WEBVTT 00:00:00.000 --> 00:00:00.946 align:middle line:90% [SQUEAKING] 00:00:00.946 --> 00:00:01.446 align:middle line:90% [RUSTLING] 00:00:01.446 --> 00:00:04.820 align:middle line:90% [CLICKING] 00:00:04.820 --> 00:00:11.400 align:middle line:90% 00:00:11.400 --> 00:00:17.040 align:middle line:84% PROFESSOR: So last time, we more or less finished 00:00:17.040 --> 00:00:24.990 align:middle line:84% discussing quantization of the free scalar field theory. 00:00:24.990 --> 00:00:30.210 align:middle line:84% And then we showed that the excitations of that theory 00:00:30.210 --> 00:00:35.880 align:middle line:84% essentially give rise to relativistic particles. 00:00:35.880 --> 00:00:39.510 align:middle line:84% And so we also discussed various other things-- 00:00:39.510 --> 00:00:43.200 align:middle line:84% so structure of the Hilbert space, 00:00:43.200 --> 00:00:45.360 align:middle line:90% conserved quantities, et cetera. 00:00:45.360 --> 00:00:48.760 align:middle line:90% 00:00:48.760 --> 00:00:53.050 align:middle line:84% So at the end, we talked about how 00:00:53.050 --> 00:01:04.890 align:middle line:84% to understand the conserved charge at the quantum level. 00:01:04.890 --> 00:01:19.940 align:middle line:90% 00:01:19.940 --> 00:01:26.390 align:middle line:84% So classically-- so we have a, say, symmetric transformation-- 00:01:26.390 --> 00:01:32.180 align:middle line:84% means that if you take phi a to some phi a prime-- 00:01:32.180 --> 00:01:37.230 align:middle line:84% phi a prime, normally, we write it in the infinitesimal way-- 00:01:37.230 --> 00:01:41.030 align:middle line:84% introduce some parameter a-- some parameter epsilon. 00:01:41.030 --> 00:01:43.710 align:middle line:84% And I can also have multiple symmetries. 00:01:43.710 --> 00:01:47.200 align:middle line:84% So for each transformation-- say I have epsilon. 00:01:47.200 --> 00:01:53.420 align:middle line:84% So alpha labels different transformations, and then-- 00:01:53.420 --> 00:01:57.080 align:middle line:84% so this is the transformation, which can depend on phi a. 00:01:57.080 --> 00:02:01.890 align:middle line:84% Also depends on its derivatives, et cetera. 00:02:01.890 --> 00:02:04.370 align:middle line:84% And then if you have such a symmetric transformation 00:02:04.370 --> 00:02:09.030 align:middle line:84% we discussed before, then that generates a conserved current, 00:02:09.030 --> 00:02:10.889 align:middle line:90% and that is labeled by alpha. 00:02:10.889 --> 00:02:13.710 align:middle line:84% Alpha just denotes different conserved current. 00:02:13.710 --> 00:02:16.710 align:middle line:84% And then yeah, so it is conserved. 00:02:16.710 --> 00:02:21.480 align:middle line:90% 00:02:21.480 --> 00:02:25.060 align:middle line:84% So this is the classical statement. 00:02:25.060 --> 00:02:27.370 align:middle line:84% Oh, yeah, also Noether's theorem tells us 00:02:27.370 --> 00:02:29.450 align:middle line:84% how to write down such a current explicitly. 00:02:29.450 --> 00:02:32.280 align:middle line:90% 00:02:32.280 --> 00:02:35.220 align:middle line:90% So we can just write it as-- 00:02:35.220 --> 00:02:37.050 align:middle line:84% the current can be written explicitly. 00:02:37.050 --> 00:02:48.800 align:middle line:90% 00:02:48.800 --> 00:02:55.580 align:middle line:84% So k mu is the transformation of the Lagrangian density. 00:02:55.580 --> 00:02:57.800 align:middle line:84% Suppose under this transformation the Lagrangian 00:02:57.800 --> 00:03:04.290 align:middle line:84% density transform as, say, alpha partial mu, k mu alpha. 00:03:04.290 --> 00:03:06.830 align:middle line:90% 00:03:06.830 --> 00:03:08.300 align:middle line:90% So alpha, it doesn't matter. 00:03:08.300 --> 00:03:11.030 align:middle line:84% Alpha just label different symmetries-- the index. 00:03:11.030 --> 00:03:15.970 align:middle line:84% It doesn't matter whether it's up or down. 00:03:15.970 --> 00:03:21.908 align:middle line:84% And then the zeroth component of the charge-- 00:03:21.908 --> 00:03:24.200 align:middle line:84% zeroth component of the current when you integrate over 00:03:24.200 --> 00:03:29.220 align:middle line:84% all space then give rise to a conserved charge. 00:03:29.220 --> 00:03:40.620 align:middle line:84% So this is conserved charge times the-- 00:03:40.620 --> 00:03:42.720 align:middle line:90% yeah. 00:03:42.720 --> 00:03:43.920 align:middle line:90% So now, let's discuss-- 00:03:43.920 --> 00:03:46.570 align:middle line:90% 00:03:46.570 --> 00:03:49.340 align:middle line:84% so when you go to quantum level-- 00:03:49.340 --> 00:03:52.390 align:middle line:84% so when you quantize the theory, and this just becomes 00:03:52.390 --> 00:03:55.060 align:middle line:84% quantum operator, and the whole current 00:03:55.060 --> 00:03:57.040 align:middle line:90% becomes a quantum operator. 00:03:57.040 --> 00:04:02.080 align:middle line:84% And then Q is also a quantum operator. 00:04:02.080 --> 00:04:03.010 align:middle line:90% So the quantum level-- 00:04:03.010 --> 00:04:10.080 align:middle line:90% 00:04:10.080 --> 00:04:11.970 align:middle line:90% and now Q alpha-- 00:04:11.970 --> 00:04:14.850 align:middle line:84% because it's conserved, so this becomes 00:04:14.850 --> 00:04:16.410 align:middle line:90% time-independent operator. 00:04:16.410 --> 00:04:22.930 align:middle line:90% 00:04:22.930 --> 00:04:27.310 align:middle line:84% We see explicitly in the case of the, say, Hamiltonian dens-- 00:04:27.310 --> 00:04:30.020 align:middle line:90% Hamiltonian and the momentum. 00:04:30.020 --> 00:04:34.420 align:middle line:84% So we showed explicitly they actually don't depend on time. 00:04:34.420 --> 00:04:36.130 align:middle line:84% But at the quantum mechanical level, 00:04:36.130 --> 00:04:38.980 align:middle line:84% actually, this Q can play a more important role. 00:04:38.980 --> 00:04:43.100 align:middle line:90% 00:04:43.100 --> 00:04:46.950 align:middle line:84% In fact, it can be viewed as a generator of symmetries. 00:04:46.950 --> 00:04:47.630 align:middle line:90% So we can show-- 00:04:47.630 --> 00:04:51.080 align:middle line:90% 00:04:51.080 --> 00:04:54.660 align:middle line:84% we will show the following statement-- 00:04:54.660 --> 00:04:58.860 align:middle line:84% that if you consider so epsilon-- 00:04:58.860 --> 00:05:00.510 align:middle line:90% so consider such a quantity-- 00:05:00.510 --> 00:05:04.050 align:middle line:90% 00:05:04.050 --> 00:05:07.870 align:middle line:84% can you see the commutator of Q with phi a. 00:05:07.870 --> 00:05:12.110 align:middle line:90% 00:05:12.110 --> 00:05:14.000 align:middle line:84% So now, let's continue over to here. 00:05:14.000 --> 00:05:17.030 align:middle line:90% 00:05:17.030 --> 00:05:19.050 align:middle line:90% Maybe just let me write it here. 00:05:19.050 --> 00:05:26.040 align:middle line:84% And that generates minus epsilon alpha f alpha a. 00:05:26.040 --> 00:05:31.590 align:middle line:84% So the commutator of this Q alpha epsilon alpha with phi a 00:05:31.590 --> 00:05:35.790 align:middle line:84% essentially generates this transformation, essentially, 00:05:35.790 --> 00:05:38.250 align:middle line:84% generate this infinitesimal transformation. 00:05:38.250 --> 00:05:41.490 align:middle line:90% 00:05:41.490 --> 00:05:45.240 align:middle line:84% So that's why we call the Q alpha also generator 00:05:45.240 --> 00:05:46.410 align:middle line:90% of symmetries. 00:05:46.410 --> 00:05:54.660 align:middle line:90% 00:05:54.660 --> 00:05:57.960 align:middle line:84% So this is the statement I'm going to show very soon. 00:05:57.960 --> 00:06:01.390 align:middle line:84% So before I do that, do you have any questions? 00:06:01.390 --> 00:06:02.080 align:middle line:90% Yes? 00:06:02.080 --> 00:06:04.360 align:middle line:84% AUDIENCE: In the conserved current equation, 00:06:04.360 --> 00:06:07.840 align:middle line:84% how can f have an alpha in the upper index 00:06:07.840 --> 00:06:10.803 align:middle line:84% when J has an alpha in the lower index, its not meaningful? 00:06:10.803 --> 00:06:11.470 align:middle line:90% PROFESSOR: Yeah. 00:06:11.470 --> 00:06:14.200 align:middle line:84% Just alpha-- don't worry about the location of alpha index. 00:06:14.200 --> 00:06:19.750 align:middle line:84% Alpha just label a different current and just went-- yeah, 00:06:19.750 --> 00:06:22.060 align:middle line:84% here, we don't worry about its notation. 00:06:22.060 --> 00:06:25.660 align:middle line:84% I just put it up and down just for convenience. 00:06:25.660 --> 00:06:28.910 align:middle line:90% 00:06:28.910 --> 00:06:31.310 align:middle line:90% Other questions? 00:06:31.310 --> 00:06:32.315 align:middle line:90% Yes? 00:06:32.315 --> 00:06:33.940 align:middle line:84% AUDIENCE: Can you [INAUDIBLE] positions 00:06:33.940 --> 00:06:36.020 align:middle line:90% is that a sum over alpha? 00:06:36.020 --> 00:06:37.550 align:middle line:90% PROFESSOR: Sorry? 00:06:37.550 --> 00:06:40.340 align:middle line:84% AUDIENCE: In the proposed, var L is 00:06:40.340 --> 00:06:43.730 align:middle line:84% epsilon alpha partial to mu alpha, is that a sum 00:06:43.730 --> 00:06:44.720 align:middle line:90% over all alpha? 00:06:44.720 --> 00:06:47.540 align:middle line:84% PROFESSOR: Yeah, that's sum over all alphas. 00:06:47.540 --> 00:06:50.990 align:middle line:90% So any repeated indices is sum. 00:06:50.990 --> 00:06:52.880 align:middle line:90% So if you only have one-- 00:06:52.880 --> 00:06:55.710 align:middle line:84% so for example, if you have one epsilon-- of course, 00:06:55.710 --> 00:06:58.740 align:middle line:84% there is no sum, but if you have two different charges, 00:06:58.740 --> 00:06:59.990 align:middle line:90% and then this is sum. 00:06:59.990 --> 00:07:02.600 align:middle line:90% 00:07:02.600 --> 00:07:03.590 align:middle line:90% Other questions? 00:07:03.590 --> 00:07:08.760 align:middle line:90% 00:07:08.760 --> 00:07:12.500 align:middle line:84% So we are going to show this statement. 00:07:12.500 --> 00:07:14.050 align:middle line:84% And so this statement is very easy 00:07:14.050 --> 00:07:19.210 align:middle line:90% to show in the general case. 00:07:19.210 --> 00:07:22.480 align:middle line:84% So let me call this equation star. 00:07:22.480 --> 00:07:26.410 align:middle line:84% So let's first consider the case-- 00:07:26.410 --> 00:07:29.665 align:middle line:84% first, let's consider the sub case that the-- 00:07:29.665 --> 00:07:33.360 align:middle line:90% 00:07:33.360 --> 00:07:44.100 align:middle line:84% let's first suppose f a alpha-- just this transformation 00:07:44.100 --> 00:07:56.580 align:middle line:84% of phi a does not contain the time derivative of phi a. 00:07:56.580 --> 00:08:05.070 align:middle line:84% And also the zeroth component of the k is equal to 0. 00:08:05.070 --> 00:08:10.560 align:middle line:90% Let's consider a first spec-- 00:08:10.560 --> 00:08:11.685 align:middle line:90% a more limited case. 00:08:11.685 --> 00:08:16.830 align:middle line:90% 00:08:16.830 --> 00:08:18.890 align:middle line:84% So this more limited case actually 00:08:18.890 --> 00:08:22.510 align:middle line:90% covers many, many examples. 00:08:22.510 --> 00:08:26.280 align:middle line:84% It actually covers a majority of examples. 00:08:26.280 --> 00:08:29.580 align:middle line:84% For example, if you look at the translation symmetry, 00:08:29.580 --> 00:08:32.510 align:middle line:84% which gives rise to the stress tensor-- 00:08:32.510 --> 00:08:36.690 align:middle line:84% so this will-- so this applies to the case I mu-- 00:08:36.690 --> 00:08:38.650 align:middle line:90% T mu i. 00:08:38.650 --> 00:08:41.380 align:middle line:84% So this corresponding to the spatial-- 00:08:41.380 --> 00:08:44.890 align:middle line:84% so this is the conserved current corresponding to the spatial. 00:08:44.890 --> 00:08:50.950 align:middle line:84% So this is the current for spatial translations. 00:08:50.950 --> 00:08:56.820 align:middle line:90% 00:08:56.820 --> 00:08:59.710 align:middle line:84% When we do spatial translations, then of course, 00:08:59.710 --> 00:09:04.590 align:middle line:84% this f given by the spatial derivatives, 00:09:04.590 --> 00:09:07.890 align:middle line:84% and then it does not involve time derivative. 00:09:07.890 --> 00:09:12.120 align:middle line:84% And in that case, the k also does not involve in-- 00:09:12.120 --> 00:09:13.350 align:middle line:90% k 0 component's 0. 00:09:13.350 --> 00:09:15.590 align:middle line:84% Yeah, just I try to remind myself 00:09:15.590 --> 00:09:17.820 align:middle line:90% of what you worked in your pset. 00:09:17.820 --> 00:09:24.330 align:middle line:84% And so for this example, i is the counterpart-- 00:09:24.330 --> 00:09:25.770 align:middle line:90% is the alpha here. 00:09:25.770 --> 00:09:30.390 align:middle line:84% So for each direction, you have a translation, and so this-- 00:09:30.390 --> 00:09:34.170 align:middle line:84% so the alpha-- of course, i is the alpha here 00:09:34.170 --> 00:09:35.940 align:middle line:90% and the mu is the J mu here. 00:09:35.940 --> 00:09:39.670 align:middle line:90% 00:09:39.670 --> 00:09:44.610 align:middle line:84% And also this applies to the case which-- 00:09:44.610 --> 00:09:49.200 align:middle line:90% for all internal symmetries. 00:09:49.200 --> 00:09:58.000 align:middle line:90% 00:09:58.000 --> 00:10:01.360 align:middle line:84% By internal symmetries, we mean the transformations don't 00:10:01.360 --> 00:10:04.660 align:middle line:90% involve space-time coordinates. 00:10:04.660 --> 00:10:07.390 align:middle line:84% For example, you did in your pset 00:10:07.390 --> 00:10:11.395 align:middle line:84% the complex scalar field which you can just rotate by phase. 00:10:11.395 --> 00:10:13.270 align:middle line:84% And that's called the internal transformation 00:10:13.270 --> 00:10:15.220 align:middle line:84% because that transformation does not 00:10:15.220 --> 00:10:18.910 align:middle line:90% involve spacetime coordinates. 