 --> 00:38:24.640 align:middle line:90% by one. 00:38:24.640 --> 00:38:28.530 align:middle line:84% So let's do this one as an example. 00:38:28.530 --> 00:38:30.360 align:middle line:90% So yeah, let me just erase here. 00:38:30.360 --> 00:38:32.325 align:middle line:90% We need space here. 00:38:32.325 --> 00:38:38.700 align:middle line:90% 00:38:38.700 --> 00:38:43.100 align:middle line:84% So here, actually, there are two possible diagrams. 00:38:43.100 --> 00:38:46.595 align:middle line:84% I can have x1 and x2 connected to x. 00:38:46.595 --> 00:38:51.470 align:middle line:84% I can also have x1, x2 connected to y. 00:38:51.470 --> 00:38:53.210 align:middle line:90% Because x and y are symmetric. 00:38:53.210 --> 00:38:55.130 align:middle line:90% So I only draw one of them. 00:38:55.130 --> 00:38:57.350 align:middle line:84% So that means there's a factor of 2. 00:38:57.350 --> 00:38:59.420 align:middle line:84% I can either connect to x or connect to y. 00:38:59.420 --> 00:39:01.490 align:middle line:90% So I have 1/2. 00:39:01.490 --> 00:39:06.920 align:middle line:84% So let me also copy this, lambda 4 factorial squared. 00:39:06.920 --> 00:39:10.610 align:middle line:90% And then I have a 2 factor. 00:39:10.610 --> 00:39:13.520 align:middle line:84% x1, x2 can either connect to x or connect to y. 00:39:13.520 --> 00:39:17.310 align:middle line:90% 00:39:17.310 --> 00:39:22.110 align:middle line:84% So x1 have four possibilities to connect to x, 00:39:22.110 --> 00:39:26.920 align:middle line:84% and x2 have three possibilities to connect to x. 00:39:26.920 --> 00:39:30.495 align:middle line:84% And then the remaining phi x then have four possibilities 00:39:30.495 --> 00:39:33.430 align:middle line:90% to connect to y. 00:39:33.430 --> 00:39:37.030 align:middle line:84% And the other remaining phi x have three possibility 00:39:37.030 --> 00:39:37.870 align:middle line:90% to connect to y. 00:39:37.870 --> 00:39:42.220 align:middle line:84% So essentially, that's your prefactors, 00:39:42.220 --> 00:39:43.820 align:middle line:90% combinatorial factors. 00:39:43.820 --> 00:39:48.940 align:middle line:84% And so if you worked it out, so that give you minus lambda 00:39:48.940 --> 00:39:51.750 align:middle line:90% squared 1/2 squared. 00:39:51.750 --> 00:39:54.410 align:middle line:90% 00:39:54.410 --> 00:39:55.040 align:middle line:90% Yes? 00:39:55.040 --> 00:39:57.012 align:middle line:84% AUDIENCE: Why were there four-- so after we 00:39:57.012 --> 00:39:59.215 align:middle line:84% x1 and x2 to x, why were there four remaining? 00:39:59.215 --> 00:39:59.840 align:middle line:90% HONG LIU: Yeah. 00:39:59.840 --> 00:40:04.220 align:middle line:84% It's because now you have two phi remaining, 00:40:04.220 --> 00:40:06.020 align:middle line:90% two phi x remaining. 00:40:06.020 --> 00:40:10.110 align:middle line:84% So two phi x remaining have to connect with y. 00:40:10.110 --> 00:40:13.140 align:middle line:84% So the first phi, we have four possibility to connect y 00:40:13.140 --> 00:40:15.100 align:middle line:84% because there are four y's there. 00:40:15.100 --> 00:40:16.560 align:middle line:90% So that's this factor of 4. 00:40:16.560 --> 00:40:18.790 align:middle line:84% And then the other phi x have three possibilities 00:40:18.790 --> 00:40:19.620 align:middle line:90% to connect with y. 00:40:19.620 --> 00:40:21.090 align:middle line:90% And so that's the factor of 3. 00:40:21.090 --> 00:40:23.250 align:middle line:84% And once you do that, then this y just 00:40:23.250 --> 00:40:24.870 align:middle line:90% have to connect with itself. 00:40:24.870 --> 00:40:26.010 align:middle line:90% Yes. 00:40:26.010 --> 00:40:31.520 align:middle line:84% AUDIENCE: Do we have a diagram x1 to x, to x to y, y to x2, 00:40:31.520 --> 00:40:34.650 align:middle line:90% and x loop and y loop? 00:40:34.650 --> 00:40:37.160 align:middle line:90% HONG LIU: [LAUGHS] Which one? 00:40:37.160 --> 00:40:37.940 align:middle line:90% Say it again. 00:40:37.940 --> 00:40:40.400 align:middle line:84% AUDIENCE: Wait, so you should just draw it on the board. 00:40:40.400 --> 00:40:41.840 align:middle line:90% AUDIENCE: x1 to x. 00:40:41.840 --> 00:40:44.090 align:middle line:90% HONG LIU: x1 to x. 00:40:44.090 --> 00:40:48.860 align:middle line:84% AUDIENCE: And then one loop of x, x to y. 00:40:48.860 --> 00:40:50.420 align:middle line:90% Loop at y. 00:40:50.420 --> 00:40:52.850 align:middle line:90% And y to x2. 00:40:52.850 --> 00:40:55.724 align:middle line:90% HONG LIU: And y to x2. 00:40:55.724 --> 00:40:57.560 align:middle line:84% AUDIENCE: Oh, we didn't include the-- 00:40:57.560 --> 00:41:00.950 align:middle line:90% HONG LIU: Yeah, let me see. 00:41:00.950 --> 00:41:02.385 align:middle line:90% Yes. 00:41:02.385 --> 00:41:04.010 align:middle line:84% I should also have a diagram like this. 00:41:04.010 --> 00:41:05.510 align:middle line:90% Yeah, it's possible. 00:41:05.510 --> 00:41:07.970 align:middle line:90% I think I just forgot it. 00:41:07.970 --> 00:41:08.480 align:middle line:90% Let me see. 00:41:08.480 --> 00:41:10.010 align:middle line:90% Do I have something like this? 00:41:10.010 --> 00:41:12.530 align:middle line:84% No, actually-- yeah, I think forgot it. 00:41:12.530 --> 00:41:14.150 align:middle line:90% Good, good, good. 00:41:14.150 --> 00:41:15.470 align:middle line:90% This diagram is here. 00:41:15.470 --> 00:41:17.660 align:middle line:90% OK, so this is b prime. 00:41:17.660 --> 00:41:19.280 align:middle line:90% [LAUGHTER] 00:41:19.280 --> 00:41:20.150 align:middle line:90% Thank you. 00:41:20.150 --> 00:41:21.830 align:middle line:90% [LAUGHS] Thank you. 00:41:21.830 --> 00:41:24.530 align:middle line:90% So we can also have that. 00:41:24.530 --> 00:41:25.190 align:middle line:90% Yeah. 00:41:25.190 --> 00:41:27.870 align:middle line:84% Here, it's really for the illustration. 00:41:27.870 --> 00:41:30.980 align:middle line:84% I don't-- actually, I should claim at the beginning, 00:41:30.980 --> 00:41:36.380 align:middle line:84% I didn't aim for completeness and just for illustration. 00:41:36.380 --> 00:41:39.780 align:middle line:84% Because if we try to be too complete, 00:41:39.780 --> 00:41:43.020 align:middle line:84% then that may take too much time. 00:41:43.020 --> 00:41:43.790 align:middle line:90% OK, good, good. 00:41:43.790 --> 00:41:45.670 align:middle line:90% This is also a possibility. 00:41:45.670 --> 00:41:50.140 align:middle line:90% Yes, good job. 00:41:50.140 --> 00:41:56.920 align:middle line:84% And so now let's look at this one. 00:41:56.920 --> 00:42:00.010 align:middle line:90% So now let's look at this one. 00:42:00.010 --> 00:42:07.040 align:middle line:84% So again, let's copy this 1/2 minus i divided by 4 factorial 00:42:07.040 --> 00:42:08.840 align:middle line:90% squared-- 00:42:08.840 --> 00:42:13.660 align:middle line:84% and then the combinatorial factors. 