WEBVTT 00:00:00.000 --> 00:00:06.986 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:06.986 --> 00:00:10.990 align:middle line:90% 00:00:10.990 --> 00:00:17.200 align:middle line:84% PROFESSOR: Last time, we started talking about the quantize-- 00:00:17.200 --> 00:00:20.590 align:middle line:84% the Maxwell theory in the Lorenz gauge. 00:00:20.590 --> 00:00:26.185 align:middle line:84% So in the Lorenz gauge, we consider the following action. 00:00:26.185 --> 00:00:35.010 align:middle line:90% 00:00:35.010 --> 00:00:35.850 align:middle line:90% OK. 00:00:35.850 --> 00:00:40.530 align:middle line:84% So we showed earlier that the Lorentz gauge 00:00:40.530 --> 00:00:44.140 align:middle line:84% can be ensured just following the equation of motion. 00:00:44.140 --> 00:00:48.900 align:middle line:84% So you just get from here, the equation of motion, and then 00:00:48.900 --> 00:00:50.440 align:middle line:90% the equation of motion-- 00:00:50.440 --> 00:00:53.700 align:middle line:84% yeah, so the equation of motion will 00:00:53.700 --> 00:01:00.240 align:middle line:84% lead to partial square, partial mu A mu equal to 0. 00:01:00.240 --> 00:01:03.730 align:middle line:84% So this is ensures partial mu a mu equal to 0. 00:01:03.730 --> 00:01:05.910 align:middle line:84% You just have to make sure your boundary 00:01:05.910 --> 00:01:10.425 align:middle line:84% conditions, such as the partial mu A mu equal to O. OK? 00:01:10.425 --> 00:01:12.960 align:middle line:90% 00:01:12.960 --> 00:01:15.690 align:middle line:84% And in particular, the action is particularly 00:01:15.690 --> 00:01:22.330 align:middle line:84% simple for xi equal to 1, in which case we just have-- 00:01:22.330 --> 00:01:23.950 align:middle line:84% actually we just have [INAUDIBLE].. 00:01:23.950 --> 00:01:30.550 align:middle line:90% 00:01:30.550 --> 00:01:31.450 align:middle line:90% OK. 00:01:31.450 --> 00:01:35.600 align:middle line:84% So as if you just have 4 decoupled massless scalar. 00:01:35.600 --> 00:01:36.100 align:middle line:90% OK? 00:01:36.100 --> 00:01:39.520 align:middle line:84% As if you just have 4 massless scalar. 00:01:39.520 --> 00:01:42.070 align:middle line:84% And the equation of motion is actually all very simple. 00:01:42.070 --> 00:01:46.390 align:middle line:90% 00:01:46.390 --> 00:01:47.860 align:middle line:90% So we can just proceed. 00:01:47.860 --> 00:01:56.190 align:middle line:84% Copy our result before for massless scalar. 00:01:56.190 --> 00:01:59.040 align:middle line:84% And just treat each A mu as a massless scalar, 00:01:59.040 --> 00:02:01.420 align:middle line:90% and then we can just do it. 00:02:01.420 --> 00:02:08.970 align:middle line:84% For example, the canonical momentum conjugate to A mu 00:02:08.970 --> 00:02:15.085 align:middle line:84% would be just a mu dot, just A mu dot. 00:02:15.085 --> 00:02:15.585 align:middle line:90% OK. 00:02:15.585 --> 00:02:19.230 align:middle line:90% 00:02:19.230 --> 00:02:23.805 align:middle line:84% So the canonical quantization condition, then, is given by-- 00:02:23.805 --> 00:02:43.540 align:middle line:90% 00:02:43.540 --> 00:02:48.670 align:middle line:84% and we can also just straightforwardly write down 00:02:48.670 --> 00:02:56.020 align:middle line:84% A mu, the expansion for-- the operator expansion for A mu. 00:02:56.020 --> 00:03:07.485 align:middle line:90% 00:03:07.485 --> 00:03:07.985 align:middle line:90% OK. 00:03:07.985 --> 00:03:11.020 align:middle line:90% 00:03:11.020 --> 00:03:13.460 align:middle line:90% So we have four of them. 00:03:13.460 --> 00:03:16.450 align:middle line:90% So again, we will-- 00:03:16.450 --> 00:03:19.810 align:middle line:84% instead of writing them as 4 massless scalar, 00:03:19.810 --> 00:03:24.250 align:middle line:84% we will-- as we did before, for the Coulomb gauge case, 00:03:24.250 --> 00:03:27.640 align:middle line:84% we introduce a polarization vector. 00:03:27.640 --> 00:03:29.810 align:middle line:84% So we will introduce a polarization vector. 00:03:29.810 --> 00:03:31.510 align:middle line:84% So there are four possible things. 00:03:31.510 --> 00:03:33.380 align:middle line:90% There are four components. 00:03:33.380 --> 00:03:37.940 align:middle line:84% So there are four possible polarizations-- 00:03:37.940 --> 00:03:38.440 align:middle line:90% times. 00:03:38.440 --> 00:04:12.070 align:middle line:90% 00:04:12.070 --> 00:04:13.810 align:middle line:90% OK. 00:04:13.810 --> 00:04:16.089 align:middle line:90% So we just get that. 00:04:16.089 --> 00:04:25.320 align:middle line:84% So this epsilon mu alpha for alpha equal to 0, 1, 2, 3-- 00:04:25.320 --> 00:04:27.640 align:middle line:84% so these are four polarization vectors. 00:04:27.640 --> 00:04:28.140 align:middle line:90% OK? 00:04:28.140 --> 00:04:36.690 align:middle line:90% 00:04:36.690 --> 00:04:37.680 align:middle line:90% You can choose. 00:04:37.680 --> 00:04:44.250 align:middle line:84% You can just choose, say, 1, 0, 0, 0, 0, 1, 0, 0, et cetera. 00:04:44.250 --> 00:04:47.670 align:middle line:84% And then this just like four decoupled scalar, OK? 00:04:47.670 --> 00:04:51.390 align:middle line:84% But this way introduces a polarization vector, 00:04:51.390 --> 00:04:56.240 align:middle line:84% allowing to write them in a more general way. 00:04:56.240 --> 00:04:57.445 align:middle line:90% So we normally pick-- 00:04:57.445 --> 00:05:05.070 align:middle line:90% 00:05:05.070 --> 00:05:09.870 align:middle line:84% for example, the 0-th component, the 0-th polarization vector 00:05:09.870 --> 00:05:14.080 align:middle line:84% just to be along the time direction, OK? 00:05:14.080 --> 00:05:17.090 align:middle line:84% So just along the time direction. 00:05:17.090 --> 00:05:21.980 align:middle line:84% And then we also introduce another-- 00:05:21.980 --> 00:05:25.490 align:middle line:84% so the number 3, we introduce to be 00:05:25.490 --> 00:05:28.940 align:middle line:84% proportional to the direction of the momentum. 00:05:28.940 --> 00:05:33.910 align:middle line:90% 00:05:33.910 --> 00:05:35.470 align:middle line:90% So this is proportional to the-- 00:05:35.470 --> 00:05:39.100 align:middle line:84% parallel to the momentum, to the spatial momentum. 00:05:39.100 --> 00:05:43.990 align:middle line:90% So the k mu here would be-- 00:05:43.990 --> 00:05:46.490 align:middle line:90% because this is massless. 00:05:46.490 --> 00:05:47.650 align:middle line:90% So we just have this. 00:05:47.650 --> 00:05:49.510 align:middle line:84% OK so so the 0-th component is just 00:05:49.510 --> 00:05:53.210 align:middle line:90% equal to the magnitude of k. 00:05:53.210 --> 00:06:02.870 align:middle line:84% And then we take the epsilon 1, 2 epsilon mu, then 1, 2 to be, 00:06:02.870 --> 00:06:06.725 align:middle line:84% let's say, orthogonal to the momentum. 00:06:06.725 --> 00:06:09.790 align:middle line:90% 00:06:09.790 --> 00:06:18.940 align:middle line:84% So these are called the transverse polarizations. 00:06:18.940 --> 00:06:25.000 align:middle line:90% 00:06:25.000 --> 00:06:30.280 align:middle line:84% So we take them also to be orthogonal to 0-th component. 00:06:30.280 --> 00:06:32.620 align:middle line:90% That means there's no-- 00:06:32.620 --> 00:06:35.650 align:middle line:84% the time component for 1, 2 will be 0, too, 00:06:35.650 --> 00:06:39.010 align:middle line:84% and then also it should be proportional-- it should also 00:06:39.010 --> 00:06:40.720 align:middle line:90% be orthogonal to the momentum. 00:06:40.720 --> 00:06:45.730 align:middle line:90% 00:06:45.730 --> 00:06:51.260 align:middle line:84% So we take, also, them to be orthogonal to each other. 00:06:51.260 --> 00:06:58.278 align:middle line:90% 00:06:58.278 --> 00:06:59.022 align:middle line:90% Eta mu nu. 00:06:59.022 --> 00:07:00.855 align:middle line:84% We take them to be orthogonal to each other. 00:07:00.855 --> 00:07:06.370 align:middle line:90% 00:07:06.370 --> 00:07:10.930 align:middle line:84% This is the normal, orthonormal condition. 00:07:10.930 --> 00:07:18.060 align:middle line:90% 00:07:18.060 --> 00:07:19.200 align:middle line:90% OK? 00:07:19.200 --> 00:07:21.240 align:middle line:84% Take these spaces to be orthonormal. 00:07:21.240 --> 00:07:23.400 align:middle line:90% And then this is also complete. 00:07:23.400 --> 00:07:27.900 align:middle line:84% Means that if I sum over different alpha and beta-- 00:07:27.900 --> 00:07:34.130 align:middle line:90% 00:07:34.130 --> 00:07:36.650 align:middle line:84% and actually I can get back to eta mu nu. 00:07:36.650 --> 00:07:39.160 align:middle line:90% OK? 00:07:39.160 --> 00:07:41.620 align:middle line:84% So this is called the completeness. 00:07:41.620 --> 00:07:44.440 align:middle line:90% So they form a complete basis. 00:07:44.440 --> 00:07:48.234 align:middle line:84% So any Lorentz vector can be expanded in terms of it. 00:07:48.234 --> 00:07:53.340 align:middle line:90% 00:07:53.340 --> 00:08:00.950 align:middle line:84% So if we take, say, k to be the z-direction, 00:08:00.950 --> 00:08:03.980 align:middle line:84% then the simplest choice would be, say, you just do-- 00:08:03.980 --> 00:08:08.120 align:middle line:90% 00:08:08.120 --> 00:08:09.395 align:middle line:90% so we just have that. 00:08:09.395 --> 00:08:31.860 align:middle line:90% 00:08:31.860 --> 00:08:33.270 align:middle line:90% OK, so this is simplest. 00:08:33.270 --> 00:08:36.510 align:middle line:84% But you can consider more general polarization. 00:08:36.510 --> 00:08:40.210 align:middle line:90% 00:08:40.210 --> 00:08:45.760 align:middle line:84% And so now, if you plug in this expansion, 00:08:45.760 --> 00:08:48.460 align:middle line:84% as we do, for the scalar, and then you plug in 00:08:48.460 --> 00:08:51.237 align:middle line:84% back to that commutation relation, 00:08:51.237 --> 00:08:53.320 align:middle line:84% then you will get the commutation relation with a. 00:08:53.320 --> 00:08:55.460 align:middle line:90% So this is straightforward. 00:08:55.460 --> 00:08:58.540 align:middle line:84% So if we call this equation star-- 00:08:58.540 --> 00:09:01.225 align:middle line:90% star star, this equation star. 00:09:01.225 --> 00:09:03.880 align:middle line:90% 00:09:03.880 --> 00:09:20.180 align:middle line:84% So plug star star into star, and then we get the ak alpha ak 00:09:20.180 --> 00:09:32.870 align:middle line:84% prime beta dagger should be equal to beta alpha beta 2 pi 00:09:32.870 --> 00:09:40.710 align:middle line:90% cubed delta -- prime OK. 00:09:40.710 --> 00:09:45.030 align:middle line:84% And the the rest commutators are all 0. 00:09:45.030 --> 00:09:53.500 align:middle line:90% 00:09:53.500 --> 00:09:56.005 align:middle line:84% And then you can define the vacuum. 00:09:56.005 --> 00:10:00.310 align:middle line:90% 00:10:00.310 --> 00:10:10.850 align:middle line:84% Define a annihlated by all the alphas for any alpha and k. 00:10:10.850 --> 00:10:15.300 align:middle line:90% 00:10:15.300 --> 00:10:20.580 align:middle line:84% And I will introduce the Hilbert space. 00:10:20.580 --> 00:10:25.245 align:middle line:90% I call it H big to be the psi-- 00:10:25.245 --> 00:10:28.740 align:middle line:90% 00:10:28.740 --> 00:10:42.930 align:middle line:84% the collection of psi built by acting ak alpha dagger on 0. 00:10:42.930 --> 00:10:45.630 align:middle line:90% OK? 00:10:45.630 --> 00:10:50.370 align:middle line:84% So you can just add arbitrary number of them on 0, 00:10:50.370 --> 00:10:52.480 align:middle line:90% then you get your Hilbert space. 00:10:52.480 --> 00:10:56.020 align:middle line:84% So we call this Hilbert space to be my big Hilbert space. 00:10:56.020 --> 00:11:00.960 align:middle line:84% And you will see the reason we do that. 00:11:00.960 --> 00:11:01.770 align:middle line:90% OK? 00:11:01.770 --> 00:11:05.720 align:middle line:84% So later, we will see why we call this H big. 00:11:05.720 --> 00:11:09.670 align:middle line:90% So this is seemingly all fine. 00:11:09.670 --> 00:11:12.090 align:middle line:84% But of course, there are problems. 00:11:12.090 --> 00:11:18.120 align:middle line:84% This cannot be right, because the Maxwell theory, we said, 00:11:18.120 --> 00:11:21.720 align:middle line:84% only has two transverse degrees of freedom. 00:11:21.720 --> 00:11:23.040 align:middle line:90% But here we have four. 00:11:23.040 --> 00:11:25.140 align:middle line:84% We have four massless with degrees freedom. 00:11:25.140 --> 00:11:28.490 align:middle line:90% We must have too many. 00:11:28.490 --> 00:11:34.280 align:middle line:84% And indeed, you see there, here, are problems. 00:11:34.280 --> 00:11:36.340 align:middle line:90% So one problem is the following. 00:11:36.340 --> 00:11:40.210 align:middle line:90% 00:11:40.210 --> 00:11:44.218 align:middle line:90% So one problem is the following. 00:11:44.218 --> 00:11:45.135 align:middle line:90% So there are problems. 