WEBVTT 00:00:00.000 --> 00:00:01.948 align:middle line:90% [SQUEAKING] 00:00:01.948 --> 00:00:03.409 align:middle line:90% [RUSTLING] 00:00:03.409 --> 00:00:05.357 align:middle line:90% [CLICKING] 00:00:05.357 --> 00:00:11.460 align:middle line:90% 00:00:11.460 --> 00:00:14.520 align:middle line:90% HONG LIU: Let us start. 00:00:14.520 --> 00:00:18.900 align:middle line:84% First, I forgot to mention last time I will always 00:00:18.900 --> 00:00:23.280 align:middle line:84% use the metric convention for Minkowski spacetime 00:00:23.280 --> 00:00:25.630 align:middle line:90% to be given by this one. 00:00:25.630 --> 00:00:26.940 align:middle line:90% So the time component-- 00:00:26.940 --> 00:00:28.260 align:middle line:90% I have minus 1. 00:00:28.260 --> 00:00:31.785 align:middle line:84% And then the spatial component is positive 1. 00:00:31.785 --> 00:00:34.530 align:middle line:90% 00:00:34.530 --> 00:00:35.610 align:middle line:90% And yeah. 00:00:35.610 --> 00:00:38.790 align:middle line:84% So also, let me just first remind you 00:00:38.790 --> 00:00:41.480 align:middle line:90% what we did in last lecture. 00:00:41.480 --> 00:00:44.210 align:middle line:84% So in the last lecture, we talked 00:00:44.210 --> 00:00:52.330 align:middle line:84% about the principle of locality, which is a powerful principle. 00:00:52.330 --> 00:00:55.660 align:middle line:84% And so this naturally leads to the concept of fields. 00:00:55.660 --> 00:00:59.380 align:middle line:90% 00:00:59.380 --> 00:01:01.990 align:middle line:84% And so I will generally denote fields-- 00:01:01.990 --> 00:01:04.060 align:middle line:84% say, using this kind of notation. 00:01:04.060 --> 00:01:08.380 align:middle line:84% Yeah, and when we talk about fields abstractly, 00:01:08.380 --> 00:01:10.690 align:middle line:84% we will use this kind of location. 00:01:10.690 --> 00:01:16.870 align:middle line:84% And so phi_a is just a label for different fields. 00:01:16.870 --> 00:01:20.260 align:middle line:84% And also-- so the fields depend on spatial coordinates 00:01:20.260 --> 00:01:22.958 align:middle line:90% and also depend on time. 00:01:22.958 --> 00:01:25.250 align:middle line:84% So the field coordinate-- you can think of this spatial 00:01:25.250 --> 00:01:29.600 align:middle line:84% dependence as the label for the fields-- 00:01:29.600 --> 00:01:30.930 align:middle line:90% for the degrees of freedom. 00:01:30.930 --> 00:01:33.650 align:middle line:84% So at each spatial point, the degrees of freedom-- 00:01:33.650 --> 00:01:36.740 align:middle line:84% and the time describe its evolution. 00:01:36.740 --> 00:01:38.710 align:middle line:90% OK. 00:01:38.710 --> 00:01:41.780 align:middle line:90% So essentially, this is like-- 00:01:41.780 --> 00:01:45.740 align:middle line:84% so classical field is like classical mechanics degrees 00:01:45.740 --> 00:01:48.133 align:middle line:84% of freedom but now with the infinite number 00:01:48.133 --> 00:01:49.050 align:middle line:90% of degrees of freedom. 00:01:49.050 --> 00:01:49.550 align:middle line:90% OK? 00:01:49.550 --> 00:01:54.470 align:middle line:84% So the classical field theory is like classical mechanics 00:01:54.470 --> 00:01:58.990 align:middle line:84% but now with infinite number of degrees of freedom. 00:01:58.990 --> 00:02:02.110 align:middle line:84% Because now you have some finite number degrees of freedom 00:02:02.110 --> 00:02:04.550 align:middle line:90% for each point in spacetime. 00:02:04.550 --> 00:02:05.050 align:middle line:90% OK? 00:02:05.050 --> 00:02:09.030 align:middle line:90% 00:02:09.030 --> 00:02:12.060 align:middle line:84% And the principle locality also implies 00:02:12.060 --> 00:02:15.710 align:middle line:84% that the action for such dynamical variables-- 00:02:15.710 --> 00:02:17.760 align:middle line:84% so the fields are our dynamical variables-- 00:02:17.760 --> 00:02:21.570 align:middle line:84% and also implies that the action has a local form. 00:02:21.570 --> 00:02:28.380 align:middle line:90% 00:02:28.380 --> 00:02:33.440 align:middle line:84% So the action have the form that is always a single spacetime 00:02:33.440 --> 00:02:37.310 align:middle line:84% integral and some function, which 00:02:37.310 --> 00:02:40.370 align:middle line:84% is called Lagrangian density, which 00:02:40.370 --> 00:02:42.720 align:middle line:90% is a function of the fields-- 00:02:42.720 --> 00:02:46.890 align:middle line:84% say, phi at some point at the point x, which-- 00:02:46.890 --> 00:02:49.470 align:middle line:90% and then also its derivatives. 00:02:49.470 --> 00:02:52.980 align:middle line:90% 00:02:52.980 --> 00:02:57.330 align:middle line:84% And so because of locality, then the action 00:02:57.330 --> 00:03:00.010 align:middle line:90% has a very simple form. 00:03:00.010 --> 00:03:00.760 align:middle line:90% OK. 00:03:00.760 --> 00:03:03.550 align:middle line:84% And then, as in classical mechanics, 00:03:03.550 --> 00:03:05.410 align:middle line:84% you can introduce canonical momentum, 00:03:05.410 --> 00:03:08.860 align:middle line:84% conjugate to your dynamical variable. 00:03:08.860 --> 00:03:13.450 align:middle line:90% So we have momentum density. 00:03:13.450 --> 00:03:15.730 align:middle line:84% So you conjugate to phi a each point. 00:03:15.730 --> 00:03:27.440 align:middle line:90% 00:03:27.440 --> 00:03:27.950 align:middle line:90% OK? 00:03:27.950 --> 00:03:32.510 align:middle line:84% So this is the phi dot, which is the time derivative of phi. 00:03:32.510 --> 00:03:37.110 align:middle line:84% And then you can also introduce the Hamiltonian density, 00:03:37.110 --> 00:03:42.800 align:middle line:90% which is the pi a phi dot a. 00:03:42.800 --> 00:03:46.690 align:middle line:84% So a is summed and then minus L. OK? 00:03:46.690 --> 00:03:48.370 align:middle line:84% And then the Hamiltonian is obtained 00:03:48.370 --> 00:03:49.900 align:middle line:90% by integrating this over space. 00:03:49.900 --> 00:03:52.540 align:middle line:90% 00:03:52.540 --> 00:03:53.500 align:middle line:90% Yeah. 00:03:53.500 --> 00:03:56.020 align:middle line:84% And also, the equation of motion-- 00:03:56.020 --> 00:03:58.210 align:middle line:84% when you do the variation of the action 00:03:58.210 --> 00:03:59.710 align:middle line:84% and you find the equation of motion, 00:03:59.710 --> 00:04:02.470 align:middle line:84% it's given by the following general form. 00:04:02.470 --> 00:04:16.940 align:middle line:90% 00:04:16.940 --> 00:04:17.440 align:middle line:90% OK? 00:04:17.440 --> 00:04:21.490 align:middle line:84% So the equation of motion is given by this general form. 00:04:21.490 --> 00:04:25.950 align:middle line:84% And when we study field theory, as we study, say, 00:04:25.950 --> 00:04:30.600 align:middle line:84% any other subject, we always start with simplistic examples. 00:04:30.600 --> 00:04:31.150 align:middle line:90% OK? 00:04:31.150 --> 00:04:36.366 align:middle line:84% And the simplest example would be, say, a single scalar field, 00:04:36.366 --> 00:04:45.360 align:middle line:84% OK, if you can see the single scalar field, which 00:04:45.360 --> 00:04:46.290 align:middle line:90% you have low index. 00:04:46.290 --> 00:04:50.100 align:middle line:84% We just have phi, which now depends on spacetime point x. 00:04:50.100 --> 00:04:54.360 align:middle line:84% And then the simplest action for such a scalar field-- 00:04:54.360 --> 00:05:01.810 align:middle line:84% you can just write down based on a general principle-- say-- 00:05:01.810 --> 00:05:02.310 align:middle line:90% yeah. 00:05:02.310 --> 00:05:04.740 align:middle line:90% So you kind of have this form. 00:05:04.740 --> 00:05:08.400 align:middle line:84% So we discussed last time the simplest action for scalar 00:05:08.400 --> 00:05:10.770 align:middle line:90% field have the following form-- 00:05:10.770 --> 00:05:14.980 align:middle line:84% partial mu phi, partial mu phi, and then 00:05:14.980 --> 00:05:17.710 align:middle line:90% plus m-squared phi-squared. 00:05:17.710 --> 00:05:20.170 align:middle line:84% And then the sign here is determined because 00:05:20.170 --> 00:05:22.600 align:middle line:90% of the metric convention. 00:05:22.600 --> 00:05:24.010 align:middle line:90% OK. 00:05:24.010 --> 00:05:29.060 align:middle line:84% So this is the simplest scalar field theory we can write down. 00:05:29.060 --> 00:05:29.560 align:middle line:90% OK? 00:05:29.560 --> 00:05:34.240 align:middle line:84% So here, I have written here a quadratic function 00:05:34.240 --> 00:05:38.300 align:middle line:84% for phi, which we wrote as a general function last time. 00:05:38.300 --> 00:05:42.790 align:middle line:84% And here, the kinetic term comes from the Lorentz symmetry. 00:05:42.790 --> 00:05:46.060 align:middle line:84% So this is the simplest derivative term 00:05:46.060 --> 00:05:50.490 align:middle line:84% you can write down which respect Lorentz symmetry. 00:05:50.490 --> 00:05:53.600 align:middle line:84% You can have more complicated terms. 00:05:53.600 --> 00:05:55.860 align:middle line:84% You can have this term, for example, squared. 00:05:55.860 --> 00:05:56.360 align:middle line:90% OK? 00:05:56.360 --> 00:05:57.818 align:middle line:84% But that would be more complicated. 00:05:57.818 --> 00:06:01.980 align:middle line:90% So this is the simplest one. 00:06:01.980 --> 00:06:03.810 align:middle line:84% And so this will be the simplest field 00:06:03.810 --> 00:06:06.000 align:middle line:90% theory we will study first. 00:06:06.000 --> 00:06:06.690 align:middle line:90% OK? 00:06:06.690 --> 00:06:09.300 align:middle line:90% And yeah. 00:06:09.300 --> 00:06:13.776 align:middle line:84% So in this case, for this example, 00:06:13.776 --> 00:06:21.810 align:middle line:84% the momentum density just equal to phi dot-- 00:06:21.810 --> 00:06:23.550 align:middle line:90% just time derivative of phi. 00:06:23.550 --> 00:06:27.270 align:middle line:84% And then the equation of motion is 00:06:27.270 --> 00:06:30.180 align:middle line:90% given by a linear equation phi. 00:06:30.180 --> 00:06:31.160 align:middle line:90% It's given by that. 00:06:31.160 --> 00:06:34.550 align:middle line:90% 00:06:34.550 --> 00:06:38.380 align:middle line:84% So this is a very simple theory with very simple equation 00:06:38.380 --> 00:06:40.600 align:middle line:90% of motion. 00:06:40.600 --> 00:06:41.950 align:middle line:90% But later, we will see-- 00:06:41.950 --> 00:06:46.570 align:middle line:84% I actually will teach us a lot about general field theory. 00:06:46.570 --> 00:06:52.870 align:middle line:84% So it's a very quick summary of some of the main points 00:06:52.870 --> 00:06:54.790 align:middle line:90% of last lecture. 00:06:54.790 --> 00:06:56.230 align:middle line:90% Do you have any questions? 00:06:56.230 --> 00:07:03.950 align:middle line:90% 00:07:03.950 --> 00:07:04.670 align:middle line:90% Yes? 00:07:04.670 --> 00:07:06.950 align:middle line:84% AUDIENCE: You haven't specified if phi a is 00:07:06.950 --> 00:07:08.320 align:middle line:90% either complex or real. 00:07:08.320 --> 00:07:09.550 align:middle line:90% HONG LIU: Right. 00:07:09.550 --> 00:07:10.930 align:middle line:90% So phi here is a real field. 00:07:10.930 --> 00:07:13.580 align:middle line:90% Yeah. 00:07:13.580 --> 00:07:14.080 align:middle line:90% Yeah. 00:07:14.080 --> 00:07:16.190 align:middle line:90% So phi just take a real value. 00:07:16.190 --> 00:07:16.690 align:middle line:90% Yeah. 00:07:16.690 --> 00:07:19.810 align:middle line:90% 00:07:19.810 --> 00:07:20.800 align:middle line:90% Other questions? 00:07:20.800 --> 00:07:26.330 align:middle line:90% 00:07:26.330 --> 00:07:27.000 align:middle line:90% Good? 00:07:27.000 --> 00:07:27.500 align:middle line:90% OK. 00:07:27.500 --> 00:07:31.280 align:middle line:90% 00:07:31.280 --> 00:07:36.770 align:middle line:84% So now we will talk about the second topic related 00:07:36.770 --> 00:07:47.210 align:middle line:84% to our discussion of classical field theory, the symmetries 00:07:47.210 --> 00:07:48.140 align:middle line:90% and conservation laws. 00:07:48.140 --> 00:07:57.660 align:middle line:90% 00:07:57.660 --> 00:08:01.970 align:middle line:84% So when we have a theory, say, with such an action, 00:08:01.970 --> 00:08:05.910 align:middle line:84% we say the theory has a symmetry. 00:08:05.910 --> 00:08:10.960 align:middle line:84% So the definition of the symmetry is the following. 00:08:10.960 --> 00:08:21.090 align:middle line:84% So symmetry is some transformation 00:08:21.090 --> 00:08:28.700 align:middle line:84% of each field phi a into, say, some function of phi. 00:08:28.700 --> 00:08:37.970 align:middle line:84% So this new phi a can, say, depend on all possible fields. 00:08:37.970 --> 00:08:38.799 align:middle line:90% OK? 00:08:38.799 --> 00:08:44.810 align:middle line:84% And so some transformation of your dynamical variables-- 00:08:44.810 --> 00:08:47.380 align:middle line:84% so here, prime just means a different function, 00:08:47.380 --> 00:08:49.180 align:middle line:84% it does not mean derivatives, OK-- 00:08:49.180 --> 00:08:55.146 align:middle line:84% which leaves the action invariant. 00:08:55.146 --> 00:09:01.950 align:middle line:90% 00:09:01.950 --> 00:09:04.590 align:middle line:90% OK? 00:09:04.590 --> 00:09:12.220 align:middle line:84% So whenever happens-- and we say this is a symmetry. 00:09:12.220 --> 00:09:19.050 align:middle line:84% So for example, in this example, OK-- 00:09:19.050 --> 00:09:23.460 align:middle line:90% 00:09:23.460 --> 00:09:26.350 align:middle line:84% let me give you a label to this equation. 00:09:26.350 --> 00:09:29.230 align:middle line:90% Let me call this equation star. 00:09:29.230 --> 00:09:39.180 align:middle line:84% So in the example of star, say, for example, the symmetry 00:09:39.180 --> 00:09:42.090 align:middle line:90% includes, say, translation. 00:09:42.090 --> 00:09:50.520 align:middle line:90% 00:09:50.520 --> 00:09:51.020 align:middle line:90% OK? 00:09:51.020 --> 00:09:58.940 align:middle line:84% So if you imagine we take a coordinate transformation 00:09:58.940 --> 00:10:05.265 align:middle line:84% and we do a shift in coordinates, a constant shift-- 00:10:05.265 --> 00:10:07.630 align:middle line:90% 00:10:07.630 --> 00:10:08.130 align:middle line:90% OK? 00:10:08.130 --> 00:10:12.110 align:middle line:84% Let's imagine we do a constant shift of the coordinates. 00:10:12.110 --> 00:10:14.430 align:middle line:90% And a mu is a constant. 00:10:14.430 --> 00:10:16.100 align:middle line:90% So a mu are constants. 00:10:16.100 --> 00:10:22.930 align:middle line:90% 00:10:22.930 --> 00:10:23.650 align:middle line:90% OK? 00:10:23.650 --> 00:10:25.850 align:middle line:90% So it's a constant vector. 00:10:25.850 --> 00:10:28.510 align:middle line:84% And now if we assume that the phi transform 00:10:28.510 --> 00:10:29.900 align:middle line:90% as the following-- 00:10:29.900 --> 00:10:31.450 align:middle line:84% so when you change the coordinates, 00:10:31.450 --> 00:10:33.910 align:middle line:84% it essentially just changes the label. 00:10:33.910 --> 00:10:34.660 align:middle line:90% OK? 00:10:34.660 --> 00:10:38.050 align:middle line:84% It changes the label on the fields. 00:10:38.050 --> 00:10:41.780 align:middle line:84% But the value of the field should not change. 00:10:41.780 --> 00:10:47.220 align:middle line:84% So that means that the phi evaluated at the original point 00:10:47.220 --> 00:10:48.760 align:middle line:90% should be the same. 00:10:48.760 --> 00:10:52.290 align:middle line:84% So that means the new transform of the phi evaluated 00:10:52.290 --> 00:10:54.780 align:middle line:90% at the new point, OK-- 00:10:54.780 --> 00:10:57.360 align:middle line:84% new x prime point-- should be the same as the value 00:10:57.360 --> 00:10:58.960 align:middle line:90% at the original point. 