WEBVTT 00:00:00.000 --> 00:00:01.944 align:middle line:90% [SQUEAKING] 00:00:01.944 --> 00:00:03.888 align:middle line:90% [RUSTLING] 00:00:03.888 --> 00:00:05.346 align:middle line:90% [CLICKING] 00:00:05.346 --> 00:00:11.490 align:middle line:90% 00:00:11.490 --> 00:00:14.530 align:middle line:84% HONG LIU: So in the last lecture, 00:00:14.530 --> 00:00:19.590 align:middle line:84% we have concluded the discussion of fermions. 00:00:19.590 --> 00:00:23.910 align:middle line:84% And now, we go to the last missing piece 00:00:23.910 --> 00:00:27.090 align:middle line:84% before we can talk about the QED. 00:00:27.090 --> 00:00:30.540 align:middle line:84% It's how to quantize the Maxwell field, OK? 00:00:30.540 --> 00:00:34.110 align:middle line:90% How to get photon, OK? 00:00:34.110 --> 00:00:35.940 align:middle line:90% And so today, we start. 00:00:35.940 --> 00:00:49.680 align:middle line:90% 00:00:49.680 --> 00:00:50.640 align:middle line:90% OK? 00:00:50.640 --> 00:00:53.670 align:middle line:90% So this is a short chapter. 00:00:53.670 --> 00:00:55.500 align:middle line:84% I think we should be able to finish it 00:00:55.500 --> 00:00:59.550 align:middle line:90% just this week in two lectures. 00:00:59.550 --> 00:01:04.500 align:middle line:84% So first, let me just remind you of some aspects 00:01:04.500 --> 00:01:07.440 align:middle line:90% of the classical Maxwell theory. 00:01:07.440 --> 00:01:12.270 align:middle line:90% 00:01:12.270 --> 00:01:16.200 align:middle line:84% And then we will talk about this quantization, OK? 00:01:16.200 --> 00:01:19.490 align:middle line:90% 00:01:19.490 --> 00:01:25.790 align:middle line:84% So the Lagrangian, say, for the classical Maxwell 00:01:25.790 --> 00:01:29.430 align:middle line:90% theory can be written as-- 00:01:29.430 --> 00:01:32.735 align:middle line:84% Lagrangian density can be written as F mu nu. 00:01:32.735 --> 00:01:37.010 align:middle line:90% 00:01:37.010 --> 00:01:39.800 align:middle line:84% So F mu nu is just the standard-- 00:01:39.800 --> 00:01:42.380 align:middle line:90% 00:01:42.380 --> 00:01:48.770 align:middle line:84% partial mu A nu equal to partial nu A mu. 00:01:48.770 --> 00:01:51.335 align:middle line:84% And the J mu is just the electromagnetic current. 00:01:51.335 --> 00:01:57.170 align:middle line:90% 00:01:57.170 --> 00:01:59.120 align:middle line:90% OK. 00:01:59.120 --> 00:02:00.770 align:middle line:84% J mu just electromagnetic current. 00:02:00.770 --> 00:02:02.330 align:middle line:90% OK. 00:02:02.330 --> 00:02:08.259 align:middle line:84% And then so from this Lagrangian density, 00:02:08.259 --> 00:02:12.530 align:middle line:84% then you can derive the equation motion, which is just 00:02:12.530 --> 00:02:15.360 align:middle line:90% the Maxwell equation, OK? 00:02:15.360 --> 00:02:23.370 align:middle line:84% So the Maxwell equation is given by partial mu F mu 00:02:23.370 --> 00:02:28.220 align:middle line:90% nu equal to minus J mu, OK? 00:02:28.220 --> 00:02:31.260 align:middle line:90% 00:02:31.260 --> 00:02:35.140 align:middle line:84% And so this is the familiar equation 00:02:35.140 --> 00:02:40.680 align:middle line:84% which we see from classical electrodynamics, OK? 00:02:40.680 --> 00:02:51.250 align:middle line:84% And so if you act partial mu on both sides-- 00:02:51.250 --> 00:02:54.195 align:middle line:84% so now, let's consider act partial mu on both sides. 00:02:54.195 --> 00:02:59.470 align:middle line:90% 00:02:59.470 --> 00:03:04.110 align:middle line:84% Since F mu mu is, by definition, antisymmetric, 00:03:04.110 --> 00:03:06.680 align:middle line:90% so this is automatically 0. 00:03:06.680 --> 00:03:11.510 align:middle line:84% And so this tells you that by consistency, the current 00:03:11.510 --> 00:03:14.270 align:middle line:90% has to be conserved, OK? 00:03:14.270 --> 00:03:16.610 align:middle line:84% The current which is appearing on the right-hand side 00:03:16.610 --> 00:03:18.860 align:middle line:84% of the Maxwell equation has to be conserved. 00:03:18.860 --> 00:03:21.920 align:middle line:84% Otherwise, you don't have a consistent equations, OK? 00:03:21.920 --> 00:03:30.910 align:middle line:90% 00:03:30.910 --> 00:03:40.360 align:middle line:84% And so another important feature of the Maxwell equation 00:03:40.360 --> 00:03:44.190 align:middle line:90% is so-called the gauge symmetry. 00:03:44.190 --> 00:03:46.290 align:middle line:90% So the Maxwell action-- 00:03:46.290 --> 00:03:51.060 align:middle line:84% both the action and the equation motion are invariant under-- 00:03:51.060 --> 00:03:55.700 align:middle line:90% so the L, yeah the action. 00:03:55.700 --> 00:04:01.340 align:middle line:90% 00:04:01.340 --> 00:04:06.050 align:middle line:84% So you integrate this over full spacetime. 00:04:06.050 --> 00:04:15.980 align:middle line:84% So it's invariant under the following transformation-- 00:04:15.980 --> 00:04:24.080 align:middle line:84% that A mu goes to A mu prime equal to A mu plus 00:04:24.080 --> 00:04:26.930 align:middle line:90% partial mu lambda x. 00:04:26.930 --> 00:04:28.196 align:middle line:90% OK? 00:04:28.196 --> 00:04:37.620 align:middle line:84% And here the lambda x is an arbitrary function 00:04:37.620 --> 00:04:40.590 align:middle line:90% of spacetime coordinates, OK? 00:04:40.590 --> 00:04:41.715 align:middle line:90% It's an arbitrary function. 00:04:41.715 --> 00:04:44.370 align:middle line:90% 00:04:44.370 --> 00:04:49.610 align:middle line:84% So the fact that the Maxwell theory-- 00:04:49.610 --> 00:04:56.000 align:middle line:84% the action is invariant under this transformation. 00:04:56.000 --> 00:04:57.590 align:middle line:84% So you can easily check yourself. 00:04:57.590 --> 00:05:01.430 align:middle line:84% Under this transformation, F mu mu does not change, OK? 00:05:01.430 --> 00:05:04.520 align:middle line:84% So the transformation between this and this-- they 00:05:04.520 --> 00:05:06.720 align:middle line:90% just cancel each other. 00:05:06.720 --> 00:05:09.485 align:middle line:90% And so F mu nu does not change. 00:05:09.485 --> 00:05:11.610 align:middle line:84% And then you see the equation of motion, of course, 00:05:11.610 --> 00:05:13.370 align:middle line:84% is invariant under this transformation 00:05:13.370 --> 00:05:15.230 align:middle line:90% because F mu nu does not change. 00:05:15.230 --> 00:05:17.450 align:middle line:90% And J mu does not transform. 00:05:17.450 --> 00:05:20.960 align:middle line:84% And also, the action is invariant 00:05:20.960 --> 00:05:25.250 align:middle line:84% because the F mu nu does not change. 00:05:25.250 --> 00:05:30.140 align:middle line:84% And naively, this term does change under the transformation 00:05:30.140 --> 00:05:32.930 align:middle line:84% because you have this partial mu lambda, 00:05:32.930 --> 00:05:35.420 align:middle line:84% but the additional term corresponding to partial mu 00:05:35.420 --> 00:05:36.950 align:middle line:90% lambda times J mu-- 00:05:36.950 --> 00:05:41.090 align:middle line:84% but then you can do integration by parts under the integration 00:05:41.090 --> 00:05:42.840 align:middle line:90% because J mu is conserved. 00:05:42.840 --> 00:05:45.620 align:middle line:84% And so the action is invariant, OK? 00:05:45.620 --> 00:05:48.830 align:middle line:90% So the action is invariant. 00:05:48.830 --> 00:05:53.390 align:middle line:84% And so this symmetry is a little bit different from the symmetry 00:05:53.390 --> 00:05:56.600 align:middle line:84% we have discussed before because the symmetry depends 00:05:56.600 --> 00:06:02.420 align:middle line:84% on the function of arbitrary spacetime, 00:06:02.420 --> 00:06:04.250 align:middle line:90% of spacetime coordinates. 00:06:04.250 --> 00:06:10.500 align:middle line:90% 00:06:10.500 --> 00:06:11.190 align:middle line:90% OK. 00:06:11.190 --> 00:06:13.740 align:middle line:84% So for this reason and the previous symmetry 00:06:13.740 --> 00:06:17.340 align:middle line:84% we have talked about, the transformation parameters 00:06:17.340 --> 00:06:21.390 align:middle line:84% are independent of the spacetime coordinates, OK? 00:06:21.390 --> 00:06:23.040 align:middle line:90% They're constants. 00:06:23.040 --> 00:06:25.455 align:middle line:84% And so that's why this is called the local symmetry. 00:06:25.455 --> 00:06:33.400 align:middle line:90% 00:06:33.400 --> 00:06:37.060 align:middle line:84% Sometimes, it's also called, just for historical reason, 00:06:37.060 --> 00:06:37.705 align:middle line:90% gauge symmetry. 00:06:37.705 --> 00:06:44.720 align:middle line:90% 00:06:44.720 --> 00:06:45.220 align:middle line:90% OK. 00:06:45.220 --> 00:06:47.630 align:middle line:84% It's called local symmetry or gauge symmetry-- 00:06:47.630 --> 00:06:50.570 align:middle line:90% 00:06:50.570 --> 00:06:56.190 align:middle line:84% and yeah, in contrast to the global symmetry, 00:06:56.190 --> 00:06:59.390 align:middle line:84% which we encountered before, whose transformation parameters 00:06:59.390 --> 00:07:02.520 align:middle line:90% are spacetime-independent. 00:07:02.520 --> 00:07:10.420 align:middle line:84% So the gauge symmetry and the global symmetries-- 00:07:10.420 --> 00:07:12.760 align:middle line:84% even though they are all symmetries, 00:07:12.760 --> 00:07:16.960 align:middle line:84% they have very, very different physical interpretation, OK? 00:07:16.960 --> 00:07:19.810 align:middle line:84% So global symmetry is a genuine symmetry. 00:07:19.810 --> 00:07:24.910 align:middle line:84% So the theory's translation variant 00:07:24.910 --> 00:07:28.630 align:middle line:84% means that what's happening here in Boston 00:07:28.630 --> 00:07:31.750 align:middle line:84% is the same happening in, say, Washington, DC, OK? 00:07:31.750 --> 00:07:35.230 align:middle line:84% Generally, things are invariant under that. 00:07:35.230 --> 00:07:39.400 align:middle line:84% But the local symmetry is very different, OK? 00:07:39.400 --> 00:07:41.275 align:middle line:90% So the local symmetry means-- 00:07:41.275 --> 00:07:47.510 align:middle line:90% 00:07:47.510 --> 00:08:00.620 align:middle line:84% so the presence of gauge symmetry 00:08:00.620 --> 00:08:17.210 align:middle line:84% implies that the system has redundant degrees of freedom. 00:08:17.210 --> 00:08:31.380 align:middle line:90% 00:08:31.380 --> 00:08:34.000 align:middle line:90% OK? 00:08:34.000 --> 00:08:36.820 align:middle line:84% So let me just give you a cartoon 00:08:36.820 --> 00:08:39.190 align:middle line:90% to illustrate this point. 00:08:39.190 --> 00:08:43.510 align:middle line:84% So the space of A mu, of course, is 00:08:43.510 --> 00:08:44.890 align:middle line:90% an infinite dimensional space. 00:08:44.890 --> 00:08:46.510 align:middle line:90% But let me just-- 00:08:46.510 --> 00:08:51.940 align:middle line:84% imagine this blackboard is the space of A mu, OK? 00:08:51.940 --> 00:08:55.460 align:middle line:90% This plane is the space of A mu. 00:08:55.460 --> 00:08:57.500 align:middle line:84% Under this gauge transformation, it 00:08:57.500 --> 00:09:00.700 align:middle line:90% can be viewed as the following. 00:09:00.700 --> 00:09:09.250 align:middle line:84% Say for each A mu, imagine you make an arbitrary 00:09:09.250 --> 00:09:10.780 align:middle line:90% transformation-- 00:09:10.780 --> 00:09:12.370 align:middle line:90% this lambda x. 00:09:12.370 --> 00:09:13.930 align:middle line:90% And so this line-- 00:09:13.930 --> 00:09:21.250 align:middle line:84% so let's denote the space of lambda x. 00:09:21.250 --> 00:09:23.246 align:middle line:90% OK? 00:09:23.246 --> 00:09:26.553 align:middle line:84% So at each point A mu, you can make a transformation 00:09:26.553 --> 00:09:27.970 align:middle line:84% for different lambda corresponding 00:09:27.970 --> 00:09:30.820 align:middle line:90% to different points, OK? 00:09:30.820 --> 00:09:32.860 align:middle line:84% And if you start with a different A mu, 00:09:32.860 --> 00:09:35.380 align:middle line:90% then you have a different-- 00:09:35.380 --> 00:09:36.010 align:middle line:90% yeah. 00:09:36.010 --> 00:09:37.660 align:middle line:90% So each of them is called-- 00:09:37.660 --> 00:09:43.000 align:middle line:90% 00:09:43.000 --> 00:09:54.810 align:middle line:84% so each of them is called the orbit of gauge transformations. 00:09:54.810 --> 00:09:58.950 align:middle line:90% 00:09:58.950 --> 00:09:59.940 align:middle line:90% OK? 00:09:59.940 --> 00:10:03.270 align:middle line:84% And they are parameterized by lambda, OK? 00:10:03.270 --> 00:10:05.860 align:middle line:84% And they're parameterized by lambda. 00:10:05.860 --> 00:10:08.320 align:middle line:84% And the fact that the physics is invariant 00:10:08.320 --> 00:10:12.990 align:middle line:84% under this transformation means that the physics 00:10:12.990 --> 00:10:16.050 align:middle line:84% at this point for this choice of A 00:10:16.050 --> 00:10:19.470 align:middle line:84% is the same as for this choice of A, OK? 00:10:19.470 --> 00:10:22.590 align:middle line:84% And similarly, the physics on this joint-- 00:10:22.590 --> 00:10:26.610 align:middle line:84% just physics on the same orbit is the same, OK? 00:10:26.610 --> 00:10:29.300 align:middle line:90% They describe the same physics. 