WEBVTT 00:00:00.000 --> 00:00:02.435 align:middle line:90% [SQUEAKING] 00:00:02.435 --> 00:00:03.409 align:middle line:90% [RUSTLING] 00:00:03.409 --> 00:00:04.383 align:middle line:90% 00:00:04.383 --> 00:00:07.792 align:middle line:90% [CLICKING] 00:00:07.792 --> 00:00:11.510 align:middle line:90% 00:00:11.510 --> 00:00:14.610 align:middle line:84% PROFESSOR: So at the end of last lecture, 00:00:14.610 --> 00:00:19.450 align:middle line:84% we start talking about this propagator. 00:00:19.450 --> 00:00:29.260 align:middle line:90% 00:00:29.260 --> 00:00:37.030 align:middle line:84% So in non-relativistic quantum mechanics, 00:00:37.030 --> 00:00:45.420 align:middle line:84% in the Heisenberg picture, we can introduce this position 00:00:45.420 --> 00:00:52.080 align:middle line:84% eigenvector corresponding to the eigenvalue of the position 00:00:52.080 --> 00:00:58.050 align:middle line:84% operator at time t with eigenvalue x. 00:00:58.050 --> 00:01:02.130 align:middle line:84% And then from this object, you can construct a propagator. 00:01:02.130 --> 00:01:07.680 align:middle line:90% 00:01:07.680 --> 00:01:09.360 align:middle line:84% So this object is powerful if you 00:01:09.360 --> 00:01:12.180 align:middle line:84% know how to compute this object, because if you 00:01:12.180 --> 00:01:17.310 align:middle line:84% load the function at t prime, and then by-- 00:01:17.310 --> 00:01:21.630 align:middle line:84% yeah, by just integrate it with this object, 00:01:21.630 --> 00:01:25.170 align:middle line:84% and then you can find your function at a later time. 00:01:25.170 --> 00:01:29.610 align:middle line:84% And so this is a powerful object in non-relativistic quantum 00:01:29.610 --> 00:01:31.950 align:middle line:90% mechanics. 00:01:31.950 --> 00:01:36.150 align:middle line:84% So now, we can ask what are the analogous objects-- 00:01:36.150 --> 00:01:43.800 align:middle line:84% say whether they exist analogous object in field theory. 00:01:43.800 --> 00:01:55.710 align:middle line:84% But in QFT, there is no natural way 00:01:55.710 --> 00:02:15.760 align:middle line:84% to define a localized state such as-- 00:02:15.760 --> 00:02:19.750 align:middle line:84% so there's no counterpart of this object. 00:02:19.750 --> 00:02:21.910 align:middle line:84% So one reason is that now there's 00:02:21.910 --> 00:02:23.620 align:middle line:90% no position operator anymore. 00:02:23.620 --> 00:02:27.220 align:middle line:84% So remember, now, x is actually a label-- 00:02:27.220 --> 00:02:31.520 align:middle line:90% no longer a operator. 00:02:31.520 --> 00:02:34.430 align:middle line:84% It's the phi is the operator. x is just a label. 00:02:34.430 --> 00:02:37.490 align:middle line:84% So there's no-- in quantum field theory, 00:02:37.490 --> 00:02:40.840 align:middle line:90% there's no position operator. 00:02:40.840 --> 00:02:44.890 align:middle line:90% 00:02:44.890 --> 00:02:47.800 align:middle line:84% So you cannot-- and then you cannot define-- 00:02:47.800 --> 00:02:50.920 align:middle line:84% yeah, so the position operator, then you can-- 00:02:50.920 --> 00:02:54.010 align:middle line:84% no natural way to define the eigenvector 00:02:54.010 --> 00:02:58.310 align:middle line:90% associated with localized state. 00:02:58.310 --> 00:03:00.205 align:middle line:84% And there's also a fundamental barrier-- 00:03:00.205 --> 00:03:03.560 align:middle line:90% 00:03:03.560 --> 00:03:06.280 align:middle line:84% there's a fundamental contradiction 00:03:06.280 --> 00:03:08.800 align:middle line:90% with Lorentz symmetry. 00:03:08.800 --> 00:03:12.070 align:middle line:84% So there's also a fundamental barrier-- 00:03:12.070 --> 00:03:21.675 align:middle line:84% fundamental obstruction from Lorentz symmetry. 00:03:21.675 --> 00:03:34.750 align:middle line:90% 00:03:34.750 --> 00:03:42.480 align:middle line:90% So you can show that localized-- 00:03:42.480 --> 00:03:56.900 align:middle line:84% you can show a localized state in space 00:03:56.900 --> 00:04:00.350 align:middle line:90% is not Lorentz covariant. 00:04:00.350 --> 00:04:07.370 align:middle line:90% 00:04:07.370 --> 00:04:09.050 align:middle line:90% So this does not-- 00:04:09.050 --> 00:04:16.100 align:middle line:84% so suppose-- yeah, you can do this by proof by contradiction. 00:04:16.100 --> 00:04:28.280 align:middle line:84% Suppose they exist-- there exist states like some states-- 00:04:28.280 --> 00:04:34.260 align:middle line:90% 00:04:34.260 --> 00:04:36.750 align:middle line:90% say, x t. 00:04:36.750 --> 00:04:44.460 align:middle line:84% So by definition, a local state should satisfy in-- 00:04:44.460 --> 00:04:52.025 align:middle line:84% if you evaluate them at the same time, 00:04:52.025 --> 00:04:56.930 align:middle line:84% then that should be proportional to delta function 00:04:56.930 --> 00:04:59.480 align:middle line:84% because that's by definition what it means what 00:04:59.480 --> 00:05:01.190 align:middle line:90% we mean by a localized state-- 00:05:01.190 --> 00:05:03.760 align:middle line:90% localized at the point. 00:05:03.760 --> 00:05:06.020 align:middle line:84% And that's the property satisfied by this. 00:05:06.020 --> 00:05:08.578 align:middle line:84% If you set t prime equal to t, and then this 00:05:08.578 --> 00:05:09.620 align:middle line:90% will be a delta function. 00:05:09.620 --> 00:05:13.540 align:middle line:90% 00:05:13.540 --> 00:05:19.210 align:middle line:84% But now, suppose you can define such a state, 00:05:19.210 --> 00:05:21.640 align:middle line:84% but then this notion cannot be covariant defined. 00:05:21.640 --> 00:05:24.970 align:middle line:84% It's because this is not a covariant object. 00:05:24.970 --> 00:05:27.670 align:middle line:84% So the three-dimensional spatial delta 00:05:27.670 --> 00:05:30.850 align:middle line:84% function that does not transform nicely 00:05:30.850 --> 00:05:34.630 align:middle line:90% under Lorentz transformation. 00:05:34.630 --> 00:05:39.430 align:middle line:84% And so you can show that if you act-- 00:05:39.430 --> 00:05:48.880 align:middle line:84% suppose U lambda is the unitary operator 00:05:48.880 --> 00:05:51.530 align:middle line:84% generating a Lorentz transformation, 00:05:51.530 --> 00:05:54.670 align:middle line:90% which we discussed last time. 00:05:54.670 --> 00:05:58.430 align:middle line:90% If you have the momentum-- 00:05:58.430 --> 00:06:02.780 align:middle line:84% if you have the generated Lorentz transformation. 00:06:02.780 --> 00:06:10.180 align:middle line:84% And then you can show that the U lambda acting on x and t 00:06:10.180 --> 00:06:16.890 align:middle line:84% is not the same as the lambda x and the lambda t. 00:06:16.890 --> 00:06:19.770 align:middle line:84% So lambda x and lambda T denotes a Lorentz transformation 00:06:19.770 --> 00:06:21.750 align:middle line:90% of this vector. 00:06:21.750 --> 00:06:23.670 align:middle line:84% Here, I'm using a shorthand notation. 00:06:23.670 --> 00:06:27.580 align:middle line:90% 00:06:27.580 --> 00:06:30.720 align:middle line:84% So lambda-- x means the Lorentz transformation acts 00:06:30.720 --> 00:06:33.120 align:middle line:84% on x, and here, means the Lorentz transformation 00:06:33.120 --> 00:06:36.200 align:middle line:90% acts on t. 00:06:36.200 --> 00:06:40.430 align:middle line:84% And you actually will do this explicitly in your pset. 00:06:40.430 --> 00:06:43.660 align:middle line:90% You will check this explicitly. 00:06:43.660 --> 00:06:47.030 align:middle line:90% This does not transform. 00:06:47.030 --> 00:06:51.420 align:middle line:84% So this cannot be a covariant concept. 00:06:51.420 --> 00:06:56.100 align:middle line:84% So that means that you can most define in some frame, 00:06:56.100 --> 00:06:59.400 align:middle line:84% but if you change your frame, and then this state 00:06:59.400 --> 00:07:01.350 align:middle line:90% is no longer localized. 00:07:01.350 --> 00:07:07.800 align:middle line:84% So this concept is not the Lorentz-- 00:07:07.800 --> 00:07:11.670 align:middle line:84% it's not compatible with Lorentz symmetry. 00:07:11.670 --> 00:07:21.660 align:middle line:84% So the closest analog you can define in field theory-- 00:07:21.660 --> 00:07:25.010 align:middle line:90% 00:07:25.010 --> 00:07:27.500 align:middle line:90% closest-- sorry, not closed-- 00:07:27.500 --> 00:07:35.230 align:middle line:84% closest analog in QFT in field theory is this object. 00:07:35.230 --> 00:07:41.090 align:middle line:84% So let's define an object x could be defined by-- 00:07:41.090 --> 00:07:43.970 align:middle line:84% so again, whenever I write something like x, 00:07:43.970 --> 00:07:47.300 align:middle line:84% you should view this as a shorthand for the full vector 00:07:47.300 --> 00:07:51.150 align:middle line:90% including both space and time. 00:07:51.150 --> 00:07:55.890 align:middle line:84% And so this second, define an object by this-- 00:07:55.890 --> 00:07:59.580 align:middle line:84% acting-- let's just consider-- suppose we have a real scalar 00:07:59.580 --> 00:08:00.750 align:middle line:90% field theory. 00:08:00.750 --> 00:08:02.960 align:middle line:84% So for example, let's just consider real scalar field 00:08:02.960 --> 00:08:03.460 align:middle line:90% theory. 00:08:03.460 --> 00:08:05.830 align:middle line:84% So we can define an object like this. 00:08:05.830 --> 00:08:10.530 align:middle line:84% So this is the closest we can get into a position eigenstate. 00:08:10.530 --> 00:08:13.410 align:middle line:90% 00:08:13.410 --> 00:08:15.520 align:middle line:90% And then-- yeah, its conjugate. 00:08:15.520 --> 00:08:19.020 align:middle line:90% Then it's given by phi. 00:08:19.020 --> 00:08:20.790 align:middle line:84% So phi is a hermitian, because it's real, 00:08:20.790 --> 00:08:23.440 align:middle line:90% so it's a hermitian operator. 00:08:23.440 --> 00:08:26.790 align:middle line:84% So now, you can check this object by definition 00:08:26.790 --> 00:08:30.120 align:middle line:84% because phi is a scalar field will transforms nicely 00:08:30.120 --> 00:08:32.679 align:middle line:90% on the Lorentz transformation. 00:08:32.679 --> 00:08:37.309 align:middle line:84% And the vacuum is invariant under Lorentz transformation. 00:08:37.309 --> 00:08:40.870 align:middle line:84% And so this object actually transforms nicely. 00:08:40.870 --> 00:08:48.830 align:middle line:84% So under Lorentz transformation, x actually it goes to lambda x. 00:08:48.830 --> 00:08:53.590 align:middle line:84% So this object actually transforms nicely. 00:08:53.590 --> 00:08:56.760 align:middle line:90% 00:08:56.760 --> 00:08:59.730 align:middle line:84% But you can also ask, what is the overlap between these two 00:08:59.730 --> 00:09:03.900 align:middle line:84% such kind of position eigenstate if we call this quoted position 00:09:03.900 --> 00:09:05.520 align:middle line:90% eigenstate? 00:09:05.520 --> 00:09:08.910 align:middle line:84% So now let's consider the overlap 00:09:08.910 --> 00:09:12.210 align:middle line:90% between two such states. 00:09:12.210 --> 00:09:14.940 align:middle line:90% And then this will be given by-- 00:09:14.940 --> 00:09:20.460 align:middle line:84% from this definition will be given by 0 phi x and phi 00:09:20.460 --> 00:09:24.790 align:middle line:90% x prime on there. 00:09:24.790 --> 00:09:29.540 align:middle line:90% 00:09:29.540 --> 00:09:35.540 align:middle line:84% So it's given by the expectation value of x times phi 00:09:35.540 --> 00:09:39.950 align:middle line:84% x prime between the vacuum states 00:09:39.950 --> 00:09:42.440 align:middle line:90% just from the definition. 00:09:42.440 --> 00:09:44.410 align:middle line:84% So the reason we say this is the closest 00:09:44.410 --> 00:09:48.420 align:middle line:84% analog is because even though this state is 00:09:48.420 --> 00:09:52.320 align:middle line:84% Lorentz covariant, but this state is not localized. 00:09:52.320 --> 00:09:55.440 align:middle line:84% So if you want a state to be localized, 00:09:55.440 --> 00:09:59.550 align:middle line:84% this has to be proportional to delta function 00:09:59.550 --> 00:10:01.830 align:middle line:84% when you take the them at the same time. 00:10:01.830 --> 00:10:03.210 align:middle line:90% But now you can show-- 00:10:03.210 --> 00:10:06.870 align:middle line:84% so this is the object we can compute explicitly, 00:10:06.870 --> 00:10:10.650 align:middle line:84% because we already solved the phi completely 00:10:10.650 --> 00:10:12.060 align:middle line:90% in terms of a and a dagger. 00:10:12.060 --> 00:10:15.510 align:middle line:84% You can just plug-in here to compute this object explicitly, 00:10:15.510 --> 00:10:21.210 align:middle line:84% and we will do a towards the end of today's lecture. 00:10:21.210 --> 00:10:23.470 align:middle line:84% Here, let me just tell you the results. 00:10:23.470 --> 00:10:26.400 align:middle line:84% So you can see that this guy actually 00:10:26.400 --> 00:10:32.850 align:middle line:84% is non-zero for t equal to t prime 00:10:32.850 --> 00:10:37.740 align:middle line:90% and x not equal to x prime. 00:10:37.740 --> 00:10:41.160 align:middle line:84% So that means this is not-- this means that this cannot be 00:10:41.160 --> 00:10:42.250 align:middle line:90% a delta function. 00:10:42.250 --> 00:10:45.180 align:middle line:84% So when you evaluate them at the same time, 00:10:45.180 --> 00:10:47.140 align:middle line:84% if it's a localized state, then it should be 00:10:47.140 --> 00:10:49.240 align:middle line:90% proportional to delta function. 00:10:49.240 --> 00:10:51.520 align:middle line:90% But this is not the case. 00:10:51.520 --> 00:10:56.410 align:middle line:84% So this object is not quite localized, 00:10:56.410 --> 00:11:01.000 align:middle line:84% but we will see actually in some sense, it's localized, 00:11:01.000 --> 00:11:04.570 align:middle line:84% and we will see in what sense later when we 00:11:04.570 --> 00:11:08.210 align:middle line:90% compute this object explicitly. 00:11:08.210 --> 00:11:10.580 align:middle line:84% So right now, let's just say the conclusion 00:11:10.580 --> 00:11:13.250 align:middle line:90% that this is not localized. 