00:11:45.135 --> 00:11:48.670 align:middle line:90% 00:11:48.670 --> 00:11:50.070 align:middle line:90% So one problem, yeah-- 00:11:50.070 --> 00:11:52.740 align:middle line:84% so the zeroth order problem is that we 00:11:52.740 --> 00:12:02.940 align:middle line:84% have four massless degrees of freedom, two too many. 00:12:02.940 --> 00:12:03.440 align:middle line:90% OK? 00:12:03.440 --> 00:12:07.590 align:middle line:90% 00:12:07.590 --> 00:12:12.270 align:middle line:84% So we know that we only should only have two. 00:12:12.270 --> 00:12:16.400 align:middle line:84% And the second, you see that this is problematic. 00:12:16.400 --> 00:12:20.640 align:middle line:90% 00:12:20.640 --> 00:12:22.140 align:middle line:90% Is that, you notice-- 00:12:22.140 --> 00:12:25.840 align:middle line:84% so here, we have eta alpha beta eta mu mu. 00:12:25.840 --> 00:12:30.270 align:middle line:84% So this is all just from Lorentz covariance, OK? 00:12:30.270 --> 00:12:34.720 align:middle line:84% But this eta alpha beta is problematic 00:12:34.720 --> 00:12:45.040 align:middle line:84% because if you look at the commutator for ak 0-- 00:12:45.040 --> 00:12:46.090 align:middle line:90% OK, so the zeros-- 00:12:46.090 --> 00:12:50.410 align:middle line:90% 00:12:50.410 --> 00:13:01.450 align:middle line:84% that polarization, then you have ak 0 ak prime 0 dagger 00:13:01.450 --> 00:13:05.770 align:middle line:90% equal to minus 2 pi cubed. 00:13:05.770 --> 00:13:07.345 align:middle line:90% Say delta k minus k prime. 00:13:07.345 --> 00:13:09.910 align:middle line:90% 00:13:09.910 --> 00:13:11.110 align:middle line:90% OK, you have a minus sign. 00:13:11.110 --> 00:13:15.290 align:middle line:90% 00:13:15.290 --> 00:13:21.670 align:middle line:84% So this is problematic because if we, say, create particles-- 00:13:21.670 --> 00:13:37.850 align:middle line:84% say if we create the particles by acting ak alpha dagger 00:13:37.850 --> 00:13:46.450 align:middle line:84% on the 0, and then you find, because of this minus sign, 00:13:46.450 --> 00:14:04.980 align:middle line:84% this create the particle with polarization epsilon alpha mu. 00:14:04.980 --> 00:14:06.960 align:middle line:84% So now, because of this minus sign, 00:14:06.960 --> 00:14:13.140 align:middle line:84% you see that if we look at the overlap between the particle 00:14:13.140 --> 00:14:15.690 align:middle line:90% with the 0 polarization-- 00:14:15.690 --> 00:14:26.360 align:middle line:84% so if we look at k 0 with some k prime 0-- 00:14:26.360 --> 00:14:30.815 align:middle line:84% and then because of this minus sign, we have minus. 00:14:30.815 --> 00:14:33.680 align:middle line:90% 00:14:33.680 --> 00:14:40.940 align:middle line:84% So this will be proportional to minus 2 pi cube theta cube 00:14:40.940 --> 00:14:43.520 align:middle line:90% k minus k. 00:14:43.520 --> 00:14:46.670 align:middle line:90% So this is smaller than 0. 00:14:46.670 --> 00:14:49.835 align:middle line:84% So you see, actually, in the product-- 00:14:49.835 --> 00:14:52.550 align:middle line:90% 00:14:52.550 --> 00:14:55.790 align:middle line:84% so that means that this state actually has negative norm. 00:14:55.790 --> 00:15:04.380 align:middle line:90% 00:15:04.380 --> 00:15:06.090 align:middle line:84% That means this state is negative norm. 00:15:06.090 --> 00:15:17.560 align:middle line:90% 00:15:17.560 --> 00:15:21.350 align:middle line:84% But this is actually a good sign. 00:15:21.350 --> 00:15:24.250 align:middle line:84% So this is another way to see that we have 00:15:24.250 --> 00:15:27.130 align:middle line:90% too many degrees of freedom. 00:15:27.130 --> 00:15:30.850 align:middle line:84% So that means not all degrees of freedom here can be physical. 00:15:30.850 --> 00:15:34.390 align:middle line:84% So here, it just tells you there are some degree of freedom here 00:15:34.390 --> 00:15:38.370 align:middle line:84% that must be unphysical because they have negative norm. 00:15:38.370 --> 00:15:41.250 align:middle line:84% The states they create have negative norm. 00:15:41.250 --> 00:15:43.995 align:middle line:84% So they cannot really correspond to genuine physical states. 00:15:43.995 --> 00:15:50.100 align:middle line:90% 00:15:50.100 --> 00:15:54.090 align:middle line:84% But those problems should not worry us because, so far, 00:15:54.090 --> 00:15:57.720 align:middle line:84% what we have done is to quantize this theory. 00:15:57.720 --> 00:16:01.553 align:middle line:90% 00:16:01.553 --> 00:16:03.720 align:middle line:84% So far what we have done is to quantize this theory. 00:16:03.720 --> 00:16:06.780 align:middle line:84% But this theory is not the Maxwell theory. 00:16:06.780 --> 00:16:10.880 align:middle line:84% To get the Maxwell theory, we have to do two more steps. 00:16:10.880 --> 00:16:16.140 align:middle line:84% One more step-- one step is to impose conditions 00:16:16.140 --> 00:16:18.950 align:middle line:84% so that makes sure this partial mu a mu is equal to 0, 00:16:18.950 --> 00:16:21.860 align:middle line:84% because we haven't fixed the gauge. 00:16:21.860 --> 00:16:25.170 align:middle line:84% The second condition, we mentioned before. 00:16:25.170 --> 00:16:27.010 align:middle line:84% So there are two more things we need to. 00:16:27.010 --> 00:16:38.930 align:middle line:90% 00:16:38.930 --> 00:16:47.180 align:middle line:84% So first is that we need to ensure the gauge 00:16:47.180 --> 00:16:48.560 align:middle line:90% partial mu A mu is equal to 0. 00:16:48.560 --> 00:16:52.340 align:middle line:90% 00:16:52.340 --> 00:16:55.680 align:middle line:84% The second thing, as we mentioned before, 00:16:55.680 --> 00:16:59.070 align:middle line:84% is that after fixing the Lorenz gauge, 00:16:59.070 --> 00:17:02.580 align:middle line:84% they're still residual gauge degrees of freedom left. 00:17:02.580 --> 00:17:09.000 align:middle line:84% And we have to fix the residual gauge 00:17:09.000 --> 00:17:10.500 align:middle line:90% freedom in the Lorenz gauge. 00:17:10.500 --> 00:17:14.485 align:middle line:90% 00:17:14.485 --> 00:17:14.985 align:middle line:90% OK? 00:17:14.985 --> 00:17:22.700 align:middle line:90% 00:17:22.700 --> 00:17:28.665 align:middle line:84% So when you do this, then you will get a physical Hilbert 00:17:28.665 --> 00:17:29.165 align:middle line:90% space. 00:17:29.165 --> 00:17:31.753 align:middle line:90% 00:17:31.753 --> 00:17:33.420 align:middle line:84% You will get the physical Hilbert space. 00:17:33.420 --> 00:17:36.890 align:middle line:84% And this physical Hilbert space, we will see-- 00:17:36.890 --> 00:17:40.490 align:middle line:84% you will see-- will only contain, indeed, 00:17:40.490 --> 00:17:43.700 align:middle line:84% two transverse massless degrees of freedom. 00:17:43.700 --> 00:17:49.340 align:middle line:84% And the two degrees of freedom here are gotten rid of, 00:17:49.340 --> 00:17:51.330 align:middle line:90% I got rid of. 00:17:51.330 --> 00:17:59.030 align:middle line:84% And so in the past, I described those procedures in detail 00:17:59.030 --> 00:18:00.630 align:middle line:90% in class. 00:18:00.630 --> 00:18:04.430 align:middle line:84% But actually, that didn't have very good effect. 00:18:04.430 --> 00:18:07.910 align:middle line:90% So later, I put it in your Pset. 00:18:07.910 --> 00:18:09.660 align:middle line:84% Actually, I think it worked better, 00:18:09.660 --> 00:18:12.980 align:middle line:84% so forced you to go through it and to think through it, how 00:18:12.980 --> 00:18:16.160 align:middle line:90% to do these two steps. 00:18:16.160 --> 00:18:22.610 align:middle line:84% Because those steps, they are not difficult technically, 00:18:22.610 --> 00:18:24.680 align:middle line:90% but you need to think carefully. 00:18:24.680 --> 00:18:26.150 align:middle line:90% You need to think carefully. 00:18:26.150 --> 00:18:28.835 align:middle line:84% So let me just make some comments on this step. 00:18:28.835 --> 00:18:31.410 align:middle line:90% 00:18:31.410 --> 00:18:33.765 align:middle line:90% So recall in the Coulomb gauge-- 00:18:33.765 --> 00:18:38.410 align:middle line:90% 00:18:38.410 --> 00:18:39.685 align:middle line:90% so in the Coulomb gauge-- 00:18:39.685 --> 00:18:44.830 align:middle line:90% 00:18:44.830 --> 00:18:59.710 align:middle line:84% so recall in Coulomb gauge, we impose this condition. 00:18:59.710 --> 00:19:01.300 align:middle line:84% We essentially solve this condition. 00:19:01.300 --> 00:19:02.630 align:middle line:90% We impose this condition. 00:19:02.630 --> 00:19:04.990 align:middle line:84% So classically, we impose this condition, 00:19:04.990 --> 00:19:07.780 align:middle line:84% essentially, as part of the equation of motion. 00:19:07.780 --> 00:19:11.200 align:middle line:84% And we solve the A, which satisfies this condition. 00:19:11.200 --> 00:19:15.650 align:middle line:84% We just one into two transverse A, OK? 00:19:15.650 --> 00:19:19.100 align:middle line:84% And then quantum mechanically, because we impose it 00:19:19.100 --> 00:19:23.060 align:middle line:84% as a classical equation-- we impose it 00:19:23.060 --> 00:19:25.430 align:middle line:84% as part of the equation of motion, classically. 00:19:25.430 --> 00:19:28.050 align:middle line:84% So quantum mechanically, this becomes an operator equation, 00:19:28.050 --> 00:19:31.300 align:middle line:90% which we impose on operators. 00:19:31.300 --> 00:19:36.150 align:middle line:84% But in this case, we can no longer impose. 00:19:36.150 --> 00:19:37.995 align:middle line:90% But in Lorenz gauge-- 00:19:37.995 --> 00:19:39.645 align:middle line:84% so this leads to the operator equation. 00:19:39.645 --> 00:19:43.110 align:middle line:90% 00:19:43.110 --> 00:19:58.910 align:middle line:84% But in the Lorenz gauge, we cannot impose partial mu a mu 00:19:58.910 --> 00:20:11.960 align:middle line:84% as the operator equation, because the equation of motion 00:20:11.960 --> 00:20:16.950 align:middle line:84% already almost implies this is equal to 0. 00:20:16.950 --> 00:20:21.240 align:middle line:84% We just have to impose some boundary condition 00:20:21.240 --> 00:20:24.720 align:middle line:84% to ensure that, indeed, this equation only has 00:20:24.720 --> 00:20:28.260 align:middle line:84% solutions corresponding to partial mu and mu equal to 0. 00:20:28.260 --> 00:20:31.030 align:middle line:84% So if you impose this separately as the-- 00:20:31.030 --> 00:20:34.580 align:middle line:90% 00:20:34.580 --> 00:20:36.320 align:middle line:90% yeah. 00:20:36.320 --> 00:20:40.350 align:middle line:84% So you only need to impose them as boundary condition. 00:20:40.350 --> 00:20:44.780 align:middle line:84% So quantum mechanically, that implies we cannot impose this 00:20:44.780 --> 00:20:45.920 align:middle line:90% as an operator equation. 00:20:45.920 --> 00:20:47.780 align:middle line:84% Indeed, you are showing your pset, 00:20:47.780 --> 00:20:49.887 align:middle line:84% and it will be inconsistent if you impose 00:20:49.887 --> 00:20:51.095 align:middle line:90% this as an operator equation. 00:20:51.095 --> 00:20:53.600 align:middle line:90% 00:20:53.600 --> 00:20:57.410 align:middle line:84% So what does it mean at the common level corresponding 00:20:57.410 --> 00:21:02.280 align:middle line:84% to classically you impose the boundary condition? 00:21:02.280 --> 00:21:04.420 align:middle line:84% So classically-imposed boundary condition 00:21:04.420 --> 00:21:06.850 align:middle line:84% limits possible configurations you 00:21:06.850 --> 00:21:11.650 align:middle line:84% can have, limit possible configurations you can have 00:21:11.650 --> 00:21:13.900 align:middle line:84% at the quantum-mechanical level corresponding 00:21:13.900 --> 00:21:17.510 align:middle line:90% to restriction on the states. 00:21:17.510 --> 00:21:20.480 align:middle line:84% So the quantum level, it turns out that the right thing to do 00:21:20.480 --> 00:21:25.130 align:middle line:84% is to impose this kind of condition on the states, 00:21:25.130 --> 00:21:29.210 align:middle line:84% require your states to be annihilated 00:21:29.210 --> 00:21:32.160 align:middle line:90% by something like this. 00:21:32.160 --> 00:21:35.420 align:middle line:84% But it turns out, actually, the story is more subtle than that. 00:21:35.420 --> 00:21:39.860 align:middle line:84% If you just require the state annihilated by this thing, 00:21:39.860 --> 00:21:41.202 align:middle line:90% actually it does not work. 00:21:41.202 --> 00:21:43.160 align:middle line:84% And so you have to do a little bit more subtle. 00:21:43.160 --> 00:21:45.200 align:middle line:90% So the fun will be in your Pset. 00:21:45.200 --> 00:21:46.790 align:middle line:90% You will go through that. 00:21:46.790 --> 00:21:49.020 align:middle line:90% You'll go through that. 00:21:49.020 --> 00:21:52.830 align:middle line:90% So do you have any questions? 00:21:52.830 --> 00:21:53.820 align:middle line:90% Yes? 00:21:53.820 --> 00:21:56.275 align:middle line:84% AUDIENCE: Why is it too much to just say that is 0? 00:21:56.275 --> 00:21:58.650 align:middle line:84% I guess it makes sense that you can just fix the boundary 00:21:58.650 --> 00:22:00.770 align:middle line:90% conditions but -- 00:22:00.770 --> 00:22:04.010 align:middle line:84% PROFESSOR: Yeah, so classically, it does not matter. 00:22:04.010 --> 00:22:07.210 align:middle line:84% As a classical, it doesn't matter. 