WEBVTT 00:00:00.000 --> 00:00:01.940 align:middle line:90% [SQUEAKING] 00:00:01.940 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:05.335 align:middle line:90% [CLICKING] 00:00:05.335 --> 00:00:10.500 align:middle line:90% 00:00:10.500 --> 00:00:14.820 align:middle line:84% PROFESSOR: So first, I'll try to remind you 00:00:14.820 --> 00:00:18.040 align:middle line:84% what we did at the end of last lecture 00:00:18.040 --> 00:00:20.280 align:middle line:90% which was last Wednesday. 00:00:20.280 --> 00:00:23.370 align:middle line:84% And so we introduced these things 00:00:23.370 --> 00:00:28.170 align:middle line:84% called the Grassmann variables which are anticommuting. 00:00:28.170 --> 00:00:32.400 align:middle line:84% Classically, they anticommute-- so for example, if you have, 00:00:32.400 --> 00:00:35.940 align:middle line:84% say, a quantity called theta, and then theta 0-- 00:00:35.940 --> 00:00:38.400 align:middle line:90% theta squared equal to 0. 00:00:38.400 --> 00:00:42.450 align:middle line:84% And then two such kind of Grassmann variables 00:00:42.450 --> 00:00:44.655 align:middle line:84% commute with-- anticommute with each other. 00:00:44.655 --> 00:00:48.660 align:middle line:90% 00:00:48.660 --> 00:00:52.800 align:middle line:84% And then if you consider a function of such theta, 00:00:52.800 --> 00:00:55.140 align:middle line:84% then you just do a Taylor expansion, 00:00:55.140 --> 00:00:56.620 align:middle line:90% and you only have two terms. 00:00:56.620 --> 00:00:59.505 align:middle line:84% So the first term is a constant, and the second term 00:00:59.505 --> 00:01:02.550 align:middle line:84% is proportional to theta, because all higher power 00:01:02.550 --> 00:01:04.950 align:middle line:90% terms vanish. 00:01:04.950 --> 00:01:07.350 align:middle line:84% So the same thing with a multiple variable-- 00:01:07.350 --> 00:01:13.770 align:middle line:84% just expand until you reach the square of any variable. 00:01:13.770 --> 00:01:17.470 align:middle line:84% And then you can do a differentiation. 00:01:17.470 --> 00:01:22.120 align:middle line:84% So we always define the differentiation from the left. 00:01:22.120 --> 00:01:27.070 align:middle line:90% So this is given by f1. 00:01:27.070 --> 00:01:30.790 align:middle line:84% And we can also define the integration, 00:01:30.790 --> 00:01:33.820 align:middle line:84% and the integration is determined by two rules. 00:01:33.820 --> 00:01:45.140 align:middle line:84% The first rule is d theta, a constant equal to 0, 00:01:45.140 --> 00:01:48.670 align:middle line:84% and, also, d theta theta equal to 1. 00:01:48.670 --> 00:01:50.550 align:middle line:84% OK, so based on these two rules-- 00:01:50.550 --> 00:01:54.770 align:middle line:84% and then you can just work out the integral of any functions. 00:01:54.770 --> 00:01:58.960 align:middle line:90% 00:01:58.960 --> 00:02:05.400 align:middle line:84% So before I proceed, do you have any questions on this? 00:02:05.400 --> 00:02:10.530 align:middle line:90% 00:02:10.530 --> 00:02:11.800 align:middle line:90% Good. 00:02:11.800 --> 00:02:18.380 align:middle line:84% So for multivariable function, you just expand. 00:02:18.380 --> 00:02:23.670 align:middle line:84% Say, for example, if you have two variables, 00:02:23.670 --> 00:02:32.370 align:middle line:84% then you just have f0 plus f1 theta 1 plus f2 theta 2 plus 00:02:32.370 --> 00:02:37.170 align:middle line:90% f1,2 theta 1 theta 2. 00:02:37.170 --> 00:02:40.660 align:middle line:84% So we can also define the integra-- 00:02:40.660 --> 00:02:43.830 align:middle line:84% yeah, the differentiation of a multivariable 00:02:43.830 --> 00:02:45.040 align:middle line:90% function-- more easy. 00:02:45.040 --> 00:02:48.910 align:middle line:84% You just be a little bit careful of the direction 00:02:48.910 --> 00:02:50.920 align:middle line:90% you take the derivative. 00:02:50.920 --> 00:02:53.520 align:middle line:84% So now, let's look at the example of the integration. 00:02:53.520 --> 00:03:01.200 align:middle line:90% 00:03:01.200 --> 00:03:09.100 align:middle line:84% OK, let's look at the integral of a function of two variables. 00:03:09.100 --> 00:03:13.200 align:middle line:84% So you should keep in mind that this order is important 00:03:13.200 --> 00:03:15.510 align:middle line:84% because theta 1 and theta 2 are anticommute. 00:03:15.510 --> 00:03:19.950 align:middle line:84% So if we write d theta 1 and d theta 2, then it is d 00:03:19.950 --> 00:03:21.570 align:middle line:90% theta 1 d theta 2. 00:03:21.570 --> 00:03:24.300 align:middle line:84% It's equal to minus d theta 2 d theta 1. 00:03:24.300 --> 00:03:26.380 align:middle line:90% You want to change the order. 00:03:26.380 --> 00:03:30.150 align:middle line:84% So we can just do the integration by using this rule. 00:03:30.150 --> 00:03:41.620 align:middle line:84% You just plug in this expansion into here, 00:03:41.620 --> 00:03:44.680 align:middle line:84% and then, obviously, all these three 00:03:44.680 --> 00:03:48.345 align:middle line:90% will give you 0 because of the-- 00:03:48.345 --> 00:03:52.940 align:middle line:90% 00:03:52.940 --> 00:03:57.420 align:middle line:84% and only the last term will contribute. 00:03:57.420 --> 00:04:04.970 align:middle line:84% And then you just get f1,2, then d theta 1 d theta 2, 00:04:04.970 --> 00:04:07.070 align:middle line:90% then theta 1 times theta 2. 00:04:07.070 --> 00:04:09.700 align:middle line:90% 00:04:09.700 --> 00:04:14.130 align:middle line:84% So now if you want to do the integration, because theta 2 00:04:14.130 --> 00:04:16.649 align:middle line:90% is before the theta here. 00:04:16.649 --> 00:04:19.440 align:middle line:84% Theta 2 is closer, and the theta 2 before theta 1-- 00:04:19.440 --> 00:04:21.910 align:middle line:84% and then you need to change the order. 00:04:21.910 --> 00:04:26.730 align:middle line:84% So we can do it by doing f1,2 d theta 1, 00:04:26.730 --> 00:04:30.810 align:middle line:84% and then you have d theta 2 integral, theta 2, 00:04:30.810 --> 00:04:32.730 align:middle line:90% and then theta 1. 00:04:32.730 --> 00:04:35.910 align:middle line:84% Now we have changed the order of theta 1 theta 2, 00:04:35.910 --> 00:04:39.220 align:middle line:90% and this gives me 1. 00:04:39.220 --> 00:04:40.560 align:middle line:90% So after this, it gives me 1. 00:04:40.560 --> 00:04:41.520 align:middle line:90% This also gives me 1. 00:04:41.520 --> 00:04:45.630 align:middle line:84% So this is just equal to minus f1,2. 00:04:45.630 --> 00:04:51.450 align:middle line:84% So similarly, you can just do an arbitrary number of integrals 00:04:51.450 --> 00:04:52.680 align:middle line:90% with-- 00:04:52.680 --> 00:04:55.110 align:middle line:84% you can just do integrals with an arbitrary 00:04:55.110 --> 00:04:56.400 align:middle line:90% number of variables. 00:04:56.400 --> 00:05:04.370 align:middle line:84% Just keep in mind that d theta 1 d theta 2 theta 2 theta 1-- 00:05:04.370 --> 00:05:06.200 align:middle line:90% this is equal to 1. 00:05:06.200 --> 00:05:09.440 align:middle line:84% So you do this first, and then you do that, 00:05:09.440 --> 00:05:12.960 align:middle line:84% and then you get the right order. 00:05:12.960 --> 00:05:14.010 align:middle line:90% Any questions on this? 00:05:14.010 --> 00:05:14.610 align:middle line:90% Yes. 00:05:14.610 --> 00:05:18.120 align:middle line:84% AUDIENCE: Here, f0 and f1, are those complex numbers, 00:05:18.120 --> 00:05:20.130 align:middle line:90% or Grassmann numbers, or both? 00:05:20.130 --> 00:05:22.950 align:middle line:84% PROFESSOR: Here, we just take them to be ordinary numbers. 00:05:22.950 --> 00:05:24.450 align:middle line:84% So they can just be complex numbers. 00:05:24.450 --> 00:05:25.457 align:middle line:90% That's right, yeah. 00:05:25.457 --> 00:05:28.440 align:middle line:90% 00:05:28.440 --> 00:05:29.310 align:middle line:90% Other questions? 00:05:29.310 --> 00:05:33.760 align:middle line:90% 00:05:33.760 --> 00:05:36.040 align:middle line:90% OK, good. 00:05:36.040 --> 00:05:40.210 align:middle line:84% So now we can look at a little bit more complicated integral. 00:05:40.210 --> 00:05:42.090 align:middle line:84% So now let's look at the Gaussian. 00:05:42.090 --> 00:05:51.790 align:middle line:90% 00:05:51.790 --> 00:05:53.830 align:middle line:84% So let's look at such a Gaussian integral. 00:05:53.830 --> 00:05:56.920 align:middle line:90% 00:05:56.920 --> 00:05:58.950 align:middle line:84% So this Gaussian integral looks complicated, 00:05:58.950 --> 00:06:00.210 align:middle line:90% but, of course, it's simple. 00:06:00.210 --> 00:06:02.110 align:middle line:84% Because again, this is when you expand it. 00:06:02.110 --> 00:06:04.020 align:middle line:90% There's only a single term left. 00:06:04.020 --> 00:06:09.060 align:middle line:84% So this is just equal to d theta 1 d theta 2, 00:06:09.060 --> 00:06:14.120 align:middle line:90% and then 1 minus this guy. 00:06:14.120 --> 00:06:18.850 align:middle line:90% Yeah, so theta 1 a1,2 theta 2. 00:06:18.850 --> 00:06:21.730 align:middle line:84% Because the next term, when you do the Taylor expansion, 00:06:21.730 --> 00:06:22.780 align:middle line:90% will give you 0. 00:06:22.780 --> 00:06:23.770 align:middle line:90% A little bit. 00:06:23.770 --> 00:06:28.630 align:middle line:84% So that will involve the theta 1 square or theta 2 square. 00:06:28.630 --> 00:06:39.110 align:middle line:84% And now if you use this rule, you just get a1,2. 00:06:39.110 --> 00:06:45.590 align:middle line:84% And so you can now generalize two Gaussian integrals 00:06:45.590 --> 00:06:48.980 align:middle line:84% with an arbitrary number of variables. 00:06:48.980 --> 00:06:57.350 align:middle line:84% And so we can write a more general integral, d theta 1 d 00:06:57.350 --> 00:07:03.390 align:middle line:90% theta 2 n, because they-- 00:07:03.390 --> 00:07:09.900 align:middle line:84% so minus 1/2 theta i a ij theta j. 00:07:09.900 --> 00:07:13.970 align:middle line:84% So now here, you should assume i and j are summed. 00:07:13.970 --> 00:07:18.160 align:middle line:90% OK, so ij is from 1 to 2n. 00:07:18.160 --> 00:07:25.490 align:middle line:90% 00:07:25.490 --> 00:07:29.340 align:middle line:84% So for this integral, again, the strategy is simple. 00:07:29.340 --> 00:07:36.080 align:middle line:84% You just, again, expand it and then just do term by term. 00:07:36.080 --> 00:07:39.920 align:middle line:84% So at a certain time when you expand to a certain order, 00:07:39.920 --> 00:07:44.790 align:middle line:84% the theta will repeat and then the expansion truncates. 00:07:44.790 --> 00:07:47.340 align:middle line:84% But you can actually easily convince yourself 00:07:47.340 --> 00:07:53.720 align:middle line:84% without doing any calculation to see what the answer is. 00:07:53.720 --> 00:07:59.050 align:middle line:84% So if you don't expand to a sufficient number of theta, 00:07:59.050 --> 00:08:05.290 align:middle line:84% you will get 0 because you have to have each theta for d theta 00:08:05.290 --> 00:08:07.480 align:middle line:90% in order for this rule. 00:08:07.480 --> 00:08:11.290 align:middle line:84% So you have to have one theta for each d theta. 00:08:11.290 --> 00:08:15.160 align:middle line:84% So that means you have to expand to the order which each theta 00:08:15.160 --> 00:08:18.070 align:middle line:90% just appears exactly once. 00:08:18.070 --> 00:08:19.623 align:middle line:90% And then if you do that-- 00:08:19.623 --> 00:08:21.040 align:middle line:84% if you look at that term, you find 00:08:21.040 --> 00:08:27.920 align:middle line:84% that this just gives you square root determinant aij 00:08:27.920 --> 00:08:30.925 align:middle line:90% determinant a. 00:08:30.925 --> 00:08:32.580 align:middle line:84% You find this actually gives you-- 00:08:32.580 --> 00:08:41.770 align:middle line:90% 00:08:41.770 --> 00:08:44.003 align:middle line:84% here, a is an antisymmetric matrix-- 00:08:44.003 --> 00:08:45.670 align:middle line:84% by definition, it should be anti-- yeah, 00:08:45.670 --> 00:08:47.590 align:middle line:84% because theta i and theta j are anticommute. 00:08:47.590 --> 00:08:49.975 align:middle line:84% So a ij should be an antisymmetric matrix. 00:08:49.975 --> 00:09:08.090 align:middle line:90% 00:09:08.090 --> 00:09:09.365 align:middle line:90% OK, any questions on this? 00:09:09.365 --> 00:09:13.320 align:middle line:90% 00:09:13.320 --> 00:09:15.410 align:middle line:84% So this looks like standard Gaussian integral. 00:09:15.410 --> 00:09:17.840 align:middle line:84% Remember the standard Gaussian integral. 00:09:17.840 --> 00:09:20.420 align:middle line:90% What would you get? 00:09:20.420 --> 00:09:23.480 align:middle line:84% If you have such kind of quadratic structure 00:09:23.480 --> 00:09:27.170 align:middle line:84% for the standard Gaussian integral, for ordinary numbers, 00:09:27.170 --> 00:09:30.000 align:middle line:90% do you remember what you get? 00:09:30.000 --> 00:09:30.580 align:middle line:90% Yes. 00:09:30.580 --> 00:09:34.760 align:middle line:84% AUDIENCE: Yeah, it's like 2 pi to some power over [INAUDIBLE].. 00:09:34.760 --> 00:09:36.440 align:middle line:90% PROFESSOR: Exactly. 00:09:36.440 --> 00:09:36.980 align:middle line:90% Right. 