00:42:13.660 --> 00:42:15.070 align:middle line:90% Again, I can have the freedom-- 00:42:15.070 --> 00:42:18.970 align:middle line:84% so x1 either connect to x or connect to y. 00:42:18.970 --> 00:42:21.430 align:middle line:84% That two diagram would be the same. 00:42:21.430 --> 00:42:23.110 align:middle line:90% And so I have two possibilities. 00:42:23.110 --> 00:42:27.070 align:middle line:84% Here, I only draw one of them because I just flipped them 00:42:27.070 --> 00:42:29.200 align:middle line:90% because x and y are symmetric. 00:42:29.200 --> 00:42:31.930 align:middle line:90% So I have a factor of 2. 00:42:31.930 --> 00:42:34.600 align:middle line:84% And then the x1 will have four opportunities 00:42:34.600 --> 00:42:38.650 align:middle line:90% to connect with four of x. 00:42:38.650 --> 00:42:41.950 align:middle line:84% And x2 have four possibilities to connect four of y 00:42:41.950 --> 00:42:43.570 align:middle line:90% because x2 connect with y. 00:42:43.570 --> 00:42:46.510 align:middle line:90% So I have two factors, 4 and 4. 00:42:46.510 --> 00:42:50.590 align:middle line:90% 00:42:50.590 --> 00:42:52.550 align:middle line:90% And so after that-- 00:42:52.550 --> 00:42:55.840 align:middle line:84% so phi x have three left and phi y have three left. 00:42:55.840 --> 00:42:59.320 align:middle line:84% And then take one of the phi, then 00:42:59.320 --> 00:43:02.650 align:middle line:84% have three possibilities you can connect to y. 00:43:02.650 --> 00:43:05.470 align:middle line:90% And the other one have two. 00:43:05.470 --> 00:43:07.510 align:middle line:90% And then that's it. 00:43:07.510 --> 00:43:08.950 align:middle line:90% That's it. 00:43:08.950 --> 00:43:11.470 align:middle line:90% So that all the possibilities. 00:43:11.470 --> 00:43:13.370 align:middle line:90% And if you work it out-- 00:43:13.370 --> 00:43:18.480 align:middle line:84% so this gives you lambda squared divided by 3 factorial. 00:43:18.480 --> 00:43:21.600 align:middle line:84% And then-- so as an exercise, you should work out this one 00:43:21.600 --> 00:43:23.760 align:middle line:90% yourself. 00:43:23.760 --> 00:43:29.610 align:middle line:84% And yeah, which I didn't work out, but I did 00:43:29.610 --> 00:43:31.770 align:middle line:90% work out this one. 00:43:31.770 --> 00:43:35.440 align:middle line:84% So this one I also give you an exercise, 00:43:35.440 --> 00:43:38.400 align:middle line:84% so this is minus 1/2 lambda squared 00:43:38.400 --> 00:43:45.240 align:middle line:84% 4 factorial squared times 4 times 3 times 2 equal 00:43:45.240 --> 00:43:50.400 align:middle line:84% to minus 1/2 lambda squared divided by 4 factorial. 00:43:50.400 --> 00:43:53.750 align:middle line:90% And so this one is the-- 00:43:53.750 --> 00:44:01.350 align:middle line:84% OK, minus 1/2 lambda squared 4 factorial squared times 2 times 00:44:01.350 --> 00:44:04.050 align:middle line:90% 3 times 2 times 3. 00:44:04.050 --> 00:44:09.530 align:middle line:84% So that gives you minus lambda squared divided by 16. 00:44:09.530 --> 00:44:14.680 align:middle line:84% So this final one is minus 1/2 lambda squared 00:44:14.680 --> 00:44:23.070 align:middle line:84% divided by 4 factorial squared times 3 squared-- 00:44:23.070 --> 00:44:28.225 align:middle line:84% and then given by minus lambda squared divided by 2 1/8 00:44:28.225 --> 00:44:28.725 align:middle line:90% squared. 00:44:28.725 --> 00:44:33.740 align:middle line:90% 00:44:33.740 --> 00:44:34.250 align:middle line:90% Good? 00:44:34.250 --> 00:44:35.470 align:middle line:90% So yeah. 00:44:35.470 --> 00:44:39.640 align:middle line:84% So I write them down just for you to check yourself later. 00:44:39.640 --> 00:44:46.690 align:middle line:84% So I let you do this as an exercise for yourself, 00:44:46.690 --> 00:44:49.920 align:middle line:84% to do this exercise later yourself. 00:44:49.920 --> 00:44:54.480 align:middle line:84% So here, we can already observe some patterns 00:44:54.480 --> 00:44:56.205 align:middle line:84% So here, we can observe some patterns. 00:44:56.205 --> 00:44:58.710 align:middle line:90% 00:44:58.710 --> 00:45:06.570 align:middle line:84% And so you can do this for each individual diagrams. 00:45:06.570 --> 00:45:10.840 align:middle line:84% But if you try to count this way, 00:45:10.840 --> 00:45:13.690 align:middle line:84% if you try to count it this way, as we are doing here, 00:45:13.690 --> 00:45:17.750 align:middle line:84% even though it's not difficult to do, but it's still tedious. 00:45:17.750 --> 00:45:19.300 align:middle line:90% It's still tedious. 00:45:19.300 --> 00:45:22.360 align:middle line:84% Each diagram have to count the factor of 4, 3, et cetera. 00:45:22.360 --> 00:45:25.360 align:middle line:84% So it'd be nice if you find a better trick. 00:45:25.360 --> 00:45:27.790 align:middle line:84% So now there's a better way to do it, 00:45:27.790 --> 00:45:33.580 align:middle line:84% so the better way to do it, so better trick. 00:45:33.580 --> 00:45:37.090 align:middle line:90% 00:45:37.090 --> 00:45:39.515 align:middle line:84% So it's by notice the following-- 00:45:39.515 --> 00:45:43.320 align:middle line:90% 00:45:43.320 --> 00:45:49.040 align:middle line:84% so when you go to the lambda n-th order, so there's always-- 00:45:49.040 --> 00:45:53.360 align:middle line:84% there's an n factorial coming from expanding the exponential. 00:45:53.360 --> 00:45:56.010 align:middle line:90% 00:45:56.010 --> 00:46:00.360 align:middle line:84% So because we are expanding the exponential and the-- 00:46:00.360 --> 00:46:10.950 align:middle line:90% 00:46:10.950 --> 00:46:20.370 align:middle line:84% so there's always a factorial come from the expanding of-- 00:46:20.370 --> 00:46:24.580 align:middle line:90% so now at n-th order-- 00:46:24.580 --> 00:46:27.230 align:middle line:90% 00:46:27.230 --> 00:46:29.640 align:middle line:84% yeah, I think I erased my vertex. 00:46:29.640 --> 00:46:33.740 align:middle line:84% So at the n-th order, you have an n factorial 00:46:33.740 --> 00:46:36.590 align:middle line:84% from exponential, but you also have n vertices. 00:46:36.590 --> 00:46:40.280 align:middle line:84% Because each power, each lambda comes with a factor 00:46:40.280 --> 00:46:41.075 align:middle line:90% of these vertices. 00:46:41.075 --> 00:46:45.110 align:middle line:90% 00:46:45.110 --> 00:46:53.430 align:middle line:84% So now when you permute all these n, so when you permute 00:46:53.430 --> 00:46:55.290 align:middle line:90% the n interacting vertices-- 00:46:55.290 --> 00:47:10.250 align:middle line:90% 00:47:10.250 --> 00:47:14.690 align:middle line:84% and you also get the factor of n factorial 00:47:14.690 --> 00:47:16.280 align:middle line:90% because they're all symmetric. 