00:10:18.910 --> 00:10:21.970 align:middle line:84% In contrast, when we find the stress tensor, 00:10:21.970 --> 00:10:24.970 align:middle line:84% we will do a spacetime translation, 00:10:24.970 --> 00:10:28.480 align:middle line:90% and that does involve spacetime. 00:10:28.480 --> 00:10:31.960 align:middle line:84% So this corresponding to the spacetime transformations-- 00:10:31.960 --> 00:10:32.950 align:middle line:90% spacetime symmetries. 00:10:32.950 --> 00:10:36.070 align:middle line:84% And this is-- the phase is the internal symmetries. 00:10:36.070 --> 00:10:41.050 align:middle line:84% So for all internal symmetries, you don't-- the transformation 00:10:41.050 --> 00:10:43.020 align:middle line:84% will not contain the time derivative. 00:10:43.020 --> 00:10:46.090 align:middle line:90% 00:10:46.090 --> 00:10:51.970 align:middle line:84% So in this case, then it's easy because in this case, 00:10:51.970 --> 00:10:57.130 align:middle line:84% then we just have the one-- then the zeroth component of J mu. 00:10:57.130 --> 00:11:00.200 align:middle line:90% J just given by-- 00:11:00.200 --> 00:11:05.540 align:middle line:84% the first term by definition is just the canonical momentum 00:11:05.540 --> 00:11:11.090 align:middle line:90% conjugate to the a. 00:11:11.090 --> 00:11:16.550 align:middle line:84% So the first factor, partial L partial partial mu phi a-- when 00:11:16.550 --> 00:11:18.440 align:middle line:90% you set mu equal to 0-- 00:11:18.440 --> 00:11:20.270 align:middle line:84% so that's just the derivative with respect 00:11:20.270 --> 00:11:23.180 align:middle line:84% to the time derivative of phi a, and that, by definition, 00:11:23.180 --> 00:11:30.020 align:middle line:84% is just the canonical momentum density conjugate to phi a. 00:11:30.020 --> 00:11:37.252 align:middle line:90% And then you have f alpha a-- 00:11:37.252 --> 00:11:38.130 align:middle line:90% f a alpha. 00:11:38.130 --> 00:11:42.940 align:middle line:90% 00:11:42.940 --> 00:11:43.440 align:middle line:90% Good? 00:11:43.440 --> 00:11:51.450 align:middle line:90% 00:11:51.450 --> 00:11:57.650 align:middle line:84% So let me just make one comment on this expression. 00:11:57.650 --> 00:12:01.690 align:middle line:84% So this expression, of course, classically, you 00:12:01.690 --> 00:12:03.800 align:middle line:90% can write whatever way you want. 00:12:03.800 --> 00:12:05.350 align:middle line:84% But quantum mechanically, there's 00:12:05.350 --> 00:12:08.740 align:middle line:84% a very important subtlety because this typically 00:12:08.740 --> 00:12:13.360 align:middle line:84% will involve phi, and then this is momentum, 00:12:13.360 --> 00:12:15.430 align:middle line:84% and they don't necessarily commute. 00:12:15.430 --> 00:12:18.550 align:middle line:84% So this is the field theory version 00:12:18.550 --> 00:12:23.590 align:middle line:84% of so-called operator ordering ambiguity, which is already 00:12:23.590 --> 00:12:24.670 align:middle line:90% in quantum mechanics. 00:12:24.670 --> 00:12:28.388 align:middle line:84% In quantum mechanics, when you go from classical-- 00:12:28.388 --> 00:12:30.430 align:middle line:84% even it's already in the non-relativistic quantum 00:12:30.430 --> 00:12:31.000 align:middle line:90% mechanics. 00:12:31.000 --> 00:12:32.710 align:middle line:84% When you go to classical mechanics 00:12:32.710 --> 00:12:34.720 align:middle line:84% to quantum mechanics, and then there's 00:12:34.720 --> 00:12:37.630 align:middle line:84% an issue how you order operators when you 00:12:37.630 --> 00:12:41.200 align:middle line:90% have x and p at the same time. 00:12:41.200 --> 00:12:51.670 align:middle line:84% So here, you also have potential operator ordering ambiguity. 00:12:51.670 --> 00:12:53.080 align:middle line:90% You have to pay attention. 00:12:53.080 --> 00:12:58.910 align:middle line:90% 00:12:58.910 --> 00:13:01.640 align:middle line:84% Often, such kind of ambiguities can be resolved 00:13:01.640 --> 00:13:03.810 align:middle line:90% by physical considerations. 00:13:03.810 --> 00:13:09.350 align:middle line:84% So we will see some examples in a little bit. 00:13:09.350 --> 00:13:12.353 align:middle line:90% 00:13:12.353 --> 00:13:13.770 align:middle line:84% So now, let's just imagine there's 00:13:13.770 --> 00:13:16.560 align:middle line:84% some ordering we have chosen, and now 00:13:16.560 --> 00:13:18.060 align:middle line:90% let's look at this commutator. 00:13:18.060 --> 00:13:25.940 align:middle line:90% 00:13:25.940 --> 00:13:28.740 align:middle line:84% So we can just write down the definition. 00:13:28.740 --> 00:13:34.240 align:middle line:84% So this is just integrating over all spatial direction 00:13:34.240 --> 00:13:40.690 align:middle line:84% epsilon alpha, and then we have pi a f a alpha phi a. 00:13:40.690 --> 00:13:43.300 align:middle line:90% 00:13:43.300 --> 00:13:45.320 align:middle line:90% Yeah, let me just-- 00:13:45.320 --> 00:13:49.850 align:middle line:84% sorry-- here, let me call phi b, because the phi a-- 00:13:49.850 --> 00:13:51.180 align:middle line:90% now, they are differ-- 00:13:51.180 --> 00:13:54.400 align:middle line:90% so these two indices are summed. 00:13:54.400 --> 00:13:58.910 align:middle line:84% So let me call them b, and then you have phi a here. 00:13:58.910 --> 00:14:04.640 align:middle line:84% So now, you can use commutation relation between the momentum 00:14:04.640 --> 00:14:08.750 align:middle line:90% density and the phi phi. 00:14:08.750 --> 00:14:12.770 align:middle line:84% So we discussed in general that the-- 00:14:12.770 --> 00:14:15.320 align:middle line:90% so we discussed the scalar case. 00:14:15.320 --> 00:14:20.060 align:middle line:84% But in general, you have this commutation relation of-- 00:14:20.060 --> 00:14:24.740 align:middle line:90% which is delta 3-- 00:14:24.740 --> 00:14:30.200 align:middle line:84% say if this is the x, and this is x prime, 00:14:30.200 --> 00:14:34.690 align:middle line:84% and then this is x minus x prime, and delta ab. 00:14:34.690 --> 00:14:40.420 align:middle line:84% So each one is only have a canonical conjugate momentum 00:14:40.420 --> 00:14:42.280 align:middle line:90% with its own momentum. 00:14:42.280 --> 00:14:44.800 align:middle line:84% If they have different fields, then of course yeah. 00:14:44.800 --> 00:14:46.940 align:middle line:90% So you have that. 00:14:46.940 --> 00:14:49.900 align:middle line:90% And now, if you use this-- 00:14:49.900 --> 00:14:52.130 align:middle line:90% so here is x. 00:14:52.130 --> 00:14:57.590 align:middle line:84% Yeah, let me call this prime and this to be x. 00:14:57.590 --> 00:15:02.660 align:middle line:84% Sorry, I should label more explicitly x here, 00:15:02.660 --> 00:15:05.225 align:middle line:90% and then here would be x. 00:15:05.225 --> 00:15:09.220 align:middle line:84% And so here is x, and here will be x prime. 00:15:09.220 --> 00:15:11.750 align:middle line:84% Sorry, I didn't leave myself enough space. 00:15:11.750 --> 00:15:13.030 align:middle line:90% Let me just rewrite it. 00:15:13.030 --> 00:15:16.840 align:middle line:90% 00:15:16.840 --> 00:15:23.830 align:middle line:84% So this gives me d3 x prime epsilon alpha. 00:15:23.830 --> 00:15:37.550 align:middle line:84% Then, I will have this pi a, then x prime f alpha a x prime, 00:15:37.550 --> 00:15:40.310 align:middle line:90% and then this commutator with-- 00:15:40.310 --> 00:15:43.060 align:middle line:90% yeah, not covariant. 00:15:43.060 --> 00:15:46.340 align:middle line:90% Sorry, alpha is upstairs. 00:15:46.340 --> 00:15:47.850 align:middle line:90% b and then phi x. 00:15:47.850 --> 00:15:51.910 align:middle line:90% 00:15:51.910 --> 00:15:56.860 align:middle line:84% And yeah, so we can evaluate this, 00:15:56.860 --> 00:15:58.930 align:middle line:84% say, for example, all at t equal to 0. 00:15:58.930 --> 00:16:00.070 align:middle line:90% So I suppress the t. 00:16:00.070 --> 00:16:02.800 align:middle line:90% 00:16:02.800 --> 00:16:06.760 align:middle line:84% And then yeah-- and now, you can just trivially use this 00:16:06.760 --> 00:16:12.930 align:middle line:84% commutation relation, and then this just gives you minus i 00:16:12.930 --> 00:16:20.270 align:middle line:84% epsilon alpha f alpha a fa alpha x-- 00:16:20.270 --> 00:16:21.350 align:middle line:90% evaluate at x. 00:16:21.350 --> 00:16:25.400 align:middle line:90% 00:16:25.400 --> 00:16:29.120 align:middle line:84% So that just confirms that expression. 00:16:29.120 --> 00:16:34.130 align:middle line:90% 00:16:34.130 --> 00:16:39.620 align:middle line:84% So this is not satisfied when you have time translation. 00:16:39.620 --> 00:16:42.410 align:middle line:90% So for time translation-- 00:16:42.410 --> 00:16:47.320 align:middle line:84% so first, any questions on this in this simple case? 00:16:47.320 --> 00:16:48.890 align:middle line:90% The idea is very simple. 00:16:48.890 --> 00:16:51.750 align:middle line:84% Essentially just the first term proportional to the momentum, 00:16:51.750 --> 00:16:54.585 align:middle line:90% and the momentum-- 00:16:54.585 --> 00:16:58.790 align:middle line:84% a conjugate to this, and then you just take that. 00:16:58.790 --> 00:16:59.360 align:middle line:90% Yes? 00:16:59.360 --> 00:17:01.415 align:middle line:84% AUDIENCE: Did we not get a term with the f 00:17:01.415 --> 00:17:03.849 align:middle line:84% alpha a and phi because of our restriction? 00:17:03.849 --> 00:17:04.849 align:middle line:90% PROFESSOR: That's right. 00:17:04.849 --> 00:17:08.270 align:middle line:84% Yeah, so if there's no partial zero-- 00:17:08.270 --> 00:17:09.800 align:middle line:84% if there's no time derivative here, 00:17:09.800 --> 00:17:11.990 align:middle line:84% it means there's no momentum here. 00:17:11.990 --> 00:17:13.579 align:middle line:84% If there's no momentum here, then this 00:17:13.579 --> 00:17:14.810 align:middle line:90% will commute with this one. 00:17:14.810 --> 00:17:16.935 align:middle line:84% Yeah, no matter if it's in here, it always commute. 00:17:16.935 --> 00:17:18.530 align:middle line:90% Yeah, a very good question. 00:17:18.530 --> 00:17:21.770 align:middle line:84% Yeah, I forgot to mention this point. 00:17:21.770 --> 00:17:23.142 align:middle line:90% Other the questions? 00:17:23.142 --> 00:17:28.560 align:middle line:90% 00:17:28.560 --> 00:17:29.300 align:middle line:90% OK, good. 00:17:29.300 --> 00:17:32.760 align:middle line:90% 00:17:32.760 --> 00:17:38.200 align:middle line:90% So now, the special case-- 00:17:38.200 --> 00:17:40.000 align:middle line:84% we said if you have time translation-- 00:17:40.000 --> 00:17:46.330 align:middle line:90% 00:17:46.330 --> 00:17:49.540 align:middle line:90% when you have time translation-- 00:17:49.540 --> 00:17:54.640 align:middle line:84% for example, one example is the-- 00:17:54.640 --> 00:18:00.250 align:middle line:84% yeah, the Hamiltonian density, H is 00:18:00.250 --> 00:18:08.080 align:middle line:84% equal to pi a partial T phi a minus L. So the L here-- 00:18:08.080 --> 00:18:12.910 align:middle line:84% for the time translation L is the analog of the k, 00:18:12.910 --> 00:18:16.940 align:middle line:84% and the partial T phi A is the F here, 00:18:16.940 --> 00:18:18.820 align:middle line:90% and then here is the pi a. 00:18:18.820 --> 00:18:21.415 align:middle line:84% So here you also have time derivative. 00:18:21.415 --> 00:18:25.420 align:middle line:90% 00:18:25.420 --> 00:18:28.330 align:middle line:84% And so this by itself is also a pi. 00:18:28.330 --> 00:18:33.200 align:middle line:84% And then L may also contain time derivative. 00:18:33.200 --> 00:18:37.130 align:middle line:90% So L is the k0 here. 00:18:37.130 --> 00:18:43.290 align:middle line:84% So in this case, the k0 is non-zero, which is equal to L. 00:18:43.290 --> 00:18:45.660 align:middle line:84% And so in this case, the conserved charge 00:18:45.660 --> 00:18:55.850 align:middle line:84% is just H. So in this case, you-- 00:18:55.850 --> 00:18:57.710 align:middle line:84% our general argument here don't apply 00:18:57.710 --> 00:18:59.930 align:middle line:84% because now, you have pi here also, 00:18:59.930 --> 00:19:02.480 align:middle line:84% and you may have a time derivative here 00:19:02.480 --> 00:19:05.980 align:middle line:84% also so the story is more complicated. 00:19:05.980 --> 00:19:07.540 align:middle line:84% But this case, we know trivially it's 00:19:07.540 --> 00:19:14.290 align:middle line:84% true because it's by definition H acting on any field. 00:19:14.290 --> 00:19:17.100 align:middle line:90% 00:19:17.100 --> 00:19:20.790 align:middle line:90% It's minus i partial t phi a. 00:19:20.790 --> 00:19:24.690 align:middle line:84% And this is exactly the time translation-- 00:19:24.690 --> 00:19:26.745 align:middle line:90% transformation of fields. 00:19:26.745 --> 00:19:29.450 align:middle line:90% 00:19:29.450 --> 00:19:34.280 align:middle line:84% So in this special case, this equation is also satisfied. 00:19:34.280 --> 00:19:39.540 align:middle line:90% 00:19:39.540 --> 00:19:40.980 align:middle line:90% Got any questions on this? 00:19:40.980 --> 00:19:44.550 align:middle line:90% 00:19:44.550 --> 00:19:47.777 align:middle line:84% So this is just follow from the Heisenberg equation itself. 00:19:47.777 --> 00:19:49.860 align:middle line:84% And now, you should just understand this equation. 00:19:49.860 --> 00:19:52.950 align:middle line:90% 00:19:52.950 --> 00:19:55.820 align:middle line:84% S-- so H now can be considered as a symmetry 00:19:55.820 --> 00:19:58.610 align:middle line:84% generator for time translation symmetry, 00:19:58.610 --> 00:20:03.830 align:middle line:84% and when we act on the field and generate the time translation. 00:20:03.830 --> 00:20:07.340 align:middle line:84% Time translation just infinitesimally just a time 00:20:07.340 --> 00:20:08.240 align:middle line:90% derivative on phi. 00:20:08.240 --> 00:20:13.620 align:middle line:90% 00:20:13.620 --> 00:20:15.700 align:middle line:90% Good? 00:20:15.700 --> 00:20:22.020 align:middle line:84% So these are infinitesimal transformations, 00:20:22.020 --> 00:20:27.090 align:middle line:84% but now starting from here, you can actually 00:20:27.090 --> 00:20:31.770 align:middle line:84% obtain finite transformations on the field also using Q. 00:20:31.770 --> 00:20:35.340 align:middle line:84% So you can generate finite transformations. 