WEBVTT 00:00:00.000 --> 00:00:02.470 align:middle line:90% [SQUEAKING] 00:00:02.470 --> 00:00:03.952 align:middle line:90% [RUSTLING] 00:00:03.952 --> 00:00:07.410 align:middle line:90% [CLICKING] 00:00:07.410 --> 00:00:11.870 align:middle line:90% 00:00:11.870 --> 00:00:14.610 align:middle line:90% PROFESSOR: OK, good. 00:00:14.610 --> 00:00:20.370 align:middle line:84% So last time, we quantized this theory. 00:00:20.370 --> 00:00:25.850 align:middle line:84% So let me just write down the theory again. 00:00:25.850 --> 00:00:31.400 align:middle line:90% 00:00:31.400 --> 00:00:33.620 align:middle line:90% So we quantized this theory. 00:00:33.620 --> 00:00:36.590 align:middle line:84% And so let me also just write down 00:00:36.590 --> 00:00:41.360 align:middle line:84% its canonical momentum, density, and then the Hamiltonian 00:00:41.360 --> 00:00:45.830 align:middle line:90% density would be 1/2 pi square. 00:00:45.830 --> 00:01:00.700 align:middle line:90% 00:01:00.700 --> 00:01:02.350 align:middle line:84% And the classical equation of motion-- 00:01:02.350 --> 00:01:09.340 align:middle line:90% 00:01:09.340 --> 00:01:15.370 align:middle line:84% and so we're not repeat the quantization procedure. 00:01:15.370 --> 00:01:20.035 align:middle line:84% And we wrote down the most general solution at the level-- 00:01:20.035 --> 00:01:23.590 align:middle line:90% 00:01:23.590 --> 00:01:24.970 align:middle line:90% can be written as the following. 00:01:24.970 --> 00:01:33.670 align:middle line:90% 00:01:33.670 --> 00:01:36.020 align:middle line:90% This factor is the convention. 00:01:36.020 --> 00:01:45.730 align:middle line:84% And then we have a k u k plus a k. 00:01:45.730 --> 00:01:51.130 align:middle line:84% So from now on, I will suppress the hat. 00:01:51.130 --> 00:01:52.540 align:middle line:90% So I will suppress hat. 00:01:52.540 --> 00:01:55.360 align:middle line:84% You should always now view it as a quantum field-- 00:01:55.360 --> 00:01:56.950 align:middle line:90% quantum operators. 00:01:56.950 --> 00:02:02.380 align:middle line:90% And the plus u k star. 00:02:02.380 --> 00:02:07.580 align:middle line:84% And the u k is the complete set of solution, 00:02:07.580 --> 00:02:16.780 align:middle line:84% which is given by exponential minus i omega t plus i k x. 00:02:16.780 --> 00:02:19.480 align:middle line:90% 00:02:19.480 --> 00:02:23.810 align:middle line:84% So this is just a basis of solution. 00:02:23.810 --> 00:02:27.160 align:middle line:84% And so this is the most general-- 00:02:27.160 --> 00:02:31.990 align:middle line:84% so this is the complete solution to the operator equation 00:02:31.990 --> 00:02:35.350 align:middle line:90% for this operator phi. 00:02:35.350 --> 00:02:37.840 align:middle line:90% And then you can also find its-- 00:02:37.840 --> 00:02:41.000 align:middle line:84% find its conjugate momentum density. 00:02:41.000 --> 00:02:44.810 align:middle line:84% So the pi, you can just take the derivative straightforward. 00:02:44.810 --> 00:02:46.370 align:middle line:90% I will not write it explicitly. 00:02:46.370 --> 00:02:48.980 align:middle line:90% 00:02:48.980 --> 00:02:53.000 align:middle line:84% And then we discussed that you can 00:02:53.000 --> 00:02:57.890 align:middle line:84% impose a canonical commutation relation which 00:02:57.890 --> 00:03:12.140 align:middle line:84% is that the phi t x phi t x prime equal to pi t x, 00:03:12.140 --> 00:03:18.500 align:middle line:90% and pi t x prime should be 0. 00:03:18.500 --> 00:03:28.830 align:middle line:90% And then phi t x pi t x prime-- 00:03:28.830 --> 00:03:30.300 align:middle line:84% then given by the delta function. 00:03:30.300 --> 00:03:36.420 align:middle line:90% 00:03:36.420 --> 00:03:42.450 align:middle line:84% So that's what we did at the end of last lecture. 00:03:42.450 --> 00:03:45.750 align:middle line:84% And then you can just plug it in-- 00:03:45.750 --> 00:03:48.870 align:middle line:84% plug this-- plug those expression into here. 00:03:48.870 --> 00:03:50.670 align:middle line:84% Then you can find the commutation relations 00:03:50.670 --> 00:03:52.620 align:middle line:90% between those integrations. 00:03:52.620 --> 00:03:57.840 align:middle line:84% So a k and a k dagger integrating constant 00:03:57.840 --> 00:03:59.070 align:middle line:90% of your operator equations. 00:03:59.070 --> 00:04:01.560 align:middle line:90% They are constant operators. 00:04:01.560 --> 00:04:05.287 align:middle line:84% And so when you plug them in, then you find the canonical-- 00:04:05.287 --> 00:04:06.870 align:middle line:84% then you find the commutation relation 00:04:06.870 --> 00:04:08.710 align:middle line:90% between a k and a k dagger. 00:04:08.710 --> 00:04:22.070 align:middle line:84% And then you find ak and a k prime equal to a dagger 00:04:22.070 --> 00:04:30.140 align:middle line:84% k and a dagger k prime is equal to 0 and a k 00:04:30.140 --> 00:04:42.375 align:middle line:84% a k prime dagger is equal to delta function in the wave 00:04:42.375 --> 00:04:42.875 align:middle line:90% space. 00:04:42.875 --> 00:04:52.740 align:middle line:90% 00:04:52.740 --> 00:04:55.570 align:middle line:84% So now, if you look at those expressions-- 00:04:55.570 --> 00:04:57.930 align:middle line:90% so they essentially look-- 00:04:57.930 --> 00:04:59.820 align:middle line:84% they essentially-- we essentially 00:04:59.820 --> 00:05:03.810 align:middle line:84% find the infinite number of harmonic oscillators. 00:05:03.810 --> 00:05:10.860 align:middle line:84% And each harmonic oscillator labeled by continuous-- 00:05:10.860 --> 00:05:13.740 align:middle line:90% by a set of continuous number k. 00:05:13.740 --> 00:05:15.870 align:middle line:84% And then the commutation relation 00:05:15.870 --> 00:05:19.270 align:middle line:84% is just like a continuum generalization of standard 00:05:19.270 --> 00:05:21.660 align:middle line:90% a a dagger equal to 1. 00:05:21.660 --> 00:05:24.330 align:middle line:84% And they are independent because of this delta function. 00:05:24.330 --> 00:05:28.543 align:middle line:84% And if k is not equal to k prime, and then you get 0. 00:05:28.543 --> 00:05:29.085 align:middle line:90% They commute. 00:05:29.085 --> 00:05:33.520 align:middle line:90% 00:05:33.520 --> 00:05:38.950 align:middle line:84% So you can see the story a little bit sharper. 00:05:38.950 --> 00:05:41.260 align:middle line:84% That indeed, we just get the infinite number 00:05:41.260 --> 00:05:43.190 align:middle line:90% of harmonic oscillators. 00:05:43.190 --> 00:05:45.640 align:middle line:84% So these are just commutation relations. 00:05:45.640 --> 00:05:49.175 align:middle line:84% This still does not tell you we have a harmonic oscillator. 00:05:49.175 --> 00:05:50.800 align:middle line:84% To see harmonic oscillator, we actually 00:05:50.800 --> 00:05:53.470 align:middle line:84% need to calculate the Hamiltonian. 00:05:53.470 --> 00:05:56.560 align:middle line:84% So now let's try to calculate the Hamiltonian. 00:05:56.560 --> 00:06:03.680 align:middle line:90% 00:06:03.680 --> 00:06:07.310 align:middle line:84% So now we can try to calculate the Hamiltonian. 00:06:07.310 --> 00:06:12.590 align:middle line:90% 00:06:12.590 --> 00:06:15.260 align:middle line:90% And so this is just-- 00:06:15.260 --> 00:06:16.430 align:middle line:90% can just do it-- 00:06:16.430 --> 00:06:19.460 align:middle line:90% 00:06:19.460 --> 00:06:23.540 align:middle line:84% straightforwardly just plug that in-- 00:06:23.540 --> 00:06:28.370 align:middle line:84% just plot it plug that expression in we wrote above. 00:06:28.370 --> 00:06:30.230 align:middle line:84% And then you just plug-in the expression 00:06:30.230 --> 00:06:37.190 align:middle line:84% for the phi and the pi into that equation, 00:06:37.190 --> 00:06:40.640 align:middle line:84% and then just straightforward to calculate it. 00:06:40.640 --> 00:06:43.610 align:middle line:84% And I will not go through the details. 00:06:43.610 --> 00:06:49.380 align:middle line:84% Of Yeah, it's a straightforward calculation. 00:06:49.380 --> 00:06:55.490 align:middle line:84% So then you find maybe after some minutes-- 00:06:55.490 --> 00:06:59.000 align:middle line:90% 00:06:59.000 --> 00:07:02.320 align:middle line:84% then you find that the following answer. 00:07:02.320 --> 00:07:04.537 align:middle line:84% So now, you-- when you plug them in, 00:07:04.537 --> 00:07:06.120 align:middle line:84% then you find that you can do actually 00:07:06.120 --> 00:07:09.900 align:middle line:84% do the spatial integral because you just have the plane wave. 00:07:09.900 --> 00:07:11.560 align:middle line:84% You can just do the spatial integral. 00:07:11.560 --> 00:07:13.560 align:middle line:84% And when you do the spatial integral, 00:07:13.560 --> 00:07:16.210 align:middle line:84% then that gives you a delta function momentum space, 00:07:16.210 --> 00:07:16.870 align:middle line:90% et cetera. 00:07:16.870 --> 00:07:18.290 align:middle line:84% And then you can reduce everything 00:07:18.290 --> 00:07:19.410 align:middle line:90% into a single moment-- 00:07:19.410 --> 00:07:21.870 align:middle line:90% a single k integral. 00:07:21.870 --> 00:07:27.570 align:middle line:84% And then what you find is that you get omega k. 00:07:27.570 --> 00:07:35.130 align:middle line:84% And then you find a k dagger a k. 00:07:35.130 --> 00:07:45.230 align:middle line:90% And then plus a k dagger. 00:07:45.230 --> 00:07:46.190 align:middle line:90% So let me see. 00:07:46.190 --> 00:07:50.380 align:middle line:84% Yeah, actually-- yeah, I think 1/2. 00:07:50.380 --> 00:07:53.280 align:middle line:90% 00:07:53.280 --> 00:07:56.280 align:middle line:90% Just get that. 00:07:56.280 --> 00:07:58.470 align:middle line:84% So this is the standard expression 00:07:58.470 --> 00:08:01.290 align:middle line:90% for harmonic oscillator now. 00:08:01.290 --> 00:08:06.780 align:middle line:84% Actually, it's exactly identical to just some-- 00:08:06.780 --> 00:08:10.620 align:middle line:84% yeah, integral is like sum-- just sum over continuum 00:08:10.620 --> 00:08:12.150 align:middle line:90% harmonic oscillators. 00:08:12.150 --> 00:08:16.620 align:middle line:84% And each with frequency omega k, and each harmonic oscillator 00:08:16.620 --> 00:08:19.920 align:middle line:90% is labeled by this number k. 00:08:19.920 --> 00:08:24.540 align:middle line:84% And so we can do from the standard trick 00:08:24.540 --> 00:08:30.330 align:middle line:84% to write it actually as to write as a commutator. 00:08:30.330 --> 00:08:32.280 align:middle line:90% So we can just write it as-- 00:08:32.280 --> 00:08:41.210 align:middle line:90% 00:08:41.210 --> 00:08:44.340 align:middle line:90% and then we just have this. 00:08:44.340 --> 00:08:46.190 align:middle line:84% So we combine these two terms together 00:08:46.190 --> 00:08:48.230 align:middle line:90% and introduce a commutator. 00:08:48.230 --> 00:08:52.610 align:middle line:84% You just have a dagger a and then plus the commutator 00:08:52.610 --> 00:08:53.240 align:middle line:90% between them. 00:08:53.240 --> 00:09:03.090 align:middle line:90% 00:09:03.090 --> 00:09:07.980 align:middle line:84% So in the standard story, this is equal to 1. 00:09:07.980 --> 00:09:13.290 align:middle line:84% Then you get the standard answer that you get omega a dagger 00:09:13.290 --> 00:09:16.290 align:middle line:90% a plus 1/2. 00:09:16.290 --> 00:09:21.390 align:middle line:90% But here, we can again-- 00:09:21.390 --> 00:09:23.470 align:middle line:90% let me just write one more step. 00:09:23.470 --> 00:09:38.320 align:middle line:84% So this is k omega k a k dagger a k, then plus E0. 00:09:38.320 --> 00:09:42.250 align:middle line:84% And now E0 is the sum of all the zero point energy of all 00:09:42.250 --> 00:09:43.810 align:middle line:90% the harmonic oscillators. 00:09:43.810 --> 00:09:52.850 align:middle line:90% 00:09:52.850 --> 00:09:57.770 align:middle line:90% So 1/2 omega k-- 00:09:57.770 --> 00:10:01.880 align:middle line:84% except that the tricky thing is the following. 00:10:01.880 --> 00:10:03.690 align:middle line:84% Except now, we have this commutator 00:10:03.690 --> 00:10:05.300 align:middle line:90% of a k with a k dagger. 00:10:05.300 --> 00:10:07.760 align:middle line:84% In the standard case, this is equal to 1. 00:10:07.760 --> 00:10:14.570 align:middle line:84% But in our case, when you set k prime equal to k, 00:10:14.570 --> 00:10:16.670 align:middle line:90% you get the delta 0. 00:10:16.670 --> 00:10:18.230 align:middle line:90% So we get delta 0. 00:10:18.230 --> 00:10:25.760 align:middle line:84% So then you get that 2 pi cubed delta 3 0. 00:10:25.760 --> 00:10:30.170 align:middle line:84% And remember, this 0 is in momentum space. 00:10:30.170 --> 00:10:32.690 align:middle line:84% And so this is in k space, not in the-- 00:10:32.690 --> 00:10:34.430 align:middle line:84% we have two kinds of delta functions 00:10:34.430 --> 00:10:37.030 align:middle line:84% in the-- delta function in k space, delta function 00:10:37.030 --> 00:10:38.340 align:middle line:90% in coordinate space. 00:10:38.340 --> 00:10:40.580 align:middle line:84% So here, I put the subscript k-- here 00:10:40.580 --> 00:10:46.490 align:middle line:84% means that this is the delta function 0 in k space. 00:10:46.490 --> 00:10:48.380 align:middle line:84% And so this is the zero point energy. 00:10:48.380 --> 00:10:59.430 align:middle line:90% 00:10:59.430 --> 00:11:06.030 align:middle line:84% So this zero point energy is divergent, 00:11:06.030 --> 00:11:09.758 align:middle line:84% but we will comment on that a little bit later. 