00:11:13.250 --> 00:11:14.900 align:middle line:90% So this is not-- 00:11:14.900 --> 00:11:18.305 align:middle line:84% so this means this is not a localized state. 00:11:18.305 --> 00:11:37.020 align:middle line:90% 00:11:37.020 --> 00:11:38.630 align:middle line:90% Good. 00:11:38.630 --> 00:11:44.380 align:middle line:90% So heuristically, we can say-- 00:11:44.380 --> 00:11:52.915 align:middle line:84% yeah, so let me just make a side remark here. 00:11:52.915 --> 00:11:58.790 align:middle line:90% 00:11:58.790 --> 00:12:03.880 align:middle line:84% So if we-- so as we say, this is not a position eigenstate. 00:12:03.880 --> 00:12:07.660 align:middle line:84% But if we treat it as some kind of approximation 00:12:07.660 --> 00:12:10.270 align:middle line:84% to the position eigenstate, and then we 00:12:10.270 --> 00:12:16.180 align:middle line:84% can talk about the wave function of a particle. 00:12:16.180 --> 00:12:18.580 align:middle line:90% So now let's suppose-- 00:12:18.580 --> 00:12:31.860 align:middle line:84% so given a single particle state, psi-- 00:12:31.860 --> 00:12:35.700 align:middle line:84% so psi would be say some kind of superposition, 00:12:35.700 --> 00:12:37.190 align:middle line:90% which we wrote last time-- 00:12:37.190 --> 00:12:41.120 align:middle line:84% so some arbitrary superposition of the single particle state 00:12:41.120 --> 00:12:43.590 align:middle line:90% integrate over the momentum. 00:12:43.590 --> 00:12:48.290 align:middle line:84% And then we can define its wave function. 00:12:48.290 --> 00:12:52.420 align:middle line:90% 00:12:52.420 --> 00:12:58.980 align:middle line:84% So I put this function as a quote 00:12:58.980 --> 00:13:07.180 align:middle line:84% as psi x equal to just overlap between the psi 00:13:07.180 --> 00:13:11.950 align:middle line:90% and this x object. 00:13:11.950 --> 00:13:16.600 align:middle line:84% And then this is given by just from that definition-- 00:13:16.600 --> 00:13:23.590 align:middle line:90% just given by 0 phi x psi. 00:13:23.590 --> 00:13:29.990 align:middle line:90% So we can define a function-- 00:13:29.990 --> 00:13:33.630 align:middle line:84% so this is closest analog you can define a wave 00:13:33.630 --> 00:13:36.980 align:middle line:84% function for particle in the field theory 00:13:36.980 --> 00:13:38.360 align:middle line:90% is to use this object. 00:13:38.360 --> 00:13:41.110 align:middle line:90% 00:13:41.110 --> 00:13:43.150 align:middle line:90% And now, you can check-- 00:13:43.150 --> 00:13:47.530 align:middle line:84% so it's a very simple exercise for you to check yourself. 00:13:47.530 --> 00:13:51.370 align:middle line:84% So if I just take this to be the pure momentum state-- 00:13:51.370 --> 00:13:57.850 align:middle line:84% say, for example, if I take psi to be the pure momentum 00:13:57.850 --> 00:13:59.260 align:middle line:90% state of k-- 00:13:59.260 --> 00:14:04.810 align:middle line:84% remember, this notation k is the square root 2 omega 00:14:04.810 --> 00:14:08.950 align:middle line:90% k on this state. 00:14:08.950 --> 00:14:11.320 align:middle line:84% So this is the pure momentum state. 00:14:11.320 --> 00:14:16.540 align:middle line:84% And then if you calculate this object, and then you get-- 00:14:16.540 --> 00:14:23.230 align:middle line:84% then k x then just becomes 0 phi x k. 00:14:23.230 --> 00:14:24.460 align:middle line:90% Now, you can again-- 00:14:24.460 --> 00:14:26.980 align:middle line:84% you know explicitly how phi expressed 00:14:26.980 --> 00:14:29.110 align:middle line:84% in terms of a and a dagger, and you 00:14:29.110 --> 00:14:32.230 align:middle line:84% know how this is obtained from the vacuum 00:14:32.230 --> 00:14:33.610 align:middle line:90% by acting a dagger on it. 00:14:33.610 --> 00:14:35.680 align:middle line:84% You can just compute this object. 00:14:35.680 --> 00:14:38.020 align:middle line:90% So take a couple of minutes. 00:14:38.020 --> 00:14:42.400 align:middle line:84% And you can also just essentially guess the answer. 00:14:42.400 --> 00:14:44.065 align:middle line:84% So this just gives you the plane wave-- 00:14:44.065 --> 00:14:47.470 align:middle line:90% 00:14:47.470 --> 00:14:49.720 align:middle line:84% just gives you an initial plane wave. 00:14:49.720 --> 00:14:54.640 align:middle line:84% So with your energy are given by omega k. 00:14:54.640 --> 00:14:57.080 align:middle line:84% So you just get the plane wave function. 00:14:57.080 --> 00:14:59.287 align:middle line:84% So in this sense, it's like a wave function. 00:14:59.287 --> 00:15:01.120 align:middle line:84% It makes sense to call it the wave function. 00:15:01.120 --> 00:15:06.915 align:middle line:90% 00:15:06.915 --> 00:15:08.550 align:middle line:90% Yes? 00:15:08.550 --> 00:15:12.090 align:middle line:84% STUDENT: So in this state here, we call this the wave function. 00:15:12.090 --> 00:15:13.770 align:middle line:84% If we take it, and you multiply it 00:15:13.770 --> 00:15:16.040 align:middle line:90% by its own complex transpose-- 00:15:16.040 --> 00:15:18.480 align:middle line:84% can we interpret it as a probability? 00:15:18.480 --> 00:15:21.420 align:middle line:84% PROFESSOR: Yeah, this is the comment I'm going to make now. 00:15:21.420 --> 00:15:24.210 align:middle line:90% Yeah, this is a good question. 00:15:24.210 --> 00:15:26.610 align:middle line:90% So we cannot really-- 00:15:26.610 --> 00:15:31.560 align:middle line:90% so we cannot-- so in general-- 00:15:31.560 --> 00:15:49.850 align:middle line:84% so psi x prime, in general, cannot be interpreted as giving 00:15:49.850 --> 00:16:05.445 align:middle line:84% the probability of the particle at x. 00:16:05.445 --> 00:16:12.060 align:middle line:90% 00:16:12.060 --> 00:16:15.270 align:middle line:84% So the reason we cannot do that is because this is not-- 00:16:15.270 --> 00:16:18.995 align:middle line:84% because x is not the genuine position eigenstate. 00:16:18.995 --> 00:16:20.370 align:middle line:84% And also in quantum field theory, 00:16:20.370 --> 00:16:22.730 align:middle line:90% we don't really have this-- 00:16:22.730 --> 00:16:26.450 align:middle line:84% we don't really have this proper sense of talking about-- 00:16:26.450 --> 00:16:30.470 align:middle line:84% you cannot really-- in general, you cannot separate the single 00:16:30.470 --> 00:16:33.230 align:middle line:84% particle with multiple particle exactly, in general, 00:16:33.230 --> 00:16:34.910 align:middle line:90% in quantum field theory. 00:16:34.910 --> 00:16:40.320 align:middle line:84% In free theory, you can because there is no interaction. 00:16:40.320 --> 00:16:44.390 align:middle line:90% But in general, you cannot. 00:16:44.390 --> 00:16:46.280 align:middle line:84% So the reason this is a side remark 00:16:46.280 --> 00:16:49.160 align:middle line:84% is that this is often used by people, 00:16:49.160 --> 00:16:53.390 align:middle line:84% and they just call it a wave function. 00:16:53.390 --> 00:16:54.920 align:middle line:84% But you should keep in mind what's 00:16:54.920 --> 00:16:58.640 align:middle line:84% the content of this object, and it's not 00:16:58.640 --> 00:17:02.510 align:middle line:84% something you can actually rigorously define 00:17:02.510 --> 00:17:06.380 align:middle line:84% both mathematically and physically as a wave function. 00:17:06.380 --> 00:17:11.290 align:middle line:90% 00:17:11.290 --> 00:17:12.405 align:middle line:90% Any questions on this? 00:17:12.405 --> 00:17:14.905 align:middle line:90% 00:17:14.905 --> 00:17:16.280 align:middle line:84% STUDENT: So could you explain how 00:17:16.280 --> 00:17:19.430 align:middle line:90% you got e to the negative i? 00:17:19.430 --> 00:17:21.819 align:middle line:84% PROFESSOR: Oh, you just calculate. 00:17:21.819 --> 00:17:24.609 align:middle line:90% 00:17:24.609 --> 00:17:28.300 align:middle line:84% But it's very easy to see how you get that because remember, 00:17:28.300 --> 00:17:30.070 align:middle line:90% phi x is-- 00:17:30.070 --> 00:17:36.440 align:middle line:84% phi x can be expanded in terms of a u plus a dagger u star. 00:17:36.440 --> 00:17:39.790 align:middle line:84% And the k is obtained by a dagger acting on 0. 00:17:39.790 --> 00:17:44.710 align:middle line:84% Essentially, just a and a dagger part, and give you this object, 00:17:44.710 --> 00:17:48.500 align:middle line:84% and a part proportional to u, and that's how you get it. 00:17:48.500 --> 00:17:52.570 align:middle line:90% 00:17:52.570 --> 00:17:53.070 align:middle line:90% Good? 00:17:53.070 --> 00:17:57.780 align:middle line:90% 00:17:57.780 --> 00:18:02.310 align:middle line:84% So when we compute this object explicitly, 00:18:02.310 --> 00:18:05.700 align:middle line:84% you will get a more precise sense 00:18:05.700 --> 00:18:10.470 align:middle line:84% what I mean that this is an approximation. 00:18:10.470 --> 00:18:12.340 align:middle line:84% You will get a more precise sense. 00:18:12.340 --> 00:18:14.610 align:middle line:90% And so this-- 00:18:14.610 --> 00:18:16.548 align:middle line:84% I'll now defer that a little bit because I 00:18:16.548 --> 00:18:18.215 align:middle line:84% want to talk about something else first. 00:18:18.215 --> 00:18:21.690 align:middle line:90% 00:18:21.690 --> 00:18:22.190 align:middle line:90% Good. 00:18:22.190 --> 00:18:23.030 align:middle line:90% Other questions? 00:18:23.030 --> 00:18:27.400 align:middle line:90% 00:18:27.400 --> 00:18:32.760 align:middle line:84% So here-- so this actually is a very important object. 00:18:32.760 --> 00:18:34.680 align:middle line:90% Forget about this definition. 00:18:34.680 --> 00:18:38.730 align:middle line:84% So this object-- yeah, let me just write this object again. 00:18:38.730 --> 00:18:41.570 align:middle line:90% So this G plus-- 00:18:41.570 --> 00:18:49.010 align:middle line:84% so this object 0 phi x and phi x prime 0-- 00:18:49.010 --> 00:18:58.190 align:middle line:84% so we motivated as this overlap between these two 00:18:58.190 --> 00:19:00.050 align:middle line:90% localized states-- 00:19:00.050 --> 00:19:02.697 align:middle line:84% these two approximate localized states. 00:19:02.697 --> 00:19:04.280 align:middle line:84% But in quantum field theory, actually, 00:19:04.280 --> 00:19:09.380 align:middle line:84% this is a very important object, and actually, we normally 00:19:09.380 --> 00:19:13.760 align:middle line:84% assign a rotation G plus with this object. 00:19:13.760 --> 00:19:16.910 align:middle line:84% And so the application of this object, 00:19:16.910 --> 00:19:19.430 align:middle line:84% we will not talk too much about it. 00:19:19.430 --> 00:19:21.860 align:middle line:90% But in condensed matter-- 00:19:21.860 --> 00:19:24.140 align:middle line:84% for example, say in phi, describes 00:19:24.140 --> 00:19:26.990 align:middle line:84% some approximate-- some continuum description of a spin 00:19:26.990 --> 00:19:27.770 align:middle line:90% system. 00:19:27.770 --> 00:19:30.500 align:middle line:84% And then this would be measure the correlation 00:19:30.500 --> 00:19:33.470 align:middle line:84% between the spins at different locations-- 00:19:33.470 --> 00:19:35.450 align:middle line:84% at different locations and different time. 00:19:35.450 --> 00:19:37.733 align:middle line:84% So this is what people call the correlation functions. 00:19:37.733 --> 00:19:40.400 align:middle line:84% And so in condensed matter, this actually plays a very important 00:19:40.400 --> 00:19:43.520 align:middle line:84% role, and matches the correlation 00:19:43.520 --> 00:19:47.450 align:middle line:84% between different physical objects. 00:19:47.450 --> 00:19:50.825 align:middle line:90% And so here-- so when x-- 00:19:50.825 --> 00:19:54.340 align:middle line:90% 00:19:54.340 --> 00:19:59.920 align:middle line:84% for x and x prime, at non-equal time-- 00:19:59.920 --> 00:20:05.460 align:middle line:90% 00:20:05.460 --> 00:20:11.670 align:middle line:84% in general, phi x and phi x prime do not commute. 00:20:11.670 --> 00:20:16.270 align:middle line:90% 00:20:16.270 --> 00:20:18.460 align:middle line:84% Remember our commutation relation-- 00:20:18.460 --> 00:20:20.800 align:middle line:84% canonical commutation relation is phi-- 00:20:20.800 --> 00:20:23.050 align:middle line:84% phi x and phi x prime, they commute 00:20:23.050 --> 00:20:25.300 align:middle line:90% when they at equal time. 00:20:25.300 --> 00:20:29.620 align:middle line:84% When they are not equal time, in general, 00:20:29.620 --> 00:20:32.320 align:middle line:90% they don't have to commute. 00:20:32.320 --> 00:20:35.740 align:middle line:84% So the ordering here is actually important. 00:20:35.740 --> 00:20:45.940 align:middle line:84% So the ordering of phi in G plus is actually important. 00:20:45.940 --> 00:20:52.120 align:middle line:90% 00:20:52.120 --> 00:20:56.410 align:middle line:90% So if I define some-- 00:20:56.410 --> 00:21:00.380 align:middle line:84% I can consider a lot of objects with a different ordering. 00:21:00.380 --> 00:21:03.550 align:middle line:84% Say if I put the phi x prime first, 00:21:03.550 --> 00:21:07.720 align:middle line:84% I can also consider this object for the phi x prime first. 00:21:07.720 --> 00:21:10.720 align:middle line:90% 00:21:10.720 --> 00:21:12.460 align:middle line:90% So this is normally called-- 00:21:12.460 --> 00:21:15.830 align:middle line:84% now, this is a different function in general 00:21:15.830 --> 00:21:18.860 align:middle line:84% because they don't commute and we call it G minus. 00:21:18.860 --> 00:21:25.090 align:middle line:90% 00:21:25.090 --> 00:21:29.570 align:middle line:84% So this is also appropriately just x prime x. 00:21:29.570 --> 00:21:34.300 align:middle line:84% So this is the x prime x and this is 00:21:34.300 --> 00:21:39.950 align:middle line:90% using that notation is x prime. 00:21:39.950 --> 00:21:43.865 align:middle line:84% And so in general, G plus and G minus, they are not the same. 00:21:43.865 --> 00:21:46.520 align:middle line:90% 00:21:46.520 --> 00:21:51.940 align:middle line:84% So you can also define some other functions-- 00:21:51.940 --> 00:21:57.520 align:middle line:84% say since the phi and phi x and phi x prime don't commute, 00:21:57.520 --> 00:21:58.720 align:middle line:90% then you say-- 00:21:58.720 --> 00:22:01.810 align:middle line:84% then we can consider more general orderings. 