00:22:07.210 --> 00:22:10.960 align:middle line:84% But quantum mechanically, you will show you 00:22:10.960 --> 00:22:14.860 align:middle line:84% a P set if you impose this as an operator equation. 00:22:14.860 --> 00:22:18.880 align:middle line:84% Then that's incompatible with this canonical quantization 00:22:18.880 --> 00:22:19.720 align:middle line:90% condition. 00:22:19.720 --> 00:22:21.910 align:middle line:90% AUDIENCE: Yeah. 00:22:21.910 --> 00:22:24.700 align:middle line:84% PROFESSOR: Yeah, here, I just motivate that classically 00:22:24.700 --> 00:22:27.850 align:middle line:84% in order to ensure this unity to impose the boundary condition, 00:22:27.850 --> 00:22:29.900 align:middle line:90% not as the equation of motion. 00:22:29.900 --> 00:22:32.140 align:middle line:84% So quantum mechanically, we just-- yeah. 00:22:32.140 --> 00:22:36.253 align:middle line:90% 00:22:36.253 --> 00:22:36.920 align:middle line:90% Other questions? 00:22:36.920 --> 00:22:37.970 align:middle line:90% Yes? 00:22:37.970 --> 00:22:40.766 align:middle line:84% AUDIENCE: Is there a reason why the subspace with positive norm 00:22:40.766 --> 00:22:42.474 align:middle line:84% is like closed -- a closed Hilbert space? 00:22:42.474 --> 00:22:45.780 align:middle line:90% 00:22:45.780 --> 00:22:47.030 align:middle line:90% PROFESSOR: Yeah, you will see. 00:22:47.030 --> 00:22:48.710 align:middle line:90% You will see. 00:22:48.710 --> 00:22:50.450 align:middle line:84% You will see what's happening, yeah. 00:22:50.450 --> 00:22:52.100 align:middle line:90% So it turns out-- 00:22:52.100 --> 00:22:55.830 align:middle line:84% yeah, let me just say some words. 00:22:55.830 --> 00:22:59.150 align:middle line:84% So it turns out when you do this step, 00:22:59.150 --> 00:23:02.400 align:middle line:90% and then you will eliminate-- 00:23:02.400 --> 00:23:12.285 align:middle line:84% so the step one, we eliminate the negative norm state. 00:23:12.285 --> 00:23:17.000 align:middle line:90% 00:23:17.000 --> 00:23:18.550 align:middle line:90% AUDIENCE: OK. 00:23:18.550 --> 00:23:23.260 align:middle line:84% PROFESSOR: So you find that once you impose properly this kind 00:23:23.260 --> 00:23:25.750 align:middle line:84% of condition on the states, and you actually 00:23:25.750 --> 00:23:28.810 align:middle line:84% eliminate the negative norm state. 00:23:28.810 --> 00:23:31.420 align:middle line:84% But then you find after you eliminate the negative norm 00:23:31.420 --> 00:23:34.775 align:middle line:90% states, in this Hilbert space-- 00:23:34.775 --> 00:23:37.970 align:middle line:90% 00:23:37.970 --> 00:23:44.090 align:middle line:84% in this Hilbert space, there's still states of zero-norm. 00:23:44.090 --> 00:23:45.920 align:middle line:84% There's still states of zero-norm. 00:23:45.920 --> 00:23:50.940 align:middle line:84% And then when you fix the second, and then 00:23:50.940 --> 00:23:52.785 align:middle line:84% you eliminate those zero-norm states. 00:23:52.785 --> 00:23:56.160 align:middle line:90% 00:23:56.160 --> 00:24:02.250 align:middle line:84% And then you get your physical Hilbert space, 00:24:02.250 --> 00:24:04.440 align:middle line:84% because, remember, in the Coulomb gauge, 00:24:04.440 --> 00:24:08.850 align:middle line:84% everything is just like harmonic oscillator. 00:24:08.850 --> 00:24:11.340 align:middle line:90% There's no zero-norm state. 00:24:11.340 --> 00:24:13.140 align:middle line:90% You fix the gauge completely. 00:24:13.140 --> 00:24:16.410 align:middle line:84% But here, because you didn't fix the gauge completely-- 00:24:16.410 --> 00:24:19.170 align:middle line:84% and even after you eliminate those negative norm states, 00:24:19.170 --> 00:24:20.610 align:middle line:90% you have zero-norm states. 00:24:20.610 --> 00:24:24.360 align:middle line:84% And then they correspond to remaining gauge freedom. 00:24:24.360 --> 00:24:28.804 align:middle line:84% And once you get rid of them, and then you have your-- 00:24:28.804 --> 00:24:29.304 align:middle line:90% yeah. 00:24:29.304 --> 00:24:31.990 align:middle line:90% 00:24:31.990 --> 00:24:33.290 align:middle line:90% Good. 00:24:33.290 --> 00:24:35.110 align:middle line:90% Any questions on this? 00:24:35.110 --> 00:24:35.662 align:middle line:90% Yes? 00:24:35.662 --> 00:24:37.620 align:middle line:84% AUDIENCE: Instead of doing Maxwell theory, what 00:24:37.620 --> 00:24:40.390 align:middle line:84% if we're just trying to do four massless scalars? 00:24:40.390 --> 00:24:42.472 align:middle line:90% PROFESSOR: Yeah. 00:24:42.472 --> 00:24:44.680 align:middle line:84% AUDIENCE: And if you try to do it with these vectors, 00:24:44.680 --> 00:24:46.314 align:middle line:84% you'd still get these negative norms. 00:24:46.314 --> 00:24:48.923 align:middle line:90% 00:24:48.923 --> 00:24:49.840 align:middle line:90% What's going on there? 00:24:49.840 --> 00:24:51.273 align:middle line:90% PROFESSOR: Good, good, good. 00:24:51.273 --> 00:24:52.690 align:middle line:84% That's a very good question, which 00:24:52.690 --> 00:24:56.350 align:middle line:90% I'm waiting for you to ask. 00:24:56.350 --> 00:24:57.950 align:middle line:90% So the key thing-- 00:24:57.950 --> 00:25:00.970 align:middle line:84% so you say we can certainly consider four massless scalar 00:25:00.970 --> 00:25:02.200 align:middle line:90% field. 00:25:02.200 --> 00:25:04.510 align:middle line:84% If I consider four Maxwell scalar field, 00:25:04.510 --> 00:25:07.000 align:middle line:84% there should not be a problem, why 00:25:07.000 --> 00:25:09.400 align:middle line:84% there should be negative norm state or zero-norm state. 00:25:09.400 --> 00:25:14.955 align:middle line:84% But the key is that here, the states 00:25:14.955 --> 00:25:17.820 align:middle line:84% are contract-- this A mu A mu are contracted 00:25:17.820 --> 00:25:21.410 align:middle line:90% using the Lorentzian metric. 00:25:21.410 --> 00:25:24.400 align:middle line:84% And so if you look at this Lagrangian, 00:25:24.400 --> 00:25:29.350 align:middle line:84% the 0-th components actually have the opposite sign 00:25:29.350 --> 00:25:35.570 align:middle line:84% to the standard massless scalar field, and because of that. 00:25:35.570 --> 00:25:37.930 align:middle line:84% So this is not your ordinary massless scalar field. 00:25:37.930 --> 00:25:40.770 align:middle line:84% They actually have opposite signs in your Lagrangian. 00:25:40.770 --> 00:25:42.630 align:middle line:84% But you say, oh, can we just change 00:25:42.630 --> 00:25:43.920 align:middle line:90% the sign for that component? 00:25:43.920 --> 00:25:46.875 align:middle line:84% You cannot because we have Lorentz symmetry. 00:25:46.875 --> 00:25:48.660 align:middle line:84% Because we have Lorentz symmetry. 00:25:48.660 --> 00:25:51.630 align:middle line:84% PROFESSOR: So the Lorentz symmetry forces you, somehow, 00:25:51.630 --> 00:25:54.210 align:middle line:84% when you reduce the Maxwell theory to this four massless 00:25:54.210 --> 00:25:56.550 align:middle line:84% scalar, one of them have the wrong sign. 00:25:56.550 --> 00:25:59.430 align:middle line:84% So that's why-- so that the wrong sign is related to why 00:25:59.430 --> 00:26:00.780 align:middle line:90% you have the wrong sign here. 00:26:00.780 --> 00:26:04.050 align:middle line:90% 00:26:04.050 --> 00:26:06.630 align:middle line:84% The same kind of Lorentz covariance 00:26:06.630 --> 00:26:11.217 align:middle line:84% tells you here is eta alpha beta rather than delta alpha beta, 00:26:11.217 --> 00:26:12.300 align:middle line:90% and then delta alpha beta. 00:26:12.300 --> 00:26:16.650 align:middle line:90% 00:26:16.650 --> 00:26:18.330 align:middle line:90% Good? 00:26:18.330 --> 00:26:20.700 align:middle line:90% OK. 00:26:20.700 --> 00:26:24.930 align:middle line:84% So this concludes our discussion of the canonical quantization 00:26:24.930 --> 00:26:27.870 align:middle line:90% for the Maxwell series. 00:26:27.870 --> 00:26:29.818 align:middle line:84% So here I will quickly describe how 00:26:29.818 --> 00:26:31.110 align:middle line:90% to do it for the path integral. 00:26:31.110 --> 00:26:34.080 align:middle line:90% 00:26:34.080 --> 00:26:37.890 align:middle line:84% So I will just do it very quickly, 00:26:37.890 --> 00:26:42.250 align:middle line:84% because many of the elements are familiar before. 00:26:42.250 --> 00:26:44.310 align:middle line:84% And so I will just point out things which 00:26:44.310 --> 00:26:49.270 align:middle line:90% are new for the Maxwell case. 00:26:49.270 --> 00:26:49.770 align:middle line:90% OK? 00:26:49.770 --> 00:27:04.520 align:middle line:90% 00:27:04.520 --> 00:27:10.040 align:middle line:84% So we already familiar how to do the path integral. 00:27:10.040 --> 00:27:12.960 align:middle line:84% So Maxwell theory is a free theory. 00:27:12.960 --> 00:27:15.300 align:middle line:84% So in principle, we know how to do it. 00:27:15.300 --> 00:27:17.920 align:middle line:90% So let's just-- yeah. 00:27:17.920 --> 00:27:22.300 align:middle line:84% So here, let's go back without fixing the gauge first, OK? 00:27:22.300 --> 00:27:27.040 align:middle line:84% So this is now completely starting from the original, 00:27:27.040 --> 00:27:36.220 align:middle line:84% just starting from the Maxwell equal to minus 1/4 as with mu F 00:27:36.220 --> 00:27:39.048 align:middle line:90% mu mu. 00:27:39.048 --> 00:27:40.590 align:middle line:84% So let's just go back to this theory. 00:27:40.590 --> 00:27:43.500 align:middle line:90% 00:27:43.500 --> 00:27:46.850 align:middle line:90% Let's go back to this theory. 00:27:46.850 --> 00:27:50.120 align:middle line:84% And then let's just try to do a-- 00:27:50.120 --> 00:27:55.630 align:middle line:84% try to see how the path integral works. 00:27:55.630 --> 00:27:59.700 align:middle line:84% So remember, this is a free theory. 00:27:59.700 --> 00:28:01.980 align:middle line:84% So we can just write down the generating functional 00:28:01.980 --> 00:28:03.180 align:middle line:90% for this theory. 00:28:03.180 --> 00:28:05.190 align:middle line:84% And then we can, in principle, calculate 00:28:05.190 --> 00:28:07.170 align:middle line:84% all possible correlation functions here. 00:28:07.170 --> 00:28:10.110 align:middle line:84% Free theory is supposed to be very simple. 00:28:10.110 --> 00:28:13.050 align:middle line:84% So we just write down the generating functional, which 00:28:13.050 --> 00:28:17.380 align:middle line:90% we integrate over all A mu. 00:28:17.380 --> 00:28:19.330 align:middle line:90% And then we have i S-- 00:28:19.330 --> 00:28:22.750 align:middle line:90% so this is Maxwell theory-- 00:28:22.750 --> 00:28:23.800 align:middle line:90% then we add the source. 00:28:23.800 --> 00:28:31.550 align:middle line:90% 00:28:31.550 --> 00:28:33.260 align:middle line:84% So this J mu will have nothing to do 00:28:33.260 --> 00:28:34.630 align:middle line:90% with electromagnetic current. 00:28:34.630 --> 00:28:37.010 align:middle line:84% This is just a J mu with the generating function 00:28:37.010 --> 00:28:42.360 align:middle line:90% used to take derivatives. 00:28:42.360 --> 00:28:44.955 align:middle line:84% So this is just an external source. 00:28:44.955 --> 00:28:48.110 align:middle line:90% 00:28:48.110 --> 00:28:49.680 align:middle line:84% So now this is a Gaussian integral. 00:28:49.680 --> 00:28:54.920 align:middle line:84% So in principle, we can just do it. 00:28:54.920 --> 00:28:56.900 align:middle line:84% Because this is quadratic, and now 00:28:56.900 --> 00:28:59.780 align:middle line:84% we are familiar how to do the quadratic integral. 00:28:59.780 --> 00:29:01.610 align:middle line:90% We can just do it. 00:29:01.610 --> 00:29:11.565 align:middle line:84% And to do it inside, we always write it in the form like this. 00:29:11.565 --> 00:29:17.050 align:middle line:90% 00:29:17.050 --> 00:29:18.415 align:middle line:90% We write it in the matrix form. 00:29:18.415 --> 00:29:27.780 align:middle line:90% 00:29:27.780 --> 00:29:30.280 align:middle line:84% OK, we write it as a matrix form. 00:29:30.280 --> 00:29:34.440 align:middle line:84% And this K mu nu just can be read from here by integration 00:29:34.440 --> 00:29:35.650 align:middle line:90% by parts, et cetera. 00:29:35.650 --> 00:29:37.900 align:middle line:84% So let me just write down the answer. 00:29:37.900 --> 00:29:43.370 align:middle line:84% So this gave me mu just equal to-- 00:29:43.370 --> 00:29:50.490 align:middle line:84% so let me call this 0, call this 0 K mu nu. 00:29:50.490 --> 00:29:54.000 align:middle line:84% So this just given by partial square eta 00:29:54.000 --> 00:30:00.933 align:middle line:84% mu nu minus partial mu partial nu, and then 00:30:00.933 --> 00:30:01.725 align:middle line:90% the delta function. 00:30:01.725 --> 00:30:04.570 align:middle line:90% 00:30:04.570 --> 00:30:05.750 align:middle line:90% And the delta function. 00:30:05.750 --> 00:30:08.050 align:middle line:84% So this partial will only act on x. 00:30:08.050 --> 00:30:10.560 align:middle line:90% 00:30:10.560 --> 00:30:13.110 align:middle line:90% So naively, we can just-- 00:30:13.110 --> 00:30:14.850 align:middle line:84% so now this is just a Gaussian integral, 00:30:14.850 --> 00:30:17.835 align:middle line:84% so we can use principle just to directly write down the answer. 