00:09:36.980 --> 00:09:42.050 align:middle line:84% So if theta are ordinary variables, 00:09:42.050 --> 00:09:44.660 align:middle line:84% then you will get some constant divided 00:09:44.660 --> 00:09:46.940 align:middle line:90% by square root of determinant a. 00:09:46.940 --> 00:09:49.940 align:middle line:84% But for the Grassmann number, you just solve it, 00:09:49.940 --> 00:09:53.240 align:middle line:84% and you get the proportional to the determinant-- not divided 00:09:53.240 --> 00:09:54.704 align:middle line:90% by the determinant. 00:09:54.704 --> 00:09:58.970 align:middle line:90% 00:09:58.970 --> 00:10:03.320 align:middle line:84% Good, so we can also introduce complex Grassmann variables. 00:10:03.320 --> 00:10:14.520 align:middle line:90% 00:10:14.520 --> 00:10:19.370 align:middle line:84% So, for example, I can introduce theta over the square root 00:10:19.370 --> 00:10:23.270 align:middle line:90% 2, theta 1 plus i theta 2. 00:10:23.270 --> 00:10:25.160 align:middle line:84% So theta 1 and theta 2 are considered 00:10:25.160 --> 00:10:28.170 align:middle line:84% to be 2 real Grassmann variables. 00:10:28.170 --> 00:10:33.280 align:middle line:84% i is just ordinary i, and then the theta star 00:10:33.280 --> 00:10:38.620 align:middle line:84% would be, of course, the theta square root 2 theta 1 minus i 00:10:38.620 --> 00:10:41.410 align:middle line:90% theta 2. 00:10:41.410 --> 00:10:44.230 align:middle line:84% And so the rule for defining the product-- 00:10:44.230 --> 00:10:48.040 align:middle line:90% 00:10:48.040 --> 00:10:49.870 align:middle line:84% the complex conjugation for the product-- 00:10:49.870 --> 00:10:54.850 align:middle line:84% is defined to be eta star theta star. 00:10:54.850 --> 00:10:57.620 align:middle line:84% So this is different from the ordinary variables. 00:10:57.620 --> 00:11:00.160 align:middle line:84% So this is defined more like for an operator. 00:11:00.160 --> 00:11:02.380 align:middle line:84% So for Grassmann variables, when we 00:11:02.380 --> 00:11:07.120 align:middle line:84% define the complex conjugate, we actually change the order-- 00:11:07.120 --> 00:11:12.790 align:middle line:84% as well, what we normally do for a Hermitian conjugate. 00:11:12.790 --> 00:11:16.900 align:middle line:84% So now, if you have a function, say, theta theta 00:11:16.900 --> 00:11:21.710 align:middle line:84% star, with theta now a complex of Grassmann variables, 00:11:21.710 --> 00:11:23.680 align:middle line:90% you just do the same thing. 00:11:23.680 --> 00:11:33.320 align:middle line:84% So you have c0 plus c1 theta plus c1 bar theta star, 00:11:33.320 --> 00:11:38.330 align:middle line:90% then plus c1,1 theta theta star. 00:11:38.330 --> 00:11:42.380 align:middle line:84% So again, you're just expanding theta and theta star. 00:11:42.380 --> 00:11:46.580 align:middle line:84% And then you truncate here, because the further terms 00:11:46.580 --> 00:11:49.100 align:middle line:84% will involve either theta square or theta star square. 00:11:49.100 --> 00:11:54.290 align:middle line:90% 00:11:54.290 --> 00:12:00.590 align:middle line:84% So here, for the integration, we define d theta d 00:12:00.590 --> 00:12:05.150 align:middle line:84% theta star and then theta star theta to be 1. 00:12:05.150 --> 00:12:07.760 align:middle line:90% So we choose this convention. 00:12:07.760 --> 00:12:14.720 align:middle line:90% 00:12:14.720 --> 00:12:16.760 align:middle line:90% So we define this to be what? 00:12:16.760 --> 00:12:20.150 align:middle line:84% So that's corresponding to a specific choice of a measure 00:12:20.150 --> 00:12:21.620 align:middle line:90% for this d theta d theta star. 00:12:21.620 --> 00:12:24.710 align:middle line:90% 00:12:24.710 --> 00:12:25.210 align:middle line:90% Good? 00:12:25.210 --> 00:12:27.745 align:middle line:84% So now we can look at the complex Gaussian integral. 00:12:27.745 --> 00:12:32.330 align:middle line:90% 00:12:32.330 --> 00:12:35.840 align:middle line:90% OK, now let's look at Gaussian. 00:12:35.840 --> 00:12:43.410 align:middle line:84% So if you have d theta star d theta, 00:12:43.410 --> 00:12:48.970 align:middle line:84% say, exponential minus theta star b theta-- 00:12:48.970 --> 00:12:52.270 align:middle line:84% so b is just some number-- some arbitrary number. 00:12:52.270 --> 00:12:56.890 align:middle line:84% And again, you can just do it by expanding this in power series. 00:12:56.890 --> 00:12:59.920 align:middle line:84% And just as in that example, you just 00:12:59.920 --> 00:13:01.346 align:middle line:90% find that this is equal to b. 00:13:01.346 --> 00:13:06.580 align:middle line:90% 00:13:06.580 --> 00:13:11.890 align:middle line:84% And you can also now do multiple variables-- 00:13:11.890 --> 00:13:15.000 align:middle line:90% so multiple variable Gaussian. 00:13:15.000 --> 00:13:25.640 align:middle line:84% So suppose you have now j equal to 1 to n d theta j star 00:13:25.640 --> 00:13:28.280 align:middle line:90% d theta j-- 00:13:28.280 --> 00:13:30.090 align:middle line:90% the product of all of them. 00:13:30.090 --> 00:13:40.370 align:middle line:84% So if you have minus theta i star A ij theta j-- 00:13:40.370 --> 00:13:45.110 align:middle line:84% again, i and the j should be assumed to be summed. 00:13:45.110 --> 00:13:49.930 align:middle line:84% And then when you expand it, again, you 00:13:49.930 --> 00:13:53.470 align:middle line:90% expand precisely to the order-- 00:13:53.470 --> 00:13:58.030 align:middle line:84% only one term in the expansion contribute. 00:13:58.030 --> 00:14:01.240 align:middle line:84% It's the expansion in which each theta and theta 00:14:01.240 --> 00:14:03.670 align:middle line:90% star appear exactly once. 00:14:03.670 --> 00:14:06.520 align:middle line:84% So when you look at that term, you 00:14:06.520 --> 00:14:12.320 align:middle line:84% find that this precisely gives you det A. 00:14:12.320 --> 00:14:14.840 align:middle line:84% So again, this, should be contrasted 00:14:14.840 --> 00:14:17.630 align:middle line:90% with the complex integral-- 00:14:17.630 --> 00:14:19.850 align:middle line:84% ordinary complex Gaussian integral. 00:14:19.850 --> 00:14:23.180 align:middle line:84% So if you have an ordinary Gaussian integral 00:14:23.180 --> 00:14:25.310 align:middle line:84% for complex variables, again, you 00:14:25.310 --> 00:14:27.650 align:middle line:84% will get 1 over determinant A. It's 00:14:27.650 --> 00:14:29.420 align:middle line:84% some constant over determinant A. 00:14:29.420 --> 00:14:34.820 align:middle line:84% So for Grassmann, you just get determinant A. 00:14:34.820 --> 00:14:41.210 align:middle line:84% And this feature is very key in distinguishing fermions 00:14:41.210 --> 00:14:44.180 align:middle line:84% and the bosons, and plays a very important role. 00:14:44.180 --> 00:14:47.590 align:middle line:90% 00:14:47.590 --> 00:14:53.680 align:middle line:84% So we can also consider more general integrals, 00:14:53.680 --> 00:14:55.570 align:middle line:84% more general Gaussian integrals like this. 00:14:55.570 --> 00:14:58.610 align:middle line:84% So now, let me just introduce a little bit of notation. 00:14:58.610 --> 00:15:07.760 align:middle line:84% So let me call the theta equal to theta 1 theta, say, n. 00:15:07.760 --> 00:15:10.010 align:middle line:90% Let me introduce eta. 00:15:10.010 --> 00:15:20.130 align:middle line:84% Eta is some other Grassmann variables, eta 1 eta n. 00:15:20.130 --> 00:15:25.830 align:middle line:84% And then we can consider an integral 00:15:25.830 --> 00:15:31.640 align:middle line:84% like this which we will encounter. 00:15:31.640 --> 00:15:34.180 align:middle line:84% Later, we will encounter integrals like this. 00:15:34.180 --> 00:15:39.230 align:middle line:84% Again, d theta star j d theta j-- 00:15:39.230 --> 00:15:45.130 align:middle line:84% then you have exponential, so minus theta dagger. 00:15:45.130 --> 00:15:48.100 align:middle line:84% So now, the dagger is including both the transpose 00:15:48.100 --> 00:15:50.560 align:middle line:90% and taking the star. 00:15:50.560 --> 00:15:53.260 align:middle line:90% And A, now, is just the matrix. 00:15:53.260 --> 00:15:57.980 align:middle line:84% Again, just write this in the matrix notation. 00:15:57.980 --> 00:16:00.530 align:middle line:84% But now, suppose that here I only have a quadratic term. 00:16:00.530 --> 00:16:03.270 align:middle line:84% But suppose now I have some linear term-- 00:16:03.270 --> 00:16:09.890 align:middle line:84% say, eta dagger theta and theta dagger eta. 00:16:09.890 --> 00:16:11.180 align:middle line:90% OK, suppose now I have this. 00:16:11.180 --> 00:16:14.470 align:middle line:90% 00:16:14.470 --> 00:16:18.130 align:middle line:84% And again, you can just complete the square 00:16:18.130 --> 00:16:21.700 align:middle line:84% through the Grassmann version of the complete-- the square. 00:16:21.700 --> 00:16:25.385 align:middle line:84% And then you can convince yourself 00:16:25.385 --> 00:16:27.550 align:middle line:90% that this is given by det A-- 00:16:27.550 --> 00:16:30.660 align:middle line:90% 00:16:30.660 --> 00:16:33.735 align:middle line:90% determinant of A-- matrix A-- 00:16:33.735 --> 00:16:45.440 align:middle line:84% and then exponential eta dagger A minus 1 eta. 00:16:45.440 --> 00:16:51.160 align:middle line:84% So up to the sign and i, et cetera-- 00:16:51.160 --> 00:16:54.130 align:middle line:84% this is exactly what you'd normally expect 00:16:54.130 --> 00:16:55.400 align:middle line:90% after you complete the square. 00:16:55.400 --> 00:16:58.510 align:middle line:84% You get A minus 1 here, and then you get this eta. 00:16:58.510 --> 00:17:01.360 align:middle line:90% 00:17:01.360 --> 00:17:02.770 align:middle line:90% Any questions on this? 00:17:02.770 --> 00:17:03.998 align:middle line:90% Yes. 00:17:03.998 --> 00:17:06.569 align:middle line:84% AUDIENCE: Is A still antisymmetric? 00:17:06.569 --> 00:17:08.569 align:middle line:84% PROFESSOR: It does not have to be antisymmetric. 00:17:08.569 --> 00:17:09.069 align:middle line:90% Yeah. 00:17:09.069 --> 00:17:14.220 align:middle line:90% 00:17:14.220 --> 00:17:16.965 align:middle line:84% So it can be some complex matrix, in principle. 00:17:16.965 --> 00:17:20.930 align:middle line:84% Yeah-- some general complex matrix. 00:17:20.930 --> 00:17:29.290 align:middle line:84% So I will also denote this by I eta and eta star. 00:17:29.290 --> 00:17:40.000 align:middle line:90% 00:17:40.000 --> 00:17:44.350 align:middle line:84% OK, so we only need the one last formula, 00:17:44.350 --> 00:17:47.380 align:middle line:84% and then we can talk about the path 00:17:47.380 --> 00:17:51.820 align:middle line:90% integral for the Dirac fields. 00:17:51.820 --> 00:17:55.460 align:middle line:84% So one last formula is that we, say-- 00:17:55.460 --> 00:18:03.390 align:middle line:84% let's try to calculate the following integral-- 00:18:03.390 --> 00:18:07.128 align:middle line:90% 1 over I0,0. 00:18:07.128 --> 00:18:09.480 align:middle line:84% OK, I0,0 means just this integral. 00:18:09.480 --> 00:18:14.160 align:middle line:84% Just this integral is I0,0 with both eta equal to zero. 00:18:14.160 --> 00:18:16.400 align:middle line:90% So 1 over I0,0-- 00:18:16.400 --> 00:18:22.524 align:middle line:84% suppose let's consider this integral, j from 1 to n 00:18:22.524 --> 00:18:27.440 align:middle line:90% d theta j star d theta j. 00:18:27.440 --> 00:18:31.990 align:middle line:84% Now suppose we have theta k star-- 00:18:31.990 --> 00:18:36.580 align:middle line:84% theta k-- and theta l star in downstairs, 00:18:36.580 --> 00:18:38.830 align:middle line:84% and then you have the exponential minus 00:18:38.830 --> 00:18:41.705 align:middle line:90% theta dagger A theta. 00:18:41.705 --> 00:18:43.330 align:middle line:84% Suppose you have an integral like this. 00:18:43.330 --> 00:18:45.798 align:middle line:90% 00:18:45.798 --> 00:18:48.090 align:middle line:84% So the integral like this can be obtained from this one 00:18:48.090 --> 00:18:49.330 align:middle line:90% by taking derivatives. 00:18:49.330 --> 00:18:51.640 align:middle line:84% So this is like a generating functional. 00:18:51.640 --> 00:18:55.370 align:middle line:84% So then this can be considered as 1 over-- 00:18:55.370 --> 00:18:57.360 align:middle line:84% yeah, let me just suppress these two zeros-- 00:18:57.360 --> 00:19:08.740 align:middle line:84% 1 over I. This can have minus partial partial eta star k, 00:19:08.740 --> 00:19:11.610 align:middle line:90% and then yeah-- 00:19:11.610 --> 00:19:16.630 align:middle line:90% partial partial l. 00:19:16.630 --> 00:19:22.150 align:middle line:84% OK, then take derivative on I eta and eta star-- 00:19:22.150 --> 00:19:25.102 align:middle line:84% taking derivative on this, and then 00:19:25.102 --> 00:19:27.310 align:middle line:84% after you take the derivative on each side of the eta 00:19:27.310 --> 00:19:28.810 align:middle line:90% equal to eta star equal to 0. 00:19:28.810 --> 00:19:32.190 align:middle line:90% 00:19:32.190 --> 00:19:35.160 align:middle line:84% So when you take these two derivatives, 00:19:35.160 --> 00:19:39.600 align:middle line:84% you can see that we bring down the theta k and theta l. 00:19:39.600 --> 00:19:45.030 align:middle line:84% And this sign-- just make sure you get the sign correctly. 