00:20:35.340 --> 00:20:42.490 align:middle line:90% 00:20:42.490 --> 00:20:46.620 align:middle line:84% So let's consider, say, U lambda alpha-- 00:20:46.620 --> 00:20:49.642 align:middle line:84% so lambda alpha are just some parameters. 00:20:49.642 --> 00:20:50.850 align:middle line:90% Let's consider this quantity. 00:20:50.850 --> 00:20:54.030 align:middle line:90% 00:20:54.030 --> 00:20:58.860 align:middle line:84% Now I exponentiate this Q. So Q is the operator, 00:20:58.860 --> 00:21:02.340 align:middle line:84% and lambda is just some numbers-- 00:21:02.340 --> 00:21:04.590 align:middle line:84% lambda is some constant parameters. 00:21:04.590 --> 00:21:06.180 align:middle line:90% And again, the alpha is summed. 00:21:06.180 --> 00:21:06.830 align:middle line:90% Yes? 00:21:06.830 --> 00:21:08.580 align:middle line:84% AUDIENCE: So I understand how this follows 00:21:08.580 --> 00:21:11.400 align:middle line:84% from the Heisenberg equation of motion, but here, 00:21:11.400 --> 00:21:13.440 align:middle line:84% how would you evaluate this commutator 00:21:13.440 --> 00:21:17.130 align:middle line:84% not knowing the form of L mu because you said it can depend 00:21:17.130 --> 00:21:20.400 align:middle line:90% on time derivatives as well. 00:21:20.400 --> 00:21:22.920 align:middle line:84% PROFESSOR: Yeah, here, just this general argument 00:21:22.920 --> 00:21:25.770 align:middle line:90% don't apply anymore. 00:21:25.770 --> 00:21:28.710 align:middle line:84% So in this case, you really have to-- 00:21:28.710 --> 00:21:31.260 align:middle line:84% you can check this explicitly-- you can just write down 00:21:31.260 --> 00:21:32.860 align:middle line:90% the explicit value-- 00:21:32.860 --> 00:21:38.340 align:middle line:84% explicit form of h, and then work it out explicitly. 00:21:38.340 --> 00:21:40.650 align:middle line:84% But you don't have to do it because we 00:21:40.650 --> 00:21:44.730 align:middle line:84% know this has to be true just by self-consistency 00:21:44.730 --> 00:21:48.708 align:middle line:84% because we solved the Heisenberg equa-- yeah, because when we 00:21:48.708 --> 00:21:51.000 align:middle line:84% quantize it, essentially, we are solving the Heisenberg 00:21:51.000 --> 00:21:51.500 align:middle line:90% equation. 00:21:51.500 --> 00:21:57.740 align:middle line:90% 00:21:57.740 --> 00:22:01.280 align:middle line:84% Yeah, but you can check it explicitly yourself 00:22:01.280 --> 00:22:02.150 align:middle line:90% that this is true. 00:22:02.150 --> 00:22:07.140 align:middle line:90% 00:22:07.140 --> 00:22:09.850 align:middle line:90% Good? 00:22:09.850 --> 00:22:15.390 align:middle line:84% So here-- so lambda alpha here just some collection 00:22:15.390 --> 00:22:23.390 align:middle line:84% of finite transformation parameters. 00:22:23.390 --> 00:22:37.360 align:middle line:90% 00:22:37.360 --> 00:22:42.740 align:middle line:84% So this can be used to generate the finite transformations. 00:22:42.740 --> 00:22:51.000 align:middle line:84% So if you act-- if u lambda alpha phi-- 00:22:51.000 --> 00:22:59.280 align:middle line:84% phi a and u lambda alpha dagger, and then 00:22:59.280 --> 00:23:00.750 align:middle line:90% you just get phi a prime. 00:23:00.750 --> 00:23:03.370 align:middle line:84% And now, this is a finite transformation-- 00:23:03.370 --> 00:23:06.920 align:middle line:84% a finite symmetry transformation of phi a. 00:23:06.920 --> 00:23:08.540 align:middle line:90% And when you expand-- 00:23:08.540 --> 00:23:11.600 align:middle line:84% with lambda, alpha is infinitesimal, 00:23:11.600 --> 00:23:14.000 align:middle line:84% and then you can just expand this exponential, 00:23:14.000 --> 00:23:17.810 align:middle line:84% then the leading term becomes this one-- 00:23:17.810 --> 00:23:21.020 align:middle line:84% the leading term becomes this one. 00:23:21.020 --> 00:23:26.890 align:middle line:84% Yeah, so when lambda alpha equal to epsilon alpha 00:23:26.890 --> 00:23:32.680 align:middle line:84% goes to 0, and this equation star, and then star, 00:23:32.680 --> 00:23:46.100 align:middle line:84% star reduces to star as the leading nontrivial order 00:23:46.100 --> 00:23:47.765 align:middle line:90% when you expand the exponential. 00:23:47.765 --> 00:23:53.270 align:middle line:90% 00:23:53.270 --> 00:24:00.950 align:middle line:84% So this can be done in general just-- 00:24:00.950 --> 00:24:03.800 align:middle line:90% so the reason this is true-- 00:24:03.800 --> 00:24:05.840 align:middle line:84% without even doing any calculation, 00:24:05.840 --> 00:24:08.840 align:middle line:84% you can just imagine you can build this U-- 00:24:08.840 --> 00:24:11.540 align:middle line:84% this finite transformation by infinite number 00:24:11.540 --> 00:24:14.250 align:middle line:84% of infinitesimal transformations. 00:24:14.250 --> 00:24:17.290 align:middle line:84% And the infinite-- so when you build them up, 00:24:17.290 --> 00:24:22.290 align:middle line:84% and then you will just generate the finite transformation. 00:24:22.290 --> 00:24:26.400 align:middle line:84% So you already worked out the example in your pset-- 00:24:26.400 --> 00:24:32.430 align:middle line:90% 00:24:32.430 --> 00:24:35.760 align:middle line:84% so you already worked out the example in your pset that-- 00:24:35.760 --> 00:24:51.870 align:middle line:84% if you can see the Ht minus I P i x i acting on phi 0 minus i H 00:24:51.870 --> 00:24:57.870 align:middle line:90% t plus i P i x i. 00:24:57.870 --> 00:25:01.680 align:middle line:84% And then this just gives you phi t x i-- 00:25:01.680 --> 00:25:06.110 align:middle line:90% t x-- just give you phi t x. 00:25:06.110 --> 00:25:17.180 align:middle line:84% And so this is-- so here, H t minus P i x i 00:25:17.180 --> 00:25:21.560 align:middle line:84% is our lambda alpha Q alpha here. 00:25:21.560 --> 00:25:26.960 align:middle line:84% So alpha here runs 0 in i direction. 00:25:26.960 --> 00:25:30.320 align:middle line:84% So in the 0 direction, the t now is just 00:25:30.320 --> 00:25:39.976 align:middle line:84% some parameter, and the lambda alpha equal to t and minus x i. 00:25:39.976 --> 00:25:43.370 align:middle line:90% And then Q alpha just-- 00:25:43.370 --> 00:25:47.840 align:middle line:84% and Q alpha here is equal to-- yeah, t x i. 00:25:47.840 --> 00:25:50.840 align:middle line:90% Let me just do t x i. 00:25:50.840 --> 00:25:55.460 align:middle line:84% And the Q alpha for here is just H minus P i. 00:25:55.460 --> 00:26:00.950 align:middle line:90% 00:26:00.950 --> 00:26:01.450 align:middle line:90% Yeah. 00:26:01.450 --> 00:26:12.300 align:middle line:90% 00:26:12.300 --> 00:26:12.840 align:middle line:90% Good? 00:26:12.840 --> 00:26:14.660 align:middle line:84% So this is just an example of that. 00:26:14.660 --> 00:26:18.020 align:middle line:90% This is just an example of that. 00:26:18.020 --> 00:26:20.790 align:middle line:84% And also you can consider Lorentz transformation. 00:26:20.790 --> 00:26:23.477 align:middle line:84% So another example is a Lorentz transformation. 00:26:23.477 --> 00:26:27.300 align:middle line:90% 00:26:27.300 --> 00:26:32.440 align:middle line:84% So let's consider-- so in your pset, you already-- 00:26:32.440 --> 00:26:36.570 align:middle line:84% so in the Lorentz transformation, 00:26:36.570 --> 00:26:39.550 align:middle line:90% the conserved charge is-- 00:26:39.550 --> 00:26:41.740 align:middle line:90% actually, there are six of them. 00:26:41.740 --> 00:26:50.310 align:middle line:90% M mu nu going from 0 to 3. 00:26:50.310 --> 00:26:52.333 align:middle line:84% So there are all together six conserved charges 00:26:52.333 --> 00:26:54.000 align:middle line:84% associated with Lorentz transformation-- 00:26:54.000 --> 00:26:57.600 align:middle line:84% three rotations and three boosts. 00:26:57.600 --> 00:27:04.290 align:middle line:84% And then so when you write the finite transformation, and then 00:27:04.290 --> 00:27:05.970 align:middle line:90% you will have six parameters. 00:27:05.970 --> 00:27:10.010 align:middle line:90% 00:27:10.010 --> 00:27:11.960 align:middle line:90% You have six parameters. 00:27:11.960 --> 00:27:15.470 align:middle line:90% And omega mu mu-- 00:27:15.470 --> 00:27:19.320 align:middle line:84% and because mu mu is antisymmetric, 00:27:19.320 --> 00:27:21.700 align:middle line:84% which you worked out in your pset, 00:27:21.700 --> 00:27:26.050 align:middle line:84% so this parameter should also be antisymmetric. 00:27:26.050 --> 00:27:32.065 align:middle line:90% And so there are finite angles. 00:27:32.065 --> 00:27:37.815 align:middle line:90% 00:27:37.815 --> 00:27:40.880 align:middle line:90% Oh, sorry, not finite angles-- 00:27:40.880 --> 00:27:46.810 align:middle line:90% finite angles and boosts. 00:27:46.810 --> 00:27:51.100 align:middle line:84% So essentially, they give the parameters 00:27:51.100 --> 00:27:54.550 align:middle line:84% for the finite rotation angles and the boost. 00:27:54.550 --> 00:27:58.390 align:middle line:84% And so that generates a Lorentzian transformation. 00:27:58.390 --> 00:28:02.380 align:middle line:84% That generates a Lorentz transformation. 00:28:02.380 --> 00:28:04.980 align:middle line:90% And so for example, you-- 00:28:04.980 --> 00:28:13.530 align:middle line:84% say if you act on u lambda on phi x, and u lambda-- 00:28:13.530 --> 00:28:20.900 align:middle line:84% you can check dagger on phi x-- what you get is phi prime x. 00:28:20.900 --> 00:28:27.050 align:middle line:84% And phi prime x is also the same as phi lambda 1 on x. 00:28:27.050 --> 00:28:33.130 align:middle line:84% So here, lambda-- so here, lambda 1 x is the Lorentz 00:28:33.130 --> 00:28:49.470 align:middle line:84% transformation of x mu with parameter determined by-- 00:28:49.470 --> 00:28:51.730 align:middle line:90% yeah, just omega mu nu. 00:28:51.730 --> 00:28:54.160 align:middle line:84% So this capital lambda is just the corresponding Lorentz 00:28:54.160 --> 00:28:55.335 align:middle line:90% transformation. 00:28:55.335 --> 00:28:58.070 align:middle line:90% 00:28:58.070 --> 00:28:59.970 align:middle line:90% Is this clear? 00:28:59.970 --> 00:29:01.250 align:middle line:90% Yes? 00:29:01.250 --> 00:29:05.540 align:middle line:84% AUDIENCE: So you assumed that you can represent, 00:29:05.540 --> 00:29:08.570 align:middle line:84% I guess, the operators that perform these transformations 00:29:08.570 --> 00:29:09.090 align:middle line:90% like this. 00:29:09.090 --> 00:29:11.590 align:middle line:84% But don't you have to think about 00:29:11.590 --> 00:29:14.090 align:middle line:84% if it follows in the algebra such 00:29:14.090 --> 00:29:16.700 align:middle line:84% that you can represent it like this, or like the group 00:29:16.700 --> 00:29:18.650 align:middle line:90% of transformations? 00:29:18.650 --> 00:29:19.730 align:middle line:90% PROFESSOR: Yeah, indeed. 00:29:19.730 --> 00:29:22.490 align:middle line:90% So here, it's not-- 00:29:22.490 --> 00:29:23.330 align:middle line:90% here, it's not said. 00:29:23.330 --> 00:29:24.230 align:middle line:90% I assume. 00:29:24.230 --> 00:29:27.840 align:middle line:84% Here, I'm just directly telling you the result. 00:29:27.840 --> 00:29:29.180 align:middle line:90% Just when you build-- 00:29:29.180 --> 00:29:36.760 align:middle line:84% yeah, so if you want to generate the-- 00:29:36.760 --> 00:29:38.510 align:middle line:84% yeah, I'm just directly-- already directly 00:29:38.510 --> 00:29:39.507 align:middle line:90% telling you the answer. 00:29:39.507 --> 00:29:41.090 align:middle line:84% AUDIENCE: So it's the composition of-- 00:29:41.090 --> 00:29:42.320 align:middle line:90% infinitesmal transformation? 00:29:42.320 --> 00:29:43.112 align:middle line:90% PROFESSOR: Exactly. 00:29:43.112 --> 00:29:46.790 align:middle line:84% If you do the composition of the infinitesimal transformation, 00:29:46.790 --> 00:29:48.530 align:middle line:90% you will get that. 00:29:48.530 --> 00:29:51.527 align:middle line:84% Yeah, the reason I write down this explicitly-- 00:29:51.527 --> 00:29:53.360 align:middle line:84% immediately, because you should have already 00:29:53.360 --> 00:29:55.430 align:middle line:84% seen this in quantum mechanics because that's 00:29:55.430 --> 00:29:57.325 align:middle line:90% how you do the angular momentum. 00:29:57.325 --> 00:29:58.700 align:middle line:84% In quantum mechanics, when you do 00:29:58.700 --> 00:30:00.980 align:middle line:90% the angular momentum, the way-- 00:30:00.980 --> 00:30:02.030 align:middle line:90% it's the same thing. 00:30:02.030 --> 00:30:08.660 align:middle line:84% Yeah, here, we just generalize it to general symmetries. 00:30:08.660 --> 00:30:10.130 align:middle line:90% Yes? 00:30:10.130 --> 00:30:14.070 align:middle line:84% AUDIENCE: So it was some similar question, but just like-- 00:30:14.070 --> 00:30:15.640 align:middle line:84% so this is like a general procedure 00:30:15.640 --> 00:30:17.640 align:middle line:84% if you have a symmetry to get the representation 00:30:17.640 --> 00:30:20.700 align:middle line:84% on your Hilbert space, just exponentiate it like this? 00:30:20.700 --> 00:30:23.490 align:middle line:84% But like in QM, we usually just postulate 00:30:23.490 --> 00:30:25.140 align:middle line:84% that there is a unitary operator, 00:30:25.140 --> 00:30:27.900 align:middle line:84% and then demand that the group law holds. 00:30:27.900 --> 00:30:30.740 align:middle line:84% So don't we need to show that those things gives-- 00:30:30.740 --> 00:30:33.960 align:middle line:84% that this preserves the group law, or is that not-- 00:30:33.960 --> 00:30:35.850 align:middle line:84% PROFESSOR: Yeah, so this is guaranteed. 00:30:35.850 --> 00:30:41.220 align:middle line:84% So yeah, I didn't go into that, but this is guaranteed. 