00:10:29.300 --> 00:10:34.820 align:middle line:84% So in other words, only a cross-section-- 00:10:34.820 --> 00:10:37.310 align:middle line:90% so you can have many orbits, OK? 00:10:37.310 --> 00:10:42.950 align:middle line:84% So in each orbit, we can just choose a representative point. 00:10:42.950 --> 00:10:44.810 align:middle line:84% And then the physics only depends 00:10:44.810 --> 00:10:49.340 align:middle line:84% on a cross-section of all these different orbits, OK? 00:10:49.340 --> 00:11:07.150 align:middle line:84% So physics only depends on a cross-section, OK? 00:11:07.150 --> 00:11:10.930 align:middle line:84% So a cross-section of this orbit, OK? 00:11:10.930 --> 00:11:11.680 align:middle line:90% Yes? 00:11:11.680 --> 00:11:13.525 align:middle line:90% AUDIENCE: What's lambda? 00:11:13.525 --> 00:11:14.400 align:middle line:90% HONG LIU: Yeah, yeah. 00:11:14.400 --> 00:11:15.800 align:middle line:90% Lambda. 00:11:15.800 --> 00:11:17.270 align:middle line:84% So different points corresponding 00:11:17.270 --> 00:11:19.690 align:middle line:90% to different lambda. 00:11:19.690 --> 00:11:23.060 align:middle line:84% So you view this, really, as an infinite dimensional space. 00:11:23.060 --> 00:11:27.160 align:middle line:84% And so each point corresponding to a different choice of lambda 00:11:27.160 --> 00:11:30.190 align:middle line:84% because lambda can be arbitrary function, OK? 00:11:30.190 --> 00:11:33.430 align:middle line:90% And yeah. 00:11:33.430 --> 00:11:34.000 align:middle line:90% So yeah. 00:11:34.000 --> 00:11:44.320 align:middle line:84% Let me just parameterize by lambda x. 00:11:44.320 --> 00:11:44.820 align:middle line:90% OK. 00:11:44.820 --> 00:11:54.876 align:middle line:90% 00:11:54.876 --> 00:12:02.700 align:middle line:84% So the fact that we can choose this lambda arbitrarily-- 00:12:02.700 --> 00:12:10.250 align:middle line:84% so this means in this case, there's 00:12:10.250 --> 00:12:17.260 align:middle line:90% one scalar degree of freedom. 00:12:17.260 --> 00:12:19.440 align:middle line:84% So lambda is just a scalar field. 00:12:19.440 --> 00:12:25.620 align:middle line:84% It can be viewed as a scalar field in spacetime coordinates. 00:12:25.620 --> 00:12:32.000 align:middle line:84% So this one scalar degree of freedom as parameterized 00:12:32.000 --> 00:12:45.890 align:middle line:90% by lambda is redundant. 00:12:45.890 --> 00:12:48.220 align:middle line:90% OK? 00:12:48.220 --> 00:12:54.380 align:middle line:84% Because any point on this orbit-- 00:12:54.380 --> 00:12:57.770 align:middle line:84% they carry exactly the same physics. 00:12:57.770 --> 00:13:01.500 align:middle line:84% They carry the exact same physics. 00:13:01.500 --> 00:13:08.665 align:middle line:90% And so any questions on this? 00:13:08.665 --> 00:13:12.420 align:middle line:90% 00:13:12.420 --> 00:13:13.760 align:middle line:90% OK. 00:13:13.760 --> 00:13:17.450 align:middle line:84% So if you remember in classic E&M, 00:13:17.450 --> 00:13:28.800 align:middle line:84% when you solve the Maxwell equation, the physical object 00:13:28.800 --> 00:13:31.450 align:middle line:84% or the physical quantities-- they 00:13:31.450 --> 00:13:34.420 align:middle line:84% are all independent of the choice. 00:13:34.420 --> 00:13:34.920 align:middle line:90% Yeah. 00:13:34.920 --> 00:13:38.460 align:middle line:84% So each choice of this section is called a gauge, OK? 00:13:38.460 --> 00:13:40.470 align:middle line:90% Choice of a gauge. 00:13:40.470 --> 00:13:43.387 align:middle line:84% And so the choice of a section is 00:13:43.387 --> 00:13:44.595 align:middle line:90% called the choice of a gauge. 00:13:44.595 --> 00:13:49.370 align:middle line:90% 00:13:49.370 --> 00:13:51.380 align:middle line:84% So when you solve the Maxwell equation, 00:13:51.380 --> 00:13:53.780 align:middle line:84% typically, you choose such a section. 00:13:53.780 --> 00:13:55.400 align:middle line:90% You choose a gauge. 00:13:55.400 --> 00:13:58.997 align:middle line:84% And then you can just solve the Maxwell equation 00:13:58.997 --> 00:14:01.580 align:middle line:84% within that gauge so that you no longer have redundant degrees 00:14:01.580 --> 00:14:03.110 align:middle line:90% of freedom, OK? 00:14:03.110 --> 00:14:05.600 align:middle line:84% So often, this gauge is chosen so that you 00:14:05.600 --> 00:14:09.380 align:middle line:84% can solve the Maxwell equation, simplify 00:14:09.380 --> 00:14:14.620 align:middle line:84% the task of solving the Maxwell equation, OK? 00:14:14.620 --> 00:14:28.260 align:middle line:84% So we often solve the Maxwell equation by fixing a gauge. 00:14:28.260 --> 00:14:34.930 align:middle line:90% 00:14:34.930 --> 00:14:35.830 align:middle line:90% OK. 00:14:35.830 --> 00:14:44.070 align:middle line:84% So there are two common examples which we use 00:14:44.070 --> 00:14:46.600 align:middle line:90% to solve the Maxwell equations. 00:14:46.600 --> 00:14:48.210 align:middle line:84% So one is called the Coulomb gauge. 00:14:48.210 --> 00:14:55.630 align:middle line:90% 00:14:55.630 --> 00:14:56.130 align:middle line:90% OK? 00:14:56.130 --> 00:15:04.760 align:middle line:84% So in this gauge, you require the spatial components of A mu 00:15:04.760 --> 00:15:07.190 align:middle line:90% satisfy this equation, OK? 00:15:07.190 --> 00:15:11.150 align:middle line:84% So I will use the notation that A mu 00:15:11.150 --> 00:15:14.240 align:middle line:84% equal to phi-- so the zeroth component I call phi. 00:15:14.240 --> 00:15:18.950 align:middle line:84% And then the spatial component, the time component 00:15:18.950 --> 00:15:21.930 align:middle line:84% I call phi and the spatial component I just call A vector. 00:15:21.930 --> 00:15:27.800 align:middle line:84% And also, I often write J mu as the rho, which 00:15:27.800 --> 00:15:33.272 align:middle line:84% is a charge density and then the electric current j, OK? 00:15:33.272 --> 00:15:36.968 align:middle line:90% 00:15:36.968 --> 00:15:41.750 align:middle line:84% So in this Coulomb gauge, then the Maxwell equation 00:15:41.750 --> 00:15:43.290 align:middle line:90% is simplified. 00:15:43.290 --> 00:15:43.790 align:middle line:90% OK? 00:15:43.790 --> 00:15:45.410 align:middle line:90% So you look at this equation. 00:15:45.410 --> 00:15:49.420 align:middle line:84% And you look at different components, OK? 00:15:49.420 --> 00:15:50.980 align:middle line:84% And so we're not going into detail. 00:15:50.980 --> 00:15:52.397 align:middle line:84% Let me just write down the answer. 00:15:52.397 --> 00:15:53.870 align:middle line:84% When you impose this Coulomb gauge, 00:15:53.870 --> 00:15:57.380 align:middle line:84% you find the Maxwell equation becomes the following. 00:15:57.380 --> 00:16:03.150 align:middle line:90% So let me call this equation 1. 00:16:03.150 --> 00:16:04.650 align:middle line:90% Call this equation star. 00:16:04.650 --> 00:16:07.500 align:middle line:84% So in the Coulomb gauge, you find 00:16:07.500 --> 00:16:09.090 align:middle line:90% the equation star becomes-- 00:16:09.090 --> 00:16:11.610 align:middle line:90% 00:16:11.610 --> 00:16:17.640 align:middle line:84% so the time component becomes the so-called Poisson equation. 00:16:17.640 --> 00:16:20.390 align:middle line:90% 00:16:20.390 --> 00:16:25.760 align:middle line:84% And then the spatial component becomes like this. 00:16:25.760 --> 00:16:37.680 align:middle line:90% 00:16:37.680 --> 00:16:38.180 align:middle line:90% OK? 00:16:38.180 --> 00:16:40.770 align:middle line:90% 00:16:40.770 --> 00:16:42.660 align:middle line:90% So you get these two equations. 00:16:42.660 --> 00:16:47.040 align:middle line:90% 00:16:47.040 --> 00:16:50.630 align:middle line:84% So let's just remind you a little bit of the physics 00:16:50.630 --> 00:16:53.750 align:middle line:84% encoded in these two equations, OK? 00:16:53.750 --> 00:17:07.650 align:middle line:84% So this is the equation in which we 00:17:07.650 --> 00:17:12.810 align:middle line:84% can use rho, the charge density distribution, 00:17:12.810 --> 00:17:15.869 align:middle line:84% to solve for the electric potential, 00:17:15.869 --> 00:17:18.390 align:middle line:90% for the scalar potential. 00:17:18.390 --> 00:17:20.579 align:middle line:84% But notice that this equation does not 00:17:20.579 --> 00:17:22.170 align:middle line:90% involve time derivatives. 00:17:22.170 --> 00:17:37.920 align:middle line:90% 00:17:37.920 --> 00:17:40.500 align:middle line:90% OK? 00:17:40.500 --> 00:17:42.510 align:middle line:90% So this is not dynamical. 00:17:42.510 --> 00:17:44.100 align:middle line:90% So phi is not dynamical. 00:17:44.100 --> 00:17:54.050 align:middle line:90% 00:17:54.050 --> 00:17:56.270 align:middle line:84% It's determined by rho instantly. 00:17:56.270 --> 00:18:09.800 align:middle line:90% 00:18:09.800 --> 00:18:10.730 align:middle line:90% OK? 00:18:10.730 --> 00:18:16.850 align:middle line:84% So this means that the configuration of phi at a given 00:18:16.850 --> 00:18:21.020 align:middle line:84% time is just determined by the configuration of rho 00:18:21.020 --> 00:18:24.090 align:middle line:90% exactly at that given time, OK? 00:18:24.090 --> 00:18:27.130 align:middle line:84% And there's no dynamics in the phi itself, OK? 00:18:27.130 --> 00:18:28.610 align:middle line:84% There's no time derivative in phi. 00:18:28.610 --> 00:18:30.152 align:middle line:84% There's no dynamic in the phi itself. 00:18:30.152 --> 00:18:33.000 align:middle line:90% 00:18:33.000 --> 00:18:34.145 align:middle line:90% So this is first. 00:18:34.145 --> 00:18:36.650 align:middle line:90% 00:18:36.650 --> 00:18:43.470 align:middle line:84% The second thing-- is that let's see what this Coulomb gauge 00:18:43.470 --> 00:18:45.640 align:middle line:90% condition means. 00:18:45.640 --> 00:18:47.220 align:middle line:84% So if you look at this condition, 00:18:47.220 --> 00:18:49.300 align:middle line:90% when you go to momentum space-- 00:18:49.300 --> 00:18:51.630 align:middle line:84% so this Fourier transform goes to momentum space. 00:18:51.630 --> 00:18:58.580 align:middle line:84% And then that just become k dot A equal to 0, OK? 00:18:58.580 --> 00:19:00.260 align:middle line:90% So that just means that-- 00:19:00.260 --> 00:19:00.830 align:middle line:90% yes? 00:19:00.830 --> 00:19:02.205 align:middle line:84% AUDIENCE: This is a technicality, 00:19:02.205 --> 00:19:04.990 align:middle line:84% but in the expression for A, shouldn't it 00:19:04.990 --> 00:19:10.210 align:middle line:84% be minus phi because it's A sub u, right? 00:19:10.210 --> 00:19:11.860 align:middle line:84% HONG LIU: Which should be minus phi? 00:19:11.860 --> 00:19:16.293 align:middle line:84% AUDIENCE: A sub u equals minus phi comma A. Because in the-- 00:19:16.293 --> 00:19:16.960 align:middle line:90% HONG LIU: Sorry. 00:19:16.960 --> 00:19:18.340 align:middle line:90% Which equation do you say is-- 00:19:18.340 --> 00:19:20.300 align:middle line:84% AUDIENCE: Oh, right under Coulomb gauge. 00:19:20.300 --> 00:19:20.930 align:middle line:90% In i. 00:19:20.930 --> 00:19:21.640 align:middle line:90% No, you're there. 00:19:21.640 --> 00:19:22.640 align:middle line:90% HONG LIU: You mean here? 00:19:22.640 --> 00:19:23.400 align:middle line:90% AUDIENCE: Yeah, above that. 00:19:23.400 --> 00:19:23.940 align:middle line:90% Yeah. 00:19:23.940 --> 00:19:25.760 align:middle line:84% HONG LIU: No, this is my convention, right? 00:19:25.760 --> 00:19:26.105 align:middle line:90% AUDIENCE: Oh. 00:19:26.105 --> 00:19:26.730 align:middle line:90% HONG LIU: Yeah. 00:19:26.730 --> 00:19:27.647 align:middle line:90% This is my convention. 00:19:27.647 --> 00:19:28.188 align:middle line:90% AUDIENCE: OK. 00:19:28.188 --> 00:19:28.900 align:middle line:90% HONG LIU: Yeah. 00:19:28.900 --> 00:19:30.100 align:middle line:90% You can call it minus phi. 00:19:30.100 --> 00:19:31.430 align:middle line:90% You can call it positive phi. 00:19:31.430 --> 00:19:31.930 align:middle line:90% Yeah. 00:19:31.930 --> 00:19:33.310 align:middle line:90% [CHUCKLING] 00:19:33.310 --> 00:19:34.690 align:middle line:90% 00:19:34.690 --> 00:19:40.250 align:middle line:84% So in momentum space, you can just write it as this. 00:19:40.250 --> 00:19:47.350 align:middle line:84% So this just means that the component of A parallel 00:19:47.350 --> 00:19:50.470 align:middle line:90% to the momentum is 0, OK? 00:19:50.470 --> 00:19:54.890 align:middle line:84% A only has component which is perpendicular to the momentum 00:19:54.890 --> 00:19:57.010 align:middle line:90% and to the spatial momentum. 