WEBVTT 00:00:00.000 --> 00:00:01.984 align:middle line:90% [SQUEAKING] 00:00:01.984 --> 00:00:03.472 align:middle line:90% [RUSTLING] 00:00:03.472 --> 00:00:06.944 align:middle line:90% [CLICKING] 00:00:06.944 --> 00:00:10.420 align:middle line:90% 00:00:10.420 --> 00:00:11.570 align:middle line:90% HONG LIU: Good. 00:00:11.570 --> 00:00:12.070 align:middle line:90% Yeah. 00:00:12.070 --> 00:00:18.220 align:middle line:84% So last lecture, we discussed Noether's theorem. 00:00:18.220 --> 00:00:22.510 align:middle line:84% For every symmetry, for every continuous symmetry, 00:00:22.510 --> 00:00:25.540 align:middle line:90% there's a conserved current. 00:00:25.540 --> 00:00:29.230 align:middle line:84% And then we also started talking about relativistic quantum 00:00:29.230 --> 00:00:31.780 align:middle line:90% mechanics. 00:00:31.780 --> 00:00:36.370 align:middle line:84% How we want to unify special relativity 00:00:36.370 --> 00:00:37.990 align:middle line:90% in quantum mechanics. 00:00:37.990 --> 00:00:40.840 align:middle line:84% So the most immediate idea for that 00:00:40.840 --> 00:00:44.770 align:middle line:84% is what's called the relativistic quantum mechanics. 00:00:44.770 --> 00:00:49.300 align:middle line:84% And the most immediate generalization 00:00:49.300 --> 00:00:51.560 align:middle line:90% of the Schrodinger equation-- 00:00:51.560 --> 00:00:52.480 align:middle line:90% so if you have-- 00:00:52.480 --> 00:00:55.110 align:middle line:90% 00:00:55.110 --> 00:00:56.950 align:middle line:84% so at the end of last lecture, we 00:00:56.950 --> 00:00:59.950 align:middle line:84% talked about, say, the most immediate generalization 00:00:59.950 --> 00:01:05.010 align:middle line:84% of the Schrodinger equation, which-- 00:01:05.010 --> 00:01:10.830 align:middle line:84% so if I have E equal to, say, p squared divided by 2m, 00:01:10.830 --> 00:01:16.410 align:middle line:84% and then you go to non-relativistic quantum 00:01:16.410 --> 00:01:17.985 align:middle line:90% mechanics Schrodinger equation. 00:01:17.985 --> 00:01:22.070 align:middle line:90% 00:01:22.070 --> 00:01:24.500 align:middle line:84% And now if you have E squared equal to p 00:01:24.500 --> 00:01:29.240 align:middle line:84% squared plus m squared for relativistic particle, 00:01:29.240 --> 00:01:31.790 align:middle line:84% and then you get what's called the Klein-Gordon equation. 00:01:31.790 --> 00:01:38.610 align:middle line:90% 00:01:38.610 --> 00:01:41.580 align:middle line:84% And again, this side has the implication 00:01:41.580 --> 00:01:44.580 align:middle line:90% of the wave function. 00:01:44.580 --> 00:01:47.620 align:middle line:84% So this describe-- and then this-- 00:01:47.620 --> 00:01:50.100 align:middle line:84% so as a generalization of this, then 00:01:50.100 --> 00:01:53.395 align:middle line:84% this means to describe the quantum 00:01:53.395 --> 00:01:59.250 align:middle line:84% mechanics of relativistic free particle, say, of mass m, 00:01:59.250 --> 00:02:01.470 align:middle line:90% of mass m. 00:02:01.470 --> 00:02:09.930 align:middle line:84% So here, the psi pi t, x is the wave function 00:02:09.930 --> 00:02:11.249 align:middle line:90% of a relativistic particle. 00:02:11.249 --> 00:02:17.150 align:middle line:90% 00:02:17.150 --> 00:02:22.550 align:middle line:90% Relativistic particle of mass m. 00:02:22.550 --> 00:02:30.240 align:middle line:90% 00:02:30.240 --> 00:02:32.610 align:middle line:84% And we also notice that this equation actually 00:02:32.610 --> 00:02:37.560 align:middle line:84% is the same as the simplest field theory equation. 00:02:37.560 --> 00:02:45.320 align:middle line:84% So we also talked about a simplest scalar field theory. 00:02:45.320 --> 00:02:45.980 align:middle line:90% Classical. 00:02:45.980 --> 00:02:49.460 align:middle line:90% So here is a simple-- 00:02:49.460 --> 00:02:52.505 align:middle line:84% a classic simplest scalar field theory. 00:02:52.505 --> 00:02:56.640 align:middle line:90% 00:02:56.640 --> 00:03:02.385 align:middle line:84% So this theory you can write down the action of the form. 00:03:02.385 --> 00:03:09.850 align:middle line:90% 00:03:09.850 --> 00:03:14.490 align:middle line:84% So this is the simplest theory you can write down. 00:03:14.490 --> 00:03:17.310 align:middle line:84% And then relativistic environment theory. 00:03:17.310 --> 00:03:19.930 align:middle line:84% And then the equation of motion of this-- 00:03:19.930 --> 00:03:23.250 align:middle line:84% so this is-- you view this as a classical field. 00:03:23.250 --> 00:03:25.230 align:middle line:84% And again, this has the equation of motion 00:03:25.230 --> 00:03:29.550 align:middle line:84% have exactly the same form as this equation. 00:03:29.550 --> 00:03:34.590 align:middle line:84% But now, here, phi, again, is a function of t, x. 00:03:34.590 --> 00:03:37.510 align:middle line:84% Now has a completely different physical interpretation. 00:03:37.510 --> 00:03:41.895 align:middle line:84% So here is-- this is a classical field. 00:03:41.895 --> 00:03:44.440 align:middle line:90% 00:03:44.440 --> 00:03:44.940 align:middle line:90% OK. 00:03:44.940 --> 00:03:47.140 align:middle line:90% So this is a classical field. 00:03:47.140 --> 00:03:47.640 align:middle line:90% OK. 00:03:47.640 --> 00:03:50.880 align:middle line:84% So in this case, the interpretation 00:03:50.880 --> 00:03:56.310 align:middle line:84% of the x in here, and in here, it's very different. 00:03:56.310 --> 00:04:00.030 align:middle line:84% So not only phi and the psi, the physical interpretation 00:04:00.030 --> 00:04:01.045 align:middle line:90% are different. 00:04:01.045 --> 00:04:03.420 align:middle line:84% The physical interpretation of x also here are different. 00:04:03.420 --> 00:04:05.400 align:middle line:90% Here, x is just a label. 00:04:05.400 --> 00:04:10.250 align:middle line:84% Is a label for the location in the space which 00:04:10.250 --> 00:04:12.070 align:middle line:90% we define this field. 00:04:12.070 --> 00:04:14.820 align:middle line:84% But here, the x is the eigenvalue 00:04:14.820 --> 00:04:20.640 align:middle line:84% of the position operator for this relativistic particle. 00:04:20.640 --> 00:04:25.980 align:middle line:84% And so they have very different physical interpretation. 00:04:25.980 --> 00:04:29.390 align:middle line:84% And so let me just label this equation by 1 00:04:29.390 --> 00:04:33.290 align:middle line:84% and label this by 2 and this by 2 prime. 00:04:33.290 --> 00:04:37.470 align:middle line:90% 00:04:37.470 --> 00:04:41.490 align:middle line:84% So we also mentioned that this one has-- 00:04:41.490 --> 00:04:44.070 align:middle line:84% the interpretation of this as the wave function 00:04:44.070 --> 00:04:46.230 align:middle line:90% for relativistic-- yeah, for-- 00:04:46.230 --> 00:04:48.660 align:middle line:84% yeah, so the equation for relativistic quantum mechanics 00:04:48.660 --> 00:04:50.160 align:middle line:90% has a number of difficulties. 00:04:50.160 --> 00:04:58.040 align:middle line:90% 00:04:58.040 --> 00:05:06.040 align:middle line:84% So the first is that, as you will show in your pset 2, 00:05:06.040 --> 00:05:08.170 align:middle line:90% there's no sensible-- 00:05:08.170 --> 00:05:23.840 align:middle line:84% no sensible way to define a positive definite probability 00:05:23.840 --> 00:05:25.370 align:middle line:90% density. 00:05:25.370 --> 00:05:33.260 align:middle line:84% So if you want to interpret this as a wave equation, 00:05:33.260 --> 00:05:34.492 align:middle line:90% then you must have a wave-- 00:05:34.492 --> 00:05:36.200 align:middle line:84% then you must have a probability density, 00:05:36.200 --> 00:05:38.390 align:middle line:84% because in quantum mechanics, probability 00:05:38.390 --> 00:05:41.560 align:middle line:90% should be conserved. 00:05:41.560 --> 00:05:45.970 align:middle line:84% And the second difficulty is that there's a negative energy 00:05:45.970 --> 00:05:49.570 align:middle line:90% state because of the square. 00:05:49.570 --> 00:05:51.430 align:middle line:84% Because when you take the square root, 00:05:51.430 --> 00:05:54.370 align:middle line:84% then you get the minus sign, and then there's a negative energy 00:05:54.370 --> 00:06:01.540 align:middle line:84% state, which you cannot avoid in quantum mechanics even though 00:06:01.540 --> 00:06:05.260 align:middle line:84% classically you can just throw them away by hand. 00:06:05.260 --> 00:06:13.050 align:middle line:84% And the third thing we mentioned at the end, 00:06:13.050 --> 00:06:17.670 align:middle line:84% we said for a relativistic wave equation, 00:06:17.670 --> 00:06:23.660 align:middle line:84% you can describe a fixed number of particles. 00:06:23.660 --> 00:06:29.320 align:middle line:90% 00:06:29.320 --> 00:06:32.050 align:middle line:84% So the particle number cannot change. 00:06:32.050 --> 00:06:37.930 align:middle line:84% So this equation describe a single particle. 00:06:37.930 --> 00:06:41.020 align:middle line:84% And if you want to describe two particles, 00:06:41.020 --> 00:06:44.110 align:middle line:84% then you need to write down a separate equation 00:06:44.110 --> 00:06:47.000 align:middle line:90% for a different wave function. 00:06:47.000 --> 00:06:49.240 align:middle line:84% So this is for the two-particle wave function will 00:06:49.240 --> 00:06:51.760 align:middle line:90% be like this and et cetera. 00:06:51.760 --> 00:06:52.870 align:middle line:90% OK. 00:06:52.870 --> 00:06:55.480 align:middle line:84% But this does not really make sense 00:06:55.480 --> 00:06:58.480 align:middle line:84% in the relativistic system because we 00:06:58.480 --> 00:07:03.710 align:middle line:84% know that in the relativistic system, E equal to mc square, 00:07:03.710 --> 00:07:05.730 align:middle line:84% in any case, you have enough energy, 00:07:05.730 --> 00:07:08.150 align:middle line:84% then you should be able to create particles. 00:07:08.150 --> 00:07:11.480 align:middle line:84% And then that means the number of particles in the given 00:07:11.480 --> 00:07:13.890 align:middle line:90% process is not conserved. 00:07:13.890 --> 00:07:16.670 align:middle line:84% So if you want to use your quantum mechanics to describe 00:07:16.670 --> 00:07:18.950 align:middle line:90% a process, and then that-- 00:07:18.950 --> 00:07:22.220 align:middle line:84% you cannot have a formalism, which the number of particles 00:07:22.220 --> 00:07:25.070 align:middle line:84% is fixed, which you cannot change. 00:07:25.070 --> 00:07:29.750 align:middle line:84% And so this is actually the most fundamental difficulty, 00:07:29.750 --> 00:07:33.890 align:middle line:84% is that you cannot change the number of particles. 00:07:33.890 --> 00:07:39.900 align:middle line:84% And related to this difficulty is this interpretation. 00:07:39.900 --> 00:07:42.930 align:middle line:84% Here, we say-- now if you want to-- 00:07:42.930 --> 00:07:46.980 align:middle line:84% we're saying here, there's a fundamental asymmetry 00:07:46.980 --> 00:07:48.990 align:middle line:90% between the t and x. 00:07:48.990 --> 00:07:51.820 align:middle line:84% Also, yeah, maybe let me put it as 4, 00:07:51.820 --> 00:07:53.340 align:middle line:90% there's also fundamental-- 00:07:53.340 --> 00:07:56.040 align:middle line:90% 00:07:56.040 --> 00:07:57.890 align:middle line:84% an additional difficulty, there's 00:07:57.890 --> 00:08:08.055 align:middle line:84% a fundamental asymmetry between t and x. 00:08:08.055 --> 00:08:11.780 align:middle line:90% 00:08:11.780 --> 00:08:14.930 align:middle line:84% So here, in the wave equation, t is just 00:08:14.930 --> 00:08:20.010 align:middle line:84% a parameter, which will be describing the evolution, 00:08:20.010 --> 00:08:28.110 align:middle line:84% but the x is the eigenvalue of quantum operators-- 00:08:28.110 --> 00:08:30.750 align:middle line:84% eigenvalues of quantum operators. 00:08:30.750 --> 00:08:33.570 align:middle line:90% Say, corresponding to, say, hat. 00:08:33.570 --> 00:08:37.030 align:middle line:84% By putting a hat, we denote the corresponding a quantum 00:08:37.030 --> 00:08:37.530 align:middle line:90% operator. 00:08:37.530 --> 00:08:40.710 align:middle line:84% Yeah, so this is the eigenvalue of position operators. 00:08:40.710 --> 00:08:43.440 align:middle line:84% And this asymmetry become even more pronounced, say, 00:08:43.440 --> 00:08:45.630 align:middle line:84% if you can see the two particles. 00:08:45.630 --> 00:08:48.055 align:middle line:84% You have two x's here, but the only one t. 00:08:48.055 --> 00:08:48.555 align:middle line:90% OK. 00:08:48.555 --> 00:08:52.270 align:middle line:90% 00:08:52.270 --> 00:08:56.650 align:middle line:84% So those-- because of those fundamental difficulties-- 00:08:56.650 --> 00:09:05.580 align:middle line:84% so if you connect this i to iv, so we conclude 00:09:05.580 --> 00:09:13.630 align:middle line:84% that the relativistic quantum mechanics defined in the sense 00:09:13.630 --> 00:09:15.640 align:middle line:84% that you write down a wave equation, 00:09:15.640 --> 00:09:18.560 align:middle line:84% and for wave function, don't even-- 00:09:18.560 --> 00:09:23.736 align:middle line:90% it does not-- that's not-- 00:09:23.736 --> 00:09:31.338 align:middle line:84% cannot be the fundamental description. 00:09:31.338 --> 00:09:33.130 align:middle line:84% But yeah, but relativistic quantum mechanic 00:09:33.130 --> 00:09:34.880 align:middle line:84% just refers to this kind of wave equation. 00:09:34.880 --> 00:09:43.930 align:middle line:90% 00:09:43.930 --> 00:09:50.550 align:middle line:84% So at most, this can be approximated approximation 00:09:50.550 --> 00:09:51.050 align:middle line:90% at most. 00:09:51.050 --> 00:09:54.980 align:middle line:90% 00:09:54.980 --> 00:10:08.510 align:middle line:84% This is approximate description in situations, say, 00:10:08.510 --> 00:10:16.280 align:middle line:84% there's no particle creation or annihilation. 