00:30:17.835 --> 00:30:26.590 align:middle line:90% 00:30:26.590 --> 00:30:28.090 align:middle line:90% Actually, let me keep this here. 00:30:28.090 --> 00:30:46.380 align:middle line:90% 00:30:46.380 --> 00:30:49.010 align:middle line:84% So in principle, we can just directly write down the answer, 00:30:49.010 --> 00:30:57.920 align:middle line:84% just this Z J equal to some number, some infinite number, 00:30:57.920 --> 00:31:03.365 align:middle line:84% which we never care, and the exponential i 00:31:03.365 --> 00:31:14.660 align:middle line:90% d 4x d 4y then J mu x. 00:31:14.660 --> 00:31:27.410 align:middle line:90% Then the K 0 mu nu minus 1. 00:31:27.410 --> 00:31:29.225 align:middle line:84% Yeah, so let me just write it like this. 00:31:29.225 --> 00:31:32.340 align:middle line:90% 00:31:32.340 --> 00:31:35.290 align:middle line:90% K0 minus 1 mu mu. 00:31:35.290 --> 00:31:39.030 align:middle line:90% 00:31:39.030 --> 00:31:42.762 align:middle line:90% x minus y J mu. 00:31:42.762 --> 00:31:43.262 align:middle line:90% OK? 00:31:43.262 --> 00:31:46.430 align:middle line:90% 00:31:46.430 --> 00:31:51.580 align:middle line:84% So this is just almost exactly the same as we do before. 00:31:51.580 --> 00:31:55.330 align:middle line:84% So this is like the kernel for your Gaussian integral. 00:31:55.330 --> 00:31:58.100 align:middle line:84% And you just take the inverse of it. 00:31:58.100 --> 00:32:01.150 align:middle line:84% And then you contract it with the source. 00:32:01.150 --> 00:32:05.920 align:middle line:84% OK, so you just know the only difference from the scalar 00:32:05.920 --> 00:32:13.660 align:middle line:84% case, or the spinor case, other than the functional space of x 00:32:13.660 --> 00:32:16.630 align:middle line:84% and y, now you also have a matrix 00:32:16.630 --> 00:32:20.745 align:middle line:84% in terms of Lorentz index, mu nu. 00:32:20.745 --> 00:32:22.120 align:middle line:84% Everything is the same as before. 00:32:22.120 --> 00:32:25.780 align:middle line:90% 00:32:25.780 --> 00:32:30.260 align:middle line:84% So it looks like, then, we already solved this theory, OK? 00:32:30.260 --> 00:32:33.060 align:middle line:90% But you should feel uneasy. 00:32:33.060 --> 00:32:34.710 align:middle line:84% You should feel easy because when 00:32:34.710 --> 00:32:37.410 align:middle line:84% we quantize it using the canonical method, 00:32:37.410 --> 00:32:40.750 align:middle line:84% we do have to go through some trouble. 00:32:40.750 --> 00:32:42.820 align:middle line:84% So how come, somehow, path integral just do it 00:32:42.820 --> 00:32:43.990 align:middle line:90% immediately? 00:32:43.990 --> 00:32:47.560 align:middle line:84% Somehow the trouble has to be conserved, 00:32:47.560 --> 00:32:51.850 align:middle line:84% no matter what method you use, no matter what method you use. 00:32:51.850 --> 00:32:56.750 align:middle line:84% Indeed, can someone guess what would be the potential trouble 00:32:56.750 --> 00:32:57.250 align:middle line:90% here? 00:32:57.250 --> 00:32:57.790 align:middle line:90% Yes? 00:32:57.790 --> 00:33:01.690 align:middle line:84% AUDIENCE: Are you overcounting, because when 00:33:01.690 --> 00:33:05.500 align:middle line:84% you integrate over, like, DA, you could have two-- 00:33:05.500 --> 00:33:07.740 align:middle line:90% two A's differ only by gauge? 00:33:07.740 --> 00:33:10.480 align:middle line:90% PROFESSOR: Yeah, right. 00:33:10.480 --> 00:33:14.830 align:middle line:84% AUDIENCE: Yeah, so then you're integrating over too many? 00:33:14.830 --> 00:33:17.290 align:middle line:90% PROFESSOR: Yeah, indeed. 00:33:17.290 --> 00:33:19.870 align:middle line:90% That's the correct statement. 00:33:19.870 --> 00:33:22.060 align:middle line:84% But if I just do it straightforwardly, 00:33:22.060 --> 00:33:25.345 align:middle line:84% somehow, then what's wrong with that? 00:33:25.345 --> 00:33:31.370 align:middle line:90% 00:33:31.370 --> 00:33:33.810 align:middle line:84% So, yeah, what you said is conceptually-- conceptually, 00:33:33.810 --> 00:33:36.930 align:middle line:84% you should suspect there's something wrong here. 00:33:36.930 --> 00:33:39.050 align:middle line:84% So whenever you conceptually suspect 00:33:39.050 --> 00:33:40.670 align:middle line:84% there's something wrong here, then you 00:33:40.670 --> 00:33:44.030 align:middle line:84% should technically be looking for some mathematical problem, 00:33:44.030 --> 00:33:46.130 align:middle line:84% because the conceptual mistake will always 00:33:46.130 --> 00:33:48.770 align:middle line:84% reflect in some mathematical difficulty. 00:33:48.770 --> 00:33:49.270 align:middle line:90% Yeah. 00:33:49.270 --> 00:33:50.383 align:middle line:90% Yes? 00:33:50.383 --> 00:33:51.800 align:middle line:84% AUDIENCE: So you have to integrate 00:33:51.800 --> 00:33:55.763 align:middle line:90% to measure [INAUDIBLE] 00:33:55.763 --> 00:33:56.430 align:middle line:90% PROFESSOR: Yeah. 00:33:56.430 --> 00:33:57.360 align:middle line:90% Yeah, you're right. 00:33:57.360 --> 00:33:59.563 align:middle line:84% It's just you integrate more things. 00:33:59.563 --> 00:34:01.980 align:middle line:84% You integrate more things, then you just get more infinite 00:34:01.980 --> 00:34:03.420 align:middle line:90% constant, but we don't care. 00:34:03.420 --> 00:34:06.210 align:middle line:90% 00:34:06.210 --> 00:34:09.090 align:middle line:84% AUDIENCE: Maybe like a sign will be off, 00:34:09.090 --> 00:34:10.860 align:middle line:84% because the sign was off earlier, right? 00:34:10.860 --> 00:34:14.730 align:middle line:84% So when you do the eigenvalues of the determinant or whatever, 00:34:14.730 --> 00:34:15.810 align:middle line:90% maybe it messes that up. 00:34:15.810 --> 00:34:16.800 align:middle line:90% PROFESSOR: Yeah. 00:34:16.800 --> 00:34:20.280 align:middle line:84% Yeah, but we have i, so it doesn't matter the sign. 00:34:20.280 --> 00:34:20.937 align:middle line:90% Yes? 00:34:20.937 --> 00:34:23.520 align:middle line:84% AUDIENCE: Does the K end up not being invertible or something? 00:34:23.520 --> 00:34:24.690 align:middle line:90% PROFESSOR: Yeah, exactly. 00:34:24.690 --> 00:34:29.048 align:middle line:84% So the key is that the K is actually not invertible. 00:34:29.048 --> 00:34:31.320 align:middle line:90% Turns out K minus 1-- 00:34:31.320 --> 00:34:38.639 align:middle line:84% so K 0 actually is not invertible. 00:34:38.639 --> 00:34:47.699 align:middle line:90% 00:34:47.699 --> 00:34:50.330 align:middle line:90% So the reason is simple. 00:34:50.330 --> 00:34:52.550 align:middle line:84% It's that because the gauge freedom 00:34:52.550 --> 00:34:55.949 align:middle line:90% in the original theory-- 00:34:55.949 --> 00:34:58.950 align:middle line:84% because the gauge freedom in the original theory, 00:34:58.950 --> 00:35:03.360 align:middle line:84% by gauge freedom, we mean that when you make a transformation 00:35:03.360 --> 00:35:08.850 align:middle line:84% by some arbitrary function, your action does not change. 00:35:08.850 --> 00:35:13.440 align:middle line:84% Then that means that this kernel must be invariant 00:35:13.440 --> 00:35:17.070 align:middle line:90% under some transformation. 00:35:17.070 --> 00:35:19.350 align:middle line:84% It's invariant under some transformation means 00:35:19.350 --> 00:35:22.170 align:middle line:84% that this must have zero eigenvector. 00:35:22.170 --> 00:35:25.290 align:middle line:84% When it has zero eigenvector, then they 00:35:25.290 --> 00:35:27.900 align:middle line:90% must not be invertible. 00:35:27.900 --> 00:35:28.920 align:middle line:90% OK? 00:35:28.920 --> 00:35:32.715 align:middle line:84% So now let's just see it explicitly.] To see it 00:35:32.715 --> 00:35:37.100 align:middle line:84% explicitly, it's easier to see this in momentum space. 00:35:37.100 --> 00:35:39.020 align:middle line:84% So if we see this in momentum space-- 00:35:39.020 --> 00:35:48.052 align:middle line:90% 00:35:48.052 --> 00:35:50.260 align:middle line:84% so in momentum space, we can just directly write down 00:35:50.260 --> 00:35:53.470 align:middle line:90% the K0 mu mu. 00:35:53.470 --> 00:35:59.220 align:middle line:84% It's just equal to k square, just k mu k 00:35:59.220 --> 00:36:04.240 align:middle line:90% nu minus k square eta mu mu. 00:36:04.240 --> 00:36:10.310 align:middle line:90% 00:36:10.310 --> 00:36:14.180 align:middle line:84% So now it's obvious this has a zero eigenvector 00:36:14.180 --> 00:36:19.020 align:middle line:84% because this is precisely the minus k square, 00:36:19.020 --> 00:36:20.700 align:middle line:90% the transverse projector. 00:36:20.700 --> 00:36:25.030 align:middle line:90% 00:36:25.030 --> 00:36:28.180 align:middle line:84% So P mu nu T is the projector into the direction 00:36:28.180 --> 00:36:30.820 align:middle line:90% perpendicular to k mu. 00:36:30.820 --> 00:36:35.370 align:middle line:84% So this satisfies the property that pT mu 00:36:35.370 --> 00:36:44.490 align:middle line:84% mu is equal to eta mu mu minus k mu k mu divided by k square, 00:36:44.490 --> 00:36:50.905 align:middle line:84% and satisfies the property p mu mu T k nu is always equal to 0. 00:36:50.905 --> 00:36:51.405 align:middle line:90% OK? 00:36:51.405 --> 00:36:56.770 align:middle line:90% 00:36:56.770 --> 00:36:58.480 align:middle line:84% So you can easily see it, because if you 00:36:58.480 --> 00:37:01.540 align:middle line:84% contract with this k mu, and k mu contract with here 00:37:01.540 --> 00:37:04.840 align:middle line:84% and give you k square, and cancel with k square here, 00:37:04.840 --> 00:37:07.940 align:middle line:84% and you have k mu, and then k mu contract with this one, 00:37:07.940 --> 00:37:08.710 align:middle line:90% you just get k mu. 00:37:08.710 --> 00:37:11.110 align:middle line:90% So they get canceled. 00:37:11.110 --> 00:37:13.450 align:middle line:84% So this is the projector to the transverse space. 00:37:13.450 --> 00:37:16.750 align:middle line:84% Of course, it has zero eigenvalue-- 00:37:16.750 --> 00:37:18.020 align:middle line:90% zero eigenvectors. 00:37:18.020 --> 00:37:21.490 align:middle line:90% 00:37:21.490 --> 00:37:25.000 align:middle line:84% So then we see that this is not invertible in momentum space, 00:37:25.000 --> 00:37:27.460 align:middle line:84% and then it will not be invertible in coordinate space. 00:37:27.460 --> 00:37:30.530 align:middle line:84% It will not be invertible in coordinate space. 00:37:30.530 --> 00:37:35.530 align:middle line:84% And now, in coordinate space, now we 00:37:35.530 --> 00:37:39.270 align:middle line:84% can see with this understanding, then 00:37:39.270 --> 00:37:42.740 align:middle line:84% now we can immediately see in coordinate space 00:37:42.740 --> 00:37:45.290 align:middle line:84% the eigenvector of zero eigenvalue 00:37:45.290 --> 00:37:48.240 align:middle line:84% precisely corresponding to a gauge transformation. 00:37:48.240 --> 00:37:56.950 align:middle line:90% 00:37:56.950 --> 00:38:04.990 align:middle line:84% It's because in coordinate space, k mu translate-- 00:38:04.990 --> 00:38:09.930 align:middle line:84% when you translate to coordinate space, means partial mu. 00:38:09.930 --> 00:38:16.640 align:middle line:84% So k mu to coordinate space means partial mu. 00:38:16.640 --> 00:38:28.760 align:middle line:84% So that means that for any function, partial mu lambda-- 00:38:28.760 --> 00:38:34.250 align:middle line:84% this is K0 mu nu, partial nu lambda must be 0. 00:38:34.250 --> 00:38:38.140 align:middle line:90% 00:38:38.140 --> 00:38:41.010 align:middle line:90% And you can check explicitly. 00:38:41.010 --> 00:38:44.970 align:middle line:84% So if you contract with partial mu here, 00:38:44.970 --> 00:38:48.870 align:middle line:84% and then you have a partial square and you have partial mu, 00:38:48.870 --> 00:38:52.800 align:middle line:84% and then eta mu mu will give you partial mu, again, they cancel. 00:38:52.800 --> 00:38:54.900 align:middle line:84% And of course, this is just precisely the gauge 00:38:54.900 --> 00:38:57.000 align:middle line:90% transformation. 00:38:57.000 --> 00:38:59.073 align:middle line:84% This is precisely the gauge transformation. 00:38:59.073 --> 00:39:03.150 align:middle line:90% 00:39:03.150 --> 00:39:05.230 align:middle line:84% OK, so the gauge transformation precisely 00:39:05.230 --> 00:39:09.330 align:middle line:84% is the zero eigenvector of this kernel. 00:39:09.330 --> 00:39:13.090 align:middle line:84% It's the eigenvector of zero eigenvalue of this kernel. 00:39:13.090 --> 00:39:15.090 align:middle line:84% So that's why this is not invertible. 00:39:15.090 --> 00:39:28.530 align:middle line:90% 00:39:28.530 --> 00:39:35.650 align:middle line:84% So the way to fix is simple because remember this picture. 00:39:35.650 --> 00:39:41.580 align:middle line:84% So this is the space of the full configurations of A mu. 00:39:41.580 --> 00:39:44.760 align:middle line:84% And A mu along those trajectories are equivalent. 