00:19:45.030 --> 00:19:49.830 align:middle line:84% And then when you do this derivative here-- 00:19:49.830 --> 00:19:53.088 align:middle line:84% and then you find that this is given by A minus 1 kl. 00:19:53.088 --> 00:19:58.630 align:middle line:90% 00:19:58.630 --> 00:20:02.062 align:middle line:84% So this is actually the same as the bosonic case. 00:20:02.062 --> 00:20:04.270 align:middle line:84% Remember, in the bosonic case, if you have a Gaussian 00:20:04.270 --> 00:20:06.760 align:middle line:84% integral, and if you have the two 00:20:06.760 --> 00:20:08.860 align:middle line:84% variables in the downstairs, then 00:20:08.860 --> 00:20:13.430 align:middle line:84% that gives you the inverse of the matrix up to a constant. 00:20:13.430 --> 00:20:19.030 align:middle line:84% Yeah, so this aspect is actually similar to the bosonic case. 00:20:19.030 --> 00:20:22.110 align:middle line:90% So do you have any questions? 00:20:22.110 --> 00:20:23.010 align:middle line:90% Yes. 00:20:23.010 --> 00:20:25.635 align:middle line:84% AUDIENCE: Sorry if I missed it, but is there a name for that i? 00:20:25.635 --> 00:20:30.142 align:middle line:90% 00:20:30.142 --> 00:20:31.350 align:middle line:90% PROFESSOR: Just the integral. 00:20:31.350 --> 00:20:31.760 align:middle line:90% Yeah. 00:20:31.760 --> 00:20:32.880 align:middle line:84% AUDIENCE: OK, it's just that integral? 00:20:32.880 --> 00:20:33.650 align:middle line:90% PROFESSOR: Yeah. 00:20:33.650 --> 00:20:37.210 align:middle line:90% Yeah, just a shorthand notation. 00:20:37.210 --> 00:20:40.610 align:middle line:84% Yeah, you can call it the generating function. 00:20:40.610 --> 00:20:42.880 align:middle line:84% Yeah, so this is a generating function 00:20:42.880 --> 00:20:47.170 align:middle line:84% for arbitrary powers of this kind of integral, 00:20:47.170 --> 00:20:49.750 align:middle line:90% because any powers of theta-- 00:20:49.750 --> 00:20:51.700 align:middle line:84% any combination of theta k, et cetera, 00:20:51.700 --> 00:20:57.570 align:middle line:84% you can just obtain by taking derivative on this I. Yeah, 00:20:57.570 --> 00:21:00.070 align:middle line:90% mm-hmm. 00:21:00.070 --> 00:21:01.070 align:middle line:90% Yes. 00:21:01.070 --> 00:21:06.887 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:21:06.887 --> 00:21:07.970 align:middle line:90% PROFESSOR: Yeah, you can-- 00:21:07.970 --> 00:21:14.450 align:middle line:90% 00:21:14.450 --> 00:21:18.487 align:middle line:90% here, the complex conjugate-- 00:21:18.487 --> 00:21:20.570 align:middle line:84% in some sense, they are independent of each other. 00:21:20.570 --> 00:21:22.910 align:middle line:84% Yeah, they are independent variables. 00:21:22.910 --> 00:21:25.730 align:middle line:84% Theta and theta star, because they depend on two variables-- 00:21:25.730 --> 00:21:28.421 align:middle line:84% and you can just view them as independent variables. 00:21:28.421 --> 00:21:31.510 align:middle line:90% 00:21:31.510 --> 00:21:34.220 align:middle line:90% OK, good. 00:21:34.220 --> 00:21:39.190 align:middle line:84% So now, finally, we have the preparation 00:21:39.190 --> 00:21:41.365 align:middle line:84% to do the path integral for the Dirac fields. 00:21:41.365 --> 00:21:52.490 align:middle line:90% 00:21:52.490 --> 00:21:56.480 align:middle line:84% So remember, Dirac fields is a vector-- 00:21:56.480 --> 00:21:59.690 align:middle line:84% a spinor field with four components. 00:21:59.690 --> 00:22:04.760 align:middle line:90% 00:22:04.760 --> 00:22:06.500 align:middle line:84% There's four components, and they depend 00:22:06.500 --> 00:22:08.150 align:middle line:90% on space-time coordinates. 00:22:08.150 --> 00:22:11.550 align:middle line:90% 00:22:11.550 --> 00:22:19.620 align:middle line:84% So normally, we just say this is the ordinary functions. 00:22:19.620 --> 00:22:24.320 align:middle line:84% But now, in order to correctly capture 00:22:24.320 --> 00:22:27.290 align:middle line:84% the fermionic nature of the Dirac fields, 00:22:27.290 --> 00:22:33.050 align:middle line:90% I now require that psi alpha x. 00:22:33.050 --> 00:22:35.750 align:middle line:90% So alpha is equal to 1, 2, 3, 4. 00:22:35.750 --> 00:22:45.690 align:middle line:84% Take values as Grassmann numbers. 00:22:45.690 --> 00:22:52.900 align:middle line:90% 00:22:52.900 --> 00:22:54.710 align:middle line:90% So what this means-- 00:22:54.710 --> 00:22:58.540 align:middle line:84% so x is still our ordinary coordinate. 00:22:58.540 --> 00:23:02.640 align:middle line:84% For any choice of space-time location, x-- 00:23:02.640 --> 00:23:07.520 align:middle line:84% so this psi alpha gives give you a Grassmann number. 00:23:07.520 --> 00:23:11.630 align:middle line:84% And just for any choice of x, this just means that. 00:23:11.630 --> 00:23:15.380 align:middle line:84% And this is a function from an ordinary space-time variable 00:23:15.380 --> 00:23:17.480 align:middle line:84% to the space of the Grassmann numbers. 00:23:17.480 --> 00:23:20.930 align:middle line:90% 00:23:20.930 --> 00:23:28.880 align:middle line:84% And in particular, this means that psi alpha x psi beta y-- 00:23:28.880 --> 00:23:30.590 align:middle line:90% they anticommute. 00:23:30.590 --> 00:23:38.040 align:middle line:90% 00:23:38.040 --> 00:23:40.990 align:middle line:90% So this is the rule. 00:23:40.990 --> 00:23:44.170 align:middle line:84% So now, that's the only difference. 00:23:44.170 --> 00:23:48.160 align:middle line:84% And now we can just do the path integral. 00:23:48.160 --> 00:23:51.145 align:middle line:84% So the path integral is just completely parallel as before. 00:23:51.145 --> 00:23:54.620 align:middle line:90% 00:23:54.620 --> 00:24:10.870 align:middle line:84% So otherwise, exactly the same as before for what we 00:24:10.870 --> 00:24:13.690 align:middle line:90% did before for a scalar field. 00:24:13.690 --> 00:24:18.160 align:middle line:84% So the only difference is now the things are integrating. 00:24:18.160 --> 00:24:21.070 align:middle line:90% 00:24:21.070 --> 00:24:25.730 align:middle line:84% So in particular, we can write any correlation functions. 00:24:25.730 --> 00:24:28.550 align:middle line:90% 00:24:28.550 --> 00:24:34.870 align:middle line:84% So, for example, again, let's use 00:24:34.870 --> 00:24:38.620 align:middle line:84% omega to denote the vacuum state. 00:24:38.620 --> 00:24:42.170 align:middle line:84% And the time-ordered correlation function-- 00:24:42.170 --> 00:24:44.840 align:middle line:84% the x denotes some product of operators. 00:24:44.840 --> 00:24:47.080 align:middle line:90% Yeah, our previous notation-- 00:24:47.080 --> 00:24:52.400 align:middle line:84% and the omega-- and divide it by omega. 00:24:52.400 --> 00:24:57.520 align:middle line:84% So this has a path integral in the present description 00:24:57.520 --> 00:25:04.530 align:middle line:84% in terms of psi bar D psi, and then x in the integrand, 00:25:04.530 --> 00:25:15.780 align:middle line:84% and then exponential i S. And then so everything is exactly 00:25:15.780 --> 00:25:19.700 align:middle line:84% the same as before, except the variables 00:25:19.700 --> 00:25:25.550 align:middle line:84% that you are integrating now are Grassmann variables, 00:25:25.550 --> 00:25:28.820 align:middle line:84% and the S is just whatever your action is. 00:25:28.820 --> 00:25:35.575 align:middle line:84% So let me explain a little bit this notation, psi. 00:25:35.575 --> 00:25:38.470 align:middle line:90% 00:25:38.470 --> 00:25:44.530 align:middle line:84% So D psi just means you have, first, you 00:25:44.530 --> 00:25:53.590 align:middle line:84% have alpha from 1 to 4, and then you have D psi alpha t and x. 00:25:53.590 --> 00:25:56.830 align:middle line:84% And this notation is exactly the same as before. 00:25:56.830 --> 00:25:59.440 align:middle line:84% When I write D psi, you should imagine I take 00:25:59.440 --> 00:26:01.660 align:middle line:90% the product of all components. 00:26:01.660 --> 00:26:05.140 align:middle line:84% And then for each of them, it's just exactly the same 00:26:05.140 --> 00:26:09.130 align:middle line:84% as we do for the bosonic field definition. 00:26:09.130 --> 00:26:12.100 align:middle line:84% The only difference is that now this is Grassmann variables. 00:26:12.100 --> 00:26:15.040 align:middle line:90% 00:26:15.040 --> 00:26:26.230 align:middle line:84% So yeah, also, the x here is, say, some product of psi's-- 00:26:26.230 --> 00:26:27.850 align:middle line:90% so arbitrary product of psis-- 00:26:27.850 --> 00:26:31.000 align:middle line:90% 00:26:31.000 --> 00:26:34.450 align:middle line:84% then everything previous takes over. 00:26:34.450 --> 00:26:36.160 align:middle line:84% You can just write down the expression. 00:26:36.160 --> 00:26:39.400 align:middle line:90% 00:26:39.400 --> 00:26:43.540 align:middle line:84% So again, in order to do calculations-- for example, 00:26:43.540 --> 00:26:45.600 align:middle line:84% to calculate the propagator, et cetera-- 00:26:45.600 --> 00:26:48.450 align:middle line:84% it's convenient to use the generating functional 00:26:48.450 --> 00:26:51.750 align:middle line:84% as we do here for this Gaussian integral. 00:26:51.750 --> 00:26:58.710 align:middle line:84% So it's convenient to introduce this generating functional eta 00:26:58.710 --> 00:27:01.510 align:middle line:90% eta bar. 00:27:01.510 --> 00:27:08.870 align:middle line:84% So this is defined to be the D psi bar D psi. 00:27:08.870 --> 00:27:12.310 align:middle line:84% And then you have exponential i s, 00:27:12.310 --> 00:27:22.870 align:middle line:84% and then you write, say, eta bar psi psi bar eta. 00:27:22.870 --> 00:27:27.470 align:middle line:84% So again, when you take derivative of eta, 00:27:27.470 --> 00:27:30.810 align:middle line:84% then you can bring down the powers of psi of psi bar. 00:27:30.810 --> 00:27:35.780 align:middle line:84% And then you use as a trick if you know this one-- 00:27:35.780 --> 00:27:37.775 align:middle line:84% and then you can just calculate any x. 00:27:37.775 --> 00:27:43.180 align:middle line:90% 00:27:43.180 --> 00:27:45.100 align:middle line:90% Any questions on this? 00:27:45.100 --> 00:27:46.259 align:middle line:90% Yes. 00:27:46.259 --> 00:27:57.233 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:27:57.233 --> 00:27:57.900 align:middle line:90% PROFESSOR: Yeah. 00:27:57.900 --> 00:28:12.540 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:28:12.540 --> 00:28:14.110 align:middle line:90% PROFESSOR: Oh. 00:28:14.110 --> 00:28:17.300 align:middle line:90% No, you don't need to-- 00:28:17.300 --> 00:28:19.560 align:middle line:84% you can view this as a definition. 00:28:19.560 --> 00:28:23.420 align:middle line:90% 00:28:23.420 --> 00:28:26.120 align:middle line:84% We just carry over what you used as the definition. 00:28:26.120 --> 00:28:28.730 align:middle line:84% We carry over what we did before, 00:28:28.730 --> 00:28:30.440 align:middle line:84% and then we just replace everything 00:28:30.440 --> 00:28:32.058 align:middle line:90% by Grassmann variables. 00:28:32.058 --> 00:28:41.820 align:middle line:90% 00:28:41.820 --> 00:28:42.850 align:middle line:90% Any questions? 00:28:42.850 --> 00:28:45.650 align:middle line:90% Any other questions? 00:28:45.650 --> 00:28:47.510 align:middle line:90% Good. 00:28:47.510 --> 00:28:52.120 align:middle line:84% OK, so let's just try to calculate this in free theory. 00:28:52.120 --> 00:28:56.880 align:middle line:84% So in the free theory, we will eventually consider the-- 00:28:56.880 --> 00:28:59.640 align:middle line:84% so far, we just consider the Dirac theory. 00:28:59.640 --> 00:29:04.620 align:middle line:84% So let's just first consider Dirac theory. 00:29:04.620 --> 00:29:06.480 align:middle line:90% So this is general. 00:29:06.480 --> 00:29:08.040 align:middle line:90% This is completely general. 00:29:08.040 --> 00:29:11.220 align:middle line:84% S can be anything, but for the Dirac theory, 00:29:11.220 --> 00:29:15.450 align:middle line:90% we have S0 is equal to minus i. 00:29:15.450 --> 00:29:24.850 align:middle line:90% 00:29:24.850 --> 00:29:28.120 align:middle line:84% So for the Dirac theory, it's simple, 00:29:28.120 --> 00:29:30.970 align:middle line:84% because it's just quadratic in psi, 00:29:30.970 --> 00:29:33.400 align:middle line:84% and then we just have a Gaussian integral here. 00:29:33.400 --> 00:29:46.070 align:middle line:90% 00:29:46.070 --> 00:29:49.520 align:middle line:84% So the first thing to do is to write, again, 00:29:49.520 --> 00:29:52.940 align:middle line:90% this as a form of a matrix. 00:29:52.940 --> 00:29:54.960 align:middle line:84% So we have already done this before. 00:29:54.960 --> 00:30:01.730 align:middle line:84% So this has just become i d4x d4y, 00:30:01.730 --> 00:30:10.750 align:middle line:84% and then you have psi bar alpha x A alpha beta x minus y 00:30:10.750 --> 00:30:13.600 align:middle line:90% psi beta y. 00:30:13.600 --> 00:30:17.980 align:middle line:90% So this A alpha beta x minus y-- 00:30:17.980 --> 00:30:21.700 align:middle line:84% this is a matrix both in the spinor space 00:30:21.700 --> 00:30:24.640 align:middle line:84% and in the space of ordinary function-- 00:30:24.640 --> 00:30:27.680 align:middle line:90% in the space of functions. 