00:10:16.280 --> 00:10:22.360 align:middle line:90% 00:10:22.360 --> 00:10:28.980 align:middle line:84% So in case which your particle number is fixed, 00:10:28.980 --> 00:10:32.250 align:middle line:84% and then you can use this as approximation, 00:10:32.250 --> 00:10:34.740 align:middle line:84% but it cannot be a fundamental description. 00:10:34.740 --> 00:10:37.860 align:middle line:84% For example, later we will talk about the fermionic version 00:10:37.860 --> 00:10:39.130 align:middle line:90% of this wave equation. 00:10:39.130 --> 00:10:41.970 align:middle line:84% So this will describe a particle without spin. 00:10:41.970 --> 00:10:45.300 align:middle line:84% So later we will describe the analogous equation 00:10:45.300 --> 00:10:49.260 align:middle line:84% for electrons for spin half, and then that 00:10:49.260 --> 00:10:53.010 align:middle line:84% can indeed be used to describe the electron in a hydrogen atom 00:10:53.010 --> 00:10:57.220 align:middle line:84% as far as you don't create new electrons, et cetera. 00:10:57.220 --> 00:10:59.760 align:middle line:84% Anyway, so relativistic quantum mechanics 00:10:59.760 --> 00:11:01.590 align:middle line:84% can only be described as some kind of-- 00:11:01.590 --> 00:11:04.170 align:middle line:84% considered as an approximate description. 00:11:04.170 --> 00:11:11.830 align:middle line:84% But now if you want to unify special relativity and quantum 00:11:11.830 --> 00:11:15.560 align:middle line:84% mechanics together, it turns out that the right formulation 00:11:15.560 --> 00:11:18.390 align:middle line:90% is just quantum field theory. 00:11:18.390 --> 00:11:24.750 align:middle line:84% So it turns out that the quantum field theory 00:11:24.750 --> 00:11:27.420 align:middle line:90% addresses these difficulties. 00:11:27.420 --> 00:11:37.130 align:middle line:90% 00:11:37.130 --> 00:11:38.770 align:middle line:90% OK. 00:11:38.770 --> 00:11:43.600 align:middle line:84% So it turns out that the right way-- 00:11:43.600 --> 00:11:46.560 align:middle line:84% so if we want to describe quantum mechanics, 00:11:46.560 --> 00:11:49.950 align:middle line:84% say, of relativistic particles of mass m-- 00:11:49.950 --> 00:11:55.080 align:middle line:84% so we want to do here, it turns out, the proper thing to do, 00:11:55.080 --> 00:11:57.500 align:middle line:84% which is a little bit unintuitive at first sight, 00:11:57.500 --> 00:12:01.400 align:middle line:84% is to start with this field theory, which seemingly have 00:12:01.400 --> 00:12:03.260 align:middle line:84% nothing to do with relativistic particles, 00:12:03.260 --> 00:12:05.900 align:middle line:84% but to start with this classical field theory 00:12:05.900 --> 00:12:08.420 align:middle line:90% and then quantize it. 00:12:08.420 --> 00:12:12.800 align:middle line:84% It turns out, once you treat this theory as a quantum field 00:12:12.800 --> 00:12:15.560 align:middle line:84% theory, and this becomes a theory 00:12:15.560 --> 00:12:18.380 align:middle line:84% of arbitrary number of relativistic particles 00:12:18.380 --> 00:12:20.390 align:middle line:90% of mass m. 00:12:20.390 --> 00:12:22.970 align:middle line:84% And so that's the non-intuitive part, 00:12:22.970 --> 00:12:27.410 align:middle line:84% and that's one of the miracle, say, of the field theory, 00:12:27.410 --> 00:12:31.820 align:middle line:84% is that automatically gives you a formalism for treating 00:12:31.820 --> 00:12:35.090 align:middle line:90% arbitrary number of particles. 00:12:35.090 --> 00:12:36.965 align:middle line:90% And yeah. 00:12:36.965 --> 00:12:41.100 align:middle line:90% 00:12:41.100 --> 00:12:43.350 align:middle line:90% And also, in field theory-- 00:12:43.350 --> 00:12:47.550 align:middle line:84% so both t and x are parameters even though x only 00:12:47.550 --> 00:12:51.070 align:middle line:90% labels your location. 00:12:51.070 --> 00:12:53.700 align:middle line:84% So both t and x are parameters, and so you 00:12:53.700 --> 00:12:56.760 align:middle line:84% can easily to make them to be on equal ground 00:12:56.760 --> 00:12:59.778 align:middle line:84% to be compatible with special relativity. 00:12:59.778 --> 00:13:01.770 align:middle line:90% OK. 00:13:01.770 --> 00:13:03.620 align:middle line:90% Good? 00:13:03.620 --> 00:13:04.925 align:middle line:90% So any questions on this? 00:13:04.925 --> 00:13:09.360 align:middle line:90% 00:13:09.360 --> 00:13:09.860 align:middle line:90% OK. 00:13:09.860 --> 00:13:13.020 align:middle line:84% So we will see that the right framework is quantum field 00:13:13.020 --> 00:13:13.520 align:middle line:90% theory. 00:13:13.520 --> 00:13:15.050 align:middle line:90% OK. 00:13:15.050 --> 00:13:19.640 align:middle line:84% So finally, as a last motivation for quantum field theory, 00:13:19.640 --> 00:13:23.210 align:middle line:90% we quickly describe the last-- 00:13:23.210 --> 00:13:33.020 align:middle line:84% so the field theory can also arise at the limit 00:13:33.020 --> 00:13:33.980 align:middle line:90% of discrete systems. 00:13:33.980 --> 00:13:44.410 align:middle line:90% 00:13:44.410 --> 00:13:49.613 align:middle line:84% And this is the most relevant for condensed matter physics. 00:13:49.613 --> 00:13:50.155 align:middle line:90% For example-- 00:13:50.155 --> 00:13:53.360 align:middle line:90% 00:13:53.360 --> 00:13:56.660 align:middle line:84% So let's just consider, say, some-- 00:13:56.660 --> 00:14:00.290 align:middle line:84% yeah, let's consider 8.03 example. 00:14:00.290 --> 00:14:05.300 align:middle line:84% So let's imagine you have this number of particles, 00:14:05.300 --> 00:14:09.650 align:middle line:84% a number of the atoms, say, on the chain. 00:14:09.650 --> 00:14:12.740 align:middle line:84% And then they're connected by some springs between them. 00:14:12.740 --> 00:14:15.960 align:middle line:90% 00:14:15.960 --> 00:14:18.750 align:middle line:90% So this is the-- 00:14:18.750 --> 00:14:21.330 align:middle line:84% yeah, consider this to be infinite. 00:14:21.330 --> 00:14:26.430 align:middle line:84% And the spacing between them, say, is a. 00:14:26.430 --> 00:14:29.910 align:middle line:84% So atom, I fixed on some lattice points. 00:14:29.910 --> 00:14:33.930 align:middle line:90% And the lattice spacing is a. 00:14:33.930 --> 00:14:38.080 align:middle line:90% So yeah, so we can label the-- 00:14:38.080 --> 00:14:41.470 align:middle line:84% say, each particle by their position. 00:14:41.470 --> 00:14:45.220 align:middle line:84% For example, this is x0, this is x1, this is x2, et cetera. 00:14:45.220 --> 00:14:45.910 align:middle line:90% OK. 00:14:45.910 --> 00:14:47.770 align:middle line:84% And and the typical particle is xn. 00:14:47.770 --> 00:14:50.930 align:middle line:90% 00:14:50.930 --> 00:14:54.470 align:middle line:84% At the location of a n-th particle is xn. 00:14:54.470 --> 00:14:59.000 align:middle line:84% And so we can also introduce the deviation 00:14:59.000 --> 00:15:01.950 align:middle line:84% between the equilibrium position of each particle. 00:15:01.950 --> 00:15:04.250 align:middle line:90% So let's call it eta n. 00:15:04.250 --> 00:15:06.440 align:middle line:84% So now let's consider the dynamics 00:15:06.440 --> 00:15:09.420 align:middle line:90% of eta n for this theory. 00:15:09.420 --> 00:15:11.930 align:middle line:84% And so this is just a deviation of the n-th particle 00:15:11.930 --> 00:15:13.720 align:middle line:90% from its equilibrium position. 00:15:13.720 --> 00:15:16.620 align:middle line:84% So eta n 0 is equilibrium position. 00:15:16.620 --> 00:15:21.150 align:middle line:84% So now so now if you write down the Lagrangian for this system, 00:15:21.150 --> 00:15:23.630 align:middle line:84% so we can easily do, you just write T minus V. 00:15:23.630 --> 00:15:29.550 align:middle line:84% So T is the kinetic energy and V is the potential energy. 00:15:29.550 --> 00:15:36.050 align:middle line:84% So we can just write it as sum over n over all particles. 00:15:36.050 --> 00:15:38.240 align:middle line:84% And then let's assume they have the same mass, 00:15:38.240 --> 00:15:42.590 align:middle line:90% let's write mu eta n dot square. 00:15:42.590 --> 00:15:44.630 align:middle line:84% So this is a kinetic term, so the mu 00:15:44.630 --> 00:15:46.760 align:middle line:90% is the mass for each particle. 00:15:46.760 --> 00:15:50.570 align:middle line:84% And then their potential-- yeah, let's assume at each point, 00:15:50.570 --> 00:15:52.400 align:middle line:90% there is also-- 00:15:52.400 --> 00:15:55.156 align:middle line:84% yeah, let's just-- yeah, then there's 00:15:55.156 --> 00:15:58.310 align:middle line:84% a interaction because each particle are connected 00:15:58.310 --> 00:16:03.020 align:middle line:84% by the spring, and so they are a harmonic force 00:16:03.020 --> 00:16:08.260 align:middle line:90% between neighboring particles. 00:16:08.260 --> 00:16:10.060 align:middle line:84% And now let's imagine also there's 00:16:10.060 --> 00:16:15.400 align:middle line:84% a harmonic potential which trap this particle itself 00:16:15.400 --> 00:16:17.950 align:middle line:90% at each location. 00:16:17.950 --> 00:16:22.480 align:middle line:84% So this is a very simple spring and the particle problem which 00:16:22.480 --> 00:16:25.900 align:middle line:90% you encounter, say, in 8.03. 00:16:25.900 --> 00:16:28.900 align:middle line:90% Is this problem clear? 00:16:28.900 --> 00:16:29.400 align:middle line:90% OK. 00:16:29.400 --> 00:16:33.460 align:middle line:84% I assume most of you have seen this problem before. 00:16:33.460 --> 00:16:36.100 align:middle line:84% And your task in 8.03 is actually 00:16:36.100 --> 00:16:42.400 align:middle line:84% to find the local modes, say, of this system. 00:16:42.400 --> 00:16:45.220 align:middle line:84% And in 8.03, you also described that we can-- 00:16:45.220 --> 00:16:48.320 align:middle line:90% and a go to zero limit. 00:16:48.320 --> 00:16:51.530 align:middle line:84% So if the lattice spacing is very small. 00:16:51.530 --> 00:16:54.250 align:middle line:84% And if we are only interested in the behavior 00:16:54.250 --> 00:16:56.410 align:middle line:84% of the system at a very large distance, 00:16:56.410 --> 00:16:58.870 align:middle line:84% say the distance is much larger than a equals to-- 00:16:58.870 --> 00:17:02.020 align:middle line:84% much larger than a, then you can essentially treat this system 00:17:02.020 --> 00:17:02.950 align:middle line:90% as a continuum. 00:17:02.950 --> 00:17:05.690 align:middle line:84% You don't have to resolve individual particles. 00:17:05.690 --> 00:17:07.150 align:middle line:90% And so we can just-- 00:17:07.150 --> 00:17:08.740 align:middle line:90% equal to newer limit. 00:17:08.740 --> 00:17:17.970 align:middle line:84% So it can be true to the chain as a continuum of particles. 00:17:17.970 --> 00:17:28.030 align:middle line:90% 00:17:28.030 --> 00:17:36.430 align:middle line:84% And so each eta n t, you replace it by eta x, t. 00:17:36.430 --> 00:17:40.090 align:middle line:90% So x label is position. 00:17:40.090 --> 00:17:43.420 align:middle line:84% So x label is position, t describe the dynamics. 00:17:43.420 --> 00:17:47.500 align:middle line:84% So eta is the deviation at the location x, 00:17:47.500 --> 00:17:49.090 align:middle line:90% and it's depend on t. 00:17:49.090 --> 00:17:51.280 align:middle line:90% So this is the oscillator. 00:17:51.280 --> 00:17:56.830 align:middle line:84% And then sum over n in the Lagrangian, 00:17:56.830 --> 00:18:02.480 align:middle line:84% then we can replace it by integral over dx. 00:18:02.480 --> 00:18:06.440 align:middle line:84% And now you just treat this as a one-dimensional continuum. 00:18:06.440 --> 00:18:08.930 align:middle line:90% So integration over dx. 00:18:08.930 --> 00:18:13.430 align:middle line:84% But of course, here, there's a label-- a lattice spacing, 00:18:13.430 --> 00:18:18.740 align:middle line:84% so the so the infinitesimal here, the element is a. 00:18:18.740 --> 00:18:23.560 align:middle line:84% The a times the sum over n, you can replace it by dx. 00:18:23.560 --> 00:18:26.250 align:middle line:90% 00:18:26.250 --> 00:18:28.530 align:middle line:90% That is the lattice spacing. 00:18:28.530 --> 00:18:31.560 align:middle line:84% And now you can just write this Lagrangian 00:18:31.560 --> 00:18:35.160 align:middle line:90% in terms of continuum theory. 00:18:35.160 --> 00:18:36.660 align:middle line:84% Now you can write this Lagrangian 00:18:36.660 --> 00:18:41.170 align:middle line:84% in terms of continuum theory, and then let's just do it. 00:18:41.170 --> 00:18:44.540 align:middle line:84% So we can write it-- yeah, let me just write one more step. 00:18:44.540 --> 00:18:46.720 align:middle line:84% So you can write it as sum over a. 00:18:46.720 --> 00:18:49.930 align:middle line:84% We take the a factor out because the a factor out has 00:18:49.930 --> 00:18:52.900 align:middle line:90% to be changing to integration. 00:18:52.900 --> 00:18:57.370 align:middle line:84% And then you have 1/2 mu divided by a eta n 00:18:57.370 --> 00:19:01.410 align:middle line:90% squared minus 1/2 lambda a. 00:19:01.410 --> 00:19:06.930 align:middle line:90% 00:19:06.930 --> 00:19:14.910 align:middle line:84% So I just slightly rewrite this Lagrangian 00:19:14.910 --> 00:19:18.180 align:middle line:84% so that it's easy to take the continuum limit. 00:19:18.180 --> 00:19:23.780 align:middle line:84% So we have taken the factor of a out, but for this term, 00:19:23.780 --> 00:19:26.210 align:middle line:84% because this concerns the difference between the two, 00:19:26.210 --> 00:19:29.240 align:middle line:84% and we also divided by a in the downstairs, 00:19:29.240 --> 00:19:31.340 align:middle line:84% and then we need to multiply a upstairs, 00:19:31.340 --> 00:19:33.860 align:middle line:90% and then there's a in the front. 