00:39:44.760 --> 00:39:47.820 align:middle line:90% 00:39:47.820 --> 00:39:50.430 align:middle line:84% To quantize it, we need to just-- 00:39:50.430 --> 00:39:52.470 align:middle line:84% the physical configuration corresponding 00:39:52.470 --> 00:39:54.940 align:middle line:90% to a section of it. 00:39:54.940 --> 00:39:59.140 align:middle line:84% So what this says is that K0 is not 00:39:59.140 --> 00:40:02.440 align:middle line:84% invertible in this full space because you have 00:40:02.440 --> 00:40:05.253 align:middle line:84% zero eigenvectors along this direction, 00:40:05.253 --> 00:40:06.670 align:middle line:84% OK, zero eigenvector corresponding 00:40:06.670 --> 00:40:07.490 align:middle line:90% to this direction. 00:40:07.490 --> 00:40:12.370 align:middle line:84% But if we restrict to a cross-section, and then 00:40:12.370 --> 00:40:15.220 align:middle line:84% this zero eigenvector no longer exists. 00:40:15.220 --> 00:40:17.350 align:middle line:90% And then it will be invertible. 00:40:17.350 --> 00:40:21.370 align:middle line:84% OK, so once we fix the gauge, then we expect-- 00:40:21.370 --> 00:40:24.760 align:middle line:84% then the k mu after we fix the gauge should be invertible. 00:40:24.760 --> 00:40:29.400 align:middle line:90% 00:40:29.400 --> 00:40:31.820 align:middle line:90% Good? 00:40:31.820 --> 00:40:33.380 align:middle line:90% So now let's fix the gauge. 00:40:33.380 --> 00:40:37.340 align:middle line:90% 00:40:37.340 --> 00:40:48.390 align:middle line:84% So K0 mu nu restricted to a section 00:40:48.390 --> 00:40:50.960 align:middle line:90% should it be invertible. 00:40:50.960 --> 00:40:55.070 align:middle line:90% 00:40:55.070 --> 00:40:59.135 align:middle line:90% OK, so we need to fix the gauge. 00:40:59.135 --> 00:41:06.700 align:middle line:90% 00:41:06.700 --> 00:41:09.660 align:middle line:84% So we will do this for the Lorenz gauge. 00:41:09.660 --> 00:41:13.830 align:middle line:84% You can do similar things for the Coulomb gauge. 00:41:13.830 --> 00:41:15.450 align:middle line:84% The reason we do the Lorenz gauge, 00:41:15.450 --> 00:41:20.650 align:middle line:84% just the Lorenz gauge, later when we introduce interactions, 00:41:20.650 --> 00:41:23.040 align:middle line:84% we will always work with the Lorenz gauge. 00:41:23.040 --> 00:41:24.855 align:middle line:84% And when we discuss QED, we will always 00:41:24.855 --> 00:41:25.980 align:middle line:90% work with the Lorenz gauge. 00:41:25.980 --> 00:41:28.860 align:middle line:84% So here I will elaborate in the Lorenz gauge. 00:41:28.860 --> 00:41:29.730 align:middle line:90% Yes? 00:41:29.730 --> 00:41:31.180 align:middle line:84% AUDIENCE: Can you explain what you 00:41:31.180 --> 00:41:33.090 align:middle line:84% mean by when you do the cross-section, 00:41:33.090 --> 00:41:35.160 align:middle line:84% the eigenvectors here doesn't exist anymore? 00:41:35.160 --> 00:41:36.960 align:middle line:90% PROFESSOR: Oh, yeah. 00:41:36.960 --> 00:41:39.060 align:middle line:90% So this zero eigenvector-- 00:41:39.060 --> 00:41:43.110 align:middle line:84% so this is a zero eigenvector of the K0, right? 00:41:43.110 --> 00:41:46.530 align:middle line:84% So at the partial mu, partial mu lambda, just 00:41:46.530 --> 00:41:50.120 align:middle line:90% parameterize this direction. 00:41:50.120 --> 00:41:54.580 align:middle line:84% But if you restrict to a section of it, and then 00:41:54.580 --> 00:41:58.000 align:middle line:84% you are not allowed to worry away from it. 00:41:58.000 --> 00:42:00.070 align:middle line:84% And then this direction just gone, 00:42:00.070 --> 00:42:03.700 align:middle line:84% so you no longer have that zero eigenvector anymore. 00:42:03.700 --> 00:42:05.000 align:middle line:90% Yeah. 00:42:05.000 --> 00:42:05.690 align:middle line:90% Yes? 00:42:05.690 --> 00:42:08.510 align:middle line:84% AUDIENCE: Should there be a convolution in the equation 00:42:08.510 --> 00:42:11.480 align:middle line:90% on the bottom left? 00:42:11.480 --> 00:42:12.320 align:middle line:90% PROFESSOR: Yeah. 00:42:12.320 --> 00:42:13.190 align:middle line:90% Yeah, good. 00:42:13.190 --> 00:42:15.980 align:middle line:90% 00:42:15.980 --> 00:42:17.570 align:middle line:84% Indeed, you have to integrate-- yeah, 00:42:17.570 --> 00:42:18.950 align:middle line:90% you have to integrate over y. 00:42:18.950 --> 00:42:20.300 align:middle line:90% Yeah. 00:42:20.300 --> 00:42:21.710 align:middle line:90% Thank you. 00:42:21.710 --> 00:42:23.720 align:middle line:84% Yeah, if I write this precisely, you 00:42:23.720 --> 00:42:27.560 align:middle line:84% have to integrate over this y and y. 00:42:27.560 --> 00:42:29.180 align:middle line:90% Yeah, that's right. 00:42:29.180 --> 00:42:32.540 align:middle line:84% So this is the proper way to write it. 00:42:32.540 --> 00:42:34.030 align:middle line:90% Good. 00:42:34.030 --> 00:42:35.760 align:middle line:84% AUDIENCE: Why does this issue only 00:42:35.760 --> 00:42:37.710 align:middle line:90% happen for gauge symmetries? 00:42:37.710 --> 00:42:41.000 align:middle line:84% Like, why don't the other symmetries need to do the same? 00:42:41.000 --> 00:42:42.110 align:middle line:90% PROFESSOR: Yeah. 00:42:42.110 --> 00:42:48.580 align:middle line:84% Because other symmetries-- you have local symmetries. 00:42:48.580 --> 00:42:50.460 align:middle line:84% So the local symmetry tells you there's 00:42:50.460 --> 00:42:52.980 align:middle line:84% really some degree of freedom redundant. 00:42:52.980 --> 00:42:55.530 align:middle line:84% So this is really corresponding to-- 00:42:55.530 --> 00:43:00.240 align:middle line:84% so this lambda x, so all possible choice of lambda, 00:43:00.240 --> 00:43:04.290 align:middle line:84% is really corresponding to some finite trajectory 00:43:04.290 --> 00:43:06.370 align:middle line:90% in your configuration space. 00:43:06.370 --> 00:43:07.890 align:middle line:84% So if you have a global symmetry, 00:43:07.890 --> 00:43:10.870 align:middle line:84% global symmetry is independent of coordinates. 00:43:10.870 --> 00:43:13.470 align:middle line:84% So from this point of view, a global symmetry 00:43:13.470 --> 00:43:16.050 align:middle line:84% just relates to the configuration at this point 00:43:16.050 --> 00:43:20.400 align:middle line:84% to the configuration at that point and at that point. 00:43:20.400 --> 00:43:23.820 align:middle line:84% A global symmetry, because it's independent of spacetime, 00:43:23.820 --> 00:43:28.040 align:middle line:84% that transformation doesn't change your number 00:43:28.040 --> 00:43:29.590 align:middle line:90% of degrees of freedom. 00:43:29.590 --> 00:43:31.930 align:middle line:84% They don't correspond to actually a trajectory 00:43:31.930 --> 00:43:33.700 align:middle line:90% in your configuration space. 00:43:33.700 --> 00:43:37.540 align:middle line:90% 00:43:37.540 --> 00:43:38.260 align:middle line:90% Good. 00:43:38.260 --> 00:43:41.330 align:middle line:90% Other questions? 00:43:41.330 --> 00:43:43.180 align:middle line:90% OK, very good. 00:43:43.180 --> 00:43:45.710 align:middle line:90% So now we will fix the gauge. 00:43:45.710 --> 00:43:50.800 align:middle line:84% So in the path integral, to fix the gauge, in principle, 00:43:50.800 --> 00:43:53.320 align:middle line:90% is straightforward. 00:43:53.320 --> 00:43:54.865 align:middle line:84% So we will consider Lorenz gauge. 00:43:54.865 --> 00:43:59.350 align:middle line:90% 00:43:59.350 --> 00:44:00.100 align:middle line:90% OK. 00:44:00.100 --> 00:44:04.540 align:middle line:84% You can do the similar thing with the Coulomb gauge. 00:44:04.540 --> 00:44:08.620 align:middle line:84% Just Lorenz gauge, normally, we work with more. 00:44:08.620 --> 00:44:11.120 align:middle line:90% So how do we fix the gauge? 00:44:11.120 --> 00:44:15.130 align:middle line:84% So in the path integral, we do it easily. 00:44:15.130 --> 00:44:18.810 align:middle line:90% So we have this integral. 00:44:18.810 --> 00:44:24.120 align:middle line:84% Then we insert the delta function corresponding 00:44:24.120 --> 00:44:25.470 align:middle line:90% to the Lorenz gauge. 00:44:25.470 --> 00:44:30.810 align:middle line:90% 00:44:30.810 --> 00:44:37.120 align:middle line:84% So this should be considered as a delta function. 00:44:37.120 --> 00:44:38.860 align:middle line:84% So this is a delta function at one point, 00:44:38.860 --> 00:44:41.350 align:middle line:84% and then you take the product over all points. 00:44:41.350 --> 00:44:44.740 align:middle line:84% So this is the delta function in the functional space. 00:44:44.740 --> 00:44:50.460 align:middle line:84% And then we will have i S, this S Maxwell, 00:44:50.460 --> 00:44:55.680 align:middle line:84% and then you can have A J. You can consider the generating 00:44:55.680 --> 00:44:57.490 align:middle line:90% function. 00:44:57.490 --> 00:45:10.820 align:middle line:84% So this will restrict you on some cross-section, 00:45:10.820 --> 00:45:12.780 align:middle line:90% on some cross-section. 00:45:12.780 --> 00:45:13.280 align:middle line:90% OK. 00:45:13.280 --> 00:45:20.090 align:middle line:90% 00:45:20.090 --> 00:45:21.455 align:middle line:90% So let me make a remark here. 00:45:21.455 --> 00:45:24.017 align:middle line:90% 00:45:24.017 --> 00:45:26.475 align:middle line:84% So here, you actually have to be a little bit more careful. 00:45:26.475 --> 00:45:29.270 align:middle line:90% 00:45:29.270 --> 00:45:32.840 align:middle line:84% So what we actually really want to do-- 00:45:32.840 --> 00:45:37.835 align:middle line:84% OK, what we really want to do is we just want to insert here. 00:45:37.835 --> 00:45:41.720 align:middle line:90% 00:45:41.720 --> 00:45:46.370 align:middle line:84% We just want to insert here delta A mu 00:45:46.370 --> 00:45:48.740 align:middle line:90% equal to A mu gauge fixed. 00:45:48.740 --> 00:45:54.410 align:middle line:90% 00:45:54.410 --> 00:45:56.900 align:middle line:84% What we want, what we really want, to do 00:45:56.900 --> 00:46:01.940 align:middle line:84% is we want just to restrict A mu to some cross-section. 00:46:01.940 --> 00:46:05.507 align:middle line:90% 00:46:05.507 --> 00:46:07.340 align:middle line:84% But for example, here, for the Lorenz gauge, 00:46:07.340 --> 00:46:09.410 align:middle line:84% we can normally not solve this condition. 00:46:09.410 --> 00:46:11.930 align:middle line:84% So that's why we do here, because we 00:46:11.930 --> 00:46:15.560 align:middle line:84% don't know how to write this a mu fixed explicitly. 00:46:15.560 --> 00:46:17.540 align:middle line:90% OK? 00:46:17.540 --> 00:46:20.630 align:middle line:84% But going from here to here, actually there's 00:46:20.630 --> 00:46:26.190 align:middle line:84% a non-trivial Jacobian, because this is a non-trivial 00:46:26.190 --> 00:46:27.750 align:middle line:90% in the functional space. 00:46:27.750 --> 00:46:34.800 align:middle line:84% And then this is a non-trivial function on A mu. 00:46:34.800 --> 00:46:39.908 align:middle line:90% So in principle-- the related-- 00:46:39.908 --> 00:46:41.325 align:middle line:84% in principle, there is a Jacobian. 00:46:41.325 --> 00:46:47.122 align:middle line:90% 00:46:47.122 --> 00:46:48.580 align:middle line:84% So let me just write this explicit. 00:46:48.580 --> 00:46:52.760 align:middle line:90% 00:46:52.760 --> 00:47:00.820 align:middle line:84% A mu x are related by some Jacobian, then 00:47:00.820 --> 00:47:04.690 align:middle line:84% delta A mu minus delta A mu fixed-- 00:47:04.690 --> 00:47:11.660 align:middle line:90% uh minus A mu fixed, OK? 00:47:11.660 --> 00:47:16.550 align:middle line:90% 00:47:16.550 --> 00:47:20.000 align:middle line:84% But now, notice that this function is a linear function 00:47:20.000 --> 00:47:23.980 align:middle line:84% of A. So remember, when you do the Jacobian, 00:47:23.980 --> 00:47:25.480 align:middle line:90% you have to take the derivative. 00:47:25.480 --> 00:47:28.530 align:middle line:90% 00:47:28.530 --> 00:47:30.950 align:middle line:84% Yeah, just remember this function. 00:47:30.950 --> 00:47:33.950 align:middle line:90% Remember this formula. 00:47:33.950 --> 00:47:43.450 align:middle line:84% Delta f x is equal to 1 over f prime x delta x minus x 0. 00:47:43.450 --> 00:47:45.680 align:middle line:84% So x0 is a solution for f x equal to 0. 00:47:45.680 --> 00:47:46.180 align:middle line:90% OK? 00:47:46.180 --> 00:47:49.390 align:middle line:90% 00:47:49.390 --> 00:47:54.130 align:middle line:84% And so you have-- yeah, so this is a Jacobian when 00:47:54.130 --> 00:47:57.730 align:middle line:90% you convert the delta function. 00:47:57.730 --> 00:48:00.458 align:middle line:84% If you have multiple variables, then this 00:48:00.458 --> 00:48:01.375 align:middle line:90% would be the Jacobian. 00:48:01.375 --> 00:48:04.260 align:middle line:90% 00:48:04.260 --> 00:48:07.980 align:middle line:84% So this course, you take the derivative on this function. 