00:10:58.960 --> 00:10:59.460 align:middle line:90% OK? 00:10:59.460 --> 00:11:02.340 align:middle line:90% So this is the-- 00:11:02.340 --> 00:11:02.850 align:middle line:90% yeah. 00:11:02.850 --> 00:11:05.260 align:middle line:84% So suppose the phi transform like this. 00:11:05.260 --> 00:11:12.340 align:middle line:84% So the scalar field should transform 00:11:12.340 --> 00:11:20.500 align:middle line:90% like this under a general-- 00:11:20.500 --> 00:11:24.310 align:middle line:84% so this is the example of a spacetime transformation, OK, 00:11:24.310 --> 00:11:26.560 align:middle line:84% and the general spacetime transformation. 00:11:26.560 --> 00:11:36.390 align:middle line:90% 00:11:36.390 --> 00:11:38.220 align:middle line:90% as, say, this equation. 00:11:38.220 --> 00:11:39.270 align:middle line:90% OK? 00:11:39.270 --> 00:11:42.720 align:middle line:90% So you change the label. 00:11:42.720 --> 00:11:44.920 align:middle line:84% But the value of phi should not change. 00:11:44.920 --> 00:11:45.420 align:middle line:90% OK? 00:11:45.420 --> 00:11:49.350 align:middle line:84% So the new phi evaluated at the new location 00:11:49.350 --> 00:11:51.530 align:middle line:84% should be the same as the value of phi 00:11:51.530 --> 00:11:53.670 align:middle line:90% at the original location. 00:11:53.670 --> 00:11:55.260 align:middle line:90% OK. 00:11:55.260 --> 00:11:58.550 align:middle line:84% And so you can easily check it yourself. 00:11:58.550 --> 00:11:59.050 align:middle line:90% OK? 00:11:59.050 --> 00:12:00.550 align:middle line:84% So you can easily check it yourself. 00:12:00.550 --> 00:12:02.140 align:middle line:90% But I will not do it here. 00:12:02.140 --> 00:12:05.980 align:middle line:84% So this exercise for yourself-- that actually this action 00:12:05.980 --> 00:12:10.270 align:middle line:84% is invariant under this transformation 00:12:10.270 --> 00:12:13.510 align:middle line:90% with this change of x. 00:12:13.510 --> 00:12:15.885 align:middle line:90% OK. 00:12:15.885 --> 00:12:17.320 align:middle line:90% Any questions on this? 00:12:17.320 --> 00:12:21.810 align:middle line:90% 00:12:21.810 --> 00:12:25.800 align:middle line:84% So second transformation-- which this is invariant-- 00:12:25.800 --> 00:12:27.570 align:middle line:90% is a Lorentz symmetry. 00:12:27.570 --> 00:12:32.660 align:middle line:90% 00:12:32.660 --> 00:12:34.700 align:middle line:84% So Lorentz transformation is another kind 00:12:34.700 --> 00:12:37.670 align:middle line:84% of spacetime transformation on the spacetime coordinates. 00:12:37.670 --> 00:12:48.870 align:middle line:84% If you take x mu to x mu prime, you go to lambda mu nu x nu. 00:12:48.870 --> 00:12:57.170 align:middle line:84% So lambda mu nu is the constant Lorentz transformation matrix. 00:12:57.170 --> 00:12:57.670 align:middle line:90% OK? 00:12:57.670 --> 00:13:01.440 align:middle line:90% 00:13:01.440 --> 00:13:01.940 align:middle line:90% OK? 00:13:01.940 --> 00:13:03.940 align:middle line:84% It's the constant Lorentz transformation matrix. 00:13:03.940 --> 00:13:07.390 align:middle line:84% And again, under such a coordinate transformation, 00:13:07.390 --> 00:13:08.140 align:middle line:90% phi-- 00:13:08.140 --> 00:13:10.200 align:middle line:90% will transform this way. 00:13:10.200 --> 00:13:11.890 align:middle line:90% Phi should transform this way. 00:13:11.890 --> 00:13:15.380 align:middle line:84% And now you can check it yourself, 00:13:15.380 --> 00:13:18.830 align:middle line:84% OK-- and because of this contraction 00:13:18.830 --> 00:13:20.270 align:middle line:90% of partial mu, partial mu. 00:13:20.270 --> 00:13:22.350 align:middle line:84% And then this is actually a symmetry. 00:13:22.350 --> 00:13:22.850 align:middle line:90% OK? 00:13:22.850 --> 00:13:25.980 align:middle line:84% Again, this is left as an exercise for yourself. 00:13:25.980 --> 00:13:28.425 align:middle line:84% And in fact, I think in your Pset problem, 00:13:28.425 --> 00:13:29.675 align:middle line:90% you will do something similar. 00:13:29.675 --> 00:13:33.520 align:middle line:90% 00:13:33.520 --> 00:13:34.555 align:middle line:90% Any questions on this? 00:13:34.555 --> 00:13:39.555 align:middle line:90% 00:13:39.555 --> 00:13:40.055 align:middle line:90% Good? 00:13:40.055 --> 00:13:43.140 align:middle line:90% 00:13:43.140 --> 00:13:48.780 align:middle line:84% So examples like this kind of transformation for translation 00:13:48.780 --> 00:13:50.590 align:middle line:84% and then for Lorentz transformation-- 00:13:50.590 --> 00:13:56.040 align:middle line:84% so this is what we call the continuous symmetries. 00:13:56.040 --> 00:13:59.210 align:middle line:90% So continuous symmetries-- 00:13:59.210 --> 00:14:12.460 align:middle line:90% 00:14:12.460 --> 00:14:12.960 align:middle line:90% OK? 00:14:12.960 --> 00:14:16.440 align:middle line:84% So these are the transformations-- 00:14:16.440 --> 00:14:19.360 align:middle line:84% yeah, I should also give a label to this equation. 00:14:19.360 --> 00:14:22.050 align:middle line:84% So let me call it just star-star-star. 00:14:22.050 --> 00:14:27.060 align:middle line:84% So-- [LIGHT CHUCKLE]---- continuous symmetries are 00:14:27.060 --> 00:14:35.850 align:middle line:84% transformations, like star-star-star, 00:14:35.850 --> 00:14:46.620 align:middle line:90% involving continuous parameters. 00:14:46.620 --> 00:14:51.470 align:middle line:90% 00:14:51.470 --> 00:14:53.690 align:middle line:90% OK? 00:14:53.690 --> 00:15:09.350 align:middle line:84% And so both one and the two are clearly continuous symmetries. 00:15:09.350 --> 00:15:13.800 align:middle line:90% 00:15:13.800 --> 00:15:17.130 align:middle line:84% Yeah, they are continuous symmetries. 00:15:17.130 --> 00:15:20.680 align:middle line:90% 00:15:20.680 --> 00:15:22.740 align:middle line:84% So you can easily see from here-- 00:15:22.740 --> 00:15:25.905 align:middle line:84% so here, the transformation is a. 00:15:25.905 --> 00:15:27.080 align:middle line:90% And a is four numbers. 00:15:27.080 --> 00:15:27.580 align:middle line:90% OK? 00:15:27.580 --> 00:15:28.170 align:middle line:90% It's a vector. 00:15:28.170 --> 00:15:29.440 align:middle line:90% It's a four vector. 00:15:29.440 --> 00:15:31.680 align:middle line:84% And a can be arbitrarily changed. 00:15:31.680 --> 00:15:33.040 align:middle line:90% So it's a continuous parameter. 00:15:33.040 --> 00:15:33.540 align:middle line:90% OK? 00:15:33.540 --> 00:15:35.160 align:middle line:90% You can take it to be 0. 00:15:35.160 --> 00:15:38.860 align:middle line:84% You can take it to be 0.1 in all directions, et cetera. 00:15:38.860 --> 00:15:41.500 align:middle line:84% So it's something that can be continuously changed. 00:15:41.500 --> 00:15:46.420 align:middle line:84% And similarly, the Lorentz transformation 00:15:46.420 --> 00:15:48.550 align:middle line:90% contains continuous parameters. 00:15:48.550 --> 00:15:50.860 align:middle line:84% So do you remember how many continuous parameters 00:15:50.860 --> 00:15:55.120 align:middle line:84% does the Lorentz transformation contain in four dimensions? 00:15:55.120 --> 00:15:55.900 align:middle line:90% Yes? 00:15:55.900 --> 00:15:56.777 align:middle line:90% AUDIENCE: 16. 00:15:56.777 --> 00:15:57.360 align:middle line:90% HONG LIU: Hmm? 00:15:57.360 --> 00:15:59.598 align:middle line:90% AUDIENCE: 16. 00:15:59.598 --> 00:16:00.140 align:middle line:90% HONG LIU: No. 00:16:00.140 --> 00:16:02.350 align:middle line:90% [LIGHT CHUCKLE] Yes? 00:16:02.350 --> 00:16:03.140 align:middle line:90% AUDIENCE: Six. 00:16:03.140 --> 00:16:04.970 align:middle line:90% HONG LIU: Yeah, six. 00:16:04.970 --> 00:16:07.250 align:middle line:84% It's because you have three rotations. 00:16:07.250 --> 00:16:10.340 align:middle line:84% Because you have three spatial directions, 00:16:10.340 --> 00:16:11.960 align:middle line:90% you have three rotations. 00:16:11.960 --> 00:16:14.030 align:middle line:84% And then you also have three posts-- 00:16:14.030 --> 00:16:15.560 align:middle line:90% yeah, so altogether six. 00:16:15.560 --> 00:16:18.080 align:middle line:84% So here, there's six continuous parameters. 00:16:18.080 --> 00:16:23.940 align:middle line:90% 00:16:23.940 --> 00:16:26.140 align:middle line:90% And here, there's a 4. 00:16:26.140 --> 00:16:26.640 align:middle line:90% OK? 00:16:26.640 --> 00:16:27.540 align:middle line:90% So here is 4. 00:16:27.540 --> 00:16:29.490 align:middle line:84% So both are continuous symmetries. 00:16:29.490 --> 00:16:31.750 align:middle line:90% OK? 00:16:31.750 --> 00:16:36.920 align:middle line:84% And in contrast, this Lagrangian have-- 00:16:36.920 --> 00:16:40.890 align:middle line:84% this action has a lot of symmetry 00:16:40.890 --> 00:16:42.740 align:middle line:90% which is not continuous. 00:16:42.740 --> 00:16:44.540 align:middle line:84% Can you say what that symmetry is? 00:16:44.540 --> 00:16:47.840 align:middle line:90% 00:16:47.840 --> 00:16:49.310 align:middle line:90% Good. 00:16:49.310 --> 00:16:54.440 align:middle line:90% Priority is a good answer. 00:16:54.440 --> 00:16:55.430 align:middle line:90% Yeah. 00:16:55.430 --> 00:16:58.890 align:middle line:84% But there's still something slightly simpler. 00:16:58.890 --> 00:16:59.390 align:middle line:90% Yeah? 00:16:59.390 --> 00:17:09.509 align:middle line:90% 00:17:09.509 --> 00:17:14.520 align:middle line:84% So notice that here, it's all quadratic in phi. 00:17:14.520 --> 00:17:16.050 align:middle line:90% And what do you observe? 00:17:16.050 --> 00:17:21.589 align:middle line:90% 00:17:21.589 --> 00:17:24.500 align:middle line:90% So here, there is symmetry. 00:17:24.500 --> 00:17:29.120 align:middle line:90% A lot of the symmetry is phi. 00:17:29.120 --> 00:17:32.785 align:middle line:84% It goes to minus phi, OK, because it is quadratic in phi. 00:17:32.785 --> 00:17:34.410 align:middle line:84% And also, of course, there's a symmetry 00:17:34.410 --> 00:17:40.560 align:middle line:84% called a parity, which in the case of x goes to minus x. 00:17:40.560 --> 00:17:43.260 align:middle line:84% So the spatial direction x goes to minus x. 00:17:43.260 --> 00:17:45.340 align:middle line:84% And then phi, again, transforms as this. 00:17:45.340 --> 00:17:45.840 align:middle line:90% OK? 00:17:45.840 --> 00:17:48.990 align:middle line:84% And then you can also check-- this is also a symmetry. 00:17:48.990 --> 00:17:51.290 align:middle line:84% So in both of these cases, they don't 00:17:51.290 --> 00:17:53.670 align:middle line:90% involve continuous parameters. 00:17:53.670 --> 00:17:56.730 align:middle line:84% And so these are called discrete symmetries. 00:17:56.730 --> 00:18:00.310 align:middle line:90% 00:18:00.310 --> 00:18:00.810 align:middle line:90% OK? 00:18:00.810 --> 00:18:03.120 align:middle line:84% So there's no continuous parameters. 00:18:03.120 --> 00:18:05.940 align:middle line:90% And yeah. 00:18:05.940 --> 00:18:09.930 align:middle line:84% In this case, both of them are called Z2. 00:18:09.930 --> 00:18:13.200 align:middle line:84% And this Z2 means that if you do the transformation once, 00:18:13.200 --> 00:18:14.930 align:middle line:90% it goes back to the identity. 00:18:14.930 --> 00:18:15.430 align:middle line:90% OK? 00:18:15.430 --> 00:18:18.087 align:middle line:90% So here, if you do the twice-- 00:18:18.087 --> 00:18:19.920 align:middle line:84% yeah, if you do it twice, it goes back to -- 00:18:19.920 --> 00:18:21.450 align:middle line:84% if you do it twice, it goes back to identity. 00:18:21.450 --> 00:18:22.325 align:middle line:90% And this is the same. 00:18:22.325 --> 00:18:23.620 align:middle line:90% OK? 00:18:23.620 --> 00:18:26.550 align:middle line:90% So here is a Z2 transformation. 00:18:26.550 --> 00:18:29.160 align:middle line:84% Also, symmetries can be separated 00:18:29.160 --> 00:18:37.370 align:middle line:84% into continuous symmetry or discrete symmetries depending 00:18:37.370 --> 00:18:40.730 align:middle line:84% on whether you have continuous parameters or not. 00:18:40.730 --> 00:18:42.770 align:middle line:84% But symmetries can also be separated 00:18:42.770 --> 00:18:46.870 align:middle line:84% into other different-- there's another classification 00:18:46.870 --> 00:18:48.250 align:middle line:90% of the symmetry. 00:18:48.250 --> 00:18:53.760 align:middle line:84% It's called the gauge symmetry and the global symmetries. 00:18:53.760 --> 00:18:55.460 align:middle line:90% So there are global symmetries-- 00:18:55.460 --> 00:19:01.760 align:middle line:90% 00:19:01.760 --> 00:19:12.830 align:middle line:84% corresponding to the transformation parameters 00:19:12.830 --> 00:19:14.210 align:middle line:90% are spacetime independent. 00:19:14.210 --> 00:19:25.140 align:middle line:90% 00:19:25.140 --> 00:19:25.650 align:middle line:90% OK? 00:19:25.650 --> 00:19:30.810 align:middle line:84% So both of these are examples of global symmetries 00:19:30.810 --> 00:19:32.340 align:middle line:90% because they are just constant. 00:19:32.340 --> 00:19:32.840 align:middle line:90% OK? 00:19:32.840 --> 00:19:35.100 align:middle line:84% They don't-- a mu don't depend on spacetime. 00:19:35.100 --> 00:19:37.890 align:middle line:84% And the lambda don't depend on spacetime. 00:19:37.890 --> 00:19:38.700 align:middle line:90% OK. 00:19:38.700 --> 00:19:49.470 align:middle line:84% And you can also have local symmetries, in which case 00:19:49.470 --> 00:20:03.570 align:middle line:84% the transformation parameters are spacetime dependent. 00:20:03.570 --> 00:20:04.070 align:middle line:90% OK? 00:20:04.070 --> 00:20:13.280 align:middle line:90% 00:20:13.280 --> 00:20:17.320 align:middle line:84% So an example of the local transformation which you may 00:20:17.320 --> 00:20:24.130 align:middle line:84% remember is the so-called gauge transformation in E&M. So E&M, 00:20:24.130 --> 00:20:25.780 align:middle line:90% you can-- 00:20:25.780 --> 00:20:27.190 align:middle line:90% yeah, anyway. 00:20:27.190 --> 00:20:28.910 align:middle line:84% So later, we will see an example. 00:20:28.910 --> 00:20:30.610 align:middle line:90% We will go back to E&M again. 00:20:30.610 --> 00:20:33.340 align:middle line:84% And you will see examples of this local symmetry. 00:20:33.340 --> 00:20:37.710 align:middle line:90% 00:20:37.710 --> 00:20:40.150 align:middle line:90% Any questions? 00:20:40.150 --> 00:20:40.953 align:middle line:90% Yes? 00:20:40.953 --> 00:20:42.370 align:middle line:84% AUDIENCE: So if the transformation 00:20:42.370 --> 00:20:44.980 align:middle line:84% changes the action but doesn't change the equations of motion, 00:20:44.980 --> 00:20:46.815 align:middle line:90% is that still a symmetry? 00:20:46.815 --> 00:20:47.440 align:middle line:90% HONG LIU: Yeah. 00:20:47.440 --> 00:20:50.440 align:middle line:84% Yeah, phi deals with the action-- 00:20:50.440 --> 00:20:51.950 align:middle line:84% yeah, that's a very good question. 00:20:51.950 --> 00:20:58.270 align:middle line:84% So by definition, if it leave the action invariant, 00:20:58.270 --> 00:21:00.010 align:middle line:90% we say it's a symmetry. 00:21:00.010 --> 00:21:02.840 align:middle line:84% And in general-- actually, almost always, 00:21:02.840 --> 00:21:04.385 align:middle line:84% if it leaves the action invariant, 00:21:04.385 --> 00:21:06.760 align:middle line:84% actually it also leaves the equation of motion invariant. 00:21:06.760 --> 00:21:07.260 align:middle line:90% Yeah. 00:21:07.260 --> 00:21:08.720 align:middle line:90% Yeah, but it's not guaranteed. 00:21:08.720 --> 00:21:09.220 align:middle line:90% Yeah. 00:21:09.220 --> 00:21:11.260 align:middle line:84% Just purely from a mathematical point of view, 00:21:11.260 --> 00:21:12.460 align:middle line:90% it's not guaranteed. 00:21:12.460 --> 00:21:18.370 align:middle line:84% But for all physical examples, it's almost always the same. 00:21:18.370 --> 00:21:19.360 align:middle line:90% Other questions? 00:21:19.360 --> 00:21:23.840 align:middle line:90% 00:21:23.840 --> 00:21:24.340 align:middle line:90% OK. 00:21:24.340 --> 00:21:26.260 align:middle line:90% Good. 00:21:26.260 --> 00:21:32.820 align:middle line:90% And now there's a very simple-- 00:21:32.820 --> 00:21:37.800 align:middle line:84% it's simple but a very deep connection between the symmetry 00:21:37.800 --> 00:21:39.750 align:middle line:90% and the conservation laws. 00:21:39.750 --> 00:21:45.020 align:middle line:84% So that you may have already learned it in-- 00:21:45.020 --> 00:21:48.330 align:middle line:84% yeah, actually I forgot when I learned it myself. 