00:30:41.220 --> 00:30:43.350 align:middle line:84% Essentially, that's how you-- yeah, 00:30:43.350 --> 00:30:45.570 align:middle line:84% this is how you build the algebra, 00:30:45.570 --> 00:30:47.520 align:middle line:90% because the algebra is also-- 00:30:47.520 --> 00:30:50.490 align:middle line:84% the finite transformation is also 00:30:50.490 --> 00:30:53.520 align:middle line:84% built up by infinitesimal transformations. 00:30:53.520 --> 00:30:55.950 align:middle line:90% And so essentially guaranteed-- 00:30:55.950 --> 00:30:59.280 align:middle line:84% yeah, so there is a theory of groups 00:30:59.280 --> 00:31:01.340 align:middle line:90% and the algebra behind this. 00:31:01.340 --> 00:31:03.920 align:middle line:90% Yes, I didn't go into that. 00:31:03.920 --> 00:31:05.810 align:middle line:84% So some of you may already seen a little bit 00:31:05.810 --> 00:31:08.640 align:middle line:84% of this when you talk about angular momentum. 00:31:08.640 --> 00:31:11.870 align:middle line:84% So in quantum mechanics, so it's exactly the same story there. 00:31:11.870 --> 00:31:13.010 align:middle line:90% Yes? 00:31:13.010 --> 00:31:15.410 align:middle line:84% AUDIENCE: So I know that rotations correspond 00:31:15.410 --> 00:31:17.960 align:middle line:84% to angular momentum being conserved, 00:31:17.960 --> 00:31:20.270 align:middle line:84% but what is the intuition for the thing that's 00:31:20.270 --> 00:31:22.580 align:middle line:90% preserved in boosts? 00:31:22.580 --> 00:31:25.250 align:middle line:90% PROFESSOR: Yeah, it's a-- 00:31:25.250 --> 00:31:29.480 align:middle line:84% yeah, indeed, there's no very intuitive way 00:31:29.480 --> 00:31:30.230 align:middle line:90% to think about it. 00:31:30.230 --> 00:31:32.750 align:middle line:90% Also it's not that useful. 00:31:32.750 --> 00:31:34.640 align:middle line:90% You can define a boost charge. 00:31:34.640 --> 00:31:38.910 align:middle line:84% So if you consider all the spatial components, 00:31:38.910 --> 00:31:41.810 align:middle line:84% and if you consider all the edge here, 00:31:41.810 --> 00:31:43.980 align:middle line:84% then these are the angular momentum operator, 00:31:43.980 --> 00:31:46.130 align:middle line:84% and this is your standard angular momentum 00:31:46.130 --> 00:31:47.510 align:middle line:90% operator, which you have seen. 00:31:47.510 --> 00:31:50.540 align:middle line:84% And if you look at 0 i, and then that's what 00:31:50.540 --> 00:31:52.430 align:middle line:90% sometimes we call boost charge. 00:31:52.430 --> 00:31:54.350 align:middle line:90% It generates the boost. 00:31:54.350 --> 00:31:59.300 align:middle line:84% But in terms of the conserved number, 00:31:59.300 --> 00:32:04.377 align:middle line:84% we don't use it in the essential way-- say, in other contexts. 00:32:04.377 --> 00:32:05.210 align:middle line:90% AUDIENCE: Thank you. 00:32:05.210 --> 00:32:05.870 align:middle line:90% PROFESSOR: Yes? 00:32:05.870 --> 00:32:07.328 align:middle line:84% AUDIENCE: But to that question, you 00:32:07.328 --> 00:32:09.500 align:middle line:84% can think of it like a center-of-mass velocity 00:32:09.500 --> 00:32:11.060 align:middle line:90% [INAUDIBLE]. 00:32:11.060 --> 00:32:12.440 align:middle line:84% PROFESSOR: Yeah, it just does not 00:32:12.440 --> 00:32:15.620 align:middle line:90% give you something additional. 00:32:15.620 --> 00:32:18.928 align:middle line:84% Just normally, when you deal with physical problems-- say 00:32:18.928 --> 00:32:20.720 align:middle line:84% angular momentum, conservation of momentum, 00:32:20.720 --> 00:32:21.800 align:middle line:84% conservation of energy, conservation-- 00:32:21.800 --> 00:32:22.633 align:middle line:90% it's already enough. 00:32:22.633 --> 00:32:25.412 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:32:25.412 --> 00:32:26.870 align:middle line:84% PROFESSOR: Yeah, you normally don't 00:32:26.870 --> 00:32:29.180 align:middle line:84% need to use a boost charge conservation 00:32:29.180 --> 00:32:30.590 align:middle line:90% to solve your problem. 00:32:30.590 --> 00:32:34.130 align:middle line:84% Yeah, so that's why we don't see this object very often, say 00:32:34.130 --> 00:32:39.220 align:middle line:84% when you discuss special relativity or other things. 00:32:39.220 --> 00:32:43.820 align:middle line:84% But here, it plays a very important role 00:32:43.820 --> 00:32:45.770 align:middle line:84% that in quantum field theory, this 00:32:45.770 --> 00:32:50.990 align:middle line:84% is the operator which generates the boost symmetry. 00:32:50.990 --> 00:32:56.270 align:middle line:84% If you want to do a boost in your Hilbert space, 00:32:56.270 --> 00:32:57.930 align:middle line:90% that's the operator to do it. 00:32:57.930 --> 00:33:01.060 align:middle line:90% 00:33:01.060 --> 00:33:02.050 align:middle line:90% Other questions? 00:33:02.050 --> 00:33:10.180 align:middle line:90% 00:33:10.180 --> 00:33:11.670 align:middle line:90% Good. 00:33:11.670 --> 00:33:13.320 align:middle line:90% Yeah, so in your pset-- 00:33:13.320 --> 00:33:15.960 align:middle line:90% 00:33:15.960 --> 00:33:18.580 align:middle line:84% yeah, another example is the phase rotation. 00:33:18.580 --> 00:33:20.770 align:middle line:84% Let me just very quickly write here. 00:33:20.770 --> 00:33:25.362 align:middle line:84% And so another example is the phase rotation 00:33:25.362 --> 00:33:26.820 align:middle line:84% if you have a complex scalar field. 00:33:26.820 --> 00:33:48.962 align:middle line:90% 00:33:48.962 --> 00:33:51.295 align:middle line:84% So there, we show there's a Noether charge corresponding 00:33:51.295 --> 00:33:54.970 align:middle line:84% to the phase rotation, there's a Noether current corresponding 00:33:54.970 --> 00:33:57.520 align:middle line:84% to rotating the phase to the field. 00:33:57.520 --> 00:33:58.720 align:middle line:90% That's a symmetry. 00:33:58.720 --> 00:34:05.302 align:middle line:84% And so that will generate the charge Q corresponding 00:34:05.302 --> 00:34:06.010 align:middle line:90% to the first two. 00:34:06.010 --> 00:34:08.699 align:middle line:90% And then if you have alpha Q-- 00:34:08.699 --> 00:34:13.110 align:middle line:84% and then when you act on phi, and then you 00:34:13.110 --> 00:34:15.389 align:middle line:84% can just check explicitly that will generate the phase 00:34:15.389 --> 00:34:16.469 align:middle line:90% rotation in phi. 00:34:16.469 --> 00:34:22.580 align:middle line:90% 00:34:22.580 --> 00:34:24.710 align:middle line:84% And we're generating phase rotation phi. 00:34:24.710 --> 00:34:39.960 align:middle line:90% 00:34:39.960 --> 00:34:40.994 align:middle line:90% Any questions on this? 00:34:40.994 --> 00:34:46.504 align:middle line:90% 00:34:46.504 --> 00:34:47.004 align:middle line:90% Good. 00:34:47.004 --> 00:34:56.820 align:middle line:90% 00:34:56.820 --> 00:35:06.025 align:middle line:84% Yeah, so let's conclude our discussion of the real scalar 00:35:06.025 --> 00:35:06.525 align:middle line:90% field. 00:35:06.525 --> 00:35:09.740 align:middle line:90% 00:35:09.740 --> 00:35:12.005 align:middle line:84% And now, we can move to a new part. 00:35:12.005 --> 00:35:15.180 align:middle line:90% 00:35:15.180 --> 00:35:18.650 align:middle line:84% So before we move to new part, your last chance 00:35:18.650 --> 00:35:20.060 align:middle line:84% to ask questions about this part. 00:35:20.060 --> 00:35:25.850 align:middle line:90% 00:35:25.850 --> 00:35:28.940 align:middle line:90% OK, great. 00:35:28.940 --> 00:35:33.395 align:middle line:84% So now, let's very quickly talk about-- 00:35:33.395 --> 00:35:51.500 align:middle line:90% 00:35:51.500 --> 00:35:54.365 align:middle line:84% so let's very quickly talk about complex scalar field. 00:35:54.365 --> 00:36:11.200 align:middle line:90% 00:36:11.200 --> 00:36:15.160 align:middle line:84% And you can see the complex scalar field, 00:36:15.160 --> 00:36:16.090 align:middle line:90% we get something new. 00:36:16.090 --> 00:36:19.207 align:middle line:84% We get the concept of antiparticle. 00:36:19.207 --> 00:36:21.540 align:middle line:84% So now, let's consider the following Lagrangian density. 00:36:21.540 --> 00:36:25.340 align:middle line:90% 00:36:25.340 --> 00:36:28.680 align:middle line:84% Again, we only consider the quadratic-- 00:36:28.680 --> 00:36:34.736 align:middle line:84% the simplest-- only quadratic Lagrangian, but now the phi 00:36:34.736 --> 00:36:35.236 align:middle line:90% is complex. 00:36:35.236 --> 00:36:44.336 align:middle line:90% 00:36:44.336 --> 00:36:45.470 align:middle line:90% Phi is complex. 00:36:45.470 --> 00:36:50.840 align:middle line:90% 00:36:50.840 --> 00:36:55.205 align:middle line:84% So complex is also the same as 2 real. 00:36:55.205 --> 00:36:59.350 align:middle line:90% 00:36:59.350 --> 00:37:04.880 align:middle line:84% So you can decompose phi into its real part 00:37:04.880 --> 00:37:06.930 align:middle line:90% and imaginary part. 00:37:06.930 --> 00:37:10.550 align:middle line:84% And if you plug them in here, and then 00:37:10.550 --> 00:37:13.220 align:middle line:84% you find you just get two separate pieces, 00:37:13.220 --> 00:37:16.880 align:middle line:84% and one is the real scalar field for the real part 00:37:16.880 --> 00:37:21.410 align:middle line:84% and the one is the free scalar field for the imaginary part, 00:37:21.410 --> 00:37:24.612 align:middle line:84% because each one-- yeah, if you write phi equal to a plus ib, 00:37:24.612 --> 00:37:27.050 align:middle line:84% and here, it just become a squared plus b squared, 00:37:27.050 --> 00:37:30.740 align:middle line:84% here, it's also a plus b squared. 00:37:30.740 --> 00:37:33.090 align:middle line:90% But there's a reason we write-- 00:37:33.090 --> 00:37:35.420 align:middle line:84% yeah, so essentially, we just get two copies 00:37:35.420 --> 00:37:37.880 align:middle line:90% of our previous theory. 00:37:37.880 --> 00:37:40.310 align:middle line:84% We just get two copies of previous theory. 00:37:40.310 --> 00:37:41.813 align:middle line:84% So you say, oh, then it's trivial. 00:37:41.813 --> 00:37:43.730 align:middle line:84% Why should we just consider this-- we just get 00:37:43.730 --> 00:37:46.010 align:middle line:90% two copies of previous theory? 00:37:46.010 --> 00:37:50.535 align:middle line:84% But this advantage-- but there's some concept-- 00:37:50.535 --> 00:37:55.145 align:middle line:84% there's something conceptually new when we write in this form. 00:37:55.145 --> 00:37:56.770 align:middle line:84% there's advantage to write in this form 00:37:56.770 --> 00:38:01.180 align:middle line:84% rather than write as two separate identical scalar 00:38:01.180 --> 00:38:02.680 align:middle line:90% field theory. 00:38:02.680 --> 00:38:04.360 align:middle line:90% It's precisely in this form-- 00:38:04.360 --> 00:38:08.270 align:middle line:90% 00:38:08.270 --> 00:38:12.500 align:middle line:90% this phase symmetry is manifest. 00:38:12.500 --> 00:38:15.530 align:middle line:84% If I just write it as two separate real scalar 00:38:15.530 --> 00:38:19.940 align:middle line:84% field, then this phase symmetry is not manifest. 00:38:19.940 --> 00:38:22.280 align:middle line:84% And so here, in this form, there's 00:38:22.280 --> 00:38:24.140 align:middle line:90% a phase symmetry manifest. 00:38:24.140 --> 00:38:28.910 align:middle line:84% So phi goes to exponential i alpha phi. 00:38:28.910 --> 00:38:33.050 align:middle line:84% And so you see this-- you immediately see that symmetry. 00:38:33.050 --> 00:38:35.000 align:middle line:84% So this-- when you write it in terms 00:38:35.000 --> 00:38:38.000 align:middle line:84% of real and imaginary part, this corresponding 00:38:38.000 --> 00:38:40.970 align:middle line:84% to a rotation between the real and imaginary part. 00:38:40.970 --> 00:38:47.660 align:middle line:90% And yeah-- good? 00:38:47.660 --> 00:38:50.270 align:middle line:84% And the equation of motion is the same 00:38:50.270 --> 00:38:51.590 align:middle line:90% because it's just the-- 00:38:51.590 --> 00:38:54.770 align:middle line:90% 00:38:54.770 --> 00:38:57.150 align:middle line:84% because you can easily find the equation of motion. 00:38:57.150 --> 00:38:57.740 align:middle line:90% It's the same. 00:38:57.740 --> 00:39:00.350 align:middle line:90% 00:39:00.350 --> 00:39:06.760 align:middle line:84% And yeah, because just two copies of your previous theory. 00:39:06.760 --> 00:39:09.340 align:middle line:84% Of course, the equation of motion should be the same. 00:39:09.340 --> 00:39:12.480 align:middle line:90% And the Lagrangian density-- 00:39:12.480 --> 00:39:17.110 align:middle line:84% the momentum density for phi then 00:39:17.110 --> 00:39:24.930 align:middle line:84% becomes partial 0 phi star, and the partial phi star 00:39:24.930 --> 00:39:27.040 align:middle line:90% becomes partial 0 phi. 00:39:27.040 --> 00:39:30.250 align:middle line:84% So we treat-- we can treat phi and the phi star 00:39:30.250 --> 00:39:32.680 align:middle line:90% as two independent fields. 00:39:32.680 --> 00:39:35.530 align:middle line:84% And then the canonical momentum-- 00:39:35.530 --> 00:39:38.470 align:middle line:84% say if you take derivative with partial 0 phi, 00:39:38.470 --> 00:39:42.040 align:middle line:84% then you get the partial 0 phi star and vice versa. 00:39:42.040 --> 00:39:45.850 align:middle line:84% So the complex conjugate momentum like that. 00:39:45.850 --> 00:39:50.580 align:middle line:90% 00:39:50.580 --> 00:39:52.170 align:middle line:84% So we can now write down-- again, 00:39:52.170 --> 00:39:56.870 align:middle line:84% we can just write down the most general solutions. 00:39:56.870 --> 00:39:59.160 align:middle line:84% So the basis of solutions is the same. 00:39:59.160 --> 00:40:02.100 align:middle line:90% 00:40:02.100 --> 00:40:10.950 align:middle line:90% Same set of solutions as before. 00:40:10.950 --> 00:40:27.640 align:middle line:84% We just have u k x equal to minus i omega k t plus i k x, 00:40:27.640 --> 00:40:33.380 align:middle line:90% and then you have u k x star. 00:40:33.380 --> 00:40:37.257 align:middle line:84% Because it's the same equation, so the basis of the equation, 00:40:37.257 --> 00:40:38.840 align:middle line:84% of course, is also the same as before. 