00:48:07.980 --> 00:48:10.870 align:middle line:90% But this is a linear-- 00:48:10.870 --> 00:48:13.185 align:middle line:90% this thing is linear in A mu. 00:48:13.185 --> 00:48:14.560 align:middle line:84% When you take the derivative, you 00:48:14.560 --> 00:48:17.380 align:middle line:84% will get something which is independent of a mu. 00:48:17.380 --> 00:48:21.660 align:middle line:84% So this Jacobian will just involve in some-- 00:48:21.660 --> 00:48:25.050 align:middle line:84% it will be independent of a mu in the function space. 00:48:25.050 --> 00:48:28.158 align:middle line:84% Something independent of a mu is just some constant. 00:48:28.158 --> 00:48:30.075 align:middle line:84% So this just gives you some infinite constant. 00:48:30.075 --> 00:48:35.150 align:middle line:90% 00:48:35.150 --> 00:48:37.240 align:middle line:84% And so we always throw infinite numbers 00:48:37.240 --> 00:48:40.090 align:middle line:84% infinite number of ways, so we don't care about it. 00:48:40.090 --> 00:48:44.060 align:middle line:84% And so we can just write down this. 00:48:44.060 --> 00:48:45.680 align:middle line:90% So this is just a remark. 00:48:45.680 --> 00:48:52.550 align:middle line:90% 00:48:52.550 --> 00:48:56.040 align:middle line:84% So now we have to evaluate this guy. 00:48:56.040 --> 00:48:59.290 align:middle line:90% 00:48:59.290 --> 00:49:02.840 align:middle line:84% So now we have to evaluate this guy. 00:49:02.840 --> 00:49:05.810 align:middle line:84% It turns out that this guy is still not easy to solve. 00:49:05.810 --> 00:49:08.967 align:middle line:84% And for the same, it's not easy to do, just 00:49:08.967 --> 00:49:09.800 align:middle line:90% for the same reason. 00:49:09.800 --> 00:49:11.930 align:middle line:84% We don't know how to solve this A mu fixed. 00:49:11.930 --> 00:49:15.260 align:middle line:84% OK, we don't know how to solve this condition explicitly. 00:49:15.260 --> 00:49:18.290 align:middle line:84% So now I'm going to use some trick. 00:49:18.290 --> 00:49:21.920 align:middle line:90% I will use two tricks. 00:49:21.920 --> 00:49:25.010 align:middle line:84% OK, I will use two tricks to convert this 00:49:25.010 --> 00:49:30.900 align:middle line:84% into something manageable, to convert into this manageable. 00:49:30.900 --> 00:49:36.950 align:middle line:84% And so this trick applying to the Maxwell 00:49:36.950 --> 00:49:42.280 align:middle line:84% is like killing a little bird with a big cannon. 00:49:42.280 --> 00:49:47.350 align:middle line:84% And so, yeah, it works here, but it's real, 00:49:47.350 --> 00:49:50.270 align:middle line:84% genuine use is actually in the long Abelian gauge theory. 00:49:50.270 --> 00:49:52.630 align:middle line:84% So when you quantize the Yang-Mills theory, 00:49:52.630 --> 00:49:56.590 align:middle line:84% that becomes essential, because it will be very complicated 00:49:56.590 --> 00:49:58.120 align:middle line:90% quantizing using other methods. 00:49:58.120 --> 00:50:03.160 align:middle line:84% And so quantizing Yang-Mills theory become essential. 00:50:03.160 --> 00:50:05.620 align:middle line:84% But nevertheless, let me just tell you these two tricks, 00:50:05.620 --> 00:50:06.890 align:middle line:90% how to treat this here. 00:50:06.890 --> 00:50:07.390 align:middle line:90% OK? 00:50:07.390 --> 00:50:19.500 align:middle line:90% 00:50:19.500 --> 00:50:23.420 align:middle line:84% So this method is called Faddeev-Popov method, 00:50:23.420 --> 00:50:30.510 align:middle line:84% and it was invented by two Russians, former Soviet, 00:50:30.510 --> 00:50:33.720 align:middle line:90% Faddeev, Popov, in the '60s. 00:50:33.720 --> 00:50:38.640 align:middle line:90% 00:50:38.640 --> 00:50:42.350 align:middle line:84% So actually, there was a story behind it. 00:50:42.350 --> 00:50:45.290 align:middle line:84% So when they invented this method, 00:50:45.290 --> 00:50:49.160 align:middle line:84% they actually wanted to quantize not this Maxwell 00:50:49.160 --> 00:50:54.320 align:middle line:84% theory, this non-Abelian Yang-Mills theory, which later 00:50:54.320 --> 00:50:57.090 align:middle line:90% will become the standard model. 00:50:57.090 --> 00:51:00.440 align:middle line:84% It becomes standard model of particle physics, electroweak 00:51:00.440 --> 00:51:02.520 align:middle line:90% theory, QCD, et cetera. 00:51:02.520 --> 00:51:04.370 align:middle line:84% But when they worked on Yang-Mills theory, 00:51:04.370 --> 00:51:08.690 align:middle line:84% Yang-Mills theory was some small corner of mathematical physics. 00:51:08.690 --> 00:51:10.595 align:middle line:90% Nobody cared. 00:51:10.595 --> 00:51:12.290 align:middle line:84% So when they invented this method, 00:51:12.290 --> 00:51:15.140 align:middle line:90% nobody really paid attention. 00:51:15.140 --> 00:51:17.190 align:middle line:90% Nobody really paid attention. 00:51:17.190 --> 00:51:23.090 align:middle line:84% But then, in 1971, 't Hooft, who was a 21-year-old, 00:51:23.090 --> 00:51:25.970 align:middle line:84% like a graduate student, then he used 00:51:25.970 --> 00:51:30.920 align:middle line:84% that, which quantized the non-Abelian gauge 00:51:30.920 --> 00:51:33.620 align:middle line:90% theory coupled to Higgs. 00:51:33.620 --> 00:51:37.370 align:middle line:84% So before that, people thought that theory was inconsistent. 00:51:37.370 --> 00:51:39.200 align:middle line:84% But then he used this path integral 00:51:39.200 --> 00:51:41.340 align:middle line:90% to quantize that theory. 00:51:41.340 --> 00:51:44.340 align:middle line:84% And so that was a triumph, for which 't Hooft got a Nobel 00:51:44.340 --> 00:51:46.270 align:middle line:90% Prize a number of years ago. 00:51:46.270 --> 00:51:50.010 align:middle line:84% And when 't Hooft came out using this method, 00:51:50.010 --> 00:51:52.380 align:middle line:84% nobody could understand it in US. 00:51:52.380 --> 00:51:54.930 align:middle line:90% 00:51:54.930 --> 00:52:01.610 align:middle line:84% People like Weinberg, who wrote the electroweak theory, which 00:52:01.610 --> 00:52:05.470 align:middle line:84% 't Hooft quantized, he couldn't understand it. 00:52:05.470 --> 00:52:08.340 align:middle line:84% And there's only one person, one person who 00:52:08.340 --> 00:52:11.340 align:middle line:90% could understand it in the US. 00:52:11.340 --> 00:52:17.550 align:middle line:84% He was an assistant professor at Stony Brook. 00:52:17.550 --> 00:52:20.790 align:middle line:84% So he started his assistant professor going to Stony Brook, 00:52:20.790 --> 00:52:24.510 align:middle line:84% and in Stony Brook, there was a big shot, the Nobel Prize 00:52:24.510 --> 00:52:27.550 align:middle line:84% winner, C.N. Yang, who invented this Yang-Mills theory. 00:52:27.550 --> 00:52:31.710 align:middle line:84% So he was asking Yang, oh, what should I do for my research, 00:52:31.710 --> 00:52:35.100 align:middle line:84% ask for a Nobel Prize winner for advice. 00:52:35.100 --> 00:52:37.950 align:middle line:84% And Yang said, maybe you can look at this Faddeev-Popov 00:52:37.950 --> 00:52:39.010 align:middle line:90% stuff. 00:52:39.010 --> 00:52:41.280 align:middle line:84% And then it turned out later only 00:52:41.280 --> 00:52:43.560 align:middle line:84% himself could understand 't Hooft's paper. 00:52:43.560 --> 00:52:45.150 align:middle line:90% And then he made it-- 00:52:45.150 --> 00:52:47.430 align:middle line:84% of course, he also did some other good stuff. 00:52:47.430 --> 00:52:54.400 align:middle line:84% But immediately, he became the only person everybody went to 00:52:54.400 --> 00:52:56.590 align:middle line:84% because he really understood the story. 00:52:56.590 --> 00:52:57.310 align:middle line:90% Yeah, anyway. 00:52:57.310 --> 00:53:00.265 align:middle line:84% Yeah, so it's an actually very interesting story behind this. 00:53:00.265 --> 00:53:03.460 align:middle line:90% 00:53:03.460 --> 00:53:07.960 align:middle line:84% So first, we have to actually modify this Lorenz gauge 00:53:07.960 --> 00:53:11.030 align:middle line:90% condition a little bit. 00:53:11.030 --> 00:53:15.230 align:middle line:84% So rather than writing as partial mu A mu, 00:53:15.230 --> 00:53:27.080 align:middle line:84% let's try to impose the gauge partial mu A mu equal 00:53:27.080 --> 00:53:34.280 align:middle line:84% to some arbitrary function Bx, because partial mu a mu 00:53:34.280 --> 00:53:38.000 align:middle line:90% will give some trajectory here. 00:53:38.000 --> 00:53:41.422 align:middle line:84% But if I choose some arbitrary function Bx, 00:53:41.422 --> 00:53:44.570 align:middle line:84% it will just give you some other trajectory. 00:53:44.570 --> 00:53:46.760 align:middle line:90% It's still some trajectory. 00:53:46.760 --> 00:53:51.140 align:middle line:84% So what Bx you choose doesn't matter. 00:53:51.140 --> 00:53:56.270 align:middle line:84% So you can just replace this delta function partial mu 00:53:56.270 --> 00:54:04.240 align:middle line:84% A mu here by delta partial mu A mu Bx. 00:54:04.240 --> 00:54:06.580 align:middle line:84% So this is the first non-trivial step. 00:54:06.580 --> 00:54:09.400 align:middle line:90% OK? 00:54:09.400 --> 00:54:12.830 align:middle line:84% So this will give you some other trajectory. 00:54:12.830 --> 00:54:17.360 align:middle line:84% So now, since the Bx does not matter-- 00:54:17.360 --> 00:54:19.310 align:middle line:84% OK, since the Bx does not matter, 00:54:19.310 --> 00:54:23.080 align:middle line:84% we can actually integrate over Bx. 00:54:23.080 --> 00:54:27.910 align:middle line:90% And then the next step is to-- 00:54:27.910 --> 00:54:31.580 align:middle line:90% so first you replace that. 00:54:31.580 --> 00:54:33.020 align:middle line:90% So this is the first step. 00:54:33.020 --> 00:54:36.520 align:middle line:84% And then the next step is that you replace Z 00:54:36.520 --> 00:54:56.700 align:middle line:84% by you integrate over Bx with a measure B squared x, and then 00:54:56.700 --> 00:55:01.560 align:middle line:90% original Z. So original ZJ. 00:55:01.560 --> 00:55:06.100 align:middle line:90% 00:55:06.100 --> 00:55:13.950 align:middle line:84% We can now depend on B because I have replaced this condition. 00:55:13.950 --> 00:55:19.660 align:middle line:84% And now since this B does not matter, 00:55:19.660 --> 00:55:22.290 align:middle line:84% so I can just integrate over B with some arbitrary measure 00:55:22.290 --> 00:55:23.940 align:middle line:90% you want. 00:55:23.940 --> 00:55:25.410 align:middle line:90% OK? 00:55:25.410 --> 00:55:30.310 align:middle line:84% The only thing this does is just give you an infinite constant. 00:55:30.310 --> 00:55:33.510 align:middle line:84% Again, we don't care-- infinite constant. 00:55:33.510 --> 00:55:42.570 align:middle line:84% OK, so this gives you some infinite constant 00:55:42.570 --> 00:55:47.370 align:middle line:84% since this thing is supposed to not be dependent on B. 00:55:47.370 --> 00:55:53.340 align:middle line:84% But this, actually, is very useful because the reason 00:55:53.340 --> 00:55:57.390 align:middle line:84% we do this is because now, when we do this, the path integral 00:55:57.390 --> 00:56:02.200 align:middle line:84% will become the following, become DB DA mu. 00:56:02.200 --> 00:56:07.780 align:middle line:84% Now we have delta partial mu A mu minus Bx. 00:56:07.780 --> 00:56:10.890 align:middle line:90% 00:56:10.890 --> 00:56:17.840 align:middle line:84% And then we have exponential i S, then minus this i xi 1/2 00:56:17.840 --> 00:56:18.675 align:middle line:90% B square. 00:56:18.675 --> 00:56:22.530 align:middle line:90% 00:56:22.530 --> 00:56:33.190 align:middle line:84% And then I have this i J dot A. So now you see the benefit, 00:56:33.190 --> 00:56:36.454 align:middle line:84% because now I have integration over B. 00:56:36.454 --> 00:56:41.330 align:middle line:84% But the appears very simply in the delta function. 00:56:41.330 --> 00:56:43.750 align:middle line:84% So now I can evaluate the delta function 00:56:43.750 --> 00:56:51.440 align:middle line:84% by using the integral of B, by using the integral of B. 00:56:51.440 --> 00:56:53.780 align:middle line:84% So now I can just straightforwardly do 00:56:53.780 --> 00:56:54.440 align:middle line:90% the B integral. 00:56:54.440 --> 00:56:59.050 align:middle line:90% 00:56:59.050 --> 00:57:00.255 align:middle line:90% So now I get a new action. 00:57:00.255 --> 00:57:06.186 align:middle line:90% 00:57:06.186 --> 00:57:16.650 align:middle line:84% i S xi plus i J dot A. And now this i S xi-- 00:57:16.650 --> 00:57:26.280 align:middle line:84% so this S xi just given by your original Maxwell, then minus xi 00:57:26.280 --> 00:57:31.110 align:middle line:90% over 2 partial mu a mu squared. 00:57:31.110 --> 00:57:34.230 align:middle line:90% 00:57:34.230 --> 00:57:36.150 align:middle line:90% OK? 00:57:36.150 --> 00:57:39.950 align:middle line:84% So this is the reason we put B square here, 00:57:39.950 --> 00:57:42.810 align:middle line:84% is that when we evaluate the B integral, then 00:57:42.810 --> 00:57:45.770 align:middle line:90% we get something quadratic. 