00:21:48.330 --> 00:21:51.580 align:middle line:84% So some of may have learned it in high school. 00:21:51.580 --> 00:21:54.066 align:middle line:84% Some of you may have learned it in 8.01, et cetera. 00:21:54.066 --> 00:21:58.050 align:middle line:84% Anyway, so there's a connection between the symmetry 00:21:58.050 --> 00:22:00.330 align:middle line:90% and the conservation laws. 00:22:00.330 --> 00:22:03.780 align:middle line:84% And so any conservation laws can be understood 00:22:03.780 --> 00:22:05.950 align:middle line:90% as a consequence of symmetry. 00:22:05.950 --> 00:22:09.300 align:middle line:84% So in classical mechanics, you may remember 00:22:09.300 --> 00:22:10.980 align:middle line:90% that the time translation-- 00:22:10.980 --> 00:22:17.920 align:middle line:90% 00:22:17.920 --> 00:22:19.930 align:middle line:84% this leads to energy conservation. 00:22:19.930 --> 00:22:26.390 align:middle line:90% 00:22:26.390 --> 00:22:32.640 align:middle line:84% And the spatial translation symmetry 00:22:32.640 --> 00:22:35.095 align:middle line:84% then leads to the momentum conservation. 00:22:35.095 --> 00:22:40.320 align:middle line:90% 00:22:40.320 --> 00:22:48.030 align:middle line:84% And if it's a rotational symmetry, 00:22:48.030 --> 00:22:50.589 align:middle line:84% it then leads to the angular momentum conservation. 00:22:50.589 --> 00:22:59.880 align:middle line:90% 00:22:59.880 --> 00:23:00.900 align:middle line:90% OK? 00:23:00.900 --> 00:23:04.350 align:middle line:84% So this should be something you already know, say, 00:23:04.350 --> 00:23:08.400 align:middle line:90% from your classical mechanics. 00:23:08.400 --> 00:23:11.740 align:middle line:84% But in classical field theory, this can be generalized. 00:23:11.740 --> 00:23:12.270 align:middle line:90% OK? 00:23:12.270 --> 00:23:18.020 align:middle line:84% So here, there's the Noether theorem-- 00:23:18.020 --> 00:23:23.075 align:middle line:84% so first discovered by Emmy Noether, the Noether theorem. 00:23:23.075 --> 00:23:26.170 align:middle line:90% 00:23:26.170 --> 00:23:41.400 align:middle line:84% She said any continuous global symmetry 00:23:41.400 --> 00:23:42.898 align:middle line:90% leads to conservation laws. 00:23:42.898 --> 00:23:43.398 align:middle line:90% OK? 00:23:43.398 --> 00:23:49.720 align:middle line:90% 00:23:49.720 --> 00:23:50.220 align:middle line:90% OK. 00:23:50.220 --> 00:23:53.530 align:middle line:90% It leads to a conservation. 00:23:53.530 --> 00:23:57.370 align:middle line:84% And so no matter what kind of symmetry, OK-- 00:23:57.370 --> 00:23:59.980 align:middle line:84% in addition to those things we are familiar with, 00:23:59.980 --> 00:24:02.500 align:middle line:84% but no matter what kind of symmetry-- but any time 00:24:02.500 --> 00:24:07.480 align:middle line:84% you have a continuous global symmetry in your system, then 00:24:07.480 --> 00:24:08.560 align:middle line:90% you have conserved. 00:24:08.560 --> 00:24:11.030 align:middle line:90% You have conservation. 00:24:11.030 --> 00:24:11.530 align:middle line:90% OK. 00:24:11.530 --> 00:24:17.710 align:middle line:84% So now let me give a proof of this Noether theorem. 00:24:17.710 --> 00:24:18.210 align:middle line:90% OK? 00:24:18.210 --> 00:24:20.168 align:middle line:84% So before I prove it, do you have any questions 00:24:20.168 --> 00:24:21.300 align:middle line:90% regarding its statement? 00:24:21.300 --> 00:24:25.050 align:middle line:90% 00:24:25.050 --> 00:24:26.166 align:middle line:90% Yes? 00:24:26.166 --> 00:24:29.910 align:middle line:84% AUDIENCE: So what's the definition of-- 00:24:29.910 --> 00:24:31.400 align:middle line:84% so you said a continuous symmetry 00:24:31.400 --> 00:24:33.488 align:middle line:84% is one that depends continuously on a parameter. 00:24:33.488 --> 00:24:35.030 align:middle line:84% But if you wanted to write out, like, 00:24:35.030 --> 00:24:38.300 align:middle line:84% equation star-star-star in a way that specializes to continuous 00:24:38.300 --> 00:24:40.630 align:middle line:84% symmetry, is there a way to do that-- 00:24:40.630 --> 00:24:45.118 align:middle line:84% to show just how to check if something is a continuous-- 00:24:45.118 --> 00:24:46.160 align:middle line:90% HONG LIU: Oh, yeah, yeah. 00:24:46.160 --> 00:24:50.295 align:middle line:84% You just check whether there's a continuous parameter. 00:24:50.295 --> 00:24:52.420 align:middle line:84% You just check whether there is a parameter or not. 00:24:52.420 --> 00:24:54.860 align:middle line:90% 00:24:54.860 --> 00:24:55.360 align:middle line:90% Yeah. 00:24:55.360 --> 00:25:00.590 align:middle line:90% 00:25:00.590 --> 00:25:03.730 align:middle line:90% Does this answer your question? 00:25:03.730 --> 00:25:04.408 align:middle line:90% Yeah. 00:25:04.408 --> 00:25:06.700 align:middle line:84% Normally, you can just see whether there's a parameter. 00:25:06.700 --> 00:25:06.970 align:middle line:90% Yeah. 00:25:06.970 --> 00:25:07.720 align:middle line:90% [LIGHT CHUCKLE] 00:25:07.720 --> 00:25:09.940 align:middle line:84% If you write down the transformation explicitly, 00:25:09.940 --> 00:25:11.170 align:middle line:90% you will be able to just see. 00:25:11.170 --> 00:25:13.810 align:middle line:84% If we go over here, there's no transformation parameter. 00:25:13.810 --> 00:25:14.950 align:middle line:90% You can just see. 00:25:14.950 --> 00:25:17.980 align:middle line:84% And then there, for the Lorentz transformation and translation, 00:25:17.980 --> 00:25:21.570 align:middle line:90% you can just see it explicitly. 00:25:21.570 --> 00:25:23.550 align:middle line:90% OK. 00:25:23.550 --> 00:25:31.620 align:middle line:84% So an important aspect of a continuous symmetry 00:25:31.620 --> 00:25:34.790 align:middle line:84% is that, because you have a continuous parameter-- 00:25:34.790 --> 00:25:38.060 align:middle line:84% so you can continuously relate it to identity. 00:25:38.060 --> 00:25:41.810 align:middle line:84% So trivial transformation-- you just 00:25:41.810 --> 00:25:43.100 align:middle line:90% don't transform at all, right? 00:25:43.100 --> 00:25:46.450 align:middle line:84% So in this case, the a mu is equal to 0. 00:25:46.450 --> 00:25:48.700 align:middle line:84% And so in this case, the trivial transformation-- just 00:25:48.700 --> 00:25:51.500 align:middle line:84% the lambda mu nu-- is the identity matrix. 00:25:51.500 --> 00:25:53.430 align:middle line:84% And then you can imagine a slightly-- 00:25:53.430 --> 00:25:56.146 align:middle line:90% 00:25:56.146 --> 00:25:59.620 align:middle line:84% a thinning on the infinitesimal rotation 00:25:59.620 --> 00:26:01.270 align:middle line:90% or infinitesimal boost. 00:26:01.270 --> 00:26:04.210 align:middle line:84% And then that's corresponding to a so-called infinitesimal 00:26:04.210 --> 00:26:05.020 align:middle line:90% transformation. 00:26:05.020 --> 00:26:05.540 align:middle line:90% OK? 00:26:05.540 --> 00:26:08.230 align:middle line:84% So here, you can also just translate a mu a little bit, 00:26:08.230 --> 00:26:10.990 align:middle line:84% OK-- so very close to the identity. 00:26:10.990 --> 00:26:13.195 align:middle line:90% And so essentially, for any-- 00:26:13.195 --> 00:26:16.710 align:middle line:90% 00:26:16.710 --> 00:26:17.210 align:middle line:90% OK? 00:26:17.210 --> 00:26:32.000 align:middle line:84% So for any global continuous symmetry, 00:26:32.000 --> 00:26:33.575 align:middle line:90% it has an infinitesimal form. 00:26:33.575 --> 00:26:46.590 align:middle line:90% 00:26:46.590 --> 00:26:47.090 align:middle line:90% OK? 00:26:47.090 --> 00:26:50.510 align:middle line:84% It's just when your transformation parameter 00:26:50.510 --> 00:26:55.190 align:middle line:84% is very close to the identity, OK, very close to the 0. 00:26:55.190 --> 00:26:58.490 align:middle line:84% And so for example, for a general transformation 00:26:58.490 --> 00:27:02.820 align:middle line:84% like this, if the transformation is close to the identity, 00:27:02.820 --> 00:27:06.180 align:middle line:84% then we can always write it in the following form-- 00:27:06.180 --> 00:27:14.420 align:middle line:84% phi a goes to phi a prime, which is equal to phi a, and with 00:27:14.420 --> 00:27:18.330 align:middle line:84% some parameter-- small parameter epsilon-- 00:27:18.330 --> 00:27:20.100 align:middle line:90% and some function-- f_a-- 00:27:20.100 --> 00:27:23.330 align:middle line:84% and then, say, phi b, OK, and the derivative of phi 00:27:23.330 --> 00:27:25.544 align:middle line:90% b, et cetera. 00:27:25.544 --> 00:27:26.044 align:middle line:90% OK? 00:27:26.044 --> 00:27:31.540 align:middle line:84% So this epsilon is the infinitesimal transformation 00:27:31.540 --> 00:27:32.230 align:middle line:90% parameter. 00:27:32.230 --> 00:27:38.710 align:middle line:90% 00:27:38.710 --> 00:27:39.210 align:middle line:90% OK? 00:27:39.210 --> 00:27:43.830 align:middle line:84% So if you have a continuous symmetry, 00:27:43.830 --> 00:27:47.670 align:middle line:84% there's always, say, an infinitesimal transformation 00:27:47.670 --> 00:27:50.510 align:middle line:90% which are close to the identity. 00:27:50.510 --> 00:27:55.510 align:middle line:84% So this epsilon corresponding to this case for a mu 00:27:55.510 --> 00:27:57.530 align:middle line:90% is very small. 00:27:57.530 --> 00:27:59.827 align:middle line:84% Or for the Lorentz transformation case 00:27:59.827 --> 00:28:01.910 align:middle line:84% corresponding to rotation angle, it's very small-- 00:28:01.910 --> 00:28:03.710 align:middle line:90% or the boost is very small. 00:28:03.710 --> 00:28:05.060 align:middle line:90% OK? 00:28:05.060 --> 00:28:05.560 align:middle line:90% Yeah. 00:28:05.560 --> 00:28:07.690 align:middle line:84% So the epsilon can be any of those parameters. 00:28:07.690 --> 00:28:10.240 align:middle line:90% 00:28:10.240 --> 00:28:12.820 align:middle line:84% And then this f, it can be some arbitrary function. 00:28:12.820 --> 00:28:13.860 align:middle line:90% OK? 00:28:13.860 --> 00:28:16.850 align:middle line:84% So this f can be some arbitrary functions of your fields. 00:28:16.850 --> 00:28:17.350 align:middle line:90% OK. 00:28:17.350 --> 00:28:23.090 align:middle line:90% 00:28:23.090 --> 00:28:23.750 align:middle line:90% Good? 00:28:23.750 --> 00:28:30.290 align:middle line:84% So in this case, yeah, the data phi, OK-- 00:28:30.290 --> 00:28:32.490 align:middle line:84% yeah, so the symmetry transformation-- 00:28:32.490 --> 00:28:36.650 align:middle line:84% so we'll use the rotation delta-hat phi a. 00:28:36.650 --> 00:28:41.435 align:middle line:84% So this symmetry was then given by the epsilon f a OK? 00:28:41.435 --> 00:28:44.190 align:middle line:90% 00:28:44.190 --> 00:28:49.710 align:middle line:84% So delta-hat here implies an infinitesimal symmetry 00:28:49.710 --> 00:28:51.660 align:middle line:90% transformation. 00:28:51.660 --> 00:28:52.160 align:middle line:90% OK. 00:28:52.160 --> 00:28:56.710 align:middle line:90% 00:28:56.710 --> 00:29:03.250 align:middle line:84% So now let's consider during a variation of S 00:29:03.250 --> 00:29:12.090 align:middle line:84% under this delta-hat, OK, since by definition of the symmetry-- 00:29:12.090 --> 00:29:18.330 align:middle line:84% because this is a symmetry, that means that delta-hat S should 00:29:18.330 --> 00:29:21.120 align:middle line:90% be equal to 0. 00:29:21.120 --> 00:29:22.410 align:middle line:90% OK? 00:29:22.410 --> 00:29:24.240 align:middle line:84% Because of the variation of the action, 00:29:24.240 --> 00:29:29.900 align:middle line:84% it should be invariant under the symmetry. 00:29:29.900 --> 00:29:33.790 align:middle line:84% And so now let's look at what's the consequence of it. 00:29:33.790 --> 00:29:38.810 align:middle line:84% So now we can just vary this L. Then the-- 00:29:38.810 --> 00:29:43.805 align:middle line:84% so this means that the variation of the L, 00:29:43.805 --> 00:29:51.510 align:middle line:84% the Lagrangian density, must be a total derivative, OK, 00:29:51.510 --> 00:29:54.570 align:middle line:84% since every transformation is proportional to-- 00:29:54.570 --> 00:29:55.680 align:middle line:90% yeah. 00:29:55.680 --> 00:29:57.815 align:middle line:90% So epsilon is a small parameter. 00:29:57.815 --> 00:29:59.190 align:middle line:84% So when we do the transformation, 00:29:59.190 --> 00:30:04.850 align:middle line:84% we only need to keep track in the order in epsilon. 00:30:04.850 --> 00:30:07.170 align:middle line:84% And so the transformation of the L 00:30:07.170 --> 00:30:08.880 align:middle line:84% must also be proportional to epsilon. 00:30:08.880 --> 00:30:10.560 align:middle line:90% OK? 00:30:10.560 --> 00:30:13.590 align:middle line:84% So since this must be invariant, that 00:30:13.590 --> 00:30:17.380 align:middle line:84% means the variation of the L must be a total derivative. 00:30:17.380 --> 00:30:19.660 align:middle line:90% So k mu would be just some-- 00:30:19.660 --> 00:30:21.910 align:middle line:90% so for some k mu. 00:30:21.910 --> 00:30:22.990 align:middle line:90% OK? 00:30:22.990 --> 00:30:23.740 align:middle line:90% For some k mu. 00:30:23.740 --> 00:30:25.600 align:middle line:90% So there must exist some k mu. 00:30:25.600 --> 00:30:28.660 align:middle line:84% And then this k mu can also be 0. 00:30:28.660 --> 00:30:30.940 align:middle line:84% In that case, the Lagrangian density is invariant. 00:30:30.940 --> 00:30:31.850 align:middle line:90% Yes? 00:30:31.850 --> 00:30:33.350 align:middle line:84% AUDIENCE: Is delta-hat the variation 00:30:33.350 --> 00:30:34.790 align:middle line:90% under your transformation? 00:30:34.790 --> 00:30:35.540 align:middle line:90% HONG LIU: Sorry? 00:30:35.540 --> 00:30:37.220 align:middle line:84% AUDIENCE: Is delta-hat the variation 00:30:37.220 --> 00:30:40.250 align:middle line:90% under the transformation? 00:30:40.250 --> 00:30:44.520 align:middle line:84% HONG LIU: Delta-hat just denotes such a transformation. 00:30:44.520 --> 00:30:46.910 align:middle line:84% It just denotes such a transformation. 00:30:46.910 --> 00:30:50.250 align:middle line:84% This is not the general variation. 00:30:50.250 --> 00:30:51.380 align:middle line:90% This is very specific. 00:30:51.380 --> 00:30:54.820 align:middle line:84% This is a symmetric transformation. 00:30:54.820 --> 00:30:57.240 align:middle line:84% And so under the symmetric transformation, 00:30:57.240 --> 00:31:00.240 align:middle line:84% your action should be invariant by definition 00:31:00.240 --> 00:31:01.650 align:middle line:90% because this is a symmetry. 00:31:01.650 --> 00:31:03.180 align:middle line:90% And that in turn-- 00:31:03.180 --> 00:31:06.390 align:middle line:84% because the S is the integration of L, 00:31:06.390 --> 00:31:08.160 align:middle line:84% then that means that the transformation 00:31:08.160 --> 00:31:12.510 align:middle line:84% of L, the Lagrangian density, must be a total derivative. 00:31:12.510 --> 00:31:15.480 align:middle line:84% And here, we keep track of everything 00:31:15.480 --> 00:31:16.920 align:middle line:90% in the order in epsilon. 00:31:16.920 --> 00:31:19.395 align:middle line:84% And so this must be proportional to epsilon 00:31:19.395 --> 00:31:20.520 align:middle line:90% times the total derivative. 00:31:20.520 --> 00:31:21.020 align:middle line:90% OK? 00:31:21.020 --> 00:31:27.450 align:middle line:84% And so we must have this structure, yeah, for some k mu, 00:31:27.450 --> 00:31:38.010 align:middle line:84% OK, which can be 0, which can be 0. 00:31:38.010 --> 00:31:39.510 align:middle line:90% OK? 00:31:39.510 --> 00:31:40.320 align:middle line:90% Good? 00:31:40.320 --> 00:31:43.920 align:middle line:84% So now let's look at what's the implications of this equation. 