00:19:57.010 --> 00:20:04.870 align:middle line:84% So we can separate, write A as the longitudinal 00:20:04.870 --> 00:20:08.140 align:middle line:90% plus A transverse, OK? 00:20:08.140 --> 00:20:11.080 align:middle line:84% The longitudinal is defined to be the component of A, 00:20:11.080 --> 00:20:13.390 align:middle line:90% which is proportional to k. 00:20:13.390 --> 00:20:15.460 align:middle line:90% So A i L-- 00:20:15.460 --> 00:20:18.010 align:middle line:84% when you go to the Fourier space, 00:20:18.010 --> 00:20:21.580 align:middle line:84% it will be proportional to k, OK? 00:20:21.580 --> 00:20:30.980 align:middle line:84% And then the A i T will be transverse to k, 00:20:30.980 --> 00:20:34.310 align:middle line:84% means that A iT dot ki is equal to 0. 00:20:34.310 --> 00:20:35.900 align:middle line:90% Then the Coulomb gauge means-- 00:20:35.900 --> 00:20:47.150 align:middle line:90% 00:20:47.150 --> 00:20:54.330 align:middle line:84% this means that A is equal to A transpose, OK? 00:20:54.330 --> 00:20:56.940 align:middle line:84% Just A transpose-- just longitudinal part is 0, OK? 00:20:56.940 --> 00:20:58.890 align:middle line:84% The part proportional to the momentum is 0. 00:20:58.890 --> 00:21:01.810 align:middle line:90% 00:21:01.810 --> 00:21:07.330 align:middle line:84% So now, we can already see what-- 00:21:07.330 --> 00:21:10.450 align:middle line:84% so I just combine these two features. 00:21:10.450 --> 00:21:13.870 align:middle line:84% Then how many dynamical degrees freedom 00:21:13.870 --> 00:21:16.705 align:middle line:90% does the Maxwell theory have? 00:21:16.705 --> 00:21:21.500 align:middle line:90% 00:21:21.500 --> 00:21:22.207 align:middle line:90% Yes? 00:21:22.207 --> 00:21:22.790 align:middle line:90% AUDIENCE: Two. 00:21:22.790 --> 00:21:24.260 align:middle line:90% HONG LIU: Why is it two? 00:21:24.260 --> 00:21:26.060 align:middle line:84% AUDIENCE: Because phi is not dynamical, 00:21:26.060 --> 00:21:27.020 align:middle line:90% so you don't care about that. 00:21:27.020 --> 00:21:27.395 align:middle line:90% HONG LIU: Yeah. 00:21:27.395 --> 00:21:28.820 align:middle line:84% AUDIENCE: You have two transverse. 00:21:28.820 --> 00:21:29.778 align:middle line:90% HONG LIU: That's right. 00:21:29.778 --> 00:21:31.220 align:middle line:90% The phi is not dynamical. 00:21:31.220 --> 00:21:32.030 align:middle line:90% We don't care. 00:21:32.030 --> 00:21:34.070 align:middle line:90% And then this Ai is 0. 00:21:34.070 --> 00:21:36.080 align:middle line:84% And so you only have the transverse component. 00:21:36.080 --> 00:21:39.420 align:middle line:84% The transverse component is perpendicular to the momentum 00:21:39.420 --> 00:21:39.920 align:middle line:90% direction. 00:21:39.920 --> 00:21:42.770 align:middle line:84% And you only have two independent components, OK? 00:21:42.770 --> 00:21:53.660 align:middle line:84% So we find there are only two transverse dynamical degrees 00:21:53.660 --> 00:21:54.160 align:middle line:90% freedom. 00:21:54.160 --> 00:22:00.760 align:middle line:90% 00:22:00.760 --> 00:22:04.090 align:middle line:84% And indeed, this is the two polarizations. 00:22:04.090 --> 00:22:11.540 align:middle line:84% Classically, this is just corresponding two polarizations 00:22:11.540 --> 00:22:13.220 align:middle line:90% of electromagnetic wave, OK? 00:22:13.220 --> 00:22:18.020 align:middle line:84% So if you look at the EM wave, you only 00:22:18.020 --> 00:22:20.210 align:middle line:84% have two independent polarizations. 00:22:20.210 --> 00:22:23.650 align:middle line:90% 00:22:23.650 --> 00:22:26.430 align:middle line:84% You only have two independent polarizations. 00:22:26.430 --> 00:22:31.450 align:middle line:84% So this is the story for the Coulomb gauge. 00:22:31.450 --> 00:22:33.220 align:middle line:84% So Coulomb gauge is very convenient 00:22:33.220 --> 00:22:36.880 align:middle line:84% and has been widely used-- in particular, 00:22:36.880 --> 00:22:40.630 align:middle line:84% in the nonrelativistic situations when you don't 00:22:40.630 --> 00:22:44.530 align:middle line:90% involve very fast velocity. 00:22:44.530 --> 00:22:48.025 align:middle line:84% But the Coulomb gauge, of course, is not perfect. 00:22:48.025 --> 00:22:50.620 align:middle line:90% 00:22:50.620 --> 00:22:54.430 align:middle line:84% So there are some drawbacks of the Coulomb gauge. 00:22:54.430 --> 00:22:59.340 align:middle line:84% So first, there is no manifest Lorentz symmetry. 00:22:59.340 --> 00:23:03.540 align:middle line:90% 00:23:03.540 --> 00:23:09.720 align:middle line:84% So the Maxwell equation is manifestly Lorentz covariant, 00:23:09.720 --> 00:23:19.770 align:middle line:84% but the Coulomb gauge is not because this condition 00:23:19.770 --> 00:23:22.320 align:middle line:90% certainly is not covariant, OK? 00:23:22.320 --> 00:23:23.280 align:middle line:90% It is not covariant. 00:23:23.280 --> 00:23:27.060 align:middle line:84% It's not the same in different frames, OK? 00:23:27.060 --> 00:23:33.630 align:middle line:84% And the one consequence of this loss of the manifest Lorentz 00:23:33.630 --> 00:23:41.740 align:middle line:84% covariance is this phi equation is determined instantaneously. 00:23:41.740 --> 00:23:46.150 align:middle line:84% So we know that when you have the relativistic theory, 00:23:46.150 --> 00:23:49.480 align:middle line:84% then everything should propagate smaller 00:23:49.480 --> 00:23:51.490 align:middle line:90% than the speed of light, OK? 00:23:51.490 --> 00:23:56.810 align:middle line:84% And so you cannot have such instant action at a distance. 00:23:56.810 --> 00:23:59.980 align:middle line:84% So this implies the instant action at a distance, OK? 00:23:59.980 --> 00:24:05.337 align:middle line:84% And this is an artifact of the Coulomb gauge. 00:24:05.337 --> 00:24:07.045 align:middle line:84% This is an artifact of the Coulomb gauge. 00:24:07.045 --> 00:24:09.940 align:middle line:84% And the underlying physics is certainly not, OK? 00:24:09.940 --> 00:24:14.020 align:middle line:84% So for example, one consequence of this 00:24:14.020 --> 00:24:18.340 align:middle line:84% is that the causality is not manifest. 00:24:18.340 --> 00:24:24.610 align:middle line:90% 00:24:24.610 --> 00:24:25.360 align:middle line:90% OK? 00:24:25.360 --> 00:24:32.720 align:middle line:84% Because you have instant action at a distance. 00:24:32.720 --> 00:24:45.490 align:middle line:90% 00:24:45.490 --> 00:24:49.230 align:middle line:84% But remember that phi itself is not physically observable. 00:24:49.230 --> 00:24:54.270 align:middle line:84% So we cannot observe phi physically. 00:24:54.270 --> 00:25:00.600 align:middle line:84% So it's OK for phi to have an instant action. 00:25:00.600 --> 00:25:04.223 align:middle line:84% But of course, the electric field, the magnetic field-- 00:25:04.223 --> 00:25:04.890 align:middle line:90% they are causal. 00:25:04.890 --> 00:25:05.640 align:middle line:90% Yes? 00:25:05.640 --> 00:25:09.126 align:middle line:84% AUDIENCE: So if we impose a Lorentz invariant condition 00:25:09.126 --> 00:25:12.485 align:middle line:84% on the gauge, do we not get any [INAUDIBLE]?? 00:25:12.485 --> 00:25:13.610 align:middle line:90% HONG LIU: Yeah, we are not. 00:25:13.610 --> 00:25:14.630 align:middle line:90% Yeah, yeah. 00:25:14.630 --> 00:25:15.620 align:middle line:90% We'll be manifest. 00:25:15.620 --> 00:25:16.610 align:middle line:90% Yeah. 00:25:16.610 --> 00:25:19.730 align:middle line:84% This also will not break the causality. 00:25:19.730 --> 00:25:23.960 align:middle line:84% Just the causality is not manifest 00:25:23.960 --> 00:25:28.050 align:middle line:84% because this phi is not physical, directly observable. 00:25:28.050 --> 00:25:28.550 align:middle line:90% Yeah. 00:25:28.550 --> 00:25:31.390 align:middle line:90% 00:25:31.390 --> 00:25:31.890 align:middle line:90% OK. 00:25:31.890 --> 00:25:34.080 align:middle line:90% So then that motivates. 00:25:34.080 --> 00:25:37.570 align:middle line:84% So that's why the Coulomb gauge are 00:25:37.570 --> 00:25:40.340 align:middle line:84% mostly suitable in the case involving 00:25:40.340 --> 00:25:44.555 align:middle line:84% low velocities in the nonrelativistic situation. 00:25:44.555 --> 00:25:45.055 align:middle line:90% Yeah. 00:25:45.055 --> 00:25:48.830 align:middle line:90% 00:25:48.830 --> 00:25:56.960 align:middle line:84% So the second-- but in a more relativistic situation, 00:25:56.960 --> 00:26:02.840 align:middle line:84% people often consider so-called Lorentz gauge. 00:26:02.840 --> 00:26:05.900 align:middle line:84% So the Lorentz gauge is defined to be 00:26:05.900 --> 00:26:13.060 align:middle line:90% partial mu A nu equal to 0, OK? 00:26:13.060 --> 00:26:14.530 align:middle line:90% So the partial mu x-- 00:26:14.530 --> 00:26:19.390 align:middle line:84% now, this equation is covariant because index are contracted. 00:26:19.390 --> 00:26:22.330 align:middle line:84% So this is a covariant equation, OK? 00:26:22.330 --> 00:26:24.460 align:middle line:90% So this is a covariant equation. 00:26:24.460 --> 00:26:29.800 align:middle line:84% And so the Lorentz covariance is manifest. 00:26:29.800 --> 00:26:34.120 align:middle line:84% Then the Maxwell equation-- so under this gauge, then 00:26:34.120 --> 00:26:43.140 align:middle line:84% the equation star now becomes very simple. 00:26:43.140 --> 00:26:48.340 align:middle line:90% 00:26:48.340 --> 00:26:48.840 align:middle line:90% OK? 00:26:48.840 --> 00:26:50.860 align:middle line:90% It just becomes that. 00:26:50.860 --> 00:26:51.360 align:middle line:90% OK? 00:26:51.360 --> 00:26:54.540 align:middle line:84% So you see, this equation is indeed covariant, OK? 00:26:54.540 --> 00:26:56.580 align:middle line:90% So the derivative contracted. 00:26:56.580 --> 00:26:58.620 align:middle line:90% And then you have J mu, OK? 00:26:58.620 --> 00:27:02.630 align:middle line:84% So now, this is something we are familiar-- 00:27:02.630 --> 00:27:07.130 align:middle line:84% this is like you have four independent scalar fields-- 00:27:07.130 --> 00:27:12.410 align:middle line:84% massless scalar fields-- but each of them has a source, OK? 00:27:12.410 --> 00:27:15.330 align:middle line:84% So that's the situation which you actually see in your pset, 00:27:15.330 --> 00:27:17.330 align:middle line:90% OK? 00:27:17.330 --> 00:27:19.710 align:middle line:84% In Pset, you look at the scalar field with the source. 00:27:19.710 --> 00:27:26.000 align:middle line:84% And so this is like four independent scalar fields. 00:27:26.000 --> 00:27:28.730 align:middle line:90% Each is sourced by J mu. 00:27:28.730 --> 00:27:30.560 align:middle line:84% But of course, they are not independent 00:27:30.560 --> 00:27:32.780 align:middle line:84% because we have this gauge condition, OK? 00:27:32.780 --> 00:27:35.050 align:middle line:90% We have this gauge condition. 00:27:35.050 --> 00:27:36.830 align:middle line:84% And also, this gauge condition, of course, 00:27:36.830 --> 00:27:39.850 align:middle line:84% is compatible with this equation because if you 00:27:39.850 --> 00:27:45.670 align:middle line:84% act partial mu on both sides, and then because of this gauge 00:27:45.670 --> 00:27:49.150 align:middle line:84% condition, the left side is 0 and the right-hand side is also 00:27:49.150 --> 00:27:51.445 align:middle line:90% 0 from the current conservation. 00:27:51.445 --> 00:28:04.320 align:middle line:90% 00:28:04.320 --> 00:28:14.280 align:middle line:84% So now, let me also make some remarks on the Lorentz gauge. 00:28:14.280 --> 00:28:17.350 align:middle line:84% The Lorentz gauge is very convenient because it gives you 00:28:17.350 --> 00:28:22.570 align:middle line:84% seemingly decoupled equations between different A mu, OK? 00:28:22.570 --> 00:28:24.246 align:middle line:90% Between different A mu. 00:28:24.246 --> 00:28:27.592 align:middle line:90% 00:28:27.592 --> 00:28:33.680 align:middle line:84% So the Lorentz gauge also suffers 00:28:33.680 --> 00:28:35.180 align:middle line:90% a little bit of a problem, OK? 00:28:35.180 --> 00:28:36.500 align:middle line:90% Inconvenience. 00:28:36.500 --> 00:28:40.610 align:middle line:84% Because the Lorentz gauge does not fix the gauge completely, 00:28:40.610 --> 00:28:41.510 align:middle line:90% OK? 00:28:41.510 --> 00:29:04.660 align:middle line:84% So after-- Lorentz gauge does not fix the gauge freedom 00:29:04.660 --> 00:29:07.720 align:middle line:90% completely. 00:29:07.720 --> 00:29:12.250 align:middle line:84% So there's still some residual gauge symmetry left, OK? 00:29:12.250 --> 00:29:17.600 align:middle line:90% So consider A mu. 00:29:17.600 --> 00:29:24.020 align:middle line:84% Goes to A mu plus partial mu phi, OK? 00:29:24.020 --> 00:29:34.415 align:middle line:84% And if phi satisfies partial mu, partial nu equal to 0-- 00:29:34.415 --> 00:29:37.880 align:middle line:90% 00:29:37.880 --> 00:29:43.970 align:middle line:84% if phi satisfies this equation, then this preserves the gauge-- 00:29:43.970 --> 00:29:49.700 align:middle line:90% 00:29:49.700 --> 00:29:51.140 align:middle line:90% preserves the condition. 