00:30:27.680 --> 00:30:34.910 align:middle line:90% So this is x minus y slash y. 00:30:34.910 --> 00:30:37.000 align:middle line:84% Yeah, so this means you take derivative on y. 00:30:37.000 --> 00:30:41.490 align:middle line:90% 00:30:41.490 --> 00:30:43.110 align:middle line:84% So once you write it in this form-- 00:30:43.110 --> 00:30:46.260 align:middle line:84% and then, again, we have this just in the Gaussian form. 00:30:46.260 --> 00:30:49.230 align:middle line:84% So now this path integral is just in the Gaussian form. 00:30:49.230 --> 00:30:53.280 align:middle line:84% And, again, we can just generalize what we did before. 00:30:53.280 --> 00:30:55.500 align:middle line:84% Just keep in mind that those integrals-- 00:30:55.500 --> 00:30:59.790 align:middle line:84% now this is given by determinant A, 00:30:59.790 --> 00:31:04.050 align:middle line:90% and then we just find that the-- 00:31:04.050 --> 00:31:09.240 align:middle line:84% so, for example, now this Z eta-- 00:31:09.240 --> 00:31:11.520 align:middle line:90% Z0. 00:31:11.520 --> 00:31:15.870 align:middle line:84% So this means that we can see the Dirac theory. 00:31:15.870 --> 00:31:18.960 align:middle line:84% So this just can be directly evaluated, 00:31:18.960 --> 00:31:22.230 align:middle line:84% and then this is given by determinant A. 00:31:22.230 --> 00:31:24.900 align:middle line:84% So now this determinant should be understood as 00:31:24.900 --> 00:31:30.420 align:middle line:84% both in the spinor space and in the space of functions. 00:31:30.420 --> 00:31:38.520 align:middle line:84% Yeah, and then we have the analog of this term, 00:31:38.520 --> 00:31:46.830 align:middle line:84% and then we have exponential eta bar, then dot D. 00:31:46.830 --> 00:31:53.110 align:middle line:84% So I denote the A minus 1 as D in the motivation. 00:31:53.110 --> 00:31:56.240 align:middle line:90% And then the D-- 00:31:56.240 --> 00:32:06.270 align:middle line:84% so this should be considered as a shorthand notation for this. 00:32:06.270 --> 00:32:08.570 align:middle line:84% So this should be considered as-- 00:32:08.570 --> 00:32:10.228 align:middle line:90% let me just write it here. 00:32:10.228 --> 00:32:12.020 align:middle line:84% So this should be considered as a shorthand 00:32:12.020 --> 00:32:27.930 align:middle line:84% notation for d4x d4y eta bar alpha x D0 alpha beta x minus y 00:32:27.930 --> 00:32:30.840 align:middle line:90% eta beta y. 00:32:30.840 --> 00:32:36.000 align:middle line:84% So this thing is a shorthand notation for this. 00:32:36.000 --> 00:32:42.650 align:middle line:84% And this D alpha beta 0 x minus y 00:32:42.650 --> 00:32:50.380 align:middle line:84% is just the inverse of this matrix A. 00:32:50.380 --> 00:32:53.940 align:middle line:84% You just should view it as an inverse of the matrix A. 00:32:53.940 --> 00:32:55.940 align:middle line:84% And that's the same thing as what 00:32:55.940 --> 00:32:59.510 align:middle line:84% we did in the bosonic case, and this will be 00:32:59.510 --> 00:33:01.760 align:middle line:90% corresponding to the Feynman. 00:33:01.760 --> 00:33:03.560 align:middle line:84% Using the same argument, you can show 00:33:03.560 --> 00:33:07.010 align:middle line:84% that this is corresponding to the Feynman propagator-- 00:33:07.010 --> 00:33:12.020 align:middle line:90% time-ordered propagator of psi. 00:33:12.020 --> 00:33:16.630 align:middle line:90% 00:33:16.630 --> 00:33:17.770 align:middle line:90% So everything is similar. 00:33:17.770 --> 00:33:19.714 align:middle line:90% Yes. 00:33:19.714 --> 00:33:22.593 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:33:22.593 --> 00:33:23.260 align:middle line:90% PROFESSOR: Yeah. 00:33:23.260 --> 00:33:26.165 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:33:26.165 --> 00:33:29.380 align:middle line:84% PROFESSOR: Yeah, so you can just view-- 00:33:29.380 --> 00:33:31.882 align:middle line:90% 00:33:31.882 --> 00:33:33.930 align:middle line:84% you don't have to think about gamma 0 here. 00:33:33.930 --> 00:33:35.400 align:middle line:84% You can just think about the psi-- 00:33:35.400 --> 00:33:39.620 align:middle line:84% eta bar just as an independent variable of eta. 00:33:39.620 --> 00:33:40.120 align:middle line:90% Yeah. 00:33:40.120 --> 00:33:41.120 align:middle line:90% Yeah, it doesn't matter. 00:33:41.120 --> 00:33:46.030 align:middle line:84% You can just treat it as some independent variable. 00:33:46.030 --> 00:33:47.734 align:middle line:90% Yes. 00:33:47.734 --> 00:33:52.180 align:middle line:84% AUDIENCE: Are we treating eta as a vector of Grassman variables? 00:33:52.180 --> 00:33:55.423 align:middle line:90% 00:33:55.423 --> 00:33:56.715 align:middle line:90% PROFESSOR: No, eta is Grassman. 00:33:56.715 --> 00:33:59.930 align:middle line:90% 00:33:59.930 --> 00:34:01.740 align:middle line:90% Yeah. 00:34:01.740 --> 00:34:03.380 align:middle line:90% Yeah, eta is Grassman. 00:34:03.380 --> 00:34:06.620 align:middle line:84% And in particular, eta depends on space time 00:34:06.620 --> 00:34:09.320 align:middle line:84% because we want to be able to take a derivative 00:34:09.320 --> 00:34:10.969 align:middle line:90% to get the psi x. 00:34:10.969 --> 00:34:14.860 align:middle line:90% And so there are-- 00:34:14.860 --> 00:34:16.115 align:middle line:90% yeah, eta has the same-- 00:34:16.115 --> 00:34:18.699 align:middle line:90% 00:34:18.699 --> 00:34:23.739 align:middle line:90% the eta is the eta 1x eta 4x. 00:34:23.739 --> 00:34:28.000 align:middle line:90% 00:34:28.000 --> 00:34:29.732 align:middle line:90% Yeah, same with eta bar. 00:34:29.732 --> 00:34:30.232 align:middle line:90% Yeah. 00:34:30.232 --> 00:34:45.270 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:34:45.270 --> 00:34:47.469 align:middle line:90% PROFESSOR: Sorry, say it again. 00:34:47.469 --> 00:34:51.670 align:middle line:84% AUDIENCE: Are integrals always complex numbers at the end? 00:34:51.670 --> 00:34:52.790 align:middle line:90% PROFESSOR: No, no, no. 00:34:52.790 --> 00:34:55.760 align:middle line:84% The integrals are just some functions 00:34:55.760 --> 00:34:59.510 align:middle line:90% of the Grassmann variables. 00:34:59.510 --> 00:35:01.788 align:middle line:84% Yeah, just some functions of Grassman variables. 00:35:01.788 --> 00:35:06.950 align:middle line:90% 00:35:06.950 --> 00:35:09.410 align:middle line:90% Good. 00:35:09.410 --> 00:35:16.530 align:middle line:84% And then from here, from this expression, 00:35:16.530 --> 00:35:18.822 align:middle line:84% you can just find all the correlation 00:35:18.822 --> 00:35:20.030 align:middle line:90% functions in the free theory. 00:35:20.030 --> 00:35:22.630 align:middle line:90% 00:35:22.630 --> 00:35:28.350 align:middle line:84% So yeah, let me just quickly write it here. 00:35:28.350 --> 00:35:32.250 align:middle line:84% But I don't have-- yeah, I think there is not enough space. 00:35:32.250 --> 00:35:33.583 align:middle line:90% Yes. 00:35:33.583 --> 00:35:35.625 align:middle line:84% AUDIENCE: You said that the result of integrating 00:35:35.625 --> 00:35:38.620 align:middle line:84% over a Grassmann variable is another Grassmann variable, 00:35:38.620 --> 00:35:39.120 align:middle line:90% right? 00:35:39.120 --> 00:35:42.985 align:middle line:90% [INAUDIBLE] 00:35:42.985 --> 00:35:43.860 align:middle line:90% PROFESSOR: No, no no. 00:35:43.860 --> 00:35:45.770 align:middle line:90% It doesn't have to be. 00:35:45.770 --> 00:35:47.570 align:middle line:90% It depends on the situation. 00:35:47.570 --> 00:35:49.550 align:middle line:90% Here, you get a constant. 00:35:49.550 --> 00:35:52.340 align:middle line:84% Here, you get the function of a Grassman variable. 00:35:52.340 --> 00:35:54.032 align:middle line:84% It just depends on the situation. 00:35:54.032 --> 00:35:54.740 align:middle line:90% AUDIENCE: Oh, OK. 00:35:54.740 --> 00:36:00.080 align:middle line:84% So in the path integral, it'll all work out 00:36:00.080 --> 00:36:02.310 align:middle line:90% to give a C number at the end. 00:36:02.310 --> 00:36:03.500 align:middle line:90% PROFESSOR: It depends. 00:36:03.500 --> 00:36:05.650 align:middle line:90% If you have eta-- 00:36:05.650 --> 00:36:08.355 align:middle line:84% so if you don't have eta, then you get a C number. 00:36:08.355 --> 00:36:11.650 align:middle line:90% 00:36:11.650 --> 00:36:14.500 align:middle line:84% But here, if you have eta, then you get the function of eta. 00:36:14.500 --> 00:36:16.940 align:middle line:90% And eta are Grassmann variables. 00:36:16.940 --> 00:36:17.914 align:middle line:90% Yeah, mm-hmm. 00:36:17.914 --> 00:36:20.820 align:middle line:90% 00:36:20.820 --> 00:36:21.570 align:middle line:90% Good. 00:36:21.570 --> 00:36:28.950 align:middle line:84% So now, you can just obtain any correlation functions 00:36:28.950 --> 00:36:29.890 align:middle line:90% in Dirac theory. 00:36:29.890 --> 00:36:32.730 align:middle line:84% You can just take derivative with respect 00:36:32.730 --> 00:37:01.280 align:middle line:84% to this Z. So, say, psi x1 psi xn 0-- 00:37:01.280 --> 00:37:03.470 align:middle line:84% and, again, with the previous convention, 00:37:03.470 --> 00:37:07.430 align:middle line:84% always use 0 to denote the vacuum of the free theory 00:37:07.430 --> 00:37:12.710 align:middle line:84% and then use this omega to denote the vacuum, 00:37:12.710 --> 00:37:14.270 align:middle line:84% say, if you have interaction theory. 00:37:14.270 --> 00:37:16.890 align:middle line:90% 00:37:16.890 --> 00:37:30.890 align:middle line:84% And so, again, you just get Z0 with eta set to be 0, 00:37:30.890 --> 00:37:33.500 align:middle line:84% and then you just do derivatives. 00:37:33.500 --> 00:37:35.130 align:middle line:90% Up to a sign-- 00:37:35.130 --> 00:37:42.650 align:middle line:84% just delta eta x1 delta eta xn, and then 00:37:42.650 --> 00:37:51.510 align:middle line:84% you take derivative on this Z. Yeah, 00:37:51.510 --> 00:37:54.970 align:middle line:84% so it depends on whether you do the bar or not the bar. 00:37:54.970 --> 00:37:57.690 align:middle line:84% So, for example, here is the bar. 00:37:57.690 --> 00:38:00.390 align:middle line:90% So here, psi-- yeah. 00:38:00.390 --> 00:38:03.840 align:middle line:84% And if it's sub psi, then you take derivative with eta bar. 00:38:03.840 --> 00:38:07.080 align:middle line:84% If it's psi bar, you take derivative with eta. 00:38:07.080 --> 00:38:12.060 align:middle line:84% Anyway, and then you do psi eta equal to eta bar equals 0-- 00:38:12.060 --> 00:38:14.430 align:middle line:90% exactly the same as before. 00:38:14.430 --> 00:38:21.570 align:middle line:84% And then again, because these have this kind of structure, 00:38:21.570 --> 00:38:24.786 align:middle line:84% you always pair eta bar with eta-- 00:38:24.786 --> 00:38:27.130 align:middle line:90% with D-- in here. 00:38:27.130 --> 00:38:33.150 align:middle line:84% So when you do that, you just get all possible contractions-- 00:38:33.150 --> 00:38:35.400 align:middle line:84% sum of all possible contractions. 00:38:35.400 --> 00:38:52.040 align:middle line:90% 00:38:52.040 --> 00:38:55.510 align:middle line:84% So each contraction is just a propagator. 00:38:55.510 --> 00:39:01.910 align:middle line:84% So if you have x1 bar x2, each contraction 00:39:01.910 --> 00:39:07.490 align:middle line:84% is just giving you D0 x1 minus x2. 00:39:07.490 --> 00:39:09.785 align:middle line:84% So now I suppress the spinor indices. 00:39:09.785 --> 00:39:15.310 align:middle line:90% 00:39:15.310 --> 00:39:18.970 align:middle line:90% So now you have to be careful. 00:39:18.970 --> 00:39:21.690 align:middle line:84% So one thing you have to be careful-- 00:39:21.690 --> 00:39:25.870 align:middle line:84% you say now all this becomes anticommuting. 00:39:25.870 --> 00:39:31.780 align:middle line:84% So when you take the derivative, you have to be careful-- 00:39:31.780 --> 00:39:33.880 align:middle line:90% when you do the contraction-- 00:39:33.880 --> 00:39:37.390 align:middle line:90% about the order of the psi's. 00:39:37.390 --> 00:39:45.720 align:middle line:84% So, for example, if you look at the 2-point function, 00:39:45.720 --> 00:39:57.635 align:middle line:84% if you have psi x1 psi bar x2 and psi bar x3 psi x4-- 00:39:57.635 --> 00:40:00.290 align:middle line:90% 00:40:00.290 --> 00:40:10.940 align:middle line:84% if you have this, then there are two possible contractions. 00:40:10.940 --> 00:40:13.280 align:middle line:84% You can have this contract with that, 00:40:13.280 --> 00:40:16.850 align:middle line:84% this contract with that, or this contract with that, 00:40:16.850 --> 00:40:18.740 align:middle line:90% and this contract with that. 00:40:18.740 --> 00:40:22.850 align:middle line:84% So you have to be careful about the sign. 00:40:22.850 --> 00:40:25.520 align:middle line:90% And if you are careful-- 00:40:25.520 --> 00:40:31.490 align:middle line:84% so you find you will get minus D0. 00:40:31.490 --> 00:40:42.000 align:middle line:84% So x1 minus x2 and the dx4 minus x3-- 00:40:42.000 --> 00:40:44.670 align:middle line:84% so the minus, then, comes from-- you need to exchange these two 00:40:44.670 --> 00:40:49.250 align:middle line:84% so that you have the form of psi and the psi bar-- 00:40:49.250 --> 00:40:51.440 align:middle line:90% have psi and psi bar. 00:40:51.440 --> 00:41:00.380 align:middle line:84% And then you can also have the D0 x1 minus x3 00:41:00.380 --> 00:41:04.670 align:middle line:90% and D0 x4 minus x2. 