00:31:43.920 --> 00:31:44.420 align:middle line:90% OK. 00:31:44.420 --> 00:31:47.030 align:middle line:84% So this equation is the constraint 00:31:47.030 --> 00:31:49.860 align:middle line:90% imposed by the symmetry. 00:31:49.860 --> 00:31:50.360 align:middle line:90% OK? 00:31:50.360 --> 00:31:52.250 align:middle line:84% Now let's look at the implications 00:31:52.250 --> 00:31:53.360 align:middle line:90% of this constraint. 00:31:53.360 --> 00:31:56.270 align:middle line:90% 00:31:56.270 --> 00:32:01.080 align:middle line:84% So now we can just do the variation of L. 00:32:01.080 --> 00:32:08.440 align:middle line:84% Now we can just do the variation of L. Yeah. 00:32:08.440 --> 00:32:10.405 align:middle line:90% Maybe I will keep that equation. 00:32:10.405 --> 00:32:11.905 align:middle line:84% And then I will do this board first. 00:32:11.905 --> 00:32:25.140 align:middle line:90% 00:32:25.140 --> 00:32:26.432 align:middle line:90% AUDIENCE: So I have a question. 00:32:26.432 --> 00:32:26.940 align:middle line:90% Does this problem-- 00:32:26.940 --> 00:32:27.810 align:middle line:90% HONG LIU: Yeah? 00:32:27.810 --> 00:32:32.970 align:middle line:84% AUDIENCE: --show if something is a symmetry of your action 00:32:32.970 --> 00:32:38.250 align:middle line:84% for your system, would you directly vary the thing-- 00:32:38.250 --> 00:32:40.630 align:middle line:84% like, parameter and then show that your action doesn't 00:32:40.630 --> 00:32:41.130 align:middle line:90% change? 00:32:41.130 --> 00:32:46.140 align:middle line:84% Or would you do that infinitesimal form 00:32:46.140 --> 00:32:48.488 align:middle line:90% and then show that-- 00:32:48.488 --> 00:32:49.530 align:middle line:90% HONG LIU: So we want to-- 00:32:49.530 --> 00:32:49.785 align:middle line:90% AUDIENCE: Yeah, I don't know. 00:32:49.785 --> 00:32:50.040 align:middle line:90% I was just-- 00:32:50.040 --> 00:32:50.820 align:middle line:90% HONG LIU: Right. 00:32:50.820 --> 00:32:57.890 align:middle line:84% So we want to use the fact that such a transformation is 00:32:57.890 --> 00:33:05.980 align:middle line:84% a symmetry to derive what is the constraint on the Lagrangian, 00:33:05.980 --> 00:33:08.590 align:middle line:84% OK, what is the constraint on the theory. 00:33:08.590 --> 00:33:12.550 align:middle line:84% Say, we assume that somehow the theory has a symmetry. 00:33:12.550 --> 00:33:15.940 align:middle line:84% And then we want to see what is the constraint this 00:33:15.940 --> 00:33:19.560 align:middle line:90% put on your theory. 00:33:19.560 --> 00:33:20.400 align:middle line:90% Yeah, that's right. 00:33:20.400 --> 00:33:21.470 align:middle line:84% We are proving the laws of the theorem. 00:33:21.470 --> 00:33:21.970 align:middle line:90% Yeah. 00:33:21.970 --> 00:33:25.330 align:middle line:90% 00:33:25.330 --> 00:33:26.830 align:middle line:90% Good? 00:33:26.830 --> 00:33:27.850 align:middle line:90% Other questions? 00:33:27.850 --> 00:33:33.410 align:middle line:90% 00:33:33.410 --> 00:33:33.910 align:middle line:90% OK. 00:33:33.910 --> 00:33:41.230 align:middle line:84% So that equation is the implication-- 00:33:41.230 --> 00:33:43.810 align:middle line:84% is the constraint imposed by the symmetry. 00:33:43.810 --> 00:33:46.030 align:middle line:84% And we want to see what this equation tells us. 00:33:46.030 --> 00:33:47.530 align:middle line:90% OK? 00:33:47.530 --> 00:33:51.060 align:middle line:84% And for this purpose, we just do the variation. 00:33:51.060 --> 00:33:55.140 align:middle line:90% So the delta-hat L-- 00:33:55.140 --> 00:33:58.680 align:middle line:84% so because L is just a function of phi-- 00:33:58.680 --> 00:34:05.916 align:middle line:84% and then we have partial L, partial phi a, delta-hat phi a; 00:34:05.916 --> 00:34:10.710 align:middle line:84% and then we have L, partial-partial mu 00:34:10.710 --> 00:34:17.060 align:middle line:84% phi a; and then partial mu delta-hat phi a. 00:34:17.060 --> 00:34:18.699 align:middle line:90% OK? 00:34:18.699 --> 00:34:22.710 align:middle line:90% So we just did the variation. 00:34:22.710 --> 00:34:25.020 align:middle line:90% And now-- OK? 00:34:25.020 --> 00:34:27.375 align:middle line:90% And so-- yeah. 00:34:27.375 --> 00:34:29.500 align:middle line:84% And now we are going to use the equation of motion. 00:34:29.500 --> 00:34:30.000 align:middle line:90% OK? 00:34:30.000 --> 00:34:33.150 align:middle line:90% So from the equation of motion-- 00:34:33.150 --> 00:34:37.070 align:middle line:84% I think I erased-- yeah, from the equation of motion, then 00:34:37.070 --> 00:34:39.230 align:middle line:84% partial L, partial phi a-- this is 00:34:39.230 --> 00:34:46.460 align:middle line:84% just equal to partial mu, partial L, partial-partial mu 00:34:46.460 --> 00:34:47.870 align:middle line:90% phi a. 00:34:47.870 --> 00:34:48.650 align:middle line:90% OK? 00:34:48.650 --> 00:34:51.719 align:middle line:90% So we just plug this into here. 00:34:51.719 --> 00:34:54.230 align:middle line:84% So when you plug this into here, now you 00:34:54.230 --> 00:34:57.950 align:middle line:84% notice this becomes a total derivative. 00:34:57.950 --> 00:35:00.460 align:middle line:90% 00:35:00.460 --> 00:35:00.960 align:middle line:90% OK? 00:35:00.960 --> 00:35:15.350 align:middle line:84% So you partial L, partial mu phi a, and delta-hat phi a. 00:35:15.350 --> 00:35:15.850 align:middle line:90% OK? 00:35:15.850 --> 00:35:19.540 align:middle line:84% So you find-- after using the equation of motion, 00:35:19.540 --> 00:35:22.450 align:middle line:84% this variation of L have the following form, which 00:35:22.450 --> 00:35:23.982 align:middle line:90% is the total derivative. 00:35:23.982 --> 00:35:26.700 align:middle line:90% 00:35:26.700 --> 00:35:28.980 align:middle line:84% And this should be equal to the right-hand side. 00:35:28.980 --> 00:35:31.620 align:middle line:90% OK? 00:35:31.620 --> 00:35:33.790 align:middle line:84% It should be equal to the right-hand side. 00:35:33.790 --> 00:35:34.290 align:middle line:90% Yeah. 00:35:34.290 --> 00:35:36.750 align:middle line:90% So yeah. 00:35:36.750 --> 00:35:38.520 align:middle line:90% And then here we find-- 00:35:38.520 --> 00:35:43.300 align:middle line:84% so this should be equal to epsilon partial mu k mu. 00:35:43.300 --> 00:35:43.800 align:middle line:90% OK? 00:35:43.800 --> 00:35:45.960 align:middle line:84% So we combine both sides together, 00:35:45.960 --> 00:35:50.625 align:middle line:84% plug in that this is equal to epsilon times fa. 00:35:50.625 --> 00:36:01.920 align:middle line:84% And then we conclude, OK, partial mu, partial L, 00:36:01.920 --> 00:36:12.740 align:middle line:84% partial-partial mu phi a, and fa minus k mu, we see equal to 0. 00:36:12.740 --> 00:36:14.140 align:middle line:90% OK? 00:36:14.140 --> 00:36:20.560 align:middle line:84% So if we call this J mu and then we 00:36:20.560 --> 00:36:22.260 align:middle line:90% have a conservation equation-- 00:36:22.260 --> 00:36:26.600 align:middle line:90% partial mu J mu equal to 0. 00:36:26.600 --> 00:36:27.100 align:middle line:90% OK? 00:36:27.100 --> 00:36:28.350 align:middle line:90% So we have a conservation law. 00:36:28.350 --> 00:36:31.250 align:middle line:90% 00:36:31.250 --> 00:36:33.520 align:middle line:84% So remember, conservation law is just 00:36:33.520 --> 00:36:40.852 align:middle line:84% to have a vector which satisfies this equation and then-- 00:36:40.852 --> 00:36:43.820 align:middle line:90% yeah. 00:36:43.820 --> 00:36:46.618 align:middle line:84% So more explicitly-- yeah, so the Jmu 00:36:46.618 --> 00:36:47.660 align:middle line:90% is just this combination. 00:36:47.660 --> 00:36:48.710 align:middle line:90% OK? 00:36:48.710 --> 00:36:50.280 align:middle line:90% Jmu is just this combination. 00:36:50.280 --> 00:36:52.400 align:middle line:90% So any questions on this? 00:36:52.400 --> 00:36:53.150 align:middle line:90% Yes? 00:36:53.150 --> 00:36:55.400 align:middle line:84% AUDIENCE: Is this partially with the shift in presence 00:36:55.400 --> 00:36:58.300 align:middle line:84% because it's the derivative with respect to spacetime here, 00:36:58.300 --> 00:37:00.890 align:middle line:84% but if we have a conservation law 00:37:00.890 --> 00:37:02.940 align:middle line:84% from the derivative with respect to time 00:37:02.940 --> 00:37:04.575 align:middle line:90% equal to 0 but not space? 00:37:04.575 --> 00:37:05.450 align:middle line:90% HONG LIU: Yeah, yeah. 00:37:05.450 --> 00:37:06.670 align:middle line:90% I will-- yeah. 00:37:06.670 --> 00:37:10.610 align:middle line:84% Yeah, this is the stronger version than your normally-- 00:37:10.610 --> 00:37:11.390 align:middle line:90% yeah. 00:37:11.390 --> 00:37:13.820 align:middle line:84% So this is the spacetime version of the conservation 00:37:13.820 --> 00:37:16.560 align:middle line:84% law from your normally classical mechanics. 00:37:16.560 --> 00:37:17.060 align:middle line:90% Yeah. 00:37:17.060 --> 00:37:19.190 align:middle line:84% So this is a field theory version of that. 00:37:19.190 --> 00:37:19.700 align:middle line:90% Yeah. 00:37:19.700 --> 00:37:23.360 align:middle line:84% Yeah, I will elaborate that equation a little bit. 00:37:23.360 --> 00:37:25.010 align:middle line:90% Yeah. 00:37:25.010 --> 00:37:25.790 align:middle line:90% Yes? 00:37:25.790 --> 00:37:32.570 align:middle line:84% AUDIENCE: So if you only transform the field, not the x 00:37:32.570 --> 00:37:33.830 align:middle line:90% and t for the-- 00:37:33.830 --> 00:37:35.090 align:middle line:90% HONG LIU: Yeah. 00:37:35.090 --> 00:37:37.160 align:middle line:84% AUDIENCE: --the effect is to eliminate 00:37:37.160 --> 00:37:40.940 align:middle line:84% that derivative of your operator for everything? 00:37:40.940 --> 00:37:42.920 align:middle line:90% Is that right? 00:37:42.920 --> 00:37:44.000 align:middle line:90% HONG LIU: Say it again? 00:37:44.000 --> 00:37:47.450 align:middle line:84% AUDIENCE: I mean, the reason you include 00:37:47.450 --> 00:37:50.180 align:middle line:84% the derivative of the field is because you 00:37:50.180 --> 00:37:53.250 align:middle line:84% are including the effect of transforming 00:37:53.250 --> 00:37:56.160 align:middle line:90% the spacetime coordinates. 00:37:56.160 --> 00:37:58.110 align:middle line:90% HONG LIU: So here-- 00:37:58.110 --> 00:37:58.860 align:middle line:90% yeah, yeah. 00:37:58.860 --> 00:38:02.107 align:middle line:84% So here is a general formulation. 00:38:02.107 --> 00:38:03.690 align:middle line:84% So this transformation can be general. 00:38:03.690 --> 00:38:07.590 align:middle line:84% It doesn't have to be, say, a spacetime transformation-- 00:38:07.590 --> 00:38:09.400 align:middle line:90% we said earlier. 00:38:09.400 --> 00:38:12.780 align:middle line:84% This is just some abstract symmetry, just 00:38:12.780 --> 00:38:16.438 align:middle line:90% some arbitrary transformations. 00:38:16.438 --> 00:38:17.920 align:middle line:90% AUDIENCE: Yeah. 00:38:17.920 --> 00:38:22.810 align:middle line:84% I mean, if you only go through the symmetries that have 00:38:22.810 --> 00:38:26.880 align:middle line:84% changed in the field but not the symmetries that haven't? 00:38:26.880 --> 00:38:28.630 align:middle line:84% HONG LIU: Yeah, because spacetime variable 00:38:28.630 --> 00:38:29.590 align:middle line:90% is a dummy variable. 00:38:29.590 --> 00:38:33.850 align:middle line:84% In the action, you can always get rid of that. 00:38:33.850 --> 00:38:38.780 align:middle line:84% The x change-- x is just a dummy variable in your action 00:38:38.780 --> 00:38:41.820 align:middle line:90% because you integrate over x. 00:38:41.820 --> 00:38:42.340 align:middle line:90% Yeah, yeah. 00:38:42.340 --> 00:38:45.100 align:middle line:90% 00:38:45.100 --> 00:38:49.050 align:middle line:84% AUDIENCE: So is it possible to interpret higher order 00:38:49.050 --> 00:38:53.925 align:middle line:84% derivatives not including that pathway? 00:38:53.925 --> 00:38:55.050 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:38:55.050 --> 00:38:56.820 align:middle line:90% This definitely include the-- 00:38:56.820 --> 00:39:00.580 align:middle line:84% normally-- yeah, you can in principle 00:39:00.580 --> 00:39:03.210 align:middle line:84% have as many derivatives as you want. 00:39:03.210 --> 00:39:06.720 align:middle line:84% But normally, if we have an action which only contains 00:39:06.720 --> 00:39:09.885 align:middle line:84% first derivatives, then your symmetry transformation 00:39:09.885 --> 00:39:12.950 align:middle line:84% will only involve first derivatives. 00:39:12.950 --> 00:39:16.320 align:middle line:84% AUDIENCE: So, though, we're just not discussing 00:39:16.320 --> 00:39:19.626 align:middle line:90% the case of transforming-- 00:39:19.626 --> 00:39:23.236 align:middle line:90% I don't know. 00:39:23.236 --> 00:39:23.895 align:middle line:90% We're-- 00:39:23.895 --> 00:39:25.020 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:39:25.020 --> 00:39:26.370 align:middle line:90% In principle, you can have it. 00:39:26.370 --> 00:39:28.050 align:middle line:90% In principle, you can have it. 00:39:28.050 --> 00:39:30.010 align:middle line:84% Yeah, I'm just writing this for simplicity. 00:39:30.010 --> 00:39:30.510 align:middle line:90% Yeah. 00:39:30.510 --> 00:39:35.500 align:middle line:90% 00:39:35.500 --> 00:39:36.000 align:middle line:90% Yeah, good. 00:39:36.000 --> 00:39:40.110 align:middle line:84% Maybe you can ask after the class, yeah, if it's not clear. 00:39:40.110 --> 00:39:42.530 align:middle line:90% Other questions? 00:39:42.530 --> 00:39:43.030 align:middle line:90% OK. 00:39:43.030 --> 00:39:43.990 align:middle line:90% Good. 00:39:43.990 --> 00:39:47.920 align:middle line:84% So let me elaborate a little bit on this equation. 00:39:47.920 --> 00:39:53.250 align:middle line:84% So this equation you may have seen in E&M, which you often 00:39:53.250 --> 00:39:56.190 align:middle line:90% call the continuity equation. 00:39:56.190 --> 00:39:59.160 align:middle line:84% If you write this equation separately, 00:39:59.160 --> 00:40:00.960 align:middle line:90% it has the following form-- 00:40:00.960 --> 00:40:08.560 align:middle line:84% partial 0 J0 plus partial I, or the divergence 00:40:08.560 --> 00:40:10.070 align:middle line:90% of the spatial component. 00:40:10.070 --> 00:40:16.050 align:middle line:84% So J mu-- if we write it in terms of the-- 00:40:16.050 --> 00:40:18.615 align:middle line:84% OK, explicitly, then you have this form. 00:40:18.615 --> 00:40:21.390 align:middle line:90% 00:40:21.390 --> 00:40:25.020 align:middle line:84% So this is so-called the continuity equation-- just 00:40:25.020 --> 00:40:29.650 align:middle line:84% the time variation of the density. 00:40:29.650 --> 00:40:30.150 align:middle line:90% OK? 00:40:30.150 --> 00:40:31.730 align:middle line:84% The 0 component that you consider 00:40:31.730 --> 00:40:33.240 align:middle line:90% as some kind of density-- 00:40:33.240 --> 00:40:35.640 align:middle line:84% it's the same as the divergence of the current. 00:40:35.640 --> 00:40:37.110 align:middle line:90% OK? 00:40:37.110 --> 00:40:38.850 align:middle line:84% And in particular, you can define 00:40:38.850 --> 00:40:45.390 align:middle line:84% a charge, which is the total spatial integration of J0. 00:40:45.390 --> 00:40:45.890 align:middle line:90% OK? 00:40:45.890 --> 00:40:51.740 align:middle line:84% So if you integrate both sides over the total volume-- 00:40:51.740 --> 00:40:56.140 align:middle line:84% and then this just becomes partial 0 Q. 00:40:56.140 --> 00:40:59.320 align:middle line:84% And this term becomes a total derivative. 00:40:59.320 --> 00:41:02.080 align:middle line:84% And then you can convert it into a boundary 00:41:02.080 --> 00:41:04.780 align:middle line:90% term using the Gauss's law. 00:41:04.780 --> 00:41:08.860 align:middle line:84% And then, normally, the current vanishes at infinity. 00:41:08.860 --> 00:41:14.020 align:middle line:84% And then you just have the charge conservation. 