00:19:33.860 --> 00:19:37.143 align:middle line:84% And now the continuum limit, you can just 00:19:37.143 --> 00:19:38.310 align:middle line:90% replace this by an integral. 00:19:38.310 --> 00:19:41.080 align:middle line:90% 00:19:41.080 --> 00:19:41.880 align:middle line:90% And now I just-- 00:19:41.880 --> 00:19:44.890 align:middle line:90% 00:19:44.890 --> 00:19:50.230 align:middle line:84% can just write it as 1/2 mu tilde eta dot squared. 00:19:50.230 --> 00:19:55.680 align:middle line:84% So now eta n just replaced by eta x, t. 00:19:55.680 --> 00:19:59.930 align:middle line:84% And here, let me call it lambda tilde. 00:19:59.930 --> 00:20:02.150 align:middle line:90% Partial x eta square. 00:20:02.150 --> 00:20:03.740 align:middle line:84% And this term, you can just replace it 00:20:03.740 --> 00:20:05.600 align:middle line:90% by the derivative of eta. 00:20:05.600 --> 00:20:09.590 align:middle line:84% And then this has just become 1/2 sigma tilde eta square. 00:20:09.590 --> 00:20:12.320 align:middle line:90% 00:20:12.320 --> 00:20:18.050 align:middle line:84% And the mu tilde, of course, is mu divided by a. 00:20:18.050 --> 00:20:21.500 align:middle line:90% Lambda tilde is lambda times a. 00:20:21.500 --> 00:20:29.830 align:middle line:84% And sigma tilde, this is sigma divided by a. 00:20:29.830 --> 00:20:32.920 align:middle line:84% So the continuum limit is that this quantity has to be fixed. 00:20:32.920 --> 00:20:34.940 align:middle line:84% The tilde, the quantity has to be fixed. 00:20:34.940 --> 00:20:37.850 align:middle line:84% And then we have a continuum Lagrangian. 00:20:37.850 --> 00:20:40.140 align:middle line:84% And then we have a classical field theory. 00:20:40.140 --> 00:20:47.480 align:middle line:84% And this theory is essentially the same as that theory. 00:20:47.480 --> 00:20:52.230 align:middle line:84% So if you take this factor mu tilde out, 00:20:52.230 --> 00:20:56.560 align:middle line:84% if we take this vector mu tilde out-- 00:20:56.560 --> 00:20:58.370 align:middle line:84% so let me just take this factor of mu 00:20:58.370 --> 00:21:02.210 align:middle line:84% tilde out in the front, just up to a wall factor. 00:21:02.210 --> 00:21:05.470 align:middle line:84% And here is lambda tilde divided by mu tilde. 00:21:05.470 --> 00:21:08.700 align:middle line:84% We call it-- let's call it v square. 00:21:08.700 --> 00:21:12.000 align:middle line:84% And this becomes sigma tilde divided by mu tilde, 00:21:12.000 --> 00:21:14.030 align:middle line:90% let's call it m squared. 00:21:14.030 --> 00:21:20.510 align:middle line:84% So the V squared is equal to mu tilde is equal to the lambda 00:21:20.510 --> 00:21:27.530 align:middle line:84% tilde divided by mu tilde, and m squared is equal to sigma 00:21:27.530 --> 00:21:29.180 align:middle line:90% tilde squared by mu tilde. 00:21:29.180 --> 00:21:31.960 align:middle line:90% 00:21:31.960 --> 00:21:33.760 align:middle line:84% And then this is just essentially 00:21:33.760 --> 00:21:39.790 align:middle line:84% identical to that theory when V equal to 1. 00:21:39.790 --> 00:21:47.900 align:middle line:84% So when V equal to 1, become the same, is just 2, 00:21:47.900 --> 00:21:49.833 align:middle line:90% the equation 2. 00:21:49.833 --> 00:21:51.500 align:middle line:84% So we could do, of course, corresponding 00:21:51.500 --> 00:21:53.810 align:middle line:84% to relativistic case, speed of light, 00:21:53.810 --> 00:21:57.560 align:middle line:84% but in general, this describe-- can-- yeah. 00:21:57.560 --> 00:22:02.000 align:middle line:84% This can describe-- but in general, this 00:22:02.000 --> 00:22:03.610 align:middle line:84% is a non-relativistic-- in general, 00:22:03.610 --> 00:22:05.360 align:middle line:84% this can be just a long relativistic field 00:22:05.360 --> 00:22:09.620 align:middle line:90% theory for other values of V. 00:22:09.620 --> 00:22:15.770 align:middle line:84% So this is-- so even though this example is very simple, 00:22:15.770 --> 00:22:21.170 align:middle line:84% but this is actually a very general way that we can treat 00:22:21.170 --> 00:22:25.370 align:middle line:84% many condensed matter systems, which often involve a lattice, 00:22:25.370 --> 00:22:28.910 align:middle line:84% say, because the solid, you can imagine all the atoms are 00:22:28.910 --> 00:22:30.390 align:middle line:90% on the lattice, et cetera. 00:22:30.390 --> 00:22:35.460 align:middle line:84% And if you're only interested in the very macroscopic behavior, 00:22:35.460 --> 00:22:39.260 align:middle line:84% then you can treat solid as a continuum. 00:22:39.260 --> 00:22:42.650 align:middle line:84% And now you can now if you're interested in the mechanics 00:22:42.650 --> 00:22:45.650 align:middle line:84% of such a system, then the quantum field theory then 00:22:45.650 --> 00:22:46.670 align:middle line:90% naturally arises. 00:22:46.670 --> 00:22:47.170 align:middle line:90% OK. 00:22:47.170 --> 00:22:50.580 align:middle line:90% 00:22:50.580 --> 00:22:51.080 align:middle line:90% Good. 00:22:51.080 --> 00:22:52.915 align:middle line:90% Any questions on this example? 00:22:52.915 --> 00:22:56.620 align:middle line:90% 00:22:56.620 --> 00:22:57.750 align:middle line:90% Yes? 00:22:57.750 --> 00:23:01.960 align:middle line:84% STUDENT: So that, it goes to 1 is the same as lambda 00:23:01.960 --> 00:23:03.680 align:middle line:90% and mu being comparable? 00:23:03.680 --> 00:23:06.513 align:middle line:84% So what does that, I guess, physically mean? 00:23:06.513 --> 00:23:07.180 align:middle line:90% HONG LIU: Sorry? 00:23:07.180 --> 00:23:08.350 align:middle line:90% STUDENT: So the limit-- 00:23:08.350 --> 00:23:09.785 align:middle line:84% HONG LIU: Yeah, yeah, yeah, yeah. 00:23:09.785 --> 00:23:12.155 align:middle line:84% STUDENT: Like, the same lambda as the same as mu? 00:23:12.155 --> 00:23:13.030 align:middle line:90% HONG LIU: Yeah, yeah. 00:23:13.030 --> 00:23:14.738 align:middle line:84% STUDENT: So what is that physically mean? 00:23:14.738 --> 00:23:16.145 align:middle line:90% Like, the strength of-- 00:23:16.145 --> 00:23:17.020 align:middle line:90% HONG LIU: Yeah, yeah. 00:23:17.020 --> 00:23:22.090 align:middle line:84% This goes right into the case that the-- 00:23:22.090 --> 00:23:25.180 align:middle line:84% yeah, it just tells you that the relativistic limit is special. 00:23:25.180 --> 00:23:27.200 align:middle line:90% Happens at very special point. 00:23:27.200 --> 00:23:29.847 align:middle line:84% STUDENT: But I guess, why is that the relativistic-- 00:23:29.847 --> 00:23:31.930 align:middle line:84% To me, lambda is like the strength of your spring, 00:23:31.930 --> 00:23:32.380 align:middle line:90% and then-- 00:23:32.380 --> 00:23:32.830 align:middle line:90% HONG LIU: Yeah. 00:23:32.830 --> 00:23:33.872 align:middle line:90% STUDENT: --use your mass. 00:23:33.872 --> 00:23:34.660 align:middle line:90% HONG LIU: Right. 00:23:34.660 --> 00:23:37.000 align:middle line:84% STUDENT: Without that being comparable, how does that-- 00:23:37.000 --> 00:23:37.300 align:middle line:90% HONG LIU: Yeah. 00:23:37.300 --> 00:23:38.038 align:middle line:84% STUDENT: Because it's not the same thing 00:23:38.038 --> 00:23:38.530 align:middle line:90% as the relativistic-- 00:23:38.530 --> 00:23:38.740 align:middle line:90% HONG LIU: Yeah. 00:23:38.740 --> 00:23:40.660 align:middle line:84% There's not much you can read from here. 00:23:40.660 --> 00:23:42.720 align:middle line:90% Yeah, yeah. 00:23:42.720 --> 00:23:45.280 align:middle line:84% It's just like when you choose some special parameters, 00:23:45.280 --> 00:23:50.730 align:middle line:84% then you can have a relativistic limit, yeah. 00:23:50.730 --> 00:23:51.470 align:middle line:90% Other questions? 00:23:51.470 --> 00:23:52.322 align:middle line:90% Yes? 00:23:52.322 --> 00:23:53.780 align:middle line:84% STUDENT: Yeah, so you said that you 00:23:53.780 --> 00:23:56.072 align:middle line:84% could use this to treat some condensed matter problems? 00:23:56.072 --> 00:23:56.700 align:middle line:90% HONG LIU: Yeah. 00:23:56.700 --> 00:24:00.980 align:middle line:84% STUDENT: Yeah, so these are all scalars, right? 00:24:00.980 --> 00:24:02.660 align:middle line:90% HONG LIU: You can also have-- 00:24:02.660 --> 00:24:04.800 align:middle line:90% you can also-- you mean-- 00:24:04.800 --> 00:24:07.170 align:middle line:84% you can also have tensors or vectors, yeah. 00:24:07.170 --> 00:24:07.670 align:middle line:90% STUDENT: OK. 00:24:07.670 --> 00:24:08.810 align:middle line:90% HONG LIU: Yeah. 00:24:08.810 --> 00:24:11.495 align:middle line:84% STUDENT: So like what would you treat with this? 00:24:11.495 --> 00:24:12.242 align:middle line:90% Like phonons. 00:24:12.242 --> 00:24:13.700 align:middle line:84% HONG LIU: Oh, you can treat-- yeah. 00:24:13.700 --> 00:24:15.410 align:middle line:84% For example, you can treat phonons. 00:24:15.410 --> 00:24:19.170 align:middle line:84% You can also treat spins and say, for example, 00:24:19.170 --> 00:24:22.940 align:middle line:84% if you have an Ising model, just consider the lattice of spins, 00:24:22.940 --> 00:24:25.550 align:middle line:84% and then the average spin, and then you 00:24:25.550 --> 00:24:27.217 align:middle line:84% can treat it as a scalar field, and then 00:24:27.217 --> 00:24:29.008 align:middle line:84% again, you can write down the field theory. 00:24:29.008 --> 00:24:29.630 align:middle line:90% Yeah. 00:24:29.630 --> 00:24:35.870 align:middle line:84% And actually, the breakthrough of the phase transition 00:24:35.870 --> 00:24:38.660 align:middle line:84% in condensed matter physics to understand 00:24:38.660 --> 00:24:40.550 align:middle line:84% what phase transition is really about 00:24:40.550 --> 00:24:43.190 align:middle line:84% and describe the behavior of the phase transition, 00:24:43.190 --> 00:24:46.850 align:middle line:84% and precisely coincided with the development of field theory. 00:24:46.850 --> 00:24:50.130 align:middle line:84% And yeah, actually, increased our understanding 00:24:50.130 --> 00:24:51.420 align:middle line:90% of quantum field theory, yeah. 00:24:51.420 --> 00:24:54.270 align:middle line:90% 00:24:54.270 --> 00:24:55.500 align:middle line:90% Good. 00:24:55.500 --> 00:24:56.340 align:middle line:90% Other questions? 00:24:56.340 --> 00:24:58.880 align:middle line:90% 00:24:58.880 --> 00:25:00.430 align:middle line:90% OK, good. 00:25:00.430 --> 00:25:09.610 align:middle line:84% Just to summarize what we have discussed so far, 00:25:09.610 --> 00:25:16.300 align:middle line:90% all paths leads to QFT. 00:25:16.300 --> 00:25:19.990 align:middle line:84% So we have described three paths, 00:25:19.990 --> 00:25:21.550 align:middle line:90% but they are pretty general. 00:25:21.550 --> 00:25:23.930 align:middle line:84% First is that the quantum dynamics-- 00:25:23.930 --> 00:25:30.200 align:middle line:84% we often interested in quantum dynamics 00:25:30.200 --> 00:25:39.630 align:middle line:84% of some classical fields, say, such as, say, 00:25:39.630 --> 00:25:43.350 align:middle line:84% electric magnetic field or spacetime metric 00:25:43.350 --> 00:25:50.440 align:middle line:84% if you are interested in gravity, et cetera. 00:25:50.440 --> 00:25:54.950 align:middle line:84% So in this case, we already have the classical field theory, 00:25:54.950 --> 00:25:56.740 align:middle line:84% but we know the world is quantum and we 00:25:56.740 --> 00:25:59.510 align:middle line:84% want to understand what's the quantum version of it. 00:25:59.510 --> 00:26:05.680 align:middle line:84% And the second is that it unifies special relativity 00:26:05.680 --> 00:26:07.930 align:middle line:90% plus quantum mechanics. 00:26:07.930 --> 00:26:10.820 align:middle line:84% So you need the field theory to unify them. 00:26:10.820 --> 00:26:25.220 align:middle line:84% And the third way is that it's the large distance description 00:26:25.220 --> 00:26:26.490 align:middle line:90% of these grid systems. 00:26:26.490 --> 00:26:33.840 align:middle line:90% 00:26:33.840 --> 00:26:34.650 align:middle line:90% OK. 00:26:34.650 --> 00:26:40.190 align:middle line:84% So yeah, just combining all three elements together, 00:26:40.190 --> 00:26:42.735 align:middle line:84% they cover many, many areas of physics. 00:26:42.735 --> 00:26:44.360 align:middle line:84% They cover many, many areas of physics. 00:26:44.360 --> 00:26:48.520 align:middle line:90% 00:26:48.520 --> 00:26:49.020 align:middle line:90% Good. 00:26:49.020 --> 00:26:53.660 align:middle line:84% So now we can say a little bit about the plan 00:26:53.660 --> 00:26:57.000 align:middle line:90% for the whole semester. 00:26:57.000 --> 00:27:00.330 align:middle line:90% So here is the plan. 00:27:00.330 --> 00:27:04.740 align:middle line:84% So this is like just rephrase of the outline, which-- 00:27:04.740 --> 00:27:12.450 align:middle line:90% 00:27:12.450 --> 00:27:16.110 align:middle line:84% So the first thing we do in chapter 2-- so here 00:27:16.110 --> 00:27:16.890 align:middle line:90% is chapter 1. 00:27:16.890 --> 00:27:26.790 align:middle line:84% Chapter 2, we discussed the simplest field theory just 00:27:26.790 --> 00:27:30.330 align:middle line:90% to squash this equation 2. 00:27:30.330 --> 00:27:32.190 align:middle line:90% The theory of 2. 00:27:32.190 --> 00:27:34.800 align:middle line:90% 2 and 2 prime. 00:27:34.800 --> 00:27:36.580 align:middle line:90% Prime is its equation of motion. 00:27:36.580 --> 00:27:40.110 align:middle line:84% So yeah, we-- in physics, we always 00:27:40.110 --> 00:27:42.330 align:middle line:90% start with the simplest example. 00:27:42.330 --> 00:27:44.440 align:middle line:84% We always start with the simplest example. 00:27:44.440 --> 00:27:49.100 align:middle line:84% And so that is the one we will start with. 00:27:49.100 --> 00:27:51.820 align:middle line:84% So what we will see is that this describes-- 00:27:51.820 --> 00:27:55.720 align:middle line:90% 00:27:55.720 --> 00:27:58.690 align:middle line:84% that field theory describe spin is, 00:27:58.690 --> 00:28:04.270 align:middle line:90% is both free massive particles. 