00:11:09.758 --> 00:11:11.175 align:middle line:84% Do you have any questions on this? 00:11:11.175 --> 00:11:16.980 align:middle line:90% 00:11:16.980 --> 00:11:17.480 align:middle line:90% Good. 00:11:17.480 --> 00:11:25.630 align:middle line:84% So remark-- let me just make a couple of remarks. 00:11:25.630 --> 00:11:32.220 align:middle line:84% First is that-- so something interesting happened. 00:11:32.220 --> 00:11:36.060 align:middle line:90% So when you plug-- 00:11:36.060 --> 00:11:41.220 align:middle line:84% when you look at the expression for phi and the pi, 00:11:41.220 --> 00:11:44.820 align:middle line:84% both phi and the pi depend on time explicitly. 00:11:44.820 --> 00:11:47.610 align:middle line:84% So there's a time dependence here. 00:11:47.610 --> 00:11:51.030 align:middle line:84% And here, we only integrate over spatial direction. 00:11:51.030 --> 00:11:54.480 align:middle line:90% 00:11:54.480 --> 00:11:58.470 align:middle line:84% So in terms of integration, we don't do anything with time. 00:11:58.470 --> 00:12:01.860 align:middle line:90% But what do you notice here? 00:12:01.860 --> 00:12:04.930 align:middle line:90% 00:12:04.930 --> 00:12:06.030 align:middle line:90% What do you notice there? 00:12:06.030 --> 00:12:08.760 align:middle line:90% 00:12:08.760 --> 00:12:10.710 align:middle line:84% STUDENT: So you drop the time dependence? 00:12:10.710 --> 00:12:13.030 align:middle line:84% PROFESSOR: Yes, it's not to say we drop time dependence 00:12:13.030 --> 00:12:16.608 align:middle line:84% because time dependence disappeared. 00:12:16.608 --> 00:12:17.275 align:middle line:90% Do you know why? 00:12:17.275 --> 00:12:21.805 align:middle line:90% 00:12:21.805 --> 00:12:23.930 align:middle line:84% STUDENT: I mean, the Hamiltonian doesn't explicitly 00:12:23.930 --> 00:12:26.030 align:middle line:90% depend on time. 00:12:26.030 --> 00:12:27.880 align:middle line:90% PROFESSOR: No, the Hamiltonian-- 00:12:27.880 --> 00:12:30.790 align:middle line:84% phi is time dependent, pi is time dependent. 00:12:30.790 --> 00:12:37.540 align:middle line:84% If you plug-- if you plug those expression into here, 00:12:37.540 --> 00:12:39.370 align:middle line:90% certainly this-- 00:12:39.370 --> 00:12:43.660 align:middle line:84% certainly inside there is depend on time explicitly. 00:12:43.660 --> 00:12:46.060 align:middle line:84% This guy is depend on time explicitly. 00:12:46.060 --> 00:12:48.637 align:middle line:90% 00:12:48.637 --> 00:12:50.470 align:middle line:84% But in the end, after you do the integration 00:12:50.470 --> 00:12:52.630 align:middle line:84% over spatial directions, then you 00:12:52.630 --> 00:12:54.550 align:middle line:84% find in the end the time dependence actually 00:12:54.550 --> 00:12:58.300 align:middle line:90% cancel during the calculation. 00:12:58.300 --> 00:13:01.930 align:middle line:84% So do you have a guess why somehow this should cancel? 00:13:01.930 --> 00:13:02.500 align:middle line:90% Yes? 00:13:02.500 --> 00:13:03.850 align:middle line:84% STUDENT: Energy conservation or something? 00:13:03.850 --> 00:13:05.140 align:middle line:90% PROFESSOR: Yeah, exactly. 00:13:05.140 --> 00:13:10.220 align:middle line:84% So you have in your pset that the time translation-- 00:13:10.220 --> 00:13:14.950 align:middle line:84% so this system is time translation symmetric. 00:13:14.950 --> 00:13:16.870 align:middle line:84% It means that you should have the-- 00:13:16.870 --> 00:13:21.850 align:middle line:84% it means that the H is a conserved number. 00:13:21.850 --> 00:13:23.780 align:middle line:90% It's a conserved quantity. 00:13:23.780 --> 00:13:29.560 align:middle line:84% And by-- so indeed, you see that this is independent of time. 00:13:29.560 --> 00:13:31.060 align:middle line:90% This is a conserved quantity. 00:13:31.060 --> 00:13:33.220 align:middle line:90% We see it explicitly. 00:13:33.220 --> 00:13:44.410 align:middle line:84% So there's no time dependence since H 00:13:44.410 --> 00:13:45.950 align:middle line:90% is a conserved quantity. 00:13:45.950 --> 00:13:50.900 align:middle line:84% So this integration is absolutely crucial, 00:13:50.900 --> 00:13:55.010 align:middle line:84% because if you don't do integration, 00:13:55.010 --> 00:13:56.990 align:middle line:90% Hamiltonian density is not-- 00:13:56.990 --> 00:13:59.430 align:middle line:84% it's only the total Hamiltonian is conserved. 00:13:59.430 --> 00:14:02.120 align:middle line:90% 00:14:02.120 --> 00:14:03.125 align:middle line:90% So this is a comment. 00:14:03.125 --> 00:14:06.610 align:middle line:90% 00:14:06.610 --> 00:14:10.190 align:middle line:90% And the second -- 00:14:10.190 --> 00:14:11.920 align:middle line:84% so since I only have two comments, 00:14:11.920 --> 00:14:15.160 align:middle line:84% let me just call it one rather than call it 0. 00:14:15.160 --> 00:14:19.270 align:middle line:84% So for the second comment is that-- 00:14:19.270 --> 00:14:20.290 align:middle line:90% yeah, just repeat. 00:14:20.290 --> 00:14:26.820 align:middle line:84% We already said-- you just have OFT of phi. 00:14:26.820 --> 00:14:31.060 align:middle line:84% And we see this QFT of phi reduce this 00:14:31.060 --> 00:14:44.050 align:middle line:84% to essentially a system of a continuum 00:14:44.050 --> 00:15:08.700 align:middle line:84% of harmonic oscillators labeled by k with frequency omega k. 00:15:08.700 --> 00:15:10.440 align:middle line:84% So we call this omega k is defined 00:15:10.440 --> 00:15:13.230 align:middle line:90% to be k squared plus m square. 00:15:13.230 --> 00:15:22.990 align:middle line:90% 00:15:22.990 --> 00:15:28.480 align:middle line:84% So as we mentioned last time, the fact 00:15:28.480 --> 00:15:31.360 align:middle line:84% that we actually see infinite number of harmonic oscillator 00:15:31.360 --> 00:15:35.350 align:middle line:84% is actually not surprising from the perspective 00:15:35.350 --> 00:15:40.300 align:middle line:84% that this field theory can be written as the continuum 00:15:40.300 --> 00:15:42.460 align:middle line:90% limit of a chain-- 00:15:42.460 --> 00:15:46.600 align:middle line:84% say, of atoms connected by springs. 00:15:46.600 --> 00:15:50.470 align:middle line:84% So when you have a chain, we wrote down 00:15:50.470 --> 00:15:52.390 align:middle line:90% a one-dimensional system. 00:15:52.390 --> 00:15:55.480 align:middle line:84% You can easily generalize to three-dimensional. 00:15:55.480 --> 00:16:01.270 align:middle line:84% Yeah, so you have those chains of atoms connected by spring, 00:16:01.270 --> 00:16:03.520 align:middle line:84% and they are just coupled oscillators. 00:16:03.520 --> 00:16:06.898 align:middle line:84% And then you can just diagonalize them, and find 00:16:06.898 --> 00:16:08.440 align:middle line:84% the normal spectrum, and each of them 00:16:08.440 --> 00:16:10.340 align:middle line:90% is just a harmonic oscillator. 00:16:10.340 --> 00:16:13.900 align:middle line:84% And so when we find those solution by doing Fourier 00:16:13.900 --> 00:16:15.040 align:middle line:90% transform-- 00:16:15.040 --> 00:16:18.700 align:middle line:84% and essentially, we are just diagonalize those interactions 00:16:18.700 --> 00:16:22.360 align:middle line:90% between those springs. 00:16:22.360 --> 00:16:26.180 align:middle line:84% And yeah, so that's why we get infinite harmonic oscillator. 00:16:26.180 --> 00:16:30.190 align:middle line:84% So from that way, if you think about it, 00:16:30.190 --> 00:16:32.620 align:middle line:84% it's totally unsurprising we get the infinite number 00:16:32.620 --> 00:16:34.240 align:middle line:90% of harmonic oscillators. 00:16:34.240 --> 00:16:36.988 align:middle line:84% But it does surprising-- it is surprising 00:16:36.988 --> 00:16:39.280 align:middle line:84% if you think about it from the point of view of a field 00:16:39.280 --> 00:16:40.210 align:middle line:90% theory. 00:16:40.210 --> 00:16:43.150 align:middle line:84% So if you find when you quantize a field theory, 00:16:43.150 --> 00:16:44.680 align:middle line:84% and then you find in the end you get 00:16:44.680 --> 00:16:47.080 align:middle line:90% a bunch of harmonic oscillator. 00:16:47.080 --> 00:16:52.480 align:middle line:84% And before doing that, it's hard to-- 00:16:52.480 --> 00:16:55.060 align:middle line:84% before you actually carry out the quantization, 00:16:55.060 --> 00:16:58.240 align:middle line:84% it's actually hard to anticipate that if you 00:16:58.240 --> 00:17:01.195 align:middle line:84% don't have this intuition from this discrete system. 00:17:01.195 --> 00:17:03.713 align:middle line:90% 00:17:03.713 --> 00:17:05.130 align:middle line:84% Do you have any questions on this? 00:17:05.130 --> 00:17:06.060 align:middle line:90% Yes? 00:17:06.060 --> 00:17:06.930 align:middle line:84% STUDENT: So when you say that there's 00:17:06.930 --> 00:17:08.762 align:middle line:84% an infinite amount of harmonic oscillators, 00:17:08.762 --> 00:17:11.220 align:middle line:84% are you saying that there's an infinite amount at any given 00:17:11.220 --> 00:17:13.230 align:middle line:84% point in space, or that each given 00:17:13.230 --> 00:17:15.099 align:middle line:84% point in a continuum of location there 00:17:15.099 --> 00:17:16.530 align:middle line:90% is one harmonic oscillator? 00:17:16.530 --> 00:17:21.730 align:middle line:84% PROFESSOR: Yeah, that's a very good question. 00:17:21.730 --> 00:17:27.390 align:middle line:84% And so it's at each point, you have a harmonic oscillator. 00:17:27.390 --> 00:17:29.010 align:middle line:84% Yeah, at each point-- essentially, 00:17:29.010 --> 00:17:32.280 align:middle line:84% at each point you have a harmonic oscillator, 00:17:32.280 --> 00:17:34.590 align:middle line:84% and then they connected in space. 00:17:34.590 --> 00:17:37.780 align:middle line:90% 00:17:37.780 --> 00:17:38.500 align:middle line:90% Yes? 00:17:38.500 --> 00:17:40.292 align:middle line:84% STUDENT: Is there any reason why you didn't 00:17:40.292 --> 00:17:43.570 align:middle line:90% compute the integral in E0? 00:17:43.570 --> 00:17:46.240 align:middle line:84% You have a delta function in there? 00:17:46.240 --> 00:17:47.620 align:middle line:90% PROFESSOR: Sorry, say it again? 00:17:47.620 --> 00:17:49.840 align:middle line:84% STUDENT: So the integral if E0 equals. 00:17:49.840 --> 00:17:51.130 align:middle line:90% Why didn't you just-- 00:17:51.130 --> 00:17:53.710 align:middle line:84% PROFESSOR: Oh, this-- we cannot do the integral. 00:17:53.710 --> 00:17:54.820 align:middle line:90% STUDENT: No, up there-- 00:17:54.820 --> 00:17:55.900 align:middle line:90% the zero point energy. 00:17:55.900 --> 00:17:56.775 align:middle line:90% PROFESSOR: Oh, right. 00:17:56.775 --> 00:18:01.030 align:middle line:84% Yeah, we will talk a little bit later, 00:18:01.030 --> 00:18:03.790 align:middle line:84% because later, we are going to elaborate 00:18:03.790 --> 00:18:06.880 align:middle line:84% on this a little bit later because we are going to try 00:18:06.880 --> 00:18:10.110 align:middle line:90% to give interpretation of this. 00:18:10.110 --> 00:18:11.250 align:middle line:90% Yes? 00:18:11.250 --> 00:18:15.750 align:middle line:84% STUDENT: In homework 1, we found that a hat was time dependent. 00:18:15.750 --> 00:18:19.530 align:middle line:84% Is a hat not time dependent here, 00:18:19.530 --> 00:18:22.860 align:middle line:84% and is that why the Hamiltonian is not time dependent? 00:18:22.860 --> 00:18:25.860 align:middle line:84% PROFESSOR: Yeah, a hat is a constant operator. 00:18:25.860 --> 00:18:31.290 align:middle line:84% Yeah, because a hat is an integration constant 00:18:31.290 --> 00:18:32.790 align:middle line:90% of your operator equation. 00:18:32.790 --> 00:18:36.960 align:middle line:90% 00:18:36.960 --> 00:18:37.593 align:middle line:90% Yes? 00:18:37.593 --> 00:18:39.510 align:middle line:84% STUDENT: Is there an analog of coherent states 00:18:39.510 --> 00:18:41.880 align:middle line:84% for each harmonic oscillator in the QFT? 00:18:41.880 --> 00:18:45.030 align:middle line:84% PROFESSOR: Yeah, there's an analog of coherent state. 00:18:45.030 --> 00:18:45.750 align:middle line:90% Yeah, indeed. 00:18:45.750 --> 00:18:46.662 align:middle line:84% STUDENT: Does it tell you anything useful if you are 00:18:46.662 --> 00:18:48.150 align:middle line:90% in Heisenberg interpretation. 00:18:48.150 --> 00:18:50.430 align:middle line:90% PROFESSOR: Yeah, you can-- 00:18:50.430 --> 00:18:54.750 align:middle line:90% yeah, we will-- you will see-- 00:18:54.750 --> 00:18:56.460 align:middle line:84% yeah, this question will become clearer 00:18:56.460 --> 00:18:58.543 align:middle line:84% when we talk a little bit about the Hilbert space. 00:18:58.543 --> 00:19:02.430 align:middle line:90% 00:19:02.430 --> 00:19:03.420 align:middle line:90% Other questions? 00:19:03.420 --> 00:19:07.010 align:middle line:90% 00:19:07.010 --> 00:19:10.720 align:middle line:90% OK, good. 00:19:10.720 --> 00:19:16.150 align:middle line:84% Now, let's-- before we talk about the Hilbert space, 00:19:16.150 --> 00:19:18.505 align:middle line:84% we need to talk about two things. 00:19:18.505 --> 00:19:22.610 align:middle line:90% 00:19:22.610 --> 00:19:26.500 align:middle line:84% So here-- yeah, here, I have a dot. 00:19:26.500 --> 00:19:28.100 align:middle line:90% And the second thing-- 00:19:28.100 --> 00:19:31.460 align:middle line:84% so Hamiltonian is an important quantity. 