00:40:38.840 --> 00:40:39.760 align:middle line:90% Yes? 00:40:39.760 --> 00:40:43.900 align:middle line:84% AUDIENCE: So if the field is an operator, doesn't it-- 00:40:43.900 --> 00:40:48.010 align:middle line:84% if it's an observable, doesn't it have to be Hermitian? 00:40:48.010 --> 00:40:50.110 align:middle line:90% PROFESSOR: No, the fields-- 00:40:50.110 --> 00:40:51.550 align:middle line:90% it depends. 00:40:51.550 --> 00:40:58.060 align:middle line:84% So yeah, indeed if it's something directly observable, 00:40:58.060 --> 00:41:02.530 align:middle line:84% it has to be Hermitian, but that can have two observables. 00:41:02.530 --> 00:41:05.860 align:middle line:84% For example, I can combine them into a real variable. 00:41:05.860 --> 00:41:07.510 align:middle line:84% Yeah, just not necessarily the phi 00:41:07.510 --> 00:41:09.340 align:middle line:84% itself have to be directly observable. 00:41:09.340 --> 00:41:10.105 align:middle line:90% It's just a field. 00:41:10.105 --> 00:41:13.340 align:middle line:90% 00:41:13.340 --> 00:41:15.170 align:middle line:84% Yeah, the field itself don't have to always 00:41:15.170 --> 00:41:16.700 align:middle line:90% be observable by itself. 00:41:16.700 --> 00:41:19.243 align:middle line:84% AUDIENCE: So then the conjugate momentum also is not-- 00:41:19.243 --> 00:41:19.910 align:middle line:90% PROFESSOR: Yeah. 00:41:19.910 --> 00:41:21.890 align:middle line:84% AUDIENCE: The momentum is not our three momentum? 00:41:21.890 --> 00:41:23.640 align:middle line:84% PROFESSOR: Yeah, this is not the momentum. 00:41:23.640 --> 00:41:25.220 align:middle line:84% This is just the canonical momentum 00:41:25.220 --> 00:41:29.490 align:middle line:90% for the canonical quantization. 00:41:29.490 --> 00:41:31.680 align:middle line:84% Yeah, this is not the spacetime momentum. 00:41:31.680 --> 00:41:34.080 align:middle line:84% AUDIENCE: Then is the adjoint of the field operator 00:41:34.080 --> 00:41:36.450 align:middle line:90% the same thing as the star? 00:41:36.450 --> 00:41:39.150 align:middle line:84% PROFESSOR: Yeah, here, we are talking about the-- 00:41:39.150 --> 00:41:41.370 align:middle line:84% here, we are talking about the classical theory. 00:41:41.370 --> 00:41:43.310 align:middle line:84% And then when we go to the quantum, 00:41:43.310 --> 00:41:46.315 align:middle line:84% indeed, we will replace the star by dagger. 00:41:46.315 --> 00:41:52.610 align:middle line:90% 00:41:52.610 --> 00:41:55.370 align:middle line:84% So now, we can just write down the most general solutions 00:41:55.370 --> 00:41:56.165 align:middle line:90% to this equation. 00:41:56.165 --> 00:41:58.108 align:middle line:90% 00:41:58.108 --> 00:42:00.650 align:middle line:84% Now, I think we can just write down the most general solution 00:42:00.650 --> 00:42:02.570 align:middle line:90% to this equation. 00:42:02.570 --> 00:42:08.730 align:middle line:84% So we just-- so there's one slight difference 00:42:08.730 --> 00:42:12.700 align:middle line:84% from before-- almost the same-- identical as before 00:42:12.700 --> 00:42:14.700 align:middle line:90% except with one difference. 00:42:14.700 --> 00:42:26.220 align:middle line:90% 00:42:26.220 --> 00:42:31.530 align:middle line:84% u k-- but here, previously, we have a k dagger. 00:42:31.530 --> 00:42:33.300 align:middle line:84% We have a k star, say, classically. 00:42:33.300 --> 00:42:35.070 align:middle line:84% Let me just first write classically. 00:42:35.070 --> 00:42:38.220 align:middle line:90% So previously, we have this. 00:42:38.220 --> 00:42:40.500 align:middle line:84% Previously, why we have a k star here 00:42:40.500 --> 00:42:41.955 align:middle line:84% when we have a real scalar field? 00:42:41.955 --> 00:42:47.800 align:middle line:90% 00:42:47.800 --> 00:42:50.060 align:middle line:84% It's because the field has to be real, 00:42:50.060 --> 00:42:53.290 align:middle line:84% and then we have always add to its complex conjugate. 00:42:53.290 --> 00:42:57.670 align:middle line:84% But now, the difference is that now phi is a complex, 00:42:57.670 --> 00:42:59.120 align:middle line:90% and the phi is complex. 00:42:59.120 --> 00:43:01.990 align:middle line:84% And then we no longer have the real condition. 00:43:01.990 --> 00:43:06.460 align:middle line:84% That means here, I can choose another arbitrary constant 00:43:06.460 --> 00:43:07.810 align:middle line:90% here. 00:43:07.810 --> 00:43:09.590 align:middle line:84% So this is the arbitrary constant. 00:43:09.590 --> 00:43:12.160 align:middle line:84% So by convention, let me just put a star here. 00:43:12.160 --> 00:43:12.980 align:middle line:90% It doesn't matter. 00:43:12.980 --> 00:43:15.460 align:middle line:90% It's just the name you call it. 00:43:15.460 --> 00:43:19.090 align:middle line:84% So now, the most general solution-- 00:43:19.090 --> 00:43:20.460 align:middle line:90% now, you have two sets. 00:43:20.460 --> 00:43:22.650 align:middle line:84% You have a, which is complex number. 00:43:22.650 --> 00:43:25.530 align:middle line:84% Then you have another complex number, which is b. 00:43:25.530 --> 00:43:29.000 align:middle line:84% Now, you have two sets of complex numbers. 00:43:29.000 --> 00:43:33.230 align:middle line:84% And it makes sense because when you have a complex variable, 00:43:33.230 --> 00:43:34.790 align:middle line:84% you double the degrees of freedom, 00:43:34.790 --> 00:43:36.740 align:middle line:84% and your integration constant also doubled. 00:43:36.740 --> 00:43:39.750 align:middle line:90% 00:43:39.750 --> 00:43:45.070 align:middle line:90% So b k independent of a. 00:43:45.070 --> 00:43:49.910 align:middle line:90% 00:43:49.910 --> 00:43:54.140 align:middle line:84% So now, you have two sets of variables. 00:43:54.140 --> 00:43:56.840 align:middle line:84% So you can get the phi star just by taking 00:43:56.840 --> 00:43:59.630 align:middle line:90% the complex conjugate of it-- 00:43:59.630 --> 00:44:02.510 align:middle line:84% just by taking the complex conjugate of it. 00:44:02.510 --> 00:44:19.480 align:middle line:84% And then the full set of integration constant just-- 00:44:19.480 --> 00:44:21.520 align:middle line:90% you have a k-- 00:44:21.520 --> 00:44:29.340 align:middle line:84% you have a k star, you have b k, and you have b k star. 00:44:29.340 --> 00:44:30.645 align:middle line:90% So this is the classical story. 00:44:30.645 --> 00:44:34.010 align:middle line:90% 00:44:34.010 --> 00:44:40.550 align:middle line:84% So now, for quantum, we just do exactly as we did before. 00:44:40.550 --> 00:44:44.570 align:middle line:84% You just promoted all those into operating equations. 00:44:44.570 --> 00:44:49.160 align:middle line:84% So the-- so you just call this dagger. 00:44:49.160 --> 00:44:52.970 align:middle line:84% Just call this dag-- just view every one of them as operators. 00:44:52.970 --> 00:44:56.140 align:middle line:84% And now, this just becomes constant operators. 00:44:56.140 --> 00:45:00.500 align:middle line:90% 00:45:00.500 --> 00:45:03.920 align:middle line:84% And now, here, you also just call them to be dagger. 00:45:03.920 --> 00:45:15.620 align:middle line:90% 00:45:15.620 --> 00:45:17.475 align:middle line:90% And the-- yeah. 00:45:17.475 --> 00:45:24.990 align:middle line:90% 00:45:24.990 --> 00:45:28.050 align:middle line:90% So now, equal time commutation-- 00:45:28.050 --> 00:45:35.210 align:middle line:84% still, we have to impose the equal time commutation 00:45:35.210 --> 00:45:35.710 align:middle line:90% relation. 00:45:35.710 --> 00:46:06.100 align:middle line:90% 00:46:06.100 --> 00:46:07.960 align:middle line:84% So now, we have to impose the equal time 00:46:07.960 --> 00:46:13.010 align:middle line:90% commutation relations again. 00:46:13.010 --> 00:46:15.350 align:middle line:90% So this all should be simple. 00:46:15.350 --> 00:46:18.770 align:middle line:84% So we should have phi, phi dagger. 00:46:18.770 --> 00:46:21.285 align:middle line:84% They can be-- they're considered as independent operators, 00:46:21.285 --> 00:46:22.910 align:middle line:84% so they should commute with each other. 00:46:22.910 --> 00:46:26.150 align:middle line:84% They're all field variables, and the phi actually 00:46:26.150 --> 00:46:28.260 align:middle line:90% commutes with itself. 00:46:28.260 --> 00:46:31.100 align:middle line:84% So this is all evaluated at different spatial-- yeah, 00:46:31.100 --> 00:46:34.730 align:middle line:84% just to save time, I will not write this expression, 00:46:34.730 --> 00:46:38.330 align:middle line:84% but all this should be evaluated at the same time 00:46:38.330 --> 00:46:56.830 align:middle line:84% by different spatial locations such as all these equal to 0. 00:46:56.830 --> 00:46:59.890 align:middle line:84% Except the only thing which is non-zero 00:46:59.890 --> 00:47:03.445 align:middle line:84% is the phi with its own canonical momentum. 00:47:03.445 --> 00:47:07.440 align:middle line:90% 00:47:07.440 --> 00:47:08.880 align:middle line:90% So everything else will be 0. 00:47:08.880 --> 00:47:14.835 align:middle line:84% The only thing non-zero is phi, which is canonical momentum. 00:47:14.835 --> 00:47:18.510 align:middle line:90% 00:47:18.510 --> 00:47:25.230 align:middle line:84% Again, it should be given by i delta 3 x minus x prime. 00:47:25.230 --> 00:47:27.840 align:middle line:84% And then the similar thing for the phi dagger. 00:47:27.840 --> 00:47:34.420 align:middle line:90% 00:47:34.420 --> 00:47:36.910 align:middle line:90% So again, I save effort. 00:47:36.910 --> 00:47:40.305 align:middle line:90% 00:47:40.305 --> 00:47:41.680 align:middle line:84% So that's what you should impose. 00:47:41.680 --> 00:47:44.480 align:middle line:90% 00:47:44.480 --> 00:47:47.420 align:middle line:84% So now, if you plug those things in, 00:47:47.420 --> 00:47:51.080 align:middle line:84% then you can find the commutation relation 00:47:51.080 --> 00:47:52.625 align:middle line:90% between a and a dagger. 00:47:52.625 --> 00:47:57.140 align:middle line:90% 00:47:57.140 --> 00:48:09.770 align:middle line:84% Again, you find-- and similarly, with b, 00:48:09.770 --> 00:48:12.500 align:middle line:90% you just get two sides of-- 00:48:12.500 --> 00:48:15.920 align:middle line:84% two infinite family of harmonic oscillators. 00:48:15.920 --> 00:48:20.750 align:middle line:84% Previously, we have one family, and now you get two families, 00:48:20.750 --> 00:48:22.985 align:middle line:84% and all other commutators vanish. 00:48:22.985 --> 00:48:28.270 align:middle line:90% 00:48:28.270 --> 00:48:29.530 align:middle line:90% So this part is boring. 00:48:29.530 --> 00:48:34.930 align:middle line:84% You can almost guess everything without doing any calculations. 00:48:34.930 --> 00:48:40.053 align:middle line:90% 00:48:40.053 --> 00:48:41.470 align:middle line:84% You essentially guessed everything 00:48:41.470 --> 00:48:43.168 align:middle line:90% without doing any calculation. 00:48:43.168 --> 00:48:46.920 align:middle line:90% 00:48:46.920 --> 00:48:48.950 align:middle line:84% So now, again, you can write the Hamiltonian 00:48:48.950 --> 00:48:51.080 align:middle line:90% in terms of a k and b k. 00:48:51.080 --> 00:48:55.880 align:middle line:84% Just will be as you would guess them to be. 00:48:55.880 --> 00:48:57.830 align:middle line:84% Just the harmonic oscillator corresponding 00:48:57.830 --> 00:49:00.110 align:middle line:84% to a k, harmonic oscillator corresponding to b k. 00:49:00.110 --> 00:49:03.800 align:middle line:90% 00:49:03.800 --> 00:49:08.170 align:middle line:84% And then the ground state-- again, 00:49:08.170 --> 00:49:12.250 align:middle line:84% it's the state annihilated by both of them. 00:49:12.250 --> 00:49:15.700 align:middle line:90% 00:49:15.700 --> 00:49:16.975 align:middle line:90% So this is the vacuum state. 00:49:16.975 --> 00:49:20.420 align:middle line:90% 00:49:20.420 --> 00:49:25.780 align:middle line:84% And now, you can act a k dagger on 0 00:49:25.780 --> 00:49:30.090 align:middle line:90% and b k dagger on 0, et cetera. 00:49:30.090 --> 00:49:33.290 align:middle line:90% 00:49:33.290 --> 00:49:35.540 align:middle line:84% So now, we have two kinds of particles. 00:49:35.540 --> 00:50:11.850 align:middle line:90% 00:50:11.850 --> 00:50:34.560 align:middle line:84% So now, we have two kinds of particles generated by a k 00:50:34.560 --> 00:50:40.230 align:middle line:90% dagger and b k dagger now. 00:50:40.230 --> 00:50:47.700 align:middle line:84% So now you have two kinds of particles respectively, 00:50:47.700 --> 00:50:53.310 align:middle line:84% and both of them have the on-shell condition. 00:50:53.310 --> 00:51:00.030 align:middle line:84% Both particles have the same mass. 00:51:00.030 --> 00:51:06.450 align:middle line:84% They satisfy the P squared equal to m squared -- 00:51:06.450 --> 00:51:08.130 align:middle line:90% minus m squared. 00:51:08.130 --> 00:51:10.380 align:middle line:84% So because they all come from the same equation. 00:51:10.380 --> 00:51:17.870 align:middle line:90% 00:51:17.870 --> 00:51:21.470 align:middle line:84% So now, the question is, how do we tell them apart? 00:51:21.470 --> 00:51:24.830 align:middle line:84% Now, we have two types of particles. 00:51:24.830 --> 00:51:30.420 align:middle line:84% They have the same mass, and we-- 00:51:30.420 --> 00:51:34.727 align:middle line:84% yeah, just by definition, they don't have any-- 00:51:34.727 --> 00:51:35.560 align:middle line:90% they have same mass. 00:51:35.560 --> 00:51:37.690 align:middle line:84% They also have the same spin because they're 00:51:37.690 --> 00:51:39.400 align:middle line:90% all spin 0 scalar particles. 00:51:39.400 --> 00:51:42.750 align:middle line:90% There's no directions. 00:51:42.750 --> 00:51:47.070 align:middle line:84% And yeah, the question is, how do we tell them apart? 00:51:47.070 --> 00:52:04.950 align:middle line:90% 00:52:04.950 --> 00:52:07.425 align:middle line:84% So here is this U(1) symmetry becomes useful. 