00:28:04.270 --> 00:28:09.980 align:middle line:90% 00:28:09.980 --> 00:28:12.950 align:middle line:84% OK, so we will see, when, we quantize that theory 2, 00:28:12.950 --> 00:28:17.480 align:middle line:84% and then we get the theory of free massless-- 00:28:17.480 --> 00:28:20.060 align:middle line:90% free spinless massive particles. 00:28:20.060 --> 00:28:21.080 align:middle line:90% OK. 00:28:21.080 --> 00:28:22.830 align:middle line:84% So you say, oh, that's a little bit boring 00:28:22.830 --> 00:28:25.530 align:middle line:84% because in this series, free-- the particle-- but free means 00:28:25.530 --> 00:28:26.940 align:middle line:90% they don't interact. 00:28:26.940 --> 00:28:30.220 align:middle line:84% The particle, they just don't interact. 00:28:30.220 --> 00:28:34.390 align:middle line:84% And then in chapter 3, we will add interactions. 00:28:34.390 --> 00:28:45.410 align:middle line:84% We will describe how to treat interactions. 00:28:45.410 --> 00:28:49.400 align:middle line:84% So we going to introduce interactions and tell you 00:28:49.400 --> 00:28:51.740 align:middle line:84% how to treat the interactions between those particles. 00:28:51.740 --> 00:28:55.950 align:middle line:90% 00:28:55.950 --> 00:29:01.150 align:middle line:84% Then in chapter 4, we go to the real physics. 00:29:01.150 --> 00:29:04.520 align:middle line:84% So the scalar fields is also real. 00:29:04.520 --> 00:29:08.212 align:middle line:84% Say, for example, it can be used to describe the Higgs boson. 00:29:08.212 --> 00:29:09.920 align:middle line:84% But the Higgs boson maybe is a little bit 00:29:09.920 --> 00:29:13.470 align:middle line:84% far from what we normally think about. 00:29:13.470 --> 00:29:18.020 align:middle line:84% So in chapter 4, we will go to something which is much closer. 00:29:18.020 --> 00:29:20.342 align:middle line:84% We'll talk about the theory of electron. 00:29:20.342 --> 00:29:21.800 align:middle line:84% So this is called the Dirac theory. 00:29:21.800 --> 00:29:24.620 align:middle line:90% 00:29:24.620 --> 00:29:31.820 align:middle line:84% So this theory describes free spin half particles. 00:29:31.820 --> 00:29:35.380 align:middle line:90% 00:29:35.380 --> 00:29:38.250 align:middle line:84% So this is a theory of electrons. 00:29:38.250 --> 00:29:39.970 align:middle line:90% OK. 00:29:39.970 --> 00:29:42.440 align:middle line:84% When we neglect these interactions. 00:29:42.440 --> 00:29:47.620 align:middle line:84% So this is the free spin half particles. 00:29:47.620 --> 00:30:06.715 align:middle line:84% And then we move on to the Maxwell theory. 00:30:06.715 --> 00:30:12.940 align:middle line:90% 00:30:12.940 --> 00:30:14.880 align:middle line:84% So this is the theory of the quantum 00:30:14.880 --> 00:30:16.950 align:middle line:90% electric and magnetic field. 00:30:16.950 --> 00:30:19.650 align:middle line:90% 00:30:19.650 --> 00:30:24.930 align:middle line:84% So when you quantize the Maxwell theory, say, without source, 00:30:24.930 --> 00:30:28.440 align:middle line:84% the vacuum Maxwell theory, and you find you get free-- 00:30:28.440 --> 00:30:32.280 align:middle line:90% again, there's no interaction-- 00:30:32.280 --> 00:30:38.350 align:middle line:90% massless spin 1 particle. 00:30:38.350 --> 00:30:41.200 align:middle line:84% The theory of massless spin 1 particle. 00:30:41.200 --> 00:30:43.820 align:middle line:90% 00:30:43.820 --> 00:30:46.690 align:middle line:84% So this is what we call the photon. 00:30:46.690 --> 00:30:51.520 align:middle line:84% So this is a quantum for electromagnetic fields. 00:30:51.520 --> 00:30:57.320 align:middle line:90% And then-- sorry. 00:30:57.320 --> 00:30:59.660 align:middle line:84% Did I-- so this should be chapter 5 now. 00:30:59.660 --> 00:31:01.820 align:middle line:90% I think I lost my count. 00:31:01.820 --> 00:31:06.850 align:middle line:84% So now go to chapter 6, we combine the 4 and the five 00:31:06.850 --> 00:31:15.450 align:middle line:90% together, combine electrons. 00:31:15.450 --> 00:31:20.090 align:middle line:84% So photon normally we denote by gamma. 00:31:20.090 --> 00:31:22.940 align:middle line:84% Combine the theory of electron and the photon 00:31:22.940 --> 00:31:29.480 align:middle line:84% together, and then plot the interactions between them. 00:31:29.480 --> 00:31:34.190 align:middle line:84% And then we get the so-called quantum electrodynamics. 00:31:34.190 --> 00:31:38.620 align:middle line:90% 00:31:38.620 --> 00:31:39.715 align:middle line:90% So this is called QED. 00:31:39.715 --> 00:31:42.580 align:middle line:90% 00:31:42.580 --> 00:31:44.065 align:middle line:90% So QED is very general. 00:31:44.065 --> 00:31:49.210 align:middle line:84% It essentially covers all the quantum phenomena-- 00:31:49.210 --> 00:31:52.450 align:middle line:84% yeah, a macroscopic phenomenon up to, say, weak interactions 00:31:52.450 --> 00:31:53.450 align:middle line:90% and strong interactions. 00:31:53.450 --> 00:31:57.871 align:middle line:84% If you don't go inside the nucleus 00:31:57.871 --> 00:32:00.250 align:middle line:84% or don't go to very high energy, I 00:32:00.250 --> 00:32:05.830 align:middle line:84% think that covers essentially most of the physics. 00:32:05.830 --> 00:32:07.820 align:middle line:90% And yeah. 00:32:07.820 --> 00:32:11.440 align:middle line:84% And then we will describe how to-- yeah, 00:32:11.440 --> 00:32:16.370 align:middle line:84% and then that will be the end of this course. 00:32:16.370 --> 00:32:17.950 align:middle line:84% So do you have any questions on this? 00:32:17.950 --> 00:32:21.090 align:middle line:90% 00:32:21.090 --> 00:32:21.590 align:middle line:90% OK. 00:32:21.590 --> 00:32:23.060 align:middle line:90% So this is a road map. 00:32:23.060 --> 00:32:24.142 align:middle line:90% Yes? 00:32:24.142 --> 00:32:25.850 align:middle line:84% STUDENT: Do these chapters correspond off 00:32:25.850 --> 00:32:27.200 align:middle line:84% of those chapters in the textbook 00:32:27.200 --> 00:32:28.760 align:middle line:84% or just chapters in the lecture notes? 00:32:28.760 --> 00:32:30.802 align:middle line:84% HONG LIU: The chapter in the lecture notes, yeah. 00:32:30.802 --> 00:32:31.890 align:middle line:90% Yeah. 00:32:31.890 --> 00:32:32.590 align:middle line:90% OK. 00:32:32.590 --> 00:32:33.090 align:middle line:90% Good. 00:32:33.090 --> 00:32:34.170 align:middle line:90% Other questions? 00:32:34.170 --> 00:32:34.920 align:middle line:90% Yes? 00:32:34.920 --> 00:32:36.960 align:middle line:84% STUDENT: I'm just curious if gluon's 00:32:36.960 --> 00:32:40.030 align:middle line:84% included in the free massless spin 1 particles? 00:32:40.030 --> 00:32:41.092 align:middle line:90% HONG LIU: Sorry? 00:32:41.092 --> 00:32:44.235 align:middle line:84% STUDENT: Gluons included in the free massless spin 1 particles 00:32:44.235 --> 00:32:44.860 align:middle line:90% HONG LIU: Yeah. 00:32:44.860 --> 00:32:50.110 align:middle line:84% Gluon is also massless spin 1, but gluon, actually, they 00:32:50.110 --> 00:32:52.130 align:middle line:90% interact with themselves. 00:32:52.130 --> 00:32:53.870 align:middle line:90% And so gluon is a different. 00:32:53.870 --> 00:32:56.110 align:middle line:84% So gluon-- to describe gluons, you 00:32:56.110 --> 00:32:58.480 align:middle line:84% have to wait for quantum field theory 2. 00:32:58.480 --> 00:33:01.090 align:middle line:84% And so the thing about the photon 00:33:01.090 --> 00:33:03.290 align:middle line:84% is that the photon don't interact with itself, 00:33:03.290 --> 00:33:05.500 align:middle line:84% but the gluons interact with itself. 00:33:05.500 --> 00:33:06.100 align:middle line:90% OK. 00:33:06.100 --> 00:33:10.750 align:middle line:84% Yeah, so essentially, we treat everything except gluons. 00:33:10.750 --> 00:33:11.250 align:middle line:90% Yeah. 00:33:11.250 --> 00:33:14.520 align:middle line:90% 00:33:14.520 --> 00:33:15.660 align:middle line:90% Other questions? 00:33:15.660 --> 00:33:18.410 align:middle line:90% Other questions? 00:33:18.410 --> 00:33:19.670 align:middle line:90% OK, good. 00:33:19.670 --> 00:33:22.768 align:middle line:84% So now we can just move to chapter 2. 00:33:22.768 --> 00:33:24.310 align:middle line:84% Now we are talking about this theory. 00:33:24.310 --> 00:33:24.810 align:middle line:90% OK. 00:33:24.810 --> 00:33:28.720 align:middle line:84% So actually, I should not erase it. 00:33:28.720 --> 00:33:30.120 align:middle line:84% So now we talk about this theory. 00:33:30.120 --> 00:33:32.820 align:middle line:90% 00:33:32.820 --> 00:33:35.710 align:middle line:84% Because this theory describe free particles, 00:33:35.710 --> 00:33:37.890 align:middle line:84% so we call it a free scalar field theory. 00:33:37.890 --> 00:33:47.530 align:middle line:90% 00:33:47.530 --> 00:33:52.010 align:middle line:84% So this is the theory we are interested in. 00:33:52.010 --> 00:33:54.520 align:middle line:84% So now we will describe how to quantize this theory. 00:33:54.520 --> 00:34:04.670 align:middle line:90% 00:34:04.670 --> 00:34:05.780 align:middle line:90% Good. 00:34:05.780 --> 00:34:12.790 align:middle line:84% So first, we will quickly go through the quantization 00:34:12.790 --> 00:34:16.090 align:middle line:84% of harmonic oscillator, which you should already 00:34:16.090 --> 00:34:18.880 align:middle line:90% have done in your Pset. 00:34:18.880 --> 00:34:22.670 align:middle line:84% And so we can do it relatively fast. 00:34:22.670 --> 00:34:32.469 align:middle line:84% So the quantization of harmonic oscillator 00:34:32.469 --> 00:34:33.730 align:middle line:90% in the Heisenberg picture. 00:34:33.730 --> 00:34:38.510 align:middle line:90% 00:34:38.510 --> 00:34:40.850 align:middle line:84% So we will see that once we understand 00:34:40.850 --> 00:34:45.219 align:middle line:84% this example in the right way, and then 00:34:45.219 --> 00:34:48.429 align:middle line:84% quantizing this field theory becomes trivial. 00:34:48.429 --> 00:34:52.030 align:middle line:84% And quantizing this field theory becomes trivial. 00:34:52.030 --> 00:34:56.139 align:middle line:84% OK, so let's start with quantum harmonic oscillator. 00:34:56.139 --> 00:34:59.020 align:middle line:90% 00:34:59.020 --> 00:35:01.780 align:middle line:84% For simplicity, I take the mass to be 1, 00:35:01.780 --> 00:35:03.790 align:middle line:84% and to take the frequency to be 1. 00:35:03.790 --> 00:35:06.360 align:middle line:90% 00:35:06.360 --> 00:35:06.860 align:middle line:90% Yeah. 00:35:06.860 --> 00:35:09.960 align:middle line:84% Yeah, let me put the frequency here, omega. 00:35:09.960 --> 00:35:10.460 align:middle line:90% OK. 00:35:10.460 --> 00:35:13.780 align:middle line:90% Let's take the mass to be 1. 00:35:13.780 --> 00:35:16.250 align:middle line:90% And so for this series-- 00:35:16.250 --> 00:35:18.170 align:middle line:84% so this is a simple harmonic oscillator, 00:35:18.170 --> 00:35:25.420 align:middle line:84% which you have seen it maybe for most of your intellectual life. 00:35:25.420 --> 00:35:27.360 align:middle line:90% And so P will be x dot. 00:35:27.360 --> 00:35:30.720 align:middle line:84% It's a momentum-- the conjugate momentum is x dot. 00:35:30.720 --> 00:35:37.620 align:middle line:84% And so the Hamiltonian is the P squared divided by 2 plus 1/2 00:35:37.620 --> 00:35:40.470 align:middle line:90% omega squared x squared. 00:35:40.470 --> 00:35:45.490 align:middle line:84% And the equation of motion is x dot --double dot-- 00:35:45.490 --> 00:35:46.510 align:middle line:90% equal to x squared. 00:35:46.510 --> 00:35:56.520 align:middle line:90% 00:35:56.520 --> 00:35:58.927 align:middle line:90% So let's first look at this-- 00:35:58.927 --> 00:36:01.010 align:middle line:84% look at harmonic oscillator as a classical theory. 00:36:01.010 --> 00:36:13.190 align:middle line:90% 00:36:13.190 --> 00:36:17.920 align:middle line:84% So for classical theory, we know how to solve this equation. 00:36:17.920 --> 00:36:20.720 align:middle line:84% We just need to solve this equation. 00:36:20.720 --> 00:36:21.880 align:middle line:90% So classical solution. 00:36:21.880 --> 00:36:26.560 align:middle line:90% 00:36:26.560 --> 00:36:34.420 align:middle line:84% Is given by x t equal to A cosine omega t plus B sine 00:36:34.420 --> 00:36:36.310 align:middle line:90% omega t. 00:36:36.310 --> 00:36:38.860 align:middle line:84% And A and B, just some integration constant. 00:36:38.860 --> 00:36:41.490 align:middle line:90% 00:36:41.490 --> 00:36:45.140 align:middle line:84% And for convenience, I can also write it in the complex form. 00:36:45.140 --> 00:36:50.390 align:middle line:84% Right here is following equal to a exponential minus i omega 00:36:50.390 --> 00:36:55.490 align:middle line:84% t plus a star exponential i omega t. 00:36:55.490 --> 00:36:58.640 align:middle line:90% And a is some complex constant. 00:36:58.640 --> 00:37:00.760 align:middle line:84% And again, it's an integration constant, 00:37:00.760 --> 00:37:03.760 align:middle line:84% I just rewrite the integration constants slightly differently. 00:37:03.760 --> 00:37:06.830 align:middle line:90% 00:37:06.830 --> 00:37:10.870 align:middle line:84% So these are just integration constants. 00:37:10.870 --> 00:37:14.330 align:middle line:84% So now-- yeah, so this is a complete solution 00:37:14.330 --> 00:37:16.340 align:middle line:90% of the problem. 00:37:16.340 --> 00:37:17.465 align:middle line:90% So now let's go to quantum. 00:37:17.465 --> 00:37:21.690 align:middle line:90% 00:37:21.690 --> 00:37:26.590 align:middle line:84% So when we go to quantum, and then we 00:37:26.590 --> 00:37:29.800 align:middle line:84% replace this classical dynamical variable, 00:37:29.800 --> 00:37:34.540 align:middle line:84% then become the Heisenberg operator, 00:37:34.540 --> 00:37:36.430 align:middle line:90% become the quantum operator. 