00:19:31.460 --> 00:19:34.700 align:middle line:84% So this is one of the conserved quantity. 00:19:34.700 --> 00:19:36.760 align:middle line:84% So there are other conserved quantities. 00:19:36.760 --> 00:19:48.350 align:middle line:90% 00:19:48.350 --> 00:19:52.330 align:middle line:84% So there are other conserved quantities, and one of them 00:19:52.330 --> 00:19:54.100 align:middle line:84% is the conserved quantity corresponding 00:19:54.100 --> 00:19:56.860 align:middle line:90% to spatial translation. 00:19:56.860 --> 00:20:01.750 align:middle line:84% So the spatial translation gives you conserved momentum-- 00:20:01.750 --> 00:20:05.210 align:middle line:90% momentum conservation. 00:20:05.210 --> 00:20:17.020 align:middle line:84% So from spatial translation, then you 00:20:17.020 --> 00:20:20.776 align:middle line:90% get this conserved charge p i. 00:20:20.776 --> 00:20:23.740 align:middle line:90% 00:20:23.740 --> 00:20:26.920 align:middle line:84% Again, you should have done this in your homework 00:20:26.920 --> 00:20:28.300 align:middle line:90% to have the following form-- 00:20:28.300 --> 00:20:31.210 align:middle line:90% 00:20:31.210 --> 00:20:33.410 align:middle line:90% d3 x pi. 00:20:33.410 --> 00:20:45.370 align:middle line:90% 00:20:45.370 --> 00:20:50.590 align:middle line:84% So this is the Noether charge for the conserved quantity 00:20:50.590 --> 00:20:53.470 align:middle line:84% associated with the spatial translation. 00:20:53.470 --> 00:20:58.290 align:middle line:84% And we interpret it as a spacetime momentum. 00:20:58.290 --> 00:21:00.590 align:middle line:84% This should be interpreted as spacetime momentum. 00:21:00.590 --> 00:21:04.440 align:middle line:90% 00:21:04.440 --> 00:21:07.580 align:middle line:84% So this is-- so here, we have two momentum here. 00:21:07.580 --> 00:21:12.420 align:middle line:90% You don't confuse them. 00:21:12.420 --> 00:21:18.350 align:middle line:84% So this pi is the canonical momentum conjugate 00:21:18.350 --> 00:21:20.090 align:middle line:90% to this canonical variable-- 00:21:20.090 --> 00:21:23.630 align:middle line:90% 00:21:23.630 --> 00:21:26.270 align:middle line:84% pi is the canonical momentum conjugate 00:21:26.270 --> 00:21:30.500 align:middle line:90% to this field variable phi. 00:21:30.500 --> 00:21:35.270 align:middle line:84% And this is the genuine physical spacetime momentum. 00:21:35.270 --> 00:21:37.760 align:middle line:84% It's the momentum of your full system-- 00:21:37.760 --> 00:21:39.350 align:middle line:90% of your full physical system. 00:21:39.350 --> 00:21:42.300 align:middle line:90% 00:21:42.300 --> 00:21:46.650 align:middle line:84% And again, we can just plug it in-- 00:21:46.650 --> 00:21:49.980 align:middle line:84% we can just plug in the explicit expression of pi and phi. 00:21:49.980 --> 00:21:53.340 align:middle line:84% We already know its time evolution. 00:21:53.340 --> 00:21:55.080 align:middle line:84% We already know the time evolution, 00:21:55.080 --> 00:21:58.120 align:middle line:90% and then we can plug them in. 00:21:58.120 --> 00:22:04.270 align:middle line:84% And then you find that the answer is given by-- 00:22:04.270 --> 00:22:10.410 align:middle line:84% again, after some smaller number of minutes calculation. 00:22:10.410 --> 00:22:12.890 align:middle line:84% And then you find that the answer is given by this. 00:22:12.890 --> 00:22:17.390 align:middle line:90% 00:22:17.390 --> 00:22:19.680 align:middle line:84% In this case, there is no zero point energy. 00:22:19.680 --> 00:22:21.230 align:middle line:90% You just have this expression. 00:22:21.230 --> 00:22:30.540 align:middle line:90% 00:22:30.540 --> 00:22:35.640 align:middle line:84% And furthermore, you also have a Lorentz transformation-- 00:22:35.640 --> 00:22:36.435 align:middle line:90% Lorentz symmetry. 00:22:36.435 --> 00:22:42.120 align:middle line:90% 00:22:42.120 --> 00:22:45.900 align:middle line:84% And then that will lead to, say, the conserved charge associated 00:22:45.900 --> 00:22:48.150 align:middle line:90% with Lorentz transformation. 00:22:48.150 --> 00:22:51.600 align:middle line:84% And again, you can find it explicitly. 00:22:51.600 --> 00:22:57.045 align:middle line:84% And so in your pset 2, you will find the expression of this M 00:22:57.045 --> 00:22:59.280 align:middle line:90% mu nu in terms of phi and pi. 00:22:59.280 --> 00:23:02.880 align:middle line:90% 00:23:02.880 --> 00:23:04.830 align:middle line:84% And then you can again express this guy 00:23:04.830 --> 00:23:07.500 align:middle line:90% in terms of a and a dagger. 00:23:07.500 --> 00:23:12.490 align:middle line:84% But I didn't have the guts to assign as the part of the pset, 00:23:12.490 --> 00:23:15.420 align:middle line:84% so I used it as a bonus problem because that calculation-- 00:23:15.420 --> 00:23:17.340 align:middle line:84% involved a little bit extra calculation. 00:23:17.340 --> 00:23:20.760 align:middle line:84% Yeah, it involves a little bit slightly more tedious 00:23:20.760 --> 00:23:24.030 align:middle line:84% calculation, so I used that as a bonus problem 00:23:24.030 --> 00:23:29.010 align:middle line:84% for those people who like to have some more fun. 00:23:29.010 --> 00:23:33.690 align:middle line:90% Yeah, anyway, so this you can-- 00:23:33.690 --> 00:23:35.180 align:middle line:90% will see in your pset. 00:23:35.180 --> 00:23:37.920 align:middle line:90% 00:23:37.920 --> 00:23:38.870 align:middle line:90% Good. 00:23:38.870 --> 00:23:43.280 align:middle line:84% And then again, pi, you see the explicitly 00:23:43.280 --> 00:23:46.490 align:middle line:90% that this is time independent-- 00:23:46.490 --> 00:23:48.086 align:middle line:90% it's time independent. 00:23:48.086 --> 00:23:53.440 align:middle line:90% 00:23:53.440 --> 00:23:54.820 align:middle line:90% Good. 00:23:54.820 --> 00:23:57.460 align:middle line:84% And then the next thing is let's talk 00:23:57.460 --> 00:23:58.690 align:middle line:90% about this zero point energy. 00:23:58.690 --> 00:24:02.290 align:middle line:90% 00:24:02.290 --> 00:24:05.730 align:middle line:84% So this zero point energy, we can write it as the following. 00:24:05.730 --> 00:24:08.260 align:middle line:90% 00:24:08.260 --> 00:24:12.730 align:middle line:84% So we can-- yeah, so this is a constant. 00:24:12.730 --> 00:24:15.120 align:middle line:84% So you can take it outside of this k integral. 00:24:15.120 --> 00:24:18.510 align:middle line:84% This does not depend on k because the-- 00:24:18.510 --> 00:24:21.030 align:middle line:90% yeah. 00:24:21.030 --> 00:24:27.360 align:middle line:84% So this is just 2 pi cube delta 3 0. 00:24:27.360 --> 00:24:31.470 align:middle line:84% So this is-- remember, this is a k space delta function. 00:24:31.470 --> 00:24:33.390 align:middle line:90% And then you have the d3 k. 00:24:33.390 --> 00:24:43.650 align:middle line:90% 00:24:43.650 --> 00:24:45.660 align:middle line:90% So that's the expression we get. 00:24:45.660 --> 00:24:50.750 align:middle line:90% 00:24:50.750 --> 00:24:52.510 align:middle line:84% So now, let's try to understand what's 00:24:52.510 --> 00:24:53.680 align:middle line:90% the meaning of this term? 00:24:53.680 --> 00:24:56.860 align:middle line:90% 00:24:56.860 --> 00:24:59.080 align:middle line:84% Let's try to understand the meaning of this term. 00:24:59.080 --> 00:25:00.705 align:middle line:84% To understand the meaning of that term, 00:25:00.705 --> 00:25:03.550 align:middle line:84% we need to do a little bit of a mathematical trick. 00:25:03.550 --> 00:25:06.460 align:middle line:84% We need to a little bit of a mathematical trick. 00:25:06.460 --> 00:25:12.740 align:middle line:84% So remember the definition of delta function, 00:25:12.740 --> 00:25:23.540 align:middle line:84% we have 2 pi cubed delta 3k equal to the Fourier transform 00:25:23.540 --> 00:25:26.400 align:middle line:90% of the exponential of i k x. 00:25:26.400 --> 00:25:28.773 align:middle line:84% And so this is the definition of-- essentially, 00:25:28.773 --> 00:25:30.315 align:middle line:84% the definition of the delta function. 00:25:30.315 --> 00:25:35.730 align:middle line:90% 00:25:35.730 --> 00:25:39.900 align:middle line:84% And now, here, we want to set k equal to 0. 00:25:39.900 --> 00:25:42.670 align:middle line:90% So let's set k equal to 0 here. 00:25:42.670 --> 00:25:47.910 align:middle line:84% And then we find that these 2 pi cubed theta 0-- 00:25:47.910 --> 00:25:51.450 align:middle line:84% so let's remember this again in the k space. 00:25:51.450 --> 00:25:54.540 align:middle line:84% And then we just set k equal to 0 here. 00:25:54.540 --> 00:25:56.070 align:middle line:90% And then what is this? 00:25:56.070 --> 00:25:59.040 align:middle line:84% If you set k equal to 0, you get what this? 00:25:59.040 --> 00:25:59.730 align:middle line:90% Yes? 00:25:59.730 --> 00:26:00.390 align:middle line:90% STUDENT: The volume-- 00:26:00.390 --> 00:26:01.230 align:middle line:90% PROFESSOR: Yeah, that's right. 00:26:01.230 --> 00:26:01.860 align:middle line:90% STUDENT: --of the full space. 00:26:01.860 --> 00:26:04.068 align:middle line:84% PROFESSOR: You just get the volume of the full space. 00:26:04.068 --> 00:26:07.590 align:middle line:90% 00:26:07.590 --> 00:26:10.080 align:middle line:90% Let's just imagine you-- 00:26:10.080 --> 00:26:12.150 align:middle line:84% in order to make sense of this quantity, 00:26:12.150 --> 00:26:15.990 align:middle line:84% you can put the whole universe in a big box. 00:26:15.990 --> 00:26:18.030 align:middle line:84% Imagine the whole universe is in a big box, 00:26:18.030 --> 00:26:21.570 align:middle line:84% and then this is the total volume. 00:26:21.570 --> 00:26:24.877 align:middle line:84% And now, since this is the volume, 00:26:24.877 --> 00:26:26.460 align:middle line:84% now we have a very good interpretation 00:26:26.460 --> 00:26:28.510 align:middle line:90% of this quantity. 00:26:28.510 --> 00:26:34.830 align:middle line:84% And now we can write E0 as merely volume times epsilon 0. 00:26:34.830 --> 00:26:36.723 align:middle line:84% And then this quantity epsilon 0 then 00:26:36.723 --> 00:26:38.640 align:middle line:84% have the interpretation of the energy density. 00:26:38.640 --> 00:26:46.610 align:middle line:90% 00:26:46.610 --> 00:26:47.870 align:middle line:90% So now this is energy density. 00:26:47.870 --> 00:26:56.000 align:middle line:90% 00:26:56.000 --> 00:26:59.910 align:middle line:84% So this energy density, we can write it in one more step-- 00:26:59.910 --> 00:27:02.270 align:middle line:90% so continue to here. 00:27:02.270 --> 00:27:04.490 align:middle line:84% So we can write it one more step. 00:27:04.490 --> 00:27:08.120 align:middle line:84% Plug in the expression value of omega k. 00:27:08.120 --> 00:27:09.710 align:middle line:90% So this is the-- 00:27:09.710 --> 00:27:10.460 align:middle line:90% just 1/2. 00:27:10.460 --> 00:27:13.220 align:middle line:90% 00:27:13.220 --> 00:27:23.400 align:middle line:90% Sorry, I forgot 2 pi d k. 00:27:23.400 --> 00:27:25.470 align:middle line:90% And then you have k square. 00:27:25.470 --> 00:27:28.680 align:middle line:90% 00:27:28.680 --> 00:27:30.255 align:middle line:90% Yeah, so that's what you get. 00:27:30.255 --> 00:27:32.940 align:middle line:90% 00:27:32.940 --> 00:27:34.350 align:middle line:84% So how do you like this integral? 00:27:34.350 --> 00:27:37.790 align:middle line:90% 00:27:37.790 --> 00:27:41.390 align:middle line:84% STUDENT: So just a question to go back here. 00:27:41.390 --> 00:27:44.060 align:middle line:84% Wouldn't it be quicker if you just left the delta function 00:27:44.060 --> 00:27:46.550 align:middle line:84% inside of your integral and just say 00:27:46.550 --> 00:27:48.560 align:middle line:90% that it's one for your entire-- 00:27:48.560 --> 00:27:50.180 align:middle line:90% like for all k. 00:27:50.180 --> 00:27:53.950 align:middle line:84% So why are those two pictures equivalent? 00:27:53.950 --> 00:27:56.740 align:middle line:84% Because in that picture, you don't get the volume? 00:27:56.740 --> 00:27:59.170 align:middle line:90% PROFESSOR: Sorry, say it again? 00:27:59.170 --> 00:28:00.190 align:middle line:90% What's two picture? 00:28:00.190 --> 00:28:03.100 align:middle line:84% STUDENT: If you didn't pull out your delta function from 00:28:03.100 --> 00:28:05.230 align:middle line:84% your integral-- you just did it with-- 00:28:05.230 --> 00:28:07.570 align:middle line:84% that delta function is one for all k. 00:28:07.570 --> 00:28:10.210 align:middle line:84% PROFESSOR: No, because this is just a constant. 00:28:10.210 --> 00:28:12.160 align:middle line:84% Just when you have an integral of a constant, 00:28:12.160 --> 00:28:14.830 align:middle line:90% you can always pull it out. 00:28:14.830 --> 00:28:17.380 align:middle line:84% Because delta 0 does not depend on k anymore. 00:28:17.380 --> 00:28:21.920 align:middle line:90% 00:28:21.920 --> 00:28:22.910 align:middle line:90% Good. 00:28:22.910 --> 00:28:26.020 align:middle line:84% So how do you like this integral? 00:28:26.020 --> 00:28:27.640 align:middle line:90% Can you do it? 00:28:27.640 --> 00:28:28.900 align:middle line:90% STUDENT: No, it diverges 00:28:28.900 --> 00:28:29.650 align:middle line:90% PROFESSOR: Good. 00:28:29.650 --> 00:28:32.800 align:middle line:84% So it's fruitless to do this integral 00:28:32.800 --> 00:28:34.120 align:middle line:90% because this is divergent. 