00:52:07.425 --> 00:52:10.190 align:middle line:90% 00:52:10.190 --> 00:52:13.060 align:middle line:84% Here is this U(1) symmetry becomes useful. 00:52:13.060 --> 00:52:21.490 align:middle line:84% So this U(1) so this phase rotation is a symmetry. 00:52:21.490 --> 00:52:25.610 align:middle line:84% And mathematically, this is called the U(1) transformation. 00:52:25.610 --> 00:52:29.680 align:middle line:84% So this is normally called U(1) symmetry. 00:52:29.680 --> 00:52:35.710 align:middle line:84% U(1) is just a mathematical term for phase rotation-- 00:52:35.710 --> 00:52:37.690 align:middle line:84% mathematically, it's a U(1) group. 00:52:37.690 --> 00:52:39.260 align:middle line:90% It's a mathematical term for-- 00:52:39.260 --> 00:52:45.450 align:middle line:90% 00:52:45.450 --> 00:52:50.670 align:middle line:84% so the way we take them apart is by looking at their conserved-- 00:52:50.670 --> 00:52:52.320 align:middle line:90% look at their quantum numbers. 00:52:52.320 --> 00:52:55.260 align:middle line:84% If we look at two particles, then how do we tell them apart? 00:52:55.260 --> 00:52:57.640 align:middle line:84% We look at their quantum numbers. 00:52:57.640 --> 00:52:59.500 align:middle line:84% And they're quantum numbers, so far, 00:52:59.500 --> 00:53:03.000 align:middle line:84% they look almost the same because their mass is the same. 00:53:03.000 --> 00:53:05.000 align:middle line:84% And by definition, they are scalar particles, 00:53:05.000 --> 00:53:06.545 align:middle line:90% and so they have 0-- 00:53:06.545 --> 00:53:09.960 align:middle line:84% they're spin 0-- they don't have spin. 00:53:09.960 --> 00:53:12.552 align:middle line:84% And so the only other conserved number-- 00:53:12.552 --> 00:53:14.510 align:middle line:84% yeah, if you want to talk about quantum number, 00:53:14.510 --> 00:53:17.210 align:middle line:84% their quantum number has got to be conserved, 00:53:17.210 --> 00:53:19.160 align:middle line:84% because otherwise, if it changes with time, 00:53:19.160 --> 00:53:22.400 align:middle line:84% it doesn't matter-- it doesn't make sense to use that number 00:53:22.400 --> 00:53:23.930 align:middle line:90% to label a particle. 00:53:23.930 --> 00:53:26.960 align:middle line:84% But we get one more conserved number corresponding 00:53:26.960 --> 00:53:30.350 align:middle line:90% to this symmetry. 00:53:30.350 --> 00:53:32.730 align:middle line:90% So this U(1) symmetry-- 00:53:32.730 --> 00:53:37.920 align:middle line:84% so Noether's theorem-- yeah, from the Noether's theorem, 00:53:37.920 --> 00:53:39.480 align:middle line:90% so this is U(1) symmetry-- 00:53:39.480 --> 00:53:43.470 align:middle line:90% 00:53:43.470 --> 00:53:46.240 align:middle line:84% then tell us there's a conserved charge. 00:53:46.240 --> 00:53:48.740 align:middle line:90% So if you work it out-- 00:53:48.740 --> 00:53:50.230 align:middle line:84% you work out the loss theorem, then 00:53:50.230 --> 00:53:53.170 align:middle line:84% you find the corresponding conserved charge, which you 00:53:53.170 --> 00:53:55.180 align:middle line:90% should have already worked out. 00:53:55.180 --> 00:53:59.320 align:middle line:84% So let me just write it classically. 00:53:59.320 --> 00:54:02.745 align:middle line:84% So I just write them as phi star. 00:54:02.745 --> 00:54:04.620 align:middle line:84% So classically, they have the following form. 00:54:04.620 --> 00:54:11.580 align:middle line:90% 00:54:11.580 --> 00:54:16.160 align:middle line:84% So at the quantum level, this becomes operators. 00:54:16.160 --> 00:54:19.890 align:middle line:90% But here, there's an operator-- 00:54:19.890 --> 00:54:22.580 align:middle line:84% but now, here, there's an operator 00:54:22.580 --> 00:54:26.270 align:middle line:84% of-- when we go to the quantum level, 00:54:26.270 --> 00:54:30.110 align:middle line:84% now there's an operator ordering ambiguity now. 00:54:30.110 --> 00:54:37.460 align:middle line:84% You see, here, the phi don't commute with its own conjugate 00:54:37.460 --> 00:54:39.830 align:middle line:90% momentum. 00:54:39.830 --> 00:54:42.020 align:middle line:84% And so here, we have phi multiply 00:54:42.020 --> 00:54:43.770 align:middle line:90% its conjugate momentum. 00:54:43.770 --> 00:54:46.400 align:middle line:84% And if we change the order, and then 00:54:46.400 --> 00:54:49.070 align:middle line:90% we will result a delta function. 00:54:49.070 --> 00:54:53.050 align:middle line:90% 00:54:53.050 --> 00:54:56.882 align:middle line:84% And that will give rise to infinite nu-- 00:54:56.882 --> 00:54:57.382 align:middle line:90% yeah. 00:54:57.382 --> 00:55:00.300 align:middle line:90% 00:55:00.300 --> 00:55:07.560 align:middle line:84% So at the quantum level, different ordering-- 00:55:07.560 --> 00:55:15.740 align:middle line:90% 00:55:15.740 --> 00:55:26.155 align:middle line:84% differ by some infinite constant. 00:55:26.155 --> 00:55:30.940 align:middle line:90% 00:55:30.940 --> 00:55:32.955 align:middle line:84% So you can take phi, and you can-- 00:55:32.955 --> 00:55:34.330 align:middle line:84% yeah, if you change the ordering, 00:55:34.330 --> 00:55:36.040 align:middle line:90% then you get a delta function. 00:55:36.040 --> 00:55:39.940 align:middle line:84% But delta function, again, is evaluated 00:55:39.940 --> 00:55:42.610 align:middle line:84% because these two is evaluated at the same x, 00:55:42.610 --> 00:55:45.528 align:middle line:84% and then you get the delta function evaluated at the 0. 00:55:45.528 --> 00:55:47.320 align:middle line:84% So essentially, you get infinite constants. 00:55:47.320 --> 00:55:49.910 align:middle line:90% 00:55:49.910 --> 00:56:15.840 align:middle line:84% But we can fix this ambiguity by requiring your vacuum 00:56:15.840 --> 00:56:19.050 align:middle line:84% state-- your lowest energy state should have charge 00:56:19.050 --> 00:56:22.970 align:middle line:84% 0, because by definition, when you're at the lowest energy, 00:56:22.970 --> 00:56:24.170 align:middle line:90% there's nothing there. 00:56:24.170 --> 00:56:27.440 align:middle line:90% It's a vacuum. 00:56:27.440 --> 00:56:32.780 align:middle line:84% By requiring the lowest energy state, have charge 0. 00:56:32.780 --> 00:56:38.060 align:middle line:84% So that uniquely fixes whatever infinite-- 00:56:38.060 --> 00:56:40.210 align:middle line:90% the ordering. 00:56:40.210 --> 00:56:45.700 align:middle line:84% So by requiring this equal to 0, and then you 00:56:45.700 --> 00:56:47.630 align:middle line:90% can show that the Q-- 00:56:47.630 --> 00:56:49.780 align:middle line:90% so now, you can essentially-- 00:56:49.780 --> 00:56:51.265 align:middle line:90% the Q has the following form. 00:56:51.265 --> 00:56:58.310 align:middle line:90% 00:56:58.310 --> 00:56:59.720 align:middle line:90% So you can already-- 00:56:59.720 --> 00:57:07.040 align:middle line:84% you may be able to guess the answer of a k dagger a k, then 00:57:07.040 --> 00:57:11.450 align:middle line:90% minus b k dagger b k. 00:57:11.450 --> 00:57:13.640 align:middle line:90% That's what you get. 00:57:13.640 --> 00:57:17.660 align:middle line:84% So you plug those expressions in. 00:57:17.660 --> 00:57:20.450 align:middle line:84% You plug those expressions at this time derivative. 00:57:20.450 --> 00:57:22.610 align:middle line:90% Time derivative gives you pi. 00:57:22.610 --> 00:57:24.920 align:middle line:84% You just plug those expressions in, 00:57:24.920 --> 00:57:27.290 align:middle line:84% and again, you find-- because this is conserved, 00:57:27.290 --> 00:57:30.300 align:middle line:84% you find all these time dependents cancel, et cetera. 00:57:30.300 --> 00:57:34.420 align:middle line:84% And then you get these two expressions up 00:57:34.420 --> 00:57:37.650 align:middle line:90% to some constant. 00:57:37.650 --> 00:57:43.260 align:middle line:84% And then this condition requires that that constant must be 0. 00:57:43.260 --> 00:57:46.500 align:middle line:84% Yeah, when you write in this form. 00:57:46.500 --> 00:57:48.930 align:middle line:84% You see this annihlates the Q because the a is 00:57:48.930 --> 00:57:50.730 align:middle line:84% on the right-hand side of a dagger. 00:57:50.730 --> 00:57:54.570 align:middle line:84% So this automatically annihilates the vacuum. 00:57:54.570 --> 00:58:01.950 align:middle line:90% 00:58:01.950 --> 00:58:05.330 align:middle line:84% So essentially, this is just occupation number 00:58:05.330 --> 00:58:08.210 align:middle line:84% for a and this occupation number for b k. 00:58:08.210 --> 00:58:20.050 align:middle line:90% 00:58:20.050 --> 00:58:22.050 align:middle line:90% So we essentially get Q-- 00:58:22.050 --> 00:58:28.930 align:middle line:90% 00:58:28.930 --> 00:58:31.330 align:middle line:84% essentially, the occupation number 00:58:31.330 --> 00:58:36.460 align:middle line:84% for a k and the occupation number for b k. 00:58:36.460 --> 00:58:40.890 align:middle line:84% So occupation number, I just write as N. Yes? 00:58:40.890 --> 00:58:43.290 align:middle line:84% AUDIENCE: I still don't see how this fixes anything, 00:58:43.290 --> 00:58:47.165 align:middle line:84% because if you switch them, you still get the same issue-- 00:58:47.165 --> 00:58:47.790 align:middle line:90% PROFESSOR: Huh? 00:58:47.790 --> 00:58:49.410 align:middle line:84% AUDIENCE: For this line of argument. 00:58:49.410 --> 00:58:52.110 align:middle line:84% If you permute them, you still get the infinite cost, 00:58:52.110 --> 00:58:55.920 align:middle line:84% so I'm confused why this is considered a [INAUDIBLE].. 00:58:55.920 --> 00:58:58.260 align:middle line:84% PROFESSOR: No, the way you permit them, 00:58:58.260 --> 00:59:00.270 align:middle line:84% then you no longer violate the vacuum, 00:59:00.270 --> 00:59:03.120 align:middle line:90% so that constant will cancel. 00:59:03.120 --> 00:59:05.510 align:middle line:84% So only this form with a is sitting 00:59:05.510 --> 00:59:07.100 align:middle line:84% on the right-hand side of a dagger, 00:59:07.100 --> 00:59:09.560 align:middle line:90% this will violate the vacuum. 00:59:09.560 --> 00:59:10.790 align:middle line:90% AUDIENCE: Right, but-- 00:59:10.790 --> 00:59:11.300 align:middle line:90% OK. 00:59:11.300 --> 00:59:14.810 align:middle line:84% I guess because earlier, the line of argument 00:59:14.810 --> 00:59:17.370 align:middle line:84% was, if you permute them, you would get a delta function, 00:59:17.370 --> 00:59:19.490 align:middle line:84% because you would get the same thing here, right? 00:59:19.490 --> 00:59:21.170 align:middle line:84% If you permit them, do you also-- 00:59:21.170 --> 00:59:23.540 align:middle line:84% PROFESSOR: No, no, no, no, no, no, no. 00:59:23.540 --> 00:59:26.900 align:middle line:90% Here, there is ambiguity. 00:59:26.900 --> 00:59:29.750 align:middle line:84% Here, I don't know how I should order them. 00:59:29.750 --> 00:59:32.990 align:middle line:84% Whether I should put phi to the right of pi or put pi 00:59:32.990 --> 00:59:33.740 align:middle line:90% to the left-- 00:59:33.740 --> 00:59:34.730 align:middle line:90% to the right of phi. 00:59:34.730 --> 00:59:38.180 align:middle line:90% I'm not sure how should I do. 00:59:38.180 --> 00:59:43.170 align:middle line:84% Here, there's no prescription, and here, there's 00:59:43.170 --> 00:59:44.070 align:middle line:90% no prescription. 00:59:44.070 --> 00:59:48.360 align:middle line:84% But if I impose this condition, then that fully-- 00:59:48.360 --> 00:59:50.880 align:middle line:84% then that tells you when I impose this condition, 00:59:50.880 --> 00:59:52.305 align:middle line:90% then Q must be of this form. 00:59:52.305 --> 00:59:55.790 align:middle line:90% 00:59:55.790 --> 00:59:58.040 align:middle line:84% Of course, when you permute them, it's the same thing. 00:59:58.040 --> 01:00:01.300 align:middle line:84% When you permute them, you don't change the operator. 01:00:01.300 --> 01:00:02.680 align:middle line:90% You don't change the operator. 01:00:02.680 --> 01:00:06.010 align:middle line:84% Yeah, but if you do, say-- let's do this order. 01:00:06.010 --> 01:00:08.590 align:middle line:90% 01:00:08.590 --> 01:00:11.430 align:middle line:84% If you do this order, what you get 01:00:11.430 --> 01:00:15.830 align:middle line:84% is this expression with an infinite constant. 01:00:15.830 --> 01:00:17.740 align:middle line:84% And then this condition tells you somehow 01:00:17.740 --> 01:00:22.230 align:middle line:84% you have to change the order here so that constant is 0. 01:00:22.230 --> 01:00:24.847 align:middle line:90% 01:00:24.847 --> 01:00:26.430 align:middle line:84% Yeah, when you write it in this form-- 01:00:26.430 --> 01:00:30.390 align:middle line:84% when you write it in a and b, and that constant has to be 0. 01:00:30.390 --> 01:00:31.466 align:middle line:90% Yes? 01:00:31.466 --> 01:00:34.400 align:middle line:84% AUDIENCE: So does the ordering ambiguity 01:00:34.400 --> 01:00:40.030 align:middle line:84% have any physical consequences, because besides just things 01:00:40.030 --> 01:00:43.700 align:middle line:90% that can be subtracted off? 01:00:43.700 --> 01:00:48.770 align:middle line:84% PROFESSOR: Yeah, it's-- yeah, this is a good question. 01:00:48.770 --> 01:00:51.650 align:middle line:90% Which-- it's hard to say. 01:00:51.650 --> 01:00:55.430 align:middle line:84% So most of the time, you can fix them 01:00:55.430 --> 01:00:57.560 align:middle line:90% by some physical invariant. 01:00:57.560 --> 01:01:00.020 align:middle line:84% By phys-- yeah, as here, you can fix it 01:01:00.020 --> 01:01:02.690 align:middle line:84% based on physical requirement, and then you 01:01:02.690 --> 01:01:04.400 align:middle line:84% don't have to worry about it anymore. 01:01:04.400 --> 01:01:07.580 align:middle line:90% 01:01:07.580 --> 01:01:09.680 align:middle line:84% But the ordering ambiguity requires 01:01:09.680 --> 01:01:13.280 align:middle line:84% you try to find such a physical requirement. 01:01:13.280 --> 01:01:16.670 align:middle line:84% Yeah, because otherwise, what ordering would you use? 01:01:16.670 --> 01:01:20.442 align:middle line:84% And the process of finding that requirement by itself 01:01:20.442 --> 01:01:21.650 align:middle line:90% is understanding the physics. 