00:47:16.280 --> 00:47:18.960 align:middle line:84% They're all symmetric, doing the contraction. 00:47:18.960 --> 00:47:21.230 align:middle line:84% And if you have one way to contract, and then 00:47:21.230 --> 00:47:23.720 align:middle line:84% you can have another way to contract by permuting 00:47:23.720 --> 00:47:24.950 align:middle line:90% all these different vertices. 00:47:24.950 --> 00:47:27.760 align:middle line:90% 00:47:27.760 --> 00:47:29.515 align:middle line:84% So then these two factors cancel. 00:47:29.515 --> 00:47:38.530 align:middle line:90% 00:47:38.530 --> 00:47:40.105 align:middle line:84% These two functions are also modulo. 00:47:40.105 --> 00:47:45.330 align:middle line:90% 00:47:45.330 --> 00:47:52.470 align:middle line:84% Modulo symmetries in permuting vertices-- 00:47:52.470 --> 00:48:02.310 align:middle line:90% 00:48:02.310 --> 00:48:04.440 align:middle line:84% if there are symmetries in permuting 00:48:04.440 --> 00:48:09.960 align:middle line:84% vertices, then, of course, you don't get n factorial. 00:48:09.960 --> 00:48:13.530 align:middle line:84% Because some of the permutations give you the same diagram. 00:48:13.530 --> 00:48:17.850 align:middle line:84% And so the modulo symmetries in permuting vertices-- 00:48:17.850 --> 00:48:20.310 align:middle line:90% then you have this n factorial. 00:48:20.310 --> 00:48:22.080 align:middle line:84% So this is the first observation. 00:48:22.080 --> 00:48:25.390 align:middle line:90% 00:48:25.390 --> 00:48:29.560 align:middle line:84% So the second observation is that 00:48:29.560 --> 00:48:38.860 align:middle line:84% for each vertex which you have minus i lambda divided 00:48:38.860 --> 00:48:40.915 align:middle line:90% by 4 factorial phi 4-- 00:48:40.915 --> 00:48:45.130 align:middle line:90% 00:48:45.130 --> 00:49:25.110 align:middle line:84% so if each phi in this phi 4 contracts differently, 00:49:25.110 --> 00:49:26.520 align:middle line:90% contracts differently-- 00:49:26.520 --> 00:49:31.400 align:middle line:90% 00:49:31.400 --> 00:49:34.510 align:middle line:84% so all these four phi's are symmetric with each other. 00:49:34.510 --> 00:49:37.360 align:middle line:84% There's not any phi which is special. 00:49:37.360 --> 00:49:41.440 align:middle line:84% But if each phi contracts differently, 00:49:41.440 --> 00:49:46.990 align:middle line:84% then again, you permute the phi's, it should be the same-- 00:49:46.990 --> 00:49:48.895 align:middle line:90% should lead to a-- 00:49:48.895 --> 00:49:50.500 align:middle line:84% that diagram should also be included. 00:49:50.500 --> 00:49:55.250 align:middle line:90% 00:49:55.250 --> 00:49:56.810 align:middle line:90% And then you can permute phi. 00:49:56.810 --> 00:50:02.010 align:middle line:90% 00:50:02.010 --> 00:50:11.000 align:middle line:84% Permutation of 4 phi's then leads to 4 factorial. 00:50:11.000 --> 00:50:15.030 align:middle line:90% 00:50:15.030 --> 00:50:24.200 align:middle line:84% So, again, modulo-- so this 4 factorial again 00:50:24.200 --> 00:50:27.750 align:middle line:90% cancels with this 4 factorial. 00:50:27.750 --> 00:50:46.660 align:middle line:84% So the two 4 factorial cancel modulo symmetries 00:50:46.660 --> 00:50:56.780 align:middle line:84% in permuting phi's coming from the same vertex. 00:50:56.780 --> 00:51:07.310 align:middle line:90% 00:51:07.310 --> 00:51:09.838 align:middle line:84% Again, so if these two phi's are contracted 00:51:09.838 --> 00:51:11.630 align:middle line:84% the same-- of course, when you permute them 00:51:11.630 --> 00:51:14.210 align:middle line:90% you don't generate new diagram. 00:51:14.210 --> 00:51:16.980 align:middle line:90% You don't generate new ways. 00:51:16.980 --> 00:51:21.170 align:middle line:84% So now we have two different permutations. 00:51:21.170 --> 00:51:24.290 align:middle line:84% One is to permute the vertices, and the second 00:51:24.290 --> 00:51:27.410 align:middle line:84% is to permute the phi's within each vertex. 00:51:27.410 --> 00:51:30.470 align:middle line:90% 00:51:30.470 --> 00:51:36.570 align:middle line:84% So that means both of these n factorial and these 4 00:51:36.570 --> 00:51:40.150 align:middle line:90% factorial, they cancel. 00:51:40.150 --> 00:51:41.470 align:middle line:90% So we can forget-- 00:51:41.470 --> 00:51:53.810 align:middle line:84% so this means forget about the n factorial factor 00:51:53.810 --> 00:51:54.920 align:middle line:90% from exponential. 00:51:54.920 --> 00:52:02.680 align:middle line:90% 00:52:02.680 --> 00:52:07.750 align:middle line:84% And we also forget about this 1 over 4 factorial. 00:52:07.750 --> 00:52:21.750 align:middle line:84% And then treat each vertex minus i lambda, 00:52:21.750 --> 00:52:25.320 align:middle line:84% just coming from a factor of minus i lambda. 00:52:25.320 --> 00:52:27.510 align:middle line:84% But then this way, we over-count. 00:52:27.510 --> 00:52:30.980 align:middle line:90% 00:52:30.980 --> 00:52:33.470 align:middle line:90% So this way, we over-count. 00:52:33.470 --> 00:52:34.820 align:middle line:90% And then we need to divide-- 00:52:34.820 --> 00:52:41.980 align:middle line:90% 00:52:41.980 --> 00:52:44.340 align:middle line:90% so we need to divide. 00:52:44.340 --> 00:53:09.540 align:middle line:84% And then divide by symmetry factors 00:53:09.540 --> 00:53:28.290 align:middle line:84% from permuting vertices and the legs. 00:53:28.290 --> 00:53:31.260 align:middle line:90% 00:53:31.260 --> 00:53:31.760 align:middle line:90% Yeah. 00:53:31.760 --> 00:53:36.450 align:middle line:90% 00:53:36.450 --> 00:53:36.950 align:middle line:90% OK. 00:53:36.950 --> 00:53:39.275 align:middle line:84% So now let's go back to this diagram. 00:53:39.275 --> 00:53:42.068 align:middle line:90% 00:53:42.068 --> 00:53:43.610 align:middle line:84% So now let's go back to this diagram. 00:53:43.610 --> 00:53:46.240 align:middle line:90% 00:53:46.240 --> 00:53:51.330 align:middle line:84% So in this diagram, there's no symmetry between x and y. 00:53:51.330 --> 00:53:54.840 align:middle line:90% 00:53:54.840 --> 00:53:57.870 align:middle line:84% Because x and y, they're not symmetric. 00:53:57.870 --> 00:53:59.230 align:middle line:90% So we don't need to divide-- 00:53:59.230 --> 00:54:01.230 align:middle line:84% so there's no symmetry factor associated 00:54:01.230 --> 00:54:07.960 align:middle line:90% with permuting the vertices. 00:54:07.960 --> 00:54:11.410 align:middle line:84% But for the vertex come from the x, 00:54:11.410 --> 00:54:15.310 align:middle line:90% these two legs are symmetric. 