00:29:51.140 --> 00:29:54.680 align:middle line:90% 00:29:54.680 --> 00:30:02.880 align:middle line:84% Preserves this condition if phi satisfies the equation, OK? 00:30:02.880 --> 00:30:07.940 align:middle line:84% So this tells you that the gauge freedom is not completely 00:30:07.940 --> 00:30:10.520 align:middle line:90% fixed, OK? 00:30:10.520 --> 00:30:19.340 align:middle line:84% So now, if you look at this freedom, 00:30:19.340 --> 00:30:25.830 align:middle line:84% so here it's like you can shift -- 00:30:25.830 --> 00:30:29.490 align:middle line:84% lambda is an arbitrary scalar field. 00:30:29.490 --> 00:30:31.570 align:middle line:90% It's an arbitrary function. 00:30:31.570 --> 00:30:35.400 align:middle line:84% So you can view it as an arbitrary scalar field, OK? 00:30:35.400 --> 00:30:38.010 align:middle line:90% With no constraints, OK? 00:30:38.010 --> 00:30:39.570 align:middle line:90% With no constraint. 00:30:39.570 --> 00:30:43.050 align:middle line:84% We can say it's an arbitrary off-shell scalar field, OK? 00:30:43.050 --> 00:30:46.740 align:middle line:84% But now, after you fix the Lorentz gauge, 00:30:46.740 --> 00:30:50.730 align:middle line:84% you find that there's still some remaining gauge freedom-- 00:30:50.730 --> 00:30:52.830 align:middle line:84% and the remaining gauge freedom corresponding 00:30:52.830 --> 00:30:56.710 align:middle line:84% to an on-shell massive scalar field, OK? 00:30:56.710 --> 00:30:58.960 align:middle line:90% Still, if phi satisfies this-- 00:30:58.960 --> 00:31:02.200 align:middle line:84% because this equation is like an equation of motion 00:31:02.200 --> 00:31:03.920 align:middle line:90% for a massless scalar field. 00:31:03.920 --> 00:31:05.680 align:middle line:84% And so after you fix the Lorentz gauge, 00:31:05.680 --> 00:31:10.180 align:middle line:84% there's still a freedom of the on-shell massless scalar field 00:31:10.180 --> 00:31:12.980 align:middle line:90% left, OK? 00:31:12.980 --> 00:31:16.350 align:middle line:90% So this is the first remark. 00:31:16.350 --> 00:31:21.510 align:middle line:84% So the second remark is related to-- 00:31:21.510 --> 00:31:24.000 align:middle line:90% later, we are going to-- 00:31:24.000 --> 00:31:30.990 align:middle line:84% how we treat, say, this Lorentz gauge to quantize it, OK? 00:31:30.990 --> 00:31:34.590 align:middle line:90% So this is an alternative way. 00:31:34.590 --> 00:31:36.990 align:middle line:84% So you can just fix the Lorentz gauge, say, 00:31:36.990 --> 00:31:38.247 align:middle line:90% in your equation of motion. 00:31:38.247 --> 00:31:40.830 align:middle line:84% Just in your equation of motion, anywhere, you see this thing. 00:31:40.830 --> 00:31:42.630 align:middle line:90% You set it to be 0, OK? 00:31:42.630 --> 00:31:45.570 align:middle line:84% But we can also actually impose the Lorentz gauge 00:31:45.570 --> 00:31:47.790 align:middle line:90% at the action level, OK? 00:31:47.790 --> 00:32:08.000 align:middle line:84% So we can fix the Lorentz gauge as the action level as follows. 00:32:08.000 --> 00:32:13.210 align:middle line:90% So we consider a new action-- 00:32:13.210 --> 00:32:17.680 align:middle line:84% say, a new Lagrangian related to the previous one 00:32:17.680 --> 00:32:23.950 align:middle line:90% by a shift like this, OK? 00:32:23.950 --> 00:32:26.530 align:middle line:90% And the lambda here-- 00:32:26.530 --> 00:32:27.400 align:middle line:90% yeah. 00:32:27.400 --> 00:32:28.420 align:middle line:90% Let me not use lambda. 00:32:28.420 --> 00:32:32.710 align:middle line:90% 00:32:32.710 --> 00:32:34.360 align:middle line:90% So that should be not-- 00:32:34.360 --> 00:32:35.830 align:middle line:90% yeah, just xi, OK? 00:32:35.830 --> 00:32:37.645 align:middle line:84% So xi here is arbitrary constant. 00:32:37.645 --> 00:32:46.580 align:middle line:90% 00:32:46.580 --> 00:32:47.608 align:middle line:90% OK? 00:32:47.608 --> 00:32:48.650 align:middle line:90% Xi is arbitrary constant. 00:32:48.650 --> 00:32:52.180 align:middle line:90% 00:32:52.180 --> 00:32:57.390 align:middle line:84% So now, let's look at this new theory 00:32:57.390 --> 00:33:00.500 align:middle line:84% we just obtained from the original theory 00:33:00.500 --> 00:33:02.990 align:middle line:90% by adding this term, OK? 00:33:02.990 --> 00:33:06.520 align:middle line:84% So now, let's look at this equation of motion. 00:33:06.520 --> 00:33:08.270 align:middle line:84% So if you look at this equation of motion, 00:33:08.270 --> 00:33:11.630 align:middle line:84% then you find it's given by the following form. 00:33:11.630 --> 00:33:14.380 align:middle line:90% Partial square A mu. 00:33:14.380 --> 00:33:18.270 align:middle line:84% So partial square just means the partial nu partial nu, OK? 00:33:18.270 --> 00:33:24.925 align:middle line:84% And then you have lambda minus 1 partial mu-- 00:33:24.925 --> 00:33:30.010 align:middle line:90% partial nu A mu equal J mu, OK? 00:33:30.010 --> 00:33:33.440 align:middle line:90% You find the equation like this. 00:33:33.440 --> 00:33:35.640 align:middle line:90% OK. 00:33:35.640 --> 00:33:36.140 align:middle line:90% Yeah. 00:33:36.140 --> 00:33:38.030 align:middle line:90% Sorry. 00:33:38.030 --> 00:33:38.530 align:middle line:90% Sorry. 00:33:38.530 --> 00:33:42.710 align:middle line:90% 00:33:42.710 --> 00:33:46.610 align:middle line:84% So you see an equation like this. 00:33:46.610 --> 00:33:54.750 align:middle line:84% So you get this new term because of in the Lagrangian, OK? 00:33:54.750 --> 00:33:58.335 align:middle line:84% So now, let's act partial mu on both sides. 00:33:58.335 --> 00:34:06.840 align:middle line:90% 00:34:06.840 --> 00:34:09.929 align:middle line:84% Since the current is conserved, and then 00:34:09.929 --> 00:34:13.969 align:middle line:84% when you act partial mu here, you get partial mu A mu. 00:34:13.969 --> 00:34:16.639 align:middle line:84% And then yeah, you just combine together. 00:34:16.639 --> 00:34:22.489 align:middle line:84% You find you get partial square partial mu A mu equal to 0. 00:34:22.489 --> 00:34:26.840 align:middle line:90% 00:34:26.840 --> 00:34:29.520 align:middle line:84% So if you act the partial mu on both sides, 00:34:29.520 --> 00:34:33.030 align:middle line:90% you find that this equal to 0. 00:34:33.030 --> 00:34:40.020 align:middle line:84% So now, we can impose partial mu A mu equal to 0, OK? 00:34:40.020 --> 00:35:07.110 align:middle line:84% We can enforce with star star by imposing boundary conditions so 00:35:07.110 --> 00:35:18.520 align:middle line:84% that this equation only has trivial solutions, which 00:35:18.520 --> 00:35:20.935 align:middle line:84% means that this partial mu A mu is equal to 0, OK? 00:35:20.935 --> 00:35:25.060 align:middle line:90% 00:35:25.060 --> 00:35:25.630 align:middle line:90% Sorry. 00:35:25.630 --> 00:35:28.030 align:middle line:84% I'm running out of space here, OK? 00:35:28.030 --> 00:35:31.930 align:middle line:90% So you have this equation. 00:35:31.930 --> 00:35:35.890 align:middle line:84% And now, you can enforce the Lorentz gauge 00:35:35.890 --> 00:35:39.420 align:middle line:84% by putting the boundary conditions 00:35:39.420 --> 00:35:42.710 align:middle line:84% so that this equation only has 0 solution, OK? 00:35:42.710 --> 00:35:48.310 align:middle line:84% And then equivalently, then you have imposed the Lorentz gauge. 00:35:48.310 --> 00:35:50.920 align:middle line:84% But the nice thing of this approach 00:35:50.920 --> 00:35:57.470 align:middle line:84% is that now, you can treat it as action-level. 00:35:57.470 --> 00:36:00.770 align:middle line:84% And the advantage, we will see later, OK? 00:36:00.770 --> 00:36:03.133 align:middle line:84% Because in quantum theory, the action 00:36:03.133 --> 00:36:04.550 align:middle line:84% is very important-- in particular, 00:36:04.550 --> 00:36:07.260 align:middle line:90% when you do the path integral. 00:36:07.260 --> 00:36:11.130 align:middle line:84% In particular, when you do the path integral. 00:36:11.130 --> 00:36:12.900 align:middle line:90% Any questions on this? 00:36:12.900 --> 00:36:13.680 align:middle line:90% Yes? 00:36:13.680 --> 00:36:14.340 align:middle line:84% AUDIENCE: Could you go into a little bit 00:36:14.340 --> 00:36:16.752 align:middle line:84% more detail about what the specific [INAUDIBLE]?? 00:36:16.752 --> 00:36:20.020 align:middle line:90% 00:36:20.020 --> 00:36:22.040 align:middle line:90% HONG LIU: Yeah. 00:36:22.040 --> 00:36:26.070 align:middle line:84% We will do it later in your Pset. 00:36:26.070 --> 00:36:28.200 align:middle line:90% [LAUGHTER] 00:36:28.200 --> 00:36:28.700 align:middle line:90% Right. 00:36:28.700 --> 00:36:33.500 align:middle line:90% 00:36:33.500 --> 00:36:34.900 align:middle line:90% Yeah. 00:36:34.900 --> 00:36:37.820 align:middle line:84% So we will touch on this question. 00:36:37.820 --> 00:36:41.150 align:middle line:84% We will touch on this question when we quantize it. 00:36:41.150 --> 00:36:43.840 align:middle line:84% This feature will become very important. 00:36:43.840 --> 00:36:46.460 align:middle line:84% And then you will have an opportunity 00:36:46.460 --> 00:36:50.720 align:middle line:84% to work out in detail in your Pset. 00:36:50.720 --> 00:36:53.050 align:middle line:90% Other questions? 00:36:53.050 --> 00:36:53.680 align:middle line:90% Yes? 00:36:53.680 --> 00:36:56.420 align:middle line:84% AUDIENCE: Is there a subtlety as to why you-- 00:36:56.420 --> 00:36:59.250 align:middle line:84% does it matter when you fix your gauge at the action level 00:36:59.250 --> 00:37:01.910 align:middle line:84% or later on with your equation of motion? 00:37:01.910 --> 00:37:02.535 align:middle line:90% HONG LIU: Yeah. 00:37:02.535 --> 00:37:05.170 align:middle line:90% It's just a little bit easier. 00:37:05.170 --> 00:37:06.270 align:middle line:90% Yeah. 00:37:06.270 --> 00:37:07.980 align:middle line:84% Later, when we do the path integral, 00:37:07.980 --> 00:37:09.277 align:middle line:90% you will see it's easier. 00:37:09.277 --> 00:37:09.777 align:middle line:90% Yeah. 00:37:09.777 --> 00:37:13.300 align:middle line:90% 00:37:13.300 --> 00:37:14.950 align:middle line:84% Classically, there is no difference. 00:37:14.950 --> 00:37:18.210 align:middle line:84% But quantum mechanically, we always want to do 00:37:18.210 --> 00:37:20.990 align:middle line:90% is at the action level. 00:37:20.990 --> 00:37:22.440 align:middle line:90% Yeah. 00:37:22.440 --> 00:37:24.840 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:37:24.840 --> 00:37:26.190 align:middle line:90% 00:37:26.190 --> 00:37:27.060 align:middle line:90% HONG LIU: Sorry? 00:37:27.060 --> 00:37:28.080 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:37:28.080 --> 00:37:28.890 align:middle line:90% HONG LIU: Yeah. 00:37:28.890 --> 00:37:32.870 align:middle line:84% Yeah, we have an extra constant, but the physics should not 00:37:32.870 --> 00:37:33.870 align:middle line:90% depend on this constant. 00:37:33.870 --> 00:37:39.450 align:middle line:90% 00:37:39.450 --> 00:37:42.010 align:middle line:90% Other questions? 00:37:42.010 --> 00:37:43.660 align:middle line:90% OK, good. 00:37:43.660 --> 00:37:52.060 align:middle line:84% So let me just point out-- when xi is equal to 1, 00:37:52.060 --> 00:37:56.130 align:middle line:84% the story is particularly simple. 00:37:56.130 --> 00:38:00.830 align:middle line:84% So for general xi, you get this more complicated equation, 00:38:00.830 --> 00:38:02.720 align:middle line:84% but for xi equal to 1, you notice 00:38:02.720 --> 00:38:04.670 align:middle line:90% that here this term is just 0. 00:38:04.670 --> 00:38:10.610 align:middle line:84% And then even with this new Lagrangian, 00:38:10.610 --> 00:38:13.790 align:middle line:84% you get exactly the same equation of motion 00:38:13.790 --> 00:38:19.370 align:middle line:84% as if you have imposed the Lorentz gauge 00:38:19.370 --> 00:38:22.130 align:middle line:90% in the equation of motion, OK? 00:38:22.130 --> 00:38:24.665 align:middle line:84% So xi equal to 1 is particularly simple. 00:38:24.665 --> 00:38:29.670 align:middle line:90% 00:38:29.670 --> 00:38:32.445 align:middle line:84% So xi equal to 1 is particularly simple for a reason. 00:38:32.445 --> 00:38:36.470 align:middle line:90% 00:38:36.470 --> 00:38:44.130 align:middle line:84% So if you put the xi equal to 1 here, 00:38:44.130 --> 00:38:45.880 align:middle line:84% then you will find that this term actually 00:38:45.880 --> 00:38:49.660 align:middle line:84% cancel with some of the terms in this L exactly. 00:38:49.660 --> 00:38:56.030 align:middle line:84% So actually, when xi is equal to 1, 00:38:56.030 --> 00:39:02.500 align:middle line:84% you find that the Lagrangian becomes the following. 00:39:02.500 --> 00:39:12.990 align:middle line:84% It becomes 1/2 partial mu A nu partial mu A nu minus J 00:39:12.990 --> 00:39:14.960 align:middle line:90% mu A nu, OK? 00:39:14.960 --> 00:39:17.300 align:middle line:90% It just becomes very simple. 