00:37:36.430 --> 00:37:40.800 align:middle line:84% In particular, in the Heisenberg picture, and then 00:37:40.800 --> 00:37:44.170 align:middle line:84% this operator will depend on time. 00:37:44.170 --> 00:37:49.250 align:middle line:84% And now this equation become operator equation. 00:37:49.250 --> 00:37:54.480 align:middle line:84% So now let's-- maybe I should label my equation. 00:37:54.480 --> 00:37:58.765 align:middle line:84% So now this star become an operator equation. 00:37:58.765 --> 00:38:02.324 align:middle line:90% 00:38:02.324 --> 00:38:08.180 align:middle line:84% Now star is the operator equation for x hat. 00:38:08.180 --> 00:38:09.782 align:middle line:84% So you have exactly the same equation 00:38:09.782 --> 00:38:11.990 align:middle line:84% as the classical equation, but now the interpretation 00:38:11.990 --> 00:38:13.600 align:middle line:90% is different. 00:38:13.600 --> 00:38:16.160 align:middle line:90% Now the x hat becomes the-- 00:38:16.160 --> 00:38:18.019 align:middle line:84% now the x become the operator equation. 00:38:18.019 --> 00:38:22.330 align:middle line:90% 00:38:22.330 --> 00:38:26.140 align:middle line:90% So now the solution-- 00:38:26.140 --> 00:38:27.520 align:middle line:90% now let me call this star star. 00:38:27.520 --> 00:38:30.150 align:middle line:90% 00:38:30.150 --> 00:38:32.220 align:middle line:84% So this still solves that equation. 00:38:32.220 --> 00:38:34.930 align:middle line:90% 00:38:34.930 --> 00:38:38.380 align:middle line:84% So this still solves that equation, except-- 00:38:38.380 --> 00:38:41.080 align:middle line:84% so these are just C numbers, because this is a function 00:38:41.080 --> 00:38:43.660 align:middle line:90% of t, these are C numbers. 00:38:43.660 --> 00:38:45.730 align:middle line:90% But now x becomes-- 00:38:45.730 --> 00:38:50.260 align:middle line:84% so now quantum mechanically, so these have become now, 00:38:50.260 --> 00:38:51.610 align:middle line:90% become the quantum solution. 00:38:51.610 --> 00:38:55.640 align:middle line:90% 00:38:55.640 --> 00:38:57.320 align:middle line:90% So now I have hat. 00:38:57.320 --> 00:38:59.508 align:middle line:84% So this still solves the equation. 00:38:59.508 --> 00:39:01.550 align:middle line:84% The quantum mechanic equation just becomes a hat. 00:39:01.550 --> 00:39:07.230 align:middle line:90% 00:39:07.230 --> 00:39:09.120 align:middle line:84% And so this still solves that equation, 00:39:09.120 --> 00:39:12.330 align:middle line:90% but these are C numbers. 00:39:12.330 --> 00:39:14.430 align:middle line:84% The left-hand side is the operator, 00:39:14.430 --> 00:39:17.850 align:middle line:84% and it can only be that A hat and B hat are operators. 00:39:17.850 --> 00:39:20.340 align:middle line:90% And also, a must be operators. 00:39:20.340 --> 00:39:23.370 align:middle line:84% And the star we replace it by dagger. 00:39:23.370 --> 00:39:31.760 align:middle line:84% So now, say-- now a, you just go to a hat 00:39:31.760 --> 00:39:35.060 align:middle line:90% and a star goes to a dagger. 00:39:35.060 --> 00:39:36.200 align:middle line:90% A hat dagger. 00:39:36.200 --> 00:39:40.520 align:middle line:84% OK, now these are integration constants 00:39:40.520 --> 00:39:42.660 align:middle line:90% for the operator equations. 00:39:42.660 --> 00:39:43.790 align:middle line:90% So they are just-- 00:39:43.790 --> 00:39:46.850 align:middle line:84% now they become constant operators. 00:39:46.850 --> 00:39:51.440 align:middle line:84% So they are just constant quantum operators. 00:39:51.440 --> 00:39:54.010 align:middle line:90% 00:39:54.010 --> 00:39:55.790 align:middle line:90% They're just quantum operators. 00:39:55.790 --> 00:39:59.770 align:middle line:84% So they are integration constants for your quantum 00:39:59.770 --> 00:40:01.980 align:middle line:90% operator equations. 00:40:01.980 --> 00:40:04.810 align:middle line:90% So now the solution-- 00:40:04.810 --> 00:40:08.360 align:middle line:84% so now the-- yeah, so now this is your quantum solution. 00:40:08.360 --> 00:40:12.280 align:middle line:90% 00:40:12.280 --> 00:40:16.160 align:middle line:84% So this is the form we will often use. 00:40:16.160 --> 00:40:18.740 align:middle line:84% You can also use that form, but the equivalent, 00:40:18.740 --> 00:40:22.520 align:middle line:84% but this is the form we will often use. 00:40:22.520 --> 00:40:24.600 align:middle line:84% You can also, from here, you take the derivative, 00:40:24.600 --> 00:40:26.543 align:middle line:84% you can find the P. So again, this will 00:40:26.543 --> 00:40:27.710 align:middle line:90% become an operator equation. 00:40:27.710 --> 00:40:29.283 align:middle line:84% You take the derivative of x hat-- 00:40:29.283 --> 00:40:32.630 align:middle line:84% x, and then you find p, et cetera. 00:40:32.630 --> 00:40:36.150 align:middle line:90% 00:40:36.150 --> 00:40:36.650 align:middle line:90% Yeah. 00:40:36.650 --> 00:40:42.767 align:middle line:84% So you-- p hat t, you just take the derivative. 00:40:42.767 --> 00:40:46.260 align:middle line:90% 00:40:46.260 --> 00:40:48.410 align:middle line:90% Then you can just work it out. 00:40:48.410 --> 00:40:49.035 align:middle line:90% It's very easy. 00:40:49.035 --> 00:40:52.430 align:middle line:90% 00:40:52.430 --> 00:40:57.680 align:middle line:90% So now-- so this equation just-- 00:40:57.680 --> 00:41:00.200 align:middle line:84% so we already solved the quantum problem 00:41:00.200 --> 00:41:02.600 align:middle line:84% because we find the full evolution-- full solution 00:41:02.600 --> 00:41:05.900 align:middle line:84% to the quantum operator equation. 00:41:05.900 --> 00:41:09.700 align:middle line:84% Except that we still need to impose 00:41:09.700 --> 00:41:13.620 align:middle line:84% the canonical quantization condition. 00:41:13.620 --> 00:41:16.490 align:middle line:90% So this is just equal to i. 00:41:16.490 --> 00:41:21.980 align:middle line:84% So the standard-- so if you plug in the expression for x 00:41:21.980 --> 00:41:23.120 align:middle line:90% and the t-- 00:41:23.120 --> 00:41:31.960 align:middle line:84% and the p into here, and then you find that a and a dagger, 00:41:31.960 --> 00:41:33.520 align:middle line:90% the commutator is equal to 1. 00:41:33.520 --> 00:41:38.110 align:middle line:84% So this is your familiar creation and annihilation 00:41:38.110 --> 00:41:42.400 align:middle line:84% operator for harmonic oscillator. 00:41:42.400 --> 00:41:44.600 align:middle line:90% For harmonic oscillator. 00:41:44.600 --> 00:41:50.410 align:middle line:84% And now, we can also use the a to build the Hilbert space 00:41:50.410 --> 00:41:55.020 align:middle line:84% because a are the-- yeah, because all your operator now-- 00:41:55.020 --> 00:41:57.150 align:middle line:90% because x and p-- 00:41:57.150 --> 00:42:00.210 align:middle line:84% x and the p are expressed in terms of a and a dagger. 00:42:00.210 --> 00:42:02.730 align:middle line:84% So essentially, any operator of this theory 00:42:02.730 --> 00:42:04.950 align:middle line:84% can all be expressed in terms of a and a dagger. 00:42:04.950 --> 00:42:12.010 align:middle line:90% 00:42:12.010 --> 00:42:14.770 align:middle line:84% And then you can just use a and a dagger because a and a dagger 00:42:14.770 --> 00:42:15.820 align:middle line:90% essentially-- 00:42:15.820 --> 00:42:18.580 align:middle line:84% they are fundamental building block 00:42:18.580 --> 00:42:20.200 align:middle line:90% of your full quantum theory. 00:42:20.200 --> 00:42:23.810 align:middle line:84% And then you can also use that to build the Hilbert space. 00:42:23.810 --> 00:42:28.480 align:middle line:84% So the Hilbert space is defined by the lowest 00:42:28.480 --> 00:42:42.940 align:middle line:84% state is annihilated by a, and then the higher state 00:42:42.940 --> 00:42:49.170 align:middle line:84% obtained by acting a dagger on the ground state. 00:42:49.170 --> 00:42:52.020 align:middle line:90% So this is your full theory. 00:42:52.020 --> 00:42:54.510 align:middle line:90% So this is your full theory. 00:42:54.510 --> 00:43:00.220 align:middle line:84% And so now, now you can compute anything in this theory 00:43:00.220 --> 00:43:04.390 align:middle line:84% just with those knowledge, just with those knowledge. 00:43:04.390 --> 00:43:06.340 align:middle line:84% So any questions on this regarding 00:43:06.340 --> 00:43:09.710 align:middle line:90% the harmonic oscillator? 00:43:09.710 --> 00:43:10.385 align:middle line:90% Good, OK. 00:43:10.385 --> 00:43:12.890 align:middle line:90% 00:43:12.890 --> 00:43:16.720 align:middle line:90% So let me just summarize. 00:43:16.720 --> 00:43:20.950 align:middle line:84% So this is very, very familiar, but let's summarize 00:43:20.950 --> 00:43:22.780 align:middle line:90% the rule we have been using. 00:43:22.780 --> 00:43:31.310 align:middle line:84% Summarize the steps in context of the harmonic oscillator. 00:43:31.310 --> 00:43:34.090 align:middle line:84% And then the same steps can be used 00:43:34.090 --> 00:43:38.750 align:middle line:90% to quantize the field theory. 00:43:38.750 --> 00:43:41.380 align:middle line:90% Steps of quantization. 00:43:41.380 --> 00:43:46.070 align:middle line:90% 00:43:46.070 --> 00:43:48.540 align:middle line:90% So we make it a general. 00:43:48.540 --> 00:43:54.920 align:middle line:84% So first-- so the 0 step is that the classical equation 00:43:54.920 --> 00:44:00.175 align:middle line:84% of motion becomes a quantum operator equation. 00:44:00.175 --> 00:44:03.040 align:middle line:90% 00:44:03.040 --> 00:44:07.690 align:middle line:84% Then the first step is to find the most general solution-- 00:44:07.690 --> 00:44:17.050 align:middle line:90% 00:44:17.050 --> 00:44:35.950 align:middle line:84% to find the most general solution 00:44:35.950 --> 00:44:38.925 align:middle line:84% to classical equation of motion-- 00:44:38.925 --> 00:44:40.300 align:middle line:84% yeah, just to equation of motion. 00:44:40.300 --> 00:44:43.150 align:middle line:90% 00:44:43.150 --> 00:44:50.760 align:middle line:84% And then you go to quantum, and then you 00:44:50.760 --> 00:45:01.764 align:middle line:84% just promote the integration constants 00:45:01.764 --> 00:45:08.030 align:middle line:84% in your classical solution in 1, in step 1 00:45:08.030 --> 00:45:11.690 align:middle line:90% to constant operators-- 00:45:11.690 --> 00:45:14.360 align:middle line:90% 00:45:14.360 --> 00:45:15.960 align:middle line:90% constant quantum operators. 00:45:15.960 --> 00:45:19.510 align:middle line:90% 00:45:19.510 --> 00:45:21.580 align:middle line:90% So this gives the-- 00:45:21.580 --> 00:45:31.470 align:middle line:84% then you have the full-time evolution at the quantum level. 00:45:31.470 --> 00:45:37.210 align:middle line:90% 00:45:37.210 --> 00:45:42.680 align:middle line:84% Now you know how the quantum operator evolves. 00:45:42.680 --> 00:46:03.610 align:middle line:84% And then you impose canonical quantization conditions. 00:46:03.610 --> 00:46:07.870 align:middle line:90% 00:46:07.870 --> 00:46:10.930 align:middle line:84% So that will tell you the commutators 00:46:10.930 --> 00:46:15.790 align:middle line:84% between those integration constant operators 00:46:15.790 --> 00:46:19.210 align:middle line:90% just as we do here. 00:46:19.210 --> 00:46:45.350 align:middle line:84% And then constant operators in 2, 00:46:45.350 --> 00:46:48.020 align:middle line:84% now you know also the commutation relation 00:46:48.020 --> 00:46:48.830 align:middle line:90% between them-- 00:46:48.830 --> 00:46:50.930 align:middle line:84% among them, and then now can be used 00:46:50.930 --> 00:46:58.250 align:middle line:90% to generate the Hilbert space. 00:46:58.250 --> 00:47:09.310 align:middle line:90% 00:47:09.310 --> 00:47:10.600 align:middle line:90% OK. 00:47:10.600 --> 00:47:15.120 align:middle line:84% So this step 1 to 4 are very general. 00:47:15.120 --> 00:47:19.950 align:middle line:84% And if you can do it, and then you can essentially do it-- 00:47:19.950 --> 00:47:23.320 align:middle line:90% apply it to any system. 00:47:23.320 --> 00:47:25.612 align:middle line:84% Say one-- harmonic oscillator is one degree of freedom, 00:47:25.612 --> 00:47:28.112 align:middle line:84% you can apply it two degrees freedom, three degrees freedom. 00:47:28.112 --> 00:47:30.190 align:middle line:84% And also to field theory, infinite number degrees 00:47:30.190 --> 00:47:31.970 align:middle line:90% of freedom. 00:47:31.970 --> 00:47:33.890 align:middle line:84% And now we will apply these to field theory. 00:47:33.890 --> 00:47:34.520 align:middle line:90% Yes? 00:47:34.520 --> 00:47:37.130 align:middle line:84% STUDENT: Do you get the finite dimensional Hilbert spaces 00:47:37.130 --> 00:47:39.020 align:middle line:90% from this procedure? 00:47:39.020 --> 00:47:42.230 align:middle line:84% HONG LIU: For this procedure, you cannot, 00:47:42.230 --> 00:47:45.458 align:middle line:84% but you can get the finite dimension-- yeah, 00:47:45.458 --> 00:47:47.750 align:middle line:84% because the finite dimensional Hilbert space don't have 00:47:47.750 --> 00:47:49.130 align:middle line:90% the classical analog. 00:47:49.130 --> 00:47:51.470 align:middle line:84% So here, we start with a classical system, 00:47:51.470 --> 00:47:52.820 align:middle line:90% and then we quantize it. 00:47:52.820 --> 00:47:58.400 align:middle line:84% And the quantum system is a finite dimensional Hilbert 00:47:58.400 --> 00:47:59.420 align:middle line:90% space. 00:47:59.420 --> 00:48:01.640 align:middle line:84% They're essentially intrinsically quantum. 