00:22:01.810 --> 00:22:04.840 align:middle line:84% We can consider the superposition of these two. 00:22:04.840 --> 00:22:13.050 align:middle line:84% So you can consider the retarded function so-called, 00:22:13.050 --> 00:22:18.120 align:middle line:84% which is defined to be the theta t minus t prime-- 00:22:18.120 --> 00:22:21.110 align:middle line:90% 00:22:21.110 --> 00:22:25.610 align:middle line:84% the commutator of phi x and phi x prime. 00:22:25.610 --> 00:22:28.670 align:middle line:90% 00:22:28.670 --> 00:22:33.040 align:middle line:84% So this is some other object you can define, 00:22:33.040 --> 00:22:36.220 align:middle line:84% which is the difference between the two, 00:22:36.220 --> 00:22:41.320 align:middle line:84% but it only takes value when t is greater than t prime. 00:22:41.320 --> 00:22:43.930 align:middle line:84% So when t is smaller than t prime, then it's 0. 00:22:43.930 --> 00:22:48.680 align:middle line:84% So this is another other object we often use. 00:22:48.680 --> 00:22:53.660 align:middle line:84% And so this often is denoted by theta t minus t prime, 00:22:53.660 --> 00:22:57.180 align:middle line:84% and call this object-- this commutator to be delta t, 00:22:57.180 --> 00:23:00.420 align:middle line:90% t prime. 00:23:00.420 --> 00:23:03.600 align:middle line:84% So sometimes, people also use the following object-- 00:23:03.600 --> 00:23:10.440 align:middle line:84% the G A equal to minus t, t prime minus t. 00:23:10.440 --> 00:23:13.590 align:middle line:84% So only it's non-zero when t prime is 00:23:13.590 --> 00:23:16.170 align:middle line:90% greater than t and the delta. 00:23:16.170 --> 00:23:16.850 align:middle line:90% Oh, sorry. 00:23:16.850 --> 00:23:19.310 align:middle line:90% Here, it should be x, x prime. 00:23:19.310 --> 00:23:25.440 align:middle line:90% 00:23:25.440 --> 00:23:28.590 align:middle line:84% And finally, you can define an object 00:23:28.590 --> 00:23:31.920 align:middle line:90% called G F which is half-- 00:23:31.920 --> 00:23:33.390 align:middle line:90% so this is called the retarded. 00:23:33.390 --> 00:23:35.670 align:middle line:84% This is called advanced, which is 00:23:35.670 --> 00:23:38.760 align:middle line:84% like half of the retarded and half advanced. 00:23:38.760 --> 00:23:44.450 align:middle line:84% So when t greater than t prime, you take the G plus. 00:23:44.450 --> 00:23:47.690 align:middle line:90% And when t is greater than-- 00:23:47.690 --> 00:23:50.390 align:middle line:90% t prime greater than t-- 00:23:50.390 --> 00:23:51.650 align:middle line:90% let me do it here-- 00:23:51.650 --> 00:23:58.810 align:middle line:90% 00:23:58.810 --> 00:24:00.190 align:middle line:90% you do G minus. 00:24:00.190 --> 00:24:03.700 align:middle line:90% 00:24:03.700 --> 00:24:06.670 align:middle line:84% Again, this G F is a function of x and x prime. 00:24:06.670 --> 00:24:09.850 align:middle line:90% 00:24:09.850 --> 00:24:12.340 align:middle line:84% So because they don't commute, so these are various objects 00:24:12.340 --> 00:24:14.140 align:middle line:90% you can define. 00:24:14.140 --> 00:24:15.940 align:middle line:90% So one second. 00:24:15.940 --> 00:24:20.680 align:middle line:84% So at the moment, they may seem not very intuitive to you-- 00:24:20.680 --> 00:24:23.650 align:middle line:84% why should we be worried about those objects. 00:24:23.650 --> 00:24:27.910 align:middle line:84% And later, you will see actually some of those objects 00:24:27.910 --> 00:24:33.190 align:middle line:84% play a very important role, and some of these objects 00:24:33.190 --> 00:24:36.960 align:middle line:90% will play a very important role. 00:24:36.960 --> 00:24:39.060 align:middle line:84% Yeah, for now, they're just some definitions which 00:24:39.060 --> 00:24:40.840 align:middle line:90% we will use potentially later. 00:24:40.840 --> 00:24:41.340 align:middle line:90% Yes? 00:24:41.340 --> 00:24:47.610 align:middle line:84% STUDENT: So considering spins on a lattice, would x and x 00:24:47.610 --> 00:24:52.000 align:middle line:84% prime be the positions of the spins on the lattice? 00:24:52.000 --> 00:24:56.940 align:middle line:84% So why would it physically differ one I define which-- 00:24:56.940 --> 00:25:00.430 align:middle line:84% it seems arbitrary to switch them right? 00:25:00.430 --> 00:25:03.320 align:middle line:84% PROFESSOR: Yeah, if you do them at equal time, 00:25:03.320 --> 00:25:05.260 align:middle line:90% then it doesn't matter. 00:25:05.260 --> 00:25:07.030 align:middle line:84% But remember, x and x prime, they 00:25:07.030 --> 00:25:11.230 align:middle line:84% don't have the same time, and if they're not at the same time-- 00:25:11.230 --> 00:25:13.690 align:middle line:84% and then actually, sometimes, the ordering matters. 00:25:13.690 --> 00:25:16.930 align:middle line:84% Yeah, whether you do this measurement first, and then 00:25:16.930 --> 00:25:19.150 align:middle line:84% you wait a while to do that measurement, 00:25:19.150 --> 00:25:22.060 align:middle line:84% or you do this measurement first and then-- yeah, 00:25:22.060 --> 00:25:25.230 align:middle line:90% then that can be different. 00:25:25.230 --> 00:25:26.175 align:middle line:90% Yeah? 00:25:26.175 --> 00:25:30.110 align:middle line:84% STUDENT: So G minus is x' inner product with x? 00:25:30.110 --> 00:25:31.922 align:middle line:90% PROFESSOR: Oh, yeah. 00:25:31.922 --> 00:25:34.695 align:middle line:84% STUDENT: G plus just uh, wouldn't it just be-- 00:25:34.695 --> 00:25:36.320 align:middle line:84% PROFESSOR: Yeah, just they're switched. 00:25:36.320 --> 00:25:40.140 align:middle line:84% Yeah, remember this order is important. 00:25:40.140 --> 00:25:46.690 align:middle line:84% So here, I have x first and x prime second. 00:25:46.690 --> 00:25:50.260 align:middle line:90% So this order is different-- 00:25:50.260 --> 00:25:51.190 align:middle line:90% is important. 00:25:51.190 --> 00:25:52.957 align:middle line:84% STUDENT: But is G minus then just the complex congugate 00:25:52.957 --> 00:25:53.400 align:middle line:90% of G plus? 00:25:53.400 --> 00:25:55.692 align:middle line:84% PROFESSOR: Yeah, it is the complex congugate of G plus. 00:25:55.692 --> 00:25:59.703 align:middle line:90% 00:25:59.703 --> 00:26:00.370 align:middle line:90% Other questions? 00:26:00.370 --> 00:26:00.940 align:middle line:90% Yeah? 00:26:00.940 --> 00:26:03.148 align:middle line:84% STUDENT: Yeah, so like the retarded and advanced ones 00:26:03.148 --> 00:26:08.440 align:middle line:84% aren't like Lorentz covariant because of the theta function? 00:26:08.440 --> 00:26:11.980 align:middle line:84% PROFESSOR: Yeah, we will talk about that. 00:26:11.980 --> 00:26:17.400 align:middle line:90% Yeah, they are actually. 00:26:17.400 --> 00:26:18.289 align:middle line:90% Yes? 00:26:18.289 --> 00:26:20.247 align:middle line:84% STUDENT: Can you go back and explain more about 00:26:20.247 --> 00:26:22.970 align:middle line:84% why the non-equal time non-equal position 00:26:22.970 --> 00:26:25.590 align:middle line:90% operators don't commute? 00:26:25.590 --> 00:26:30.070 align:middle line:84% PROFESSOR: Yeah, it just say our canonical-- 00:26:30.070 --> 00:26:32.110 align:middle line:84% yeah, our canonical commutation relation 00:26:32.110 --> 00:26:36.660 align:middle line:84% only requires them to commute at the equal time. 00:26:36.660 --> 00:26:39.960 align:middle line:84% And when they are not equal time, and each of them 00:26:39.960 --> 00:26:42.090 align:middle line:90% will evolve. 00:26:42.090 --> 00:26:48.060 align:middle line:84% And remember, the phi itself is-- 00:26:48.060 --> 00:26:50.080 align:middle line:90% phi itself depends on time. 00:26:50.080 --> 00:26:53.130 align:middle line:90% Yeah, so let me just-- 00:26:53.130 --> 00:26:54.472 align:middle line:90% if I just write the-- 00:26:54.472 --> 00:26:57.360 align:middle line:90% 00:26:57.360 --> 00:26:59.160 align:middle line:90% yeah, let me just write the-- 00:26:59.160 --> 00:27:10.230 align:middle line:84% so this is the k, say a k u k plus a k dagger u k dagger u k 00:27:10.230 --> 00:27:12.570 align:middle line:90% star. 00:27:12.570 --> 00:27:15.270 align:middle line:84% So this is my expression for phi. 00:27:15.270 --> 00:27:17.900 align:middle line:90% 00:27:17.900 --> 00:27:19.970 align:middle line:84% So canonical commutation relation 00:27:19.970 --> 00:27:24.620 align:middle line:84% only requires that they are equal when t equal to t prime. 00:27:24.620 --> 00:27:26.600 align:middle line:84% But when t is not equal to t prime, 00:27:26.600 --> 00:27:28.160 align:middle line:84% I just have some general expression. 00:27:28.160 --> 00:27:31.980 align:middle line:84% I do the commutator-- does not guarantee you will get 0. 00:27:31.980 --> 00:27:33.688 align:middle line:90% Yeah, you just get-- 00:27:33.688 --> 00:27:34.980 align:middle line:90% yeah, you have different terms. 00:27:34.980 --> 00:27:38.540 align:middle line:90% They may not cancel each other. 00:27:38.540 --> 00:27:42.140 align:middle line:84% So at t equal to 0, we guarantee that they cancel each other, 00:27:42.140 --> 00:27:44.647 align:middle line:84% but when t is not equal to 0, then not 00:27:44.647 --> 00:27:45.980 align:middle line:90% guaranteed to cancel each other. 00:27:45.980 --> 00:27:49.280 align:middle line:84% But we will figure out when they cancel, will not cancel. 00:27:49.280 --> 00:27:52.070 align:middle line:84% Yeah, we will talk about that more explicitly. 00:27:52.070 --> 00:27:53.117 align:middle line:90% Yes? 00:27:53.117 --> 00:27:55.367 align:middle line:84% STUDENT: Is it correct that if the point are spacelike 00:27:55.367 --> 00:27:58.510 align:middle line:84% separated then we can rotate so that t -- 00:27:58.510 --> 00:28:01.155 align:middle line:90% to a frame where t --? 00:28:01.155 --> 00:28:02.780 align:middle line:84% PROFESSOR: Yeah, we will talk about it. 00:28:02.780 --> 00:28:03.280 align:middle line:90% Yeah-- 00:28:03.280 --> 00:28:04.750 align:middle line:90% STUDENT: [INAUDIBLE] 00:28:04.750 --> 00:28:08.130 align:middle line:84% PROFESSOR: Yeah, we will talk about that. 00:28:08.130 --> 00:28:09.900 align:middle line:90% Good? 00:28:09.900 --> 00:28:13.770 align:middle line:84% So let me just emphasize again, this is something in principle 00:28:13.770 --> 00:28:15.180 align:middle line:90% you can define. 00:28:15.180 --> 00:28:19.800 align:middle line:84% And it's phys-- some of its meaning will be clear later. 00:28:19.800 --> 00:28:24.120 align:middle line:84% And in the next pset-- not in this pset-- in the next p set, 00:28:24.120 --> 00:28:27.480 align:middle line:84% you will see an example, which this thing actually 00:28:27.480 --> 00:28:28.900 align:middle line:90% play a very important role. 00:28:28.900 --> 00:28:32.700 align:middle line:84% So this is the analog of the retarded function in E and M? 00:28:32.700 --> 00:28:35.370 align:middle line:84% Do you still remember retarded function E and M? 00:28:35.370 --> 00:28:38.370 align:middle line:84% And yeah, so this is quantum version of this retarded 00:28:38.370 --> 00:28:43.080 align:middle line:84% function in E and M. And you will see application 00:28:43.080 --> 00:28:45.810 align:middle line:90% of this in the pset four. 00:28:45.810 --> 00:28:48.930 align:middle line:84% And later, we will use this thing all the time. 00:28:48.930 --> 00:28:51.630 align:middle line:84% So this G F, we will use it all the time 00:28:51.630 --> 00:28:54.960 align:middle line:84% when we consider interactions, and so that's why 00:28:54.960 --> 00:28:57.300 align:middle line:90% we want to introduce it here. 00:28:57.300 --> 00:29:02.190 align:middle line:84% And this G plus G minus are called the Wightman functions. 00:29:02.190 --> 00:29:07.558 align:middle line:90% 00:29:07.558 --> 00:29:09.350 align:middle line:84% So sometimes, called the Wightman function. 00:29:09.350 --> 00:29:11.790 align:middle line:84% The G F is called the Feynman function. 00:29:11.790 --> 00:29:15.210 align:middle line:84% So Feynman was the first person who introduced it. 00:29:15.210 --> 00:29:20.200 align:middle line:90% 00:29:20.200 --> 00:29:25.870 align:middle line:84% And yeah, so G plus G minus was introduced by other people, 00:29:25.870 --> 00:29:30.510 align:middle line:84% but Wightman made many important use of it 00:29:30.510 --> 00:29:32.260 align:middle line:84% in the early days of quantum field theory, 00:29:32.260 --> 00:29:33.802 align:middle line:84% so it's called the Wightman function. 00:29:33.802 --> 00:29:37.460 align:middle line:90% 00:29:37.460 --> 00:29:39.950 align:middle line:84% So now, let's just study a little bit the properties 00:29:39.950 --> 00:29:41.360 align:middle line:90% of those functions. 00:29:41.360 --> 00:29:46.820 align:middle line:84% So you can also think these are essentially the simplest 00:29:46.820 --> 00:29:49.130 align:middle line:90% observables of the system. 00:29:49.130 --> 00:29:51.410 align:middle line:84% And in particular, if you think of them as correlation 00:29:51.410 --> 00:29:53.660 align:middle line:84% functions, and then that just tells you 00:29:53.660 --> 00:29:56.270 align:middle line:84% the correlation between the phi at a different point. 00:29:56.270 --> 00:30:00.770 align:middle line:84% So this is in some sense, also the simplest observables 00:30:00.770 --> 00:30:03.580 align:middle line:90% in your theory. 00:30:03.580 --> 00:30:04.080 align:middle line:90% Good? 00:30:04.080 --> 00:30:07.390 align:middle line:90% Do you have other questions? 00:30:07.390 --> 00:30:07.960 align:middle line:90% Yes? 00:30:07.960 --> 00:30:09.418 align:middle line:84% STUDENT: So this is the correlation 00:30:09.418 --> 00:30:12.570 align:middle line:90% for the vacuum state, right? 00:30:12.570 --> 00:30:15.276 align:middle line:84% PROFESSOR: Yeah, this is the correlation vacuum state. 00:30:15.276 --> 00:30:19.350 align:middle line:84% STUDENT: So you have a different state? 00:30:19.350 --> 00:30:21.850 align:middle line:84% PROFESSOR: Yeah, if you're not in a vacuum state, of course, 00:30:21.850 --> 00:30:23.070 align:middle line:90% you get a different function. 