00:41:04.670 --> 00:41:07.210 align:middle line:90% 00:41:07.210 --> 00:41:11.350 align:middle line:84% So in this one, you can just change the order. 00:41:11.350 --> 00:41:13.240 align:middle line:84% So here, it should be time-ordered. 00:41:13.240 --> 00:41:17.270 align:middle line:90% 00:41:17.270 --> 00:41:20.245 align:middle line:84% And so the reason this plus sign is here 00:41:20.245 --> 00:41:24.490 align:middle line:84% is because you can shift this x2 by two positions 00:41:24.490 --> 00:41:29.320 align:middle line:84% to the right of the psi x4, and then these two 00:41:29.320 --> 00:41:32.170 align:middle line:90% will now become neighbors. 00:41:32.170 --> 00:41:35.740 align:middle line:84% And then psi 4 and psi 2 will be neighbors. 00:41:35.740 --> 00:41:38.840 align:middle line:84% So you just get the positive sign. 00:41:38.840 --> 00:41:41.710 align:middle line:84% So you just have to be careful about the sign. 00:41:41.710 --> 00:41:49.610 align:middle line:90% 00:41:49.610 --> 00:41:51.632 align:middle line:90% Any questions on this? 00:41:51.632 --> 00:41:53.715 align:middle line:84% Other than that, everything is the same as before. 00:41:53.715 --> 00:41:57.760 align:middle line:90% 00:41:57.760 --> 00:42:00.310 align:middle line:84% So now, you can also just do the interacting theory. 00:42:00.310 --> 00:42:13.100 align:middle line:90% 00:42:13.100 --> 00:42:19.490 align:middle line:84% Say, now, the correlation function in the interacting 00:42:19.490 --> 00:42:24.920 align:middle line:90% theory divided by omega-- 00:42:24.920 --> 00:42:28.100 align:middle line:84% and now, you just do perturbation theory. 00:42:28.100 --> 00:42:35.250 align:middle line:84% We just use that, and then you can write it as the correlation 00:42:35.250 --> 00:42:36.420 align:middle line:90% functions in free theory. 00:42:36.420 --> 00:42:39.770 align:middle line:90% 00:42:39.770 --> 00:42:48.940 align:middle line:84% Again, everything is the same, just now 00:42:48.940 --> 00:42:52.060 align:middle line:84% you have to be careful now the integral is over the Grassmann 00:42:52.060 --> 00:42:54.130 align:middle line:90% variables. 00:42:54.130 --> 00:42:56.080 align:middle line:90% So T-- again, here is T-- 00:42:56.080 --> 00:42:59.270 align:middle line:90% 00:42:59.270 --> 00:43:00.130 align:middle line:90% time-ordered. 00:43:00.130 --> 00:43:04.220 align:middle line:90% 00:43:04.220 --> 00:43:06.920 align:middle line:84% Other than that, everything is just the same as before. 00:43:06.920 --> 00:43:12.240 align:middle line:84% And again, you can show that a vacuum-- 00:43:12.240 --> 00:43:20.035 align:middle line:90% diagrams with vacuum bubbles-- 00:43:20.035 --> 00:43:24.670 align:middle line:90% 00:43:24.670 --> 00:43:25.170 align:middle line:90% cancel. 00:43:25.170 --> 00:43:27.780 align:middle line:90% 00:43:27.780 --> 00:43:30.480 align:middle line:84% It will cancel so you don't have to include the vacuum bubble. 00:43:30.480 --> 00:43:33.330 align:middle line:84% And then the epsilon prescription 00:43:33.330 --> 00:43:36.240 align:middle line:84% is the m goes to m minus i epsilon. 00:43:36.240 --> 00:43:37.785 align:middle line:90% OK, very similar to before. 00:43:37.785 --> 00:43:44.050 align:middle line:90% 00:43:44.050 --> 00:43:44.550 align:middle line:90% Good. 00:43:44.550 --> 00:43:45.480 align:middle line:90% Any questions? 00:43:45.480 --> 00:43:48.210 align:middle line:90% 00:43:48.210 --> 00:43:53.240 align:middle line:84% So now, we can just apply, identically, 00:43:53.240 --> 00:43:56.600 align:middle line:84% the formalism we developed before for interacting theory 00:43:56.600 --> 00:44:00.020 align:middle line:84% of scalar fields to here, except we just 00:44:00.020 --> 00:44:02.960 align:middle line:84% have to be careful about some signs, 00:44:02.960 --> 00:44:05.570 align:middle line:84% because when you exchange the fermions, you get some signs. 00:44:05.570 --> 00:44:08.210 align:middle line:84% And other than those subtleties of signs, 00:44:08.210 --> 00:44:11.760 align:middle line:84% everything else will be the same. 00:44:11.760 --> 00:44:13.810 align:middle line:84% But those signs sometimes can be annoying. 00:44:13.810 --> 00:44:17.220 align:middle line:90% 00:44:17.220 --> 00:44:21.090 align:middle line:90% So we can derive the Feynman-- 00:44:21.090 --> 00:44:23.580 align:middle line:84% we can draw the Feynman diagrams as before 00:44:23.580 --> 00:44:26.610 align:middle line:84% and write down the Feynman rules, et cetera. 00:44:26.610 --> 00:44:29.520 align:middle line:90% So now, let me just write-- 00:44:29.520 --> 00:44:34.240 align:middle line:90% 00:44:34.240 --> 00:44:35.740 align:middle line:84% yeah, just everything carries over. 00:44:35.740 --> 00:44:37.570 align:middle line:84% So let me just emphasize the difference 00:44:37.570 --> 00:44:38.560 align:middle line:90% from the scalar case. 00:44:38.560 --> 00:45:11.480 align:middle line:90% 00:45:11.480 --> 00:45:29.430 align:middle line:84% So the difference of Feynman rules from the scalar case-- 00:45:29.430 --> 00:45:40.070 align:middle line:90% 00:45:40.070 --> 00:45:43.080 align:middle line:90% so there's various differences. 00:45:43.080 --> 00:45:45.260 align:middle line:90% So first, let's talk about the-- 00:45:45.260 --> 00:45:47.840 align:middle line:84% remember, we have different Feynman rules 00:45:47.840 --> 00:45:52.790 align:middle line:84% for Green functions and also have ones 00:45:52.790 --> 00:45:54.660 align:middle line:90% for scattering amplitudes. 00:45:54.660 --> 00:45:57.140 align:middle line:84% So the scattering amplitude is the one which 00:45:57.140 --> 00:46:00.270 align:middle line:90% the external legs truncated. 00:46:00.270 --> 00:46:03.530 align:middle line:84% And in the Green function, you don't truncate external legs. 00:46:03.530 --> 00:46:07.600 align:middle line:84% So now, let me first say for the Green functions-- 00:46:07.600 --> 00:46:10.190 align:middle line:90% so for the propagator-- 00:46:10.190 --> 00:46:18.260 align:middle line:90% 00:46:18.260 --> 00:46:22.660 align:middle line:84% again, the propagator is just represented by a line. 00:46:22.660 --> 00:46:27.970 align:middle line:84% But here, we have a complex field. 00:46:27.970 --> 00:46:30.850 align:middle line:84% So as we did for the complex scalar field, now 00:46:30.850 --> 00:46:35.360 align:middle line:84% you need to assign an arrow to indicate the flow 00:46:35.360 --> 00:46:37.690 align:middle line:90% of the direction of the charge. 00:46:37.690 --> 00:46:43.560 align:middle line:84% So we always assign the arrow to flow from the bar 00:46:43.560 --> 00:46:48.120 align:middle line:90% to the unbarred side. 00:46:48.120 --> 00:46:52.500 align:middle line:84% So for that one, we will do that. 00:46:52.500 --> 00:46:54.720 align:middle line:90% So this will be x2-- 00:46:54.720 --> 00:46:57.100 align:middle line:90% will be x1. 00:46:57.100 --> 00:47:01.960 align:middle line:84% Suppose this is beta, this will be alpha. 00:47:01.960 --> 00:47:09.940 align:middle line:84% So if I put the alpha beta here, this 00:47:09.940 --> 00:47:12.760 align:middle line:84% will be alpha x1 and the beta x2, 00:47:12.760 --> 00:47:15.850 align:middle line:90% and then the arrow goes to the-- 00:47:15.850 --> 00:47:17.770 align:middle line:90% yeah, so this is a charge arrow. 00:47:17.770 --> 00:47:19.420 align:middle line:90% This is not the momentum arrow. 00:47:19.420 --> 00:47:23.620 align:middle line:84% You can also, in additionally, assign a momentum. 00:47:23.620 --> 00:47:25.600 align:middle line:84% You can additionally assign a momentum. 00:47:25.600 --> 00:47:30.310 align:middle line:84% And the momentum-- yeah, so this is in a coordinate space. 00:47:30.310 --> 00:47:32.810 align:middle line:84% Sorry, so this is in the coordinate space. 00:47:32.810 --> 00:47:39.700 align:middle line:84% And so this just gives you D0 alpha beta x1 minus x2-- 00:47:39.700 --> 00:47:41.450 align:middle line:90% so in the coordinate space. 00:47:41.450 --> 00:47:45.920 align:middle line:84% So in momentum space, we don't label the locations. 00:47:45.920 --> 00:47:51.960 align:middle line:84% So in momentum space, we just have alpha and the beta. 00:47:51.960 --> 00:47:55.440 align:middle line:84% And now you also have a momentum. 00:47:55.440 --> 00:48:02.370 align:middle line:84% So you can choose the momentum in any direction you want. 00:48:02.370 --> 00:48:04.847 align:middle line:84% So, often, for convenience, just as in the scalar case, 00:48:04.847 --> 00:48:06.930 align:middle line:84% we often choose a momentum to be in the same arrow 00:48:06.930 --> 00:48:07.990 align:middle line:90% in this direction. 00:48:07.990 --> 00:48:10.200 align:middle line:84% But in the case which you want to choose a momentum 00:48:10.200 --> 00:48:12.150 align:middle line:84% arrow to be different, you can just 00:48:12.150 --> 00:48:18.870 align:middle line:84% draw a momentum arrow which can be different. 00:48:18.870 --> 00:48:21.270 align:middle line:84% So the momentum space is the propagator. 00:48:21.270 --> 00:48:24.100 align:middle line:84% It is given by what we have already written before. 00:48:24.100 --> 00:48:32.160 align:middle line:84% So it's given by minus 1 over i slash minus n plus i 00:48:32.160 --> 00:48:35.066 align:middle line:90% epsilon alpha beta. 00:48:35.066 --> 00:48:41.690 align:middle line:90% 00:48:41.690 --> 00:48:42.230 align:middle line:90% Yes. 00:48:42.230 --> 00:48:51.740 align:middle line:84% AUDIENCE: [INAUDIBLE] I thought of it as a particle traveling 00:48:51.740 --> 00:48:52.583 align:middle line:90% through space. 00:48:52.583 --> 00:48:53.250 align:middle line:90% PROFESSOR: Yeah. 00:48:53.250 --> 00:48:54.667 align:middle line:84% AUDIENCE: Like we were contracting 00:48:54.667 --> 00:48:59.280 align:middle line:84% two different fields, so why do I still think about it 00:48:59.280 --> 00:49:01.753 align:middle line:84% as a fermion moving through space time. 00:49:01.753 --> 00:49:02.420 align:middle line:90% PROFESSOR: Yeah. 00:49:02.420 --> 00:49:05.170 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:49:05.170 --> 00:49:07.630 align:middle line:90% PROFESSOR: Yeah, you can-- 00:49:07.630 --> 00:49:11.347 align:middle line:84% no, that aspect is the same as in the complex scalar case. 00:49:11.347 --> 00:49:12.680 align:middle line:90% AUDIENCE: Right, but even then-- 00:49:12.680 --> 00:49:14.510 align:middle line:84% PROFESSOR: No different from the complex scalar case. 00:49:14.510 --> 00:49:15.335 align:middle line:90% Yeah. 00:49:15.335 --> 00:49:17.000 align:middle line:84% AUDIENCE: But why do we still think 00:49:17.000 --> 00:49:20.000 align:middle line:84% of it as the same particle propagating even though it's 00:49:20.000 --> 00:49:21.080 align:middle line:90% interacting [INAUDIBLE] 00:49:21.080 --> 00:49:21.500 align:middle line:90% PROFESSOR: Yeah. 00:49:21.500 --> 00:49:23.708 align:middle line:84% Yeah, you can consider it as the particle propagating 00:49:23.708 --> 00:49:25.797 align:middle line:84% in one direction and then the antiparticle 00:49:25.797 --> 00:49:27.380 align:middle line:84% propagating in the opposite direction. 00:49:27.380 --> 00:49:28.713 align:middle line:90% You can interpret it either way. 00:49:28.713 --> 00:49:31.270 align:middle line:90% 00:49:31.270 --> 00:49:34.900 align:middle line:84% Yeah, the importance is the direction of the charge flow. 00:49:34.900 --> 00:49:41.380 align:middle line:90% 00:49:41.380 --> 00:49:41.880 align:middle line:90% Good. 00:49:41.880 --> 00:49:47.000 align:middle line:90% 00:49:47.000 --> 00:49:49.430 align:middle line:84% Yeah, so the momentum we emphasize. 00:49:49.430 --> 00:50:00.840 align:middle line:84% So the charge arrow is not arbitrary, 00:50:00.840 --> 00:50:10.950 align:middle line:84% but the momentum arrow is arbitrary. 00:50:10.950 --> 00:50:13.470 align:middle line:84% So momentum arrow-- you can choose whatever you want. 00:50:13.470 --> 00:50:17.640 align:middle line:90% 00:50:17.640 --> 00:50:26.270 align:middle line:84% So also, the rule we do for the external-- 00:50:26.270 --> 00:50:31.910 align:middle line:90% 00:50:31.910 --> 00:50:37.690 align:middle line:84% so the b-- so if we talk about the external point, 00:50:37.690 --> 00:50:43.420 align:middle line:84% we look for the propagator for the correlation functions. 00:50:43.420 --> 00:50:47.710 align:middle line:84% Say you have those external points, and some of them 00:50:47.710 --> 00:50:51.340 align:middle line:84% will be psi, some of them will be psi bar. 00:50:51.340 --> 00:50:53.745 align:middle line:84% Again, we assign the rule as this. 00:50:53.745 --> 00:50:58.080 align:middle line:90% 00:50:58.080 --> 00:51:06.430 align:middle line:84% For each external point, if it's a psi-- 00:51:06.430 --> 00:51:12.200 align:middle line:84% if the endpoint is given by the psi bar alpha x-- 00:51:12.200 --> 00:51:16.640 align:middle line:84% from the rule, the arrow always leaves the direction 00:51:16.640 --> 00:51:17.960 align:middle line:90% from the psi bar. 00:51:17.960 --> 00:51:21.740 align:middle line:84% Then we should draw it as alpha x. 00:51:21.740 --> 00:51:25.500 align:middle line:90% So this is the external point. 