00:48:01.640 --> 00:48:03.410 align:middle line:90% And yeah. 00:48:03.410 --> 00:48:04.370 align:middle line:90% Like spin. 00:48:04.370 --> 00:48:07.550 align:middle line:84% Spin is an intrinsic in quantum, so yeah. 00:48:07.550 --> 00:48:08.428 align:middle line:90% Yes? 00:48:08.428 --> 00:48:10.970 align:middle line:84% STUDENT: This question is to do with spin operators, the fact 00:48:10.970 --> 00:48:15.450 align:middle line:84% that there are no finite dimensional representations 00:48:15.450 --> 00:48:17.620 align:middle line:90% for these dagger operators. 00:48:17.620 --> 00:48:18.205 align:middle line:90% Any dagger. 00:48:18.205 --> 00:48:19.080 align:middle line:90% HONG LIU: Yeah, yeah. 00:48:19.080 --> 00:48:21.493 align:middle line:84% Yeah, it's just because they don't have-- yeah-- 00:48:21.493 --> 00:48:23.160 align:middle line:84% yeah, the reason is just they don't have 00:48:23.160 --> 00:48:24.590 align:middle line:90% classical counterparts, yeah. 00:48:24.590 --> 00:48:27.100 align:middle line:90% 00:48:27.100 --> 00:48:27.827 align:middle line:90% Yes? 00:48:27.827 --> 00:48:30.160 align:middle line:84% STUDENT: It is it always true that the constant operator 00:48:30.160 --> 00:48:32.285 align:middle line:84% is sufficient to generate the entire Hilbert space? 00:48:32.285 --> 00:48:35.410 align:middle line:84% HONG LIU: Yeah, because if you think about it this way-- 00:48:35.410 --> 00:48:38.730 align:middle line:84% that's a very good question, because-- 00:48:38.730 --> 00:48:40.970 align:middle line:84% let's just look at this harmonic oscillator, 00:48:40.970 --> 00:48:44.180 align:middle line:84% and then you can try to generalize it. 00:48:44.180 --> 00:48:48.900 align:middle line:84% Because they are integration constant over the x and the p. 00:48:48.900 --> 00:48:52.170 align:middle line:84% Then any operator in your theory can all 00:48:52.170 --> 00:48:54.220 align:middle line:84% be expressed in terms of a and a dagger. 00:48:54.220 --> 00:48:57.180 align:middle line:84% And then and then your Hilbert space must be-- 00:48:57.180 --> 00:49:00.630 align:middle line:84% you must be able to generate the Hilbert space using them. 00:49:00.630 --> 00:49:04.450 align:middle line:84% Because they are the building block of your whole operator. 00:49:04.450 --> 00:49:05.282 align:middle line:90% Yeah, yeah. 00:49:05.282 --> 00:49:06.990 align:middle line:84% STUDENT: How do you get the vacuum state? 00:49:06.990 --> 00:49:09.180 align:middle line:84% Is there always a generalization of vacuum state? 00:49:09.180 --> 00:49:10.055 align:middle line:90% HONG LIU: Yeah, yeah. 00:49:10.055 --> 00:49:11.910 align:middle line:84% The vacuum state here is based on-- 00:49:11.910 --> 00:49:16.080 align:middle line:90% is coming from the energy. 00:49:16.080 --> 00:49:19.590 align:middle line:84% So once we solve x and p, and then we 00:49:19.590 --> 00:49:22.110 align:middle line:84% can write the Hamiltonian in terms of x and p, 00:49:22.110 --> 00:49:24.602 align:middle line:84% and then you just look for the lowest energy state. 00:49:24.602 --> 00:49:26.310 align:middle line:84% And then you find the lowest energy state 00:49:26.310 --> 00:49:27.750 align:middle line:90% that satisfy this equation. 00:49:27.750 --> 00:49:28.260 align:middle line:90% Yeah. 00:49:28.260 --> 00:49:30.810 align:middle line:84% And then from there, you can find other states. 00:49:30.810 --> 00:49:32.315 align:middle line:90% Yeah, yeah. 00:49:32.315 --> 00:49:33.690 align:middle line:84% Yeah, the same thing we are going 00:49:33.690 --> 00:49:35.148 align:middle line:84% to-- yeah, the same strategy we are 00:49:35.148 --> 00:49:38.750 align:middle line:84% going to use for the quantum field theory. 00:49:38.750 --> 00:49:39.250 align:middle line:90% OK. 00:49:39.250 --> 00:49:41.270 align:middle line:90% Good? 00:49:41.270 --> 00:49:41.800 align:middle line:90% OK, good. 00:49:41.800 --> 00:49:44.710 align:middle line:90% So now it become a mechanical. 00:49:44.710 --> 00:49:50.380 align:middle line:84% We can just apply this to this theory. 00:49:50.380 --> 00:49:52.100 align:middle line:84% We can just apply this to this theory. 00:49:52.100 --> 00:49:53.960 align:middle line:90% And now let me add here. 00:49:53.960 --> 00:49:58.495 align:middle line:84% So here, the canonical momentum density conjugate to phi is-- 00:49:58.495 --> 00:50:02.720 align:middle line:84% I called it pi before, is just the time derivative of phi. 00:50:02.720 --> 00:50:07.090 align:middle line:84% And the Hamiltonian density, you can find it explicitly, 00:50:07.090 --> 00:50:10.380 align:middle line:90% is pi squared plus 1/2. 00:50:10.380 --> 00:50:17.990 align:middle line:90% 00:50:17.990 --> 00:50:20.220 align:middle line:84% And then this is the classical equation of motion. 00:50:20.220 --> 00:50:20.720 align:middle line:90% OK. 00:50:20.720 --> 00:50:23.860 align:middle line:90% 00:50:23.860 --> 00:50:26.320 align:middle line:84% So we can-- now let's just solve this classical equation 00:50:26.320 --> 00:50:26.820 align:middle line:90% of motion. 00:50:26.820 --> 00:50:29.320 align:middle line:90% 00:50:29.320 --> 00:50:32.070 align:middle line:90% So this equation can be-- 00:50:32.070 --> 00:50:35.100 align:middle line:84% so this equation is easy to solve because 00:50:35.100 --> 00:50:38.730 align:middle line:90% of the translation symmetry. 00:50:38.730 --> 00:50:41.130 align:middle line:84% You can just do a Fourier transform. 00:50:41.130 --> 00:50:45.130 align:middle line:90% So we can Fourier transform. 00:50:45.130 --> 00:50:47.655 align:middle line:90% So now let's do can-- 00:50:47.655 --> 00:50:50.320 align:middle line:90% you can Fourier transform. 00:50:50.320 --> 00:50:56.810 align:middle line:90% So 2 prime can be solved using-- 00:50:56.810 --> 00:51:04.430 align:middle line:90% 00:51:04.430 --> 00:51:06.020 align:middle line:90% OK. 00:51:06.020 --> 00:51:11.900 align:middle line:84% So we can just write phi x equal to exponential 00:51:11.900 --> 00:51:16.730 align:middle line:90% minus iEt plus i k dot x. 00:51:16.730 --> 00:51:24.260 align:middle line:90% 00:51:24.260 --> 00:51:26.270 align:middle line:90% Yeah. 00:51:26.270 --> 00:51:27.200 align:middle line:90% OK. 00:51:27.200 --> 00:51:32.180 align:middle line:84% And then you can see then this just solves the-- 00:51:32.180 --> 00:51:48.260 align:middle line:84% so this is provide a basis of solutions to 2 prime. 00:51:48.260 --> 00:51:50.630 align:middle line:84% It's just a plane wave, just a plane wave. 00:51:50.630 --> 00:51:54.760 align:middle line:90% 00:51:54.760 --> 00:51:57.600 align:middle line:84% So now when you plug this into there, 00:51:57.600 --> 00:52:00.690 align:middle line:84% then you just get the dispersion relation. 00:52:00.690 --> 00:52:06.300 align:middle line:84% Each square should be m squared plus k squared. 00:52:06.300 --> 00:52:11.790 align:middle line:84% So we'll denote this as omega k squared. 00:52:11.790 --> 00:52:19.500 align:middle line:84% So omega k is defined to be just the k squared plus m squared. 00:52:19.500 --> 00:52:26.540 align:middle line:90% 00:52:26.540 --> 00:52:29.450 align:middle line:84% So E-- when you take the square root of E-- 00:52:29.450 --> 00:52:32.810 align:middle line:84% so you can take plus-minus omega k. 00:52:32.810 --> 00:52:34.670 align:middle line:90% Can we plus-minus omega k? 00:52:34.670 --> 00:52:39.010 align:middle line:84% So we normally call the solution-- 00:52:39.010 --> 00:52:53.050 align:middle line:84% we normally separate-- so for historical reasons, 00:52:53.050 --> 00:52:55.030 align:middle line:90% we normally put-- 00:52:55.030 --> 00:53:02.110 align:middle line:84% define u k x to be exponential minus i omega k t-- 00:53:02.110 --> 00:53:06.530 align:middle line:90% 00:53:06.530 --> 00:53:07.030 align:middle line:90% x. 00:53:07.030 --> 00:53:11.930 align:middle line:84% OK, now we have inserted the positive root of E. 00:53:11.930 --> 00:53:15.875 align:middle line:84% So this is normally called the positive energy solution. 00:53:15.875 --> 00:53:20.710 align:middle line:90% 00:53:20.710 --> 00:53:26.560 align:middle line:84% Even though this-- even though this name is a little bit 00:53:26.560 --> 00:53:27.640 align:middle line:90% misleading. 00:53:27.640 --> 00:53:33.490 align:middle line:84% So actually, this-- we don't define the energy actually-- 00:53:33.490 --> 00:53:35.290 align:middle line:84% yeah, later we will see, this is not really 00:53:35.290 --> 00:53:38.710 align:middle line:90% the energy of a particle. 00:53:38.710 --> 00:53:41.410 align:middle line:84% And so this is just a traditional name. 00:53:41.410 --> 00:53:43.210 align:middle line:90% This is just a traditional name. 00:53:43.210 --> 00:53:44.650 align:middle line:90% Conventional name. 00:53:44.650 --> 00:53:48.960 align:middle line:84% And then you can define the complex conjugate of k. 00:53:48.960 --> 00:53:51.010 align:middle line:90% And now you have-- 00:53:51.010 --> 00:53:57.035 align:middle line:84% then corresponding, you have a minus omega k in there. 00:53:57.035 --> 00:54:00.780 align:middle line:90% 00:54:00.780 --> 00:54:06.140 align:middle line:84% So we take this complex conjugate. 00:54:06.140 --> 00:54:08.265 align:middle line:84% And so this is called the negative energy solution. 00:54:08.265 --> 00:54:13.350 align:middle line:90% 00:54:13.350 --> 00:54:19.530 align:middle line:84% So altogether, they form a complete set of solutions. 00:54:19.530 --> 00:54:21.270 align:middle line:90% So complete set of solutions. 00:54:21.270 --> 00:54:26.150 align:middle line:90% 00:54:26.150 --> 00:54:29.740 align:middle line:84% So complete basis-- it's a complete set of-- 00:54:29.740 --> 00:54:45.678 align:middle line:84% complete basis is formed by u k and the u k star for all k. 00:54:45.678 --> 00:54:48.860 align:middle line:90% 00:54:48.860 --> 00:54:50.590 align:middle line:90% For all k. 00:54:50.590 --> 00:54:52.330 align:middle line:90% So these are-- when you-- 00:54:52.330 --> 00:54:54.370 align:middle line:84% these are the complete set of solutions 00:54:54.370 --> 00:54:55.900 align:middle line:90% to that wave equation. 00:54:55.900 --> 00:54:58.420 align:middle line:90% So that's just the-- 00:54:58.420 --> 00:55:01.120 align:middle line:90% yeah. 00:55:01.120 --> 00:55:02.410 align:middle line:90% Any questions on this? 00:55:02.410 --> 00:55:10.380 align:middle line:90% 00:55:10.380 --> 00:55:12.320 align:middle line:84% So this is just like a-- classically, 00:55:12.320 --> 00:55:13.880 align:middle line:90% this is like a wave. 00:55:13.880 --> 00:55:15.990 align:middle line:90% Just like a plane wave. 00:55:15.990 --> 00:55:18.380 align:middle line:84% Which you should also have seen in 8.03. 00:55:18.380 --> 00:55:18.880 align:middle line:90% Yeah. 00:55:18.880 --> 00:55:22.080 align:middle line:90% 00:55:22.080 --> 00:55:23.170 align:middle line:90% Good. 00:55:23.170 --> 00:55:24.640 align:middle line:90% So now we can find that-- 00:55:24.640 --> 00:55:27.010 align:middle line:84% so now we can write down the most general. 00:55:27.010 --> 00:55:28.100 align:middle line:90% So this is a basis. 00:55:28.100 --> 00:55:31.020 align:middle line:90% 00:55:31.020 --> 00:55:34.100 align:middle line:84% So these are the constant part of the exponential plus-minus i 00:55:34.100 --> 00:55:36.800 align:middle line:90% omega t here. 00:55:36.800 --> 00:55:39.410 align:middle line:84% So now we can write down the most general solutions 00:55:39.410 --> 00:55:43.840 align:middle line:84% by just putting in the integration constants. 00:55:43.840 --> 00:55:51.740 align:middle line:84% So the most general classical solutions. 00:55:51.740 --> 00:55:54.670 align:middle line:90% 00:55:54.670 --> 00:56:01.810 align:middle line:84% So you can just write phi x equal to integrate over 00:56:01.810 --> 00:56:06.820 align:middle line:84% all possible value of k because this is for all k, 00:56:06.820 --> 00:56:08.335 align:middle line:90% so we just sum over all of them. 00:56:08.335 --> 00:56:10.850 align:middle line:90% 00:56:10.850 --> 00:56:13.720 align:middle line:90% And so we so-- 00:56:13.720 --> 00:56:16.510 align:middle line:84% so this factor is just for convenience. 00:56:16.510 --> 00:56:17.380 align:middle line:90% OK. 00:56:17.380 --> 00:56:18.700 align:middle line:90% It's just a convention. 00:56:18.700 --> 00:56:22.510 align:middle line:84% You don't have to put it here, it's just a convention. 00:56:22.510 --> 00:56:34.810 align:middle line:84% And then we have a k u k plus a k star u k star. 00:56:34.810 --> 00:56:38.650 align:middle line:84% So this is just the most general set of solutions with a k 00:56:38.650 --> 00:56:41.170 align:middle line:84% and a k star as integration constants. 00:56:41.170 --> 00:56:51.020 align:middle line:90% 00:56:51.020 --> 00:56:53.375 align:middle line:84% So this is a full set of the integration constant. 00:56:53.375 --> 00:57:00.000 align:middle line:90% 00:57:00.000 --> 00:57:00.500 align:middle line:90% Good? 00:57:00.500 --> 00:57:03.100 align:middle line:90% 00:57:03.100 --> 00:57:09.780 align:middle line:84% So now when you go to quantum level, 00:57:09.780 --> 00:57:12.620 align:middle line:84% so now we can just follow the rule. 00:57:12.620 --> 00:57:16.075 align:middle line:84% We find the most general classical solution. 00:57:16.075 --> 00:57:17.450 align:middle line:84% And in the quantum level, we just 00:57:17.450 --> 00:57:20.610 align:middle line:90% promote this to be operator. 00:57:20.610 --> 00:57:25.490 align:middle line:84% You just put the hat there and change this to dagger. 00:57:25.490 --> 00:57:28.490 align:middle line:90% 00:57:28.490 --> 00:57:36.290 align:middle line:84% So now this become the basis of quantum operators. 00:57:36.290 --> 00:57:40.260 align:middle line:90% 00:57:40.260 --> 00:57:44.310 align:middle line:84% So these are the full set of constant quantum operators. 00:57:44.310 --> 00:57:47.740 align:middle line:90% 00:57:47.740 --> 00:57:50.730 align:middle line:90% And this solves your theory on-- 00:57:50.730 --> 00:57:53.380 align:middle line:90% 00:57:53.380 --> 00:57:56.837 align:middle line:84% so this solves the operator equation. 