00:28:34.120 --> 00:28:38.580 align:middle line:90% 00:28:38.580 --> 00:28:44.500 align:middle line:84% And there's very good reason for this divergence physically. 00:28:44.500 --> 00:28:47.530 align:middle line:84% So this is just the energy density. 00:28:47.530 --> 00:28:51.015 align:middle line:84% So essentially, this is the energy per unit volume-- 00:28:51.015 --> 00:28:53.630 align:middle line:90% 00:28:53.630 --> 00:28:56.800 align:middle line:90% say in some unit volume. 00:28:56.800 --> 00:28:58.920 align:middle line:84% So let's imagine you have a unit volume here. 00:28:58.920 --> 00:29:04.200 align:middle line:90% 00:29:04.200 --> 00:29:12.720 align:middle line:84% Remember, this-- beside-- before this, corresponding to you 00:29:12.720 --> 00:29:15.810 align:middle line:84% take a discrete system with some lattice spacing a, 00:29:15.810 --> 00:29:19.200 align:middle line:84% and then you take the lattice, and each lattice point, 00:29:19.200 --> 00:29:20.940 align:middle line:84% you have a harmonic oscillator, and then 00:29:20.940 --> 00:29:24.300 align:middle line:84% you take a lattice spacing-- a equal to 0. 00:29:24.300 --> 00:29:26.550 align:middle line:84% So that means for any unit volume, when 00:29:26.550 --> 00:29:30.090 align:middle line:84% you take a equal to 0, you have infinite number 00:29:30.090 --> 00:29:31.125 align:middle line:90% of oscillators inside. 00:29:31.125 --> 00:29:33.810 align:middle line:90% 00:29:33.810 --> 00:29:35.475 align:middle line:84% So essentially, when you-- yeah, you 00:29:35.475 --> 00:29:39.180 align:middle line:84% have the lattice of oscillators at each point. 00:29:39.180 --> 00:29:41.310 align:middle line:84% Anyway, I will not try to draw it. 00:29:41.310 --> 00:29:45.390 align:middle line:84% And then when you take a to 0, you have any unit-- 00:29:45.390 --> 00:29:49.680 align:middle line:84% any volume unit, you will have the number of oscillators 00:29:49.680 --> 00:29:53.430 align:middle line:84% go to infinity, and that's where this divergence come from. 00:29:53.430 --> 00:29:54.900 align:middle line:90% It just come from-- 00:29:54.900 --> 00:29:58.000 align:middle line:84% in field theory, you have a continuum degree of freedom. 00:29:58.000 --> 00:30:00.030 align:middle line:84% So at each point, you have a degree of freedom, 00:30:00.030 --> 00:30:02.670 align:middle line:84% and within any volume, then you definitely 00:30:02.670 --> 00:30:05.010 align:middle line:84% have infinite number of degrees of freedom. 00:30:05.010 --> 00:30:11.770 align:middle line:84% so this is just from continuum of freedom. 00:30:11.770 --> 00:30:15.430 align:middle line:84% So this is the first time you see a diverging 00:30:15.430 --> 00:30:18.160 align:middle line:84% quantity in quantum field theory, 00:30:18.160 --> 00:30:20.185 align:middle line:84% but you will soon find that this is-- 00:30:20.185 --> 00:30:22.830 align:middle line:90% 00:30:22.830 --> 00:30:26.760 align:middle line:84% soon, you find that this is normal. 00:30:26.760 --> 00:30:28.080 align:middle line:90% It will be a fact of life. 00:30:28.080 --> 00:30:31.410 align:middle line:84% Just you will see divergence very commonly-- 00:30:31.410 --> 00:30:34.140 align:middle line:84% and so-- because you have continuum degrees of freedom. 00:30:34.140 --> 00:30:37.090 align:middle line:90% 00:30:37.090 --> 00:30:41.320 align:middle line:84% So the whole thing about quantum field theory 00:30:41.320 --> 00:30:45.580 align:middle line:84% is to find a way to deal with those divergences. 00:30:45.580 --> 00:30:49.840 align:middle line:84% And one of the key differences between quantum field theory 00:30:49.840 --> 00:30:52.970 align:middle line:84% and just finite quantum mechanics or finite number 00:30:52.970 --> 00:30:56.170 align:middle line:84% degrees of freedom is because of those divergences. 00:30:56.170 --> 00:30:58.120 align:middle line:84% And the big part of quantum field theory 00:30:58.120 --> 00:31:00.790 align:middle line:84% is to understand how to treat those divergences. 00:31:00.790 --> 00:31:04.180 align:middle line:84% And they actually don't affect your physics, 00:31:04.180 --> 00:31:08.830 align:middle line:84% but you do have to develop sometimes sophisticated tricks 00:31:08.830 --> 00:31:14.090 align:middle line:90% to treat OK, 00:31:14.090 --> 00:31:14.590 align:middle line:90% Good. 00:31:14.590 --> 00:31:17.430 align:middle line:90% 00:31:17.430 --> 00:31:22.470 align:middle line:84% And this infinite answer is also closely 00:31:22.470 --> 00:31:26.670 align:middle line:84% connected to a very famous problem you may have heard. 00:31:26.670 --> 00:31:30.360 align:middle line:84% It's called the cosmological constant problem. 00:31:30.360 --> 00:31:34.740 align:middle line:84% Because this tells you that any quantum field 00:31:34.740 --> 00:31:40.480 align:middle line:84% theory have an infinite, say, 0 point energy. 00:31:40.480 --> 00:31:43.090 align:middle line:90% And so you may say, OK-- 00:31:43.090 --> 00:31:50.730 align:middle line:84% so infinite-- whenever we say something infinite, 00:31:50.730 --> 00:31:55.800 align:middle line:84% there's one thing you always do to treat it. 00:31:55.800 --> 00:31:57.620 align:middle line:84% Can you guess what is the thing you always 00:31:57.620 --> 00:31:59.150 align:middle line:90% do when you see infinities? 00:31:59.150 --> 00:32:02.690 align:middle line:90% 00:32:02.690 --> 00:32:03.440 align:middle line:90% Yes? 00:32:03.440 --> 00:32:05.550 align:middle line:84% STUDENT: Just like subtract infinity. 00:32:05.550 --> 00:32:07.600 align:middle line:84% PROFESSOR: Yeah, that's one idea. 00:32:07.600 --> 00:32:10.560 align:middle line:84% That's one idea, but to subtract infinity is very hard. 00:32:10.560 --> 00:32:14.240 align:middle line:84% In your calculus class, when you subtract infinity 00:32:14.240 --> 00:32:16.680 align:middle line:84% from infinity, you can get infinity, 00:32:16.680 --> 00:32:19.580 align:middle line:90% so you have to be very careful. 00:32:19.580 --> 00:32:20.270 align:middle line:90% Yes? 00:32:20.270 --> 00:32:22.150 align:middle line:90% STUDENT: Just ignore the term. 00:32:22.150 --> 00:32:23.900 align:middle line:84% PROFESSOR: That's a lot of very good idea. 00:32:23.900 --> 00:32:26.690 align:middle line:90% [LAUGHTER] 00:32:26.690 --> 00:32:28.220 align:middle line:90% Indeed, that's what we often do. 00:32:28.220 --> 00:32:28.820 align:middle line:90% Yes? 00:32:28.820 --> 00:32:30.980 align:middle line:90% STUDENT: Divide by. 00:32:30.980 --> 00:32:33.350 align:middle line:90% PROFESSOR: That's also-- indeed. 00:32:33.350 --> 00:32:39.220 align:middle line:84% But to do all those things, you have to do one thing first. 00:32:39.220 --> 00:32:40.090 align:middle line:90% Yes? 00:32:40.090 --> 00:32:42.673 align:middle line:84% STUDENT: You might approximate it, it's like 1 over the Planck 00:32:42.673 --> 00:32:43.300 align:middle line:90% constant. 00:32:43.300 --> 00:32:45.180 align:middle line:84% PROFESSOR: Yeah, it's also very close. 00:32:45.180 --> 00:32:46.434 align:middle line:90% You need to do-- 00:32:46.434 --> 00:32:47.267 align:middle line:90% STUDENT: [INAUDIBLE] 00:32:47.267 --> 00:32:47.694 align:middle line:90% PROFESSOR: Hmm? 00:32:47.694 --> 00:32:48.550 align:middle line:90% STUDENT: [INAUDIBLE] 00:32:48.550 --> 00:32:49.592 align:middle line:90% PROFESSOR: Yeah, exactly. 00:32:49.592 --> 00:32:52.900 align:middle line:84% You need to find a way to make it finite first, 00:32:52.900 --> 00:32:54.580 align:middle line:90% and then you can subtract it. 00:32:54.580 --> 00:32:58.420 align:middle line:84% And then you can-- yeah, just like when you sum 1 plus 1/2 00:32:58.420 --> 00:33:00.740 align:middle line:90% plus 1/3, et cetera. 00:33:00.740 --> 00:33:03.970 align:middle line:90% You get an infinite series. 00:33:03.970 --> 00:33:06.290 align:middle line:84% But if you want to estimate the outcome-- yeah, 00:33:06.290 --> 00:33:07.090 align:middle line:90% you get divergence. 00:33:07.090 --> 00:33:10.330 align:middle line:84% But you always try to cut it off the series, 00:33:10.330 --> 00:33:12.400 align:middle line:84% and then approximate definite answer. 00:33:12.400 --> 00:33:14.560 align:middle line:90% And here is similar. 00:33:14.560 --> 00:33:19.060 align:middle line:84% So here we always put some momentum cut-off. 00:33:19.060 --> 00:33:22.840 align:middle line:84% So imagine the momentum is smaller 00:33:22.840 --> 00:33:25.930 align:middle line:84% than some momentum-- say, some value lambda. 00:33:25.930 --> 00:33:27.610 align:middle line:84% And lambda corresponding to maybe 00:33:27.610 --> 00:33:30.520 align:middle line:84% to the scale which this quantum field is no longer apply, 00:33:30.520 --> 00:33:33.250 align:middle line:84% because nobody told you that this quantum field 00:33:33.250 --> 00:33:36.190 align:middle line:84% theory should apply at all length scales. 00:33:36.190 --> 00:33:39.880 align:middle line:84% Because for example, in this lattice model, 00:33:39.880 --> 00:33:41.815 align:middle line:84% this scale will be 1 over the lattice spacing. 00:33:41.815 --> 00:33:45.440 align:middle line:90% 00:33:45.440 --> 00:33:49.910 align:middle line:84% Anyway, so once you cut it off, still you 00:33:49.910 --> 00:33:52.840 align:middle line:90% get a pretty big number. 00:33:52.840 --> 00:33:57.040 align:middle line:84% And you cannot ignore it because this is a physical zero point 00:33:57.040 --> 00:33:57.730 align:middle line:90% energy. 00:33:57.730 --> 00:34:00.280 align:middle line:84% In principle, you can measure it. 00:34:00.280 --> 00:34:03.980 align:middle line:84% But in real life, we don't see it. 00:34:03.980 --> 00:34:07.810 align:middle line:84% So we have quantum fields flying around all the time, 00:34:07.810 --> 00:34:10.580 align:middle line:84% but we don't see this big vacuum energy. 00:34:10.580 --> 00:34:13.900 align:middle line:84% So this is called the cosmological constant problem. 00:34:13.900 --> 00:34:18.429 align:middle line:84% Actually, there was just a colloquium last week 00:34:18.429 --> 00:34:21.310 align:middle line:84% about this cosmological related to this cosmological constant 00:34:21.310 --> 00:34:23.800 align:middle line:90% problem. 00:34:23.800 --> 00:34:24.300 align:middle line:90% Good. 00:34:24.300 --> 00:34:25.199 align:middle line:90% Any questions? 00:34:25.199 --> 00:34:28.143 align:middle line:90% 00:34:28.143 --> 00:34:29.685 align:middle line:84% Now, we can talk about Hilbert space. 00:34:29.685 --> 00:34:43.870 align:middle line:90% 00:34:43.870 --> 00:34:47.380 align:middle line:90% So as a harmonic oscillator-- 00:34:47.380 --> 00:34:50.110 align:middle line:84% so we can first define the vacuum state. 00:34:50.110 --> 00:34:55.980 align:middle line:84% So now, what we will do is indeed in quantum field theory 00:34:55.980 --> 00:35:00.210 align:middle line:84% itself, we can just ignore this E0. 00:35:00.210 --> 00:35:02.880 align:middle line:90% 00:35:02.880 --> 00:35:06.660 align:middle line:84% From now on, we will just ignore this E0 because this just 00:35:06.660 --> 00:35:07.920 align:middle line:90% to give you overall-- 00:35:07.920 --> 00:35:13.140 align:middle line:84% just give you a constant, which does not do anything. 00:35:13.140 --> 00:35:16.860 align:middle line:84% It's like the potential energy E and M. 00:35:16.860 --> 00:35:21.470 align:middle line:84% Yeah, just like the unit-- yeah, just like the-- anyway, 00:35:21.470 --> 00:35:25.280 align:middle line:84% so from now on, we ignore this term. 00:35:25.280 --> 00:35:27.372 align:middle line:84% So often, I will just write the Hamiltonian-- 00:35:27.372 --> 00:35:28.330 align:middle line:90% I just write this term. 00:35:28.330 --> 00:35:32.180 align:middle line:90% 00:35:32.180 --> 00:35:35.663 align:middle line:84% But later, we actually see examples-- 00:35:35.663 --> 00:35:36.830 align:middle line:90% later, we will see examples. 00:35:36.830 --> 00:35:40.250 align:middle line:84% Actually, this E0 can actually have physical implications, 00:35:40.250 --> 00:35:43.040 align:middle line:84% but just for our current purpose, we ignore it. 00:35:43.040 --> 00:35:46.570 align:middle line:90% 00:35:46.570 --> 00:35:49.280 align:middle line:84% So now, the lowest energy state-- 00:35:49.280 --> 00:35:52.740 align:middle line:84% then it's clear from here because this 00:35:52.740 --> 00:35:54.310 align:middle line:90% is just a constant. 00:35:54.310 --> 00:36:01.470 align:middle line:84% So the lowest energy state ground state, which 00:36:01.470 --> 00:36:04.790 align:middle line:84% is we often call the vacuum state, 00:36:04.790 --> 00:36:09.790 align:middle line:84% means there's nothing there, is given by a k 0 equal 00:36:09.790 --> 00:36:15.130 align:middle line:90% to 0 defined to be for any k. 00:36:15.130 --> 00:36:19.730 align:middle line:84% So the ground state satisfy, which I denoted by 0-- 00:36:19.730 --> 00:36:24.610 align:middle line:84% then satisfy annihlated by this ak. 00:36:24.610 --> 00:36:27.580 align:middle line:90% And then the general state-- 00:36:27.580 --> 00:36:31.210 align:middle line:84% you can just-- so general states have the following form. 00:36:31.210 --> 00:36:36.150 align:middle line:90% 00:36:36.150 --> 00:36:38.790 align:middle line:90% Say I can write n k 1. 00:36:38.790 --> 00:36:42.090 align:middle line:84% So the k 1 oscillator excited n times. 