00:41:14.020 --> 00:41:15.922 align:middle line:90% You just have charge conversion. 00:41:15.922 --> 00:41:17.500 align:middle line:90% OK. 00:41:17.500 --> 00:41:19.630 align:middle line:90% And yeah. 00:41:19.630 --> 00:41:21.790 align:middle line:90% So this is the field-- 00:41:21.790 --> 00:41:23.530 align:middle line:84% so normally in classical mechanics, 00:41:23.530 --> 00:41:25.500 align:middle line:90% you just have this equation. 00:41:25.500 --> 00:41:26.000 align:middle line:90% OK? 00:41:26.000 --> 00:41:31.110 align:middle line:84% And this is the field theory version of your conservation. 00:41:31.110 --> 00:41:31.830 align:middle line:90% Yes? 00:41:31.830 --> 00:41:35.420 align:middle line:84% AUDIENCE: So is there a way to maybe to know what it is 00:41:35.420 --> 00:41:36.670 align:middle line:90% and how we got it? 00:41:36.670 --> 00:41:37.430 align:middle line:90% HONG LIU: Sorry? 00:41:37.430 --> 00:41:41.060 align:middle line:84% AUDIENCE: Is there a way to calculate what J represents? 00:41:41.060 --> 00:41:42.032 align:middle line:90% HONG LIU: J represents? 00:41:42.032 --> 00:41:43.490 align:middle line:84% AUDIENCE: Yeah, what it represents? 00:41:43.490 --> 00:41:44.615 align:middle line:90% HONG LIU: Yeah, yeah, yeah. 00:41:44.615 --> 00:41:48.540 align:middle line:84% If I give you a specific-- yeah, you will do it in your Pset. 00:41:48.540 --> 00:41:52.240 align:middle line:84% So if I give you a specific Lagrangian-- 00:41:52.240 --> 00:41:54.570 align:middle line:84% if you start a specific transformation, 00:41:54.570 --> 00:41:57.550 align:middle line:84% and then you will be able to find all those quantities-- 00:41:57.550 --> 00:41:59.718 align:middle line:84% you will be able to find the k explicitly. 00:41:59.718 --> 00:42:02.260 align:middle line:84% And you will be able to find all those quantities explicitly. 00:42:02.260 --> 00:42:09.730 align:middle line:84% Then you can find the J. Other questions? 00:42:09.730 --> 00:42:14.130 align:middle line:90% 00:42:14.130 --> 00:42:15.790 align:middle line:90% Yes? 00:42:15.790 --> 00:42:19.080 align:middle line:84% AUDIENCE: Minus delta L of 0 implies that delta L 00:42:19.080 --> 00:42:20.020 align:middle line:90% needs to be over here. 00:42:20.020 --> 00:42:20.687 align:middle line:90% HONG LIU: Right. 00:42:20.687 --> 00:42:24.970 align:middle line:90% It's because if we have here-- 00:42:24.970 --> 00:42:28.330 align:middle line:84% so here, if we have delta s equal to 0-- 00:42:28.330 --> 00:42:33.130 align:middle line:84% and then you can just put the delta in. 00:42:33.130 --> 00:42:37.540 align:middle line:84% And then this quantity that is integrated over that-- 00:42:37.540 --> 00:42:39.280 align:middle line:84% if this is 0, then this has to be 00:42:39.280 --> 00:42:48.475 align:middle line:84% a total derivative Other questions? 00:42:48.475 --> 00:42:48.975 align:middle line:90% Yes? 00:42:48.975 --> 00:42:52.150 align:middle line:84% AUDIENCE: So what happens if it's not a global symmetry? 00:42:52.150 --> 00:42:54.050 align:middle line:90% Where does this argument break? 00:42:54.050 --> 00:42:55.580 align:middle line:90% HONG LIU: Right, yeah. 00:42:55.580 --> 00:42:56.080 align:middle line:90% Good. 00:42:56.080 --> 00:43:00.290 align:middle line:90% This is a very good question. 00:43:00.290 --> 00:43:05.560 align:middle line:84% So yeah, I will not have time to go into here. 00:43:05.560 --> 00:43:07.700 align:middle line:84% Let me just very quickly mention. 00:43:07.700 --> 00:43:10.180 align:middle line:84% So this is called the first Noether theorem-- 00:43:10.180 --> 00:43:15.500 align:middle line:84% and when epsilon is spacetime independent. 00:43:15.500 --> 00:43:18.730 align:middle line:84% And so when epsilon is not spacetime dependent, 00:43:18.730 --> 00:43:20.780 align:middle line:84% then the story is a little bit more complicated. 00:43:20.780 --> 00:43:25.500 align:middle line:84% So the epsilon will be inside this total derivative. 00:43:25.500 --> 00:43:27.260 align:middle line:84% You cannot take it out, et cetera. 00:43:27.260 --> 00:43:29.600 align:middle line:84% And then the story changes a little bit. 00:43:29.600 --> 00:43:32.200 align:middle line:84% And then there's something called the second Noether 00:43:32.200 --> 00:43:32.950 align:middle line:90% theorem. 00:43:32.950 --> 00:43:35.950 align:middle line:84% And actually, in that case, when you have local symmetries when 00:43:35.950 --> 00:43:38.500 align:middle line:84% epsilon depends on spacetime, instead 00:43:38.500 --> 00:43:40.570 align:middle line:84% of finding conservation laws, you 00:43:40.570 --> 00:43:43.030 align:middle line:84% find that your equation of motion-- 00:43:43.030 --> 00:43:47.320 align:middle line:84% not all-- you find some parts of the equation of motion 00:43:47.320 --> 00:43:49.210 align:middle line:90% are redundant. 00:43:49.210 --> 00:43:50.660 align:middle line:90% Yeah. 00:43:50.660 --> 00:43:51.260 align:middle line:90% Yeah. 00:43:51.260 --> 00:43:55.910 align:middle line:84% And if you-- it's not difficult. But actually, in principle, I 00:43:55.910 --> 00:43:59.990 align:middle line:84% can put it in the Pset problem, if people really 00:43:59.990 --> 00:44:01.100 align:middle line:90% want to see it. 00:44:01.100 --> 00:44:01.970 align:middle line:90% Yes? 00:44:01.970 --> 00:44:03.660 align:middle line:84% AUDIENCE: So this is something to think 00:44:03.660 --> 00:44:07.200 align:middle line:84% about the transformation, the vicinity of the identity. 00:44:07.200 --> 00:44:09.720 align:middle line:84% What happens to the information of the transformation 00:44:09.720 --> 00:44:10.530 align:middle line:90% outside of-- 00:44:10.530 --> 00:44:11.910 align:middle line:90% HONG LIU: Good, good, good. 00:44:11.910 --> 00:44:18.990 align:middle line:84% So this is the beauty of physics, which you may already 00:44:18.990 --> 00:44:20.640 align:middle line:90% have seen in quantum mechanics. 00:44:20.640 --> 00:44:24.800 align:middle line:84% So whenever we see a symmetry, no matter how 00:44:24.800 --> 00:44:26.660 align:middle line:90% complicated that symmetry is-- 00:44:26.660 --> 00:44:29.500 align:middle line:90% say, some continuous symmetry-- 00:44:29.500 --> 00:44:32.740 align:middle line:84% it's enough just to understand that symmetry 00:44:32.740 --> 00:44:38.020 align:middle line:84% near the identity, near to the infinitesimal transformation. 00:44:38.020 --> 00:44:40.510 align:middle line:84% It's because any finite transformation 00:44:40.510 --> 00:44:43.960 align:middle line:84% you can build up from just adding up 00:44:43.960 --> 00:44:45.620 align:middle line:84% the infinitesimal transformations. 00:44:45.620 --> 00:44:47.703 align:middle line:84% So once you know the infinitesimal transformation, 00:44:47.703 --> 00:44:51.310 align:middle line:84% actually, essentially up to some global or topological 00:44:51.310 --> 00:44:53.200 align:middle line:84% structure, essentially it determines 00:44:53.200 --> 00:44:55.350 align:middle line:90% the full finite transformation. 00:44:55.350 --> 00:44:56.350 align:middle line:90% Yeah? 00:44:56.350 --> 00:44:58.058 align:middle line:84% AUDIENCE: So we can define transformation 00:44:58.058 --> 00:45:00.100 align:middle line:84% by taking the derivative of things that 00:45:00.100 --> 00:45:02.070 align:middle line:90% are different that obviously-- 00:45:02.070 --> 00:45:02.930 align:middle line:90% HONG LIU: No, no. 00:45:02.930 --> 00:45:04.560 align:middle line:90% It's always a symmetry, right? 00:45:04.560 --> 00:45:07.640 align:middle line:90% Because each step is a cemetery. 00:45:07.640 --> 00:45:09.270 align:middle line:90% And so you add them up. 00:45:09.270 --> 00:45:12.573 align:middle line:84% You add them up still a symmetry. 00:45:12.573 --> 00:45:13.115 align:middle line:90% AUDIENCE: OK. 00:45:13.115 --> 00:45:15.442 align:middle line:84% But a sequence of different symmetries, for example? 00:45:15.442 --> 00:45:17.150 align:middle line:84% HONG LIU: Oh, if you do different-- yeah, 00:45:17.150 --> 00:45:20.180 align:middle line:90% they're still a symmetry. 00:45:20.180 --> 00:45:23.960 align:middle line:84% Yeah, because by definition if something transforms 00:45:23.960 --> 00:45:25.835 align:middle line:84% that's invariant into another transformation, 00:45:25.835 --> 00:45:27.050 align:middle line:90% it's still invariant. 00:45:27.050 --> 00:45:32.170 align:middle line:90% 00:45:32.170 --> 00:45:32.670 align:middle line:90% Good? 00:45:32.670 --> 00:45:33.930 align:middle line:90% Other questions? 00:45:33.930 --> 00:45:34.702 align:middle line:90% Yes? 00:45:34.702 --> 00:45:36.285 align:middle line:84% AUDIENCE: For discrete symmetry, where 00:45:36.285 --> 00:45:39.060 align:middle line:84% we don't have this epsilon, can we do something similar? 00:45:39.060 --> 00:45:39.750 align:middle line:90% HONG LIU: Sorry? 00:45:39.750 --> 00:45:41.070 align:middle line:84% AUDIENCE: For discrete symmetries-- 00:45:41.070 --> 00:45:42.090 align:middle line:90% HONG LIU: For discrete symmetry? 00:45:42.090 --> 00:45:43.125 align:middle line:84% AUDIENCE: --do we have to do this analysis? 00:45:43.125 --> 00:45:45.035 align:middle line:84% Or can we can do something similar? 00:45:45.035 --> 00:45:45.660 align:middle line:90% HONG LIU: Yeah. 00:45:45.660 --> 00:45:47.490 align:middle line:84% In general for discrete symmetries, 00:45:47.490 --> 00:45:50.880 align:middle line:84% you don't have such a conservation law. 00:45:50.880 --> 00:45:52.800 align:middle line:84% You don't have such a conservation law. 00:45:52.800 --> 00:45:56.400 align:middle line:84% But you can still sometimes define some discrete quantum 00:45:56.400 --> 00:46:00.390 align:middle line:84% number which is conserved, like the parity which-- 00:46:00.390 --> 00:46:02.430 align:middle line:90% in the parity case. 00:46:02.430 --> 00:46:04.600 align:middle line:90% But you don't have a current. 00:46:04.600 --> 00:46:05.100 align:middle line:90% OK? 00:46:05.100 --> 00:46:08.962 align:middle line:84% So here, J mu is called the conserved current. 00:46:08.962 --> 00:46:10.170 align:middle line:90% But you won't have a current. 00:46:10.170 --> 00:46:15.050 align:middle line:90% 00:46:15.050 --> 00:46:15.934 align:middle line:90% Good? 00:46:15.934 --> 00:46:16.820 align:middle line:90% Good. 00:46:16.820 --> 00:46:24.570 align:middle line:84% So this concludes our very quick discussion 00:46:24.570 --> 00:46:29.445 align:middle line:84% of the important features of classical field theory. 00:46:29.445 --> 00:46:34.530 align:middle line:84% And with this preparation, now we can start talking about 00:46:34.530 --> 00:46:37.710 align:middle line:84% or start thinking about going to quantum mechanics, 00:46:37.710 --> 00:46:39.420 align:middle line:84% talking about the quantum fields. 00:46:39.420 --> 00:46:40.440 align:middle line:90% OK? 00:46:40.440 --> 00:46:45.750 align:middle line:84% And so here is a good place to pause a little bit from what 00:46:45.750 --> 00:46:51.540 align:middle line:84% we already said about the classical field theory 00:46:51.540 --> 00:46:57.300 align:middle line:84% to anticipate a little bit about the quantum fields theory. 00:46:57.300 --> 00:46:58.050 align:middle line:90% OK? 00:46:58.050 --> 00:47:09.190 align:middle line:84% So here, I will restate our goal for quantum field theory, which 00:47:09.190 --> 00:47:13.900 align:middle line:84% I said a little bit in the last lecture. 00:47:13.900 --> 00:47:15.820 align:middle line:90% OK? 00:47:15.820 --> 00:47:19.390 align:middle line:84% So now we have seen the classical fields. 00:47:19.390 --> 00:47:23.050 align:middle line:84% And our goal is to understand the quantum 00:47:23.050 --> 00:47:24.250 align:middle line:90% version of this story. 00:47:24.250 --> 00:47:24.880 align:middle line:90% OK? 00:47:24.880 --> 00:47:30.070 align:middle line:84% So classical field theory is classical mechanics 00:47:30.070 --> 00:47:32.110 align:middle line:84% with infinite number of degrees of freedom. 00:47:32.110 --> 00:47:34.373 align:middle line:84% And now we want to quantize this infinite number 00:47:34.373 --> 00:47:35.290 align:middle line:90% of degrees of freedom. 00:47:35.290 --> 00:47:37.210 align:middle line:84% And now this becomes quantum field theory. 00:47:37.210 --> 00:47:37.720 align:middle line:90% OK? 00:47:37.720 --> 00:47:40.090 align:middle line:84% And now we want to understand the quantum dynamics 00:47:40.090 --> 00:47:42.760 align:middle line:90% of such kind of phi fields. 00:47:42.760 --> 00:47:43.630 align:middle line:90% OK. 00:47:43.630 --> 00:47:48.670 align:middle line:84% And for example, an example is the E&M. 00:47:48.670 --> 00:47:51.070 align:middle line:84% We have electric and magnetic fields. 00:47:51.070 --> 00:47:51.760 align:middle line:90% OK? 00:47:51.760 --> 00:47:54.250 align:middle line:84% And you know the Maxwell equations. 00:47:54.250 --> 00:47:56.080 align:middle line:84% You can solve the Maxwell equations. 00:47:56.080 --> 00:47:59.680 align:middle line:84% And so this defines the classical field theory. 00:47:59.680 --> 00:48:05.890 align:middle line:84% And then we will tell you how to actually quantize such a system 00:48:05.890 --> 00:48:09.820 align:middle line:84% to understand the electric field and the magnetic fields 00:48:09.820 --> 00:48:11.620 align:middle line:90% are quantum mechanical. 00:48:11.620 --> 00:48:13.950 align:middle line:90% OK. 00:48:13.950 --> 00:48:15.060 align:middle line:90% Good. 00:48:15.060 --> 00:48:17.830 align:middle line:84% And that actually would be our goal. 00:48:17.830 --> 00:48:18.330 align:middle line:90% OK? 00:48:18.330 --> 00:48:20.820 align:middle line:84% So we will start with the simplest field theory-- 00:48:20.820 --> 00:48:22.590 align:middle line:90% just the scalar field theory. 00:48:22.590 --> 00:48:26.580 align:middle line:84% And then we will go to the Dirac theory, 00:48:26.580 --> 00:48:28.320 align:middle line:90% which describes the electron. 00:48:28.320 --> 00:48:32.190 align:middle line:84% And then, eventually, we will go to the theory which describes 00:48:32.190 --> 00:48:33.940 align:middle line:90% the electromagnetic field. 00:48:33.940 --> 00:48:36.480 align:middle line:90% So this is called the QED, OK-- 00:48:36.480 --> 00:48:37.950 align:middle line:90% quantum electrodynamics. 00:48:37.950 --> 00:48:41.460 align:middle line:84% So that will be the endpoint of this course. 00:48:41.460 --> 00:48:45.040 align:middle line:90% 00:48:45.040 --> 00:48:46.240 align:middle line:90% OK. 00:48:46.240 --> 00:48:52.420 align:middle line:84% So now let's say a little bit how 00:48:52.420 --> 00:48:54.700 align:middle line:84% we go through this classical field theory 00:48:54.700 --> 00:48:59.870 align:middle line:84% to quantum field theory, OK-- so just some general remarks. 00:48:59.870 --> 00:49:01.990 align:middle line:84% So before doing that, let's think a little bit 00:49:01.990 --> 00:49:04.298 align:middle line:84% how we go from classical mechanics, 00:49:04.298 --> 00:49:06.090 align:middle line:84% with a small number of degrees of freedom-- 00:49:06.090 --> 00:49:07.600 align:middle line:84% and how you go to quantum mechanics. 00:49:07.600 --> 00:49:08.890 align:middle line:90% OK? 00:49:08.890 --> 00:49:12.435 align:middle line:90% So let's recall. 00:49:12.435 --> 00:49:19.500 align:middle line:90% 00:49:19.500 --> 00:49:21.510 align:middle line:90% So in classical mechanics-- 00:49:21.510 --> 00:49:23.962 align:middle line:90% 00:49:23.962 --> 00:49:25.670 align:middle line:84% so let's just consider the simplest case, 00:49:25.670 --> 00:49:27.545 align:middle line:84% if you just have a single degree of freedom-- 00:49:27.545 --> 00:49:31.970 align:middle line:84% say x(t), OK-- just a one-dimensional particle whose 00:49:31.970 --> 00:49:34.850 align:middle line:90% motion just describes the x(t). 00:49:34.850 --> 00:49:40.320 align:middle line:84% And then, of course, the goal of classical mechanics 00:49:40.320 --> 00:49:43.470 align:middle line:84% is just to solve the equation of motion over x(t). 