00:57:56.837 --> 00:57:58.420 align:middle line:84% And this solves the operator equation. 00:57:58.420 --> 00:58:01.240 align:middle line:90% 00:58:01.240 --> 00:58:05.760 align:middle line:84% So now the next thing is to impose 00:58:05.760 --> 00:58:10.420 align:middle line:84% the canonical commutation relation. 00:58:10.420 --> 00:58:19.220 align:middle line:90% 00:58:19.220 --> 00:58:20.990 align:middle line:90% So first, we have to-- 00:58:20.990 --> 00:58:25.050 align:middle line:84% now we have to do a little bit thinking. 00:58:25.050 --> 00:58:28.380 align:middle line:84% So so far, it's just straightforward, 00:58:28.380 --> 00:58:31.830 align:middle line:84% but now we have to do a little bit thinking. 00:58:31.830 --> 00:58:38.280 align:middle line:84% So for finite, for harmonic oscillator, 00:58:38.280 --> 00:58:44.350 align:middle line:84% or for quantum systems of a single variable, 00:58:44.350 --> 00:58:46.060 align:middle line:90% you just have x-- 00:58:46.060 --> 00:58:47.065 align:middle line:90% you just have these. 00:58:47.065 --> 00:58:49.720 align:middle line:90% 00:58:49.720 --> 00:58:50.770 align:middle line:90% You just have these. 00:58:50.770 --> 00:58:54.910 align:middle line:84% And now we need to come up with a generalization of these 00:58:54.910 --> 00:58:55.660 align:middle line:90% to a field theory. 00:58:55.660 --> 00:58:58.840 align:middle line:90% 00:58:58.840 --> 00:59:02.620 align:middle line:84% So we need to come up with a generalization of that field 00:59:02.620 --> 00:59:08.550 align:middle line:90% theory corresponding to phi x. 00:59:08.550 --> 00:59:12.310 align:middle line:84% So now let me making the time and the spatial 00:59:12.310 --> 00:59:14.500 align:middle line:90% coordinates separate. 00:59:14.500 --> 00:59:17.020 align:middle line:84% And its conjugate momentum is phi-- 00:59:17.020 --> 00:59:19.540 align:middle line:90% is pi. 00:59:19.540 --> 00:59:20.695 align:middle line:90% Conjugate momentum density. 00:59:20.695 --> 00:59:24.990 align:middle line:90% 00:59:24.990 --> 00:59:26.615 align:middle line:84% So we should do them at the same time. 00:59:26.615 --> 00:59:29.820 align:middle line:90% 00:59:29.820 --> 00:59:34.120 align:middle line:84% Remember, p is the same evolution operator, 00:59:34.120 --> 00:59:36.160 align:middle line:84% so they have to be evaluated at the same time. 00:59:36.160 --> 00:59:37.720 align:middle line:90% So it's equal time. 00:59:37.720 --> 00:59:40.510 align:middle line:84% Canonical quantization conditions 00:59:40.510 --> 00:59:41.665 align:middle line:90% always are the equal time. 00:59:41.665 --> 00:59:44.540 align:middle line:90% 00:59:44.540 --> 00:59:47.160 align:middle line:90% But x is a label of operators. 00:59:47.160 --> 00:59:49.820 align:middle line:84% So x, they don't have to be the same. 00:59:49.820 --> 00:59:51.780 align:middle line:84% So hee, it can be x; here, can be x prime. 00:59:51.780 --> 00:59:52.280 align:middle line:90% OK. 00:59:52.280 --> 00:59:55.340 align:middle line:84% So now we have to come up with a generalization of what is 00:59:55.340 --> 00:59:57.020 align:middle line:90% this quantity for field theory. 00:59:57.020 --> 01:00:00.390 align:middle line:90% 01:00:00.390 --> 01:00:03.580 align:middle line:84% So now we just need to do a little bit of guesswork. 01:00:03.580 --> 01:00:04.950 align:middle line:90% You can easily guess it. 01:00:04.950 --> 01:00:07.417 align:middle line:90% 01:00:07.417 --> 01:00:09.750 align:middle line:84% So before we do that, do you have any questions on this? 01:00:09.750 --> 01:00:10.440 align:middle line:90% Yes? 01:00:10.440 --> 01:00:13.590 align:middle line:84% STUDENT: x is already the operator in this feature? 01:00:13.590 --> 01:00:15.390 align:middle line:84% HONG LIU: Yeah, yeah. x is always-- 01:00:15.390 --> 01:00:19.140 align:middle line:84% so x is always-- here, it's always just the label 01:00:19.140 --> 01:00:21.060 align:middle line:90% of the spatial location. 01:00:21.060 --> 01:00:22.770 align:middle line:84% Yeah, it's a label for the-- yeah, yeah, 01:00:22.770 --> 01:00:24.900 align:middle line:90% it's your field theory label. 01:00:24.900 --> 01:00:25.680 align:middle line:90% Yes? 01:00:25.680 --> 01:00:27.480 align:middle line:84% STUDENT: So in this procedure, I guess 01:00:27.480 --> 01:00:29.640 align:middle line:84% you always end up with your operators 01:00:29.640 --> 01:00:31.230 align:middle line:90% as being constant in time. 01:00:31.230 --> 01:00:33.780 align:middle line:84% Is there any way that you can get them 01:00:33.780 --> 01:00:36.510 align:middle line:84% where it's like the evolution is more complex rather 01:00:36.510 --> 01:00:39.335 align:middle line:84% than just a constant operator and phase factor? 01:00:39.335 --> 01:00:39.960 align:middle line:90% HONG LIU: Yeah. 01:00:39.960 --> 01:00:42.570 align:middle line:84% So normally, if you have second-order differential 01:00:42.570 --> 01:00:45.600 align:middle line:84% equation, you always have some integration constant. 01:00:45.600 --> 01:00:46.350 align:middle line:90% Yeah, that's it. 01:00:46.350 --> 01:00:46.950 align:middle line:90% Yeah. 01:00:46.950 --> 01:00:49.440 align:middle line:84% STUDENT: So that carries forward all throughout? 01:00:49.440 --> 01:00:51.090 align:middle line:90% HONG LIU: Yeah, yeah. 01:00:51.090 --> 01:00:51.590 align:middle line:90% Yeah. 01:00:51.590 --> 01:00:54.300 align:middle line:90% 01:00:54.300 --> 01:00:57.280 align:middle line:90% Other questions? 01:00:57.280 --> 01:00:57.977 align:middle line:90% Yes? 01:00:57.977 --> 01:01:00.310 align:middle line:84% STUDENT: Should it be obvious why the classical equation 01:01:00.310 --> 01:01:02.620 align:middle line:84% of motion translates directly into the operator 01:01:02.620 --> 01:01:03.573 align:middle line:90% equation of motion? 01:01:03.573 --> 01:01:04.240 align:middle line:90% HONG LIU: Right. 01:01:04.240 --> 01:01:04.750 align:middle line:90% Yeah, yeah. 01:01:04.750 --> 01:01:06.880 align:middle line:90% So that's a very good-- 01:01:06.880 --> 01:01:11.120 align:middle line:90% that's a very good question. 01:01:11.120 --> 01:01:15.040 align:middle line:84% So that's just an extension of our usual procedure 01:01:15.040 --> 01:01:16.750 align:middle line:90% for the quantum mechanics. 01:01:16.750 --> 01:01:19.300 align:middle line:84% So the usual procedure, when you-- 01:01:19.300 --> 01:01:20.920 align:middle line:84% even just for the harmonic oscillator 01:01:20.920 --> 01:01:23.883 align:middle line:84% for single-variable system, you have this correspondence 01:01:23.883 --> 01:01:26.050 align:middle line:84% between the classical system and the quantum system. 01:01:26.050 --> 01:01:27.820 align:middle line:84% When you quantize the classical system, 01:01:27.820 --> 01:01:29.362 align:middle line:84% then the classical equation of motion 01:01:29.362 --> 01:01:32.080 align:middle line:84% become a quantum operator equation. 01:01:32.080 --> 01:01:35.320 align:middle line:84% Here, we just use the same rule because quantum field theory 01:01:35.320 --> 01:01:38.110 align:middle line:84% is just the theory of infinite number of degrees freedoms. 01:01:38.110 --> 01:01:40.340 align:middle line:84% We are not changing the rule of quantum mechanics. 01:01:40.340 --> 01:01:42.640 align:middle line:90% And so that's why we just-- 01:01:42.640 --> 01:01:45.100 align:middle line:84% again, just promote the classical equation 01:01:45.100 --> 01:01:46.390 align:middle line:90% into the operator equation. 01:01:46.390 --> 01:01:50.820 align:middle line:90% 01:01:50.820 --> 01:01:52.640 align:middle line:90% Other questions? 01:01:52.640 --> 01:01:53.900 align:middle line:90% Yes? 01:01:53.900 --> 01:01:56.000 align:middle line:84% STUDENT: One way of understanding the Heisenberg 01:01:56.000 --> 01:01:58.790 align:middle line:84% equations for the quantum mechanics is really 01:01:58.790 --> 01:02:02.180 align:middle line:84% the Poisson brackets for the classical theory. 01:02:02.180 --> 01:02:06.245 align:middle line:84% Is there something like that for field theory as well? 01:02:06.245 --> 01:02:06.898 align:middle line:90% I guess the-- 01:02:06.898 --> 01:02:08.690 align:middle line:84% HONG LIU: Yeah, yeah, yeah, there is, yeah. 01:02:08.690 --> 01:02:12.260 align:middle line:84% So classically, you can define the Poisson bracket 01:02:12.260 --> 01:02:15.230 align:middle line:84% between the classical field variables. 01:02:15.230 --> 01:02:17.450 align:middle line:84% And then quantum mechanically, it just 01:02:17.450 --> 01:02:20.620 align:middle line:90% become quantum commutators. 01:02:20.620 --> 01:02:23.120 align:middle line:84% STUDENT: Is that how we could come up with those commutation 01:02:23.120 --> 01:02:23.668 align:middle line:90% relations? 01:02:23.668 --> 01:02:25.460 align:middle line:84% HONG LIU: Yeah, yeah, you can also do that. 01:02:25.460 --> 01:02:26.160 align:middle line:90% That's right. 01:02:26.160 --> 01:02:26.660 align:middle line:90% Yeah. 01:02:26.660 --> 01:02:30.290 align:middle line:84% So one way to come to this is you first describe-- first, 01:02:30.290 --> 01:02:34.410 align:middle line:84% you need to generalize your standard Poisson 01:02:34.410 --> 01:02:38.490 align:middle line:84% bracket for finite number equals freedom to classical field 01:02:38.490 --> 01:02:42.040 align:middle line:84% theory, and then you can just generalize that to the quantum. 01:02:42.040 --> 01:02:43.350 align:middle line:90% Yeah, indeed. 01:02:43.350 --> 01:02:44.760 align:middle line:90% That's one route of doing it. 01:02:44.760 --> 01:02:47.390 align:middle line:90% 01:02:47.390 --> 01:02:50.530 align:middle line:90% OK, other questions? 01:02:50.530 --> 01:02:51.190 align:middle line:90% Good? 01:02:51.190 --> 01:02:53.290 align:middle line:90% So we can just guess the answer. 01:02:53.290 --> 01:02:56.770 align:middle line:84% The answer is very easy to guess. 01:02:56.770 --> 01:03:10.920 align:middle line:84% So remember-- so if you have a single x and a p, 01:03:10.920 --> 01:03:14.580 align:middle line:90% that's what you have. 01:03:14.580 --> 01:03:27.230 align:middle line:84% But if you have more than one particles, 01:03:27.230 --> 01:03:32.020 align:middle line:84% if you have more than one particles, say just hint. 01:03:32.020 --> 01:03:34.840 align:middle line:90% 01:03:34.840 --> 01:03:44.110 align:middle line:84% Say you have multiple particles system in quantum mechanics, 01:03:44.110 --> 01:03:49.740 align:middle line:84% and then you have x a and p a as your dynamical variable. 01:03:49.740 --> 01:03:55.460 align:middle line:84% So a equal to 1 to, say, n, say, the number of particles. 01:03:55.460 --> 01:03:57.770 align:middle line:84% And then your canonical quantization condition 01:03:57.770 --> 01:04:08.134 align:middle line:84% just become x a t p b t equal to i delta a b. 01:04:08.134 --> 01:04:26.790 align:middle line:84% And that different x a commute and different p commute 01:04:26.790 --> 01:04:30.060 align:middle line:90% given the p commute. 01:04:30.060 --> 01:04:31.830 align:middle line:84% Some reason, let me just put this out. 01:04:31.830 --> 01:04:40.560 align:middle line:90% 01:04:40.560 --> 01:04:45.360 align:middle line:84% So now this a and b are essentially 01:04:45.360 --> 01:04:48.030 align:middle line:90% just replaced by x and x prime. 01:04:48.030 --> 01:04:50.940 align:middle line:84% So x and x prime are just continuum version of those 01:04:50.940 --> 01:04:52.680 align:middle line:90% a and b. 01:04:52.680 --> 01:04:55.110 align:middle line:84% Remember that we can't emphasize the x and the x prime 01:04:55.110 --> 01:04:58.750 align:middle line:84% are the labels of your degrees of freedom. 01:04:58.750 --> 01:04:59.955 align:middle line:90% So now you can just guess-- 01:04:59.955 --> 01:05:03.040 align:middle line:90% 01:05:03.040 --> 01:05:08.160 align:middle line:84% so we must have the following scene. 01:05:08.160 --> 01:05:16.980 align:middle line:84% So from here, we must have phi t, x phi t, x prime 01:05:16.980 --> 01:05:22.560 align:middle line:90% must be 0, and then pi t, x-- 01:05:22.560 --> 01:05:26.820 align:middle line:90% so pi is the analog of p here. 01:05:26.820 --> 01:05:30.000 align:middle line:90% yeah, so those are operators. 01:05:30.000 --> 01:05:34.230 align:middle line:90% t, x prime must be 0. 01:05:34.230 --> 01:05:44.380 align:middle line:84% And then phi t, x with pi t, x prime 01:05:44.380 --> 01:05:46.370 align:middle line:84% should be something-- can only be 0, 01:05:46.370 --> 01:05:50.020 align:middle line:84% when x not equal to x prime, can only 01:05:50.020 --> 01:05:51.820 align:middle line:90% be 0 when x equal to x prime. 01:05:51.820 --> 01:05:56.420 align:middle line:90% 01:05:56.420 --> 01:05:58.075 align:middle line:90% OK, as a generalization of this. 01:05:58.075 --> 01:06:01.630 align:middle line:90% 01:06:01.630 --> 01:06:06.130 align:middle line:84% And so you can-- now you can guess, so what should this be? 01:06:06.130 --> 01:06:09.400 align:middle line:90% 01:06:09.400 --> 01:06:11.490 align:middle line:90% So-- yeah? 01:06:11.490 --> 01:06:12.880 align:middle line:90% STUDENT: So direct delta? 01:06:12.880 --> 01:06:15.790 align:middle line:84% HONG LIU: Yeah, this should be just Dirac delta. 01:06:15.790 --> 01:06:17.230 align:middle line:84% But now you can ask the question, 01:06:17.230 --> 01:06:18.880 align:middle line:90% why has to be direct delta? 01:06:18.880 --> 01:06:24.290 align:middle line:84% Maybe why should not be, say, the derivatives of Dirac delta? 01:06:24.290 --> 01:06:26.990 align:middle line:84% Say, why should not be, say, 100th derivative 01:06:26.990 --> 01:06:30.000 align:middle line:90% of Dirac delta? 01:06:30.000 --> 01:06:31.710 align:middle line:84% And that question can be addressed just 01:06:31.710 --> 01:06:35.050 align:middle line:90% by from dimensional analysis. 