00:30:23.070 --> 00:30:27.460 align:middle line:84% STUDENT: So is there a way to measure for other states? 00:30:27.460 --> 00:30:29.320 align:middle line:84% PROFESSOR: Yeah, you can compute. 00:30:29.320 --> 00:30:33.220 align:middle line:84% So in this theory, we can compute in any state. 00:30:33.220 --> 00:30:37.345 align:middle line:84% But just we often use -- we probe in the vacuum state. 00:30:37.345 --> 00:30:40.970 align:middle line:90% 00:30:40.970 --> 00:30:42.550 align:middle line:90% Other questions? 00:30:42.550 --> 00:30:44.140 align:middle line:84% Yeah, for example, in the lab, if you 00:30:44.140 --> 00:30:46.390 align:middle line:84% can prepare in some whatever state 00:30:46.390 --> 00:30:54.060 align:middle line:84% you prepare, you calculate this object in that state, 00:30:54.060 --> 00:30:55.800 align:middle line:84% but the vacuum is the simplest one. 00:30:55.800 --> 00:30:56.620 align:middle line:90% It's the simplest. 00:30:56.620 --> 00:30:57.120 align:middle line:90% Yes? 00:30:57.120 --> 00:31:00.096 align:middle line:84% STUDENT: [INAUDIBLE] we can, we can 00:31:00.096 --> 00:31:02.576 align:middle line:84% define a useful localized state in space 00:31:02.576 --> 00:31:05.056 align:middle line:84% and [INAUDIBLE] other [? state ?] defined 00:31:05.056 --> 00:31:14.000 align:middle line:84% is also [INAUDIBLE],, spacetime is useful [INAUDIBLE]?? 00:31:14.000 --> 00:31:17.750 align:middle line:84% PROFESSOR: No, it's not quite in spacetime. 00:31:17.750 --> 00:31:20.230 align:middle line:84% We will see in what sense it's approximate to the local 00:31:20.230 --> 00:31:20.730 align:middle line:90% in space. 00:31:20.730 --> 00:31:27.050 align:middle line:90% 00:31:27.050 --> 00:31:27.550 align:middle line:90% Good. 00:31:27.550 --> 00:31:31.950 align:middle line:84% Now, let's just discuss the property of those objects. 00:31:31.950 --> 00:31:49.610 align:middle line:84% And so the first thing is that the-- 00:31:49.610 --> 00:31:52.145 align:middle line:84% remember the phi satisfy the following equation. 00:31:52.145 --> 00:31:54.950 align:middle line:90% 00:31:54.950 --> 00:31:59.310 align:middle line:84% So this is the operating equation satisfy phi. 00:31:59.310 --> 00:32:06.330 align:middle line:90% And so if you act this on the-- 00:32:06.330 --> 00:32:07.860 align:middle line:90% so suppose this is-- 00:32:07.860 --> 00:32:09.870 align:middle line:84% so here, you have two arg-- here, 00:32:09.870 --> 00:32:12.850 align:middle line:84% you have two arguments, x and x prime. 00:32:12.850 --> 00:32:14.730 align:middle line:84% So here, when we write differential operator, 00:32:14.730 --> 00:32:16.080 align:middle line:90% we always refer to x. 00:32:16.080 --> 00:32:19.050 align:middle line:90% It's always derivative of x. 00:32:19.050 --> 00:32:22.070 align:middle line:84% So we must satisfy this equation. 00:32:22.070 --> 00:32:25.930 align:middle line:84% So if you act this on the G plus and the G minus, 00:32:25.930 --> 00:32:27.670 align:middle line:90% you immediately get 0. 00:32:27.670 --> 00:32:30.820 align:middle line:84% So you find-- because of [INAUDIBLE] equation, 00:32:30.820 --> 00:32:35.980 align:middle line:84% then G plus/minus are just x and x prime immediately 00:32:35.980 --> 00:32:40.030 align:middle line:84% gives you 0 just because phi satisfies this equation. 00:32:40.030 --> 00:32:43.420 align:middle line:84% Oh, sorry, it should be a minus sign, I think. 00:32:43.420 --> 00:32:45.250 align:middle line:90% And similarly, with this-- 00:32:45.250 --> 00:32:48.218 align:middle line:84% similarly with delta-- so the delta is 00:32:48.218 --> 00:32:49.510 align:middle line:90% the difference between the two. 00:32:49.510 --> 00:32:50.680 align:middle line:90% It's the commutator. 00:32:50.680 --> 00:32:54.595 align:middle line:84% So the delta is essentially the difference between G plus and G 00:32:54.595 --> 00:32:57.160 align:middle line:84% minus, so the same thing with delta. 00:32:57.160 --> 00:33:15.730 align:middle line:90% 00:33:15.730 --> 00:33:17.590 align:middle line:84% So this is-- the first property is 00:33:17.590 --> 00:33:21.070 align:middle line:84% that they satisfy the same equation 00:33:21.070 --> 00:33:24.790 align:middle line:84% as the classical equations of phi. 00:33:24.790 --> 00:33:26.680 align:middle line:84% So the G plus and delta, they satisfy. 00:33:26.680 --> 00:33:29.260 align:middle line:90% 00:33:29.260 --> 00:33:30.310 align:middle line:90% And we will talk-- 00:33:30.310 --> 00:34:07.320 align:middle line:84% and then you can now also look at the G F and the G R. 00:34:07.320 --> 00:34:14.540 align:middle line:84% So now, if you act this object on G R, G A, or G F, 00:34:14.540 --> 00:34:18.150 align:middle line:84% the story is a little bit tricky because the partial square 00:34:18.150 --> 00:34:20.040 align:middle line:90% containing time derivatives. 00:34:20.040 --> 00:34:22.350 align:middle line:84% And when you add the time derivatives-- and time 00:34:22.350 --> 00:34:25.415 align:middle line:84% derivatives do act on those theta functions. 00:34:25.415 --> 00:34:27.540 align:middle line:84% And when you add time derivative on theta function, 00:34:27.540 --> 00:34:30.250 align:middle line:84% you will get some delta functions, et cetera. 00:34:30.250 --> 00:34:34.310 align:middle line:84% And so the story is a little bit more intricate. 00:34:34.310 --> 00:34:38.219 align:middle line:84% But if you just carry it through-- 00:34:38.219 --> 00:34:40.940 align:middle line:84% it just carries through-- then you can show. 00:34:40.940 --> 00:34:43.489 align:middle line:84% And so this is a simple exercise you 00:34:43.489 --> 00:34:45.020 align:middle line:90% should try to show yourself-- 00:34:45.020 --> 00:34:46.699 align:middle line:90% we'll not do it here-- 00:34:46.699 --> 00:34:55.920 align:middle line:84% that you find, you act this on the G R A, 00:34:55.920 --> 00:34:59.700 align:middle line:84% and F. You get actually the-- any of them, 00:34:59.700 --> 00:35:01.800 align:middle line:90% you get the same equation. 00:35:01.800 --> 00:35:04.710 align:middle line:84% You get the right-hand side becomes-- 00:35:04.710 --> 00:35:08.770 align:middle line:84% it's non-zero-- actually gives you a delta function. 00:35:08.770 --> 00:35:10.930 align:middle line:84% So this is a four-dimensional delta function. 00:35:10.930 --> 00:35:13.155 align:middle line:84% So in both in spatial location and in time. 00:35:13.155 --> 00:35:19.520 align:middle line:90% 00:35:19.520 --> 00:35:21.340 align:middle line:84% So heuristically, you can understand 00:35:21.340 --> 00:35:24.530 align:middle line:84% that this delta function in time comes 00:35:24.530 --> 00:35:31.670 align:middle line:84% from taking derivative on theta functions 00:35:31.670 --> 00:35:40.330 align:middle line:84% and then the derivative in the spatial direction-- 00:35:40.330 --> 00:35:43.330 align:middle line:84% oh, delta function in the spatial direction 00:35:43.330 --> 00:35:46.190 align:middle line:84% can come out in the following way. 00:35:46.190 --> 00:35:48.590 align:middle line:90% So here, you have two-- 00:35:48.590 --> 00:35:50.360 align:middle line:84% so here, you have two time derivatives 00:35:50.360 --> 00:35:51.890 align:middle line:90% right in the partial square. 00:35:51.890 --> 00:35:56.000 align:middle line:84% So imagine you have one time derivative acting on the theta 00:35:56.000 --> 00:35:58.910 align:middle line:84% function, and then give you a delta function in time. 00:35:58.910 --> 00:36:03.860 align:middle line:84% And then if you have one time derivative acting on this phi, 00:36:03.860 --> 00:36:06.510 align:middle line:90% and then taking it into pi. 00:36:06.510 --> 00:36:09.300 align:middle line:84% And then equal time commutator then 00:36:09.300 --> 00:36:13.110 align:middle line:84% give you a delta function in the spatial direction, 00:36:13.110 --> 00:36:16.370 align:middle line:90% and so that's how you get that. 00:36:16.370 --> 00:36:19.700 align:middle line:84% Yeah, but it's easy to check yourself. 00:36:19.700 --> 00:36:21.650 align:middle line:84% So this is the first property-- is that they 00:36:21.650 --> 00:36:23.370 align:middle line:90% satisfy those nice equations. 00:36:23.370 --> 00:36:26.170 align:middle line:90% 00:36:26.170 --> 00:36:32.560 align:middle line:84% The second property is that phi x and phi x prime, 00:36:32.560 --> 00:36:35.110 align:middle line:84% even though in general they don't commute, 00:36:35.110 --> 00:36:46.110 align:middle line:84% but they actually commute for space-like separations. 00:36:46.110 --> 00:36:51.420 align:middle line:90% 00:36:51.420 --> 00:36:55.190 align:middle line:84% So that means that if you look at the commutator of phi 00:36:55.190 --> 00:36:59.040 align:middle line:84% x and phi x prime, this is actually 00:36:59.040 --> 00:37:06.220 align:middle line:84% equal to 0 for x minus x prime greater than square greater 00:37:06.220 --> 00:37:10.170 align:middle line:84% than 0 for space-like separations. 00:37:10.170 --> 00:37:14.160 align:middle line:90% 00:37:14.160 --> 00:37:17.680 align:middle line:84% So this can be checked by expression computation. 00:37:17.680 --> 00:37:20.130 align:middle line:84% So we know the mode expression of phi. 00:37:20.130 --> 00:37:22.370 align:middle line:90% You can just do it commutator-- 00:37:22.370 --> 00:37:25.140 align:middle line:90% yeah, you can just check it. 00:37:25.140 --> 00:37:26.910 align:middle line:90% But you can also-- 00:37:26.910 --> 00:37:29.640 align:middle line:84% but there's also a simple argument to make it. 00:37:29.640 --> 00:37:33.200 align:middle line:90% 00:37:33.200 --> 00:37:39.590 align:middle line:84% So you can do-- you can check by expression computation. 00:37:39.590 --> 00:37:46.790 align:middle line:90% 00:37:46.790 --> 00:37:48.695 align:middle line:84% So here, I give you alternative argument. 00:37:48.695 --> 00:37:56.310 align:middle line:90% 00:37:56.310 --> 00:37:59.640 align:middle line:84% Here, I give you an alternative derivation. 00:37:59.640 --> 00:38:05.650 align:middle line:84% So yeah, it's very simple, but let me just divide it 00:38:05.650 --> 00:38:07.615 align:middle line:90% into three small steps. 00:38:07.615 --> 00:38:10.480 align:middle line:90% 00:38:10.480 --> 00:38:13.910 align:middle line:90% So essentially, by definition-- 00:38:13.910 --> 00:38:15.490 align:middle line:84% so let's just consider-- yeah, we 00:38:15.490 --> 00:38:19.840 align:middle line:84% consider the commutator, which here we call to be delta. 00:38:19.840 --> 00:38:22.850 align:middle line:90% So the delta x plus/minus-- 00:38:22.850 --> 00:38:25.570 align:middle line:84% oh, x and x prime, which is the commutator. 00:38:25.570 --> 00:38:28.210 align:middle line:90% 00:38:28.210 --> 00:38:30.010 align:middle line:90% Yeah, let me define-- 00:38:30.010 --> 00:38:34.090 align:middle line:84% yeah, sorry-- yeah, so delta here 00:38:34.090 --> 00:38:39.550 align:middle line:84% is the expectation value of 0, so let me call this delta hat. 00:38:39.550 --> 00:38:41.980 align:middle line:84% So the delta hat is defined to be the commutator. 00:38:41.980 --> 00:38:49.790 align:middle line:90% 00:38:49.790 --> 00:38:52.490 align:middle line:84% So the first claim is that this is actually a C number. 00:38:52.490 --> 00:38:57.520 align:middle line:90% 00:38:57.520 --> 00:39:03.070 align:middle line:84% So this is C number, and the reason is very simple. 00:39:03.070 --> 00:39:06.960 align:middle line:84% So just some constant-- even if it's non-zero can, at the most, 00:39:06.960 --> 00:39:08.670 align:middle line:90% be a constant. 00:39:08.670 --> 00:39:10.650 align:middle line:84% Be some-- yeah, but C number means 00:39:10.650 --> 00:39:14.490 align:middle line:90% that it's not an operator. 00:39:14.490 --> 00:39:16.530 align:middle line:84% So the reason this is C number is 00:39:16.530 --> 00:39:21.090 align:middle line:84% because if you look at the phi, it's linear in a and a dagger. 00:39:21.090 --> 00:39:25.410 align:middle line:84% If you look at the commutator between phi and phi itself, 00:39:25.410 --> 00:39:28.980 align:middle line:84% you always just get commuted between a and a dagger. 00:39:28.980 --> 00:39:32.490 align:middle line:84% a and a dagger just give you a constant, 00:39:32.490 --> 00:39:37.280 align:middle line:84% and so this just-- you get some C number. 00:39:37.280 --> 00:39:43.160 align:middle line:84% So in fact-- so this delta hat is actually the same as delta 00:39:43.160 --> 00:39:44.760 align:middle line:84% because the, if your C number, you 00:39:44.760 --> 00:39:46.760 align:middle line:84% take this expectation value doesn't do anything. 00:39:46.760 --> 00:39:49.470 align:middle line:90% 00:39:49.470 --> 00:39:56.010 align:middle line:84% So the second step is that now let's consider the Lorentz 00:39:56.010 --> 00:39:57.810 align:middle line:90% transformation-- 00:39:57.810 --> 00:40:02.190 align:middle line:84% the operator, which generates the Lorentz transformation, 00:40:02.190 --> 00:40:04.460 align:middle line:90% which is given by-- 00:40:04.460 --> 00:40:06.810 align:middle line:84% remember, we discussed last time. 00:40:06.810 --> 00:40:11.220 align:middle line:90% 00:40:11.220 --> 00:40:13.520 align:middle line:90% So this is the generator. 00:40:13.520 --> 00:40:16.900 align:middle line:84% So this is the unitary-- so this is operator 00:40:16.900 --> 00:40:19.069 align:middle line:90% of Lorentz transformation. 00:40:19.069 --> 00:40:25.670 align:middle line:90% 00:40:25.670 --> 00:40:29.770 align:middle line:90% Let's look at this object. 00:40:29.770 --> 00:40:31.330 align:middle line:90% And then by definition-- 00:40:31.330 --> 00:40:34.120 align:middle line:90% 00:40:34.120 --> 00:40:40.800 align:middle line:84% so this object acting on phi x dagger 00:40:40.800 --> 00:40:44.250 align:middle line:90% will give you phi prime x. 00:40:44.250 --> 00:40:47.340 align:middle line:84% And phi prime x transforms phi under the Lorentz 00:40:47.340 --> 00:40:48.840 align:middle line:84% transformation, and this is the same 00:40:48.840 --> 00:40:52.600 align:middle line:84% as you just do an inverse Lorentz transform 00:40:52.600 --> 00:40:54.250 align:middle line:90% on the coordinates. 