00:51:25.500 --> 00:51:32.190 align:middle line:84% And then, similarly, if you have a psi alpha x, this is-- 00:51:32.190 --> 00:51:36.580 align:middle line:90% and then the arrow will come in. 00:51:36.580 --> 00:51:45.000 align:middle line:90% 00:51:45.000 --> 00:51:49.290 align:middle line:84% So in momentum space, if we choose the momentum to be 00:51:49.290 --> 00:51:51.140 align:middle line:90% the same direction-- 00:51:51.140 --> 00:51:56.420 align:middle line:84% so we just have k, and we take k to be the same arrow. 00:51:56.420 --> 00:51:58.420 align:middle line:84% And here, again, it's the same thing. 00:51:58.420 --> 00:52:04.380 align:middle line:90% 00:52:04.380 --> 00:52:06.960 align:middle line:84% So this is the coordinate space, this is the momentum space. 00:52:06.960 --> 00:52:09.450 align:middle line:84% This is for psi bar, this for psi. 00:52:09.450 --> 00:52:12.717 align:middle line:90% 00:52:12.717 --> 00:52:14.050 align:middle line:90% Yeah, just follow the same rule. 00:52:14.050 --> 00:52:15.074 align:middle line:90% Yes. 00:52:15.074 --> 00:52:22.685 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:52:22.685 --> 00:52:24.310 align:middle line:84% PROFESSOR: Yeah, we'll talk about that. 00:52:24.310 --> 00:52:25.630 align:middle line:90% Yeah. 00:52:25.630 --> 00:52:29.770 align:middle line:84% Here, I'm talking about the correlation functions. 00:52:29.770 --> 00:52:32.560 align:middle line:84% In correlation functions, you don't talk about the particles. 00:52:32.560 --> 00:52:34.945 align:middle line:84% You just look at, what are the external points? 00:52:34.945 --> 00:52:36.820 align:middle line:84% And when we talk about scattering amplitudes, 00:52:36.820 --> 00:52:37.930 align:middle line:90% then we talk about-- 00:52:37.930 --> 00:52:41.570 align:middle line:84% I haven't talked about scattering amplitude yet. 00:52:41.570 --> 00:52:42.890 align:middle line:90% Do you have any questions? 00:52:42.890 --> 00:52:43.760 align:middle line:90% Other questions? 00:52:43.760 --> 00:52:47.720 align:middle line:90% 00:52:47.720 --> 00:52:49.870 align:middle line:90% OK, good. 00:52:49.870 --> 00:52:52.260 align:middle line:90% So now the most important-- 00:52:52.260 --> 00:52:56.030 align:middle line:90% 00:52:56.030 --> 00:52:59.630 align:middle line:84% now you have spinors, and now you have components. 00:52:59.630 --> 00:53:05.440 align:middle line:84% Now the propagator is a matrix and each spinor is a vector, 00:53:05.440 --> 00:53:09.970 align:middle line:84% so you have to be careful about the indices. 00:53:09.970 --> 00:53:12.730 align:middle line:84% So the last thing you have to be careful 00:53:12.730 --> 00:53:13.975 align:middle line:90% is that now you have to-- 00:53:13.975 --> 00:53:17.220 align:middle line:90% 00:53:17.220 --> 00:53:30.900 align:middle line:84% so the spinor indices are contracted 00:53:30.900 --> 00:53:46.800 align:middle line:84% following the arrows, and I will explain what this means. 00:53:46.800 --> 00:53:49.260 align:middle line:84% So here, it's a little bit heuristic-- 00:53:49.260 --> 00:53:51.600 align:middle line:84% a statement is a bit heuristic-- but I 00:53:51.600 --> 00:53:54.180 align:middle line:84% will explain in detail what this means when we talk 00:53:54.180 --> 00:53:56.400 align:middle line:90% about scattering amplitudes. 00:53:56.400 --> 00:53:57.760 align:middle line:90% Because the scattering ampli-- 00:53:57.760 --> 00:54:00.300 align:middle line:84% so this aspect will be similar to the scattering amplitudes 00:54:00.300 --> 00:54:02.820 align:middle line:84% that we just explained in one place. 00:54:02.820 --> 00:54:06.130 align:middle line:90% 00:54:06.130 --> 00:54:10.450 align:middle line:90% So the scattering amplitude-- 00:54:10.450 --> 00:54:13.330 align:middle line:84% so the rule follows, again, from this LSZ, 00:54:13.330 --> 00:54:15.630 align:middle line:90% which I will not derive-- 00:54:15.630 --> 00:54:21.090 align:middle line:84% this Lehmann-Symanzik-Zimmermann reduction rule. 00:54:21.090 --> 00:54:24.850 align:middle line:90% And so you consider-- 00:54:24.850 --> 00:54:27.480 align:middle line:90% say you have some initial-- 00:54:27.480 --> 00:54:31.320 align:middle line:84% you have some final particles, some initial particles 00:54:31.320 --> 00:54:33.690 align:middle line:84% scattered into some final particles. 00:54:33.690 --> 00:54:42.250 align:middle line:84% And now we specify the particles not only by their momentum, 00:54:42.250 --> 00:54:44.490 align:middle line:90% but also by their polarization. 00:54:44.490 --> 00:54:50.430 align:middle line:84% So suppose that the p1 is the momentum of one 00:54:50.430 --> 00:54:52.620 align:middle line:84% of the first incoming particles, r1 00:54:52.620 --> 00:54:59.350 align:middle line:84% would be its polarization and et cetera. 00:54:59.350 --> 00:55:04.580 align:middle line:84% And for antiparticle, we put a bar. 00:55:04.580 --> 00:55:07.260 align:middle line:84% So for the second particle, it's the antiparticle. 00:55:07.260 --> 00:55:09.990 align:middle line:90% Then we put r2 bar. 00:55:09.990 --> 00:55:14.580 align:middle line:84% OK, so that means this is an antiparticle. 00:55:14.580 --> 00:55:17.620 align:middle line:84% And similarly, for the final state, 00:55:17.620 --> 00:55:24.000 align:middle line:84% we specify it by momentum and the polarization. 00:55:24.000 --> 00:55:31.790 align:middle line:84% And if we do a bar, again, we mean the antiparticle. 00:55:31.790 --> 00:55:33.470 align:middle line:84% The scattering amplitude is where 00:55:33.470 --> 00:55:36.920 align:middle line:84% we want to calculate things like this with some initial momentum 00:55:36.920 --> 00:55:37.865 align:middle line:90% and the-- 00:55:37.865 --> 00:55:41.700 align:middle line:90% 00:55:41.700 --> 00:56:01.000 align:middle line:84% so to specify by the initial and the final momenta 00:56:01.000 --> 00:56:02.350 align:middle line:90% and the polarizations. 00:56:02.350 --> 00:56:11.680 align:middle line:90% 00:56:11.680 --> 00:56:16.530 align:middle line:84% And then the bar is for antiparticle. 00:56:16.530 --> 00:56:25.380 align:middle line:90% 00:56:25.380 --> 00:56:30.380 align:middle line:84% So between the scattering amplitude and the Green 00:56:30.380 --> 00:56:33.650 align:middle line:84% function, the only difference is how 00:56:33.650 --> 00:56:38.780 align:middle line:90% you choose the external lines. 00:56:38.780 --> 00:56:42.880 align:middle line:84% All the propagators-- these are the same. 00:56:42.880 --> 00:56:45.930 align:middle line:84% So now, remember in the scalar case, 00:56:45.930 --> 00:56:52.380 align:middle line:84% we just remove all the external lines. 00:56:52.380 --> 00:56:58.640 align:middle line:84% But here, we have to be a little bit careful because now they 00:56:58.640 --> 00:57:00.570 align:middle line:90% have the polarization. 00:57:00.570 --> 00:57:03.780 align:middle line:84% And now we should have things to specify the polarization 00:57:03.780 --> 00:57:06.230 align:middle line:90% of each particle. 00:57:06.230 --> 00:57:10.560 align:middle line:84% So now it's reasonable to expect, 00:57:10.560 --> 00:57:13.140 align:middle line:84% and the polarization of each particle 00:57:13.140 --> 00:57:16.305 align:middle line:84% is specified by those u and v functions we derived earlier. 00:57:16.305 --> 00:57:30.880 align:middle line:90% 00:57:30.880 --> 00:57:32.800 align:middle line:84% So we truncate all the propagators 00:57:32.800 --> 00:57:35.860 align:middle line:84% for the external lines, but we have 00:57:35.860 --> 00:57:43.660 align:middle line:90% to assign polarization vectors. 00:57:43.660 --> 00:57:48.040 align:middle line:90% 00:57:48.040 --> 00:58:12.390 align:middle line:84% So need to-- assign polarization vectors for each initial 00:58:12.390 --> 00:58:15.375 align:middle line:90% and final particle-- 00:58:15.375 --> 00:58:17.970 align:middle line:90% 00:58:17.970 --> 00:58:19.485 align:middle line:90% particle or antiparticle. 00:58:19.485 --> 00:58:22.780 align:middle line:90% 00:58:22.780 --> 00:58:30.190 align:middle line:84% So let me just state the rule, and then I 00:58:30.190 --> 00:58:33.630 align:middle line:90% will motivate the rule. 00:58:33.630 --> 00:58:43.950 align:middle line:84% So for the initial state, if you consider the initial state-- 00:58:43.950 --> 00:58:45.630 align:middle line:90% then if you have a particle-- 00:58:45.630 --> 00:58:51.420 align:middle line:90% 00:58:51.420 --> 00:58:56.540 align:middle line:84% so suppose this particle is polarization-- r1. 00:58:56.540 --> 00:59:01.910 align:middle line:84% And then we draw a line like this with the arrow like that, 00:59:01.910 --> 00:59:06.980 align:middle line:84% and then assign the polarization vector, assign u r1. 00:59:06.980 --> 00:59:10.400 align:middle line:84% So suppose that these have momentum p1 and p1. 00:59:10.400 --> 00:59:15.760 align:middle line:90% 00:59:15.760 --> 00:59:28.790 align:middle line:84% And for antiparticle, say, for example, r2 bar-- 00:59:28.790 --> 00:59:32.500 align:middle line:84% say we just reverse the direction of the arrow. 00:59:32.500 --> 00:59:34.480 align:middle line:84% And then the polarization vector-- 00:59:34.480 --> 00:59:39.040 align:middle line:90% I use v bar r2 and p2. 00:59:39.040 --> 00:59:43.060 align:middle line:84% I always draw the momentum to be the same as the charge arrow. 00:59:43.060 --> 00:59:46.000 align:middle line:84% So this is for the initial state, 00:59:46.000 --> 00:59:48.130 align:middle line:84% and the final state is the following. 00:59:48.130 --> 00:59:52.910 align:middle line:90% 00:59:52.910 --> 00:59:56.290 align:middle line:84% So we just first write down the rule, then I will motivate it. 00:59:56.290 --> 01:00:01.690 align:middle line:84% So the final state-- if it's a particle, 01:00:01.690 --> 01:00:03.220 align:middle line:90% then the particle will come out. 01:00:03.220 --> 01:00:06.660 align:middle line:90% 01:00:06.660 --> 01:00:12.790 align:middle line:84% And this is s1 and, say, suppose this is k1. 01:00:12.790 --> 01:00:17.170 align:middle line:84% And then it's given by u bar s1 k1. 01:00:17.170 --> 01:00:23.440 align:middle line:84% So if it's an antiparticle, and then it's 01:00:23.440 --> 01:00:26.430 align:middle line:90% an arrow which is going out-- 01:00:26.430 --> 01:00:42.295 align:middle line:84% suppose this is s2 bar, then this is given by v2 vs2 k2. 01:00:42.295 --> 01:00:45.730 align:middle line:90% 01:00:45.730 --> 01:00:49.180 align:middle line:84% So when you do the scattering amplitude, 01:00:49.180 --> 01:00:51.550 align:middle line:84% you forget about the external propagator, 01:00:51.550 --> 01:00:56.680 align:middle line:84% but you attach to each external line a polarization vector. 01:00:56.680 --> 01:00:58.692 align:middle line:84% OK, polarization vector-- just specify u and v 01:00:58.692 --> 01:00:59.650 align:middle line:90% according to this rule. 01:00:59.650 --> 01:01:03.150 align:middle line:90% 01:01:03.150 --> 01:01:05.260 align:middle line:84% So now, let's see why this is the rule. 01:01:05.260 --> 01:01:09.570 align:middle line:90% 01:01:09.570 --> 01:01:12.260 align:middle line:84% So remember-- so let's use this as an example. 01:01:12.260 --> 01:01:15.420 align:middle line:90% 01:01:15.420 --> 01:01:20.525 align:middle line:84% So this is the initial state, then that means this is a ket. 01:01:20.525 --> 01:01:23.350 align:middle line:90% 01:01:23.350 --> 01:01:26.560 align:middle line:84% So if you remember that psi has the structure-- 01:01:26.560 --> 01:01:33.160 align:middle line:90% the au plus b dagger v-- 01:01:33.160 --> 01:01:37.330 align:middle line:84% then psi bar has the structure a dagger u bar 01:01:37.330 --> 01:01:42.110 align:middle line:90% plus b, say, v bar. 01:01:42.110 --> 01:01:44.840 align:middle line:90% So remember this structure. 01:01:44.840 --> 01:01:52.300 align:middle line:84% So for the initial state, let's try to motivate this rule. 01:01:52.300 --> 01:01:55.870 align:middle line:84% So initial state, you have a p1 r1. 01:01:55.870 --> 01:01:58.310 align:middle line:84% So suppose you have a particle like this. 01:01:58.310 --> 01:02:00.160 align:middle line:90% So this is obtained by-- 01:02:00.160 --> 01:02:07.640 align:middle line:84% so don't worry about the proportional factor. 01:02:07.640 --> 01:02:18.460 align:middle line:84% So this is given by p1, then r1 dagger acting on 0. 01:02:18.460 --> 01:02:20.380 align:middle line:90% And then this you can obtain-- 01:02:20.380 --> 01:02:23.830 align:middle line:84% you see the psi is proportional-- so 01:02:23.830 --> 01:02:26.260 align:middle line:84% in order to have a dagger, you have to have psi bar. 01:02:26.260 --> 01:02:29.510 align:middle line:90% 01:02:29.510 --> 01:02:33.580 align:middle line:84% So this is given by-- it can be obtained from psi bar 01:02:33.580 --> 01:02:37.660 align:middle line:84% by multiplying psi bar, say, by something like this. 01:02:37.660 --> 01:02:44.150 align:middle line:84% So exponential minus i-- yeah, so you can verify this, 01:02:44.150 --> 01:02:46.430 align:middle line:84% but, heuristically, you will understand. 