01:06:35.050 --> 01:06:36.990 align:middle line:90% So here, we know-- 01:06:36.990 --> 01:06:38.910 align:middle line:84% somehow this must be related to Dirac delta, 01:06:38.910 --> 01:06:41.250 align:middle line:90% and now let's decide. 01:06:41.250 --> 01:06:43.950 align:middle line:84% So now you can do the dimension-- do a little bit 01:06:43.950 --> 01:06:47.020 align:middle line:90% dimensional analysis. 01:06:47.020 --> 01:06:50.490 align:middle line:84% So if you just write down the action-- 01:06:50.490 --> 01:06:52.170 align:middle line:84% yeah, the action I have just erased. 01:06:52.170 --> 01:06:53.070 align:middle line:90% Sorry. 01:06:53.070 --> 01:06:55.320 align:middle line:84% So if you look back on the action-- 01:06:55.320 --> 01:06:57.630 align:middle line:84% let me just outline the idea because I'm 01:06:57.630 --> 01:07:00.280 align:middle line:84% sure you can do dimensional analysis yourself. 01:07:00.280 --> 01:07:02.310 align:middle line:90% So if you look at the action-- 01:07:02.310 --> 01:07:04.710 align:middle line:84% so the action is dimensionless in the natural unit 01:07:04.710 --> 01:07:06.790 align:middle line:90% we are using. 01:07:06.790 --> 01:07:12.850 align:middle line:84% So from that, you can deduce the dimension of phi 01:07:12.850 --> 01:07:17.900 align:middle line:84% to be 1 over L-- so 1 over the length. 01:07:17.900 --> 01:07:20.150 align:middle line:90% And from the fact that the pi-- 01:07:20.150 --> 01:07:23.360 align:middle line:90% 01:07:23.360 --> 01:07:24.230 align:middle line:90% where's pi? 01:07:24.230 --> 01:07:27.560 align:middle line:84% And maybe I have also erased-- is equal to phi dot, 01:07:27.560 --> 01:07:32.450 align:middle line:84% means pi should be dimension 1 over L squared because you 01:07:32.450 --> 01:07:34.520 align:middle line:84% take the derivative on time, and then there's 01:07:34.520 --> 01:07:37.070 align:middle line:90% another factor of L. 01:07:37.070 --> 01:07:40.780 align:middle line:84% Then that means, on the right-hand side here, 01:07:40.780 --> 01:07:44.180 align:middle line:84% must be something 1 over L to a cube 01:07:44.180 --> 01:07:46.790 align:middle line:84% because there's no other parameters here. 01:07:46.790 --> 01:07:50.570 align:middle line:84% Yeah, because here, there should be an i. 01:07:50.570 --> 01:07:53.302 align:middle line:84% And if it's a dimension 1 over cube, 01:07:53.302 --> 01:07:55.010 align:middle line:84% there can only be the delta function, not 01:07:55.010 --> 01:07:59.120 align:middle line:84% the 100th derivative of delta functions. 01:07:59.120 --> 01:08:01.130 align:middle line:84% So this thing should be just a delta function. 01:08:01.130 --> 01:08:04.970 align:middle line:90% 01:08:04.970 --> 01:08:08.330 align:middle line:84% So this is the convention that there should be i. 01:08:08.330 --> 01:08:11.720 align:middle line:84% And then it should be just the free delta function. 01:08:11.720 --> 01:08:15.770 align:middle line:84% And this indeed have the dimension of 1 over L cube. 01:08:15.770 --> 01:08:21.109 align:middle line:90% 01:08:21.109 --> 01:08:23.060 align:middle line:90% Good? 01:08:23.060 --> 01:08:26.120 align:middle line:90% So now, you can just plug-- 01:08:26.120 --> 01:08:30.220 align:middle line:84% so you have the expression for x, for phi. 01:08:30.220 --> 01:08:33.040 align:middle line:84% You take the time derivative of this, 01:08:33.040 --> 01:08:36.970 align:middle line:84% you get the expression for pi, and now 01:08:36.970 --> 01:08:39.640 align:middle line:84% you can just plug them into here, 01:08:39.640 --> 01:08:42.399 align:middle line:84% you can just plug them into here. 01:08:42.399 --> 01:08:44.710 align:middle line:84% And then you can find the commutation relation between 01:08:44.710 --> 01:08:46.630 align:middle line:90% those a k's. 01:08:46.630 --> 01:08:49.660 align:middle line:84% And so this is a slightly tedious calculation, 01:08:49.660 --> 01:08:54.590 align:middle line:84% which is, however, a little bit fun, which, of course, 01:08:54.590 --> 01:08:58.109 align:middle line:90% I will leave you to do. 01:08:58.109 --> 01:09:02.779 align:middle line:84% So if you just plug them in, and then you can deduce that 01:09:02.779 --> 01:09:05.939 align:middle line:84% the following commutation relation between a's. 01:09:05.939 --> 01:09:09.439 align:middle line:84% So this is-- I think this is in Pset 2, 01:09:09.439 --> 01:09:11.420 align:middle line:90% but I can still change my mind. 01:09:11.420 --> 01:09:14.479 align:middle line:90% Yeah, I wanted to put in Pset 2. 01:09:14.479 --> 01:09:18.920 align:middle line:84% So you find the commutator between a and the commutator 01:09:18.920 --> 01:09:19.715 align:middle line:90% between a dagger-- 01:09:19.715 --> 01:09:23.010 align:middle line:90% 01:09:23.010 --> 01:09:25.800 align:middle line:84% yeah, so now, we will suppress the hat 01:09:25.800 --> 01:09:29.250 align:middle line:84% because if I write hat I think over and over 01:09:29.250 --> 01:09:30.029 align:middle line:90% will be too tired. 01:09:30.029 --> 01:09:33.760 align:middle line:90% 01:09:33.760 --> 01:09:36.189 align:middle line:90% So these are 0. 01:09:36.189 --> 01:09:40.074 align:middle line:84% So the commutation relation between a and 0, and a-- 01:09:40.074 --> 01:09:43.210 align:middle line:90% 01:09:43.210 --> 01:09:52.844 align:middle line:90% between-- within a and a dagger. 01:09:52.844 --> 01:09:56.020 align:middle line:90% 01:09:56.020 --> 01:10:00.980 align:middle line:84% So this gives you 2 pi cube delta function in k. 01:10:00.980 --> 01:10:03.630 align:middle line:90% 01:10:03.630 --> 01:10:05.660 align:middle line:84% So this is a three delta function k. 01:10:05.660 --> 01:10:10.590 align:middle line:90% 01:10:10.590 --> 01:10:14.880 align:middle line:84% So again, this is a straightforward generalization, 01:10:14.880 --> 01:10:18.200 align:middle line:84% so if you have multiple harmonic oscillators-- 01:10:18.200 --> 01:10:20.680 align:middle line:84% so if you have considered the multiple harmonic oscillators 01:10:20.680 --> 01:10:23.950 align:middle line:84% before, and then the a between the different harmonic 01:10:23.950 --> 01:10:25.840 align:middle line:84% oscillator because k prime are just-- 01:10:25.840 --> 01:10:28.240 align:middle line:90% here, just corresponding to-- 01:10:28.240 --> 01:10:29.860 align:middle line:90% essentially you have-- yeah. 01:10:29.860 --> 01:10:32.380 align:middle line:84% Here, this is-- essentially, you have infinite number 01:10:32.380 --> 01:10:33.700 align:middle line:90% of harmonic oscillators. 01:10:33.700 --> 01:10:36.610 align:middle line:84% And each one of them labeled by a k. 01:10:36.610 --> 01:10:38.215 align:middle line:90% So this is just like-- 01:10:38.215 --> 01:10:40.465 align:middle line:84% essentially we find-- yeah, let me just write it here. 01:10:40.465 --> 01:10:44.360 align:middle line:90% 01:10:44.360 --> 01:10:48.160 align:middle line:84% So from those commutation relation, 01:10:48.160 --> 01:10:59.450 align:middle line:84% we conclude this theory, when we quantize-- 01:10:59.450 --> 01:11:13.060 align:middle line:84% after we quantize it, become an infinite number 01:11:13.060 --> 01:11:15.580 align:middle line:84% independent harmonic oscillators, 01:11:15.580 --> 01:11:17.320 align:middle line:90% decoupled harmonic oscillators. 01:11:17.320 --> 01:11:21.720 align:middle line:90% 01:11:21.720 --> 01:11:30.900 align:middle line:84% Harmonic oscillators labeled by continuous parameter k. 01:11:30.900 --> 01:11:34.130 align:middle line:90% 01:11:34.130 --> 01:11:35.720 align:middle line:90% So k is-- yeah. 01:11:35.720 --> 01:11:36.770 align:middle line:90% k is the wave number. 01:11:36.770 --> 01:11:44.710 align:middle line:90% 01:11:44.710 --> 01:11:50.620 align:middle line:84% So for each k, there is a, and a-- 01:11:50.620 --> 01:11:54.880 align:middle line:84% so between a themselves, it's 0; between a daggers, it's 0; 01:11:54.880 --> 01:11:59.200 align:middle line:84% but a, a dagger, they not equal to 0. 01:11:59.200 --> 01:12:03.490 align:middle line:84% And so this is, again, the continuum generalization of 1. 01:12:03.490 --> 01:12:06.160 align:middle line:84% This is a continuum generalization of 1 01:12:06.160 --> 01:12:08.020 align:middle line:84% because you have a continuous variables. 01:12:08.020 --> 01:12:08.590 align:middle line:90% Yes? 01:12:08.590 --> 01:12:11.500 align:middle line:84% STUDENT: So how about the commutation relations 01:12:11.500 --> 01:12:13.740 align:middle line:90% of different times? 01:12:13.740 --> 01:12:15.960 align:middle line:84% HONG LIU: Yeah, then you cannot say for sure. 01:12:15.960 --> 01:12:18.810 align:middle line:84% STUDENT: Like by Lorentz invariance 01:12:18.810 --> 01:12:21.480 align:middle line:84% x and t be on the same plane, shouldn't we have the same-- 01:12:21.480 --> 01:12:23.460 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 01:12:23.460 --> 01:12:27.270 align:middle line:84% No, but you see the quantization condition 01:12:27.270 --> 01:12:29.050 align:middle line:90% is in quantum mechanics. 01:12:29.050 --> 01:12:32.040 align:middle line:84% Quantum mechanics, t and x are not on the same footing. 01:12:32.040 --> 01:12:34.710 align:middle line:84% You can require your action to be-- x and t 01:12:34.710 --> 01:12:36.030 align:middle line:90% to be on the same footing. 01:12:36.030 --> 01:12:38.040 align:middle line:84% Once you start quantize your theory, 01:12:38.040 --> 01:12:42.030 align:middle line:84% and then t will have a pronounced role. 01:12:42.030 --> 01:12:43.710 align:middle line:84% STUDENT: OK, so if I wanted to, I 01:12:43.710 --> 01:12:46.440 align:middle line:84% couldn't write like the commutation relation 01:12:46.440 --> 01:12:48.570 align:middle line:90% as functions of for vector x? 01:12:48.570 --> 01:12:49.112 align:middle line:90% HONG LIU: No. 01:12:49.112 --> 01:12:49.945 align:middle line:90% STUDENT: --delta x-- 01:12:49.945 --> 01:12:51.210 align:middle line:90% HONG LIU: No, no, no, no, no. 01:12:51.210 --> 01:12:52.650 align:middle line:84% The canonical commutation relation 01:12:52.650 --> 01:12:54.630 align:middle line:84% have to be imposed at equal time. 01:12:54.630 --> 01:12:59.290 align:middle line:90% 01:12:59.290 --> 01:13:01.470 align:middle line:90% Other questions? 01:13:01.470 --> 01:13:03.360 align:middle line:90% Good? 01:13:03.360 --> 01:13:08.200 align:middle line:90% So yeah. 01:13:08.200 --> 01:13:12.220 align:middle line:90% So essentially, we just get-- 01:13:12.220 --> 01:13:13.990 align:middle line:90% and now it's just trivial. 01:13:13.990 --> 01:13:16.790 align:middle line:84% So you can just build up your Hilbert space. 01:13:16.790 --> 01:13:18.540 align:middle line:84% Essentially, you just have infinite number 01:13:18.540 --> 01:13:20.610 align:middle line:90% of harmonic oscillators. 01:13:20.610 --> 01:13:22.960 align:middle line:84% You just have infinite number of harmonic oscillators. 01:13:22.960 --> 01:13:24.960 align:middle line:84% And there's no surprise, you get infinite number 01:13:24.960 --> 01:13:29.110 align:middle line:84% of harmonic oscillators because we mentioned that this field 01:13:29.110 --> 01:13:32.740 align:middle line:84% theory can be actually written as a continuum 01:13:32.740 --> 01:13:37.150 align:middle line:84% limit of these particles on the chain, 01:13:37.150 --> 01:13:39.570 align:middle line:84% which, in these 8.03 three examples, you know, 01:13:39.570 --> 01:13:41.730 align:middle line:84% that's just a harmonic oscillator. 01:13:41.730 --> 01:13:44.610 align:middle line:84% Once you find the lower modes, they're 01:13:44.610 --> 01:13:47.187 align:middle line:84% all just a bunch of harmonic oscillators. 01:13:47.187 --> 01:13:49.020 align:middle line:84% And this is just a three-dimensional version 01:13:49.020 --> 01:13:51.420 align:middle line:90% of that. 01:13:51.420 --> 01:13:53.100 align:middle line:90% And now we will-- 01:13:53.100 --> 01:13:59.200 align:middle line:84% yeah, today we are running out of time. 01:13:59.200 --> 01:14:02.370 align:middle line:84% So next time, we will see that each excitations 01:14:02.370 --> 01:14:05.640 align:middle line:84% of the harmonic oscillator can be interpreted 01:14:05.640 --> 01:14:07.950 align:middle line:90% as a spacetime particle. 01:14:07.950 --> 01:14:09.760 align:middle line:90% So that's the cool thing of it. 01:14:09.760 --> 01:14:12.640 align:middle line:84% And now you have this infinite number of harmonic oscillator, 01:14:12.640 --> 01:14:14.230 align:middle line:90% and now you can act-- 01:14:14.230 --> 01:14:17.460 align:middle line:84% now you can define the vacuum, and then act this creation 01:14:17.460 --> 01:14:18.840 align:middle line:90% operators on the vacuum. 01:14:18.840 --> 01:14:21.150 align:middle line:84% And now you find each excitation actually corresponding 01:14:21.150 --> 01:14:24.690 align:middle line:90% to a particle and has the-- 01:14:24.690 --> 01:14:26.460 align:middle line:84% corresponding to a relativistic particle. 01:14:26.460 --> 01:14:29.220 align:middle line:84% And that's how you can have actually arbitrary number 01:14:29.220 --> 01:14:31.920 align:middle line:90% of particles in this theory. 01:14:31.920 --> 01:14:36.570 align:middle line:84% And yeah, because you can excite as many times as you want. 01:14:36.570 --> 01:14:38.990 align:middle line:90% Each excitation is a particle. 01:14:38.990 --> 01:14:39.740 align:middle line:90% Good, good. 01:14:39.740 --> 01:14:42.110 align:middle line:84% OK, so I think it's a good time-- yeah, 01:14:42.110 --> 01:14:45.110 align:middle line:84% we are two minutes, I think, early, 01:14:45.110 --> 01:14:49.540 align:middle line:84% but I think it's a very good place to break. 01:14:49.540 --> 01:14:53.000 align:middle line:90%