00:36:42.090 --> 00:36:45.930 align:middle line:90% n k 2-- n k-- 00:36:45.930 --> 00:36:49.380 align:middle line:84% the k 2 oscillator excited n k 2 times, 00:36:49.380 --> 00:36:52.140 align:middle line:90% et cetera, which is given by-- 00:36:52.140 --> 00:37:00.780 align:middle line:84% which is proportional to a k 1 dagger to the power n k 1 00:37:00.780 --> 00:37:09.060 align:middle line:84% a k 2 dagger n k 2 etcetera acting on 0 00:37:09.060 --> 00:37:13.170 align:middle line:84% just like we have large number of harmonic oscillators. 00:37:13.170 --> 00:37:22.390 align:middle line:90% 00:37:22.390 --> 00:37:23.980 align:middle line:90% Questions on this? 00:37:23.980 --> 00:37:24.798 align:middle line:90% Yes? 00:37:24.798 --> 00:37:26.340 align:middle line:84% STUDENT: How do you know there exists 00:37:26.340 --> 00:37:28.757 align:middle line:84% some state that's annihilated by every single annihilation 00:37:28.757 --> 00:37:30.010 align:middle line:90% operator? 00:37:30.010 --> 00:37:34.150 align:middle line:84% PROFESSOR: You postulate, but you can actually write it 00:37:34.150 --> 00:37:37.750 align:middle line:90% down its wave function. 00:37:37.750 --> 00:37:42.970 align:middle line:84% Just as in the standard harmonic oscillator case, 00:37:42.970 --> 00:37:48.624 align:middle line:84% you can starting from a dagger annihlated by a. 00:37:48.624 --> 00:37:50.830 align:middle line:84% You can start from this equation to write down 00:37:50.830 --> 00:37:51.950 align:middle line:90% its wave function. 00:37:51.950 --> 00:37:55.060 align:middle line:84% And here, you can write down the wave function for the vacuum 00:37:55.060 --> 00:37:56.500 align:middle line:90% state to in terms of phi. 00:37:56.500 --> 00:37:59.790 align:middle line:90% 00:37:59.790 --> 00:38:01.350 align:middle line:90% Yes? 00:38:01.350 --> 00:38:03.300 align:middle line:84% STUDENT: In the notation for the ket there 00:38:03.300 --> 00:38:05.990 align:middle line:84% it looks like we have countable number of frequencies? 00:38:05.990 --> 00:38:08.738 align:middle line:90% Is that, like why is that? 00:38:08.738 --> 00:38:09.780 align:middle line:90% PROFESSOR: You mean here? 00:38:09.780 --> 00:38:10.580 align:middle line:90% STUDENT: Yeah. 00:38:10.580 --> 00:38:12.720 align:middle line:84% PROFESSOR: Yeah, here, I'm just saying it 00:38:12.720 --> 00:38:15.145 align:middle line:90% depends on which k are excited. 00:38:15.145 --> 00:38:17.763 align:middle line:90% 00:38:17.763 --> 00:38:19.180 align:middle line:84% Yeah, that's a very good question. 00:38:19.180 --> 00:38:22.140 align:middle line:84% I will comment on related issues very soon. 00:38:22.140 --> 00:38:26.660 align:middle line:84% But here, I just write down some state. 00:38:26.660 --> 00:38:30.770 align:middle line:90% Yeah, excite k1, k2 as I want. 00:38:30.770 --> 00:38:34.130 align:middle line:84% I'm not saying that this is the-- 00:38:34.130 --> 00:38:37.488 align:middle line:84% yeah, you can have as many as you want. 00:38:37.488 --> 00:38:39.780 align:middle line:84% STUDENT: Yeah, but if you use that notation it's still. 00:38:39.780 --> 00:38:40.230 align:middle line:90% PROFESSOR: That's right. 00:38:40.230 --> 00:38:40.770 align:middle line:90% It's true. 00:38:40.770 --> 00:38:43.830 align:middle line:90% Yeah, but I write it-- 00:38:43.830 --> 00:38:45.850 align:middle line:84% when I write it this way, it implies 00:38:45.850 --> 00:38:48.690 align:middle line:84% I only excite countable number of them. 00:38:48.690 --> 00:38:51.720 align:middle line:84% But in principle-- yeah, we will soon 00:38:51.720 --> 00:38:55.110 align:middle line:84% touch a point, which is related to your question. 00:38:55.110 --> 00:38:56.040 align:middle line:90% Other questions? 00:38:56.040 --> 00:38:59.200 align:middle line:90% 00:38:59.200 --> 00:39:01.470 align:middle line:90% OK, good. 00:39:01.470 --> 00:39:04.770 align:middle line:84% And so for example, the simplest excited state 00:39:04.770 --> 00:39:06.660 align:middle line:84% would be just excite one of them. 00:39:06.660 --> 00:39:09.420 align:middle line:90% 00:39:09.420 --> 00:39:12.930 align:middle line:90% So let me denote this by k. 00:39:12.930 --> 00:39:16.110 align:middle line:84% And then the simplest way you just excite two of them. 00:39:16.110 --> 00:39:24.690 align:middle line:90% 00:39:24.690 --> 00:39:26.690 align:middle line:84% So this k1 and k2 can be the same. 00:39:26.690 --> 00:39:31.820 align:middle line:84% If these k1 k2 are the same, and then just like the square. 00:39:31.820 --> 00:39:33.200 align:middle line:84% And here, also I will not be very 00:39:33.200 --> 00:39:37.180 align:middle line:90% careful about the normalization. 00:39:37.180 --> 00:39:41.380 align:middle line:84% And now, we ask, what are the physical interpretations 00:39:41.380 --> 00:39:43.060 align:middle line:90% of those states? 00:39:43.060 --> 00:39:47.816 align:middle line:84% So now let's ask what are the physical interpretations? 00:39:47.816 --> 00:39:54.980 align:middle line:90% 00:39:54.980 --> 00:40:04.850 align:middle line:84% So for this purpose, we can just look at their quantum numbers 00:40:04.850 --> 00:40:08.735 align:middle line:84% under, say, the Hamiltonian and under the spacetime momentum. 00:40:08.735 --> 00:40:11.660 align:middle line:90% 00:40:11.660 --> 00:40:14.690 align:middle line:90% So for example, H 0-- 00:40:14.690 --> 00:40:19.260 align:middle line:84% so if you act on H on 0, then of course, 00:40:19.260 --> 00:40:25.010 align:middle line:84% you get 0 assuming we throw E0 away. 00:40:25.010 --> 00:40:31.070 align:middle line:90% Say ignore E0 so far. 00:40:31.070 --> 00:40:33.980 align:middle line:84% And then so for the ground state-- 00:40:33.980 --> 00:40:35.660 align:middle line:90% and the p i acting on 0. 00:40:35.660 --> 00:40:37.430 align:middle line:90% Of course, it's 0. 00:40:37.430 --> 00:40:40.590 align:middle line:84% So you can see just because this have ak here. 00:40:40.590 --> 00:40:42.560 align:middle line:90% And so E also-- 00:40:42.560 --> 00:40:45.242 align:middle line:84% when we throw this away, you have this --yeah. 00:40:45.242 --> 00:40:47.980 align:middle line:90% 00:40:47.980 --> 00:40:50.456 align:middle line:84% And now, let's look at the excited state. 00:40:50.456 --> 00:40:54.760 align:middle line:90% 00:40:54.760 --> 00:40:59.650 align:middle line:90% So now to look at this state. 00:40:59.650 --> 00:41:02.320 align:middle line:90% So H acting on k-- 00:41:02.320 --> 00:41:05.650 align:middle line:84% so this answer is obvious because all different 00:41:05.650 --> 00:41:08.260 align:middle line:84% oscillators, they are independent of each other, 00:41:08.260 --> 00:41:11.770 align:middle line:90% and this just gives you omega k. 00:41:11.770 --> 00:41:14.110 align:middle line:84% We can just use our result for harmonic oscillator. 00:41:14.110 --> 00:41:18.090 align:middle line:90% 00:41:18.090 --> 00:41:20.250 align:middle line:84% And now, you can look at the momentum-- 00:41:20.250 --> 00:41:23.280 align:middle line:90% 00:41:23.280 --> 00:41:28.580 align:middle line:84% so this is a spacetime momentum operator acting on here. 00:41:28.580 --> 00:41:32.030 align:middle line:90% And then it has k i. 00:41:32.030 --> 00:41:37.550 align:middle line:84% If you look at this expression, when you do the-- 00:41:37.550 --> 00:41:40.730 align:middle line:90% act this on that-- 00:41:40.730 --> 00:41:43.520 align:middle line:90% 00:41:43.520 --> 00:41:48.470 align:middle line:84% you act this on that, and when you can just 00:41:48.470 --> 00:41:51.470 align:middle line:84% use the standard trick when you do the commutator. 00:41:51.470 --> 00:41:54.320 align:middle line:84% And then the particular k for this is chosen, 00:41:54.320 --> 00:41:57.775 align:middle line:84% and then the eigenvalue will be just given by that particular k 00:41:57.775 --> 00:41:58.700 align:middle line:90% i. 00:41:58.700 --> 00:42:00.950 align:middle line:84% So this is one step of calculation there, 00:42:00.950 --> 00:42:03.470 align:middle line:90% and you should do it yourself. 00:42:03.470 --> 00:42:06.830 align:middle line:90% 1-minute calculation. 00:42:06.830 --> 00:42:09.170 align:middle line:84% This a delta function will be generated, 00:42:09.170 --> 00:42:12.200 align:middle line:84% and then that will get rid of that integral, 00:42:12.200 --> 00:42:13.820 align:middle line:84% and then you will pick up this k i. 00:42:13.820 --> 00:42:18.220 align:middle line:90% 00:42:18.220 --> 00:42:20.440 align:middle line:90% So now, this equation has very-- 00:42:20.440 --> 00:42:24.790 align:middle line:84% now has an obvious physical interpretation. 00:42:24.790 --> 00:42:31.810 align:middle line:84% So that means that this state k has spacetime momentum. 00:42:31.810 --> 00:42:38.850 align:middle line:84% So now, I write down a four-momentum omega k n k. 00:42:38.850 --> 00:42:43.500 align:middle line:84% So this is exactly the momentum of a relativistic particle 00:42:43.500 --> 00:42:45.480 align:middle line:90% on shell. 00:42:45.480 --> 00:42:49.200 align:middle line:90% So this is of a relativistic-- 00:42:49.200 --> 00:43:04.510 align:middle line:84% so this is a momentum of a relativistic particle, 00:43:04.510 --> 00:43:06.500 align:middle line:84% because omega k is equal to k squared 00:43:06.500 --> 00:43:09.440 align:middle line:90% plus m square of mass m. 00:43:09.440 --> 00:43:14.190 align:middle line:90% 00:43:14.190 --> 00:43:17.560 align:middle line:84% Yeah, so this means that the p squared. 00:43:17.560 --> 00:43:19.960 align:middle line:84% So this is a four-vector squared is equal to m-- 00:43:19.960 --> 00:43:21.610 align:middle line:84% yeah, so this satisfies the p squared 00:43:21.610 --> 00:43:22.750 align:middle line:90% equal to minus m square. 00:43:22.750 --> 00:43:27.560 align:middle line:90% 00:43:27.560 --> 00:43:29.570 align:middle line:90% So we can just-- 00:43:29.570 --> 00:43:31.550 align:middle line:84% so it's very logical just to interpret this 00:43:31.550 --> 00:43:37.240 align:middle line:84% as a particle of mass m, because we can just interpret it 00:43:37.240 --> 00:43:38.240 align:middle line:90% as a particle of mass m. 00:43:38.240 --> 00:43:41.400 align:middle line:90% 00:43:41.400 --> 00:43:42.950 align:middle line:90% Good? 00:43:42.950 --> 00:43:46.080 align:middle line:90% So now, let's look at this one. 00:43:46.080 --> 00:43:49.260 align:middle line:84% So again, the calculation is very simple. 00:43:49.260 --> 00:43:50.600 align:middle line:90% So you find for that one-- 00:43:50.600 --> 00:44:28.220 align:middle line:90% 00:44:28.220 --> 00:44:31.040 align:middle line:90% so you find for that one-- 00:44:31.040 --> 00:44:40.560 align:middle line:90% for k1 k2, you find the energy-- 00:44:40.560 --> 00:44:42.820 align:middle line:84% energy just defined-- again, you find 00:44:42.820 --> 00:44:45.370 align:middle line:84% that this is an energy eigenstate of the H. 00:44:45.370 --> 00:44:49.450 align:middle line:84% It's an eigenstate of H with energy eigenvalue 00:44:49.450 --> 00:44:52.390 align:middle line:90% omega k1 plus omega k2. 00:44:52.390 --> 00:44:55.480 align:middle line:90% 00:44:55.480 --> 00:45:00.700 align:middle line:84% And it's a momentum eigenstate of P i with an eigenvalue given 00:45:00.700 --> 00:45:02.860 align:middle line:90% by k1 plus k2. 00:45:02.860 --> 00:45:09.480 align:middle line:90% 00:45:09.480 --> 00:45:27.954 align:middle line:84% So E, k are eigenvalues of H and P. 00:45:27.954 --> 00:45:28.920 align:middle line:90% STUDENT: Question. 00:45:28.920 --> 00:45:29.810 align:middle line:90% PROFESSOR: Yeah? 00:45:29.810 --> 00:45:31.890 align:middle line:84% STUDENT: Yeah, so could that mass be 0? 00:45:31.890 --> 00:45:34.830 align:middle line:90% 00:45:34.830 --> 00:45:38.540 align:middle line:84% PROFESSOR: So here-- so this mass is not 0 because omega k 00:45:38.540 --> 00:45:40.970 align:middle line:90% is defined to be this-- 00:45:40.970 --> 00:45:44.970 align:middle line:84% it's defined by my theory-- defined by my Lagrangian. 00:45:44.970 --> 00:45:48.990 align:middle line:84% So this is the parameter of your-- 00:45:48.990 --> 00:45:50.670 align:middle line:90% yeah, of your action. 00:45:50.670 --> 00:45:51.270 align:middle line:90% Yes? 00:45:51.270 --> 00:45:54.720 align:middle line:84% STUDENT: So I have a -- with the energy spectrum, 00:45:54.720 --> 00:45:55.650 align:middle line:90% does the spacing-- 00:45:55.650 --> 00:45:57.240 align:middle line:84% does it happen the same way where 00:45:57.240 --> 00:45:59.640 align:middle line:84% the spacing gets smaller as you go to higher 00:45:59.640 --> 00:46:00.750 align:middle line:90% and higher energies or? 00:46:00.750 --> 00:46:03.780 align:middle line:84% PROFESSOR: Yeah, for here, it's uniform. 00:46:03.780 --> 00:46:06.390 align:middle line:84% Here, just for each k, it's uniform. 00:46:06.390 --> 00:46:08.140 align:middle line:84% But of course, when you add them together, 00:46:08.140 --> 00:46:10.590 align:middle line:84% you get something very complicated. 00:46:10.590 --> 00:46:12.675 align:middle line:84% But for each k, you just uniform. 00:46:12.675 --> 00:46:16.020 align:middle line:90% 00:46:16.020 --> 00:46:18.270 align:middle line:84% So the very-- so the most natural way 00:46:18.270 --> 00:46:29.310 align:middle line:84% to interpret this is just have two particles 00:46:29.310 --> 00:46:45.302 align:middle line:84% of momentum of four momenta omega k1 k1 and omega k2 k2. 