00:57:45.770 --> 00:57:48.090 align:middle line:90% And now we have that. 00:57:48.090 --> 00:57:52.440 align:middle line:84% And now you see this action is precisely 00:57:52.440 --> 00:57:57.570 align:middle line:90% this action we have there. 00:57:57.570 --> 00:58:00.120 align:middle line:84% OK, it's precisely that action given there. 00:58:00.120 --> 00:58:04.080 align:middle line:84% So we also derive it using path integral. 00:58:04.080 --> 00:58:07.170 align:middle line:84% We also derive it using path integral, using this trick. 00:58:07.170 --> 00:58:12.410 align:middle line:90% 00:58:12.410 --> 00:58:13.640 align:middle line:90% So now we just have this. 00:58:13.640 --> 00:58:16.270 align:middle line:90% 00:58:16.270 --> 00:58:17.800 align:middle line:90% Now we just have this. 00:58:17.800 --> 00:58:23.178 align:middle line:84% And now we can straightforwardly calculate-- now we 00:58:23.178 --> 00:58:25.470 align:middle line:84% can straightforwardly calculate this Gaussian integral. 00:58:25.470 --> 00:58:32.740 align:middle line:90% 00:58:32.740 --> 00:58:33.250 align:middle line:90% One second. 00:58:33.250 --> 00:58:37.008 align:middle line:90% 00:58:37.008 --> 00:58:39.050 align:middle line:84% Yeah, now we can just straightforwardly calculate 00:58:39.050 --> 00:58:39.633 align:middle line:90% this integral. 00:58:39.633 --> 00:59:05.260 align:middle line:90% 00:59:05.260 --> 00:59:14.490 align:middle line:84% By the way, the reason I say the story here is trivial, 00:59:14.490 --> 00:59:17.060 align:middle line:84% it's a little bit of killing a bird with-- 00:59:17.060 --> 00:59:19.580 align:middle line:84% it's because for the Maxwell case, 00:59:19.580 --> 00:59:22.010 align:middle line:84% this Jacobian is actually trivial. 00:59:22.010 --> 00:59:23.100 align:middle line:90% It's a constant. 00:59:23.100 --> 00:59:25.340 align:middle line:84% But if you do it for a non-Abelian gauge theory, 00:59:25.340 --> 00:59:28.430 align:middle line:84% say, for the electroweak theory or for QCD, 00:59:28.430 --> 00:59:30.980 align:middle line:84% this Jacobian is highly non-trivial. 00:59:30.980 --> 00:59:34.070 align:middle line:84% And part of this trick is also how 00:59:34.070 --> 00:59:37.520 align:middle line:84% to treat this Jacobian, which we don't have to do here. 00:59:37.520 --> 00:59:40.550 align:middle line:84% So here we have a much simpler story. 00:59:40.550 --> 00:59:44.000 align:middle line:84% So now we can just look at this path integral, which 00:59:44.000 --> 00:59:47.300 align:middle line:84% now S xi become invertible, because we already know 00:59:47.300 --> 00:59:49.730 align:middle line:90% from here this is invertible. 00:59:49.730 --> 00:59:52.010 align:middle line:84% At least for xi equal to 1, given by this, 00:59:52.010 --> 00:59:54.080 align:middle line:90% this is invertible. 00:59:54.080 --> 00:59:58.290 align:middle line:90% So you can actually now invert. 00:59:58.290 --> 01:00:12.220 align:middle line:84% So now we can write this S xi is equal to 1/2 d4x A mu x K mu -- 01:00:12.220 --> 01:00:16.870 align:middle line:90% K mu nu xi x minus y A nu y. 01:00:16.870 --> 01:00:19.920 align:middle line:90% 01:00:19.920 --> 01:00:21.220 align:middle line:90% And this K mu nu-- 01:00:21.220 --> 01:00:25.680 align:middle line:90% 01:00:25.680 --> 01:00:30.630 align:middle line:84% yeah, K mu nu just whatever you get from that thing, 01:00:30.630 --> 01:00:32.370 align:middle line:84% just whatever you get from that thing. 01:00:32.370 --> 01:00:37.800 align:middle line:84% And then the K mu mu now is invertible. 01:00:37.800 --> 01:00:44.690 align:middle line:84% And then you find this thing just becomes some constant, 01:00:44.690 --> 01:00:58.060 align:middle line:84% then exponential i d4x d4y J mu x, then K xi 01:00:58.060 --> 01:01:05.780 align:middle line:90% minus 1 mu nu x minus 1 J nu y. 01:01:05.780 --> 01:01:07.550 align:middle line:90% OK? 01:01:07.550 --> 01:01:11.420 align:middle line:84% And again, this inverse gives you the Feynman function. 01:01:11.420 --> 01:01:14.230 align:middle line:90% 01:01:14.230 --> 01:01:18.070 align:middle line:84% Again, this gives you the Feynman function. 01:01:18.070 --> 01:01:21.730 align:middle line:90% So the K xi minus 1 mu mu-- 01:01:21.730 --> 01:01:24.500 align:middle line:90% 01:01:24.500 --> 01:01:25.280 align:middle line:90% or mu nu. 01:01:25.280 --> 01:01:30.770 align:middle line:90% 01:01:30.770 --> 01:01:32.630 align:middle line:84% Yeah, so we're not going into detail here. 01:01:32.630 --> 01:01:34.380 align:middle line:90% The story is straightforward. 01:01:34.380 --> 01:01:45.350 align:middle line:84% So this should correspond to the Feynman function 01:01:45.350 --> 01:01:48.290 align:middle line:90% for the two gauge fields. 01:01:48.290 --> 01:01:56.977 align:middle line:90% 01:01:56.977 --> 01:01:59.060 align:middle line:84% OK, this should correspond to the Feynman function 01:01:59.060 --> 01:02:02.250 align:middle line:90% of these two gauge fields. 01:02:02.250 --> 01:02:05.410 align:middle line:84% So now let me just write down the answer. 01:02:05.410 --> 01:02:09.910 align:middle line:84% So I urge you to check it yourself explicitly. 01:02:09.910 --> 01:02:13.900 align:middle line:84% So you can easily write down what is K mu nu here. 01:02:13.900 --> 01:02:20.050 align:middle line:84% And then you need to write down what is K mu nu here, 01:02:20.050 --> 01:02:26.920 align:middle line:84% and then you can write down what is the K minus 1. 01:02:26.920 --> 01:02:30.280 align:middle line:84% So let me just write down the answer, which we actually 01:02:30.280 --> 01:02:34.780 align:middle line:84% will use later for the Feynman diagram calculation. 01:02:34.780 --> 01:02:43.050 align:middle line:84% OK, so you can show this given by a momentum space minus i k 01:02:43.050 --> 01:02:45.090 align:middle line:90% squared minus i epsilon. 01:02:45.090 --> 01:02:47.460 align:middle line:84% Remember, now, here it's massless. 01:02:47.460 --> 01:02:51.070 align:middle line:90% And you have eta mu nu minus-- 01:02:51.070 --> 01:03:00.840 align:middle line:84% plus xi minus 1 k mu k nu divided by k squared. 01:03:00.840 --> 01:03:04.550 align:middle line:90% 01:03:04.550 --> 01:03:08.620 align:middle line:84% So you can work out the inverse given by this. 01:03:08.620 --> 01:03:16.250 align:middle line:84% And so this can also be written in terms to the minus i k 01:03:16.250 --> 01:03:17.945 align:middle line:90% squared plus i epsilon. 01:03:17.945 --> 01:03:23.160 align:middle line:90% 01:03:23.160 --> 01:03:25.770 align:middle line:84% You can write it in terms of this transverse projector, 01:03:25.770 --> 01:03:37.760 align:middle line:84% k mu mu t, then plus xi P mu nu L. 01:03:37.760 --> 01:03:45.120 align:middle line:84% So P mu nu L is the longitudinal projector. 01:03:45.120 --> 01:03:51.900 align:middle line:84% You just define it by k mu k nu divided by k squared. 01:03:51.900 --> 01:03:53.520 align:middle line:90% That's divided by k squared. 01:03:53.520 --> 01:03:56.740 align:middle line:90% 01:03:56.740 --> 01:03:58.715 align:middle line:84% So this is an answer very simple. 01:03:58.715 --> 01:04:01.310 align:middle line:90% 01:04:01.310 --> 01:04:04.860 align:middle line:90% So now the interesting thing-- 01:04:04.860 --> 01:04:06.750 align:middle line:84% the two interesting things here-- 01:04:06.750 --> 01:04:11.985 align:middle line:90% 01:04:11.985 --> 01:04:13.360 align:middle line:84% so let me just make some remarks. 01:04:13.360 --> 01:04:20.580 align:middle line:90% 01:04:20.580 --> 01:04:31.900 align:middle line:84% So first, if you look at xi equal to 1, 01:04:31.900 --> 01:04:35.530 align:middle line:84% then this term vanishes, and then 01:04:35.530 --> 01:04:40.210 align:middle line:84% you just have eta mu divided by k squared minus i epsilon. 01:04:40.210 --> 01:04:44.240 align:middle line:84% So that's exactly what we expect from here. 01:04:44.240 --> 01:04:51.050 align:middle line:84% So here, you just essentially have the massless 1 over k 01:04:51.050 --> 01:04:55.480 align:middle line:84% square propagator for the massless particle. 01:04:55.480 --> 01:05:02.410 align:middle line:84% And then you have eta mu mu from the Lorenz signature. 01:05:02.410 --> 01:05:05.605 align:middle line:84% So for eta equal to 1-- xi equal to 1, indeed we covered that. 01:05:05.605 --> 01:05:08.570 align:middle line:90% 01:05:08.570 --> 01:05:10.855 align:middle line:84% So comment is that when xi is equal to 0. 01:05:10.855 --> 01:05:14.740 align:middle line:90% 01:05:14.740 --> 01:05:17.530 align:middle line:84% so this is just proportional to the transverse projector. 01:05:17.530 --> 01:05:21.240 align:middle line:90% 01:05:21.240 --> 01:05:22.510 align:middle line:90% To the transverse projector. 01:05:22.510 --> 01:05:28.410 align:middle line:84% But remember, the original Maxwell action, 01:05:28.410 --> 01:05:29.820 align:middle line:90% we used xi equal to 0. 01:05:29.820 --> 01:05:31.290 align:middle line:90% Actually, that's not invertible. 01:05:31.290 --> 01:05:34.110 align:middle line:90% 01:05:34.110 --> 01:05:36.890 align:middle line:84% So here, we're actually doing a different order. 01:05:36.890 --> 01:05:39.180 align:middle line:90% We first take a nonzero xi. 01:05:39.180 --> 01:05:40.680 align:middle line:90% We invert it. 01:05:40.680 --> 01:05:43.350 align:middle line:84% And then you can set actually xi equal to 0. 01:05:43.350 --> 01:05:44.920 align:middle line:90% OK, that's still works. 01:05:44.920 --> 01:05:48.720 align:middle line:84% And then xi equal to 0 still works, because we already 01:05:48.720 --> 01:05:50.010 align:middle line:90% fixed the gauge. 01:05:50.010 --> 01:05:53.232 align:middle line:90% We already fixed the gauge. 01:05:53.232 --> 01:05:54.440 align:middle line:90% So this is the second remark. 01:05:54.440 --> 01:05:59.500 align:middle line:90% 01:05:59.500 --> 01:06:07.480 align:middle line:84% And then the Wick's theorem for A mu 01:06:07.480 --> 01:06:10.540 align:middle line:84% just immediately follows from here. 01:06:10.540 --> 01:06:18.391 align:middle line:84% Just as in the case for the scalars and the fermions, 01:06:18.391 --> 01:06:19.750 align:middle line:90% it immediately follows. 01:06:19.750 --> 01:06:28.240 align:middle line:90% 01:06:28.240 --> 01:06:29.090 align:middle line:90% You just contract. 01:06:29.090 --> 01:06:33.020 align:middle line:84% We all get these Feynman functions between them. 01:06:33.020 --> 01:06:33.652 align:middle line:90% Yes? 01:06:33.652 --> 01:06:35.110 align:middle line:84% AUDIENCE: How difficult would it be 01:06:35.110 --> 01:06:37.840 align:middle line:84% to show that the path integral approach 01:06:37.840 --> 01:06:41.440 align:middle line:84% and the canonical approach are equivalent for the vector? 01:06:41.440 --> 01:06:50.630 align:middle line:84% PROFESSOR: Oh, no, it's not difficult. One simple thing-- 01:06:50.630 --> 01:06:55.700 align:middle line:84% this one point I'm going to make a little bit later, 01:06:55.700 --> 01:07:01.760 align:middle line:84% in a few minutes later, and actually for-- 01:07:01.760 --> 01:07:04.430 align:middle line:84% yeah, let me make some comment a little bit later. 01:07:04.430 --> 01:07:06.410 align:middle line:84% Yeah, they are completely equivalent. 01:07:06.410 --> 01:07:08.570 align:middle line:84% AUDIENCE: It just seems a little not obvious 01:07:08.570 --> 01:07:11.290 align:middle line:90% at first glance from the gauge. 01:07:11.290 --> 01:07:12.290 align:middle line:90% PROFESSOR: Right, right. 01:07:12.290 --> 01:07:13.920 align:middle line:90% Yeah. 01:07:13.920 --> 01:07:17.940 align:middle line:84% So for what we want to do, it's actually 01:07:17.940 --> 01:07:23.560 align:middle line:84% a little bit simpler, for reasons I'm going to mention. 01:07:23.560 --> 01:07:25.160 align:middle line:90% Other questions? 01:07:25.160 --> 01:07:25.660 align:middle line:90% Yes? 01:07:25.660 --> 01:07:32.900 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:07:32.900 --> 01:07:37.180 align:middle line:84% PROFESSOR: Yeah, it's because a projector is just 01:07:37.180 --> 01:07:40.170 align:middle line:90% a very simple way to-- 01:07:40.170 --> 01:07:45.740 align:middle line:90% so this is just-- 01:07:45.740 --> 01:07:49.750 align:middle line:84% so by definition, this is symmetric, in mu nu. 01:07:49.750 --> 01:07:52.720 align:middle line:84% This definition is symmetric in mu nu, 01:07:52.720 --> 01:07:56.650 align:middle line:84% because under time ordering, you can just exchange them. 01:07:56.650 --> 01:08:00.670 align:middle line:84% That means that in terms of Lorentz indices, 01:08:00.670 --> 01:08:04.330 align:middle line:84% you must be able to expand it in some symmetric tensors built 01:08:04.330 --> 01:08:07.400 align:middle line:90% from k mu. 01:08:07.400 --> 01:08:10.420 align:middle line:84% Because k mu is the only vector here. 01:08:10.420 --> 01:08:13.270 align:middle line:84% So that means you have to build this symmetric tensor 01:08:13.270 --> 01:08:16.120 align:middle line:90% from just eta mu mu and k mu. 01:08:16.120 --> 01:08:21.399 align:middle line:84% And the transverse and the longitudinal projectors 01:08:21.399 --> 01:08:25.540 align:middle line:84% are just the tensors you can build from k mu. 01:08:25.540 --> 01:08:26.595 align:middle line:90% Yes? 01:08:26.595 --> 01:08:27.970 align:middle line:84% AUDIENCE: Until we fix the gauge, 01:08:27.970 --> 01:08:30.609 align:middle line:84% we still have these parameters xi we still 01:08:30.609 --> 01:08:32.590 align:middle line:90% have a family of actions. 