01:01:21.650 --> 01:01:25.325 align:middle line:90% 01:01:25.325 --> 01:01:25.825 align:middle line:90% Yes? 01:01:25.825 --> 01:01:27.242 align:middle line:84% AUDIENCE: Should we have done that 01:01:27.242 --> 01:01:29.225 align:middle line:84% with the-- when you've got the infinite energy 01:01:29.225 --> 01:01:31.600 align:middle line:84% in the previous lecture, because you know how it required 01:01:31.600 --> 01:01:33.350 align:middle line:84% the ground state to have 0 energy and then 01:01:33.350 --> 01:01:34.732 align:middle line:90% get some different ordering. 01:01:34.732 --> 01:01:35.440 align:middle line:90% PROFESSOR: Right. 01:01:35.440 --> 01:01:39.980 align:middle line:84% No, in that case, we don't have ordering ambiguity. 01:01:39.980 --> 01:01:46.180 align:middle line:84% So in that case, because we just have pi square itself 01:01:46.180 --> 01:01:49.960 align:middle line:84% plus phi squared, and plus phi squared. 01:01:49.960 --> 01:01:52.250 align:middle line:84% There's no ordering ambiguity here. 01:01:52.250 --> 01:01:55.640 align:middle line:84% And so whatever you get is whatever you get. 01:01:55.640 --> 01:01:57.230 align:middle line:84% And if you get infinite constant, 01:01:57.230 --> 01:01:58.572 align:middle line:90% then you get infinite constant. 01:01:58.572 --> 01:01:59.530 align:middle line:90% You don't have freedom. 01:01:59.530 --> 01:02:05.040 align:middle line:90% 01:02:05.040 --> 01:02:06.180 align:middle line:90% Other questions? 01:02:06.180 --> 01:02:06.810 align:middle line:90% Yes? 01:02:06.810 --> 01:02:09.840 align:middle line:84% AUDIENCE: So I don't also see how phi and phi dagger-- 01:02:09.840 --> 01:02:11.648 align:middle line:90% it's clear that they commute. 01:02:11.648 --> 01:02:14.190 align:middle line:84% How do you know that there's no coupling between the fields-- 01:02:14.190 --> 01:02:17.190 align:middle line:84% or when there is coupling, how do you represent it in-- 01:02:17.190 --> 01:02:19.450 align:middle line:84% PROFESSOR: Oh, this is a definition. 01:02:19.450 --> 01:02:22.895 align:middle line:90% You say where this come from? 01:02:22.895 --> 01:02:23.520 align:middle line:90% AUDIENCE: Yeah. 01:02:23.520 --> 01:02:26.280 align:middle line:84% Like, why do we know for sure that it's 0? 01:02:26.280 --> 01:02:29.610 align:middle line:84% PROFESSOR: Yeah, this is just the definition of-- 01:02:29.610 --> 01:02:32.370 align:middle line:90% this is the same. 01:02:32.370 --> 01:02:35.010 align:middle line:84% This may have nothing to do with field theory. 01:02:35.010 --> 01:02:37.940 align:middle line:84% Only this equation have to do with field theory. 01:02:37.940 --> 01:02:39.690 align:middle line:84% This have nothing to do with field theory. 01:02:39.690 --> 01:02:42.750 align:middle line:84% This is just quantum mechanics that different degrees-- 01:02:42.750 --> 01:02:46.470 align:middle line:84% different degrees freedom, they commute with each other. 01:02:46.470 --> 01:02:49.650 align:middle line:84% Yeah, just x1 commute with x2 in quantum mechanics. 01:02:49.650 --> 01:02:52.710 align:middle line:84% Yeah, phi-- here, just phi and phi dagger just 01:02:52.710 --> 01:02:54.860 align:middle line:90% is analog of x1 and x2 there. 01:02:54.860 --> 01:02:58.590 align:middle line:90% 01:02:58.590 --> 01:03:00.140 align:middle line:90% Other questions? 01:03:00.140 --> 01:03:01.682 align:middle line:84% AUDIENCE: Can every operator commutes 01:03:01.682 --> 01:03:03.140 align:middle line:84% do we still have quantum mechanics? 01:03:03.140 --> 01:03:05.730 align:middle line:90% 01:03:05.730 --> 01:03:07.590 align:middle line:84% PROFESSOR: No, it's different from that. 01:03:07.590 --> 01:03:11.430 align:middle line:90% Here, it just means that the-- 01:03:11.430 --> 01:03:13.800 align:middle line:84% here, you should view them as the variables 01:03:13.800 --> 01:03:15.300 align:middle line:90% in the configuration space. 01:03:15.300 --> 01:03:18.180 align:middle line:84% And just the-- in quantum mechanics, 01:03:18.180 --> 01:03:20.608 align:middle line:84% the variables in configuration space, they 01:03:20.608 --> 01:03:21.900 align:middle line:90% always commute with each other. 01:03:21.900 --> 01:03:28.040 align:middle line:90% 01:03:28.040 --> 01:03:29.420 align:middle line:90% Good. 01:03:29.420 --> 01:03:33.890 align:middle line:84% So now, the key thing is that here is the minus sign. 01:03:33.890 --> 01:03:41.032 align:middle line:84% So that means when you look at the commutator between Q and a, 01:03:41.032 --> 01:03:43.756 align:middle line:90% you get the a. 01:03:43.756 --> 01:03:48.214 align:middle line:84% And if you look at the commutator a with b k, 01:03:48.214 --> 01:03:52.060 align:middle line:90% you actually get minus b k. 01:03:52.060 --> 01:03:56.980 align:middle line:84% You get-- yeah, let me just do the dagger because this 01:03:56.980 --> 01:03:58.330 align:middle line:90% is used to create particle. 01:03:58.330 --> 01:04:03.440 align:middle line:90% 01:04:03.440 --> 01:04:08.510 align:middle line:84% So if you just calculate the commutator, you just find that. 01:04:08.510 --> 01:04:29.360 align:middle line:84% So that means if you look at the states defined by one particle 01:04:29.360 --> 01:04:33.200 align:middle line:84% state corresponding to a and one particle state corresponding 01:04:33.200 --> 01:04:49.580 align:middle line:84% to b, then both are eigenvectors of Q, 01:04:49.580 --> 01:04:56.480 align:middle line:84% but here with eigenvalue 1, but this one is eigenvalue minus 1. 01:04:56.480 --> 01:04:58.535 align:middle line:84% So it means they actually have opposite charge. 01:04:58.535 --> 01:05:03.540 align:middle line:90% 01:05:03.540 --> 01:05:10.510 align:middle line:90% So k and-- oh, by the way-- 01:05:10.510 --> 01:05:13.500 align:middle line:84% so let me explain a little bit of this notation. 01:05:13.500 --> 01:05:16.320 align:middle line:84% This k does not mean the magnitude of k. 01:05:16.320 --> 01:05:22.570 align:middle line:84% So this k means the-- because normally, 01:05:22.570 --> 01:05:26.550 align:middle line:84% our convention is that the four vector-- 01:05:26.550 --> 01:05:29.640 align:middle line:84% say the x mu we always just write it as x. 01:05:29.640 --> 01:05:32.310 align:middle line:90% The p mu we just write as p. 01:05:32.310 --> 01:05:35.130 align:middle line:84% So here, you should view this as a four vector. 01:05:35.130 --> 01:05:36.045 align:middle line:90% It's a four vector. 01:05:36.045 --> 01:05:39.810 align:middle line:90% 01:05:39.810 --> 01:05:42.690 align:middle line:90% It just is a shorthand notation. 01:05:42.690 --> 01:05:47.770 align:middle line:84% So k and the k bar, then they have opposite charge. 01:05:47.770 --> 01:05:50.330 align:middle line:90% 01:05:50.330 --> 01:06:00.910 align:middle line:84% Have the same mass, the same spin 0, but opposite charge. 01:06:00.910 --> 01:06:03.490 align:middle line:90% 01:06:03.490 --> 01:06:07.620 align:middle line:84% So by using this Q, we can now distinguish these two 01:06:07.620 --> 01:06:08.490 align:middle line:90% particles. 01:06:08.490 --> 01:06:11.080 align:middle line:90% 01:06:11.080 --> 01:06:14.500 align:middle line:84% So normally, because such a particle-- 01:06:14.500 --> 01:06:17.005 align:middle line:90% because in this case, which-- 01:06:17.005 --> 01:06:20.080 align:middle line:84% they are-- all quantum numbers are the same, except they 01:06:20.080 --> 01:06:21.970 align:middle line:90% have opposite charge. 01:06:21.970 --> 01:06:24.580 align:middle line:84% We call them particle and antiparticle. 01:06:24.580 --> 01:06:25.660 align:middle line:90% So this we call-- 01:06:25.660 --> 01:06:28.670 align:middle line:90% create a particle. 01:06:28.670 --> 01:06:31.060 align:middle line:84% So here, we say they create an antiparticle. 01:06:31.060 --> 01:06:40.130 align:middle line:90% 01:06:40.130 --> 01:06:47.780 align:middle line:84% So here, let me just make a remark without proving it. 01:06:47.780 --> 01:06:49.880 align:middle line:84% So in relativistic quantum field theory, 01:06:49.880 --> 01:06:55.720 align:middle line:84% you can show that any particle has an antiparticle. 01:06:55.720 --> 01:06:58.210 align:middle line:84% In any quantum field-- any relativistic quantum field 01:06:58.210 --> 01:07:02.600 align:middle line:84% theory, any particle has an antiparticle. 01:07:02.600 --> 01:07:05.603 align:middle line:84% In the case of the real scalar field, 01:07:05.603 --> 01:07:07.020 align:middle line:84% you say we only have one particle. 01:07:07.020 --> 01:07:09.010 align:middle line:84% In that case, the antiparticle itself. 01:07:09.010 --> 01:07:18.204 align:middle line:90% 01:07:18.204 --> 01:07:19.290 align:middle line:90% Good. 01:07:19.290 --> 01:07:21.810 align:middle line:90% Do you have any questions? 01:07:21.810 --> 01:07:22.620 align:middle line:90% Yes? 01:07:22.620 --> 01:07:25.037 align:middle line:84% AUDIENCE: Can we derive the fact that Q is considered just 01:07:25.037 --> 01:07:27.330 align:middle line:84% by the number in the a and the number of b 01:07:27.330 --> 01:07:29.084 align:middle line:90% is considered separately? 01:07:29.084 --> 01:07:31.620 align:middle line:84% Because I'm having trouble seeing why conservation of Q 01:07:31.620 --> 01:07:35.120 align:middle line:90% is a non-trivial statement. 01:07:35.120 --> 01:07:37.010 align:middle line:84% PROFESSOR: Yeah, the conservation of Q, 01:07:37.010 --> 01:07:39.950 align:middle line:90% it follows from the symmetry. 01:07:39.950 --> 01:07:42.592 align:middle line:84% AUDIENCE: But couldn't I just-- like, the number of a particles 01:07:42.592 --> 01:07:44.300 align:middle line:84% and the number of b particles are kind of 01:07:44.300 --> 01:07:45.150 align:middle line:90% separately conserved. 01:07:45.150 --> 01:07:46.850 align:middle line:84% So couldn'te I just write off that Q was conserved from 01:07:46.850 --> 01:07:47.810 align:middle line:90% the very beginning? 01:07:47.810 --> 01:07:51.550 align:middle line:84% PROFESSOR: Yeah, it's-- no, actually, 01:07:51.550 --> 01:07:54.860 align:middle line:84% the number of a particle and number of b particle, 01:07:54.860 --> 01:07:57.650 align:middle line:84% they're not a separately conserved. 01:07:57.650 --> 01:08:01.470 align:middle line:84% They can actually-- in principle, they can annihilate. 01:08:01.470 --> 01:08:04.830 align:middle line:84% Yeah, here, in free they cannot annihilate. 01:08:04.830 --> 01:08:08.940 align:middle line:84% But if I can see the slightly more complicated theory-- 01:08:08.940 --> 01:08:10.980 align:middle line:84% say, if they are allowed to interact, 01:08:10.980 --> 01:08:14.490 align:middle line:84% and then they can actually annihilate and then disappear. 01:08:14.490 --> 01:08:17.430 align:middle line:90% And so in that case-- 01:08:17.430 --> 01:08:19.620 align:middle line:90% but this formalism still works. 01:08:19.620 --> 01:08:22.090 align:middle line:90% This symmetry is still there. 01:08:22.090 --> 01:08:25.229 align:middle line:84% So that's why the symmetry is powerful. 01:08:25.229 --> 01:08:28.200 align:middle line:84% Also applies to the case with interactions. 01:08:28.200 --> 01:08:34.779 align:middle line:84% Yeah, indeed, what you said is correct in the free theory. 01:08:34.779 --> 01:08:35.620 align:middle line:90% Yes? 01:08:35.620 --> 01:08:38.654 align:middle line:84% AUDIENCE: So does this mean particles and antiparticles-- 01:08:38.654 --> 01:08:40.029 align:middle line:84% like, the difference between them 01:08:40.029 --> 01:08:43.390 align:middle line:84% is conserved, so they work in pairs? 01:08:43.390 --> 01:08:46.640 align:middle line:84% PROFESSOR: Yeah, essentially, their property-- 01:08:46.640 --> 01:08:48.490 align:middle line:84% it means their property almost identical 01:08:48.490 --> 01:08:51.640 align:middle line:84% except they have opposite charge. 01:08:51.640 --> 01:08:55.240 align:middle line:90% Yeah, they have essentially-- 01:08:55.240 --> 01:08:58.870 align:middle line:84% just like a real world, we have electron, we have matter, 01:08:58.870 --> 01:09:01.060 align:middle line:90% we have anti-matter. 01:09:01.060 --> 01:09:04.240 align:middle line:84% They have essentially identical properties. 01:09:04.240 --> 01:09:07.630 align:middle line:84% And if the world is made of anti-matter, 01:09:07.630 --> 01:09:12.430 align:middle line:84% everything still will behave the same, and yeah-- 01:09:12.430 --> 01:09:14.702 align:middle line:84% except just the charge is different. 01:09:14.702 --> 01:09:16.660 align:middle line:84% AUDIENCE: And the difference between the number 01:09:16.660 --> 01:09:20.710 align:middle line:84% of antiparticles and particles is conserved? 01:09:20.710 --> 01:09:22.130 align:middle line:90% PROFESSOR: It's conserved. 01:09:22.130 --> 01:09:25.434 align:middle line:90% Yeah, it's conserved. 01:09:25.434 --> 01:09:26.340 align:middle line:90% Ryan. 01:09:26.340 --> 01:09:29.550 align:middle line:84% AUDIENCE: So I guess this is the electric charge, 01:09:29.550 --> 01:09:32.069 align:middle line:90% or is this some other charge? 01:09:32.069 --> 01:09:33.630 align:middle line:84% PROFESSOR: I think some other charge. 01:09:33.630 --> 01:09:36.540 align:middle line:84% So in the case we will go to when 01:09:36.540 --> 01:09:39.850 align:middle line:84% we talk about electrons-- so electrons involving 01:09:39.850 --> 01:09:41.100 align:middle line:90% a more complicated formalism. 01:09:41.100 --> 01:09:44.130 align:middle line:84% We have to introduce particles with spin. 01:09:44.130 --> 01:09:47.310 align:middle line:84% And in that case, we also have electron and anti-electron. 01:09:47.310 --> 01:09:49.770 align:middle line:84% And in that case, it's-- the story is parallel. 01:09:49.770 --> 01:09:53.107 align:middle line:90% 01:09:53.107 --> 01:09:54.690 align:middle line:84% AUDIENCE: So this does not necessarily 01:09:54.690 --> 01:09:57.730 align:middle line:84% involve E&M or things reppelling. 