00:46:45.302 --> 00:46:48.750 align:middle line:90% 00:46:48.750 --> 00:46:54.300 align:middle line:84% And similarly, you can do this for any state like this. 00:46:54.300 --> 00:46:56.910 align:middle line:90% 00:46:56.910 --> 00:47:11.400 align:middle line:84% They are all eigenvectors of H and P. So n k 1, n k 2, 00:47:11.400 --> 00:47:12.360 align:middle line:90% you catch-- 00:47:12.360 --> 00:47:13.320 align:middle line:90% et cetera. 00:47:13.320 --> 00:47:21.105 align:middle line:84% So this corresponding to n k 1 particles of momentum. 00:47:21.105 --> 00:47:23.900 align:middle line:90% 00:47:23.900 --> 00:47:26.670 align:middle line:90% They're all on shell-- 00:47:26.670 --> 00:47:33.290 align:middle line:84% on shell particle of momentum k1 and n 00:47:33.290 --> 00:47:37.950 align:middle line:84% k 2 has two particles of momentum k2. 00:47:37.950 --> 00:47:41.490 align:middle line:90% 00:47:41.490 --> 00:47:44.190 align:middle line:90% So this tells you one thing. 00:47:44.190 --> 00:47:48.225 align:middle line:84% So now let me just make some remarks. 00:47:48.225 --> 00:47:59.960 align:middle line:90% 00:47:59.960 --> 00:48:02.180 align:middle line:84% Any questions on this before I make my remarks? 00:48:02.180 --> 00:48:05.350 align:middle line:90% 00:48:05.350 --> 00:48:06.190 align:middle line:90% Good. 00:48:06.190 --> 00:48:08.410 align:middle line:84% So the first point is that now you 00:48:08.410 --> 00:48:12.430 align:middle line:84% can see this can describe any number of particles. 00:48:12.430 --> 00:48:21.150 align:middle line:90% 00:48:21.150 --> 00:48:23.010 align:middle line:84% So mathematically, in our description, 00:48:23.010 --> 00:48:25.830 align:middle line:90% there are harmonic oscillators. 00:48:25.830 --> 00:48:27.838 align:middle line:84% But each excitation of the harmonic oscillator 00:48:27.838 --> 00:48:29.880 align:middle line:84% is from the spacetime point of view corresponding 00:48:29.880 --> 00:48:31.690 align:middle line:90% to a particle. 00:48:31.690 --> 00:48:35.190 align:middle line:84% So this is the beautiful thing of this theory. 00:48:35.190 --> 00:48:40.730 align:middle line:90% 00:48:40.730 --> 00:48:44.810 align:middle line:90% And then due to-- 00:48:44.810 --> 00:48:49.190 align:middle line:84% because of the commutation relation, a k 1 dagger, 00:48:49.190 --> 00:48:53.440 align:middle line:84% a k 2 dagger equal to 0 for any k1 k2. 00:48:53.440 --> 00:48:55.990 align:middle line:90% 00:48:55.990 --> 00:48:57.550 align:middle line:90% So you have full symmetry. 00:48:57.550 --> 00:48:59.260 align:middle line:84% So when you construct the state, you 00:48:59.260 --> 00:49:01.660 align:middle line:84% can just commute them as you want. 00:49:01.660 --> 00:49:03.640 align:middle line:90% So that means they're-- 00:49:03.640 --> 00:49:21.090 align:middle line:84% so full symmetry in permuting all these different particles 00:49:21.090 --> 00:49:22.215 align:middle line:90% in the general state. 00:49:22.215 --> 00:49:27.840 align:middle line:90% 00:49:27.840 --> 00:49:30.560 align:middle line:84% So this tells us these are bosons. 00:49:30.560 --> 00:49:43.750 align:middle line:90% 00:49:43.750 --> 00:49:50.035 align:middle line:84% And two is that all particles have positive energy. 00:49:50.035 --> 00:49:58.010 align:middle line:90% 00:49:58.010 --> 00:50:07.510 align:middle line:84% So even though that E squared equal to k 00:50:07.510 --> 00:50:10.990 align:middle line:84% squared plus m squared-- this equation have two solutions-- 00:50:10.990 --> 00:50:13.960 align:middle line:90% plus minus omega k. 00:50:13.960 --> 00:50:16.120 align:middle line:84% But when you look at physical state-- 00:50:16.120 --> 00:50:18.370 align:middle line:84% when you look at your state and look at the eigenvalue 00:50:18.370 --> 00:50:18.980 align:middle line:90% of state-- 00:50:18.980 --> 00:50:21.040 align:middle line:84% so all particles have physical energies, 00:50:21.040 --> 00:50:24.160 align:middle line:84% so you don't have this negative energy problem associated 00:50:24.160 --> 00:50:25.375 align:middle line:90% with taking the square root. 00:50:25.375 --> 00:50:43.990 align:middle line:90% 00:50:43.990 --> 00:50:55.510 align:middle line:84% Also you have total energy of a state 00:50:55.510 --> 00:51:04.690 align:middle line:84% is equal to sum of energies of all the particles. 00:51:04.690 --> 00:51:11.890 align:middle line:90% 00:51:11.890 --> 00:51:15.250 align:middle line:84% So this tells you there's no interactions between them. 00:51:15.250 --> 00:51:19.390 align:middle line:84% Because if you have potential energies between particles, 00:51:19.390 --> 00:51:21.940 align:middle line:84% and then that will change the energy. 00:51:21.940 --> 00:51:24.190 align:middle line:84% When you put the two particles together, 00:51:24.190 --> 00:51:27.130 align:middle line:84% will no longer be the same over the sum 00:51:27.130 --> 00:51:30.280 align:middle line:84% of the individual energy for each particle. 00:51:30.280 --> 00:51:33.980 align:middle line:84% And so that tells you there's no interactions. 00:51:33.980 --> 00:51:38.186 align:middle line:84% So this is a theory of free particles. 00:51:38.186 --> 00:51:47.930 align:middle line:90% 00:51:47.930 --> 00:51:50.065 align:middle line:84% Now, this is a good starting point. 00:51:50.065 --> 00:51:53.050 align:middle line:90% 00:51:53.050 --> 00:51:54.580 align:middle line:90% At least now, we have particles. 00:51:54.580 --> 00:52:05.170 align:middle line:90% 00:52:05.170 --> 00:52:07.030 align:middle line:90% Questions on this? 00:52:07.030 --> 00:52:07.720 align:middle line:90% Yes? 00:52:07.720 --> 00:52:12.640 align:middle line:84% STUDENT: So [INAUDIBLE] a k 1 let's 00:52:12.640 --> 00:52:14.710 align:middle line:84% say, and you have a particle with momentum k1 00:52:14.710 --> 00:52:17.950 align:middle line:84% now, is there a way to change this particle's momentum, 00:52:17.950 --> 00:52:20.530 align:middle line:84% or if you apply again, you-- like in this picture, 00:52:20.530 --> 00:52:23.285 align:middle line:84% it's like you have another particle momentum k1. 00:52:23.285 --> 00:52:24.910 align:middle line:84% PROFESSOR: No, there's no way to change 00:52:24.910 --> 00:52:26.380 align:middle line:90% the momentum of a particle. 00:52:26.380 --> 00:52:29.110 align:middle line:84% So once you created the particle, 00:52:29.110 --> 00:52:30.580 align:middle line:90% it just goes straight. 00:52:30.580 --> 00:52:35.080 align:middle line:84% Yeah, it just does not change anymore. 00:52:35.080 --> 00:52:37.420 align:middle line:84% Momentum for that particle is conserved. 00:52:37.420 --> 00:52:39.160 align:middle line:84% STUDENT: Is there-- if you wanted to-- 00:52:39.160 --> 00:52:40.780 align:middle line:84% if you wanted to create a theory where 00:52:40.780 --> 00:52:43.300 align:middle line:84% you can change the momentum, is there a way, or is it just-- 00:52:43.300 --> 00:52:44.870 align:middle line:90% PROFESSOR: Yeah, there is a way. 00:52:44.870 --> 00:52:46.960 align:middle line:84% So after dealing with this free theory, 00:52:46.960 --> 00:52:50.500 align:middle line:84% and then we will consider the last simplest theory, 00:52:50.500 --> 00:52:52.360 align:middle line:84% and then that will introduce interactions. 00:52:52.360 --> 00:52:52.585 align:middle line:90% STUDENT: I see. 00:52:52.585 --> 00:52:52.660 align:middle line:90% So-- 00:52:52.660 --> 00:52:54.410 align:middle line:84% PROFESSOR: And when you have interactions, 00:52:54.410 --> 00:52:56.620 align:middle line:84% then the particle momentum can change. 00:52:56.620 --> 00:52:57.120 align:middle line:90% Yes? 00:52:57.120 --> 00:52:58.578 align:middle line:84% STUDENT: So the thing we're calling 00:52:58.578 --> 00:53:00.640 align:middle line:84% particles are like localized in momentum space, 00:53:00.640 --> 00:53:02.553 align:middle line:84% but not at all in position space. 00:53:02.553 --> 00:53:03.220 align:middle line:90% PROFESSOR: Good. 00:53:03.220 --> 00:53:05.170 align:middle line:90% Yeah, these are the momentum-- 00:53:05.170 --> 00:53:09.220 align:middle line:84% yeah, it's like the plane wave in the non-relativistic-- yeah. 00:53:09.220 --> 00:53:11.713 align:middle line:84% STUDENT: And if you were to try to localize it in position 00:53:11.713 --> 00:53:13.630 align:middle line:84% space like a Gaussian wave packet or something 00:53:13.630 --> 00:53:15.100 align:middle line:84% kind of like that, would you still 00:53:15.100 --> 00:53:16.780 align:middle line:84% be able to commute things and stuff? 00:53:16.780 --> 00:53:19.390 align:middle line:84% PROFESSOR: Yeah, so the commute things 00:53:19.390 --> 00:53:23.410 align:middle line:84% don't change because everything is built by a k, 00:53:23.410 --> 00:53:26.720 align:middle line:84% and they always commute, and so that won't change. 00:53:26.720 --> 00:53:28.750 align:middle line:84% But indeed, we will talk about the wave packet 00:53:28.750 --> 00:53:31.840 align:middle line:90% to localize in space. 00:53:31.840 --> 00:53:35.090 align:middle line:90% Other questions? 00:53:35.090 --> 00:53:35.830 align:middle line:90% OK, good. 00:53:35.830 --> 00:53:52.610 align:middle line:90% 00:53:52.610 --> 00:53:55.060 align:middle line:90% So also-- so the last-- 00:53:55.060 --> 00:54:01.150 align:middle line:84% we will talk a little bit about the technical point. 00:54:01.150 --> 00:54:04.900 align:middle line:90% 00:54:04.900 --> 00:54:09.130 align:middle line:84% It's that so far, we haven't talked about the normalization 00:54:09.130 --> 00:54:10.853 align:middle line:90% of such a state. 00:54:10.853 --> 00:54:12.895 align:middle line:84% So now, let's look at the normalization of state. 00:54:12.895 --> 00:54:21.550 align:middle line:90% 00:54:21.550 --> 00:54:25.605 align:middle line:84% So let's just look at this state. 00:54:25.605 --> 00:54:26.980 align:middle line:84% The single particle state-- let's 00:54:26.980 --> 00:54:30.800 align:middle line:90% look at this normalization. 00:54:30.800 --> 00:54:33.010 align:middle line:90% So let's look at k with k prime. 00:54:33.010 --> 00:54:36.310 align:middle line:90% 00:54:36.310 --> 00:54:39.940 align:middle line:84% So now, if you take the overlap of this with the k prime, 00:54:39.940 --> 00:54:42.700 align:middle line:84% and then you can reduce this to the commutator of a k 00:54:42.700 --> 00:54:44.150 align:middle line:90% and a k prime-- 00:54:44.150 --> 00:54:46.120 align:middle line:90% a k dagger and a k prime. 00:54:46.120 --> 00:54:50.380 align:middle line:90% So you will get just this. 00:54:50.380 --> 00:54:54.560 align:middle line:84% So again, this is a five-second calculation. 00:54:54.560 --> 00:54:57.865 align:middle line:84% So you find this if you do the overlap. 00:54:57.865 --> 00:55:02.380 align:middle line:90% 00:55:02.380 --> 00:55:06.850 align:middle line:84% But you already did in your p set-- 00:55:06.850 --> 00:55:09.310 align:middle line:84% this thing is actually not Lorentz invariant. 00:55:09.310 --> 00:55:11.150 align:middle line:90% This is another good thing. 00:55:11.150 --> 00:55:16.150 align:middle line:84% This is another good object under Lorentz transformations. 00:55:16.150 --> 00:55:18.580 align:middle line:84% But when we construct state, we would 00:55:18.580 --> 00:55:21.400 align:middle line:84% like to have our state to have good properties under Lorentz 00:55:21.400 --> 00:55:23.260 align:middle line:90% transformation. 00:55:23.260 --> 00:55:28.090 align:middle line:84% So we will choose a slightly different normalization. 00:55:28.090 --> 00:55:33.400 align:middle line:84% So instead of this state, so we will 00:55:33.400 --> 00:55:35.350 align:middle line:90% define the following states. 00:55:35.350 --> 00:55:39.340 align:middle line:84% And we will define a k now without the vector 00:55:39.340 --> 00:55:46.300 align:middle line:84% to be the square root 2 omega k and this k. 00:55:46.300 --> 00:55:48.950 align:middle line:90% 00:55:48.950 --> 00:56:00.410 align:middle line:84% And so this is the 2 square root of k a k dagger acting on 0. 00:56:00.410 --> 00:56:07.130 align:middle line:84% And now, if you compute the overlap with this k and k 00:56:07.130 --> 00:56:08.610 align:middle line:90% prime-- 00:56:08.610 --> 00:56:10.940 align:middle line:90% so now, you have square root k-- 00:56:10.940 --> 00:56:13.010 align:middle line:84% square root omega k, for one of them, 00:56:13.010 --> 00:56:15.740 align:middle line:84% and then you have square root k for two of them, 00:56:15.740 --> 00:56:17.435 align:middle line:90% and then you have 2 omega k. 00:56:17.435 --> 00:56:20.030 align:middle line:90% 00:56:20.030 --> 00:56:29.680 align:middle line:84% And then you have 2 pi cubed delta 3k k minus [INAUDIBLE].. 00:56:29.680 --> 00:56:33.190 align:middle line:84% So now you recognize this object from your pset. 00:56:33.190 --> 00:56:37.030 align:middle line:84% The omega k multiply by this guy actually 00:56:37.030 --> 00:56:40.660 align:middle line:84% have good Lorentz transformation properties. 00:56:40.660 --> 00:56:46.840 align:middle line:84% So this guy actually transforms nicely 00:56:46.840 --> 00:56:50.480 align:middle line:90% under Lorentz transformations. 00:56:50.480 --> 00:56:53.170 align:middle line:84% So this will be the normalization 00:56:53.170 --> 00:56:55.480 align:middle line:90% we will use from now on. 00:56:55.480 --> 00:56:58.880 align:middle line:90% 00:56:58.880 --> 00:57:02.600 align:middle line:90% So these states have very-- 00:57:02.600 --> 00:57:05.630 align:middle line:90% so indeed, you can show-- 00:57:05.630 --> 00:57:08.630 align:middle line:84% so that will be, I think, in your pset 3. 