00:39:17.300 --> 00:39:19.100 align:middle line:84% Again, just the action now becomes 00:39:19.100 --> 00:39:22.580 align:middle line:84% like you have just a bunch of free scalar fields, OK? 00:39:22.580 --> 00:39:23.565 align:middle line:90% Massless scalar fields. 00:39:23.565 --> 00:39:27.315 align:middle line:90% 00:39:27.315 --> 00:39:27.815 align:middle line:90% OK? 00:39:27.815 --> 00:39:42.330 align:middle line:90% 00:39:42.330 --> 00:39:42.830 align:middle line:90% Good. 00:39:42.830 --> 00:39:43.820 align:middle line:90% Any questions? 00:39:43.820 --> 00:39:46.990 align:middle line:90% 00:39:46.990 --> 00:39:48.233 align:middle line:90% Yes? 00:39:48.233 --> 00:39:49.622 align:middle line:90% AUDIENCE: For the [INAUDIBLE]? 00:39:49.622 --> 00:39:52.197 align:middle line:90% 00:39:52.197 --> 00:39:53.030 align:middle line:90% HONG LIU: Oh, sorry. 00:39:53.030 --> 00:39:53.530 align:middle line:90% Yeah. 00:39:53.530 --> 00:39:56.130 align:middle line:90% 00:39:56.130 --> 00:39:58.537 align:middle line:84% Yes, so this mu should be downstairs. 00:39:58.537 --> 00:40:03.240 align:middle line:90% 00:40:03.240 --> 00:40:03.740 align:middle line:90% OK. 00:40:03.740 --> 00:40:06.950 align:middle line:84% So just even at the action level, 00:40:06.950 --> 00:40:09.435 align:middle line:84% it looks like we have four decoupled massless scalar 00:40:09.435 --> 00:40:09.935 align:middle line:90% fields. 00:40:09.935 --> 00:40:16.940 align:middle line:90% 00:40:16.940 --> 00:40:19.110 align:middle line:90% OK, good. 00:40:19.110 --> 00:40:20.630 align:middle line:90% So now, we are ready. 00:40:20.630 --> 00:40:24.720 align:middle line:84% So this is a quick review of the classical story. 00:40:24.720 --> 00:40:27.860 align:middle line:84% So now, let's discuss how to quantize it. 00:40:27.860 --> 00:40:45.110 align:middle line:90% 00:40:45.110 --> 00:40:45.610 align:middle line:90% OK. 00:40:45.610 --> 00:40:51.470 align:middle line:90% 00:40:51.470 --> 00:40:56.060 align:middle line:84% And so again, we will first do the standard canonical 00:40:56.060 --> 00:40:57.860 align:middle line:90% quantization, OK? 00:40:57.860 --> 00:41:01.070 align:middle line:84% Write down the most general solutions 00:41:01.070 --> 00:41:02.990 align:middle line:90% to the operator equations. 00:41:02.990 --> 00:41:08.368 align:middle line:84% And then we will discuss using the path integral, OK? 00:41:08.368 --> 00:41:10.160 align:middle line:84% Using the path integral-- the path integral 00:41:10.160 --> 00:41:14.660 align:middle line:84% is convenient for treating interactions. 00:41:14.660 --> 00:41:22.580 align:middle line:84% And the canonical approach is convenient for understanding 00:41:22.580 --> 00:41:26.930 align:middle line:84% what's the physical degrees freedom, to understand 00:41:26.930 --> 00:41:30.940 align:middle line:90% what's going on physically, OK? 00:41:30.940 --> 00:41:35.560 align:middle line:84% So for simplicity, let's just forget about J mu, OK? 00:41:35.560 --> 00:41:38.080 align:middle line:84% So let's just look at the Maxwell theory itself. 00:41:38.080 --> 00:41:41.660 align:middle line:90% 00:41:41.660 --> 00:41:44.200 align:middle line:84% So J mu does not do anything in terms of quantization 00:41:44.200 --> 00:41:46.980 align:middle line:90% of the Maxwell theory, OK? 00:41:46.980 --> 00:41:49.500 align:middle line:84% So we don't need to worry about it 00:41:49.500 --> 00:41:53.750 align:middle line:90% when we quantize this theory. 00:41:53.750 --> 00:41:59.180 align:middle line:84% So this theory is quadratic in A mu. 00:41:59.180 --> 00:42:04.520 align:middle line:84% So this is what we call a free theory, OK? 00:42:04.520 --> 00:42:05.960 align:middle line:90% Because it's quadratic in A mu. 00:42:05.960 --> 00:42:08.090 align:middle line:90% There's no interaction, OK? 00:42:08.090 --> 00:42:12.380 align:middle line:84% So in the absence of source, in the absence 00:42:12.380 --> 00:42:15.680 align:middle line:84% of the charge density or current density, 00:42:15.680 --> 00:42:20.450 align:middle line:84% the Maxwell theory has no self-interaction, OK? 00:42:20.450 --> 00:42:21.690 align:middle line:90% Has no self-interaction. 00:42:21.690 --> 00:42:30.280 align:middle line:84% So the photon is free in the absence of sources. 00:42:30.280 --> 00:42:31.515 align:middle line:90% So this is a free theory. 00:42:31.515 --> 00:42:39.752 align:middle line:90% 00:42:39.752 --> 00:42:41.210 align:middle line:84% But the quantization of this theory 00:42:41.210 --> 00:42:45.745 align:middle line:84% is actually very subtle due to the following reasons, OK? 00:42:45.745 --> 00:42:49.130 align:middle line:90% 00:42:49.130 --> 00:43:01.110 align:middle line:90% So subtleties in quantization. 00:43:01.110 --> 00:43:05.210 align:middle line:90% 00:43:05.210 --> 00:43:17.337 align:middle line:84% First, is that because of the gauge symmetry, 00:43:17.337 --> 00:43:18.920 align:middle line:84% there's a redundant degree of freedom. 00:43:18.920 --> 00:43:26.510 align:middle line:90% 00:43:26.510 --> 00:43:27.010 align:middle line:90% OK? 00:43:27.010 --> 00:43:28.468 align:middle line:84% They are redundant degrees freedom, 00:43:28.468 --> 00:43:31.720 align:middle line:90% which are unphysical, OK? 00:43:31.720 --> 00:43:33.160 align:middle line:90% Which are unphysical. 00:43:33.160 --> 00:43:37.900 align:middle line:90% So we want to only quantize-- 00:43:37.900 --> 00:43:43.150 align:middle line:84% should quantize-- only physical degrees freedom. 00:43:43.150 --> 00:43:50.490 align:middle line:90% 00:43:50.490 --> 00:43:51.820 align:middle line:90% OK? 00:43:51.820 --> 00:43:55.060 align:middle line:84% So this is much more subtle at the quantum 00:43:55.060 --> 00:43:57.070 align:middle line:90% level than the classical level. 00:43:57.070 --> 00:44:00.790 align:middle line:84% So at the classical level, we just fix the gauge, OK? 00:44:00.790 --> 00:44:03.380 align:middle line:90% Then that's it. 00:44:03.380 --> 00:44:06.950 align:middle line:84% But at the quantum level, things fluctuate, OK? 00:44:06.950 --> 00:44:11.330 align:middle line:84% So you have to make sure the fluctuations you look at 00:44:11.330 --> 00:44:13.760 align:middle line:84% are corresponding to physical fluctuations, not 00:44:13.760 --> 00:44:16.550 align:middle line:84% corresponding to fluctuations of those unphysical degrees 00:44:16.550 --> 00:44:17.810 align:middle line:90% freedom, OK? 00:44:17.810 --> 00:44:21.410 align:middle line:84% And so that makes the quantum theory more subtle 00:44:21.410 --> 00:44:23.150 align:middle line:90% compared with classical, OK? 00:44:23.150 --> 00:44:25.970 align:middle line:90% 00:44:25.970 --> 00:44:32.240 align:middle line:84% And the second subtlety is that F mu nu is antisymmetric. 00:44:32.240 --> 00:44:36.440 align:middle line:90% 00:44:36.440 --> 00:44:40.730 align:middle line:84% So the second one is more technical. 00:44:40.730 --> 00:44:42.950 align:middle line:90% So this one is very conceptual. 00:44:42.950 --> 00:44:48.610 align:middle line:90% 00:44:48.610 --> 00:44:49.810 align:middle line:90% It's antisymmetric. 00:44:49.810 --> 00:44:58.460 align:middle line:84% So that means there's no partial 0A0 term, OK? 00:44:58.460 --> 00:45:00.590 align:middle line:84% Because this is symmetric between the two indices. 00:45:00.590 --> 00:45:03.920 align:middle line:84% And these are antisymmetric between two indices. 00:45:03.920 --> 00:45:06.470 align:middle line:90% There's no A0 partial 0 A0. 00:45:06.470 --> 00:45:09.080 align:middle line:84% Then that means there's no time derivative term 00:45:09.080 --> 00:45:12.980 align:middle line:84% for the A0 component in your action, OK? 00:45:12.980 --> 00:45:34.780 align:middle line:84% So that means L does not contain time derivatives of A0. 00:45:34.780 --> 00:45:37.390 align:middle line:90% 00:45:37.390 --> 00:45:44.680 align:middle line:84% So this means that if we look at the momentum conjugate to A0-- 00:45:44.680 --> 00:45:49.350 align:middle line:90% 00:45:49.350 --> 00:45:53.250 align:middle line:84% we should look at the Lagrangian density 00:45:53.250 --> 00:45:56.790 align:middle line:84% divided by time derivative is 0-- this is just identically 0. 00:45:56.790 --> 00:45:59.760 align:middle line:90% 00:45:59.760 --> 00:46:02.340 align:middle line:84% So again, this is corresponding to a constrained equation. 00:46:02.340 --> 00:46:08.240 align:middle line:90% 00:46:08.240 --> 00:46:11.950 align:middle line:84% This does not express a canonical momentum 00:46:11.950 --> 00:46:15.010 align:middle line:90% in terms of the A0 dot, OK? 00:46:15.010 --> 00:46:17.613 align:middle line:84% Or the time derivative of anything. 00:46:17.613 --> 00:46:19.030 align:middle line:84% So this is a constrained equation. 00:46:19.030 --> 00:46:22.090 align:middle line:84% And again, if you want to treat properly, 00:46:22.090 --> 00:46:25.520 align:middle line:84% we have to use constrained quantization, OK? 00:46:25.520 --> 00:46:29.270 align:middle line:84% We have to use constrained quantization. 00:46:29.270 --> 00:46:32.740 align:middle line:84% And so this leads to subtleties, OK? 00:46:32.740 --> 00:46:36.780 align:middle line:90% 00:46:36.780 --> 00:46:46.470 align:middle line:84% Remember-- we said since the constrained allowing-- we 00:46:46.470 --> 00:46:48.960 align:middle line:84% already discussed for the Dirac equation. 00:46:48.960 --> 00:46:54.150 align:middle line:84% And there we avoided going into such constrained quantization 00:46:54.150 --> 00:46:55.380 align:middle line:90% by using a trick. 00:46:55.380 --> 00:46:57.330 align:middle line:84% And here we will do similarly, OK? 00:46:57.330 --> 00:46:59.730 align:middle line:84% We will not directly deal with constrained quantization, 00:46:59.730 --> 00:47:02.310 align:middle line:84% but we use some trick to go around this issue. 00:47:02.310 --> 00:47:04.390 align:middle line:90% OK? 00:47:04.390 --> 00:47:04.920 align:middle line:90% Good. 00:47:04.920 --> 00:47:08.690 align:middle line:84% And so let me just write down by passing 00:47:08.690 --> 00:47:18.090 align:middle line:84% the canonical momentum due to Ai, then this is Ai dot, OK? 00:47:18.090 --> 00:47:19.750 align:middle line:90% Dot is the time derivative. 00:47:19.750 --> 00:47:25.770 align:middle line:84% And then so this is corresponding to minus F0i. 00:47:25.770 --> 00:47:28.620 align:middle line:84% And then this gives you just Ei, OK? 00:47:28.620 --> 00:47:31.427 align:middle line:84% So the canonical momentum corresponding to Ai 00:47:31.427 --> 00:47:32.385 align:middle line:90% is just electric field. 00:47:32.385 --> 00:47:39.180 align:middle line:90% 00:47:39.180 --> 00:47:40.440 align:middle line:90% OK. 00:47:40.440 --> 00:47:43.500 align:middle line:90% So it's just electric field. 00:47:43.500 --> 00:47:45.150 align:middle line:90% OK? 00:47:45.150 --> 00:47:53.190 align:middle line:84% So with this warning, then let's proceed to quantize the theory, 00:47:53.190 --> 00:47:54.030 align:middle line:90% OK? 00:47:54.030 --> 00:47:58.210 align:middle line:84% So as we said, we want to only quantize on a cross-section, 00:47:58.210 --> 00:47:58.710 align:middle line:90% OK? 00:47:58.710 --> 00:48:03.810 align:middle line:84% We don't want to quantize those unphysical degrees freedom, OK? 00:48:03.810 --> 00:48:05.790 align:middle line:84% Those unphysical degrees freedom. 00:48:05.790 --> 00:48:08.328 align:middle line:84% And so we have to fix the gauge, OK? 00:48:08.328 --> 00:48:09.495 align:middle line:90% So we have to fix the gauge. 00:48:09.495 --> 00:48:43.450 align:middle line:90% 00:48:43.450 --> 00:48:43.950 align:middle line:90% OK? 00:48:43.950 --> 00:48:48.360 align:middle line:84% So first, let's see how to do it in the Coulomb gauge. 00:48:48.360 --> 00:48:54.960 align:middle line:84% So Coulomb gauge is conceptually simple, OK? 00:48:54.960 --> 00:48:58.790 align:middle line:90% 00:48:58.790 --> 00:49:01.970 align:middle line:84% So in this case, so in the absence of source, 00:49:01.970 --> 00:49:03.710 align:middle line:90% let's look at the two equations. 00:49:03.710 --> 00:49:09.587 align:middle line:84% And then again, we follow the same strategy as before. 00:49:09.587 --> 00:49:11.170 align:middle line:84% We follow the same strategy as before. 00:49:11.170 --> 00:49:13.990 align:middle line:84% We first find the most general classical solution. 00:49:13.990 --> 00:49:18.010 align:middle line:84% And then we turn that into the quantum operator solution, OK? 00:49:18.010 --> 00:49:20.470 align:middle line:90% The same strategy. 00:49:20.470 --> 00:49:24.510 align:middle line:84% So in the Coulomb gauge, then we have two equations. 