00:40:54.250 --> 00:40:57.420 align:middle line:90% So this is the-- 00:40:57.420 --> 00:40:59.730 align:middle line:90% so by the way, just-- 00:40:59.730 --> 00:41:03.340 align:middle line:84% do you feel comfortable about this equation? 00:41:03.340 --> 00:41:06.080 align:middle line:84% Do you know where this equation comes from? 00:41:06.080 --> 00:41:07.580 align:middle line:84% So you're all comfortable with this? 00:41:07.580 --> 00:41:08.180 align:middle line:90% OK, good. 00:41:08.180 --> 00:41:15.110 align:middle line:90% 00:41:15.110 --> 00:41:19.100 align:middle line:84% So by definition, U acts on phi as this. 00:41:19.100 --> 00:41:21.560 align:middle line:90% So now, let's just act both-- 00:41:21.560 --> 00:41:25.835 align:middle line:84% act U and U lambda dagger on both sides of this equation. 00:41:25.835 --> 00:41:32.800 align:middle line:90% 00:41:32.800 --> 00:41:35.240 align:middle line:90% And then we get-- 00:41:35.240 --> 00:41:37.320 align:middle line:90% so on the right-hand side-- 00:41:37.320 --> 00:41:41.990 align:middle line:84% so this is a unitary operator -- because the M is Hermitian, 00:41:41.990 --> 00:41:43.370 align:middle line:90% this is a unitary operator. 00:41:43.370 --> 00:41:46.100 align:middle line:84% So the right-hand side, you just go back to this C number. 00:41:46.100 --> 00:41:47.900 align:middle line:90% It does not change. 00:41:47.900 --> 00:41:49.670 align:middle line:90% So the equation does not change. 00:41:49.670 --> 00:41:53.250 align:middle line:90% And the left-hand side-- 00:41:53.250 --> 00:41:56.270 align:middle line:90% so the delta is the same. 00:41:56.270 --> 00:41:57.875 align:middle line:90% So you find that the-- 00:41:57.875 --> 00:42:01.100 align:middle line:90% 00:42:01.100 --> 00:42:06.020 align:middle line:84% yeah, so you find the delta x prime, which 00:42:06.020 --> 00:42:07.400 align:middle line:90% is equal to this C number. 00:42:07.400 --> 00:42:14.330 align:middle line:84% Then, it's the same as U Lambda phi x phi x 00:42:14.330 --> 00:42:18.050 align:middle line:90% prime and the U lambda dagger. 00:42:18.050 --> 00:42:22.410 align:middle line:84% And that gives you-- me phi lambda 00:42:22.410 --> 00:42:27.810 align:middle line:84% minus 1 x commutator with phi lambda 1-- minus 1 x prime. 00:42:27.810 --> 00:42:31.140 align:middle line:84% So that's the same as delta lambda 00:42:31.140 --> 00:42:33.240 align:middle line:90% x and the lambda x prime. 00:42:33.240 --> 00:43:09.590 align:middle line:90% 00:43:09.590 --> 00:43:10.970 align:middle line:90% Good? 00:43:10.970 --> 00:43:16.170 align:middle line:84% So now, we can look at the last step. 00:43:16.170 --> 00:43:19.730 align:middle line:84% So you see, when we make a Lorentz transformation 00:43:19.730 --> 00:43:22.580 align:middle line:84% on delta, actually nothing changes, 00:43:22.580 --> 00:43:26.400 align:middle line:84% and the value is the same because the commutator is a C 00:43:26.400 --> 00:43:26.900 align:middle line:90% number. 00:43:26.900 --> 00:43:29.610 align:middle line:90% 00:43:29.610 --> 00:43:45.140 align:middle line:84% And then-- now remember for any space-like separated x and x 00:43:45.140 --> 00:43:56.440 align:middle line:84% prime, we can always choose to find the some lambda-- 00:43:56.440 --> 00:44:03.820 align:middle line:84% some Lorentz transformation-- that the lambda 00:44:03.820 --> 00:44:14.570 align:middle line:84% x and the lambda x prime to be at the equal time. 00:44:14.570 --> 00:44:15.990 align:middle line:90% They have the same time. 00:44:15.990 --> 00:44:17.670 align:middle line:84% We can always make a transformation. 00:44:17.670 --> 00:44:21.270 align:middle line:84% Yeah, and let me call it something else-- t tilde. 00:44:21.270 --> 00:44:24.720 align:middle line:90% So your original x is for-- 00:44:24.720 --> 00:44:28.460 align:middle line:84% you can always transform it to equal time. 00:44:28.460 --> 00:44:33.470 align:middle line:84% And then according to this one, then will be identically 0. 00:44:33.470 --> 00:44:40.740 align:middle line:84% And then we find delta x and x prime equal 00:44:40.740 --> 00:44:49.860 align:middle line:84% to 0, because now, this is evaluated at equal time. 00:44:49.860 --> 00:44:53.180 align:middle line:90% So this is very intuitive. 00:44:53.180 --> 00:44:57.350 align:middle line:84% When you make a Lorentz transformation, you can-- 00:44:57.350 --> 00:45:02.060 align:middle line:84% yeah-- but this is a precise proof of the statement. 00:45:02.060 --> 00:45:04.830 align:middle line:90% 00:45:04.830 --> 00:45:07.090 align:middle line:90% Any questions on this? 00:45:07.090 --> 00:45:08.281 align:middle line:90% Yes? 00:45:08.281 --> 00:45:10.165 align:middle line:84% STUDENT: What's the x on [INAUDIBLE]?? 00:45:10.165 --> 00:45:13.302 align:middle line:90% 00:45:13.302 --> 00:45:14.010 align:middle line:90% PROFESSOR: Sorry? 00:45:14.010 --> 00:45:17.760 align:middle line:90% STUDENT: [INAUDIBLE] 00:45:17.760 --> 00:45:22.370 align:middle line:84% PROFESSOR: Yeah, for x minus x prime squared greater than 0. 00:45:22.370 --> 00:45:24.000 align:middle line:84% So this is a four distance square. 00:45:24.000 --> 00:45:25.620 align:middle line:84% You're asking about this notation? 00:45:25.620 --> 00:45:27.840 align:middle line:90% Yeah, this is just-- 00:45:27.840 --> 00:45:30.510 align:middle line:90% there are four distance square-- 00:45:30.510 --> 00:45:33.360 align:middle line:90% the separation between them. 00:45:33.360 --> 00:45:35.062 align:middle line:90% Yes? 00:45:35.062 --> 00:45:37.370 align:middle line:84% STUDENT: I have a question on the first property. 00:45:37.370 --> 00:45:40.400 align:middle line:84% So is that equation up in the corner, 00:45:40.400 --> 00:45:43.390 align:middle line:84% so that defines the-- it says that G 00:45:43.390 --> 00:45:47.133 align:middle line:84% is like the Green's function for your Klein-Gordon operator? 00:45:47.133 --> 00:45:47.800 align:middle line:90% PROFESSOR: Yeah. 00:45:47.800 --> 00:45:49.660 align:middle line:90% STUDENT: So why does that-- 00:45:49.660 --> 00:45:52.588 align:middle line:84% is there an intuition for why the correlator in the vacuum 00:45:52.588 --> 00:45:54.130 align:middle line:84% state gives you the Green's function? 00:45:54.130 --> 00:45:57.130 align:middle line:84% Because then you could use this to get any classical solution 00:45:57.130 --> 00:45:57.940 align:middle line:90% right? 00:45:57.940 --> 00:46:02.260 align:middle line:90% 00:46:02.260 --> 00:46:06.820 align:middle line:84% PROFESSOR: Yeah, so the intuition is that essentially-- 00:46:06.820 --> 00:46:10.840 align:middle line:84% yeah, so for example, this is just how we-- 00:46:10.840 --> 00:46:17.890 align:middle line:90% 00:46:17.890 --> 00:46:22.730 align:middle line:84% yeah, first is that this kind of thing is simple enough. 00:46:22.730 --> 00:46:27.370 align:middle line:84% Essentially, you can just see by observation, but in a more 00:46:27.370 --> 00:46:28.870 align:middle line:90% complicated theory. 00:46:28.870 --> 00:46:32.710 align:middle line:84% And you can actually derive it-- and the retarded Green function 00:46:32.710 --> 00:46:34.940 align:middle line:90% always have this form. 00:46:34.940 --> 00:46:37.760 align:middle line:84% Yeah, so that's a thorough more elaborate derivation. 00:46:37.760 --> 00:46:41.513 align:middle line:90% 00:46:41.513 --> 00:46:43.930 align:middle line:84% STUDENT: When you're talking about classical field theory, 00:46:43.930 --> 00:46:46.540 align:middle line:84% usually solving the Green's function is kind of hard 00:46:46.540 --> 00:46:48.774 align:middle line:90% and you have to expand-- 00:46:48.774 --> 00:46:53.580 align:middle line:84% like, here, we quantize it and then got out-- 00:46:53.580 --> 00:46:56.040 align:middle line:84% PROFESSOR: Yeah, this is different. 00:46:56.040 --> 00:46:58.500 align:middle line:84% So classically, we just define those functions 00:46:58.500 --> 00:47:01.060 align:middle line:90% using those equations. 00:47:01.060 --> 00:47:03.660 align:middle line:90% So here, we just say-- 00:47:03.660 --> 00:47:07.740 align:middle line:84% in quantum field theory, the object can be written this way. 00:47:07.740 --> 00:47:10.800 align:middle line:84% Yeah, those objects can be written this way. 00:47:10.800 --> 00:47:16.620 align:middle line:84% And this definition just follows from the standard-- yeah, 00:47:16.620 --> 00:47:22.020 align:middle line:84% if you do quantum mechanics and you-- a retarded function 00:47:22.020 --> 00:47:23.805 align:middle line:90% can always be written like this. 00:47:23.805 --> 00:47:25.680 align:middle line:84% Yeah, this is also just in quantum mechanics, 00:47:25.680 --> 00:47:28.020 align:middle line:84% but it requires a little bit of calculation to see that. 00:47:28.020 --> 00:47:31.720 align:middle line:90% 00:47:31.720 --> 00:47:33.070 align:middle line:90% Other questions? 00:47:33.070 --> 00:47:33.850 align:middle line:90% Yes? 00:47:33.850 --> 00:47:36.637 align:middle line:84% STUDENT: Should it be lambda minus 1? 00:47:36.637 --> 00:47:37.470 align:middle line:90% PROFESSOR: Oh, yeah. 00:47:37.470 --> 00:47:37.860 align:middle line:90% Sorry. 00:47:37.860 --> 00:47:39.027 align:middle line:90% Yeah, it's-- lambda minus 1. 00:47:39.027 --> 00:47:42.570 align:middle line:90% 00:47:42.570 --> 00:47:45.000 align:middle line:84% Yeah, so here also should be lambda minus 1. 00:47:45.000 --> 00:47:49.880 align:middle line:90% 00:47:49.880 --> 00:47:53.100 align:middle line:90% Other questions? 00:47:53.100 --> 00:47:53.900 align:middle line:90% OK, good. 00:47:53.900 --> 00:48:00.310 align:middle line:90% 00:48:00.310 --> 00:48:04.020 align:middle line:90% Yeah, so this also shows-- 00:48:04.020 --> 00:48:06.950 align:middle line:90% 00:48:06.950 --> 00:48:11.640 align:middle line:84% yeah, so when we talk about the canonical quantization-- 00:48:11.640 --> 00:48:15.320 align:middle line:84% so we mentioned that we need to impose this equal time 00:48:15.320 --> 00:48:19.460 align:middle line:90% commutator to be 0. 00:48:19.460 --> 00:48:21.770 align:middle line:84% That's our canonical commutation. 00:48:21.770 --> 00:48:25.130 align:middle line:84% Yeah, we impose the canonical commutation relation 00:48:25.130 --> 00:48:26.135 align:middle line:90% at a single time. 00:48:26.135 --> 00:48:29.730 align:middle line:90% 00:48:29.730 --> 00:48:34.370 align:middle line:84% And some of you were asking that actually-- 00:48:34.370 --> 00:48:36.680 align:middle line:84% does that actually break Lorentz symmetry 00:48:36.680 --> 00:48:39.500 align:middle line:84% because we have to choose a single time? 00:48:39.500 --> 00:48:41.240 align:middle line:84% But now, you can see actually this 00:48:41.240 --> 00:48:44.180 align:middle line:84% does not break Lorentz symmetry because the only 00:48:44.180 --> 00:48:48.120 align:middle line:84% space-like separated, and the commutator is always the same. 00:48:48.120 --> 00:48:53.870 align:middle line:84% So no matter what frame you choose and the-- 00:48:53.870 --> 00:48:58.070 align:middle line:84% in any frame, if it's equal time, 00:48:58.070 --> 00:49:00.920 align:middle line:84% and it's always space-like separated in some other frame. 00:49:00.920 --> 00:49:07.240 align:middle line:90% 00:49:07.240 --> 00:49:07.740 align:middle line:90% Good? 00:49:07.740 --> 00:49:14.260 align:middle line:90% 00:49:14.260 --> 00:49:19.690 align:middle line:84% So that means-- so immediate conclusion-- 00:49:19.690 --> 00:49:29.860 align:middle line:84% it means that for space-like separated, 00:49:29.860 --> 00:49:38.470 align:middle line:84% x and x prime G plus equal to G minus is equal to G F. 00:49:38.470 --> 00:49:43.540 align:middle line:84% So the time ordering does not matter, so the G plus-- 00:49:43.540 --> 00:49:44.920 align:middle line:90% then you can do G minus. 00:49:44.920 --> 00:49:47.860 align:middle line:84% And then the G F equal to the sum of them. 00:49:47.860 --> 00:49:51.130 align:middle line:84% And then when-- here, it's for t greater than t prime. 00:49:51.130 --> 00:49:53.620 align:middle line:84% Here, it's for t prime greater than t. 00:49:53.620 --> 00:49:56.140 align:middle line:84% And then this should just add equal to 1 00:49:56.140 --> 00:49:59.630 align:middle line:84% because these two become the same. 00:49:59.630 --> 00:50:07.640 align:middle line:84% And the G R and the G A equal to delta equal to 0 00:50:07.640 --> 00:50:09.630 align:middle line:90% for space-like surface case. 00:50:09.630 --> 00:50:12.660 align:middle line:84% So those functions are pretty simple. 00:50:12.660 --> 00:50:14.540 align:middle line:90% So the-- yeah. 00:50:14.540 --> 00:50:22.930 align:middle line:90% 00:50:22.930 --> 00:50:28.070 align:middle line:90% So now-- so the last property-- 00:50:28.070 --> 00:50:37.610 align:middle line:84% now, you can show due to the spacetime translation 00:50:37.610 --> 00:50:52.470 align:middle line:84% symmetry and the Lorentz symmetry of the vacuum-- 00:50:52.470 --> 00:50:55.300 align:middle line:90% 00:50:55.300 --> 00:50:57.940 align:middle line:90% of the vacuum state. 00:50:57.940 --> 00:51:00.270 align:middle line:90% So you can show-- 00:51:00.270 --> 00:51:02.850 align:middle line:90% I think this will be-- 00:51:02.850 --> 00:51:05.190 align:middle line:84% this is a pretty simple argument, 00:51:05.190 --> 00:51:07.620 align:middle line:84% but I will leave it to show yourself. 00:51:07.620 --> 00:51:12.680 align:middle line:84% You can show that any of those functions only 00:51:12.680 --> 00:51:16.580 align:middle line:84% depend on the difference between of them-- 00:51:16.580 --> 00:51:19.100 align:middle line:84% and in particular, only depend on the four distance 00:51:19.100 --> 00:51:19.790 align:middle line:90% between them. 00:51:19.790 --> 00:51:23.480 align:middle line:90% 00:51:23.480 --> 00:51:31.740 align:middle line:84% And this G here can be any of the G R, G A, G plus/minus, 00:51:31.740 --> 00:51:35.590 align:middle line:90% G F, et cetera. 