00:54:15.310 --> 00:54:18.460 align:middle line:84% So there's a factor of 2 permute them. 00:54:18.460 --> 00:54:22.240 align:middle line:84% And for this thing come from y-- again, 00:54:22.240 --> 00:54:24.820 align:middle line:84% there are two of them-- are symmetric. 00:54:24.820 --> 00:54:30.430 align:middle line:84% so That's why we divided by 2 symmetric factors, so this 1/2. 00:54:30.430 --> 00:54:35.060 align:middle line:84% And similarly, from here, again, the x and the y, 00:54:35.060 --> 00:54:36.560 align:middle line:90% they are asymmetric. 00:54:36.560 --> 00:54:38.840 align:middle line:84% Because one is contracted with x1 and one 00:54:38.840 --> 00:54:41.720 align:middle line:90% is contracted with x2. 00:54:41.720 --> 00:54:52.540 align:middle line:84% So there's no symmetry factors from commuting from vertices. 00:54:52.540 --> 00:54:58.560 align:middle line:84% But between x and y, there are three legs which are symmetric. 00:54:58.560 --> 00:55:00.840 align:middle line:84% When we permute, this three legs, and then we 00:55:00.840 --> 00:55:03.490 align:middle line:90% have three factorials. 00:55:03.490 --> 00:55:06.700 align:middle line:90% So we have three factorials. 00:55:06.700 --> 00:55:09.610 align:middle line:84% And similarly, I leave as exercise for you 00:55:09.610 --> 00:55:16.890 align:middle line:84% to do the other diagrams, to do the other diagrams. 00:55:16.890 --> 00:55:27.980 align:middle line:84% So now we are ready to just write down our rule. 00:55:27.980 --> 00:55:29.840 align:middle line:84% So now we have drawn the diagrams. 00:55:29.840 --> 00:55:33.200 align:middle line:84% And now we can use the diagram to write down expressions. 00:55:33.200 --> 00:55:55.580 align:middle line:84% So the rules of writing down expression from diagrams 00:55:55.580 --> 00:56:06.110 align:middle line:84% are called the Feynman rules, the Feynman rules. 00:56:06.110 --> 00:56:08.780 align:middle line:90% So here is the Feynman rules. 00:56:08.780 --> 00:56:16.210 align:middle line:90% So here is the Feynman rules 00:56:16.210 --> 00:56:18.000 align:middle line:90% So these are all very intuitive. 00:56:18.000 --> 00:56:21.390 align:middle line:90% 00:56:21.390 --> 00:56:25.080 align:middle line:90% So for each external point-- 00:56:25.080 --> 00:56:28.130 align:middle line:84% so here, we have some external points. 00:56:28.130 --> 00:56:40.030 align:middle line:84% So each external point, you associate it with a line. 00:56:40.030 --> 00:56:44.200 align:middle line:84% So here is the, say, for example x1. 00:56:44.200 --> 00:56:46.510 align:middle line:84% And then you just associate with a factor of 1. 00:56:46.510 --> 00:56:50.290 align:middle line:84% But there's always a line coming out of the external point 00:56:50.290 --> 00:56:52.100 align:middle line:84% because you have to contract with that one. 00:56:52.100 --> 00:57:00.340 align:middle line:90% 00:57:00.340 --> 00:57:07.550 align:middle line:84% And then for each propagator, for the propagator, 00:57:07.550 --> 00:57:09.260 align:middle line:90% we already wrote down. 00:57:09.260 --> 00:57:12.710 align:middle line:84% You just have-- say, you have two endpoint, then 00:57:12.710 --> 00:57:18.130 align:middle line:84% that's corresponding to GF 0 minus y. 00:57:18.130 --> 00:57:22.360 align:middle line:84% So each such propagator, you can just write down 00:57:22.360 --> 00:57:26.665 align:middle line:90% a factor corresponding to GF y. 00:57:26.665 --> 00:57:35.900 align:middle line:90% 00:57:35.900 --> 00:57:37.250 align:middle line:90% And then for each vertex-- 00:57:37.250 --> 00:57:46.800 align:middle line:90% 00:57:46.800 --> 00:57:48.030 align:middle line:90% I think I can erase them. 00:57:48.030 --> 00:57:56.570 align:middle line:90% 00:57:56.570 --> 00:57:57.700 align:middle line:90% So for each vertex-- 00:57:57.700 --> 00:58:08.860 align:middle line:90% 00:58:08.860 --> 00:58:11.700 align:middle line:90% so at point x-- 00:58:11.700 --> 00:58:14.685 align:middle line:84% so you associate with lambda and then integration over x. 00:58:14.685 --> 00:58:17.325 align:middle line:90% 00:58:17.325 --> 00:58:29.480 align:middle line:84% And then the last step, you divide by symmetry factors, 00:58:29.480 --> 00:58:33.240 align:middle line:90% symmetry factors for diagram. 00:58:33.240 --> 00:58:36.560 align:middle line:84% So you just draw all possible diagrams 00:58:36.560 --> 00:58:38.010 align:middle line:90% using this kind of rule. 00:58:38.010 --> 00:58:39.800 align:middle line:84% So for each propagator, you have a line, 00:58:39.800 --> 00:58:43.280 align:middle line:90% and then you have vertices. 00:58:43.280 --> 00:58:45.650 align:middle line:84% So now, from each diagram, by applying this rule, 00:58:45.650 --> 00:58:49.050 align:middle line:84% we can also write down the analytic expression. 00:58:49.050 --> 00:58:52.930 align:middle line:84% We can write down analytic expression. 00:58:52.930 --> 00:58:53.935 align:middle line:90% Any questions on this? 00:58:53.935 --> 00:59:00.650 align:middle line:90% 00:59:00.650 --> 00:59:04.850 align:middle line:84% So exercise for yourself to write down analytic expression 00:59:04.850 --> 00:59:06.140 align:middle line:90% for each of them. 00:59:06.140 --> 00:59:09.390 align:middle line:84% So yeah, let me give you an example. 00:59:09.390 --> 00:59:17.010 align:middle line:84% So for b, as an exercise, you should 00:59:17.010 --> 00:59:18.610 align:middle line:84% try to do yourself for each of them. 00:59:18.610 --> 00:59:20.780 align:middle line:84% Let me just give you one example for b. 00:59:20.780 --> 00:59:25.110 align:middle line:84% And then you have first the symmetric factors, 00:59:25.110 --> 00:59:29.570 align:middle line:84% lambda squared divided by 3 factorial. 00:59:29.570 --> 00:59:36.950 align:middle line:84% And then you have d4 x d4 y come from these two vertices. 00:59:36.950 --> 00:59:38.990 align:middle line:84% Yeah, I should just write it like this. 00:59:38.990 --> 00:59:40.880 align:middle line:90% Sorry. 00:59:40.880 --> 00:59:43.880 align:middle line:84% Yeah, let me just strictly follow this rule. 00:59:43.880 --> 00:59:44.420 align:middle line:90% OK. 00:59:44.420 --> 00:59:47.130 align:middle line:90% So we have two vertices. 00:59:47.130 --> 00:59:50.810 align:middle line:84% So each vertex, we have i lambda squared. 00:59:50.810 --> 00:59:53.370 align:middle line:90% Then you have d4 x and d4 y. 00:59:53.370 --> 00:59:56.000 align:middle line:90% 00:59:56.000 --> 01:00:01.840 align:middle line:84% And then for each propagator, we associate these. 