00:49:24.510 --> 00:49:30.350 align:middle line:84% And then we just have phi squared equal to 0, 00:49:30.350 --> 00:49:39.190 align:middle line:84% and now partial mu A mu, partial mu, partial mu Ai 00:49:39.190 --> 00:49:43.780 align:middle line:90% now equal to partial phi, OK? 00:49:43.780 --> 00:49:44.860 align:middle line:90% We have these two. 00:49:44.860 --> 00:49:48.070 align:middle line:84% So they're two equations with J equal to 0-- 00:49:48.070 --> 00:49:50.290 align:middle line:84% and then become the two equations, OK? 00:49:50.290 --> 00:49:53.670 align:middle line:90% 00:49:53.670 --> 00:50:00.520 align:middle line:84% And also, we have the gauge condition, 00:50:00.520 --> 00:50:03.570 align:middle line:84% which means that Ai must be transverse, OK? 00:50:03.570 --> 00:50:07.270 align:middle line:90% 00:50:07.270 --> 00:50:07.770 align:middle line:90% OK? 00:50:07.770 --> 00:50:13.120 align:middle line:84% So here are the equations we have to satisfy. 00:50:13.120 --> 00:50:13.620 align:middle line:90% OK. 00:50:13.620 --> 00:50:15.370 align:middle line:84% Here are the equations we have to satisfy. 00:50:15.370 --> 00:50:18.580 align:middle line:90% 00:50:18.580 --> 00:50:25.220 align:middle line:84% So here in this gauge, we don't have this problem, 00:50:25.220 --> 00:50:29.250 align:middle line:84% which is a constrained quantization because phi is not 00:50:29.250 --> 00:50:31.530 align:middle line:90% a dynamic variable, OK? 00:50:31.530 --> 00:50:35.520 align:middle line:84% So we don't have to worry about quantizing it, OK? 00:50:35.520 --> 00:50:38.680 align:middle line:84% And in fact, if you look at this equation-- 00:50:38.680 --> 00:50:41.020 align:middle line:84% so this is so-called elliptic equation. 00:50:41.020 --> 00:50:43.090 align:middle line:84% And then you can impose the boundary condition. 00:50:43.090 --> 00:50:46.020 align:middle line:84% So if you require phi to go to 0 at infinity-- 00:50:46.020 --> 00:50:48.150 align:middle line:84% spatial infinity-- and then this equation 00:50:48.150 --> 00:50:50.850 align:middle line:90% only has identical 0 solution. 00:50:50.850 --> 00:50:54.600 align:middle line:84% So the phi-- you can just set it to be 0. 00:50:54.600 --> 00:50:56.310 align:middle line:90% So phi is just 0, OK? 00:50:56.310 --> 00:50:59.700 align:middle line:84% And so now, since the phi is just 0, 00:50:59.700 --> 00:51:03.060 align:middle line:84% we don't have to worry about this canonical-- 00:51:03.060 --> 00:51:03.630 align:middle line:90% yeah. 00:51:03.630 --> 00:51:05.190 align:middle line:84% We just don't have to quantize phi. 00:51:05.190 --> 00:51:08.760 align:middle line:90% 00:51:08.760 --> 00:51:17.100 align:middle line:84% And now, the equation for Ai becomes very simple. 00:51:17.100 --> 00:51:23.270 align:middle line:84% You just have partial mu, partial mu Ai equal to 0. 00:51:23.270 --> 00:51:27.330 align:middle line:84% And now, this just becomes massless scalar field equation. 00:51:27.330 --> 00:51:27.830 align:middle line:90% OK? 00:51:27.830 --> 00:51:31.212 align:middle line:84% This is just like you have a massless scalar field equation. 00:51:31.212 --> 00:51:33.520 align:middle line:90% OK? 00:51:33.520 --> 00:51:36.170 align:middle line:84% So now, we can just proceed with the quantization. 00:51:36.170 --> 00:51:36.670 align:middle line:90% OK? 00:51:36.670 --> 00:51:41.180 align:middle line:84% But remember-- Ai is only transverse, OK? 00:51:41.180 --> 00:51:43.370 align:middle line:90% Ai has to be transverse. 00:51:43.370 --> 00:51:43.870 align:middle line:90% OK. 00:51:43.870 --> 00:51:46.370 align:middle line:84% So now, we impose this as an operator equation. 00:51:46.370 --> 00:51:48.400 align:middle line:84% So all this should be interpreted as an operator 00:51:48.400 --> 00:51:51.710 align:middle line:90% equation when we quantize it. 00:51:51.710 --> 00:51:54.970 align:middle line:84% And so that means that Ai as an operator 00:51:54.970 --> 00:51:58.850 align:middle line:84% can only have a transverse component. 00:51:58.850 --> 00:52:00.725 align:middle line:84% So now, we can just quantize the theory. 00:52:00.725 --> 00:52:06.130 align:middle line:90% 00:52:06.130 --> 00:52:06.630 align:middle line:90% OK? 00:52:06.630 --> 00:52:10.440 align:middle line:84% So the solution to this, of course, we already know, OK? 00:52:10.440 --> 00:52:14.080 align:middle line:84% So we can just now proceed with the quantization. 00:52:14.080 --> 00:52:15.460 align:middle line:84% So the first step is that we have 00:52:15.460 --> 00:52:19.480 align:middle line:84% to write down the canonical commutation relation. 00:52:19.480 --> 00:52:28.750 align:middle line:84% So this means that Ai with pi i, pi j, should be i delta ij, OK? 00:52:28.750 --> 00:52:32.010 align:middle line:90% We have to impose this, OK? 00:52:32.010 --> 00:52:34.530 align:middle line:84% So this equation-- in the current context, 00:52:34.530 --> 00:52:42.920 align:middle line:84% if I write it carefully, it's the Ai t xi t, x, and then 00:52:42.920 --> 00:52:48.500 align:middle line:90% Ej t, x prime. 00:52:48.500 --> 00:52:55.460 align:middle line:84% So this should be equal to i delta ij delta x minus x prime. 00:52:55.460 --> 00:52:57.820 align:middle line:90% 00:52:57.820 --> 00:52:58.320 align:middle line:90% OK? 00:52:58.320 --> 00:53:01.580 align:middle line:90% 00:53:01.580 --> 00:53:02.720 align:middle line:90% Yes? 00:53:02.720 --> 00:53:07.080 align:middle line:84% AUDIENCE: So for me to get to, to pick our gauge 00:53:07.080 --> 00:53:09.490 align:middle line:84% to start quantizing the theory, is there 00:53:09.490 --> 00:53:11.880 align:middle line:84% a reason why we are using the Coulomb gauge here? 00:53:11.880 --> 00:53:15.185 align:middle line:84% We discussed it violates causality-- 00:53:15.185 --> 00:53:16.060 align:middle line:90% HONG LIU: No, no, no. 00:53:16.060 --> 00:53:17.830 align:middle line:84% It does not violate causality, right? 00:53:17.830 --> 00:53:21.720 align:middle line:90% Just causality is not manifest. 00:53:21.720 --> 00:53:23.490 align:middle line:90% It does not violate causality. 00:53:23.490 --> 00:53:26.801 align:middle line:84% AUDIENCE: What does that mean, not manifest? 00:53:26.801 --> 00:53:28.255 align:middle line:90% HONG LIU: It means that the-- 00:53:28.255 --> 00:53:33.150 align:middle line:90% 00:53:33.150 --> 00:53:36.300 align:middle line:84% so manifest is at every step, you 00:53:36.300 --> 00:53:38.800 align:middle line:90% see the causality is preserved. 00:53:38.800 --> 00:53:43.310 align:middle line:84% And not manifest is that you have to check it to make sure 00:53:43.310 --> 00:53:45.370 align:middle line:90% causality is preserved. 00:53:45.370 --> 00:53:46.060 align:middle line:90% Yeah. 00:53:46.060 --> 00:53:50.560 align:middle line:84% So I say this equation is covariant. 00:53:50.560 --> 00:53:51.648 align:middle line:90% I don't have to check it. 00:53:51.648 --> 00:53:52.690 align:middle line:90% I know this is covariant. 00:53:52.690 --> 00:53:53.648 align:middle line:90% I look at the equation. 00:53:53.648 --> 00:53:55.830 align:middle line:90% It's covariant, OK? 00:53:55.830 --> 00:53:57.870 align:middle line:84% But then if I look at this theory, 00:53:57.870 --> 00:53:59.970 align:middle line:84% I don't know that theory is covariant. 00:53:59.970 --> 00:54:03.030 align:middle line:84% But this theory is actually covariant. 00:54:03.030 --> 00:54:05.150 align:middle line:84% So the way to check if it's covariant 00:54:05.150 --> 00:54:08.640 align:middle line:84% is you check its observable quantities. 00:54:08.640 --> 00:54:09.140 align:middle line:90% Yeah. 00:54:09.140 --> 00:54:11.150 align:middle line:84% But phi is not observable quantities. 00:54:11.150 --> 00:54:15.480 align:middle line:84% So that's why we say it's not manifestly causal. 00:54:15.480 --> 00:54:17.240 align:middle line:90% Yeah. 00:54:17.240 --> 00:54:19.750 align:middle line:84% So Coulomb gauge is a convenient gauge 00:54:19.750 --> 00:54:22.720 align:middle line:84% to use, even quantum mechanically. 00:54:22.720 --> 00:54:23.470 align:middle line:90% Yes? 00:54:23.470 --> 00:54:25.661 align:middle line:84% AUDIENCE: So the canonical momentum here 00:54:25.661 --> 00:54:28.845 align:middle line:84% is not related to the field momentum 00:54:28.845 --> 00:54:34.230 align:middle line:90% E cross B, not related to-- 00:54:34.230 --> 00:54:37.070 align:middle line:84% we have the momentum carried by the field E cross B? 00:54:37.070 --> 00:54:38.260 align:middle line:90% HONG LIU: No, no, no, no. 00:54:38.260 --> 00:54:40.140 align:middle line:84% No, it has nothing to do with that. 00:54:40.140 --> 00:54:40.640 align:middle line:90% No, no. 00:54:40.640 --> 00:54:44.150 align:middle line:84% This is the canonical momentum conjugated to the field. 00:54:44.150 --> 00:54:46.340 align:middle line:84% And that momentum you're talking about 00:54:46.340 --> 00:54:49.130 align:middle line:84% is the momentum carried by the electromagnetic field. 00:54:49.130 --> 00:54:51.200 align:middle line:84% It's the spacetime momentum carried 00:54:51.200 --> 00:54:56.280 align:middle line:90% by the electromagnetic field. 00:54:56.280 --> 00:54:58.730 align:middle line:84% So that's the analog of the Noether charge. 00:54:58.730 --> 00:55:02.080 align:middle line:84% That momentum is the analog of the Noether charge. 00:55:02.080 --> 00:55:02.800 align:middle line:90% Yeah. 00:55:02.800 --> 00:55:04.310 align:middle line:90% For the scalar case, yeah. 00:55:04.310 --> 00:55:04.810 align:middle line:90% Yeah. 00:55:04.810 --> 00:55:07.330 align:middle line:84% Remember-- in the scalar case, there are also two momentums. 00:55:07.330 --> 00:55:07.830 align:middle line:90% Yeah. 00:55:07.830 --> 00:55:10.910 align:middle line:90% 00:55:10.910 --> 00:55:13.320 align:middle line:90% Other questions? 00:55:13.320 --> 00:55:14.648 align:middle line:90% OK, good. 00:55:14.648 --> 00:55:20.220 align:middle line:90% So this is wrong, OK? 00:55:20.220 --> 00:55:22.680 align:middle line:84% So if we just naively write down this canonical commutation 00:55:22.680 --> 00:55:25.320 align:middle line:84% relation, this is wrong because this equation 00:55:25.320 --> 00:55:38.210 align:middle line:84% is incompatible with the Lorentz gauge condition, OK? 00:55:38.210 --> 00:55:46.850 align:middle line:90% 00:55:46.850 --> 00:55:51.140 align:middle line:84% So imagine you just add the derivative on this-- 00:55:51.140 --> 00:55:54.020 align:middle line:90% to the derivative on x. 00:55:54.020 --> 00:55:58.250 align:middle line:84% And then since this is operator equation-- and then yeah. 00:55:58.250 --> 00:56:00.530 align:middle line:90% Yeah. 00:56:00.530 --> 00:56:01.030 align:middle line:90% Yeah. 00:56:01.030 --> 00:56:04.390 align:middle line:90% So let me call this equation 1. 00:56:04.390 --> 00:56:12.050 align:middle line:84% So if act partial i on 1, then the left-hand side, 00:56:12.050 --> 00:56:24.620 align:middle line:90% I have A t, x and Ej t, x prime. 00:56:24.620 --> 00:56:29.430 align:middle line:84% On the right-hand side, then I have the delta function. 00:56:29.430 --> 00:56:33.710 align:middle line:90% Let me just-- partial i delta. 00:56:33.710 --> 00:56:37.450 align:middle line:90% 00:56:37.450 --> 00:56:39.500 align:middle line:90% OK? 00:56:39.500 --> 00:56:42.170 align:middle line:84% And the right-hand side is non-0. 00:56:42.170 --> 00:56:44.090 align:middle line:84% It's just a derivative of the delta function. 00:56:44.090 --> 00:56:46.010 align:middle line:84% But the left-hand side, according to this, 00:56:46.010 --> 00:56:47.750 align:middle line:90% should be 0, OK? 00:56:47.750 --> 00:56:49.457 align:middle line:90% So we have a contradiction. 00:56:49.457 --> 00:56:55.660 align:middle line:90% 00:56:55.660 --> 00:56:56.920 align:middle line:90% OK? 00:56:56.920 --> 00:57:00.720 align:middle line:84% So this equation cannot be right. 00:57:00.720 --> 00:57:02.460 align:middle line:90% And the reason is simple. 00:57:02.460 --> 00:57:05.970 align:middle line:84% It's because this equation also includes 00:57:05.970 --> 00:57:08.670 align:middle line:84% the longitudinal degrees freedom. 00:57:08.670 --> 00:57:11.970 align:middle line:84% But we said-- Ai can only be transverse. 00:57:11.970 --> 00:57:15.165 align:middle line:84% But this equation actually is three components. 00:57:15.165 --> 00:57:18.600 align:middle line:84% It also included the longitudinal. 00:57:18.600 --> 00:57:21.300 align:middle line:84% So we should not include that, OK? 00:57:21.300 --> 00:57:23.715 align:middle line:90% So what we should do-- 00:57:23.715 --> 00:57:28.090 align:middle line:90% 00:57:28.090 --> 00:57:33.815 align:middle line:84% we should only look at the Ai T, OK? 00:57:33.815 --> 00:58:03.380 align:middle line:90% 00:58:03.380 --> 00:58:12.170 align:middle line:84% So in fact, we should look at Ai T t, x 00:58:12.170 --> 00:58:14.400 align:middle line:84% and its conjugate momentum, which 00:58:14.400 --> 00:58:19.940 align:middle line:90% is called pi j T t, x prime. 