00:51:35.590 --> 00:51:38.900 align:middle line:84% So all of them have very nice properties. 00:51:38.900 --> 00:51:43.430 align:middle line:84% Even naively depend on two arguments. 00:51:43.430 --> 00:51:46.570 align:middle line:90% So x have four components. 00:51:46.570 --> 00:51:49.010 align:middle line:90% x prime have four components. 00:51:49.010 --> 00:51:52.890 align:middle line:84% So naively, this is a function of eight variables, 00:51:52.890 --> 00:51:56.760 align:middle line:84% but this tells you once you use all the symmetries, 00:51:56.760 --> 00:52:00.580 align:middle line:84% this is actually a function only a single variable. 00:52:00.580 --> 00:52:02.440 align:middle line:84% This is a very powerful statement. 00:52:02.440 --> 00:52:05.290 align:middle line:84% So they have very-- yeah, so this answers one 00:52:05.290 --> 00:52:08.140 align:middle line:90% of the question some-- 00:52:08.140 --> 00:52:09.700 align:middle line:90% one of you asked earlier-- 00:52:09.700 --> 00:52:13.510 align:middle line:84% that the-- despite the theta functions, 00:52:13.510 --> 00:52:16.330 align:middle line:84% you can show that they still have very nice properties 00:52:16.330 --> 00:52:17.740 align:middle line:90% under Lorentz transformation. 00:52:17.740 --> 00:52:19.930 align:middle line:84% You can show they have a nice properties 00:52:19.930 --> 00:52:21.628 align:middle line:90% under Lorentz transformation. 00:52:21.628 --> 00:52:26.080 align:middle line:90% 00:52:26.080 --> 00:52:26.580 align:middle line:90% Good. 00:52:26.580 --> 00:52:43.940 align:middle line:90% 00:52:43.940 --> 00:52:44.930 align:middle line:90% Other questions? 00:52:44.930 --> 00:52:48.260 align:middle line:90% 00:52:48.260 --> 00:52:48.890 align:middle line:90% Yes? 00:52:48.890 --> 00:52:51.530 align:middle line:90% STUDENT: [INAUDIBLE] 00:52:51.530 --> 00:52:55.760 align:middle line:84% PROFESSOR: Yeah, so this is very easy. 00:52:55.760 --> 00:53:00.230 align:middle line:84% When you have a translation symmetry, and then 00:53:00.230 --> 00:53:02.240 align:middle line:84% the reference point you choose is not-- 00:53:02.240 --> 00:53:05.550 align:middle line:84% you can choose any point to be the reference point. 00:53:05.550 --> 00:53:11.180 align:middle line:84% And so you can just choose x, and then only 00:53:11.180 --> 00:53:13.648 align:middle line:84% the separation between x and the x prime matters, 00:53:13.648 --> 00:53:15.690 align:middle line:84% because you can just choose x prime, for example, 00:53:15.690 --> 00:53:16.760 align:middle line:90% to be at the origin-- 00:53:16.760 --> 00:53:18.920 align:middle line:90% and then only the separation. 00:53:18.920 --> 00:53:21.830 align:middle line:84% And then the Lorentz symmetry tells you 00:53:21.830 --> 00:53:24.780 align:middle line:90% that only the distance matters. 00:53:24.780 --> 00:53:27.740 align:middle line:90% It doesn't matter the direction. 00:53:27.740 --> 00:53:28.240 align:middle line:90% Yes? 00:53:28.240 --> 00:53:30.407 align:middle line:84% STUDENT: Does the commutator have this property too, 00:53:30.407 --> 00:53:31.350 align:middle line:90% or is it just G? 00:53:31.350 --> 00:53:33.683 align:middle line:84% PROFESSOR: Yeah, the commutator also have this property. 00:53:33.683 --> 00:53:35.655 align:middle line:84% Yeah, the delta also have this property. 00:53:35.655 --> 00:53:40.780 align:middle line:90% 00:53:40.780 --> 00:53:42.415 align:middle line:90% Yes? 00:53:42.415 --> 00:53:44.665 align:middle line:84% STUDENT: Is this only true when x and x' are spacelike 00:53:44.665 --> 00:53:46.470 align:middle line:90% separated? 00:53:46.470 --> 00:53:50.010 align:middle line:84% PROFESSOR: No, this is true in general. 00:53:50.010 --> 00:53:54.340 align:middle line:84% This is always true, but it's only in the vacuum state. 00:53:54.340 --> 00:53:56.892 align:middle line:84% So the key is that you have to use 00:53:56.892 --> 00:53:58.350 align:middle line:84% that the vacuum state are invariant 00:53:58.350 --> 00:54:00.015 align:middle line:90% under those transformations. 00:54:00.015 --> 00:54:05.600 align:middle line:90% 00:54:05.600 --> 00:54:08.160 align:middle line:84% Yeah, it's a-- once you understand how to do it, 00:54:08.160 --> 00:54:11.750 align:middle line:84% it's a very simple argument, so that's why I will leave it 00:54:11.750 --> 00:54:14.780 align:middle line:90% into one of the exercises. 00:54:14.780 --> 00:54:19.620 align:middle line:90% 00:54:19.620 --> 00:54:22.020 align:middle line:90% Other questions? 00:54:22.020 --> 00:54:22.950 align:middle line:90% OK, good. 00:54:22.950 --> 00:54:25.100 align:middle line:84% So now, let's compute this object G plus. 00:54:25.100 --> 00:54:29.160 align:middle line:90% 00:54:29.160 --> 00:54:30.850 align:middle line:90% We have been talking about it. 00:54:30.850 --> 00:54:32.580 align:middle line:84% And now, finally, after discussing 00:54:32.580 --> 00:54:34.690 align:middle line:90% all these general properties-- 00:54:34.690 --> 00:54:36.750 align:middle line:84% so let's compute them explicitly. 00:54:36.750 --> 00:55:14.950 align:middle line:90% 00:55:14.950 --> 00:55:18.630 align:middle line:84% So let's just first compute the G plus. 00:55:18.630 --> 00:55:21.510 align:middle line:84% Did I just erase the definition of G plus? 00:55:21.510 --> 00:55:22.620 align:middle line:90% Yeah. 00:55:22.620 --> 00:55:28.450 align:middle line:90% So this is easy to do. 00:55:28.450 --> 00:55:33.030 align:middle line:84% So those properties you can just get by symmetries. 00:55:33.030 --> 00:55:37.890 align:middle line:84% Of course, you can also get by doing explicit calculations, 00:55:37.890 --> 00:55:40.410 align:middle line:84% but you can get them without doing any explicit calculation 00:55:40.410 --> 00:55:42.165 align:middle line:90% just based on symmetries. 00:55:42.165 --> 00:55:44.580 align:middle line:90% So here, we do-- 00:55:44.580 --> 00:55:46.080 align:middle line:84% calculate this object explicitly, 00:55:46.080 --> 00:55:48.380 align:middle line:84% we will see-- satisfy those properties. 00:55:48.380 --> 00:55:52.600 align:middle line:90% 00:55:52.600 --> 00:56:01.900 align:middle line:84% So we just put in the expansion of phi and phi 00:56:01.900 --> 00:56:03.040 align:middle line:90% x and the phi x prime. 00:56:03.040 --> 00:56:14.950 align:middle line:90% 00:56:14.950 --> 00:56:19.480 align:middle line:84% And then-- yeah, then we have the 0, 00:56:19.480 --> 00:56:26.080 align:middle line:84% and then we have a k u k evaluated 00:56:26.080 --> 00:56:33.790 align:middle line:84% at x plus a k dagger u k star evaluated 00:56:33.790 --> 00:56:42.820 align:middle line:84% at x times a k prime u k prime evaluated 00:56:42.820 --> 00:56:52.570 align:middle line:84% at x prime plus a k prime dagger u k prime star at x prime, 00:56:52.570 --> 00:56:55.240 align:middle line:90% and then acting on 0. 00:56:55.240 --> 00:56:59.470 align:middle line:84% So this decay with 2 omega k, and this 00:56:59.470 --> 00:57:01.610 align:middle line:90% essentially gives you phi. 00:57:01.610 --> 00:57:04.060 align:middle line:84% And the other one gives you the phi-- 00:57:04.060 --> 00:57:07.090 align:middle line:84% so this gives you the phi x, and this one 00:57:07.090 --> 00:57:09.430 align:middle line:90% gives you the phi x prime. 00:57:09.430 --> 00:57:11.110 align:middle line:90% I just calculated this. 00:57:11.110 --> 00:57:15.630 align:middle line:90% 00:57:15.630 --> 00:57:17.230 align:middle line:90% So now, this is easy to do. 00:57:17.230 --> 00:57:19.470 align:middle line:84% This is just now become an exercise 00:57:19.470 --> 00:57:21.700 align:middle line:90% in harmonic oscillator. 00:57:21.700 --> 00:57:23.880 align:middle line:90% So only this term-- 00:57:23.880 --> 00:57:25.080 align:middle line:90% multiply this term. 00:57:25.080 --> 00:57:27.060 align:middle line:90% It's non-zero. 00:57:27.060 --> 00:57:31.620 align:middle line:84% The rest of terms are 0, because this will annihilate 00:57:31.620 --> 00:57:35.430 align:middle line:84% the vacuum, and this 2a dagger together 00:57:35.430 --> 00:57:37.560 align:middle line:90% will annihilate that vacuum. 00:57:37.560 --> 00:57:41.310 align:middle line:84% So the only term not vanishing is this term 00:57:41.310 --> 00:57:42.930 align:middle line:90% combined with this term. 00:57:42.930 --> 00:57:45.920 align:middle line:90% So you just get-- 00:57:45.920 --> 00:57:47.390 align:middle line:84% and there's a delta function when 00:57:47.390 --> 00:57:50.600 align:middle line:90% you evaluate them between 0. 00:57:50.600 --> 00:57:51.980 align:middle line:90% And so you find-- 00:57:51.980 --> 00:58:03.460 align:middle line:90% 00:58:03.460 --> 00:58:05.063 align:middle line:90% And then you have -- 00:58:05.063 --> 00:58:10.900 align:middle line:90% 00:58:10.900 --> 00:58:12.130 align:middle line:90% t minus t prime. 00:58:12.130 --> 00:58:19.970 align:middle line:90% 00:58:19.970 --> 00:58:22.940 align:middle line:90% So this is the object. 00:58:22.940 --> 00:58:26.480 align:middle line:90% 00:58:26.480 --> 00:58:30.650 align:middle line:84% So we can do this integral further, 00:58:30.650 --> 00:58:36.130 align:middle line:84% and then we also rewrite this in a slightly different form. 00:58:36.130 --> 00:58:40.740 align:middle line:84% So you can also write this as integrating-- so here, 00:58:40.740 --> 00:58:45.510 align:middle line:84% omega k is on shell, so I can also write it in a more general 00:58:45.510 --> 00:58:46.740 align:middle line:90% form-- 00:58:46.740 --> 00:58:48.780 align:middle line:90% in the form like this-- 00:58:48.780 --> 00:58:56.810 align:middle line:84% 2 pi theta k 0 delta k squared plus m square 00:58:56.810 --> 00:59:04.360 align:middle line:90% exponential i k dot minus x. 00:59:04.360 --> 00:59:07.870 align:middle line:84% So in this form, the k is unconstrained, 00:59:07.870 --> 00:59:11.800 align:middle line:84% but I have a delta function to enforce it on shell. 00:59:11.800 --> 00:59:16.570 align:middle line:84% And then I put the theta k 0 to make it to be positive. 00:59:16.570 --> 00:59:19.900 align:middle line:84% And then you can see that this will lead to that. 00:59:19.900 --> 00:59:23.820 align:middle line:90% 00:59:23.820 --> 00:59:24.782 align:middle line:90% You can see that. 00:59:24.782 --> 00:59:27.680 align:middle line:90% 00:59:27.680 --> 00:59:30.620 align:middle line:90% And similarly, you can find-- 00:59:30.620 --> 00:59:32.750 align:middle line:84% similarly, you can find the integral expressions 00:59:32.750 --> 00:59:42.290 align:middle line:84% for G R, G A, and G F. I will not give the expression here. 00:59:42.290 --> 00:59:49.580 align:middle line:84% Yeah, we will-- yeah, you can easily find them yourself. 00:59:49.580 --> 00:59:55.060 align:middle line:90% So let me just make one remark. 00:59:55.060 --> 00:59:57.650 align:middle line:90% 00:59:57.650 --> 01:00:00.170 align:middle line:84% So you can now actually calculate this object. 01:00:00.170 --> 01:00:02.700 align:middle line:90% 01:00:02.700 --> 01:00:04.320 align:middle line:84% So now, let's calculate this object. 01:00:04.320 --> 01:00:09.600 align:middle line:84% Remember earlier, we say this object is the overlap 01:00:09.600 --> 01:00:13.050 align:middle line:90% between x and x prime. 01:00:13.050 --> 01:00:17.640 align:middle line:84% So let's see in what sense this is like a delta function. 01:00:17.640 --> 01:00:20.280 align:middle line:84% In this sense, it's like a delta function when 01:00:20.280 --> 01:00:22.650 align:middle line:90% we say them to be equal time. 01:00:22.650 --> 01:00:27.600 align:middle line:84% So now, note-- so let's consider the equal time 01:00:27.600 --> 01:00:28.545 align:middle line:90% t equal to t prime. 01:00:28.545 --> 01:00:32.160 align:middle line:90% 01:00:32.160 --> 01:00:33.690 align:middle line:90% So for t equal to t prime-- 01:00:33.690 --> 01:00:36.380 align:middle line:90% 01:00:36.380 --> 01:00:39.800 align:middle line:84% so the integral becomes following-- 01:00:39.800 --> 01:00:43.685 align:middle line:84% G plus x and x prime just equal to-- 01:00:43.685 --> 01:00:58.370 align:middle line:90% 01:00:58.370 --> 01:01:01.230 align:middle line:90% just become this. 01:01:01.230 --> 01:01:04.460 align:middle line:84% And this integral clearly is non-zero. 01:01:04.460 --> 01:01:09.830 align:middle line:84% So that's why we said earlier, this is not a localized state. 01:01:09.830 --> 01:01:14.330 align:middle line:84% So this is equal to this x and x prime. 01:01:14.330 --> 01:01:17.570 align:middle line:84% So this is non-zero as equal to, and so this is not 01:01:17.570 --> 01:01:20.580 align:middle line:90% a localized state. 01:01:20.580 --> 01:01:22.080 align:middle line:84% Yeah, so this is not proportional 01:01:22.080 --> 01:01:26.520 align:middle line:84% to a delta function if the thing is here. 01:01:26.520 --> 01:01:28.020 align:middle line:84% If you don't have this factor, then 01:01:28.020 --> 01:01:31.870 align:middle line:84% it's a delta function, but this one, it's not. 01:01:31.870 --> 01:01:36.000 align:middle line:84% But you can nevertheless you can calculate this integral 01:01:36.000 --> 01:01:36.500 align:middle line:90% exactly. 01:01:36.500 --> 01:01:41.010 align:middle line:90% 01:01:41.010 --> 01:01:43.410 align:middle line:84% And also just based on the here-- 01:01:43.410 --> 01:01:46.093 align:middle line:84% based on the spherical symmetry-- 01:01:46.093 --> 01:01:47.760 align:middle line:84% rotational symmetry, you can immediately 01:01:47.760 --> 01:01:52.010 align:middle line:84% see that only depend on the distance between them. 01:01:52.010 --> 01:01:58.070 align:middle line:84% Only-- so this G plus will only be a G plus a function of r, 01:01:58.070 --> 01:02:02.150 align:middle line:84% and r is the distance between the-- yeah, just based 01:02:02.150 --> 01:02:04.220 align:middle line:84% on symmetry, you can easily convince yourself. 01:02:04.220 --> 01:02:07.725 align:middle line:84% And this only depends on the distance between them 01:02:07.725 --> 01:02:09.350 align:middle line:84% just from the symmetry of the integral. 