01:08:32.590 --> 01:08:34.257 align:middle line:84% Is that the residual gauge freedom? 01:08:34.257 --> 01:08:34.840 align:middle line:90% PROFESSOR: No. 01:08:34.840 --> 01:08:36.590 align:middle line:84% No, that's not the residual gauge freedom. 01:08:36.590 --> 01:08:39.340 align:middle line:84% Just so you can show that the physics are independent of xi. 01:08:39.340 --> 01:08:41.665 align:middle line:84% Yeah, you can take xi to be any value. 01:08:41.665 --> 01:08:45.250 align:middle line:90% 01:08:45.250 --> 01:08:46.420 align:middle line:90% OK, good. 01:08:46.420 --> 01:08:47.800 align:middle line:90% Other questions? 01:08:47.800 --> 01:08:48.300 align:middle line:90% Yes? 01:08:48.300 --> 01:08:51.720 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:08:51.720 --> 01:08:53.819 align:middle line:84% PROFESSOR: Good, good, good, good. 01:08:53.819 --> 01:08:56.310 align:middle line:84% Yeah, so that's the comment I'm going to make later. 01:08:56.310 --> 01:08:57.420 align:middle line:90% Yeah, that's coming. 01:08:57.420 --> 01:09:02.670 align:middle line:84% That's a good question, which I'm going to make later. 01:09:02.670 --> 01:09:05.149 align:middle line:84% Yeah, let me just make this comment 01:09:05.149 --> 01:09:07.520 align:middle line:90% and then we'll be clear. 01:09:07.520 --> 01:09:09.130 align:middle line:84% I need to make one comment first. 01:09:09.130 --> 01:09:13.170 align:middle line:90% 01:09:13.170 --> 01:09:18.060 align:middle line:84% So now, for various physical processes 01:09:18.060 --> 01:09:22.649 align:middle line:84% which we will consider later, we are 01:09:22.649 --> 01:09:26.640 align:middle line:84% interested in the following thing. 01:09:26.640 --> 01:09:29.130 align:middle line:84% So now we can just generalize immediately. 01:09:29.130 --> 01:09:41.729 align:middle line:90% 01:09:41.729 --> 01:09:56.570 align:middle line:84% Now we can just consider an interacting theory of A mu psi 01:09:56.570 --> 01:09:58.790 align:middle line:90% and phi. 01:09:58.790 --> 01:10:01.410 align:middle line:90% OK, so A mu will be the Maxwell. 01:10:01.410 --> 01:10:02.660 align:middle line:90% This will be some Dirac field. 01:10:02.660 --> 01:10:06.510 align:middle line:84% This will be some scalar field or even multiples of them. 01:10:06.510 --> 01:10:13.460 align:middle line:84% So for any such theory, we know how to compute the vacuum 01:10:13.460 --> 01:10:14.335 align:middle line:90% correlation function. 01:10:14.335 --> 01:10:17.200 align:middle line:90% 01:10:17.200 --> 01:10:22.420 align:middle line:84% Then just what we did before just immediately follows, 01:10:22.420 --> 01:10:34.910 align:middle line:84% just D phi then x exponential i S phi, and then 01:10:34.910 --> 01:10:37.820 align:middle line:90% D phi exponential i S. 01:10:37.820 --> 01:10:41.960 align:middle line:84% OK, so again, I always use omega to denote the vacuum 01:10:41.960 --> 01:10:44.360 align:middle line:90% of the interacting theory. 01:10:44.360 --> 01:10:46.100 align:middle line:84% And then, again, you can rewrite this 01:10:46.100 --> 01:10:48.980 align:middle line:90% as a theory of free theory. 01:10:48.980 --> 01:10:51.170 align:middle line:84% Now this becomes, in the free theory, 01:10:51.170 --> 01:11:03.487 align:middle line:84% 0 Tx exponential i interacting part 0, and the 0 exponential T 01:11:03.487 --> 01:11:07.280 align:middle line:90% SI 0. 01:11:07.280 --> 01:11:09.680 align:middle line:84% So now, again, for any interacting theory, 01:11:09.680 --> 01:11:11.690 align:middle line:84% we can just use the same procedure, 01:11:11.690 --> 01:11:14.675 align:middle line:84% and then we can just do Feynman diagrams. 01:11:14.675 --> 01:11:16.050 align:middle line:84% You can just do Feynman diagrams. 01:11:16.050 --> 01:11:24.770 align:middle line:90% 01:11:24.770 --> 01:11:25.270 align:middle line:90% OK? 01:11:25.270 --> 01:11:31.020 align:middle line:90% 01:11:31.020 --> 01:11:36.120 align:middle line:90% And then the Feynman rule-- 01:11:36.120 --> 01:11:39.845 align:middle line:90% we can just do Feynman diagrams. 01:11:39.845 --> 01:11:42.910 align:middle line:90% 01:11:42.910 --> 01:11:44.470 align:middle line:84% And the Feynman rules just follow. 01:11:44.470 --> 01:11:56.910 align:middle line:90% 01:11:56.910 --> 01:12:01.890 align:middle line:84% So now, as you already asked, this 01:12:01.890 --> 01:12:06.460 align:middle line:84% sounds a little bit too simple, because we just 01:12:06.460 --> 01:12:07.495 align:middle line:90% look at this action-- 01:12:07.495 --> 01:12:10.980 align:middle line:90% 01:12:10.980 --> 01:12:14.110 align:middle line:84% we just look at this action, but we 01:12:14.110 --> 01:12:19.680 align:middle line:84% haven't done what we promised you would do in your Pset. 01:12:19.680 --> 01:12:21.870 align:middle line:84% Because by doing this, we say you haven't-- 01:12:21.870 --> 01:12:27.360 align:middle line:84% you really have not completely fixed the imposed partial mu 01:12:27.360 --> 01:12:30.000 align:middle line:90% A mu yet. 01:12:30.000 --> 01:12:34.090 align:middle line:84% So remember, we said that when you have this, 01:12:34.090 --> 01:12:37.135 align:middle line:84% that will lead to this kind of equation. 01:12:37.135 --> 01:12:43.230 align:middle line:90% 01:12:43.230 --> 01:12:46.110 align:middle line:84% You still have to impose the boundary conditions to impose 01:12:46.110 --> 01:12:48.150 align:middle line:90% the partial mu A mu equal to 0. 01:12:48.150 --> 01:12:51.690 align:middle line:84% But now it seems like we are not doing anything like that. 01:12:51.690 --> 01:12:53.190 align:middle line:84% We are not doing anything like that. 01:12:53.190 --> 01:12:56.190 align:middle line:84% And also, it seems like we are not fixing the residual gauge 01:12:56.190 --> 01:12:58.110 align:middle line:90% freedom. 01:12:58.110 --> 01:13:00.070 align:middle line:90% Why don't we do it? 01:13:00.070 --> 01:13:04.980 align:middle line:84% So we don't have to do it because, here, we 01:13:04.980 --> 01:13:09.000 align:middle line:84% are only interested in calculating the vacuum 01:13:09.000 --> 01:13:10.810 align:middle line:90% correlation functions. 01:13:10.810 --> 01:13:15.580 align:middle line:84% And the vacuum is automatically a physical state. 01:13:15.580 --> 01:13:17.290 align:middle line:84% As you will see in your Pset, the vacuum 01:13:17.290 --> 01:13:19.090 align:middle line:90% is automatic physical state. 01:13:19.090 --> 01:13:23.740 align:middle line:84% And so it's already a physical state. 01:13:23.740 --> 01:13:26.390 align:middle line:84% So in the vacuum, all those unphysical degrees 01:13:26.390 --> 01:13:28.480 align:middle line:84% of freedon just automatically will decouple. 01:13:28.480 --> 01:13:31.180 align:middle line:84% You don't have to worry about them. 01:13:31.180 --> 01:13:33.130 align:middle line:84% And so that's why-- but if you're interested 01:13:33.130 --> 01:13:36.400 align:middle line:84% in the excited states, then you have to go through the exactly 01:13:36.400 --> 01:13:39.070 align:middle line:84% the same kind of procedure to get rid of-- 01:13:39.070 --> 01:13:44.020 align:middle line:84% to do a more complicated thing, as you would do in your Pset. 01:13:44.020 --> 01:13:48.910 align:middle line:84% But here, since we are computing just a special class 01:13:48.910 --> 01:13:52.270 align:middle line:84% of physical quantities, then just by doing this, 01:13:52.270 --> 01:13:53.500 align:middle line:90% it's already enough. 01:13:53.500 --> 01:13:56.570 align:middle line:84% You don't have to worry about other subtle stuff. 01:13:56.570 --> 01:14:00.230 align:middle line:84% You don't have to worry about other subtle stuff. 01:14:00.230 --> 01:14:06.970 align:middle line:84% OK, so I think we are done for the quantizing 01:14:06.970 --> 01:14:09.250 align:middle line:90% the Maxwell theory. 01:14:09.250 --> 01:14:11.220 align:middle line:90% So now we have all our elements. 01:14:11.220 --> 01:14:11.720 align:middle line:90% Yes? 01:14:11.720 --> 01:14:15.580 align:middle line:84% AUDIENCE: So when you have like multiple internal vertices, 01:14:15.580 --> 01:14:17.300 align:middle line:90% we used to-- 01:14:17.300 --> 01:14:20.690 align:middle line:84% the first interacting theory, when we had just 01:14:20.690 --> 01:14:21.950 align:middle line:90% expanded the [INAUDIBLE]. 01:14:21.950 --> 01:14:25.655 align:middle line:90% 01:14:25.655 --> 01:14:27.030 align:middle line:84% PROFESSOR: You do the same thing. 01:14:27.030 --> 01:14:27.940 align:middle line:90% You do the same thing, right? 01:14:27.940 --> 01:14:28.815 align:middle line:90% You just expand them. 01:14:28.815 --> 01:14:30.780 align:middle line:84% AUDIENCE: --each coupling, or like -- 01:14:30.780 --> 01:14:32.010 align:middle line:90% PROFESSOR: Just expand S i. 01:14:32.010 --> 01:14:35.820 align:middle line:84% Whatever is in S i, you just treat them together. 01:14:35.820 --> 01:14:37.350 align:middle line:84% AUDIENCE: I guess, I think, like, 01:14:37.350 --> 01:14:40.610 align:middle line:84% earlier in order of whatever [INAUDIBLE] whatever. 01:14:40.610 --> 01:14:41.420 align:middle line:90% PROFESSOR: Yeah. 01:14:41.420 --> 01:14:44.900 align:middle line:84% AUDIENCE: But now is it order of which-- 01:14:44.900 --> 01:14:47.690 align:middle line:90% PROFESSOR: Yeah, yeah. 01:14:47.690 --> 01:14:51.185 align:middle line:84% Indeed, so normally, we assume all the coupling 01:14:51.185 --> 01:14:54.890 align:middle line:84% S I are of the same order, and then you just expand the S I 01:14:54.890 --> 01:14:56.420 align:middle line:90% and then-- yeah. 01:14:56.420 --> 01:14:59.670 align:middle line:84% But indeed, you can see the special situation. 01:14:59.670 --> 01:15:01.400 align:middle line:84% You may want to expand some couplings, 01:15:01.400 --> 01:15:03.080 align:middle line:90% not expand some other couplings. 01:15:03.080 --> 01:15:05.000 align:middle line:84% Then that depends on specific situation. 01:15:05.000 --> 01:15:08.540 align:middle line:90% 01:15:08.540 --> 01:15:09.500 align:middle line:90% Other questions? 01:15:09.500 --> 01:15:12.120 align:middle line:90% 01:15:12.120 --> 01:15:14.300 align:middle line:90% OK, good. 01:15:14.300 --> 01:15:17.780 align:middle line:84% So let's conclude our discussion of quantizing 01:15:17.780 --> 01:15:19.460 align:middle line:90% of Maxwell's theory. 01:15:19.460 --> 01:15:26.420 align:middle line:84% And so now-- so we have quantized the photon. 01:15:26.420 --> 01:15:29.930 align:middle line:84% We have quantized the Maxwell's theory using the Coulomb gauge. 01:15:29.930 --> 01:15:32.630 align:middle line:84% Then we see the two-photon degrees of freedom. 01:15:32.630 --> 01:15:35.240 align:middle line:84% And then we also did canonical quantization 01:15:35.240 --> 01:15:37.760 align:middle line:84% for the Lorenz gauge, which you will finish in your pset. 01:15:37.760 --> 01:15:41.420 align:middle line:84% And then we also discussed how to treat it in path integral 01:15:41.420 --> 01:15:45.710 align:middle line:84% so that you can calculate these kind of questions, 01:15:45.710 --> 01:15:48.460 align:middle line:84% so that we can treat these kind of questions. 01:15:48.460 --> 01:15:53.100 align:middle line:84% And so now, the next goal-- now we have all the technicality, 01:15:53.100 --> 01:15:55.800 align:middle line:90% all the tools we need. 01:15:55.800 --> 01:15:58.380 align:middle line:90% Now we can tackle QED. 01:15:58.380 --> 01:16:01.890 align:middle line:84% So now, finally, we can tackle QED. 01:16:01.890 --> 01:16:08.250 align:middle line:84% So this is the theory of photon and the electron. 01:16:08.250 --> 01:16:10.670 align:middle line:84% This is the theory of photon and electron, 01:16:10.670 --> 01:16:14.840 align:middle line:84% and so we can consider the physical process 01:16:14.840 --> 01:16:17.150 align:middle line:84% in this theory, which A mu and psi are now 01:16:17.150 --> 01:16:19.160 align:middle line:90% interacting with each other. 01:16:19.160 --> 01:16:22.880 align:middle line:84% A mu and psi are now interacting with each other. 01:16:22.880 --> 01:16:25.640 align:middle line:90% So we will start doing it. 01:16:25.640 --> 01:16:27.130 align:middle line:90% I think we are out-- 01:16:27.130 --> 01:16:28.820 align:middle line:84% yeah, we only have two minutes left. 01:16:28.820 --> 01:16:33.290 align:middle line:84% So maybe we will finish a little bit early today. 01:16:33.290 --> 01:16:34.310 align:middle line:90% OK, yeah. 01:16:34.310 --> 01:16:36.520 align:middle line:90% So that's all for today. 01:16:36.520 --> 01:16:41.000 align:middle line:90%