00:49:43.470 --> 00:49:44.430 align:middle line:90% OK? 00:49:44.430 --> 00:49:48.870 align:middle line:84% So if I write down the equation for x(t) and I solve it, 00:49:48.870 --> 00:49:50.440 align:middle line:90% then I'm done. 00:49:50.440 --> 00:49:50.940 align:middle line:90% OK? 00:49:50.940 --> 00:49:52.110 align:middle line:90% I'm done. 00:49:52.110 --> 00:49:54.690 align:middle line:84% I solved the classical mechanics. 00:49:54.690 --> 00:50:03.510 align:middle line:84% So when we go to quantum mechanics, what do you do? 00:50:03.510 --> 00:50:06.480 align:middle line:84% So this x(t) then becomes an operator. 00:50:06.480 --> 00:50:09.590 align:middle line:90% 00:50:09.590 --> 00:50:10.090 align:middle line:90% OK? 00:50:10.090 --> 00:50:11.350 align:middle line:90% So it becomes an operator. 00:50:11.350 --> 00:50:13.750 align:middle line:84% So the classical dynamical variable 00:50:13.750 --> 00:50:16.340 align:middle line:90% now becomes a quantum operator. 00:50:16.340 --> 00:50:20.060 align:middle line:84% And the equation of motion of x(t), if you remember-- 00:50:20.060 --> 00:50:21.560 align:middle line:84% and do you remember what it becomes? 00:50:21.560 --> 00:50:29.650 align:middle line:90% 00:50:29.650 --> 00:50:30.262 align:middle line:90% Yeah, exactly. 00:50:30.262 --> 00:50:31.720 align:middle line:84% It becomes the Heisenberg equation. 00:50:31.720 --> 00:50:35.150 align:middle line:90% 00:50:35.150 --> 00:50:42.290 align:middle line:84% So it becomes the Heisenberg equation for this operator x(t) 00:50:42.290 --> 00:50:43.420 align:middle line:90% now. 00:50:43.420 --> 00:50:45.250 align:middle line:90% OK? 00:50:45.250 --> 00:50:46.450 align:middle line:90% Yeah. 00:50:46.450 --> 00:50:50.830 align:middle line:84% So in quantum mechanics, there are normally two ways 00:50:50.830 --> 00:50:52.220 align:middle line:90% to describe it. 00:50:52.220 --> 00:50:53.875 align:middle line:84% So first is the Schrodinger picture. 00:50:53.875 --> 00:51:00.240 align:middle line:90% 00:51:00.240 --> 00:51:08.115 align:middle line:84% In this case, you have a wave function, which 00:51:08.115 --> 00:51:09.240 align:middle line:90% is the function of x and t. 00:51:09.240 --> 00:51:11.400 align:middle line:84% And then you have an operator, which is x. 00:51:11.400 --> 00:51:13.184 align:middle line:90% OK? 00:51:13.184 --> 00:51:17.380 align:middle line:84% And then, of course, you also have conjugate momentum, 00:51:17.380 --> 00:51:20.930 align:middle line:84% et cetera, which is defined to be x dot. 00:51:20.930 --> 00:51:22.430 align:middle line:90% OK? 00:51:22.430 --> 00:51:27.330 align:middle line:84% And so the wave function is a function 00:51:27.330 --> 00:51:30.460 align:middle line:90% of the eigenvalues of-- 00:51:30.460 --> 00:51:35.860 align:middle line:84% so x here should be understood as eigenvalues of this operator 00:51:35.860 --> 00:51:37.300 align:middle line:90% x-hat. 00:51:37.300 --> 00:51:39.100 align:middle line:90% OK? 00:51:39.100 --> 00:51:41.530 align:middle line:90% It's the eigenvalue of x-hat. 00:51:41.530 --> 00:51:43.270 align:middle line:84% So in the Schrodinger picture, you 00:51:43.270 --> 00:51:46.810 align:middle line:90% solve the evolution for psi. 00:51:46.810 --> 00:51:49.470 align:middle line:90% You solve the equation for psi. 00:51:49.470 --> 00:51:51.045 align:middle line:84% And then once you find psi, then you 00:51:51.045 --> 00:51:52.770 align:middle line:84% can calculate any quantities you want. 00:51:52.770 --> 00:51:53.270 align:middle line:90% OK? 00:51:53.270 --> 00:51:56.910 align:middle line:84% You can calculate any expectation values, 00:51:56.910 --> 00:52:00.340 align:middle line:90% any amplitude, et cetera. 00:52:00.340 --> 00:52:02.070 align:middle line:84% But the second way of approaching it 00:52:02.070 --> 00:52:03.270 align:middle line:90% is the Heisenberg picture. 00:52:03.270 --> 00:52:13.010 align:middle line:90% 00:52:13.010 --> 00:52:16.400 align:middle line:90% In this case, you-- 00:52:16.400 --> 00:52:19.910 align:middle line:84% the dynamical quantity is your operator. 00:52:19.910 --> 00:52:22.020 align:middle line:84% Now you're assuming the operator depend on time. 00:52:22.020 --> 00:52:23.600 align:middle line:84% So in the Schrodinger picture, your x 00:52:23.600 --> 00:52:26.570 align:middle line:84% is just some constant operator that does not depend on time. 00:52:26.570 --> 00:52:29.570 align:middle line:84% But in the Heisenberg picture, now your operator 00:52:29.570 --> 00:52:32.180 align:middle line:90% now becomes time-dependent. 00:52:32.180 --> 00:52:32.690 align:middle line:90% OK? 00:52:32.690 --> 00:52:34.580 align:middle line:90% And the equation of motion-- 00:52:34.580 --> 00:52:36.770 align:middle line:84% and then you solve the Heisenberg equation. 00:52:36.770 --> 00:52:37.460 align:middle line:90% OK? 00:52:37.460 --> 00:52:40.020 align:middle line:90% And then the state is invariant. 00:52:40.020 --> 00:52:40.520 align:middle line:90% OK? 00:52:40.520 --> 00:52:42.470 align:middle line:84% The state does not evolve with time. 00:52:42.470 --> 00:52:43.970 align:middle line:90% OK. 00:52:43.970 --> 00:52:47.490 align:middle line:84% So you solve for the Heisenberg equation for those operators. 00:52:47.490 --> 00:52:49.650 align:middle line:84% And then once you solve the Heisenberg equation, 00:52:49.650 --> 00:52:51.710 align:middle line:84% and then you can evaluate, again, 00:52:51.710 --> 00:52:54.560 align:middle line:84% your expectation value in any state 00:52:54.560 --> 00:52:57.620 align:middle line:84% you are interested in, et cetera. 00:52:57.620 --> 00:52:59.820 align:middle line:90% OK. 00:52:59.820 --> 00:53:01.250 align:middle line:90% So any questions on this? 00:53:01.250 --> 00:53:02.910 align:middle line:84% So this is a very quick review of what 00:53:02.910 --> 00:53:07.200 align:middle line:84% you did to go from classical mechanics to quantum mechanics. 00:53:07.200 --> 00:53:08.630 align:middle line:90% OK. 00:53:08.630 --> 00:53:15.940 align:middle line:84% So now in field theory, OK, we do the similar thing. 00:53:15.940 --> 00:53:18.680 align:middle line:90% 00:53:18.680 --> 00:53:29.260 align:middle line:84% So similarly, in field theory, we have classical fields. 00:53:29.260 --> 00:53:29.760 align:middle line:90% OK? 00:53:29.760 --> 00:53:32.060 align:middle line:90% We have classical fields. 00:53:32.060 --> 00:53:38.560 align:middle line:84% And then when you go to quantum, this become quantum operators. 00:53:38.560 --> 00:53:41.150 align:middle line:90% 00:53:41.150 --> 00:53:41.650 align:middle line:90% OK? 00:53:41.650 --> 00:53:46.560 align:middle line:84% So this is now quantum operators. 00:53:46.560 --> 00:53:50.520 align:middle line:84% Remember, in field theory, this x is always 00:53:50.520 --> 00:53:52.980 align:middle line:90% just labels of the space points. 00:53:52.980 --> 00:53:53.940 align:middle line:90% OK? 00:53:53.940 --> 00:53:55.570 align:middle line:90% It's not a dynamical variable. 00:53:55.570 --> 00:53:56.070 align:middle line:90% OK? 00:53:56.070 --> 00:53:58.420 align:middle line:90% The notation is a little bit-- 00:53:58.420 --> 00:53:58.920 align:middle line:90% yeah. 00:53:58.920 --> 00:54:02.320 align:middle line:84% Here, in classical mechanics, this is a dynamical variable. 00:54:02.320 --> 00:54:04.210 align:middle line:84% But in field theory, this is just labels. 00:54:04.210 --> 00:54:04.710 align:middle line:90% OK? 00:54:04.710 --> 00:54:06.720 align:middle line:90% So they're not dynamical. 00:54:06.720 --> 00:54:08.920 align:middle line:84% The dynamical variable are phi itself. 00:54:08.920 --> 00:54:11.760 align:middle line:90% So phi becomes an operator. 00:54:11.760 --> 00:54:13.290 align:middle line:90% OK? 00:54:13.290 --> 00:54:15.210 align:middle line:84% And then now the classical equation 00:54:15.210 --> 00:54:22.770 align:middle line:84% of motion for phi, which we derived-- 00:54:22.770 --> 00:54:35.680 align:middle line:84% and then become the Heisenberg equation, OK, for phi-hat. 00:54:35.680 --> 00:54:38.290 align:middle line:90% 00:54:38.290 --> 00:54:43.192 align:middle line:84% So again, here, you can do two pictures. 00:54:43.192 --> 00:54:44.650 align:middle line:84% You can do the Schrodinger picture, 00:54:44.650 --> 00:54:46.108 align:middle line:84% or you can do a Heisenberg picture. 00:54:46.108 --> 00:54:49.150 align:middle line:90% 00:54:49.150 --> 00:54:56.890 align:middle line:84% So for QFT, the Schrodinger picture-- 00:54:56.890 --> 00:54:57.390 align:middle line:90% OK. 00:54:57.390 --> 00:54:59.190 align:middle line:84% And you look at the wave functions 00:54:59.190 --> 00:55:02.850 align:middle line:84% of your dynamical variables, the eigenvalue 00:55:02.850 --> 00:55:04.290 align:middle line:90% of dynamical variables. 00:55:04.290 --> 00:55:09.800 align:middle line:90% 00:55:09.800 --> 00:55:13.850 align:middle line:84% So what is the generalization of this psi xt? 00:55:13.850 --> 00:55:18.130 align:middle line:84% So remember, here, psi is a function of eigenvalue 00:55:18.130 --> 00:55:19.580 align:middle line:90% of your dynamical variables. 00:55:19.580 --> 00:55:20.080 align:middle line:90% OK? 00:55:20.080 --> 00:55:22.730 align:middle line:90% 00:55:22.730 --> 00:55:25.760 align:middle line:84% So now, if we push this analog further, 00:55:25.760 --> 00:55:28.250 align:middle line:84% when you go to quantum field theory in the Schrodinger 00:55:28.250 --> 00:55:35.040 align:middle line:84% picture, now the wave function should be a function 00:55:35.040 --> 00:55:40.410 align:middle line:90% of phi (x) and the t. 00:55:40.410 --> 00:55:40.910 align:middle line:90% OK? 00:55:40.910 --> 00:55:53.110 align:middle line:84% So this phi x are eigenvalues of phi-hat x. 00:55:53.110 --> 00:55:53.710 align:middle line:90% OK? 00:55:53.710 --> 00:55:55.780 align:middle line:84% So in the Schrodinger picture, phi-hat 00:55:55.780 --> 00:55:58.990 align:middle line:90% does not evolve with time. 00:55:58.990 --> 00:56:01.290 align:middle line:84% And so you have the wave function, 00:56:01.290 --> 00:56:04.590 align:middle line:90% which is the function-- 00:56:04.590 --> 00:56:07.560 align:middle line:90% now become a functional-- 00:56:07.560 --> 00:56:09.960 align:middle line:84% because this itself is a function of space. 00:56:09.960 --> 00:56:10.710 align:middle line:90% OK? 00:56:10.710 --> 00:56:14.960 align:middle line:90% And yeah. 00:56:14.960 --> 00:56:17.730 align:middle line:84% And then you solve the Schrodinger equation for this. 00:56:17.730 --> 00:56:22.890 align:middle line:90% And then you have the operator. 00:56:22.890 --> 00:56:24.520 align:middle line:90% And then you have the operator. 00:56:24.520 --> 00:56:25.020 align:middle line:90% Yeah. 00:56:25.020 --> 00:56:26.220 align:middle line:90% And so you have-- 00:56:26.220 --> 00:56:27.690 align:middle line:84% and then the dynamical-- and then 00:56:27.690 --> 00:56:31.090 align:middle line:90% you have operator phi-hat x. 00:56:31.090 --> 00:56:31.870 align:middle line:90% OK? 00:56:31.870 --> 00:56:35.970 align:middle line:84% So that's what you do in the Schrodinger picture, 00:56:35.970 --> 00:56:38.650 align:middle line:90% in the Schrodinger picture. 00:56:38.650 --> 00:56:44.690 align:middle line:84% And in the Heisenberg picture, you 00:56:44.690 --> 00:56:46.190 align:middle line:90% forget about the wave function. 00:56:46.190 --> 00:56:47.450 align:middle line:90% OK? 00:56:47.450 --> 00:56:49.820 align:middle line:84% You look at the evolution of operators. 00:56:49.820 --> 00:57:11.513 align:middle line:90% 00:57:11.513 --> 00:57:13.180 align:middle line:84% So the Heisenberg for the field theory-- 00:57:13.180 --> 00:57:18.100 align:middle line:90% 00:57:18.100 --> 00:57:21.310 align:middle line:84% your Heisenberg picture for the field theory 00:57:21.310 --> 00:57:25.870 align:middle line:84% is that you look at the evolution of x and t. 00:57:25.870 --> 00:57:28.790 align:middle line:84% So now this just obeys the Heisenberg equations, 00:57:28.790 --> 00:57:31.900 align:middle line:84% which is just the quantum version 00:57:31.900 --> 00:57:34.120 align:middle line:84% of the classical equation for phi. 00:57:34.120 --> 00:57:37.340 align:middle line:84% So you look at the evolution of this. 00:57:37.340 --> 00:57:39.550 align:middle line:84% And then the state does not change with time. 00:57:39.550 --> 00:57:40.360 align:middle line:90% OK? 00:57:40.360 --> 00:57:41.860 align:middle line:84% The state does not change with time. 00:57:41.860 --> 00:57:45.030 align:middle line:90% 00:57:45.030 --> 00:57:45.553 align:middle line:90% Good. 00:57:45.553 --> 00:57:46.470 align:middle line:90% Any questions on this? 00:57:46.470 --> 00:57:50.650 align:middle line:90% 00:57:50.650 --> 00:57:52.342 align:middle line:90% Yes? 00:57:52.342 --> 00:57:53.710 align:middle line:90% AUDIENCE: When it comes to-- 00:57:53.710 --> 00:57:55.930 align:middle line:90% so we have the quantities-- 00:57:55.930 --> 00:57:56.710 align:middle line:90% HONG LIU: Yeah. 00:57:56.710 --> 00:57:59.000 align:middle line:84% AUDIENCE: --what happens to time in the Schrodinger picture? 00:57:59.000 --> 00:58:00.708 align:middle line:84% HONG LIU: So in the Schrodinger picture-- 00:58:00.708 --> 00:58:06.720 align:middle line:84% so remember, the x is a label for phi. 00:58:06.720 --> 00:58:08.700 align:middle line:90% But time is the evolution. 00:58:08.700 --> 00:58:11.440 align:middle line:84% But in the Schrodinger picture, the operators don't evolve. 00:58:11.440 --> 00:58:13.470 align:middle line:90% And so there's no time here. 00:58:13.470 --> 00:58:14.970 align:middle line:90% So we don't have time here. 00:58:14.970 --> 00:58:19.140 align:middle line:84% And we just have the analog of the x here in the Schrodinger 00:58:19.140 --> 00:58:23.640 align:middle line:84% picture for classical-- yeah, for quantum mechanics. 00:58:23.640 --> 00:58:28.320 align:middle line:84% And then the time dependence is in your wave function. 00:58:28.320 --> 00:58:30.210 align:middle line:84% So the wave function is a function 00:58:30.210 --> 00:58:35.990 align:middle line:84% of possible values, eigenvalues of this operator phi. 00:58:35.990 --> 00:58:37.890 align:middle line:90% Yeah. 00:58:37.890 --> 00:58:39.433 align:middle line:90% OK? 00:58:39.433 --> 00:58:41.100 align:middle line:84% And then the Heisenberg picture-- again, 00:58:41.100 --> 00:58:43.690 align:middle line:84% then you just focus on the operator equations. 00:58:43.690 --> 00:58:46.020 align:middle line:84% And once you solve the operator equations, 00:58:46.020 --> 00:58:50.760 align:middle line:84% and then you can calculate the expectation values in any state 00:58:50.760 --> 00:58:51.960 align:middle line:90% you want. 00:58:51.960 --> 00:58:53.820 align:middle line:90% OK. 00:58:53.820 --> 00:58:57.780 align:middle line:84% So now you can already maybe see a difference a little bit. 00:58:57.780 --> 00:59:01.060 align:middle line:84% So if you do the Schrodinger picture, 00:59:01.060 --> 00:59:02.800 align:middle line:84% you have to deal with this beast, 00:59:02.800 --> 00:59:05.740 align:middle line:84% OK, which is the wave functional, 00:59:05.740 --> 00:59:10.410 align:middle line:84% though, of all possible values of some function in space. 00:59:10.410 --> 00:59:11.310 align:middle line:90% OK? 00:59:11.310 --> 00:59:13.080 align:middle line:84% And if you have multiple fields, then this 00:59:13.080 --> 00:59:14.700 align:middle line:90% is a hugely complicated thing. 00:59:14.700 --> 00:59:16.860 align:middle line:84% And you need to write down the Schrodinger equation 00:59:16.860 --> 00:59:18.030 align:middle line:90% for it, et cetera. 