01:00:01.840 --> 01:00:09.060 align:middle line:84% And then we have GF 0 x minus y, then cube. 01:00:09.060 --> 01:00:11.610 align:middle line:84% And then finally, we divide by symmetry factors 01:00:11.610 --> 01:00:14.710 align:middle line:84% because there's three permutations of them. 01:00:14.710 --> 01:00:18.360 align:middle line:90% So we divide by 3. 01:00:18.360 --> 01:00:21.150 align:middle line:84% So this is the analytic expression corresponding 01:00:21.150 --> 01:00:21.870 align:middle line:90% to that diagram. 01:00:21.870 --> 01:00:24.530 align:middle line:90% 01:00:24.530 --> 01:00:30.050 align:middle line:84% So later, you don't even have to write down such expression 01:00:30.050 --> 01:00:31.910 align:middle line:84% in the top line I wrote down there. 01:00:31.910 --> 01:00:34.880 align:middle line:84% You just, at each order, you have certain number 01:00:34.880 --> 01:00:36.350 align:middle line:90% of external points. 01:00:36.350 --> 01:00:38.510 align:middle line:84% And then you just have a number of vertices 01:00:38.510 --> 01:00:39.630 align:middle line:90% come from n-th order. 01:00:39.630 --> 01:00:40.672 align:middle line:90% You just have n vertices. 01:00:40.672 --> 01:00:43.970 align:middle line:84% So you just try to connect all possible lines between them. 01:00:43.970 --> 01:00:47.590 align:middle line:84% And then that's all possible contributions. 01:00:47.590 --> 01:00:53.660 align:middle line:90% 01:00:53.660 --> 01:00:54.530 align:middle line:90% Good? 01:00:54.530 --> 01:00:58.795 align:middle line:84% So this is the Feynman rule in coordinate space. 01:00:58.795 --> 01:01:04.560 align:middle line:90% 01:01:04.560 --> 01:01:08.960 align:middle line:84% It's also convenient to go to the momentum space. 01:01:08.960 --> 01:01:12.360 align:middle line:84% So it's also convenient to go to momentum space. 01:01:12.360 --> 01:01:15.230 align:middle line:84% So to go to momentum space, then we 01:01:15.230 --> 01:01:16.820 align:middle line:84% can define the Fourier transform. 01:01:16.820 --> 01:01:19.660 align:middle line:90% 01:01:19.660 --> 01:01:22.280 align:middle line:84% So let's define the Fourier transform. 01:01:22.280 --> 01:01:26.500 align:middle line:84% So suppose we have an n-point function. 01:01:26.500 --> 01:01:32.280 align:middle line:84% Suppose we have an n-point function Gn. 01:01:32.280 --> 01:01:42.410 align:middle line:90% 01:01:42.410 --> 01:01:46.190 align:middle line:84% Suppose we have an n-point function x1, xn. 01:01:46.190 --> 01:01:58.340 align:middle line:90% 01:01:58.340 --> 01:02:01.400 align:middle line:84% So this is just the standard Fourier transform. 01:02:01.400 --> 01:02:04.130 align:middle line:90% 01:02:04.130 --> 01:02:07.900 align:middle line:84% So because of the translation symmetry-- 01:02:07.900 --> 01:02:15.140 align:middle line:84% so we expect all the piece must be conserved. 01:02:15.140 --> 01:02:18.010 align:middle line:84% Just from the conservation, I will derive it. 01:02:18.010 --> 01:02:20.620 align:middle line:84% But you should also be able to expect on the general grounds 01:02:20.620 --> 01:02:22.190 align:middle line:90% from the translation symmetry. 01:02:22.190 --> 01:02:28.330 align:middle line:84% So we expect that there must be a factor of the momentum 01:02:28.330 --> 01:02:35.460 align:middle line:90% conservation coming from here. 01:02:35.460 --> 01:02:45.650 align:middle line:84% And then we call that coefficient Gn P. 01:02:45.650 --> 01:02:49.800 align:middle line:84% So our convention is that for momentum space correlation 01:02:49.800 --> 01:02:53.257 align:middle line:84% functions, we defined without this factor, 01:02:53.257 --> 01:02:54.090 align:middle line:90% without this factor. 01:02:54.090 --> 01:02:54.600 align:middle line:90% Yes? 01:02:54.600 --> 01:02:58.090 align:middle line:84% AUDIENCE: For the Feynman rules, shouldn't it also 01:02:58.090 --> 01:02:59.958 align:middle line:90% restrict to connected diagrams? 01:02:59.958 --> 01:03:00.750 align:middle line:90% HONG LIU: Oh, yeah. 01:03:00.750 --> 01:03:02.250 align:middle line:84% Yeah, we will talk about that later. 01:03:02.250 --> 01:03:04.940 align:middle line:84% Yeah, we'll talk about that later. 01:03:04.940 --> 01:03:05.440 align:middle line:90% Yeah. 01:03:05.440 --> 01:03:08.560 align:middle line:84% So here, I'm just talk about the general rule. 01:03:08.560 --> 01:03:12.550 align:middle line:84% And then we will talk about the more fine points, 01:03:12.550 --> 01:03:15.910 align:middle line:84% how to enumerate all possible diagrams. 01:03:15.910 --> 01:03:18.380 align:middle line:90% 01:03:18.380 --> 01:03:18.880 align:middle line:90% OK. 01:03:18.880 --> 01:03:23.950 align:middle line:84% So when we define the momentum space correlation functions, 01:03:23.950 --> 01:03:27.490 align:middle line:90% we extract out this factor. 01:03:27.490 --> 01:03:33.100 align:middle line:84% So the reason for to extract this out 01:03:33.100 --> 01:03:36.290 align:middle line:84% is from the momentum conservation. 01:03:36.290 --> 01:03:40.750 align:middle line:84% So to see this, to see that there is a factor like this, 01:03:40.750 --> 01:03:43.340 align:middle line:90% it's very easy. 01:03:43.340 --> 01:03:46.040 align:middle line:84% So because from the translation symmetry-- 01:03:46.040 --> 01:03:54.450 align:middle line:90% 01:03:54.450 --> 01:04:00.860 align:middle line:84% so the same argument that the two-point function, the x1, x2 01:04:00.860 --> 01:04:04.640 align:middle line:84% should only be a function of x1 minus x2. 01:04:04.640 --> 01:04:07.640 align:middle line:84% So for the n-point function from the translation symmetry-- 01:04:07.640 --> 01:04:12.180 align:middle line:90% 01:04:12.180 --> 01:04:19.067 align:middle line:84% so this, you can just take one point 01:04:19.067 --> 01:04:20.150 align:middle line:90% to be the reference point. 01:04:20.150 --> 01:04:29.410 align:middle line:84% You can subtract, say, all coordinate by the value of xn. 01:04:29.410 --> 01:04:33.460 align:middle line:84% And then the last argument becomes 0. 01:04:33.460 --> 01:04:36.250 align:middle line:84% So you can choose one point that's a reference point 01:04:36.250 --> 01:04:38.500 align:middle line:84% and subtract any other point from it. 01:04:38.500 --> 01:04:40.055 align:middle line:90% And that should be equivalent. 01:04:40.055 --> 01:04:42.680 align:middle line:84% Because it doesn't matter where you choose the reference point. 01:04:42.680 --> 01:04:46.180 align:middle line:84% So here, let's choose the reference point to be xn. 01:04:46.180 --> 01:04:47.740 align:middle line:90% So this should be equal to that. 01:04:47.740 --> 01:04:51.230 align:middle line:90% 01:04:51.230 --> 01:04:55.920 align:middle line:84% So now we do a Fourier transform. 