00:57:08.630 --> 00:57:10.580 align:middle line:90% You should look forward to it. 00:57:10.580 --> 00:57:16.940 align:middle line:84% So you can show if you act a Lorentz transformation-- 00:57:16.940 --> 00:57:18.740 align:middle line:84% so lambda is a Lorentz transformation, 00:57:18.740 --> 00:57:22.460 align:middle line:84% and u Lambda is the operator to generate that Lorentz 00:57:22.460 --> 00:57:24.590 align:middle line:90% transformation-- 00:57:24.590 --> 00:57:28.370 align:middle line:84% on such a state k, and then you just get lambda k. 00:57:28.370 --> 00:57:30.620 align:middle line:84% And lambda k is the Lorentz transformation 00:57:30.620 --> 00:57:32.900 align:middle line:90% acting on that k. 00:57:32.900 --> 00:57:35.060 align:middle line:84% It's the Lorentz transform the k. 00:57:35.060 --> 00:57:37.880 align:middle line:90% 00:57:37.880 --> 00:57:46.720 align:middle line:84% But yeah, you should see your maybe p set 3, so-- 00:57:46.720 --> 00:57:47.815 align:middle line:90% if I remember it. 00:57:47.815 --> 00:57:53.950 align:middle line:90% 00:57:53.950 --> 00:57:54.470 align:middle line:90% Good. 00:57:54.470 --> 00:57:55.510 align:middle line:90% Any questions on this? 00:57:55.510 --> 00:58:01.520 align:middle line:90% 00:58:01.520 --> 00:58:03.890 align:middle line:90% Yes? 00:58:03.890 --> 00:58:05.570 align:middle line:84% STUDENT: So I'm not sure in this case 00:58:05.570 --> 00:58:08.940 align:middle line:84% here why we assume that omega k is the same for both k and k 00:58:08.940 --> 00:58:09.440 align:middle line:90% prime? 00:58:09.440 --> 00:58:12.320 align:middle line:84% PROFESSOR: Oh, because you have a delta function here. 00:58:12.320 --> 00:58:15.005 align:middle line:84% Yeah, because omega k only depend on k-- 00:58:15.005 --> 00:58:16.505 align:middle line:84% it only depends on the spatial part. 00:58:16.505 --> 00:58:19.313 align:middle line:90% 00:58:19.313 --> 00:58:19.980 align:middle line:90% Other questions? 00:58:19.980 --> 00:58:20.490 align:middle line:90% Yes? 00:58:20.490 --> 00:58:22.657 align:middle line:84% STUDENT: Yeah, so the inner product of the k k prime 00:58:22.657 --> 00:58:24.270 align:middle line:90% is not Lorentz invariant. 00:58:24.270 --> 00:58:30.590 align:middle line:84% Is that just if you're listing one of the states? 00:58:30.590 --> 00:58:34.353 align:middle line:84% PROFESSOR: No, just this guy-- when you-- 00:58:34.353 --> 00:58:36.270 align:middle line:84% it's not about whether it's Lorentz invariant. 00:58:36.270 --> 00:58:37.395 align:middle line:90% It's not Lorentz covariant. 00:58:37.395 --> 00:58:40.790 align:middle line:84% Just when you transform it, just this object-- 00:58:40.790 --> 00:58:44.660 align:middle line:84% when you transform it, it's very awkward. 00:58:44.660 --> 00:58:46.890 align:middle line:84% Does not have very-- you can transform it. 00:58:46.890 --> 00:58:49.520 align:middle line:84% You can write down a transformation for this object. 00:58:49.520 --> 00:58:52.190 align:middle line:84% It just does not transform nicely. 00:58:52.190 --> 00:58:55.070 align:middle line:84% Then that means if this does not transform nicely, 00:58:55.070 --> 00:58:57.800 align:middle line:84% it means each of them don't transform nicely, 00:58:57.800 --> 00:59:00.600 align:middle line:90% and then just not convenient. 00:59:00.600 --> 00:59:02.990 align:middle line:84% So this one have the property that when 00:59:02.990 --> 00:59:06.770 align:middle line:84% I act u on this thing, and then actually, 00:59:06.770 --> 00:59:09.740 align:middle line:84% you can just directly corresponding to the state 00:59:09.740 --> 00:59:13.550 align:middle line:84% with Lorentz transform with the momentum. 00:59:13.550 --> 00:59:15.410 align:middle line:90% Yeah, and then it's very easy. 00:59:15.410 --> 00:59:18.140 align:middle line:84% And this property does not apply to this one. 00:59:18.140 --> 00:59:19.520 align:middle line:90% It does not apply to this k. 00:59:19.520 --> 00:59:24.560 align:middle line:90% 00:59:24.560 --> 00:59:27.720 align:middle line:90% OK, good. 00:59:27.720 --> 00:59:33.590 align:middle line:84% So you can also talk about a wave packet. 00:59:33.590 --> 00:59:36.210 align:middle line:90% So this is the plane wave. 00:59:36.210 --> 00:59:38.010 align:middle line:90% So this is like a plane wave. 00:59:38.010 --> 00:59:40.760 align:middle line:90% This is a momentum eigenstate. 00:59:40.760 --> 00:59:42.530 align:middle line:84% So we can also consider general-- 00:59:42.530 --> 00:59:48.370 align:middle line:90% 00:59:48.370 --> 00:59:50.815 align:middle line:84% say, single particle state will have the following form. 00:59:50.815 --> 01:00:03.130 align:middle line:90% 01:00:03.130 --> 01:00:04.460 align:middle line:90% So we have the following form. 01:00:04.460 --> 01:00:09.520 align:middle line:84% We can write it as psi just as a superposition of this k. 01:00:09.520 --> 01:00:10.960 align:middle line:84% Again, you only need to integrate 01:00:10.960 --> 01:00:14.440 align:middle line:84% over spatial momentum, because only the spatial momentum are 01:00:14.440 --> 01:00:16.970 align:middle line:90% independent. 01:00:16.970 --> 01:00:21.650 align:middle line:84% And then some arbitrary function of k, and then on this k. 01:00:21.650 --> 01:00:23.840 align:middle line:84% So any single particle state, you 01:00:23.840 --> 01:00:25.370 align:middle line:84% should be able to write it this way. 01:00:25.370 --> 01:00:27.980 align:middle line:90% 01:00:27.980 --> 01:00:34.340 align:middle line:84% So in particular, by choosing appropriate f k, 01:00:34.340 --> 01:00:36.905 align:middle line:84% you can construct a local wave packet. 01:00:36.905 --> 01:01:09.920 align:middle line:90% 01:01:09.920 --> 01:01:15.800 align:middle line:84% So you can-- so choose k-- you can 01:01:15.800 --> 01:01:28.295 align:middle line:84% choose f k to construct a localized wave packet in space. 01:01:28.295 --> 01:01:37.350 align:middle line:90% 01:01:37.350 --> 01:01:39.690 align:middle line:84% So again, we will have some exercises 01:01:39.690 --> 01:01:42.630 align:middle line:90% like this in your pset 3. 01:01:42.630 --> 01:01:45.510 align:middle line:90% 01:01:45.510 --> 01:01:56.250 align:middle line:84% And also earlier, you could have objected when I called this 01:01:56.250 --> 01:01:57.675 align:middle line:90% to be a two-particle state. 01:01:57.675 --> 01:02:01.230 align:middle line:90% 01:02:01.230 --> 01:02:03.720 align:middle line:90% Because each is a plane wave. 01:02:03.720 --> 01:02:06.570 align:middle line:84% So essentially, they are not localized. 01:02:06.570 --> 01:02:08.400 align:middle line:90% They cover all space. 01:02:08.400 --> 01:02:12.660 align:middle line:84% And in one sense, they corresponding to two particles. 01:02:12.660 --> 01:02:16.830 align:middle line:84% So now, you can solve this problem, say, 01:02:16.830 --> 01:02:29.100 align:middle line:84% by constructing using d k1 2 pi cube d k2. 01:02:29.100 --> 01:02:32.100 align:middle line:90% 01:02:32.100 --> 01:02:40.890 align:middle line:84% So by choosing appropriate function k1 k2, this k1 k2, 01:02:40.890 --> 01:02:49.935 align:middle line:84% you can construct two widely separated wave packets. 01:02:49.935 --> 01:03:00.190 align:middle line:90% 01:03:00.190 --> 01:03:05.420 align:middle line:84% Again, it's in space, so you can really talk about-- 01:03:05.420 --> 01:03:09.450 align:middle line:84% so these are the genuine two particle states. 01:03:09.450 --> 01:03:12.310 align:middle line:84% And so that confirms actually this k 01:03:12.310 --> 01:03:17.020 align:middle line:84% should be a interpreted as the plane wave 01:03:17.020 --> 01:03:18.610 align:middle line:84% version of the two particle states. 01:03:18.610 --> 01:03:35.260 align:middle line:90% 01:03:35.260 --> 01:03:35.760 align:middle line:90% Good. 01:03:35.760 --> 01:03:38.340 align:middle line:90% Any questions? 01:03:38.340 --> 01:03:39.240 align:middle line:90% Yes? 01:03:39.240 --> 01:03:41.820 align:middle line:84% STUDENT: So now that we don't have a position operator 01:03:41.820 --> 01:03:45.360 align:middle line:84% like we did before, what's the conjugate, I guess, 01:03:45.360 --> 01:03:48.270 align:middle line:84% operator to our momentum operator, or is there-- 01:03:48.270 --> 01:03:49.020 align:middle line:90% PROFESSOR: Good. 01:03:49.020 --> 01:03:51.390 align:middle line:90% That's a very good question. 01:03:51.390 --> 01:03:56.790 align:middle line:84% Indeed, there will be a problem in your pset 3 which asks you 01:03:56.790 --> 01:04:01.027 align:middle line:84% to show there's no eigenstate-- or there's no eigenstate -- 01:04:01.027 --> 01:04:01.860 align:middle line:90% there is no perfect. 01:04:01.860 --> 01:04:04.470 align:middle line:84% Yeah, so in non-relativistic quantum mechanics, 01:04:04.470 --> 01:04:05.940 align:middle line:90% you have this state. 01:04:05.940 --> 01:04:09.480 align:middle line:84% You have this wave function, so you can localize at the point, 01:04:09.480 --> 01:04:12.780 align:middle line:84% but there's no analog of such kind of state 01:04:12.780 --> 01:04:15.780 align:middle line:84% in the relativistic case precisely 01:04:15.780 --> 01:04:18.210 align:middle line:84% because there's no position operator anymore. 01:04:18.210 --> 01:04:22.710 align:middle line:84% Yeah, so you will explore this a little bit more later yourself. 01:04:22.710 --> 01:04:26.950 align:middle line:84% Just with this formalism, you can explore all these questions 01:04:26.950 --> 01:04:28.210 align:middle line:90% yourself. 01:04:28.210 --> 01:04:29.020 align:middle line:90% That's the key. 01:04:29.020 --> 01:04:32.125 align:middle line:84% Actually, you already have the power to do that. 01:04:32.125 --> 01:04:40.300 align:middle line:90% 01:04:40.300 --> 01:04:43.270 align:middle line:84% So now, let's say a little bit more 01:04:43.270 --> 01:04:46.280 align:middle line:84% on the structure of the Hilbert space. 01:04:46.280 --> 01:04:48.340 align:middle line:84% So, here the structure of the Hilbert space 01:04:48.340 --> 01:04:57.800 align:middle line:84% is a little bit special because there's no interaction. 01:04:57.800 --> 01:05:00.400 align:middle line:84% So you can just separate it into-- 01:05:00.400 --> 01:05:04.750 align:middle line:84% so the full Hilbert space, which I write as script H. 01:05:04.750 --> 01:05:07.960 align:middle line:84% Then, you can separate it into the vacuum state, 01:05:07.960 --> 01:05:11.068 align:middle line:84% and then you have the one particle Hilbert space, 01:05:11.068 --> 01:05:13.110 align:middle line:84% and then you have the two particle Hilbert space, 01:05:13.110 --> 01:05:13.990 align:middle line:90% and they're all-- 01:05:13.990 --> 01:05:16.030 align:middle line:84% they don't have anything to do with each other 01:05:16.030 --> 01:05:17.695 align:middle line:90% because there's no interaction. 01:05:17.695 --> 01:05:24.200 align:middle line:90% 01:05:24.200 --> 01:05:26.840 align:middle line:84% So in principle, you can have infinite number of them. 01:05:26.840 --> 01:05:29.640 align:middle line:84% You can have infinite number of them. 01:05:29.640 --> 01:05:36.050 align:middle line:84% So if you restrict to always a finite number of particles-- 01:05:36.050 --> 01:05:37.550 align:middle line:90% so the set of states-- 01:05:37.550 --> 01:05:42.420 align:middle line:90% 01:05:42.420 --> 01:05:49.470 align:middle line:84% so this is one particle, this is two particle, et cetera. 01:05:49.470 --> 01:05:54.140 align:middle line:84% So that's of states with a finite number of particles. 01:05:54.140 --> 01:06:03.970 align:middle line:90% 01:06:03.970 --> 01:06:05.095 align:middle line:90% It's called the Fock space. 01:06:05.095 --> 01:06:13.480 align:middle line:90% 01:06:13.480 --> 01:06:16.240 align:middle line:84% In the Fock space, we only have state 01:06:16.240 --> 01:06:17.680 align:middle line:90% of finite number of particles. 01:06:17.680 --> 01:06:20.920 align:middle line:84% One can have arbitrary number of them-- 01:06:20.920 --> 01:06:22.570 align:middle line:90% as large as you want. 01:06:22.570 --> 01:06:26.485 align:middle line:84% 10 billion, 100 billion, but it will be finite. 01:06:26.485 --> 01:06:30.850 align:middle line:90% 01:06:30.850 --> 01:06:41.150 align:middle line:90% So finally, those k-- 01:06:41.150 --> 01:06:43.580 align:middle line:84% either this k or this k, they are just 01:06:43.580 --> 01:06:46.250 align:middle line:84% plane wave normalized because they're 01:06:46.250 --> 01:06:48.330 align:middle line:90% normalize by delta function. 01:06:48.330 --> 01:06:52.520 align:middle line:84% So strictly speaking, they are not normalizable states. 01:06:52.520 --> 01:06:54.440 align:middle line:84% And so strictly speaking, they are not 01:06:54.440 --> 01:06:56.780 align:middle line:90% in the physical Hilbert space. 01:06:56.780 --> 01:07:06.550 align:middle line:84% So strictly, speaking k or k or any of those states of k-- 01:07:06.550 --> 01:07:10.580 align:middle line:90% or k1 k2 are not normalizable. 01:07:10.580 --> 01:07:12.410 align:middle line:84% They are only plane wave normalizable. 01:07:12.410 --> 01:07:19.820 align:middle line:90% 01:07:19.820 --> 01:07:27.410 align:middle line:84% So just like-- just as psi is equal to exponential ik 01:07:27.410 --> 01:07:31.970 align:middle line:84% x in non-relativistic quantum mechanics is not normalizable. 