00:59:18.030 --> 00:59:18.990 align:middle line:90% OK? 00:59:18.990 --> 00:59:22.170 align:middle line:84% But here-- but in the Heisenberg picture, 00:59:22.170 --> 00:59:26.680 align:middle line:84% we just solve the analog of the classical equation of motion, 00:59:26.680 --> 00:59:28.650 align:middle line:84% which we have already written down. 00:59:28.650 --> 00:59:32.103 align:middle line:84% And you just now interpret it as a quantum operator equation. 00:59:32.103 --> 00:59:33.645 align:middle line:84% So which one do you think is simpler? 00:59:33.645 --> 00:59:38.000 align:middle line:90% 00:59:38.000 --> 00:59:40.570 align:middle line:84% So that's why in quantum field theory 00:59:40.570 --> 00:59:43.280 align:middle line:84% we almost always use Heisenberg picture. 00:59:43.280 --> 00:59:43.780 align:middle line:90% OK? 00:59:43.780 --> 00:59:45.400 align:middle line:84% We always use Heisenberg picture. 00:59:45.400 --> 00:59:47.380 align:middle line:90% And we don't even-- 00:59:47.380 --> 00:59:49.960 align:middle line:84% rarely think about the wave function, 00:59:49.960 --> 00:59:54.040 align:middle line:84% even though sometimes this can be useful in some problems. 00:59:54.040 --> 00:59:56.750 align:middle line:84% But for most of the time, this is much easier. 00:59:56.750 --> 00:59:57.250 align:middle line:90% OK? 00:59:57.250 --> 00:59:59.000 align:middle line:84% So that's what we are going to do. 00:59:59.000 --> 00:59:59.500 align:middle line:90% OK? 00:59:59.500 --> 01:00:01.420 align:middle line:84% And so in quantum field theory, we 01:00:01.420 --> 01:00:05.140 align:middle line:84% will just use the Heisenberg picture almost all the time. 01:00:05.140 --> 01:00:06.470 align:middle line:90% OK? 01:00:06.470 --> 01:00:10.800 align:middle line:84% From now on, I will not talk about the Schrodinger picture. 01:00:10.800 --> 01:00:11.300 align:middle line:90% OK. 01:00:11.300 --> 01:00:15.960 align:middle line:90% 01:00:15.960 --> 01:00:16.500 align:middle line:90% Good. 01:00:16.500 --> 01:00:21.430 align:middle line:84% So it's very important you remind yourself 01:00:21.430 --> 01:00:25.510 align:middle line:84% about quantum mechanics in the Heisenberg picture, OK-- 01:00:25.510 --> 01:00:30.430 align:middle line:84% And because most of your quantum mechanical classes 01:00:30.430 --> 01:00:33.760 align:middle line:84% before maybe is all in the Schrodinger picture, 01:00:33.760 --> 01:00:37.370 align:middle line:84% solving the Schrodinger equation, et cetera. 01:00:37.370 --> 01:00:39.310 align:middle line:84% But now you have to change the perspective 01:00:39.310 --> 01:00:42.550 align:middle line:84% to think of everything in terms of the Heisenberg picture. 01:00:42.550 --> 01:00:44.800 align:middle line:90% And then that will make you-- 01:00:44.800 --> 01:00:46.360 align:middle line:84% quantum field theory much easier. 01:00:46.360 --> 01:00:46.860 align:middle line:90% OK? 01:00:46.860 --> 01:00:51.440 align:middle line:90% 01:00:51.440 --> 01:00:51.940 align:middle line:90% Good. 01:00:51.940 --> 01:00:52.950 align:middle line:90% Any questions on this? 01:00:52.950 --> 01:00:59.400 align:middle line:90% 01:00:59.400 --> 01:01:00.630 align:middle line:90% Yes? 01:01:00.630 --> 01:01:03.090 align:middle line:84% AUDIENCE: So in the Heisenberg picture, 01:01:03.090 --> 01:01:06.600 align:middle line:84% if we're not concerned about the wave function and all, 01:01:06.600 --> 01:01:10.680 align:middle line:84% then how do we determine, for example, the probability 01:01:10.680 --> 01:01:13.740 align:middle line:84% or the expectation value of where 01:01:13.740 --> 01:01:16.755 align:middle line:84% x is at some point from the momentum? 01:01:16.755 --> 01:01:17.880 align:middle line:90% HONG LIU: Good, good, good. 01:01:17.880 --> 01:01:20.310 align:middle line:90% That's a very good question. 01:01:20.310 --> 01:01:25.410 align:middle line:84% So this is actually also related to the kind 01:01:25.410 --> 01:01:28.960 align:middle line:84% of questions we want to solve in quantum field theory. 01:01:28.960 --> 01:01:31.320 align:middle line:84% So in quantum field theory, we often 01:01:31.320 --> 01:01:33.190 align:middle line:90% work with the vacuum state. 01:01:33.190 --> 01:01:37.210 align:middle line:90% So for example, here-- 01:01:37.210 --> 01:01:37.710 align:middle line:90% yeah. 01:01:37.710 --> 01:01:39.127 align:middle line:84% So when you can see that the QED-- 01:01:39.127 --> 01:01:44.100 align:middle line:84% most of the time in the quantum electric or magnetic field, 01:01:44.100 --> 01:01:45.750 align:middle line:90% they're in the vacuum state. 01:01:45.750 --> 01:01:49.860 align:middle line:84% And so we just consider the psi in the vacuum state. 01:01:49.860 --> 01:01:53.160 align:middle line:84% And so we don't have to consider-- 01:01:53.160 --> 01:01:53.940 align:middle line:90% yeah. 01:01:53.940 --> 01:01:54.600 align:middle line:90% Right. 01:01:54.600 --> 01:01:55.330 align:middle line:90% Yeah. 01:01:55.330 --> 01:01:57.240 align:middle line:84% And so normally in quantum field theory, 01:01:57.240 --> 01:02:02.510 align:middle line:84% there are preferred states we are interested in. 01:02:02.510 --> 01:02:06.387 align:middle line:84% But that's a lot of the reason that the Heisenberg picture 01:02:06.387 --> 01:02:06.970 align:middle line:90% is convenient. 01:02:06.970 --> 01:02:08.090 align:middle line:90% Yeah. 01:02:08.090 --> 01:02:11.140 align:middle line:84% You don't have to consider the general state, in general. 01:02:11.140 --> 01:02:12.112 align:middle line:90% Yes? 01:02:12.112 --> 01:02:14.320 align:middle line:84% AUDIENCE: So what is the physical meaning of the wave 01:02:14.320 --> 01:02:15.835 align:middle line:90% function as it goes beyond the-- 01:02:15.835 --> 01:02:16.990 align:middle line:90% HONG LIU: Hmm? 01:02:16.990 --> 01:02:17.650 align:middle line:90% Sorry? 01:02:17.650 --> 01:02:20.420 align:middle line:84% AUDIENCE: What is the physical meaning of psi here? 01:02:20.420 --> 01:02:23.600 align:middle line:84% HONG LIU: Oh, this is just some state you are interested in. 01:02:23.600 --> 01:02:25.540 align:middle line:90% Still, you have a Hilbert space. 01:02:25.540 --> 01:02:27.430 align:middle line:90% But now you just don't-- 01:02:27.430 --> 01:02:32.230 align:middle line:84% when you study the evolution, you only evolve the operator. 01:02:32.230 --> 01:02:33.370 align:middle line:90% You don't evolve states. 01:02:33.370 --> 01:02:35.620 align:middle line:84% Yeah, so that's the difference between the Schrodinger 01:02:35.620 --> 01:02:37.500 align:middle line:90% and the Heisenberg picture. 01:02:37.500 --> 01:02:40.670 align:middle line:84% AUDIENCE: But for example, in quantum mechanics, 01:02:40.670 --> 01:02:44.588 align:middle line:84% we get the expectation values of some operators with respect 01:02:44.588 --> 01:02:45.130 align:middle line:90% to the state. 01:02:45.130 --> 01:02:47.783 align:middle line:90% And we get, like, measurements-- 01:02:47.783 --> 01:02:48.700 align:middle line:90% the results and stuff. 01:02:48.700 --> 01:02:50.410 align:middle line:90% So what about here? 01:02:50.410 --> 01:02:51.910 align:middle line:84% HONG LIU: Yeah, it's the same thing. 01:02:51.910 --> 01:02:53.050 align:middle line:90% Yeah, same thing. 01:02:53.050 --> 01:02:53.890 align:middle line:90% Yeah. 01:02:53.890 --> 01:02:55.390 align:middle line:84% At this level, there's no difference 01:02:55.390 --> 01:02:57.760 align:middle line:84% between quantum field theory and quantum mechanics. 01:02:57.760 --> 01:02:59.680 align:middle line:84% Yeah, just think about quantum mechanics 01:02:59.680 --> 01:03:01.120 align:middle line:90% in terms of Heisenberg picture. 01:03:01.120 --> 01:03:03.560 align:middle line:84% Translate everything you learned about the Schrodinger 01:03:03.560 --> 01:03:07.340 align:middle line:84% evolution, et cetera, in terms of Heisenberg picture. 01:03:07.340 --> 01:03:07.998 align:middle line:90% Yes? 01:03:07.998 --> 01:03:10.040 align:middle line:84% AUDIENCE: And now the open state is conventional. 01:03:10.040 --> 01:03:10.932 align:middle line:90% HONG LIU: Hmm? 01:03:10.932 --> 01:03:12.260 align:middle line:90% AUDIENCE: The open state is-- 01:03:12.260 --> 01:03:12.980 align:middle line:90% HONG LIU: Yeah, that's right. 01:03:12.980 --> 01:03:13.522 align:middle line:90% That's right. 01:03:13.522 --> 01:03:14.420 align:middle line:90% Yeah. 01:03:14.420 --> 01:03:15.256 align:middle line:90% Yes? 01:03:15.256 --> 01:03:17.630 align:middle line:84% AUDIENCE: Am I right that the state we're interested in 01:03:17.630 --> 01:03:19.310 align:middle line:90% is generally the vacuum state. 01:03:19.310 --> 01:03:22.735 align:middle line:90% And what sorts of measurements-- 01:03:22.735 --> 01:03:23.360 align:middle line:90% HONG LIU: Yeah. 01:03:23.360 --> 01:03:28.940 align:middle line:90% So it's-- again, so-- 01:03:28.940 --> 01:03:31.410 align:middle line:90% yeah, it's a very good question. 01:03:31.410 --> 01:03:33.710 align:middle line:84% What I should have said is that the states 01:03:33.710 --> 01:03:35.720 align:middle line:84% that we are interested in are states which 01:03:35.720 --> 01:03:37.630 align:middle line:90% are close to the vacuum state. 01:03:37.630 --> 01:03:40.670 align:middle line:84% So you excite the vacuum a little bit. 01:03:40.670 --> 01:03:44.060 align:middle line:84% And yeah, it's not-- yeah, of course, in the vacuum, 01:03:44.060 --> 01:03:45.210 align:middle line:90% you don't have anything. 01:03:45.210 --> 01:03:47.700 align:middle line:84% And we are since close to the vacuum state. 01:03:47.700 --> 01:03:48.200 align:middle line:90% Yeah. 01:03:48.200 --> 01:03:50.711 align:middle line:84% And later, when we discuss things, you will see. 01:03:50.711 --> 01:03:51.211 align:middle line:90% Yeah. 01:03:51.211 --> 01:03:54.645 align:middle line:90% 01:03:54.645 --> 01:03:55.145 align:middle line:90% Good. 01:03:55.145 --> 01:03:55.640 align:middle line:90% Good. 01:03:55.640 --> 01:03:56.140 align:middle line:90% OK. 01:03:56.140 --> 01:03:59.390 align:middle line:90% 01:03:59.390 --> 01:04:16.610 align:middle line:84% So before going into quantum field theory, 01:04:16.610 --> 01:04:21.020 align:middle line:84% let's also make some remarks, have a short discussion 01:04:21.020 --> 01:04:23.490 align:middle line:84% on relativistic quantum mechanics. 01:04:23.490 --> 01:04:24.950 align:middle line:90% OK? 01:04:24.950 --> 01:04:38.210 align:middle line:84% So naively, if you have special relativity 01:04:38.210 --> 01:04:42.480 align:middle line:90% plus quantum mechanics-- 01:04:42.480 --> 01:04:45.050 align:middle line:84% so if you want to generalized-- so most of the quantum 01:04:45.050 --> 01:04:48.050 align:middle line:84% mechanics you learned is the non-relativistic quantum 01:04:48.050 --> 01:04:48.630 align:middle line:90% mechanics. 01:04:48.630 --> 01:04:49.130 align:middle line:90% OK? 01:04:49.130 --> 01:04:51.800 align:middle line:84% But now if you want to incorporate special-- 01:04:51.800 --> 01:04:53.840 align:middle line:84% we want to combine the special relativity 01:04:53.840 --> 01:04:55.370 align:middle line:90% with quantum mechanics. 01:04:55.370 --> 01:04:58.355 align:middle line:90% And then what should you get? 01:04:58.355 --> 01:04:59.730 align:middle line:84% What do you think you should get? 01:04:59.730 --> 01:05:04.221 align:middle line:90% 01:05:04.221 --> 01:05:06.097 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:06.097 --> 01:05:07.011 align:middle line:90% HONG LIU: Hmm? 01:05:07.011 --> 01:05:09.220 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:09.220 --> 01:05:10.840 align:middle line:90% HONG LIU: I expect-- 01:05:10.840 --> 01:05:13.979 align:middle line:84% yeah, just whatever come to your mind. 01:05:13.979 --> 01:05:14.854 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:14.854 --> 01:05:15.437 align:middle line:90% HONG LIU: Hmm? 01:05:15.437 --> 01:05:17.125 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:17.125 --> 01:05:18.080 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:05:18.080 --> 01:05:18.580 align:middle line:90% [LAUGHTER] 01:05:18.580 --> 01:05:21.610 align:middle line:84% HONG LIU: That's the right answer. 01:05:21.610 --> 01:05:22.990 align:middle line:90% That's the correct answer. 01:05:22.990 --> 01:05:26.830 align:middle line:84% But I was hoping some of you will say it's 01:05:26.830 --> 01:05:28.300 align:middle line:90% relativistic quantum mechanics. 01:05:28.300 --> 01:05:28.990 align:middle line:90% [LAUGHTER] 01:05:28.990 --> 01:05:30.610 align:middle line:90% OK? 01:05:30.610 --> 01:05:37.210 align:middle line:84% So naively, when you combine these two-- 01:05:37.210 --> 01:05:40.870 align:middle line:84% in particular, if you read some old quantum mechanics books, 01:05:40.870 --> 01:05:44.490 align:middle line:84% they do discuss relativistic quantum mechanics. 01:05:44.490 --> 01:05:44.990 align:middle line:90% OK? 01:05:44.990 --> 01:05:49.240 align:middle line:84% So naively, that's what you get, OK, when you combine these two. 01:05:49.240 --> 01:05:52.030 align:middle line:84% And you say, oh, we just get the relativistic quantum mechanics. 01:05:52.030 --> 01:05:53.647 align:middle line:90% OK? 01:05:53.647 --> 01:05:55.480 align:middle line:84% But that's actually not a correct statement. 01:05:55.480 --> 01:06:00.610 align:middle line:84% Actually, strictly speaking, the reason now 01:06:00.610 --> 01:06:04.510 align:middle line:84% you don't actually learn much about the relativistic quantum 01:06:04.510 --> 01:06:06.260 align:middle line:84% mechanics is because, strictly speaking, 01:06:06.260 --> 01:06:08.990 align:middle line:84% relativistic quantum mechanics does not exist. 01:06:08.990 --> 01:06:09.680 align:middle line:90% OK? 01:06:09.680 --> 01:06:13.400 align:middle line:84% And whenever-- if you want to combine quantum mechanics 01:06:13.400 --> 01:06:16.165 align:middle line:84% with special relativity, actually, 01:06:16.165 --> 01:06:17.540 align:middle line:84% you get the quantum field theory. 01:06:17.540 --> 01:06:19.165 align:middle line:84% So the quantum field theory is actually 01:06:19.165 --> 01:06:23.720 align:middle line:84% forced on us if you want to unify special relativity 01:06:23.720 --> 01:06:24.710 align:middle line:90% and quantum mechanics. 01:06:24.710 --> 01:06:28.340 align:middle line:84% OK-- and even if you don't want to talk about the fields. 01:06:28.340 --> 01:06:28.877 align:middle line:90% OK? 01:06:28.877 --> 01:06:30.710 align:middle line:84% Even if you don't want to talk about fields, 01:06:30.710 --> 01:06:33.380 align:middle line:84% if you just want to talk about particles, still, 01:06:33.380 --> 01:06:35.990 align:middle line:84% if you want to unify these two, then it actually 01:06:35.990 --> 01:06:38.930 align:middle line:84% automatically leads to quantum field theory. 01:06:38.930 --> 01:06:43.460 align:middle line:84% So now let me just explain why this is the case, OK, 01:06:43.460 --> 01:06:48.425 align:middle line:84% so that you have some better appreciation of the quantum 01:06:48.425 --> 01:06:51.350 align:middle line:90% field theory. 01:06:51.350 --> 01:06:51.950 align:middle line:90% Good? 01:06:51.950 --> 01:06:56.180 align:middle line:84% So let's try to apply what you did 01:06:56.180 --> 01:06:59.120 align:middle line:84% for non-relativistic quantum mechanics, 01:06:59.120 --> 01:07:02.430 align:middle line:84% try to generalize it to derive some relativistic quantum 01:07:02.430 --> 01:07:02.930 align:middle line:90% mechanics. 01:07:02.930 --> 01:07:03.430 align:middle line:90% OK? 01:07:03.430 --> 01:07:06.800 align:middle line:84% Suppose you are the people in 1926. 01:07:06.800 --> 01:07:07.550 align:middle line:90% OK? 01:07:07.550 --> 01:07:09.920 align:middle line:84% Quantum mechanics just was discovered. 01:07:09.920 --> 01:07:11.940 align:middle line:84% And you say, oh, people have understood 01:07:11.940 --> 01:07:13.680 align:middle line:84% the non-relativistic quantum mechanics. 01:07:13.680 --> 01:07:17.010 align:middle line:84% Now let's generalize to special relativity. 