01:02:46.430 --> 01:02:56.410 align:middle line:90% Psi bar x gamma zero u r1 p1-- 01:02:56.410 --> 01:02:58.360 align:middle line:90% acting on 0. 01:02:58.360 --> 01:03:02.920 align:middle line:84% So because psi bar has lots of other things, 01:03:02.920 --> 01:03:05.290 align:middle line:84% you need to multiply this by this polarization vector 01:03:05.290 --> 01:03:09.340 align:middle line:84% so that it will extract the a p1 r1 piece. 01:03:09.340 --> 01:03:12.340 align:middle line:84% And so this is the polarization vector 01:03:12.340 --> 01:03:14.650 align:middle line:90% you need to include here. 01:03:14.650 --> 01:03:17.140 align:middle line:84% And the reason this is the arrow which is going out 01:03:17.140 --> 01:03:19.040 align:middle line:90% is because this is psi bar. 01:03:19.040 --> 01:03:22.210 align:middle line:84% So remember, the things always come out from the psi bar-- 01:03:22.210 --> 01:03:25.810 align:middle line:90% 01:03:25.810 --> 01:03:28.600 align:middle line:84% from this rule, to come out of this from psi bar. 01:03:28.600 --> 01:03:35.380 align:middle line:84% So similarly, you can understand the other rules. 01:03:35.380 --> 01:03:37.160 align:middle line:90% Any questions? 01:03:37.160 --> 01:03:37.850 align:middle line:90% Yes. 01:03:37.850 --> 01:03:43.752 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:03:43.752 --> 01:03:45.460 align:middle line:84% PROFESSOR: Yeah, because you have a spin. 01:03:45.460 --> 01:03:49.280 align:middle line:90% Yeah, a spin has polarization. 01:03:49.280 --> 01:03:52.210 align:middle line:84% This is to characterize the polarization of the spin. 01:03:52.210 --> 01:03:52.710 align:middle line:90% Yeah. 01:03:52.710 --> 01:03:54.793 align:middle line:84% AUDIENCE: I'm just trying to think back-- is there 01:03:54.793 --> 01:03:58.085 align:middle line:84% a symmetry that causes us to need 01:03:58.085 --> 01:03:59.293 align:middle line:90% to specify this polarization? 01:03:59.293 --> 01:04:03.250 align:middle line:90% 01:04:03.250 --> 01:04:03.987 align:middle line:90% PROFESSOR: Sorry? 01:04:03.987 --> 01:04:06.070 align:middle line:84% AUDIENCE: So this is always the case for fermions. 01:04:06.070 --> 01:04:07.960 align:middle line:90% PROFESSOR: Yeah. 01:04:07.960 --> 01:04:08.590 align:middle line:90% Right, right. 01:04:08.590 --> 01:04:10.510 align:middle line:90% Yeah, this is always-- yeah. 01:04:10.510 --> 01:04:14.773 align:middle line:84% AUDIENCE: When I'm thinking back to the scalar piece [INAUDIBLE] 01:04:14.773 --> 01:04:16.815 align:middle line:84% PROFESSOR: Yeah, because there's no polarization. 01:04:16.815 --> 01:04:20.190 align:middle line:90% 01:04:20.190 --> 01:04:22.440 align:middle line:84% Yeah, for scalar, all this just becomes one 01:04:22.440 --> 01:04:24.118 align:middle line:90% because there's no polarization. 01:04:24.118 --> 01:04:27.050 align:middle line:90% 01:04:27.050 --> 01:04:30.290 align:middle line:90% Other questions? 01:04:30.290 --> 01:04:36.040 align:middle line:84% OK, so these are just rules you can understand. 01:04:36.040 --> 01:04:38.410 align:middle line:84% And then this antiparticle initial state 01:04:38.410 --> 01:04:42.040 align:middle line:84% will be created by psi, and then you 01:04:42.040 --> 01:04:46.600 align:middle line:84% need to multiply by psi-- by v bar on this term 01:04:46.600 --> 01:04:49.390 align:middle line:90% to extract this b dagger term. 01:04:49.390 --> 01:04:53.890 align:middle line:84% Because the v v bar, they're contracted. 01:04:53.890 --> 01:04:56.830 align:middle line:84% And then you can extract this v plus. 01:04:56.830 --> 01:05:00.760 align:middle line:84% Yeah, so that's where this will come. 01:05:00.760 --> 01:05:01.480 align:middle line:90% Yes. 01:05:01.480 --> 01:05:09.150 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:09.150 --> 01:05:13.530 align:middle line:84% PROFESSOR: Yeah, in the real experiment, it's often-- 01:05:13.530 --> 01:05:17.580 align:middle line:84% it's not easy to observe the polarized. 01:05:17.580 --> 01:05:21.480 align:middle line:90% You have to have very special-- 01:05:21.480 --> 01:05:23.670 align:middle line:90% yeah. 01:05:23.670 --> 01:05:25.560 align:middle line:84% In general, you observe unpolarized. 01:05:25.560 --> 01:05:28.800 align:middle line:90% 01:05:28.800 --> 01:05:32.030 align:middle line:90% Other questions? 01:05:32.030 --> 01:05:33.410 align:middle line:90% Good. 01:05:33.410 --> 01:05:43.030 align:middle line:84% So again, now I can specify more precisely this spinor indices 01:05:43.030 --> 01:05:49.890 align:middle line:84% rule for the scattering amplitude case. 01:05:49.890 --> 01:05:55.640 align:middle line:84% So now you see the initial state, 01:05:55.640 --> 01:05:59.140 align:middle line:84% so now you have this u bar and you have u. 01:05:59.140 --> 01:06:01.810 align:middle line:84% So the rule for the spinor indices contraction 01:06:01.810 --> 01:06:03.830 align:middle line:90% are the following. 01:06:03.830 --> 01:06:14.320 align:middle line:84% So the spinor indices are contracted 01:06:14.320 --> 01:06:22.120 align:middle line:84% by starting at the one end, so at one 01:06:22.120 --> 01:06:23.470 align:middle line:90% end of a fermionic line-- 01:06:23.470 --> 01:06:38.040 align:middle line:90% 01:06:38.040 --> 01:06:50.270 align:middle line:84% this external factor, either u bar or v bar. 01:06:50.270 --> 01:06:53.120 align:middle line:90% 01:06:53.120 --> 01:06:54.770 align:middle line:90% So you start from-- 01:06:54.770 --> 01:07:13.220 align:middle line:84% and then you go back, and then you 01:07:13.220 --> 01:07:29.610 align:middle line:84% go along the complete line following the arrow backwards. 01:07:29.610 --> 01:07:42.920 align:middle line:90% 01:07:42.920 --> 01:07:47.030 align:middle line:84% So right now, it's a little bit abstract. 01:07:47.030 --> 01:07:50.555 align:middle line:84% Now let me just explain this rule using an example. 01:07:50.555 --> 01:07:51.665 align:middle line:90% It would be very clear. 01:07:51.665 --> 01:07:55.300 align:middle line:90% 01:07:55.300 --> 01:08:01.760 align:middle line:84% Now, we'll explain this rule using an example. 01:08:01.760 --> 01:08:05.450 align:middle line:84% So now let's consider the following example. 01:08:05.450 --> 01:08:10.090 align:middle line:84% So a very important example in the development of particle 01:08:10.090 --> 01:08:16.270 align:middle line:84% physics is this called Yukawa theory, which Yukawa proposed, 01:08:16.270 --> 01:08:19.240 align:middle line:84% I think, around the 1930s, something-- 01:08:19.240 --> 01:08:22.140 align:middle line:84% and which he got a Nobel Prize for it. 01:08:22.140 --> 01:08:27.220 align:middle line:84% So let's consider you have a scalar field. 01:08:27.220 --> 01:08:30.479 align:middle line:84% So let's just say you have the Klein-Gordon-- 01:08:30.479 --> 01:08:33.000 align:middle line:84% Lagrangian density for scalar fields, 01:08:33.000 --> 01:08:37.094 align:middle line:84% and then you have a Dirac Lagrangian for psi. 01:08:37.094 --> 01:08:40.260 align:middle line:90% 01:08:40.260 --> 01:08:47.470 align:middle line:84% And then, supposed they intact, a term like this-- minus g phi 01:08:47.470 --> 01:08:48.580 align:middle line:90% psi bar psi. 01:08:48.580 --> 01:08:51.939 align:middle line:90% 01:08:51.939 --> 01:08:53.950 align:middle line:84% So this clearly is Lorentz invariance. 01:08:53.950 --> 01:08:58.630 align:middle line:90% 01:08:58.630 --> 01:09:09.490 align:middle line:84% And so, essentially, now you can just draw the propagator. 01:09:09.490 --> 01:09:17.100 align:middle line:84% So the propagators-- say, let's denote this to be the phi, 01:09:17.100 --> 01:09:20.700 align:middle line:84% and then these have the standard minus i, 01:09:20.700 --> 01:09:24.189 align:middle line:84% say, k squared plus m squared minus i epsilon. 01:09:24.189 --> 01:09:26.460 align:middle line:90% So if this has momentum k-- 01:09:26.460 --> 01:09:29.100 align:middle line:90% 01:09:29.100 --> 01:09:31.840 align:middle line:90% so we also have fermionic lines. 01:09:31.840 --> 01:09:36.029 align:middle line:84% I will draw it using the real line 01:09:36.029 --> 01:09:40.970 align:middle line:84% so this can have the form ik slash 01:09:40.970 --> 01:09:44.890 align:middle line:90% plus m minus m plus i epsilon. 01:09:44.890 --> 01:09:47.979 align:middle line:90% 01:09:47.979 --> 01:09:50.290 align:middle line:84% And then we have interaction vertices-- 01:09:50.290 --> 01:09:57.040 align:middle line:90% 01:09:57.040 --> 01:09:58.530 align:middle line:90% so minus ig. 01:09:58.530 --> 01:10:02.810 align:middle line:90% 01:10:02.810 --> 01:10:05.840 align:middle line:84% So because here, I have a psi and a psi bar-- 01:10:05.840 --> 01:10:08.480 align:middle line:84% one of the lines coming out, one of the lines coming in. 01:10:08.480 --> 01:10:13.020 align:middle line:90% 01:10:13.020 --> 01:10:15.420 align:middle line:84% So now, let's just consider a scattering process. 01:10:15.420 --> 01:10:22.710 align:middle line:90% 01:10:22.710 --> 01:10:25.897 align:middle line:84% When Yukawa considered this, and the psi corresponding 01:10:25.897 --> 01:10:27.810 align:middle line:90% to say, the proton-- 01:10:27.810 --> 01:10:31.020 align:middle line:84% and then the phi corresponding to pion. 01:10:31.020 --> 01:10:34.020 align:middle line:84% And he used this theory to explain the nuclear force 01:10:34.020 --> 01:10:35.265 align:middle line:90% between the proton. 01:10:35.265 --> 01:10:40.320 align:middle line:90% 01:10:40.320 --> 01:10:42.390 align:middle line:84% So now let's just consider the proc-- yeah, 01:10:42.390 --> 01:10:47.550 align:middle line:84% but you can also consider, say, psi is an electron or phi 01:10:47.550 --> 01:10:49.710 align:middle line:90% is a Higgs field, et cetera. 01:10:49.710 --> 01:10:54.390 align:middle line:84% So now let's just consider the process of this p. 01:10:54.390 --> 01:11:02.510 align:middle line:84% So let's denote the psi particle by p, go to p. 01:11:02.510 --> 01:11:06.560 align:middle line:90% So you have two initial states. 01:11:06.560 --> 01:11:08.870 align:middle line:84% Particles in final state are also particles. 01:11:08.870 --> 01:11:11.780 align:middle line:90% 01:11:11.780 --> 01:11:18.490 align:middle line:84% And now you have the one obvious diagram you can draw, 01:11:18.490 --> 01:11:22.540 align:middle line:84% because now the internal-- so the incoming line 01:11:22.540 --> 01:11:28.570 align:middle line:84% should be-- because the particles should go in, 01:11:28.570 --> 01:11:33.510 align:middle line:84% and then the final state, the particle should come out. 01:11:33.510 --> 01:11:35.700 align:middle line:84% So one possible diagram is like this. 01:11:35.700 --> 01:11:40.920 align:middle line:90% 01:11:40.920 --> 01:11:42.510 align:middle line:90% So suppose this is the p1. 01:11:42.510 --> 01:11:50.070 align:middle line:90% 01:11:50.070 --> 01:11:57.330 align:middle line:84% Polarization s1-- suppose this is p2 s2, and then this is k1. 01:11:57.330 --> 01:12:02.790 align:middle line:84% So let's call it p1 prime s1 prime and p2 prime s2 prime. 01:12:02.790 --> 01:12:07.180 align:middle line:84% So this is one of the diagrams which can do this, 01:12:07.180 --> 01:12:10.690 align:middle line:84% but we can also have this internal line connect 01:12:10.690 --> 01:12:13.220 align:middle line:84% with that line and this line connect that line. 01:12:13.220 --> 01:12:15.265 align:middle line:84% So you can also have a structure like this. 01:12:15.265 --> 01:12:23.530 align:middle line:90% 01:12:23.530 --> 01:12:35.640 align:middle line:84% So we have p1 s1 and p2 s2, then p1 prime s1 prime p2 prime s2 01:12:35.640 --> 01:12:38.430 align:middle line:90% prime. 01:12:38.430 --> 01:12:40.770 align:middle line:84% OK, so these are the two possible diagrams, 01:12:40.770 --> 01:12:41.730 align:middle line:90% at lowest order. 01:12:41.730 --> 01:12:44.240 align:middle line:90% 01:12:44.240 --> 01:12:46.430 align:middle line:84% So now let's write down the expression corresponding 01:12:46.430 --> 01:12:47.435 align:middle line:90% to these two diagrams. 01:12:47.435 --> 01:13:00.240 align:middle line:90% 01:13:00.240 --> 01:13:01.920 align:middle line:84% So the first thing we can write down-- 01:13:01.920 --> 01:13:19.450 align:middle line:90% 01:13:19.450 --> 01:13:22.510 align:middle line:84% so each thing should give you a number in the end. 01:13:22.510 --> 01:13:25.830 align:middle line:90% 01:13:25.830 --> 01:13:36.980 align:middle line:84% So that's why-- but remember, all these are vectors. 01:13:36.980 --> 01:13:39.090 align:middle line:90% All these are spinors. 01:13:39.090 --> 01:13:41.370 align:middle line:84% So in the end, all the spinor indices 01:13:41.370 --> 01:13:43.050 align:middle line:84% should be contracted with each other. 01:13:43.050 --> 01:13:48.070 align:middle line:90% 01:13:48.070 --> 01:13:51.900 align:middle line:84% So that's why when we do this rule, 01:13:51.900 --> 01:13:54.180 align:middle line:84% you always start with u bar and the v bar, 01:13:54.180 --> 01:13:58.770 align:middle line:84% because these are the row vectors. 