01:07:31.970 --> 01:07:37.430 align:middle line:90% 01:07:37.430 --> 01:07:41.960 align:middle line:84% So they're convenient for various mathematical 01:07:41.960 --> 01:07:44.720 align:middle line:84% operations, but they don't correspond 01:07:44.720 --> 01:07:47.750 align:middle line:90% to genuine physical states. 01:07:47.750 --> 01:07:51.980 align:middle line:90% So not genuine physical states. 01:07:51.980 --> 01:07:55.340 align:middle line:90% 01:07:55.340 --> 01:08:03.330 align:middle line:84% So physical state, we have to consider for example 01:08:03.330 --> 01:08:05.280 align:middle line:90% this kind of state. 01:08:05.280 --> 01:08:11.400 align:middle line:84% And then you choose f k so that is normalizable. 01:08:11.400 --> 01:08:15.380 align:middle line:90% So if we take this, and then-- 01:08:15.380 --> 01:08:22.960 align:middle line:90% 01:08:22.960 --> 01:08:33.529 align:middle line:84% so with psi, given by that, then the normalization of psi, 01:08:33.529 --> 01:08:37.670 align:middle line:84% you can calculate it, and then this is given by-- 01:08:37.670 --> 01:08:41.779 align:middle line:90% 01:08:41.779 --> 01:08:45.750 align:middle line:84% so you can easily guess the answer. 01:08:45.750 --> 01:08:47.960 align:middle line:84% So when you do the overlap exactly, 01:08:47.960 --> 01:08:50.930 align:middle line:84% you find that this given by the modulus of f k squared. 01:08:50.930 --> 01:08:56.050 align:middle line:90% 01:08:56.050 --> 01:08:57.850 align:middle line:84% So if you do the calculation to calculate 01:08:57.850 --> 01:09:00.430 align:middle line:84% the norm of that state, you find that given by that. 01:09:00.430 --> 01:09:02.859 align:middle line:84% And then we can choose this to be finite, 01:09:02.859 --> 01:09:05.650 align:middle line:84% and then this will be then will be a normalized state. 01:09:05.650 --> 01:09:14.229 align:middle line:90% 01:09:14.229 --> 01:09:19.210 align:middle line:84% So for this reason, it's sometimes-- 01:09:19.210 --> 01:09:20.410 align:middle line:90% we don't do it very often-- 01:09:20.410 --> 01:09:22.930 align:middle line:90% 01:09:22.930 --> 01:09:33.905 align:middle line:84% can choose just a basis of f x, say, of f i. 01:09:33.905 --> 01:09:36.720 align:middle line:90% 01:09:36.720 --> 01:09:39.300 align:middle line:90% So i runs some number. 01:09:39.300 --> 01:09:58.920 align:middle line:84% And then this alpha-- then you can define alpha i and then 01:09:58.920 --> 01:10:00.000 align:middle line:90% this will provide-- 01:10:00.000 --> 01:10:07.290 align:middle line:84% the alpha i will provide a basis of the single particle 01:10:07.290 --> 01:10:08.280 align:middle line:90% normalizable states-- 01:10:08.280 --> 01:10:11.130 align:middle line:90% 01:10:11.130 --> 01:10:13.150 align:middle line:90% a single particle state. 01:10:13.150 --> 01:10:18.500 align:middle line:84% So here is a fun fact related to the earlier question was asked. 01:10:18.500 --> 01:10:23.140 align:middle line:90% So naively, here, you have-- 01:10:23.140 --> 01:10:26.320 align:middle line:90% if you look at the k-- 01:10:26.320 --> 01:10:28.000 align:middle line:90% so k form a basis. 01:10:28.000 --> 01:10:30.670 align:middle line:84% Let's just look at the single particle state. 01:10:30.670 --> 01:10:36.490 align:middle line:84% So naively, this k forms a continuum-- uncountable 01:10:36.490 --> 01:10:38.140 align:middle line:90% continuum of basis of states. 01:10:38.140 --> 01:10:41.170 align:middle line:90% 01:10:41.170 --> 01:10:44.710 align:middle line:84% But once we impose this normalizable condition, 01:10:44.710 --> 01:10:47.920 align:middle line:84% means you need to choose the normalizable f. 01:10:47.920 --> 01:10:51.100 align:middle line:84% And then you can show the basis of this normalizable f actually 01:10:51.100 --> 01:10:52.900 align:middle line:90% is countable. 01:10:52.900 --> 01:10:54.750 align:middle line:84% So in the end, the Hilbert space-- 01:10:54.750 --> 01:10:56.910 align:middle line:84% the one-dimensional the one particle Hilbert 01:10:56.910 --> 01:11:00.600 align:middle line:84% space is actually generated by countable basis 01:11:00.600 --> 01:11:03.540 align:middle line:84% just like in your ordinary quantum mechanics. 01:11:03.540 --> 01:11:07.860 align:middle line:84% It's actually rather than uncountable basis like in k 01:11:07.860 --> 01:11:09.552 align:middle line:84% once you restrict normalizability. 01:11:09.552 --> 01:11:11.850 align:middle line:84% And so this is a fun mathematical fact. 01:11:11.850 --> 01:11:16.450 align:middle line:90% 01:11:16.450 --> 01:11:17.820 align:middle line:90% Good. 01:11:17.820 --> 01:11:18.810 align:middle line:90% Questions on this? 01:11:18.810 --> 01:11:29.450 align:middle line:90% 01:11:29.450 --> 01:11:41.270 align:middle line:84% So the last few minutes, we talk about conserved charges. 01:11:41.270 --> 01:11:46.220 align:middle line:90% 01:11:46.220 --> 01:11:52.090 align:middle line:84% Quantum-- the role of conserved charges. 01:11:52.090 --> 01:11:58.480 align:middle line:90% 01:11:58.480 --> 01:12:05.410 align:middle line:84% So classically, we say if an action is 01:12:05.410 --> 01:12:07.450 align:middle line:90% invariant under some symmetry-- 01:12:07.450 --> 01:12:10.690 align:middle line:84% some transformation-- some continuous transformation, 01:12:10.690 --> 01:12:20.880 align:middle line:84% say, of the infinitesimal form, so the alpha are 01:12:20.880 --> 01:12:23.910 align:middle line:90% the infinitesimal parameters. 01:12:23.910 --> 01:12:27.060 align:middle line:84% So the alpha label different transformations, 01:12:27.060 --> 01:12:28.950 align:middle line:90% and a label different fields. 01:12:28.950 --> 01:12:31.140 align:middle line:84% Alpha label different transformations. 01:12:31.140 --> 01:12:35.820 align:middle line:84% And then you will get the conserved current J 01:12:35.820 --> 01:12:39.810 align:middle line:84% mu labeled by this alpha, so alpha label differently 01:12:39.810 --> 01:12:40.740 align:middle line:90% for each. 01:12:40.740 --> 01:12:44.970 align:middle line:84% So alpha label different symmetries for each symmetry, 01:12:44.970 --> 01:12:47.760 align:middle line:84% and then you have a conserved current, 01:12:47.760 --> 01:12:51.345 align:middle line:84% which is satisfy the J mu alpha equal to 0. 01:12:51.345 --> 01:12:54.590 align:middle line:90% 01:12:54.590 --> 01:12:57.350 align:middle line:84% The alpha equal to 1, 2, et cetera-- 01:12:57.350 --> 01:12:58.940 align:middle line:90% label different symmetries. 01:12:58.940 --> 01:13:06.800 align:middle line:90% 01:13:06.800 --> 01:13:10.760 align:middle line:84% And then you can write down your Noether current. 01:13:10.760 --> 01:13:13.490 align:middle line:84% So we derive a general formula for the Noether current 01:13:13.490 --> 01:13:16.475 align:middle line:84% is given by the form, say, partial L-- 01:13:16.475 --> 01:13:19.250 align:middle line:90% 01:13:19.250 --> 01:13:23.446 align:middle line:90% partial partial mu phi a. 01:13:23.446 --> 01:13:25.040 align:middle line:90% And then f a alpha. 01:13:25.040 --> 01:13:26.790 align:middle line:84% So this is essentially the same as before. 01:13:26.790 --> 01:13:29.670 align:middle line:84% Essentially, I just add the index alpha now to label. 01:13:29.670 --> 01:13:32.070 align:middle line:84% You can have more than one symmetries. 01:13:32.070 --> 01:13:34.620 align:middle line:90% And then minus k mu alpha. 01:13:34.620 --> 01:13:38.730 align:middle line:84% So k mu alpha is the derivative-- 01:13:38.730 --> 01:13:43.410 align:middle line:84% total derivative -- suppose -- delta L is given by epsilon 01:13:43.410 --> 01:13:47.280 align:middle line:90% alpha partial mu k mu alpha. 01:13:47.280 --> 01:13:51.070 align:middle line:84% So the k mu alpha is the total derivative corresponding 01:13:51.070 --> 01:13:53.280 align:middle line:90% to that respect -- 01:13:53.280 --> 01:13:57.270 align:middle line:84% corresponding to that transformation. 01:13:57.270 --> 01:14:00.420 align:middle line:84% And then the q alpha, that will be 01:14:00.420 --> 01:14:08.130 align:middle line:84% the conserved charge for that current, so zeroth component. 01:14:08.130 --> 01:14:09.810 align:middle line:90% So this is the classical story. 01:14:09.810 --> 01:14:13.080 align:middle line:84% Now, I'm using a little bit more general notation 01:14:13.080 --> 01:14:15.730 align:middle line:90% for each alpha. 01:14:15.730 --> 01:14:16.995 align:middle line:90% So now, the key thing-- 01:14:16.995 --> 01:14:21.115 align:middle line:90% 01:14:21.115 --> 01:14:23.240 align:middle line:84% so now the key thing is that at the quantum level-- 01:14:23.240 --> 01:14:34.768 align:middle line:90% 01:14:34.768 --> 01:14:35.810 align:middle line:90% so at the quantum level-- 01:14:35.810 --> 01:14:43.910 align:middle line:90% 01:14:43.910 --> 01:14:45.500 align:middle line:84% first, as we already seen in the case 01:14:45.500 --> 01:14:48.320 align:middle line:84% of the Hamiltonian and the momentum operator, 01:14:48.320 --> 01:14:52.640 align:middle line:90% so q alpha is time independent-- 01:14:52.640 --> 01:14:54.170 align:middle line:90% is a constant operator. 01:14:54.170 --> 01:15:00.260 align:middle line:90% 01:15:00.260 --> 01:15:03.050 align:middle line:84% So this is the fact that this is conserved. 01:15:03.050 --> 01:15:08.350 align:middle line:90% 01:15:08.350 --> 01:15:13.610 align:middle line:84% And also another important property of q 01:15:13.610 --> 01:15:19.520 align:middle line:84% is that when you act q alpha on your field, 01:15:19.520 --> 01:15:22.255 align:middle line:84% if you look at the commutator between this q alpha 01:15:22.255 --> 01:15:27.560 align:middle line:84% on your field, it actually generates 01:15:27.560 --> 01:15:30.760 align:middle line:90% finite transformation. 01:15:30.760 --> 01:15:40.480 align:middle line:84% So turns out-- yeah, so if we have put 01:15:40.480 --> 01:15:42.325 align:middle line:84% a parameter here-- epsilon alpha. 01:15:42.325 --> 01:15:45.190 align:middle line:90% 01:15:45.190 --> 01:15:47.770 align:middle line:84% Yeah, so it doesn't matter for Q-- 01:15:47.770 --> 01:15:50.680 align:middle line:90% say, it's upstairs, downstairs. 01:15:50.680 --> 01:15:51.715 align:middle line:90% And then you get-- 01:15:51.715 --> 01:15:56.060 align:middle line:90% 01:15:56.060 --> 01:15:59.150 align:middle line:84% so you just generate the transformation for f-- 01:15:59.150 --> 01:16:01.280 align:middle line:90% for phi. 01:16:01.280 --> 01:16:03.120 align:middle line:90% So this can be shown explicitly. 01:16:03.120 --> 01:16:05.900 align:middle line:90% 01:16:05.900 --> 01:16:08.520 align:middle line:90% So you can show this explicitly. 01:16:08.520 --> 01:16:10.260 align:middle line:90% We're running out of time now. 01:16:10.260 --> 01:16:14.540 align:middle line:84% So you can show explicitly just using this expression. 01:16:14.540 --> 01:16:20.310 align:middle line:84% So let me just quickly outline the way to do it. 01:16:20.310 --> 01:16:21.995 align:middle line:90% So if you, say-- 01:16:21.995 --> 01:16:26.510 align:middle line:90% 01:16:26.510 --> 01:16:31.956 align:middle line:84% so let's look at the zeroth component of this J. 01:16:31.956 --> 01:16:46.530 align:middle line:84% So the zeroth component of the J just equal to partial L, 01:16:46.530 --> 01:16:52.155 align:middle line:84% partial, partial, 0 phi a and f a alpha. 01:16:52.155 --> 01:16:55.200 align:middle line:90% 01:16:55.200 --> 01:16:56.700 align:middle line:90% So let's consider the-- 01:16:56.700 --> 01:17:00.300 align:middle line:90% 01:17:00.300 --> 01:17:04.080 align:middle line:84% yeah, I think today we will not have time to finish anyway. 01:17:04.080 --> 01:17:08.640 align:middle line:84% Yeah, let's just leave this to the next time. 01:17:08.640 --> 01:17:11.640 align:middle line:84% Yeah, let me just say that the quantum level, 01:17:11.640 --> 01:17:15.160 align:middle line:84% this will generate the infinitesimal transformation. 01:17:15.160 --> 01:17:17.880 align:middle line:84% And then you can also exponentiate 01:17:17.880 --> 01:17:24.930 align:middle line:84% this Q. You can introduce, say, exponential i lambda Q. 01:17:24.930 --> 01:17:28.620 align:middle line:84% So if q is the conserved charge-- 01:17:28.620 --> 01:17:32.460 align:middle line:84% and then you can exponentiate Q like this. 01:17:32.460 --> 01:17:38.700 align:middle line:84% And then this operator then generates-- 01:17:38.700 --> 01:17:45.970 align:middle line:84% when it acts on phi, then generates 01:17:45.970 --> 01:17:47.380 align:middle line:90% finite transformations. 01:17:47.380 --> 01:17:58.960 align:middle line:90% 01:17:58.960 --> 01:18:02.050 align:middle line:84% So this takes you to phi a prime-- 01:18:02.050 --> 01:18:03.400 align:middle line:90% the finite transformations. 01:18:03.400 --> 01:18:07.450 align:middle line:84% Anyway, so in your pset, for pset 2, 01:18:07.450 --> 01:18:12.290 align:middle line:84% you actually check yourself a couple of simple examples, 01:18:12.290 --> 01:18:15.800 align:middle line:90% and you will see this works. 01:18:15.800 --> 01:18:21.100 align:middle line:84% And in the next lecture, we will give you a simple derivation 01:18:21.100 --> 01:18:23.620 align:middle line:90% to show this actually works. 01:18:23.620 --> 01:18:25.740 align:middle line:90% It works in general. 01:18:25.740 --> 01:18:31.000 align:middle line:90%