01:04:55.920 --> 01:04:57.630 align:middle line:90% So now do a Fourier transform. 01:04:57.630 --> 01:05:00.080 align:middle line:84% So you can easily convince yourself-- 01:05:00.080 --> 01:05:02.540 align:middle line:84% so when you do a Fourier transform of this object, 01:05:02.540 --> 01:05:06.420 align:middle line:84% when you plug this into here, when 01:05:06.420 --> 01:05:10.110 align:middle line:84% you plug this into here, and then you get a delta function. 01:05:10.110 --> 01:05:14.490 align:middle line:84% So this as a simple exercise for yourself to do. 01:05:14.490 --> 01:05:17.192 align:middle line:84% And so if you plug in here, you find that there's always 01:05:17.192 --> 01:05:17.900 align:middle line:90% a delta function. 01:05:17.900 --> 01:05:21.860 align:middle line:90% 01:05:21.860 --> 01:05:24.070 align:middle line:84% So now I record that the momentum 01:05:24.070 --> 01:05:30.180 align:middle line:84% space, the Feynman propagator for the free theory 01:05:30.180 --> 01:05:35.730 align:middle line:84% in the momentum space, if you say k, 01:05:35.730 --> 01:05:40.470 align:middle line:84% then it's given by minus i k squared plus 01:05:40.470 --> 01:05:42.700 align:middle line:90% m squared minus i epsilon. 01:05:42.700 --> 01:05:47.730 align:middle line:84% So now you have a line, but now it's labeled by momentum k. 01:05:47.730 --> 01:05:49.680 align:middle line:90% Line labeled by momentum k. 01:05:49.680 --> 01:05:52.050 align:middle line:84% So this k denotes the flow of the direction 01:05:52.050 --> 01:05:53.640 align:middle line:90% of the momentum, this arrow. 01:05:53.640 --> 01:05:58.750 align:middle line:90% 01:05:58.750 --> 01:05:59.250 align:middle line:90% Good. 01:05:59.250 --> 01:06:05.220 align:middle line:84% So now, with this rule, you can easily write down what is the-- 01:06:05.220 --> 01:06:18.510 align:middle line:84% now you can just translate right. 01:06:18.510 --> 01:06:21.600 align:middle line:90% 01:06:21.600 --> 01:06:22.380 align:middle line:90% OK. 01:06:22.380 --> 01:06:26.370 align:middle line:84% So this is the one point and another point. 01:06:26.370 --> 01:06:28.727 align:middle line:90% And the third point-- 01:06:28.727 --> 01:06:30.810 align:middle line:84% so let me just-- in order to go to momentum space, 01:06:30.810 --> 01:06:34.020 align:middle line:84% let's just mention several points. 01:06:34.020 --> 01:06:38.910 align:middle line:84% So yeah, anyway, here is one point and then here 01:06:38.910 --> 01:06:40.720 align:middle line:90% is another point. 01:06:40.720 --> 01:06:42.355 align:middle line:84% And also, now let's look at the vertex. 01:06:42.355 --> 01:06:45.600 align:middle line:90% 01:06:45.600 --> 01:06:53.720 align:middle line:84% So at each vertex, we have integration. 01:06:53.720 --> 01:06:56.610 align:middle line:90% 01:06:56.610 --> 01:06:58.900 align:middle line:84% So suppose we have a vertex like this. 01:06:58.900 --> 01:07:05.580 align:middle line:84% So here is x And this is contracted with y1, y2, y3, y4. 01:07:05.580 --> 01:07:08.280 align:middle line:84% So let's imagine-- so we have a vertex like this, 01:07:08.280 --> 01:07:13.140 align:middle line:84% these four phi's contracted with four different y's. 01:07:13.140 --> 01:07:18.770 align:middle line:84% And then at such a vertex, then we 01:07:18.770 --> 01:07:27.960 align:middle line:84% will have expression like this, GF 0 x minus y1 GF 0 01:07:27.960 --> 01:07:39.220 align:middle line:84% x minus y2 and GF 0 x minus y3 GF 0 x minus y4. 01:07:39.220 --> 01:07:45.220 align:middle line:90% 01:07:45.220 --> 01:07:49.090 align:middle line:84% And now again-- so now when you plug in the expression-- 01:07:49.090 --> 01:07:54.150 align:middle line:90% 01:07:54.150 --> 01:07:57.150 align:middle line:84% so now if you plug in the expression-- so 01:07:57.150 --> 01:07:58.710 align:middle line:84% I will not write all the lines here. 01:07:58.710 --> 01:08:02.145 align:middle line:84% I think you will just see the basic idea. 01:08:02.145 --> 01:08:05.450 align:middle line:90% 01:08:05.450 --> 01:08:07.810 align:middle line:90% So this one, if you write down-- 01:08:07.810 --> 01:08:10.420 align:middle line:84% let's write each of them in terms of momentum space. 01:08:10.420 --> 01:08:12.940 align:middle line:90% 01:08:12.940 --> 01:08:15.220 align:middle line:84% So this, if you write in terms of momentum space, 01:08:15.220 --> 01:08:22.630 align:middle line:84% then you get a d 4 k1 exponential i k1 times 01:08:22.630 --> 01:08:29.830 align:middle line:90% x minus y1, then GF 0 k1-- 01:08:29.830 --> 01:08:32.229 align:middle line:84% and similarly for that and for that, for that. 01:08:32.229 --> 01:08:37.050 align:middle line:90% 01:08:37.050 --> 01:08:39.520 align:middle line:84% And now these have exponential i k1. 01:08:39.520 --> 01:08:41.560 align:middle line:84% So these have the exponential i k2. 01:08:41.560 --> 01:08:43.660 align:middle line:84% So these have exponential i k3 x. 01:08:43.660 --> 01:08:46.620 align:middle line:84% And these have exponential i k4 x. 01:08:46.620 --> 01:08:49.260 align:middle line:84% And there's no other x dependence. 01:08:49.260 --> 01:08:52.439 align:middle line:90% And then you have a d4x here. 01:08:52.439 --> 01:08:54.855 align:middle line:84% So what do you get then is just get a delta function. 01:08:54.855 --> 01:09:06.939 align:middle line:90% 01:09:06.939 --> 01:09:10.189 align:middle line:84% So you just get a delta function. 01:09:10.189 --> 01:09:13.689 align:middle line:84% So when you do that, when you go to momentum space, 01:09:13.689 --> 01:09:16.359 align:middle line:84% then you find that this should always be proportional. 01:09:16.359 --> 01:09:25.840 align:middle line:90% 01:09:25.840 --> 01:09:40.549 align:middle line:84% So that means momentum is conserved at each vertex. 01:09:40.549 --> 01:09:46.310 align:middle line:90% 01:09:46.310 --> 01:09:47.389 align:middle line:90% So if I draw-- 01:09:47.389 --> 01:09:50.750 align:middle line:90% 01:09:50.750 --> 01:09:57.460 align:middle line:90% so k1 here, k2, k3-- 01:09:57.460 --> 01:09:59.963 align:middle line:84% so I use the same convention for all of them. 01:09:59.963 --> 01:10:02.380 align:middle line:84% It doesn't matter whether you draw it outgoing or ingoing. 01:10:02.380 --> 01:10:05.440 align:middle line:84% But we draw the same convention for all of them. 01:10:05.440 --> 01:10:08.620 align:middle line:84% And then they should have to be-- 01:10:08.620 --> 01:10:11.500 align:middle line:84% then they should be proportional to a factor of this. 01:10:11.500 --> 01:10:15.410 align:middle line:90% 01:10:15.410 --> 01:10:18.500 align:middle line:84% And here, it doesn't matter what is yi. 01:10:18.500 --> 01:10:20.870 align:middle line:84% So y1, y2, y3, y4, they can either 01:10:20.870 --> 01:10:23.535 align:middle line:84% be some external point or some other internal point. 