01:13:58.770 --> 01:14:02.370 align:middle line:84% In the end, you want to contract with column vectors. 01:14:02.370 --> 01:14:06.430 align:middle line:84% So that's why you always start with u bar and the v bar. 01:14:06.430 --> 01:14:11.170 align:middle line:84% So u bar and the v bar-- if you look at this rule here-- 01:14:11.170 --> 01:14:13.530 align:middle line:84% so the u bar either corresponding 01:14:13.530 --> 01:14:17.420 align:middle line:84% to the final particles, v bar corresponding 01:14:17.420 --> 01:14:19.730 align:middle line:90% to initial antiparticles. 01:14:19.730 --> 01:14:25.200 align:middle line:84% In both cases, the arrow is always going toward the point. 01:14:25.200 --> 01:14:27.660 align:middle line:84% If you start from here, you have to go backwards. 01:14:27.660 --> 01:14:31.620 align:middle line:90% 01:14:31.620 --> 01:14:33.640 align:middle line:84% So now let's look at this example. 01:14:33.640 --> 01:14:35.490 align:middle line:84% So we have two fermionic lines here. 01:14:35.490 --> 01:14:39.390 align:middle line:90% 01:14:39.390 --> 01:14:42.500 align:middle line:84% So one is this one and one is this one. 01:14:42.500 --> 01:14:44.030 align:middle line:90% OK, we have two fermionic lines. 01:14:44.030 --> 01:14:47.730 align:middle line:84% So for this expanded line, we should start with the bar. 01:14:47.730 --> 01:14:49.850 align:middle line:84% So this is the outgoing particle, 01:14:49.850 --> 01:14:56.030 align:middle line:84% so that's corresponding to u bar here, s1 prime, 01:14:56.030 --> 01:15:02.500 align:middle line:84% and here corresponding to u bar s2 prime. 01:15:02.500 --> 01:15:04.480 align:middle line:84% And then this is corresponding to-- this 01:15:04.480 --> 01:15:07.540 align:middle line:84% is the initial particle, then this is corresponding to u s1, 01:15:07.540 --> 01:15:11.150 align:middle line:84% and this is corresponding to u s2. 01:15:11.150 --> 01:15:14.100 align:middle line:90% And so we just go backwards. 01:15:14.100 --> 01:15:17.460 align:middle line:84% So now we can just write it down very easily. 01:15:17.460 --> 01:15:21.880 align:middle line:84% So we have minus ig squared corresponding to two vertices, 01:15:21.880 --> 01:15:26.880 align:middle line:84% and for the first fermionic line, we have-- 01:15:26.880 --> 01:15:34.220 align:middle line:84% for the first diagram, we have u s1 prime bar p1 prime and then 01:15:34.220 --> 01:15:36.140 align:middle line:90% u s1 p1. 01:15:36.140 --> 01:15:38.630 align:middle line:84% And so these combine into a sca-- 01:15:38.630 --> 01:15:39.130 align:middle line:90% number. 01:15:39.130 --> 01:15:46.028 align:middle line:90% 01:15:46.028 --> 01:15:47.320 align:middle line:90% And now we have the propagator. 01:15:47.320 --> 01:15:54.160 align:middle line:84% A propagator is just a number minus i p1 prime minus p2 01:15:54.160 --> 01:15:58.380 align:middle line:84% squared plus m squared minus i epsilon. 01:15:58.380 --> 01:16:00.380 align:middle line:84% So in principle, they can have a different mass. 01:16:00.380 --> 01:16:02.290 align:middle line:90% OK, sorry. 01:16:02.290 --> 01:16:05.880 align:middle line:90% For example, here is m tilde. 01:16:05.880 --> 01:16:08.250 align:middle line:84% And then we multiply the other fermionic line. 01:16:08.250 --> 01:16:12.540 align:middle line:90% 01:16:12.540 --> 01:16:23.590 align:middle line:84% So this will be u s2 prime bar p2 prime and then u s2 p2. 01:16:23.590 --> 01:16:26.400 align:middle line:90% 01:16:26.400 --> 01:16:30.720 align:middle line:84% And now, for this diagram, we just follow the same rule. 01:16:30.720 --> 01:16:33.480 align:middle line:84% But now, this one is contracted with that, 01:16:33.480 --> 01:16:35.760 align:middle line:84% and this one is contracted with that-- 01:16:35.760 --> 01:16:40.295 align:middle line:84% so except we still have minus ig squared. 01:16:40.295 --> 01:16:43.020 align:middle line:90% 01:16:43.020 --> 01:16:52.750 align:middle line:84% So now this u s1 prime p1 prime is contracted with u s2 p2. 01:16:52.750 --> 01:16:54.390 align:middle line:90% OK, they are connected. 01:16:54.390 --> 01:16:57.590 align:middle line:84% Now they become a connected line. 01:16:57.590 --> 01:17:01.640 align:middle line:84% And, again, you have these indices, 01:17:01.640 --> 01:17:03.440 align:middle line:84% and now you work out the momentum 01:17:03.440 --> 01:17:07.430 align:middle line:84% to become p2 prime minus p1 squared 01:17:07.430 --> 01:17:10.680 align:middle line:90% plus m squared minus i epsilon. 01:17:10.680 --> 01:17:14.310 align:middle line:84% And then you have an s2 prime p2 prime, 01:17:14.310 --> 01:17:17.270 align:middle line:90% and now you have u s1 p1. 01:17:17.270 --> 01:17:21.490 align:middle line:90% 01:17:21.490 --> 01:17:26.290 align:middle line:84% Except one final thing, there's a relative minus sign 01:17:26.290 --> 01:17:29.050 align:middle line:90% between them. 01:17:29.050 --> 01:17:32.320 align:middle line:84% The reason there's a relative minus sign between them 01:17:32.320 --> 01:17:37.210 align:middle line:84% is because between these two diagrams, it's equivalent. 01:17:37.210 --> 01:17:40.830 align:middle line:84% I exchanged the order of the two external legs. 01:17:40.830 --> 01:17:42.450 align:middle line:84% So remember, if you want to exchange 01:17:42.450 --> 01:17:45.870 align:middle line:84% the order of the fermions, you have to have a minus sign. 01:17:45.870 --> 01:17:49.040 align:middle line:84% And then you have to have a minus sign between the two. 01:17:49.040 --> 01:17:49.540 align:middle line:90% Yes. 01:17:49.540 --> 01:17:52.920 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:17:52.920 --> 01:17:53.625 align:middle line:90% PROFESSOR: Oh. 01:17:53.625 --> 01:17:54.330 align:middle line:90% Yeah, of course. 01:17:54.330 --> 01:17:57.750 align:middle line:84% But we never write them down explicitly. 01:17:57.750 --> 01:18:00.550 align:middle line:84% So always assume momentum conservation. 01:18:00.550 --> 01:18:01.260 align:middle line:90% Yeah. 01:18:01.260 --> 01:18:03.990 align:middle line:84% So for this diagram, the momentum-- 01:18:03.990 --> 01:18:09.390 align:middle line:84% if I draw momentum here, it will be p prime prime minus p1. 01:18:09.390 --> 01:18:13.230 align:middle line:84% So this momentum will be p2 prime minus p1. 01:18:13.230 --> 01:18:17.210 align:middle line:90% 01:18:17.210 --> 01:18:17.990 align:middle line:90% Yes. 01:18:17.990 --> 01:18:24.860 align:middle line:84% AUDIENCE: So it's probably [INAUDIBLE] 01:18:24.860 --> 01:18:26.390 align:middle line:84% PROFESSOR: Minus p1-- that's right. 01:18:26.390 --> 01:18:27.200 align:middle line:90% Yeah, that's right. 01:18:27.200 --> 01:18:33.100 align:middle line:90% 01:18:33.100 --> 01:18:34.470 align:middle line:90% Good? 01:18:34.470 --> 01:18:39.820 align:middle line:84% So just like that, you just follow this rule. 01:18:39.820 --> 01:18:43.770 align:middle line:90% You just start with the line. 01:18:43.770 --> 01:18:46.080 align:middle line:84% Always start with u bar and v bar, 01:18:46.080 --> 01:18:48.510 align:middle line:84% and you just follow the arrow backwards. 01:18:48.510 --> 01:18:54.420 align:middle line:84% And keep track all the gamma matrices and the spinor 01:18:54.420 --> 01:18:56.080 align:middle line:90% indices, et cetera. 01:18:56.080 --> 01:19:00.060 align:middle line:84% And then, eventually, you will just get a number. 01:19:00.060 --> 01:19:06.600 align:middle line:84% You will multiply with a column vector, and then you are done. 01:19:06.600 --> 01:19:10.050 align:middle line:90% So here, we are considering-- 01:19:10.050 --> 01:19:12.300 align:middle line:90% here, the vertex is very simple. 01:19:12.300 --> 01:19:15.100 align:middle line:90% It's just a number. 01:19:15.100 --> 01:19:17.950 align:middle line:84% And later, we will consider a more complicated vertex. 01:19:17.950 --> 01:19:20.217 align:middle line:84% And actually, the vertex can contain matrices. 01:19:20.217 --> 01:19:22.300 align:middle line:84% And so that will be a little bit more complicated. 01:19:22.300 --> 01:19:24.670 align:middle line:90% Also, this is a simple-- 01:19:24.670 --> 01:19:26.050 align:middle line:90% here is a simple example. 01:19:26.050 --> 01:19:28.090 align:middle line:84% We don't have a fermionic propagator. 01:19:28.090 --> 01:19:29.980 align:middle line:84% We only have a bosonic propagator here. 01:19:29.980 --> 01:19:32.960 align:middle line:90% 01:19:32.960 --> 01:19:33.590 align:middle line:90% Yes. 01:19:33.590 --> 01:19:36.690 align:middle line:84% AUDIENCE: How do you know which one's which? 01:19:36.690 --> 01:19:39.030 align:middle line:84% PROFESSOR: Yeah, it doesn't matter 01:19:39.030 --> 01:19:41.010 align:middle line:84% because the overall signs don't matter. 01:19:41.010 --> 01:19:43.050 align:middle line:84% Because in the end, we will square it. 01:19:43.050 --> 01:19:45.450 align:middle line:84% Yeah, so when you calculate the cross section, 01:19:45.450 --> 01:19:48.335 align:middle line:84% you have to take the-- yeah, in order 01:19:48.335 --> 01:19:49.710 align:middle line:84% to calculate the probability, you 01:19:49.710 --> 01:19:53.090 align:middle line:84% have to take the amplitude square. 01:19:53.090 --> 01:19:56.990 align:middle line:84% So overall, signs we don't normally bother to track. 01:19:56.990 --> 01:20:01.510 align:middle line:84% Yeah, you only need to worry about relative signs. 01:20:01.510 --> 01:20:02.470 align:middle line:90% Other questions? 01:20:02.470 --> 01:20:05.850 align:middle line:90% 01:20:05.850 --> 01:20:06.690 align:middle line:90% Yes. 01:20:06.690 --> 01:20:11.760 align:middle line:84% AUDIENCE: Where did the use of Grassmann variables [INAUDIBLE] 01:20:11.760 --> 01:20:12.990 align:middle line:90% PROFESSOR: Yeah, the sign-- 01:20:12.990 --> 01:20:14.880 align:middle line:84% so the relative sign between them. 01:20:14.880 --> 01:20:17.830 align:middle line:90% 01:20:17.830 --> 01:20:19.238 align:middle line:90% That's the only place. 01:20:19.238 --> 01:20:20.905 align:middle line:84% For this example, that's the only place. 01:20:20.905 --> 01:20:23.470 align:middle line:90% 01:20:23.470 --> 01:20:25.030 align:middle line:90% Yeah, so that's the minus sign. 01:20:25.030 --> 01:20:28.900 align:middle line:84% And, of course, the spinor later is 01:20:28.900 --> 01:20:31.510 align:middle line:84% reflected that you always have to keep 01:20:31.510 --> 01:20:34.706 align:middle line:90% track of the spinor indices. 01:20:34.706 --> 01:20:40.240 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:20:40.240 --> 01:20:41.136 align:middle line:90% PROFESSOR: Yeah. 01:20:41.136 --> 01:20:44.330 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:20:44.330 --> 01:20:47.990 align:middle line:90% PROFESSOR: Yeah, indeed. 01:20:47.990 --> 01:20:52.183 align:middle line:84% And then you just miss this minus sign, essentially. 01:20:52.183 --> 01:20:54.600 align:middle line:84% AUDIENCE: Could you have gotten the minus sign if you just 01:20:54.600 --> 01:20:56.368 align:middle line:90% used the operators, right? 01:20:56.368 --> 01:20:57.910 align:middle line:84% PROFESSOR: Yeah, but you have to know 01:20:57.910 --> 01:21:00.550 align:middle line:84% that you have to exchange them to get the minus sign. 01:21:00.550 --> 01:21:01.990 align:middle line:90% You have to have that. 01:21:01.990 --> 01:21:05.170 align:middle line:90% Yeah, if you don't have that-- 01:21:05.170 --> 01:21:07.570 align:middle line:84% I think when Yukawa proposed it, he actually 01:21:07.570 --> 01:21:09.790 align:middle line:90% didn't know all this detail. 01:21:09.790 --> 01:21:11.800 align:middle line:90% He just estimated. 01:21:11.800 --> 01:21:14.080 align:middle line:90% And then he predicted the-- 01:21:14.080 --> 01:21:16.120 align:middle line:84% yeah, he wanted to explain the nuclear force 01:21:16.120 --> 01:21:20.920 align:middle line:84% between the proton, and then he said, oh, maybe the two protons 01:21:20.920 --> 01:21:23.620 align:middle line:84% will exchange all between proton or neutron. 01:21:23.620 --> 01:21:25.810 align:middle line:84% Or maybe they exchange a scalar particle. 01:21:25.810 --> 01:21:28.600 align:middle line:84% And he just postulated the exchange of a scalar particle. 01:21:28.600 --> 01:21:32.410 align:middle line:84% And then from the strength of the nuclear interaction, 01:21:32.410 --> 01:21:36.510 align:middle line:84% he estimated the mass of the particle-- 01:21:36.510 --> 01:21:38.100 align:middle line:90% the scalar particle-- 01:21:38.100 --> 01:21:39.985 align:middle line:90% and then they found it. 01:21:39.985 --> 01:21:40.485 align:middle line:90% Yeah. 01:21:40.485 --> 01:21:42.720 align:middle line:84% [LAUGHS] Yeah, they found such a scalar particle 01:21:42.720 --> 01:21:44.343 align:middle line:90% which was the pion. 01:21:44.343 --> 01:21:46.260 align:middle line:84% And that's how they first discovered the pion. 01:21:46.260 --> 01:21:48.120 align:middle line:90% Yeah. 01:21:48.120 --> 01:21:49.140 align:middle line:90% OK, good. 01:21:49.140 --> 01:21:51.380 align:middle line:90% Let's stop here today. 01:21:51.380 --> 01:21:55.000 align:middle line:90%