01:09:57.730 --> 01:09:58.470 align:middle line:90% PROFESSOR: No. 01:09:58.470 --> 01:10:02.740 align:middle line:84% Yeah, but this is the analog of that. 01:10:02.740 --> 01:10:06.760 align:middle line:84% So the charge here, they can be some other kind of charge. 01:10:06.760 --> 01:10:10.620 align:middle line:90% Does not have to be -- yeah. 01:10:10.620 --> 01:10:12.410 align:middle line:84% AUDIENCE: So how can you get something 01:10:12.410 --> 01:10:15.650 align:middle line:84% that's physical that's conserved from a symmetry that's 01:10:15.650 --> 01:10:16.850 align:middle line:90% not really physical? 01:10:16.850 --> 01:10:20.000 align:middle line:84% Like, this is not like you can translate in space 01:10:20.000 --> 01:10:21.462 align:middle line:90% or translate in time. 01:10:21.462 --> 01:10:22.670 align:middle line:90% This is-- I don't understand. 01:10:22.670 --> 01:10:23.990 align:middle line:84% I don't know how to think about it. 01:10:23.990 --> 01:10:26.365 align:middle line:84% PROFESSOR: Yeah, that's why we call it internal symmetry. 01:10:26.365 --> 01:10:28.860 align:middle line:90% 01:10:28.860 --> 01:10:33.750 align:middle line:84% So for things that we don't have good intuition about, 01:10:33.750 --> 01:10:35.610 align:middle line:84% you just invent the new name for it-- 01:10:35.610 --> 01:10:36.690 align:middle line:90% [LAUGHTER] 01:10:36.690 --> 01:10:39.150 align:middle line:84% --and once you get used to this new name, 01:10:39.150 --> 01:10:42.060 align:middle line:90% you say, oh, I understand it. 01:10:42.060 --> 01:10:45.960 align:middle line:84% Yeah, it is more or less getting used to it. 01:10:45.960 --> 01:10:49.350 align:middle line:84% And indeed, at first sight, it's not very intuitive 01:10:49.350 --> 01:10:51.660 align:middle line:84% because it's not something you can-- 01:10:51.660 --> 01:10:55.500 align:middle line:84% yeah, from your normal experiences, et cetera. 01:10:55.500 --> 01:11:00.180 align:middle line:90% But yeah, this is the-- 01:11:00.180 --> 01:11:03.778 align:middle line:84% but this is-- at the fundamental level, 01:11:03.778 --> 01:11:05.820 align:middle line:84% it's not that different from energy conservation. 01:11:05.820 --> 01:11:09.010 align:middle line:84% It's just some symmetry, and then leads 01:11:09.010 --> 01:11:11.350 align:middle line:90% to some conserved quantities. 01:11:11.350 --> 01:11:14.170 align:middle line:90% 01:11:14.170 --> 01:11:14.950 align:middle line:90% Yes? 01:11:14.950 --> 01:11:16.867 align:middle line:84% AUDIENCE: If we have another conserved charge, 01:11:16.867 --> 01:11:20.386 align:middle line:84% like Q and now electric charge and-- 01:11:20.386 --> 01:11:24.310 align:middle line:84% which ones do we call, I guess, particle/antiparticle pairs? 01:11:24.310 --> 01:11:25.480 align:middle line:90% Does that make sense? 01:11:25.480 --> 01:11:26.730 align:middle line:90% PROFESSOR: Yeah, that's right. 01:11:26.730 --> 01:11:29.760 align:middle line:84% They can, in principle, have multiple charges, 01:11:29.760 --> 01:11:31.510 align:middle line:84% but the particle and antiparticle, they're 01:11:31.510 --> 01:11:33.720 align:middle line:90% always opposite. 01:11:33.720 --> 01:11:36.600 align:middle line:84% Yeah, in terms of charges they're always opposite-- 01:11:36.600 --> 01:11:38.340 align:middle line:90% whatever charges you have. 01:11:38.340 --> 01:11:40.650 align:middle line:84% AUDIENCE: What if there's two different charges 01:11:40.650 --> 01:11:43.240 align:middle line:84% that each one can have, like a set of four things? 01:11:43.240 --> 01:11:44.670 align:middle line:90% Is it a different language? 01:11:44.670 --> 01:11:45.810 align:middle line:90% PROFESSOR: Yeah, I think-- 01:11:45.810 --> 01:11:49.510 align:middle line:90% 01:11:49.510 --> 01:11:51.273 align:middle line:90% yeah, you can have them-- 01:11:51.273 --> 01:11:52.690 align:middle line:84% but for particle and antiparticle, 01:11:52.690 --> 01:11:54.648 align:middle line:90% they're always opposite. 01:11:54.648 --> 01:11:56.190 align:middle line:84% Yeah, they can have multiple charges, 01:11:56.190 --> 01:11:57.615 align:middle line:90% but they're always opposite. 01:11:57.615 --> 01:12:00.230 align:middle line:90% 01:12:00.230 --> 01:12:01.130 align:middle line:90% Other questions? 01:12:01.130 --> 01:12:04.610 align:middle line:90% 01:12:04.610 --> 01:12:06.240 align:middle line:90% Good. 01:12:06.240 --> 01:12:12.520 align:middle line:84% So this concludes our discussion of the complex field. 01:12:12.520 --> 01:12:15.460 align:middle line:90% 01:12:15.460 --> 01:12:17.910 align:middle line:90% So we have a few minutes-- 01:12:17.910 --> 01:12:21.480 align:middle line:90% then let's do to the next topic. 01:12:21.480 --> 01:12:24.460 align:middle line:90% 01:12:24.460 --> 01:12:59.110 align:middle line:84% So hopefully-- so now, let's talk about the concept 01:12:59.110 --> 01:13:01.860 align:middle line:90% called the propagators. 01:13:01.860 --> 01:13:05.370 align:middle line:84% So again, let's recall in non-relativistic quantum 01:13:05.370 --> 01:13:08.170 align:middle line:90% mechanics-- 01:13:08.170 --> 01:13:12.190 align:middle line:84% So this position vector plays a key role. 01:13:12.190 --> 01:13:19.280 align:middle line:84% So this is the eigenvector of this x, 01:13:19.280 --> 01:13:21.670 align:middle line:90% which is the position operator. 01:13:21.670 --> 01:13:23.350 align:middle line:90% So again, I will-- 01:13:23.350 --> 01:13:26.110 align:middle line:84% yeah, let me just put a hat here to emphasize 01:13:26.110 --> 01:13:28.450 align:middle line:90% this is the operator. 01:13:28.450 --> 01:13:31.180 align:middle line:84% And so this is the eigenvector of x. 01:13:31.180 --> 01:13:34.337 align:middle line:84% And because this plays a key role-- 01:13:34.337 --> 01:13:36.670 align:middle line:84% because we need to use this to define the wave function. 01:13:36.670 --> 01:13:40.900 align:middle line:90% 01:13:40.900 --> 01:13:44.840 align:middle line:84% And then the amplitude of the wave function-- 01:13:44.840 --> 01:13:47.260 align:middle line:84% the square gives us the probability 01:13:47.260 --> 01:13:49.740 align:middle line:90% of a particle at location x. 01:13:49.740 --> 01:13:54.620 align:middle line:90% So this is a crucial quantity. 01:13:54.620 --> 01:13:59.860 align:middle line:84% So this language-- normally, we talk 01:13:59.860 --> 01:14:02.050 align:middle line:90% about the Schrodinger picture. 01:14:02.050 --> 01:14:08.050 align:middle line:84% You start with psi x at t equal to 0, 01:14:08.050 --> 01:14:10.340 align:middle line:90% and then you try to find-- 01:14:10.340 --> 01:14:10.840 align:middle line:90% yeah. 01:14:10.840 --> 01:14:14.455 align:middle line:84% Or let me say, you start at psi x-- 01:14:14.455 --> 01:14:17.140 align:middle line:90% 01:14:17.140 --> 01:14:21.860 align:middle line:84% say, t prime-- at some t prime, you 01:14:21.860 --> 01:14:26.060 align:middle line:90% want to find psi x at some t. 01:14:26.060 --> 01:14:29.420 align:middle line:84% So this is the question of quantum mechanics. 01:14:29.420 --> 01:14:31.910 align:middle line:84% So if you know the wave function at the initial time 01:14:31.910 --> 01:14:35.060 align:middle line:84% and you want to find the wave function at some future time. 01:14:35.060 --> 01:14:38.520 align:middle line:90% 01:14:38.520 --> 01:14:43.160 align:middle line:84% So now let's imagine we consider this question-- 01:14:43.160 --> 01:14:44.000 align:middle line:90% Heisenberg picture. 01:14:44.000 --> 01:14:49.590 align:middle line:90% 01:14:49.590 --> 01:14:57.420 align:middle line:84% So in the Schrödinger picture, there's just one eigenvector-- 01:14:57.420 --> 01:14:59.430 align:middle line:84% yeah, just one set of eigenvectors 01:14:59.430 --> 01:15:03.170 align:middle line:90% for x for different eigenvalues. 01:15:03.170 --> 01:15:09.440 align:middle line:84% But for the Heisenberg picture now, your position operator 01:15:09.440 --> 01:15:12.200 align:middle line:90% now depends on time. 01:15:12.200 --> 01:15:14.690 align:middle line:84% So the position operator at different times, 01:15:14.690 --> 01:15:16.610 align:middle line:90% they are different. 01:15:16.610 --> 01:15:20.110 align:middle line:84% So now, if you want to talk about the position eigenvector, 01:15:20.110 --> 01:15:22.480 align:middle line:90% now you need the label. 01:15:22.480 --> 01:15:26.860 align:middle line:84% So now, the position eigenvector-- 01:15:26.860 --> 01:15:29.020 align:middle line:90% it's a t label. 01:15:29.020 --> 01:15:32.380 align:middle line:84% This t does not mean that this vector evolves with time. 01:15:32.380 --> 01:15:34.180 align:middle line:90% It's just this is the label. 01:15:34.180 --> 01:15:36.220 align:middle line:84% This means that this is the eigenvector 01:15:36.220 --> 01:15:41.600 align:middle line:90% of position vector at time t. 01:15:41.600 --> 01:15:46.580 align:middle line:90% 01:15:46.580 --> 01:15:49.430 align:middle line:84% This just tells you this is the eigenvector at time t. 01:15:49.430 --> 01:15:59.100 align:middle line:90% 01:15:59.100 --> 01:16:01.980 align:middle line:84% And now, the question of-- this question of-- this is 01:16:01.980 --> 01:16:05.140 align:middle line:84% the question of quantum mechanics-- 01:16:05.140 --> 01:16:09.550 align:middle line:84% then can be reformulated into a different equation. 01:16:09.550 --> 01:16:18.570 align:middle line:84% So in this story, the question of quantum mechanics 01:16:18.570 --> 01:16:20.100 align:middle line:90% becomes the following-- 01:16:20.100 --> 01:16:24.630 align:middle line:84% what is the value of this object? 01:16:24.630 --> 01:16:29.990 align:middle line:90% 01:16:29.990 --> 01:16:34.910 align:middle line:84% Suppose you are in the eigenstate-- 01:16:34.910 --> 01:16:38.960 align:middle line:90% you are the eigenvector-- 01:16:38.960 --> 01:16:40.850 align:middle line:84% you are in the position eigenvector-- 01:16:40.850 --> 01:16:45.170 align:middle line:84% you are in the position eigenstate at time t 01:16:45.170 --> 01:16:47.810 align:middle line:90% prime with the value x prime. 01:16:47.810 --> 01:16:54.250 align:middle line:84% And what's the amplitude for you to go to location x at time t? 01:16:54.250 --> 01:16:57.910 align:middle line:84% And I remember, this does not involve time evolution. 01:16:57.910 --> 01:16:59.620 align:middle line:84% This is just-- because one is labeled 01:16:59.620 --> 01:17:03.950 align:middle line:84% in different eigenvector at different time. 01:17:03.950 --> 01:17:06.440 align:middle line:84% So if you know this object, this is called-- 01:17:06.440 --> 01:17:17.680 align:middle line:84% this will be normally denoted G. G-- 01:17:17.680 --> 01:17:19.000 align:middle line:90% this is called the propagator. 01:17:19.000 --> 01:17:24.310 align:middle line:90% 01:17:24.310 --> 01:17:25.810 align:middle line:90% This is called the propagator. 01:17:25.810 --> 01:17:31.930 align:middle line:84% So if you know the propagator of a quantum system, 01:17:31.930 --> 01:17:33.920 align:middle line:84% then you have full knowledge of that system, 01:17:33.920 --> 01:17:36.310 align:middle line:84% so you already solved that system. 01:17:36.310 --> 01:17:37.560 align:middle line:90% Why? 01:17:37.560 --> 01:17:43.440 align:middle line:84% Because say let's imagine we want to find psi t x wave 01:17:43.440 --> 01:17:49.410 align:middle line:84% function at time t, and then by definition, 01:17:49.410 --> 01:17:50.980 align:middle line:90% this is given by this. 01:17:50.980 --> 01:17:55.190 align:middle line:84% So this is the Heisenberg picture definition. 01:17:55.190 --> 01:18:00.580 align:middle line:84% It's the psi overlap with the position vector at time t. 01:18:00.580 --> 01:18:03.595 align:middle line:84% And now, this is-- now, we can insert a complete set of state. 01:18:03.595 --> 01:18:19.933 align:middle line:90% 01:18:19.933 --> 01:18:21.350 align:middle line:84% I just insert a complete set here. 01:18:21.350 --> 01:18:24.590 align:middle line:90% Just this is the identity. 01:18:24.590 --> 01:18:31.450 align:middle line:90% And then this object is just G. 01:18:31.450 --> 01:18:39.980 align:middle line:84% So you just get G x t x prime t prime, then psi t prime x 01:18:39.980 --> 01:18:40.480 align:middle line:90% prime. 01:18:40.480 --> 01:18:43.920 align:middle line:90% 01:18:43.920 --> 01:18:46.150 align:middle line:90% So if you know-- 01:18:46.150 --> 01:18:50.410 align:middle line:84% so yeah, so this just gave you the wave function at t prime. 01:18:50.410 --> 01:18:53.010 align:middle line:90% So now, if you load this-- 01:18:53.010 --> 01:18:57.720 align:middle line:84% given the initial wave function, if you know the propagator, 01:18:57.720 --> 01:18:59.670 align:middle line:90% you just need to do an integral. 01:18:59.670 --> 01:19:01.840 align:middle line:84% Then, you find the wave function here. 01:19:01.840 --> 01:19:04.980 align:middle line:84% So the full knowledge of your quantum mechanical system 01:19:04.980 --> 01:19:08.130 align:middle line:84% essentially is encoded in this propagator. 01:19:08.130 --> 01:19:11.190 align:middle line:90% 01:19:11.190 --> 01:19:13.320 align:middle line:90% It's encoded in this propagator. 01:19:13.320 --> 01:19:18.210 align:middle line:84% So yeah, now, we want to talk about what 01:19:18.210 --> 01:19:21.960 align:middle line:84% is the analog of this object, say, in relativistic quantum 01:19:21.960 --> 01:19:24.060 align:middle line:90% field theory? 01:19:24.060 --> 01:19:27.180 align:middle line:84% In quantum field theory, how do we define a propagator, 01:19:27.180 --> 01:19:32.280 align:middle line:84% and what is the analog of the position eigenstate, 01:19:32.280 --> 01:19:33.990 align:middle line:90% say, in quantum field theory? 01:19:33.990 --> 01:19:36.950 align:middle line:90% And we will discuss next time. 01:19:36.950 --> 01:19:42.000 align:middle line:90%