01:10:23.535 --> 01:10:28.730 align:middle line:90% 01:10:28.730 --> 01:10:29.836 align:middle line:90% Yes? 01:10:29.836 --> 01:10:38.520 align:middle line:84% AUDIENCE: With Gn of t object, the n-point function in space, 01:10:38.520 --> 01:10:40.770 align:middle line:84% can you get that by writing expectation values 01:10:40.770 --> 01:10:47.410 align:middle line:84% of the momentum field operators, like pi k1, pi x1, pi x2? 01:10:47.410 --> 01:10:47.910 align:middle line:90% Or is that-- 01:10:47.910 --> 01:10:48.060 align:middle line:90% HONG LIU: Yeah. 01:10:48.060 --> 01:10:48.720 align:middle line:90% Yeah, you can. 01:10:48.720 --> 01:10:49.475 align:middle line:90% Yeah, you can. 01:10:49.475 --> 01:10:51.600 align:middle line:84% AUDIENCE: And how would you do time ordering there? 01:10:51.600 --> 01:10:52.470 align:middle line:90% Because then-- 01:10:52.470 --> 01:10:55.830 align:middle line:90% HONG LIU: Yeah, you-- 01:10:55.830 --> 01:11:01.260 align:middle line:84% so this is strictly defined in terms of the-- 01:11:01.260 --> 01:11:01.770 align:middle line:90% yeah. 01:11:01.770 --> 01:11:06.190 align:middle line:84% So no, you cannot write that way in the sense that-- 01:11:06.190 --> 01:11:07.210 align:middle line:90% not as a whole momentum. 01:11:07.210 --> 01:11:07.710 align:middle line:90% Yeah. 01:11:07.710 --> 01:11:08.880 align:middle line:90% Yeah, no, you can not that way. 01:11:08.880 --> 01:11:09.480 align:middle line:90% Yeah. 01:11:09.480 --> 01:11:10.932 align:middle line:90% AUDIENCE: So you to find-- 01:11:10.932 --> 01:11:13.182 align:middle line:84% HONG LIU: You really have to find the final expression 01:11:13.182 --> 01:11:14.140 align:middle line:90% and then do the-- 01:11:14.140 --> 01:11:14.640 align:middle line:90% yeah. 01:11:14.640 --> 01:11:17.490 align:middle line:84% Yeah, for the Wightman function, you can, not for the time 01:11:17.490 --> 01:11:18.690 align:middle line:90% ordered. 01:11:18.690 --> 01:11:20.480 align:middle line:84% For the Wightman function, you can. 01:11:20.480 --> 01:11:20.980 align:middle line:90% Yeah. 01:11:20.980 --> 01:11:27.350 align:middle line:90% 01:11:27.350 --> 01:11:28.590 align:middle line:90% OK, good. 01:11:28.590 --> 01:11:34.580 align:middle line:84% So now we can write down the momentum space Feynman rule. 01:11:34.580 --> 01:11:39.160 align:middle line:90% 01:11:39.160 --> 01:11:44.560 align:middle line:84% And we will not have time to do an example. 01:11:44.560 --> 01:11:46.480 align:middle line:84% Let me just write them down, and then we 01:11:46.480 --> 01:11:49.490 align:middle line:84% will look at the example next time. 01:11:49.490 --> 01:11:56.410 align:middle line:90% So yeah. 01:11:56.410 --> 01:12:06.430 align:middle line:84% So here is the rule to compute this Gn p1 pn. 01:12:06.430 --> 01:12:10.870 align:middle line:90% We use the following rule. 01:12:10.870 --> 01:12:13.735 align:middle line:84% So first, for each external point-- 01:12:13.735 --> 01:12:24.660 align:middle line:90% 01:12:24.660 --> 01:12:30.370 align:middle line:84% and now you just can associate, say, a momentum-- 01:12:30.370 --> 01:12:31.680 align:middle line:90% respect to that momentum. 01:12:31.680 --> 01:12:35.840 align:middle line:84% And then you can associate with a factor of 1. 01:12:35.840 --> 01:12:36.930 align:middle line:90% And then you draw a line. 01:12:36.930 --> 01:12:38.340 align:middle line:90% You still draw a line. 01:12:38.340 --> 01:12:41.515 align:middle line:84% But now you associate a momentum with this line. 01:12:41.515 --> 01:12:43.390 align:middle line:84% Because when you Fourier transform this line, 01:12:43.390 --> 01:12:46.320 align:middle line:90% you will have momentum. 01:12:46.320 --> 01:12:52.600 align:middle line:84% And for each propagator, the propagator is given by this. 01:12:52.600 --> 01:12:54.510 align:middle line:90% So now you have a momentum. 01:12:54.510 --> 01:12:57.600 align:middle line:84% And then this is given by the momentum space. 01:12:57.600 --> 01:13:02.730 align:middle line:84% p squared plus m squared minus epsilon. 01:13:02.730 --> 01:13:16.170 align:middle line:84% And then now for each vertex, you have a factor of i lambda. 01:13:16.170 --> 01:13:18.510 align:middle line:84% And then you impose a momentum conservation. 01:13:18.510 --> 01:13:28.470 align:middle line:90% 01:13:28.470 --> 01:13:29.250 align:middle line:90% at each vertex. 01:13:29.250 --> 01:13:33.600 align:middle line:90% 01:13:33.600 --> 01:13:38.940 align:middle line:84% So now, remember, when we do a Fourier transform, 01:13:38.940 --> 01:13:44.190 align:middle line:84% each propagator, each line, will have a momentum integration 01:13:44.190 --> 01:13:49.440 align:middle line:84% because each one of them is integrated over momentum. 01:13:49.440 --> 01:13:51.300 align:middle line:84% So each one of them is associated 01:13:51.300 --> 01:13:53.280 align:middle line:90% with a momentum integration. 01:13:53.280 --> 01:13:55.860 align:middle line:84% And now, some of those momentum integration 01:13:55.860 --> 01:13:58.680 align:middle line:84% can be get rid of by imposing the delta function 01:13:58.680 --> 01:14:00.120 align:middle line:90% at the vertex. 01:14:00.120 --> 01:14:02.730 align:middle line:84% So you can get rid of some of them 01:14:02.730 --> 01:14:06.090 align:middle line:84% by momentum conservation at the vertex using 01:14:06.090 --> 01:14:08.250 align:middle line:90% the delta functions. 01:14:08.250 --> 01:14:12.000 align:middle line:84% But now you will have, say, a number of them left. 01:14:12.000 --> 01:14:21.180 align:middle line:84% So then you need to integrate over 01:14:21.180 --> 01:14:23.910 align:middle line:90% each undetermined momentum. 01:14:23.910 --> 01:14:36.940 align:middle line:90% 01:14:36.940 --> 01:14:40.000 align:middle line:84% So each undetermined momentum you 01:14:40.000 --> 01:14:41.560 align:middle line:84% need to integrate with this factor. 01:14:41.560 --> 01:14:44.770 align:middle line:90% 01:14:44.770 --> 01:14:46.930 align:middle line:84% And then the same thing, you need to divide it 01:14:46.930 --> 01:14:48.950 align:middle line:90% by symmetry factor. 01:14:48.950 --> 01:14:51.700 align:middle line:84% So symmetry factor is the same in coordinate 01:14:51.700 --> 01:14:54.640 align:middle line:90% or in momentum space. 