01:02:09.350 --> 01:02:09.850 align:middle line:90% Yes? 01:02:09.850 --> 01:02:14.280 align:middle line:84% STUDENT: So when we added the 2 omega k [INAUDIBLE]?? 01:02:14.280 --> 01:02:20.090 align:middle line:90% 01:02:20.090 --> 01:02:24.665 align:middle line:90% PROFESSOR: Yeah, it's for the-- 01:02:24.665 --> 01:02:27.710 align:middle line:84% for example, for calculating the-- 01:02:27.710 --> 01:02:32.640 align:middle line:90% you wan a, a dagger equal to 1. 01:02:32.640 --> 01:02:35.430 align:middle line:84% Yeah, because that's the normalization for the a dagger. 01:02:35.430 --> 01:02:40.040 align:middle line:90% 01:02:40.040 --> 01:02:42.118 align:middle line:90% Yeah, that's a good question. 01:02:42.118 --> 01:02:43.160 align:middle line:90% I forgot to mention that. 01:02:43.160 --> 01:02:46.470 align:middle line:90% 01:02:46.470 --> 01:02:49.010 align:middle line:90% Other questions? 01:02:49.010 --> 01:02:51.870 align:middle line:84% So this object at t equal to 0 only 01:02:51.870 --> 01:02:53.670 align:middle line:84% depends on the distance between them, 01:02:53.670 --> 01:02:56.580 align:middle line:84% because this is a spherical integral because this one only 01:02:56.580 --> 01:02:58.590 align:middle line:90% depends on the magnitude of k. 01:02:58.590 --> 01:03:03.470 align:middle line:84% And then this is just a standard Fourier transform. 01:03:03.470 --> 01:03:07.855 align:middle line:84% So now, you can just evaluate this integral explicitly. 01:03:07.855 --> 01:03:09.820 align:middle line:90% I urge you to do it yourself. 01:03:09.820 --> 01:03:13.330 align:middle line:84% And then you can show this is actually proportional-- 01:03:13.330 --> 01:03:16.610 align:middle line:84% you can actually evaluate this using Bessel function. 01:03:16.610 --> 01:03:19.540 align:middle line:84% So this gives you a modified Bessel function-- 01:03:19.540 --> 01:03:22.720 align:middle line:90% K0 times m r. 01:03:22.720 --> 01:03:25.070 align:middle line:90% So do any of you-- 01:03:25.070 --> 01:03:26.975 align:middle line:84% are any of you expert of Bessel function? 01:03:26.975 --> 01:03:30.492 align:middle line:90% 01:03:30.492 --> 01:03:32.325 align:middle line:84% So do you know the behavior of this function 01:03:32.325 --> 01:03:34.050 align:middle line:90% for when r is large. 01:03:34.050 --> 01:03:34.878 align:middle line:90% STUDENT: Goes to 0. 01:03:34.878 --> 01:03:42.290 align:middle line:90% 01:03:42.290 --> 01:03:43.540 align:middle line:90% PROFESSOR: Yeah, it does to 0. 01:03:43.540 --> 01:03:46.625 align:middle line:84% That's a good intuition, but it goes to 0 in what way? 01:03:46.625 --> 01:03:49.150 align:middle line:90% 01:03:49.150 --> 01:03:51.970 align:middle line:90% Actually, exponentially. 01:03:51.970 --> 01:03:55.720 align:middle line:84% So it actually goes to exponential minus mr, 01:03:55.720 --> 01:03:57.280 align:middle line:90% so for r large. 01:03:57.280 --> 01:04:01.150 align:middle line:90% For m r greater than 1. 01:04:01.150 --> 01:04:03.310 align:middle line:84% Now, you can see in what sense this is 01:04:03.310 --> 01:04:06.570 align:middle line:90% approximately localized state. 01:04:06.570 --> 01:04:11.480 align:middle line:84% You see this is not 0, but this is 01:04:11.480 --> 01:04:14.010 align:middle line:90% pretty small at large distance. 01:04:14.010 --> 01:04:17.060 align:middle line:84% So at distance r, it's much greater than 1 over m. 01:04:17.060 --> 01:04:35.560 align:middle line:90% 01:04:35.560 --> 01:04:38.860 align:middle line:84% So when r is much greater than 1 of m-- 01:04:38.860 --> 01:04:42.295 align:middle line:84% so that object is very small, so this exponential is small. 01:04:42.295 --> 01:04:51.710 align:middle line:90% 01:04:51.710 --> 01:04:54.500 align:middle line:84% So this tells you that even though this is not 01:04:54.500 --> 01:04:57.830 align:middle line:84% a perfectly localized state, this 01:04:57.830 --> 01:05:05.480 align:middle line:84% is localized to the distance of 1 over m. 01:05:05.480 --> 01:05:09.620 align:middle line:84% So if these two points are separated more than 1 01:05:09.620 --> 01:05:13.610 align:middle line:84% over m, and then the overlap becomes very small. 01:05:13.610 --> 01:05:15.360 align:middle line:90% And so this is-- 01:05:15.360 --> 01:05:21.690 align:middle line:84% yeah, so sometimes, we call this xi. 01:05:21.690 --> 01:05:24.090 align:middle line:84% And then can in condensed matter-- 01:05:24.090 --> 01:05:27.210 align:middle line:84% language which, when you interpret this as a correlation 01:05:27.210 --> 01:05:29.060 align:middle line:90% function-- 01:05:29.060 --> 01:05:33.000 align:middle line:84% which the correlation between the x and the phi x prime-- 01:05:33.000 --> 01:05:37.040 align:middle line:84% and then this is also called the correlation length. 01:05:37.040 --> 01:05:38.120 align:middle line:90% So when the phi-- 01:05:38.120 --> 01:05:40.640 align:middle line:90% the distance beyond 1 over m-- 01:05:40.640 --> 01:05:45.440 align:middle line:84% or beyond this xi and then the phi no longer correlated. 01:05:45.440 --> 01:05:49.150 align:middle line:90% 01:05:49.150 --> 01:05:51.460 align:middle line:84% So this a sense-- this is approximately localized 01:05:51.460 --> 01:05:55.030 align:middle line:84% object, and it's essentially the Compton wavelength 01:05:55.030 --> 01:05:56.857 align:middle line:90% of the particle. 01:05:56.857 --> 01:05:58.690 align:middle line:84% It's the Compton wavelength of the particle. 01:05:58.690 --> 01:06:01.030 align:middle line:84% And once you go outside the Compton wavelength 01:06:01.030 --> 01:06:05.890 align:middle line:84% of the particle, and then the overlap becomes very small. 01:06:05.890 --> 01:06:09.055 align:middle line:84% So this makes perfect physical sense. 01:06:09.055 --> 01:06:10.870 align:middle line:90% It makes perfect physical sense. 01:06:10.870 --> 01:06:13.650 align:middle line:90% 01:06:13.650 --> 01:06:15.473 align:middle line:90% There was a hand? 01:06:15.473 --> 01:06:17.640 align:middle line:84% STUDENT: Yeah, I had a question, but then I realized 01:06:17.640 --> 01:06:19.110 align:middle line:90% that my thinking was wrong. 01:06:19.110 --> 01:06:20.030 align:middle line:90% PROFESSOR: OK, good. 01:06:20.030 --> 01:06:29.370 align:middle line:90% 01:06:29.370 --> 01:06:31.620 align:middle line:84% Yeah, so this is just [INAUDIBLE] 01:06:31.620 --> 01:06:32.655 align:middle line:90% of what we said earlier. 01:06:32.655 --> 01:06:37.440 align:middle line:90% 01:06:37.440 --> 01:06:40.560 align:middle line:84% So you can similarly get such kind of integral expression 01:06:40.560 --> 01:06:43.830 align:middle line:90% for those things. 01:06:43.830 --> 01:06:48.510 align:middle line:84% But rather than-- the coordinate space expression is actually 01:06:48.510 --> 01:06:51.540 align:middle line:84% often-- will be often-- actually lead to the expression momentum 01:06:51.540 --> 01:06:52.590 align:middle line:90% space. 01:06:52.590 --> 01:06:56.940 align:middle line:90% So it often leads there. 01:06:56.940 --> 01:07:12.860 align:middle line:84% So often-- in the future, we'll often lead the expressions 01:07:12.860 --> 01:07:13.800 align:middle line:90% momentum space. 01:07:13.800 --> 01:07:21.010 align:middle line:90% 01:07:21.010 --> 01:07:23.230 align:middle line:84% So we can just consider the Fourier transform. 01:07:23.230 --> 01:07:26.520 align:middle line:90% 01:07:26.520 --> 01:07:30.310 align:middle line:84% So here is the convention of the Fourier transform. 01:07:30.310 --> 01:07:35.850 align:middle line:84% So dx-- so any object or function of x-- 01:07:35.850 --> 01:07:42.570 align:middle line:84% again, this is a four vector, and the minus i k x. 01:07:42.570 --> 01:07:45.660 align:middle line:84% So I use the same notation, G, to denote 01:07:45.660 --> 01:07:48.120 align:middle line:84% its Fourier transform, and distinguish them 01:07:48.120 --> 01:07:52.110 align:middle line:84% only by k and x because it's just annoying to always put 01:07:52.110 --> 01:07:56.740 align:middle line:90% some tilde above it. 01:07:56.740 --> 01:08:04.660 align:middle line:84% And the inverse transformation would be the G x x prime. 01:08:04.660 --> 01:08:07.480 align:middle line:84% So you see here explicitly-- yeah, there, we already 01:08:07.480 --> 01:08:13.970 align:middle line:84% see explicitly that is only the function of their separations. 01:08:13.970 --> 01:08:16.250 align:middle line:84% And then if you use the symmetry here further, 01:08:16.250 --> 01:08:18.370 align:middle line:90% then you see the distances. 01:08:18.370 --> 01:08:25.324 align:middle line:84% So we can also just write this in terms of the-- 01:08:25.324 --> 01:08:32.170 align:middle line:84% this is i k x x minus x prime G k. 01:08:32.170 --> 01:08:36.340 align:middle line:84% So this is our convention for the Fourier transform. 01:08:36.340 --> 01:08:41.670 align:middle line:84% So from here, if you just look at this expression 01:08:41.670 --> 01:08:43.950 align:middle line:84% and compare with this expression, 01:08:43.950 --> 01:08:47.790 align:middle line:84% and then immediately we conclude for this Whiteman function 01:08:47.790 --> 01:08:56.359 align:middle line:84% G plus, k is just given by 2 pi theta k 0 delta function 01:08:56.359 --> 01:08:57.874 align:middle line:90% k squared plus m squared. 01:08:57.874 --> 01:09:02.330 align:middle line:90% 01:09:02.330 --> 01:09:06.660 align:middle line:84% So this is the expression for the G plus. 01:09:06.660 --> 01:09:07.350 align:middle line:90% Yes? 01:09:07.350 --> 01:09:12.488 align:middle line:84% STUDENT: So this localization is [INAUDIBLE],, 01:09:12.488 --> 01:09:14.280 align:middle line:84% which is greater for higher momentum, so is 01:09:14.280 --> 01:09:16.363 align:middle line:84% this a manifestation of the uncertainty principle? 01:09:16.363 --> 01:09:19.319 align:middle line:90% 01:09:19.319 --> 01:09:22.050 align:middle line:90% PROFESSOR: Yeah, this is a-- 01:09:22.050 --> 01:09:24.930 align:middle line:84% so here, this is, in some sense, the uncertainty principle. 01:09:24.930 --> 01:09:29.710 align:middle line:84% This-- here, we just consider in the vacuum state, 01:09:29.710 --> 01:09:32.069 align:middle line:90% so there's not-- 01:09:32.069 --> 01:09:35.149 align:middle line:84% the state itself does not have any energy. 01:09:35.149 --> 01:09:37.050 align:middle line:84% And so this is just the Compton wavelength 01:09:37.050 --> 01:09:40.020 align:middle line:90% of the particle itself. 01:09:40.020 --> 01:09:43.689 align:middle line:84% And it's just like a static Compton wavelength. 01:09:43.689 --> 01:09:50.040 align:middle line:90% 01:09:50.040 --> 01:09:50.760 align:middle line:90% Other questions? 01:09:50.760 --> 01:09:51.479 align:middle line:90% Yes? 01:09:51.479 --> 01:09:53.771 align:middle line:84% STUDENT: If this is the vacuum state, then what does it 01:09:53.771 --> 01:09:55.720 align:middle line:90% mean to be [INAUDIBLE]? 01:09:55.720 --> 01:10:01.120 align:middle line:90% 01:10:01.120 --> 01:10:03.750 align:middle line:84% PROFESSOR: Yeah, but still you have fluctuations. 01:10:03.750 --> 01:10:05.310 align:middle line:90% This is quantum mechanics. 01:10:05.310 --> 01:10:10.020 align:middle line:90% 01:10:10.020 --> 01:10:13.440 align:middle line:84% Yeah, for example, you can calculate an analogous object 01:10:13.440 --> 01:10:15.760 align:middle line:90% for the harmonic oscillator. 01:10:15.760 --> 01:10:17.580 align:middle line:84% So if you have a harmonic oscillator, 01:10:17.580 --> 01:10:21.300 align:middle line:84% the analogous object is this object, 01:10:21.300 --> 01:10:24.990 align:middle line:84% and you can also calculate this non-zero just because 01:10:24.990 --> 01:10:26.160 align:middle line:90% of the fluctuations. 01:10:26.160 --> 01:10:29.570 align:middle line:90% 01:10:29.570 --> 01:10:30.560 align:middle line:90% Other questions? 01:10:30.560 --> 01:10:33.800 align:middle line:90% 01:10:33.800 --> 01:10:36.220 align:middle line:90% OK, good. 01:10:36.220 --> 01:10:43.480 align:middle line:84% So this is the expression for the G plus. 01:10:43.480 --> 01:10:48.580 align:middle line:84% And for those object, we can get a momentum space expression 01:10:48.580 --> 01:10:52.700 align:middle line:84% by going through the same procedure. 01:10:52.700 --> 01:10:57.590 align:middle line:84% Just plug them in, labor through, and then write 01:10:57.590 --> 01:11:00.890 align:middle line:84% the final answer in this four momentum integral form, 01:11:00.890 --> 01:11:03.020 align:middle line:84% and then you just read the answer. 01:11:03.020 --> 01:11:05.060 align:middle line:84% For them, we can do the same thing. 01:11:05.060 --> 01:11:08.350 align:middle line:90% But it's actually much easier. 01:11:08.350 --> 01:11:10.000 align:middle line:84% Instead of going through that procedure 01:11:10.000 --> 01:11:13.090 align:middle line:84% to calculate those quantities, it's actually much easier 01:11:13.090 --> 01:11:20.310 align:middle line:84% to start from here, because they satisfy these equations. 01:11:20.310 --> 01:11:23.830 align:middle line:90% So let me call this star. 01:11:23.830 --> 01:11:38.190 align:middle line:84% So to find G F k the same with G R, A k, 01:11:38.190 --> 01:11:49.165 align:middle line:84% it's actually easier to use star. 01:11:49.165 --> 01:11:55.030 align:middle line:84% Unfortunately-- so now, if you look at this equation-- 01:11:55.030 --> 01:11:59.350 align:middle line:90% 01:11:59.350 --> 01:12:02.730 align:middle line:84% if we look at this equation, then 01:12:02.730 --> 01:12:04.480 align:middle line:84% let's use Fourier transform on both sides. 01:12:04.480 --> 01:12:20.220 align:middle line:90% 01:12:20.220 --> 01:12:22.680 align:middle line:84% We do a Fourier transform on both sides. 01:12:22.680 --> 01:12:26.150 align:middle line:84% So the left-hand side, the partial-- 01:12:26.150 --> 01:12:28.010 align:middle line:84% each derivative just gives you a k. 01:12:28.010 --> 01:12:30.900 align:middle line:84% So this is the standard rule for Fourier transform. 01:12:30.900 --> 01:12:35.390 align:middle line:84% And so here, you just get k squared plus m squared. 