00:58:19.940 --> 00:58:32.470 align:middle line:84% So this should be i delta ij delta 3 x minus x prime, OK? 00:58:32.470 --> 00:58:38.840 align:middle line:84% So we should look at only the transverse component, OK? 00:58:38.840 --> 00:58:40.246 align:middle line:90% And the pi i-- 00:58:40.246 --> 00:58:44.940 align:middle line:90% 00:58:44.940 --> 00:58:47.970 align:middle line:84% the conjugate momentum for the transpose-- 00:58:47.970 --> 00:58:54.820 align:middle line:84% Ai-- if you start from here, so you convince yourself from here 00:58:54.820 --> 00:58:57.970 align:middle line:84% you can find it from yourself, OK? 00:58:57.970 --> 00:59:02.030 align:middle line:84% This is just equal to partial 0 Ai T, OK? 00:59:02.030 --> 00:59:06.230 align:middle line:84% Just partial 0 Ai T, just the time derivative. 00:59:06.230 --> 00:59:11.920 align:middle line:84% So remember-- here Ei is equal to partial 0 Ai 00:59:11.920 --> 00:59:13.900 align:middle line:90% minus partial iA 0. 00:59:13.900 --> 00:59:17.170 align:middle line:84% And this part is 0 because phi is equal to 0. 00:59:17.170 --> 00:59:21.430 align:middle line:84% And then here if you take Ai to be transpose 00:59:21.430 --> 00:59:24.880 align:middle line:84% and then pi transpose to-- yeah, equal do that, OK? 00:59:24.880 --> 00:59:28.930 align:middle line:90% 00:59:28.930 --> 00:59:33.440 align:middle line:84% But this equation is still not correct 00:59:33.440 --> 00:59:36.620 align:middle line:84% because the left-hand side is transverse, 00:59:36.620 --> 00:59:39.350 align:middle line:84% but the right-hand side is not transverse, OK? 00:59:39.350 --> 00:59:43.660 align:middle line:84% So here we have to impose a transverse projector. 00:59:43.660 --> 00:59:48.220 align:middle line:84% So rather than write it as delta ij, what I should do 00:59:48.220 --> 00:59:50.440 align:middle line:84% is I write it as a transverse projector. 00:59:50.440 --> 00:59:53.080 align:middle line:90% 00:59:53.080 --> 00:59:53.920 align:middle line:90% OK? 00:59:53.920 --> 00:59:59.140 align:middle line:84% So Pij is that when you act the Ai T-- 00:59:59.140 --> 01:00:09.340 align:middle line:84% so Pij is if you act this on Aj, it projects into the Ai T. OK? 01:00:09.340 --> 01:00:11.830 align:middle line:84% So this is the transverse projector. 01:00:11.830 --> 01:00:21.410 align:middle line:90% 01:00:21.410 --> 01:00:23.340 align:middle line:90% OK? 01:00:23.340 --> 01:00:26.700 align:middle line:84% And then now, this is a consistent equation, OK? 01:00:26.700 --> 01:00:30.100 align:middle line:84% Now, this is a consistent equation. 01:00:30.100 --> 01:00:30.940 align:middle line:90% OK? 01:00:30.940 --> 01:00:32.620 align:middle line:84% So this transverse projector, you 01:00:32.620 --> 01:00:35.560 align:middle line:84% can easily write down in momentum space. 01:00:35.560 --> 01:00:37.960 align:middle line:84% But in coordinate space, it's a little bit 01:00:37.960 --> 01:00:39.520 align:middle line:90% awkward to write it down. 01:00:39.520 --> 01:00:41.410 align:middle line:84% So formally, in coordinate space, 01:00:41.410 --> 01:00:46.720 align:middle line:84% this Pi jT can be written as following-- 01:00:46.720 --> 01:00:55.582 align:middle line:84% as delta ij minus partial i partial j divided by delta 01:00:55.582 --> 01:00:56.082 align:middle line:90% squared. 01:00:56.082 --> 01:01:01.010 align:middle line:90% 01:01:01.010 --> 01:01:03.160 align:middle line:90% OK? 01:01:03.160 --> 01:01:10.430 align:middle line:84% And so you can easily understand the meaning of this equation 01:01:10.430 --> 01:01:12.740 align:middle line:90% because when you act on the-- 01:01:12.740 --> 01:01:14.360 align:middle line:90% yeah, yeah, yeah. 01:01:14.360 --> 01:01:15.680 align:middle line:90% Anyway, you can check yourself. 01:01:15.680 --> 01:01:17.660 align:middle line:90% This works, OK? 01:01:17.660 --> 01:01:20.840 align:middle line:90% And to understand what this-- 01:01:20.840 --> 01:01:23.630 align:middle line:90% you can check formally yourself. 01:01:23.630 --> 01:01:24.440 align:middle line:90% This works, OK? 01:01:24.440 --> 01:01:27.300 align:middle line:90% Give you to the transpose. 01:01:27.300 --> 01:01:31.180 align:middle line:84% You can understand this equation by going to momentum space. 01:01:31.180 --> 01:01:34.270 align:middle line:84% You're going to momentum-- this just becomes ki, this kj. 01:01:34.270 --> 01:01:36.520 align:middle line:90% This just becomes k squared, OK? 01:01:36.520 --> 01:01:40.360 align:middle line:84% This is just ki times kj divided by k squared, OK? 01:01:40.360 --> 01:01:43.750 align:middle line:84% And so the coordinate space definition of this 01:01:43.750 --> 01:01:45.250 align:middle line:84% is just corresponding to the Fourier 01:01:45.250 --> 01:01:49.510 align:middle line:84% transform of the momentum space expression, OK? 01:01:49.510 --> 01:01:52.090 align:middle line:84% Just formally write it as this, OK? 01:01:52.090 --> 01:01:54.920 align:middle line:90% Just formally write like this. 01:01:54.920 --> 01:01:58.150 align:middle line:90% Are there any questions on this? 01:01:58.150 --> 01:01:58.720 align:middle line:90% Yes? 01:01:58.720 --> 01:02:03.624 align:middle line:84% AUDIENCE: So we don't really understand the operation of 1 01:02:03.624 --> 01:02:04.760 align:middle line:90% over del squared. 01:02:04.760 --> 01:02:07.722 align:middle line:84% We just say this is the coordinate space representation 01:02:07.722 --> 01:02:09.525 align:middle line:90% of the momentum space? 01:02:09.525 --> 01:02:10.150 align:middle line:90% HONG LIU: Yeah. 01:02:10.150 --> 01:02:11.950 align:middle line:90% Yeah, yeah. 01:02:11.950 --> 01:02:15.400 align:middle line:90% We just use this to denote-- 01:02:15.400 --> 01:02:18.040 align:middle line:84% use notation to denote what we mean 01:02:18.040 --> 01:02:21.410 align:middle line:84% by the Fourier transform of the momentum space expression. 01:02:21.410 --> 01:02:21.910 align:middle line:90% Yeah. 01:02:21.910 --> 01:02:22.690 align:middle line:90% Yes? 01:02:22.690 --> 01:02:24.850 align:middle line:84% AUDIENCE: Do you mean the inverse of the Laplacian 01:02:24.850 --> 01:02:28.437 align:middle line:84% when you write 1 over grad squared? 01:02:28.437 --> 01:02:29.020 align:middle line:90% HONG LIU: Huh? 01:02:29.020 --> 01:02:30.750 align:middle line:84% AUDIENCE: Do you mean the inverse of the Laplacian 01:02:30.750 --> 01:02:32.317 align:middle line:84% when you write something something 01:02:32.317 --> 01:02:33.690 align:middle line:90% something over grad squared? 01:02:33.690 --> 01:02:34.800 align:middle line:90% HONG LIU: Yeah. 01:02:34.800 --> 01:02:37.860 align:middle line:84% Yeah, this is just formal notation to make-- 01:02:37.860 --> 01:02:40.470 align:middle line:90% once you can check-- 01:02:40.470 --> 01:02:43.750 align:middle line:84% say, if you have something like this partial i Aj partial-- 01:02:43.750 --> 01:02:44.250 align:middle line:90% yeah. 01:02:44.250 --> 01:02:49.110 align:middle line:84% So this is a longitudinal part of A. Yeah, yeah, yeah. 01:02:49.110 --> 01:02:52.020 align:middle line:84% So one thing you can check yourself 01:02:52.020 --> 01:02:57.360 align:middle line:84% is, say, if you act this on the partial j Ai, partial j Ai-- 01:02:57.360 --> 01:03:00.720 align:middle line:84% and then this will cancel with the downstairs. 01:03:00.720 --> 01:03:02.760 align:middle line:84% And then that gives you the transverse part. 01:03:02.760 --> 01:03:05.520 align:middle line:90% 01:03:05.520 --> 01:03:06.060 align:middle line:90% Yeah. 01:03:06.060 --> 01:03:10.530 align:middle line:84% Anyway, just try to treat this as a notation, OK? 01:03:10.530 --> 01:03:15.770 align:middle line:84% So momentum space is very easy to understand. 01:03:15.770 --> 01:03:16.270 align:middle line:90% OK. 01:03:16.270 --> 01:03:22.390 align:middle line:84% So this is now our canonical commutation relation. 01:03:22.390 --> 01:03:27.700 align:middle line:84% And now, we only quantize the physical degrees freedom, OK? 01:03:27.700 --> 01:03:31.120 align:middle line:84% We only quantize the transverse degrees freedom. 01:03:31.120 --> 01:03:33.430 align:middle line:84% And everything else-- we don't worry about it, OK? 01:03:33.430 --> 01:03:35.620 align:middle line:84% Everything else-- we don't worry about it. 01:03:35.620 --> 01:03:40.210 align:middle line:84% And the transverse part just satisfies the standard massless 01:03:40.210 --> 01:03:41.380 align:middle line:90% equation of motion, OK? 01:03:41.380 --> 01:03:44.320 align:middle line:84% So remember-- the transverse part just 01:03:44.320 --> 01:03:47.720 align:middle line:84% satisfies the standard equation of motion. 01:03:47.720 --> 01:03:49.660 align:middle line:84% So now, we can just immediately write down 01:03:49.660 --> 01:03:56.110 align:middle line:84% the most general expansion of Ai T. We just 01:03:56.110 --> 01:04:00.050 align:middle line:84% write down the most general theory that satisfies this-- 01:04:00.050 --> 01:04:00.550 align:middle line:90% yeah. 01:04:00.550 --> 01:04:04.000 align:middle line:84% This is just like a massless equation. 01:04:04.000 --> 01:04:07.540 align:middle line:84% So we can just write what we do previously. 01:04:07.540 --> 01:04:17.510 align:middle line:90% 01:04:17.510 --> 01:04:18.760 align:middle line:90% OK? 01:04:18.760 --> 01:04:23.550 align:middle line:84% And then we can just write down-- 01:04:23.550 --> 01:04:36.980 align:middle line:84% so previously, we just say ak i k x plus ak dagger minus i k x. 01:04:36.980 --> 01:04:43.580 align:middle line:84% And the omega k in this case would be just equal to k, OK? 01:04:43.580 --> 01:04:44.480 align:middle line:90% Be good for massless. 01:04:44.480 --> 01:04:47.720 align:middle line:90% 01:04:47.720 --> 01:04:52.160 align:middle line:84% But now, here we have two independent components, OK? 01:04:52.160 --> 01:04:54.360 align:middle line:84% So we have two independent components. 01:04:54.360 --> 01:04:56.030 align:middle line:84% So essentially, we have two, actually, 01:04:56.030 --> 01:04:58.430 align:middle line:90% independent solutions, OK? 01:04:58.430 --> 01:05:01.760 align:middle line:84% So now, let me parameterize the two independent solutions 01:05:01.760 --> 01:05:04.670 align:middle line:84% by a polarization vector-- to write more generally 01:05:04.670 --> 01:05:08.270 align:middle line:84% by polarization vector like this. 01:05:08.270 --> 01:05:14.470 align:middle line:84% r equal to 1 to 2 epsilon i r, OK? 01:05:14.470 --> 01:05:17.180 align:middle line:90% And then this is r. 01:05:17.180 --> 01:05:20.190 align:middle line:90% This is r. 01:05:20.190 --> 01:05:20.690 align:middle line:90% OK? 01:05:20.690 --> 01:05:22.490 align:middle line:90% So epsilon r-- 01:05:22.490 --> 01:05:33.080 align:middle line:90% 01:05:33.080 --> 01:05:51.510 align:middle line:84% 1 to 2 are the basis of transverse vectors, OK? 01:05:51.510 --> 01:05:55.650 align:middle line:84% Because Ai has to be transverse, OK? 01:05:55.650 --> 01:06:01.180 align:middle line:84% So essentially, you can treat it as polarization 01:06:01.180 --> 01:06:05.280 align:middle line:90% associated with this ak, OK? 01:06:05.280 --> 01:06:15.360 align:middle line:84% So by definition, this should satisfy, say, ir ki equal to 0. 01:06:15.360 --> 01:06:18.290 align:middle line:84% So this should be transverse, OK? 01:06:18.290 --> 01:06:20.960 align:middle line:84% So we also define them to be orthogonal to each other. 01:06:20.960 --> 01:06:27.220 align:middle line:90% 01:06:27.220 --> 01:06:31.860 align:middle line:84% So r and s here are just 1 2, OK? 01:06:31.860 --> 01:06:34.860 align:middle line:84% So this is orthogonality normal to each other. 01:06:34.860 --> 01:06:38.986 align:middle line:90% 01:06:38.986 --> 01:06:42.280 align:middle line:90% Orthogonal, OK? 01:06:42.280 --> 01:06:48.410 align:middle line:84% And so they also satisfy the condition. 01:06:48.410 --> 01:06:54.830 align:middle line:84% So if you have two independent vectors, 01:06:54.830 --> 01:07:01.400 align:middle line:84% then when you sum them together, then you 01:07:01.400 --> 01:07:05.160 align:middle line:84% should get the projection operator, OK? 01:07:05.160 --> 01:07:10.485 align:middle line:84% So you should get the Pij T k, OK? 01:07:10.485 --> 01:07:13.130 align:middle line:90% 01:07:13.130 --> 01:07:18.230 align:middle line:84% So epsilon i-- they are k dependent. 01:07:18.230 --> 01:07:19.790 align:middle line:84% They depend on the spatial momentum 01:07:19.790 --> 01:07:21.760 align:middle line:84% because they have to be orthogonal to the momentum. 01:07:21.760 --> 01:07:22.260 align:middle line:90% OK? 01:07:22.260 --> 01:07:26.510 align:middle line:84% And this is just the momentum space version of this. 01:07:26.510 --> 01:07:27.440 align:middle line:90% This is the delta ij. 01:07:27.440 --> 01:07:30.410 align:middle line:90% 01:07:30.410 --> 01:07:35.850 align:middle line:90% ki kj squared, OK? 01:07:35.850 --> 01:07:37.860 align:middle line:90% So this is orthogonal condition. 01:07:37.860 --> 01:07:39.630 align:middle line:84% This is a completeness condition. 