01:14:54.640 --> 01:15:01.550 align:middle line:90% 01:15:01.550 --> 01:15:06.350 align:middle line:84% So this is the way which you can write down the momentum space 01:15:06.350 --> 01:15:10.100 align:middle line:84% expression for each diagram, momentum space 01:15:10.100 --> 01:15:12.650 align:middle line:90% expression for each step. 01:15:12.650 --> 01:15:16.880 align:middle line:84% Actually, maybe we still have some time. 01:15:16.880 --> 01:15:19.190 align:middle line:90% So we have a couple minutes. 01:15:19.190 --> 01:15:20.975 align:middle line:90% So we can just do this example. 01:15:20.975 --> 01:15:24.030 align:middle line:90% 01:15:24.030 --> 01:15:27.030 align:middle line:90% We can just do this example. 01:15:27.030 --> 01:15:28.730 align:middle line:84% So this example, let's see what it 01:15:28.730 --> 01:15:30.140 align:middle line:90% looks like in momentum space. 01:15:30.140 --> 01:15:33.380 align:middle line:90% 01:15:33.380 --> 01:15:40.610 align:middle line:90% So in momentum space, we just-- 01:15:40.610 --> 01:15:43.170 align:middle line:90% 01:15:43.170 --> 01:15:46.541 align:middle line:90% but I erase the diagram. 01:15:46.541 --> 01:15:55.390 align:middle line:90% 01:15:55.390 --> 01:16:03.780 align:middle line:84% So this diagram looks like this in coordinate space of x1, x2 01:16:03.780 --> 01:16:06.720 align:middle line:90% and x and y. 01:16:06.720 --> 01:16:14.040 align:middle line:84% Now when we go to momentum space, the diagram is the same. 01:16:14.040 --> 01:16:16.620 align:middle line:90% Just now, we label the momentum. 01:16:16.620 --> 01:16:20.710 align:middle line:84% So we have an external line here, a terminal point here. 01:16:20.710 --> 01:16:22.710 align:middle line:90% So here, I have a momentum, p1. 01:16:22.710 --> 01:16:25.680 align:middle line:90% Here, I have a momentum, p2. 01:16:25.680 --> 01:16:30.330 align:middle line:84% So now let's just call, say, this k1, k2. 01:16:30.330 --> 01:16:32.950 align:middle line:84% You can assign momentum direction as you want. 01:16:32.950 --> 01:16:33.450 align:middle line:90% Say k3. 01:16:33.450 --> 01:16:36.210 align:middle line:90% 01:16:36.210 --> 01:16:40.950 align:middle line:84% So now we need to impose a momentum conservation. 01:16:40.950 --> 01:16:42.390 align:middle line:90% So momentum flow like that. 01:16:42.390 --> 01:16:44.420 align:middle line:84% So that means that the p1 should be 01:16:44.420 --> 01:16:47.450 align:middle line:90% equal to k1 plus k2 plus k3-- 01:16:47.450 --> 01:16:51.890 align:middle line:90% 01:16:51.890 --> 01:16:55.850 align:middle line:84% from the moment conservation at this vertex And from momentum 01:16:55.850 --> 01:16:58.400 align:middle line:84% conservation at this vertex, they all 01:16:58.400 --> 01:17:00.140 align:middle line:90% go away from this vertex. 01:17:00.140 --> 01:17:02.690 align:middle line:84% And then in this case, then the p2 01:17:02.690 --> 01:17:08.930 align:middle line:84% will be minus k1 plus k2 plus k3. 01:17:08.930 --> 01:17:12.860 align:middle line:84% So that means, actually, p2 should be equal to minus p1. 01:17:12.860 --> 01:17:16.820 align:middle line:84% So this makes perfect sense because here preserves 01:17:16.820 --> 01:17:18.650 align:middle line:90% momentum, preserves momentum. 01:17:18.650 --> 01:17:21.380 align:middle line:84% So for momentum conservation, if you have p1 here, 01:17:21.380 --> 01:17:23.540 align:middle line:90% here, you should also have p1. 01:17:23.540 --> 01:17:25.710 align:middle line:84% So p2 I drew with a negative sign. 01:17:25.710 --> 01:17:27.035 align:middle line:90% And so we have negative p1. 01:17:27.035 --> 01:17:31.250 align:middle line:90% 01:17:31.250 --> 01:17:35.780 align:middle line:84% Yeah, so using this simplified notation, 01:17:35.780 --> 01:17:38.555 align:middle line:84% then we can draw a diagram like this. 01:17:38.555 --> 01:17:46.170 align:middle line:90% 01:17:46.170 --> 01:17:49.110 align:middle line:90% So here is p. 01:17:49.110 --> 01:17:53.070 align:middle line:90% So let's call this k1, k2. 01:17:53.070 --> 01:17:56.730 align:middle line:84% And then this k3 can be solved in terms of the k1, k2. 01:17:56.730 --> 01:18:02.700 align:middle line:84% So this is just the p minus k1, k2. 01:18:02.700 --> 01:18:05.070 align:middle line:90% So this is the p minus k1, k2. 01:18:05.070 --> 01:18:07.980 align:middle line:84% So now we can just write down the momentum space expression. 01:18:07.980 --> 01:18:12.320 align:middle line:90% 01:18:12.320 --> 01:18:14.350 align:middle line:90% So we have minus i lambda. 01:18:14.350 --> 01:18:15.610 align:middle line:90% We have two vertex. 01:18:15.610 --> 01:18:18.100 align:middle line:90% We have 2 minus lambda squared. 01:18:18.100 --> 01:18:21.640 align:middle line:84% Then we have two undetermined momentum, k1, k2. 01:18:21.640 --> 01:18:24.910 align:middle line:84% Because this p should be-- external momentum 01:18:24.910 --> 01:18:26.980 align:middle line:90% should be fixed. 01:18:26.980 --> 01:18:33.220 align:middle line:90% So we have k1 and k2. 01:18:33.220 --> 01:18:37.200 align:middle line:90% 01:18:37.200 --> 01:18:40.290 align:middle line:84% And then we just write down each propagator. 01:18:40.290 --> 01:18:42.030 align:middle line:84% So we just write down each propagator. 01:18:42.030 --> 01:18:47.760 align:middle line:84% So we just have minus i k1 squared plus m 01:18:47.760 --> 01:18:50.520 align:middle line:90% squared minus i epsilon-- 01:18:50.520 --> 01:18:51.870 align:middle line:90% corresponding to this one. 01:18:51.870 --> 01:18:56.610 align:middle line:84% And then I have minus i k1, k2 squared 01:18:56.610 --> 01:19:12.130 align:middle line:84% plus m squared minus epsilon times minus i p 01:19:12.130 --> 01:19:17.905 align:middle line:84% minus k1 k2 squared plus m squared minus epsilon. 01:19:17.905 --> 01:19:21.670 align:middle line:90% 01:19:21.670 --> 01:19:25.390 align:middle line:90% And then these depend on the k. 01:19:25.390 --> 01:19:30.850 align:middle line:84% And then we have two factors, so this times these two 01:19:30.850 --> 01:19:33.260 align:middle line:90% external factors. 01:19:33.260 --> 01:19:34.540 align:middle line:90% So this is minus-- 01:19:34.540 --> 01:19:36.260 align:middle line:90% they have the same momentum. 01:19:36.260 --> 01:19:43.150 align:middle line:84% So we just have p squared plus m squared minus epsilon squared. 01:19:43.150 --> 01:19:46.120 align:middle line:84% And then we divide by 3 factorial corresponding 01:19:46.120 --> 01:19:48.850 align:middle line:90% to the symmetric factors. 01:19:48.850 --> 01:19:49.960 align:middle line:90% OK, that's it. 01:19:49.960 --> 01:19:54.910 align:middle line:84% So this is the expression for this diagram in momentum space. 01:19:54.910 --> 01:19:58.250 align:middle line:90% 01:19:58.250 --> 01:20:01.570 align:middle line:90% OK, so let's stop here. 01:20:01.570 --> 01:20:06.000 align:middle line:90%