01:12:35.390 --> 01:12:39.290 align:middle line:84% And let me just call it G, so denote any of them, 01:12:39.290 --> 01:12:42.260 align:middle line:90% equal to minus i. 01:12:42.260 --> 01:12:44.100 align:middle line:84% Because the right-hand side, when 01:12:44.100 --> 01:12:46.100 align:middle line:84% you do a Fourier transform, it just becomes one. 01:12:46.100 --> 01:12:47.700 align:middle line:90% It's a delta function. 01:12:47.700 --> 01:12:51.840 align:middle line:84% So now, we can immediately write down the expression 01:12:51.840 --> 01:12:52.700 align:middle line:90% in momentum space. 01:12:52.700 --> 01:13:03.090 align:middle line:90% 01:13:03.090 --> 01:13:06.460 align:middle line:84% But now, you see we have a problem. 01:13:06.460 --> 01:13:10.240 align:middle line:90% So what problem do we have? 01:13:10.240 --> 01:13:11.290 align:middle line:90% Yes? 01:13:11.290 --> 01:13:14.050 align:middle line:84% STUDENT: When you have k squared equals negative m squared 01:13:14.050 --> 01:13:15.850 align:middle line:90% it's like a singularity. 01:13:15.850 --> 01:13:16.930 align:middle line:90% PROFESSOR: Good. 01:13:16.930 --> 01:13:19.338 align:middle line:90% Well, we have a lot of problems. 01:13:19.338 --> 01:13:20.630 align:middle line:90% So this is one of the problems. 01:13:20.630 --> 01:13:21.635 align:middle line:90% We have another problem. 01:13:21.635 --> 01:13:22.565 align:middle line:90% Yes? 01:13:22.565 --> 01:13:23.960 align:middle line:90% STUDENT: Is this on shell? 01:13:23.960 --> 01:13:24.593 align:middle line:90% PROFESSOR: Hmm? 01:13:24.593 --> 01:13:26.958 align:middle line:90% STUDENT: [INAUDIBLE] 01:13:26.958 --> 01:13:29.800 align:middle line:90% 01:13:29.800 --> 01:13:31.780 align:middle line:84% PROFESSOR: So k is not on shell, because when 01:13:31.780 --> 01:13:36.910 align:middle line:84% we do a Fourier transform, it's for the general k. 01:13:36.910 --> 01:13:39.370 align:middle line:84% Yeah, it's defined for the general k. 01:13:39.370 --> 01:13:40.990 align:middle line:84% Indeed, for the on shell value of k, 01:13:40.990 --> 01:13:43.165 align:middle line:84% then you will have a singular behavior. 01:13:43.165 --> 01:13:45.680 align:middle line:90% 01:13:45.680 --> 01:13:48.380 align:middle line:84% But for the-- but we still have another problem. 01:13:48.380 --> 01:13:57.410 align:middle line:90% 01:13:57.410 --> 01:13:58.060 align:middle line:90% Yes? 01:13:58.060 --> 01:14:00.910 align:middle line:84% STUDENT: They're not supposed to be complex? 01:14:00.910 --> 01:14:02.230 align:middle line:90% PROFESSOR: Yeah, it is complex. 01:14:02.230 --> 01:14:04.580 align:middle line:90% It's OK. 01:14:04.580 --> 01:14:06.050 align:middle line:90% Yeah, that aspect is OK. 01:14:06.050 --> 01:14:07.445 align:middle line:90% Yeah, so since-- 01:14:07.445 --> 01:14:08.810 align:middle line:90% [INAUDIBLE] 01:14:08.810 --> 01:14:15.900 align:middle line:84% So you see here, we have defined the three different functions, 01:14:15.900 --> 01:14:19.281 align:middle line:84% but there, how many solutions we have? 01:14:19.281 --> 01:14:20.160 align:middle line:90% We only have one. 01:14:20.160 --> 01:14:22.380 align:middle line:84% We have unique solution seems like. 01:14:22.380 --> 01:14:24.635 align:middle line:90% Then we have a problem. 01:14:24.635 --> 01:14:26.010 align:middle line:84% It seems like-- that seems to say 01:14:26.010 --> 01:14:27.218 align:middle line:90% all these three are the same. 01:14:27.218 --> 01:14:31.080 align:middle line:84% But if you calculate them going through this theta 01:14:31.080 --> 01:14:35.310 align:middle line:84% function [INAUDIBLE],, they are not the same. 01:14:35.310 --> 01:14:37.670 align:middle line:84% So it turns out these two problems are related. 01:14:37.670 --> 01:14:40.205 align:middle line:90% 01:14:40.205 --> 01:14:43.320 align:middle line:84% It turns out these two problems are related because now, 01:14:43.320 --> 01:14:47.790 align:middle line:84% if you consider the coordinate space expression, which 01:14:47.790 --> 01:14:50.145 align:middle line:84% obtained by the Fourier transform of this guy-- 01:14:50.145 --> 01:15:01.070 align:middle line:90% 01:15:01.070 --> 01:15:04.180 align:middle line:84% again, by Fourier transform of this guy-- 01:15:04.180 --> 01:15:12.410 align:middle line:84% so in the-- so if you write down explicitly 1 over k squared 01:15:12.410 --> 01:15:14.300 align:middle line:90% plus-- 01:15:14.300 --> 01:15:17.670 align:middle line:84% so 1 over k squared plus m squared, 01:15:17.670 --> 01:15:19.020 align:middle line:90% if we write it explicitly. 01:15:19.020 --> 01:15:24.140 align:middle line:84% So this is 1 over minus omega squared plus k squared plus m 01:15:24.140 --> 01:15:25.670 align:middle line:90% squared-- 01:15:25.670 --> 01:15:26.910 align:middle line:90% write it explicitly. 01:15:26.910 --> 01:15:30.020 align:middle line:84% And then this is equal to minus omega 01:15:30.020 --> 01:15:33.380 align:middle line:90% squared plus omega k squared. 01:15:33.380 --> 01:15:35.015 align:middle line:84% So this is just the omega k squared. 01:15:35.015 --> 01:15:37.970 align:middle line:90% 01:15:37.970 --> 01:15:43.040 align:middle line:84% So as you said, this actually has a singularity when omega 01:15:43.040 --> 01:15:45.040 align:middle line:90% equal to plus minus omega k. 01:15:45.040 --> 01:15:49.330 align:middle line:84% When you satisfy this on shell condition. 01:15:49.330 --> 01:15:53.680 align:middle line:84% But this singularity is actually along the integration contour 01:15:53.680 --> 01:15:54.190 align:middle line:90% of omega. 01:15:54.190 --> 01:15:56.850 align:middle line:90% 01:15:56.850 --> 01:16:03.190 align:middle line:84% Because in these four integrals is given by d omega dk. 01:16:03.190 --> 01:16:06.730 align:middle line:84% So the omega-- so this from minus infinity 01:16:06.730 --> 01:16:08.790 align:middle line:90% to plus infinity. 01:16:08.790 --> 01:16:12.330 align:middle line:84% And these two singularities are the real value of omega. 01:16:12.330 --> 01:16:18.080 align:middle line:84% So they actually-- so if you look at omega axis-- 01:16:18.080 --> 01:16:21.190 align:middle line:84% so at these two points, actually, they become singular. 01:16:21.190 --> 01:16:25.730 align:middle line:90% The integrand is singular. 01:16:25.730 --> 01:16:29.340 align:middle line:84% So the integral is actually not well-defined. 01:16:29.340 --> 01:16:31.620 align:middle line:84% So the integral is not well-defined. 01:16:31.620 --> 01:16:35.020 align:middle line:84% So in order to define the integral-- 01:16:35.020 --> 01:16:38.320 align:middle line:84% in order to define this omega integral, 01:16:38.320 --> 01:16:45.250 align:middle line:84% then we have to do your standard trick in complex analysis. 01:16:45.250 --> 01:16:47.690 align:middle line:90% So what is that trick? 01:16:47.690 --> 01:16:48.470 align:middle line:90% Yes? 01:16:48.470 --> 01:16:50.120 align:middle line:90% STUDENT: [INAUDIBLE] 01:16:50.120 --> 01:16:51.120 align:middle line:90% PROFESSOR: That's right. 01:16:51.120 --> 01:16:55.150 align:middle line:90% You go around the singularity. 01:16:55.150 --> 01:16:58.940 align:middle line:84% So now, you go to the complex plane-- 01:16:58.940 --> 01:17:00.290 align:middle line:90% complex omega plane. 01:17:00.290 --> 01:17:02.930 align:middle line:90% So now this is a real omega. 01:17:02.930 --> 01:17:04.400 align:middle line:90% This is imaginary omega. 01:17:04.400 --> 01:17:05.960 align:middle line:84% You go to the complex omega plane. 01:17:05.960 --> 01:17:09.390 align:middle line:84% Now, the integration constant is along the real axis. 01:17:09.390 --> 01:17:12.900 align:middle line:84% And now, you have four different choices. 01:17:12.900 --> 01:17:19.040 align:middle line:84% You can either go up or going down for each one of them. 01:17:19.040 --> 01:17:21.030 align:middle line:84% And now you have four different choices. 01:17:21.030 --> 01:17:25.420 align:middle line:84% And this 3 equals 1 into three choices of them. 01:17:25.420 --> 01:17:28.530 align:middle line:84% Then there's a fourth one, which is not frequently used, 01:17:28.530 --> 01:17:30.920 align:middle line:84% so we normally don't give them a name. 01:17:30.920 --> 01:17:34.160 align:middle line:84% And the fourth one, sometimes, we call them G F tilde. 01:17:34.160 --> 01:17:38.150 align:middle line:84% It's just-- anyway, so there are four different choices of going 01:17:38.150 --> 01:17:41.540 align:middle line:84% around the singularity, and then that gives you 01:17:41.540 --> 01:17:44.930 align:middle line:90% four possible solutions. 01:17:44.930 --> 01:17:47.421 align:middle line:90% And then that gives you the-- 01:17:47.421 --> 01:17:51.300 align:middle line:90% 01:17:51.300 --> 01:17:54.540 align:middle line:84% yeah, so now let's talk about the G R-- 01:17:54.540 --> 01:17:56.500 align:middle line:90% do them one by one. 01:17:56.500 --> 01:17:59.430 align:middle line:90% So for G R-- 01:17:59.430 --> 01:18:03.120 align:middle line:84% so remember, G R is defined to be 01:18:03.120 --> 01:18:06.360 align:middle line:84% when theta t greater than t prime, 01:18:06.360 --> 01:18:10.420 align:middle line:84% G R should be proportional to t, to t minus t prime. 01:18:10.420 --> 01:18:16.580 align:middle line:84% So that means this should be 0 for t minus t prime 01:18:16.580 --> 01:18:17.390 align:middle line:90% smaller than 0. 01:18:17.390 --> 01:18:19.910 align:middle line:90% 01:18:19.910 --> 01:18:25.090 align:middle line:84% This should be 0 for t minus t prime smaller than 0. 01:18:25.090 --> 01:18:30.370 align:middle line:84% So how do we achieve that by going around the poles? 01:18:30.370 --> 01:18:35.500 align:middle line:84% So remember, in this integral, it's a peice minus-- 01:18:35.500 --> 01:18:41.180 align:middle line:84% the omega dependent piece is minus i omega t minus t prime. 01:18:41.180 --> 01:18:43.970 align:middle line:84% So when t minus t prime smaller than 0. 01:18:43.970 --> 01:18:48.630 align:middle line:84% If you want to do this integral using contour integration, 01:18:48.630 --> 01:18:50.810 align:middle line:90% then you can close-- 01:18:50.810 --> 01:18:53.353 align:middle line:84% so this is smaller than 0, and then this is positive. 01:18:53.353 --> 01:18:55.520 align:middle line:84% And then you can close the contour in the upper half 01:18:55.520 --> 01:18:56.960 align:middle line:90% plane. 01:18:56.960 --> 01:18:59.330 align:middle line:84% You can close the contour in the upper half plane. 01:18:59.330 --> 01:19:02.090 align:middle line:84% In order for this to be identically 0, 01:19:02.090 --> 01:19:04.550 align:middle line:84% you need your integration-- counter 01:19:04.550 --> 01:19:06.830 align:middle line:90% don't enclose any singularity. 01:19:06.830 --> 01:19:11.690 align:middle line:84% So that means for the retarded, you 01:19:11.690 --> 01:19:14.360 align:middle line:84% need to going around the singularity along which 01:19:14.360 --> 01:19:15.020 align:middle line:90% direction? 01:19:15.020 --> 01:19:18.150 align:middle line:90% 01:19:18.150 --> 01:19:19.826 align:middle line:90% We only have one minute left. 01:19:19.826 --> 01:19:22.060 align:middle line:90% [LAUGHTER] 01:19:22.060 --> 01:19:24.550 align:middle line:84% For it to be 0, when you integrate-- so 01:19:24.550 --> 01:19:27.940 align:middle line:84% you want to close in the upper half plane. 01:19:27.940 --> 01:19:31.030 align:middle line:84% You want to close it because the t minus t prime 01:19:31.030 --> 01:19:34.110 align:middle line:84% is smaller than 0, and then this is positive i, and the omega 01:19:34.110 --> 01:19:35.110 align:middle line:90% in the upper half plane. 01:19:35.110 --> 01:19:37.490 align:middle line:90% This is a decay exponential. 01:19:37.490 --> 01:19:41.690 align:middle line:84% And so you wonder, there's no singularity inside the contour, 01:19:41.690 --> 01:19:46.700 align:middle line:84% and then you want to go around the singularity this way. 01:19:46.700 --> 01:19:49.400 align:middle line:90% Going above the singularity. 01:19:49.400 --> 01:19:51.920 align:middle line:84% When you're going above the singularity, and then 01:19:51.920 --> 01:19:53.970 align:middle line:90% this contour don't include-- 01:19:53.970 --> 01:19:57.350 align:middle line:90% and then within this contour-- 01:19:57.350 --> 01:19:58.730 align:middle line:90% and then there's no singularity. 01:19:58.730 --> 01:20:01.790 align:middle line:84% Then, this is identically 0 from the Cauchy theorem. 01:20:01.790 --> 01:20:04.820 align:middle line:84% So it tells you that for the retarded Green function, 01:20:04.820 --> 01:20:08.640 align:middle line:90% you need the contour like that. 01:20:08.640 --> 01:20:15.450 align:middle line:90% So similarly, for the advanced-- 01:20:15.450 --> 01:20:19.090 align:middle line:84% because the advanced, we just change the direction. 01:20:19.090 --> 01:20:21.300 align:middle line:90% So this is for the retarded. 01:20:21.300 --> 01:20:26.200 align:middle line:84% For the advanced, it's proportional to t 01:20:26.200 --> 01:20:26.898 align:middle line:90% minus t prime. 01:20:26.898 --> 01:20:28.690 align:middle line:84% Then you just go in the opposite direction. 01:20:28.690 --> 01:20:31.870 align:middle line:90% 01:20:31.870 --> 01:20:35.410 align:middle line:84% So this is for the retarded-- for the advanced. 01:20:35.410 --> 01:20:41.070 align:middle line:84% And then, finally, if we want to do it for the Feynman-- 01:20:41.070 --> 01:20:43.800 align:middle line:84% and then you choose one of them going up 01:20:43.800 --> 01:20:44.940 align:middle line:90% and one of them going down. 01:20:44.940 --> 01:20:48.030 align:middle line:84% I always forget which one going up, which one going down. 01:20:48.030 --> 01:20:52.920 align:middle line:84% Yeah, so actually, this one you go down, and this one you 01:20:52.920 --> 01:20:54.240 align:middle line:90% go up. 01:20:54.240 --> 01:21:04.140 align:middle line:84% So this is for the Feynman for the G F. That's all for today. 01:21:04.140 --> 01:21:08.000 align:middle line:90%