01:07:39.630 --> 01:07:44.870 align:middle line:90% 01:07:44.870 --> 01:07:47.360 align:middle line:90% OK? 01:07:47.360 --> 01:07:48.610 align:middle line:90% It's a completeness condition. 01:07:48.610 --> 01:07:52.360 align:middle line:90% 01:07:52.360 --> 01:07:55.630 align:middle line:90% Any questions on this? 01:07:55.630 --> 01:08:03.680 align:middle line:84% So now, this is the expansion for Ai, OK? 01:08:03.680 --> 01:08:12.620 align:middle line:90% 01:08:12.620 --> 01:08:16.620 align:middle line:84% So the only difference from the scalar case-- 01:08:16.620 --> 01:08:19.920 align:middle line:84% is that now, here you have two independent vectors. 01:08:19.920 --> 01:08:24.160 align:middle line:84% And now, we have allowed the general polarization 01:08:24.160 --> 01:08:26.590 align:middle line:90% associated with the-- 01:08:26.590 --> 01:08:28.060 align:middle line:90% yeah. 01:08:28.060 --> 01:08:30.310 align:middle line:84% Have parameterized the general solution 01:08:30.310 --> 01:08:34.120 align:middle line:84% using two independent polarization vectors, OK? 01:08:34.120 --> 01:08:39.510 align:middle line:90% 01:08:39.510 --> 01:08:49.229 align:middle line:84% So now, you can now plug this back into here. 01:08:49.229 --> 01:08:51.750 align:middle line:84% Now, you can find the commutation relation between 01:08:51.750 --> 01:08:52.679 align:middle line:90% and A and A dagger. 01:08:52.679 --> 01:08:56.859 align:middle line:90% 01:08:56.859 --> 01:09:04.840 align:middle line:84% Then you find, as you would expect, 01:09:04.840 --> 01:09:08.640 align:middle line:90% that they satisfy the standard. 01:09:08.640 --> 01:09:14.410 align:middle line:84% So from the commutation relation, 01:09:14.410 --> 01:09:23.979 align:middle line:84% we find that the akr and ak prime s dagger 01:09:23.979 --> 01:09:26.619 align:middle line:90% is equal to delta rs. 01:09:26.619 --> 01:09:34.955 align:middle line:90% 01:09:34.955 --> 01:09:35.455 align:middle line:90% OK? 01:09:35.455 --> 01:09:52.490 align:middle line:90% 01:09:52.490 --> 01:09:55.280 align:middle line:90% With the rest 0, OK? 01:09:55.280 --> 01:09:58.010 align:middle line:90% So r and s equal to 1 2. 01:09:58.010 --> 01:09:59.615 align:middle line:90% And the others are 0. 01:09:59.615 --> 01:10:07.230 align:middle line:90% 01:10:07.230 --> 01:10:07.730 align:middle line:90% OK. 01:10:07.730 --> 01:10:11.360 align:middle line:90% 01:10:11.360 --> 01:10:11.860 align:middle line:90% OK. 01:10:11.860 --> 01:10:15.520 align:middle line:84% So now, we can write down the Hilbert space. 01:10:15.520 --> 01:10:18.700 align:middle line:84% We obtain the Hilbert space again just from what we do. 01:10:18.700 --> 01:10:29.260 align:middle line:84% So we specify a vacuum requiring this-- just ar k 01:10:29.260 --> 01:10:38.440 align:middle line:84% annihilate the vacuum for any k and r equal to 1 and 2. 01:10:38.440 --> 01:10:51.430 align:middle line:84% And then the single particle state 01:10:51.430 --> 01:11:00.130 align:middle line:84% is given by just ak r and the dagger acting on 0. 01:11:00.130 --> 01:11:09.010 align:middle line:84% So we denoted this as k and epsilon r, OK? 01:11:09.010 --> 01:11:17.260 align:middle line:84% So this describes a single photon state. 01:11:17.260 --> 01:11:29.170 align:middle line:90% 01:11:29.170 --> 01:11:44.070 align:middle line:84% This polarization vector given by epsilon r, OK? 01:11:44.070 --> 01:11:48.810 align:middle line:84% So remember-- epsilon r is the transverse vector, OK? 01:11:48.810 --> 01:11:50.880 align:middle line:84% And you have two independent of them. 01:11:50.880 --> 01:11:51.630 align:middle line:90% Yes? 01:11:51.630 --> 01:11:54.065 align:middle line:84% AUDIENCE: Should there be some normalization, 01:11:54.065 --> 01:11:58.940 align:middle line:90% like square root 2 [INAUDIBLE]? 01:11:58.940 --> 01:12:00.460 align:middle line:90% HONG LIU: Yeah. 01:12:00.460 --> 01:12:01.240 align:middle line:90% I believe so. 01:12:01.240 --> 01:12:01.840 align:middle line:90% Yeah, yeah. 01:12:01.840 --> 01:12:05.350 align:middle line:84% Let me just put the proportionality. 01:12:05.350 --> 01:12:06.550 align:middle line:90% Right, yeah. 01:12:06.550 --> 01:12:07.360 align:middle line:90% Thank you. 01:12:07.360 --> 01:12:12.220 align:middle line:90% 01:12:12.220 --> 01:12:13.480 align:middle line:90% OK. 01:12:13.480 --> 01:12:14.860 align:middle line:90% So yeah. 01:12:14.860 --> 01:12:17.620 align:middle line:84% So that concludes the story for quantization in the Coulomb 01:12:17.620 --> 01:12:18.730 align:middle line:90% gauge. 01:12:18.730 --> 01:12:30.470 align:middle line:84% So just to summarize, for the Coulomb gauge 01:12:30.470 --> 01:12:35.930 align:middle line:84% so the Coulomb gauge is conceptually pretty simple, OK? 01:12:35.930 --> 01:12:38.240 align:middle line:84% So the advantage of the Coulomb gauge-- 01:12:38.240 --> 01:12:43.630 align:middle line:90% 01:12:43.630 --> 01:12:54.290 align:middle line:84% we said we can just directly quantize physical degrees 01:12:54.290 --> 01:12:57.135 align:middle line:90% of freedom, OK? 01:12:57.135 --> 01:12:58.760 align:middle line:84% Because the unphysical degrees freedom, 01:12:58.760 --> 01:13:02.270 align:middle line:84% phi, just automatically is equal to 0. 01:13:02.270 --> 01:13:04.240 align:middle line:84% And then the longitudinal part, we 01:13:04.240 --> 01:13:07.490 align:middle line:84% can just throw away by hand using the Coulomb gauge 01:13:07.490 --> 01:13:10.160 align:middle line:84% by promoting the Coulomb gauge condition as the operator 01:13:10.160 --> 01:13:10.880 align:middle line:90% equation. 01:13:10.880 --> 01:13:12.870 align:middle line:84% And then we can just solve it by hand. 01:13:12.870 --> 01:13:15.780 align:middle line:84% And then we just have the transverse part. 01:13:15.780 --> 01:13:18.200 align:middle line:84% And then once you have the transverse part, 01:13:18.200 --> 01:13:25.475 align:middle line:84% you can just quantize it then as a free massless field, OK? 01:13:25.475 --> 01:13:29.720 align:middle line:90% 01:13:29.720 --> 01:13:37.980 align:middle line:84% But the drawback, of course, is always not 01:13:37.980 --> 01:13:40.980 align:middle line:90% manifest Lorentz invariant, OK? 01:13:40.980 --> 01:13:43.620 align:middle line:90% Not manifest Lorentz covariance. 01:13:43.620 --> 01:13:52.130 align:middle line:90% 01:13:52.130 --> 01:13:52.630 align:middle line:90% OK? 01:13:52.630 --> 01:13:57.870 align:middle line:90% 01:13:57.870 --> 01:13:58.905 align:middle line:90% Any questions on this? 01:13:58.905 --> 01:14:08.950 align:middle line:90% 01:14:08.950 --> 01:14:09.450 align:middle line:90% Good. 01:14:09.450 --> 01:14:12.860 align:middle line:90% 01:14:12.860 --> 01:14:15.650 align:middle line:84% So now, let's look at the Lorentz gauge. 01:14:15.650 --> 01:14:20.420 align:middle line:90% 01:14:20.420 --> 01:14:24.680 align:middle line:84% So these two are the most commonly used gauges. 01:14:24.680 --> 01:14:28.550 align:middle line:84% And also, these are two representatives, OK? 01:14:28.550 --> 01:14:31.610 align:middle line:84% They're two representative gauges 01:14:31.610 --> 01:14:34.670 align:middle line:84% because by quantizing them, they're 01:14:34.670 --> 01:14:37.640 align:middle line:84% involving completely different procedures. 01:14:37.640 --> 01:14:44.510 align:middle line:84% And by contrasting what's happening in the Lorentz 01:14:44.510 --> 01:14:45.020 align:middle line:90% gauge-- 01:14:45.020 --> 01:14:47.240 align:middle line:84% and what's happening in the Lorentz gauge actually 01:14:47.240 --> 01:14:48.800 align:middle line:90% will be very instructive, OK? 01:14:48.800 --> 01:14:53.400 align:middle line:84% So we will just consider these two cases. 01:14:53.400 --> 01:14:55.445 align:middle line:84% So now, let's move on to the Lorentz gauge. 01:14:55.445 --> 01:15:16.650 align:middle line:90% 01:15:16.650 --> 01:15:23.610 align:middle line:84% So in the Lorentz gauge, we will use 01:15:23.610 --> 01:15:34.680 align:middle line:84% a somewhat different strategy because in the Lorentz gauge-- 01:15:34.680 --> 01:15:39.270 align:middle line:84% so first is that solving the gauge condition 01:15:39.270 --> 01:15:42.000 align:middle line:90% becomes more difficult, OK? 01:15:42.000 --> 01:15:48.880 align:middle line:84% And also, in the Lorentz gauge, there 01:15:48.880 --> 01:15:51.250 align:middle line:90% is a residual gauge freedom, OK? 01:15:51.250 --> 01:15:52.580 align:middle line:90% So there are two things. 01:15:52.580 --> 01:15:54.460 align:middle line:84% So in the Lorentz gauge, instead, we 01:15:54.460 --> 01:15:58.300 align:middle line:90% will start with this action. 01:15:58.300 --> 01:16:00.790 align:middle line:84% So we will start with this action. 01:16:00.790 --> 01:16:11.440 align:middle line:90% 01:16:11.440 --> 01:16:15.660 align:middle line:84% We will start with this action, which 01:16:15.660 --> 01:16:20.910 align:middle line:84% we say classically can be used to fix the Lorentz gauge, OK? 01:16:20.910 --> 01:16:23.630 align:middle line:90% 01:16:23.630 --> 01:16:26.930 align:middle line:84% And when psi is equal to 1, so this just 01:16:26.930 --> 01:16:34.700 align:middle line:84% becomes partial mu A nu and partial mu A nu, OK? 01:16:34.700 --> 01:16:36.440 align:middle line:90% So this is for xi to 1. 01:16:36.440 --> 01:16:41.200 align:middle line:90% 01:16:41.200 --> 01:16:45.740 align:middle line:84% So you say, this theory-- we know how to quantize. 01:16:45.740 --> 01:16:51.400 align:middle line:84% These are just four decoupled massless scalar fields, OK? 01:16:51.400 --> 01:16:55.150 align:middle line:84% For psi equal to 1, it's particularly simple. 01:16:55.150 --> 01:16:56.035 align:middle line:90% We just have four-- 01:16:56.035 --> 01:16:58.775 align:middle line:90% 01:16:58.775 --> 01:16:59.275 align:middle line:90% yeah. 01:16:59.275 --> 01:17:02.520 align:middle line:90% 01:17:02.520 --> 01:17:05.970 align:middle line:84% Oh, by the way, I should mention-- 01:17:05.970 --> 01:17:10.650 align:middle line:84% once you have that form, and once you have this commutation 01:17:10.650 --> 01:17:12.930 align:middle line:84% relation, and once you have this relation, 01:17:12.930 --> 01:17:17.190 align:middle line:84% you can find any correlation functions of A, OK? 01:17:17.190 --> 01:17:18.960 align:middle line:84% Two point functions-- propagator, 01:17:18.960 --> 01:17:20.640 align:middle line:84% Feynman propagator, retarded propagator. 01:17:20.640 --> 01:17:22.230 align:middle line:84% You can find all of them explicitly. 01:17:22.230 --> 01:17:25.310 align:middle line:90% 01:17:25.310 --> 01:17:29.240 align:middle line:84% So now, this theory seems to be very simple. 01:17:29.240 --> 01:17:33.170 align:middle line:84% This just decoupled four massive scalar fields, OK? 01:17:33.170 --> 01:17:35.570 align:middle line:84% So we can just straightforwardly write down 01:17:35.570 --> 01:17:37.670 align:middle line:90% its canonical momentum. 01:17:37.670 --> 01:17:49.520 align:middle line:84% So pi nu-- this is just equal to A nu star, A nu dot. 01:17:49.520 --> 01:17:51.570 align:middle line:90% OK? 01:17:51.570 --> 01:17:56.610 align:middle line:90% And the Hamiltonian density-- 01:17:56.610 --> 01:18:04.320 align:middle line:84% just given by 1/2 A nu dot A nu dot plus-- 01:18:04.320 --> 01:18:11.040 align:middle line:84% just like you have 4 independent massive scalar fields 01:18:11.040 --> 01:18:18.640 align:middle line:84% with the equation of motion given by this partial square A 01:18:18.640 --> 01:18:23.180 align:middle line:90% nu equal to 0, OK? 01:18:23.180 --> 01:18:27.360 align:middle line:84% So now, you can just completely take over, OK? 01:18:27.360 --> 01:18:31.400 align:middle line:84% We can just completely take over what we did 01:18:31.400 --> 01:18:34.850 align:middle line:90% for the massless scalar, OK? 01:18:34.850 --> 01:18:36.860 align:middle line:84% To write down the answer for the massless 01:18:36.860 --> 01:18:39.660 align:middle line:84% scalar-- but we're running out of time today. 01:18:39.660 --> 01:18:45.700 align:middle line:84% So next time, we will talk about-- 01:18:45.700 --> 01:18:51.380 align:middle line:84% so now, you can just treat it as four massless scalar. 01:18:51.380 --> 01:18:55.600 align:middle line:90% But then we have a problem, OK? 01:18:55.600 --> 01:19:00.670 align:middle line:84% Because we just need the Coulomb gauge. 01:19:00.670 --> 01:19:02.920 align:middle line:84% We only have two physical degrees freedom. 01:19:02.920 --> 01:19:05.320 align:middle line:90% But here we have four, OK? 01:19:05.320 --> 01:19:09.160 align:middle line:84% So somehow, we have to get rid of two, OK? 01:19:09.160 --> 01:19:12.910 align:middle line:84% And then we will find the ways to get rid of the two, OK? 01:19:12.910 --> 01:19:13.570 align:middle line:90% OK, yeah. 01:19:13.570 --> 01:19:17.070 align:middle line:84% So yeah, let's stop here for today. 01:19:17.070 --> 01:19:20.000 align:middle line:90%