01:07:17.010 --> 01:07:17.510 align:middle line:90% OK. 01:07:17.510 --> 01:07:28.960 align:middle line:84% So in non-relativistic quantum mechanics, 01:07:28.960 --> 01:07:32.680 align:middle line:84% how do we derive the Schrodinger equation? 01:07:32.680 --> 01:07:38.410 align:middle line:84% The way we do this is we start with the dispersion relation, 01:07:38.410 --> 01:07:42.380 align:middle line:84% OK, for a non-relativistic particle. 01:07:42.380 --> 01:07:44.900 align:middle line:90% And then we say this-- 01:07:44.900 --> 01:07:50.550 align:middle line:90% take E to I partial t, OK, p-- 01:07:50.550 --> 01:07:55.270 align:middle line:84% so say, if you kind of have a vector become-- 01:07:55.270 --> 01:07:57.160 align:middle line:84% yeah, so don't let me forget about h-bar. 01:07:57.160 --> 01:07:59.950 align:middle line:84% Just p then becomes spatial derivatives. 01:07:59.950 --> 01:08:03.130 align:middle line:84% And then this equation just become the Schrodinger equation 01:08:03.130 --> 01:08:04.390 align:middle line:90% for the free particles. 01:08:04.390 --> 01:08:15.680 align:middle line:90% 01:08:15.680 --> 01:08:16.180 align:middle line:90% OK. 01:08:16.180 --> 01:08:18.250 align:middle line:84% So this becomes the Schrodinger equation 01:08:18.250 --> 01:08:20.590 align:middle line:84% for free non-relativistic particles. 01:08:20.590 --> 01:08:22.548 align:middle line:84% And then you can add the potentials, et cetera. 01:08:22.548 --> 01:08:23.048 align:middle line:90% OK. 01:08:23.048 --> 01:08:25.210 align:middle line:84% So that's how you derive your Schrodinger equation 01:08:25.210 --> 01:08:28.660 align:middle line:84% for non-relativistic quantum mechanics. 01:08:28.660 --> 01:08:31.990 align:middle line:84% But if you are someone, say, in the early days of quantum 01:08:31.990 --> 01:08:37.029 align:middle line:84% mechanics, say, now let's try to generalize it 01:08:37.029 --> 01:08:40.569 align:middle line:90% to special relativity. 01:08:40.569 --> 01:08:44.724 align:middle line:84% And then it's easy to do then in the relativistic case. 01:08:44.724 --> 01:08:49.640 align:middle line:90% 01:08:49.640 --> 01:08:52.706 align:middle line:84% Then you have-- we just start with the relativistic 01:08:52.706 --> 01:08:53.539 align:middle line:90% dispersion relation. 01:08:53.539 --> 01:08:56.100 align:middle line:90% 01:08:56.100 --> 01:08:56.600 align:middle line:90% OK? 01:08:56.600 --> 01:08:59.000 align:middle line:84% You say, let's do the same thing. 01:08:59.000 --> 01:08:59.500 align:middle line:90% OK? 01:08:59.500 --> 01:09:00.649 align:middle line:90% Let's do the same thing. 01:09:00.649 --> 01:09:03.819 align:middle line:90% So this becomes I partial t. 01:09:03.819 --> 01:09:06.490 align:middle line:90% And this p becomes this one. 01:09:06.490 --> 01:09:08.904 align:middle line:84% And then we can just write this equation. 01:09:08.904 --> 01:09:20.794 align:middle line:90% 01:09:20.794 --> 01:09:21.294 align:middle line:90% OK? 01:09:21.294 --> 01:09:25.500 align:middle line:90% 01:09:25.500 --> 01:09:28.050 align:middle line:84% So now let's combine with these derivatives together-- 01:09:28.050 --> 01:09:30.720 align:middle line:84% or put all this on the same side. 01:09:30.720 --> 01:09:32.795 align:middle line:84% And then what you find is partial mu, partial mu. 01:09:32.795 --> 01:09:42.990 align:middle line:90% 01:09:42.990 --> 01:09:43.490 align:middle line:90% OK? 01:09:43.490 --> 01:09:44.705 align:middle line:90% That's what you get. 01:09:44.705 --> 01:09:46.205 align:middle line:84% So does this equation look familiar? 01:09:46.205 --> 01:09:49.310 align:middle line:90% 01:09:49.310 --> 01:09:49.810 align:middle line:90% OK? 01:09:49.810 --> 01:09:53.800 align:middle line:84% So this is the simplest scalar field theory equation of motion 01:09:53.800 --> 01:09:55.600 align:middle line:90% we have written down. 01:09:55.600 --> 01:09:57.580 align:middle line:84% But here, the interpretation is very different. 01:09:57.580 --> 01:09:58.330 align:middle line:90% OK? 01:09:58.330 --> 01:10:00.680 align:middle line:84% So earlier, we wrote down this equation. 01:10:00.680 --> 01:10:02.395 align:middle line:84% We say this is Klein-Gordon equation. 01:10:02.395 --> 01:10:06.508 align:middle line:90% 01:10:06.508 --> 01:10:08.050 align:middle line:84% So this is the Klein-Gordon equation. 01:10:08.050 --> 01:10:08.320 align:middle line:90% OK? 01:10:08.320 --> 01:10:09.910 align:middle line:84% So this is the equation of motion 01:10:09.910 --> 01:10:12.373 align:middle line:84% for the simplest free-field scalar field 01:10:12.373 --> 01:10:13.540 align:middle line:90% theory we wrote down before. 01:10:13.540 --> 01:10:16.980 align:middle line:84% But here, the interpretation is very different. 01:10:16.980 --> 01:10:20.980 align:middle line:84% So remember, the psi is not a field in quantum mechanics. 01:10:20.980 --> 01:10:26.000 align:middle line:84% It's a wave function for a single particle sitting-- 01:10:26.000 --> 01:10:33.780 align:middle line:84% so the psi is the amplitude for a particle at spatial location 01:10:33.780 --> 01:10:35.770 align:middle line:90% x at time t. 01:10:35.770 --> 01:10:36.270 align:middle line:90% OK? 01:10:36.270 --> 01:10:38.970 align:middle line:84% So this is-- and it's square gives its probability. 01:10:38.970 --> 01:10:39.810 align:middle line:90% OK? 01:10:39.810 --> 01:10:41.500 align:middle line:84% And this does not describe a field. 01:10:41.500 --> 01:10:42.000 align:middle line:90% OK? 01:10:42.000 --> 01:10:44.850 align:middle line:84% This is a wave function of a single particle, 01:10:44.850 --> 01:10:48.600 align:middle line:84% OK, even though they have the same equation as the field 01:10:48.600 --> 01:10:53.110 align:middle line:90% theory we wrote down earlier. 01:10:53.110 --> 01:10:57.580 align:middle line:84% And so that's what Klein-Gordon did. 01:10:57.580 --> 01:10:58.080 align:middle line:90% OK? 01:10:58.080 --> 01:11:00.653 align:middle line:84% So the Klein-Gordon-- we try to generalize 01:11:00.653 --> 01:11:02.070 align:middle line:84% non-relativistic quantum mechanics 01:11:02.070 --> 01:11:03.625 align:middle line:84% to relativistic quantum mechanics. 01:11:03.625 --> 01:11:05.250 align:middle line:84% And then they wrote down this equation. 01:11:05.250 --> 01:11:08.050 align:middle line:84% They say, ah, now we are immortal. 01:11:08.050 --> 01:11:08.550 align:middle line:90% OK? 01:11:08.550 --> 01:11:09.050 align:middle line:90% [LAUGHTER] 01:11:09.050 --> 01:11:11.340 align:middle line:84% Because we wrote down the first equation 01:11:11.340 --> 01:11:17.580 align:middle line:84% for relativistic quantum mechanics. 01:11:17.580 --> 01:11:25.980 align:middle line:84% And then soon realized, actually, this equation-- 01:11:25.980 --> 01:11:31.820 align:middle line:84% if you want to interpret it as the right equation for the wave 01:11:31.820 --> 01:11:34.850 align:middle line:84% function, we have actually lots of problems. 01:11:34.850 --> 01:11:37.510 align:middle line:90% 01:11:37.510 --> 01:11:38.770 align:middle line:90% There are various problems. 01:11:38.770 --> 01:11:40.468 align:middle line:90% Yeah, by saying "lots of"-- 01:11:40.468 --> 01:11:42.510 align:middle line:84% maybe it's a little bit of an exaggeration there. 01:11:42.510 --> 01:11:45.920 align:middle line:90% There are various problems. 01:11:45.920 --> 01:11:53.050 align:middle line:90% So yeah. 01:11:53.050 --> 01:11:56.680 align:middle line:84% So some-- yeah, let me call this equation-- 01:11:56.680 --> 01:12:03.040 align:middle line:90% I think I've used up my stars. 01:12:03.040 --> 01:12:05.770 align:middle line:84% I think to have a four star is a little bit too much. 01:12:05.770 --> 01:12:10.790 align:middle line:84% Let's just call this equation "1" for this section. 01:12:10.790 --> 01:12:31.160 align:middle line:84% So some immediate difficulties of interpreting 1 01:12:31.160 --> 01:12:41.980 align:middle line:84% as a wave equation, OK, for a relativistic particle-- 01:12:41.980 --> 01:12:46.480 align:middle line:90% 01:12:46.480 --> 01:12:49.340 align:middle line:90% yeah, let me just save time. 01:12:49.340 --> 01:12:49.840 align:middle line:90% OK? 01:12:49.840 --> 01:12:54.040 align:middle line:84% So this is the wave equation for a non-relativistic particle. 01:12:54.040 --> 01:12:57.275 align:middle line:84% If you want to generalize to relativistic quantum mechanics, 01:12:57.275 --> 01:12:58.900 align:middle line:84% then you would interpret this as a wave 01:12:58.900 --> 01:13:02.500 align:middle line:84% equation for a single relativistic particle. 01:13:02.500 --> 01:13:04.240 align:middle line:90% OK. 01:13:04.240 --> 01:13:16.670 align:middle line:90% So first is that-- 01:13:16.670 --> 01:13:19.800 align:middle line:90% so if you remember-- 01:13:19.800 --> 01:13:25.210 align:middle line:90% so in this equation, OK, for-- 01:13:25.210 --> 01:13:27.580 align:middle line:84% if you go back to your early days 01:13:27.580 --> 01:13:32.590 align:middle line:84% of non-relativistic quantum mechanics-- so you remember, 01:13:32.590 --> 01:13:36.340 align:middle line:84% from this equation, you can derive 01:13:36.340 --> 01:13:39.400 align:middle line:84% an equation for the conserved probability, 01:13:39.400 --> 01:13:41.680 align:middle line:90% OK, a conserved probability. 01:13:41.680 --> 01:13:46.660 align:middle line:84% And that equation tells you that psi-squared 01:13:46.660 --> 01:13:49.930 align:middle line:84% should have the interpretation of the probability. 01:13:49.930 --> 01:13:51.700 align:middle line:90% OK? 01:13:51.700 --> 01:14:09.690 align:middle line:84% So-- but for that equation, you can show there's no quantity 01:14:09.690 --> 01:14:12.190 align:middle line:90% that can be-- 01:14:12.190 --> 01:14:20.905 align:middle line:84% that no quantity that can be used as probability density. 01:14:20.905 --> 01:14:31.410 align:middle line:90% 01:14:31.410 --> 01:14:32.100 align:middle line:90% OK? 01:14:32.100 --> 01:14:34.830 align:middle line:84% So probability density by definition 01:14:34.830 --> 01:14:37.060 align:middle line:90% should satisfy two conditions. 01:14:37.060 --> 01:14:39.360 align:middle line:84% So the first condition is that it's non-negative. 01:14:39.360 --> 01:14:44.050 align:middle line:90% 01:14:44.050 --> 01:14:47.420 align:middle line:84% And the second condition is that it should be conserved. 01:14:47.420 --> 01:14:47.920 align:middle line:90% OK? 01:14:47.920 --> 01:14:49.253 align:middle line:90% The probability should conserve. 01:14:49.253 --> 01:14:52.870 align:middle line:90% Otherwise, you violate the-- 01:14:52.870 --> 01:14:54.100 align:middle line:90% yeah. 01:14:54.100 --> 01:14:58.510 align:middle line:84% So you can show that this equation does not allow. 01:14:58.510 --> 01:15:00.230 align:middle line:84% This equation allows such a quantity, 01:15:00.230 --> 01:15:02.230 align:middle line:84% but this equation does not allow. 01:15:02.230 --> 01:15:02.860 align:middle line:90% OK? 01:15:02.860 --> 01:15:06.820 align:middle line:84% And I will not go through that myself. 01:15:06.820 --> 01:15:08.710 align:middle line:84% I think that will be in your Pset 2. 01:15:08.710 --> 01:15:10.030 align:middle line:90% OK? 01:15:10.030 --> 01:15:11.230 align:middle line:90% That will be in your Pset 2. 01:15:11.230 --> 01:15:12.325 align:middle line:90% You will show it yourself. 01:15:12.325 --> 01:15:15.710 align:middle line:90% 01:15:15.710 --> 01:15:23.350 align:middle line:84% And the second difficulty is that if you square that-- 01:15:23.350 --> 01:15:24.610 align:middle line:90% if you find the energy-- 01:15:24.610 --> 01:15:27.360 align:middle line:90% 01:15:27.360 --> 01:15:30.180 align:middle line:84% if you take the square root, then the energy in principle 01:15:30.180 --> 01:15:34.800 align:middle line:90% can be, say, plus-minus. 01:15:34.800 --> 01:15:37.010 align:middle line:90% You have two solutions. 01:15:37.010 --> 01:15:37.520 align:middle line:90% OK? 01:15:37.520 --> 01:15:42.550 align:middle line:84% E equal to plus-minus this quantity. 01:15:42.550 --> 01:15:46.690 align:middle line:84% So in contrast to that equation, there's only positive energy. 01:15:46.690 --> 01:15:47.190 align:middle line:90% OK? 01:15:47.190 --> 01:15:49.360 align:middle line:84% In non-relativistic, you just have positive energy. 01:15:49.360 --> 01:15:51.190 align:middle line:84% So classically, you can just say, 01:15:51.190 --> 01:15:53.840 align:middle line:84% let's throw away the second branch. 01:15:53.840 --> 01:15:55.040 align:middle line:90% OK? 01:15:55.040 --> 01:15:58.130 align:middle line:84% We can just throw away the second branch phi-hat, 01:15:58.130 --> 01:15:58.910 align:middle line:90% classically. 01:15:58.910 --> 01:16:01.290 align:middle line:90% But quantum mechanically-- 01:16:01.290 --> 01:16:01.790 align:middle line:90% OK? 01:16:01.790 --> 01:16:13.830 align:middle line:84% So classically, just ignore the second branch, 01:16:13.830 --> 01:16:14.730 align:middle line:90% the negative branch. 01:16:14.730 --> 01:16:21.440 align:middle line:90% 01:16:21.440 --> 01:16:21.940 align:middle line:90% OK? 01:16:21.940 --> 01:16:25.630 align:middle line:84% But quantum mechanically, this is not possible 01:16:25.630 --> 01:16:27.280 align:middle line:84% because you have the equation there. 01:16:27.280 --> 01:16:29.650 align:middle line:84% Then you automatically have the negative solutions. 01:16:29.650 --> 01:16:30.190 align:middle line:90% OK? 01:16:30.190 --> 01:16:33.100 align:middle line:84% So quantum mechanically-- and then you have a dispersion 01:16:33.100 --> 01:16:33.925 align:middle line:90% relation like this. 01:16:33.925 --> 01:16:36.630 align:middle line:90% 01:16:36.630 --> 01:16:39.120 align:middle line:84% So that dispersion relation has the following form-- 01:16:39.120 --> 01:16:42.415 align:middle line:90% is phi E as a function of p. 01:16:42.415 --> 01:16:44.170 align:middle line:84% And so you have a positive branch. 01:16:44.170 --> 01:16:46.797 align:middle line:84% And then you have a negative branch. 01:16:46.797 --> 01:16:48.630 align:middle line:84% And then you remember, in quantum mechanics, 01:16:48.630 --> 01:16:52.060 align:middle line:90% you have energy levels. 01:16:52.060 --> 01:16:55.220 align:middle line:90% And you have energy levels. 01:16:55.220 --> 01:16:58.640 align:middle line:84% And such particles-- the particles here 01:16:58.640 --> 01:17:00.860 align:middle line:84% then have a higher energy than particles here. 01:17:00.860 --> 01:17:01.430 align:middle line:90% OK? 01:17:01.430 --> 01:17:04.970 align:middle line:84% And so here, the energy level is higher than the energy level 01:17:04.970 --> 01:17:05.600 align:middle line:90% here. 01:17:05.600 --> 01:17:08.270 align:middle line:84% And there's always a non-zero probability quantum 01:17:08.270 --> 01:17:11.780 align:middle line:84% mechanically for a particle to go from some higher energy 01:17:11.780 --> 01:17:13.310 align:middle line:84% level to the lower energy level, OK, 01:17:13.310 --> 01:17:14.602 align:middle line:90% just like in the hydrogen atom. 01:17:14.602 --> 01:17:20.040 align:middle line:84% If you excite it, they always go to the lower energy level. 01:17:20.040 --> 01:17:23.460 align:middle line:84% So then quantum mechanically, such a thing cannot be avoided. 01:17:23.460 --> 01:17:23.960 align:middle line:90% OK? 01:17:23.960 --> 01:17:26.210 align:middle line:84% So you cannot just throw this branch away. 01:17:26.210 --> 01:17:26.870 align:middle line:90% OK. 01:17:26.870 --> 01:17:34.610 align:middle line:84% And so this will lead to instability 01:17:34.610 --> 01:17:39.450 align:middle line:84% because the energy can be infinitely negative. 01:17:39.450 --> 01:17:39.950 align:middle line:90% OK? 01:17:39.950 --> 01:17:41.200 align:middle line:90% It can be infinitely negative. 01:17:41.200 --> 01:17:44.192 align:middle line:84% And then all your particles will all 01:17:44.192 --> 01:17:45.650 align:middle line:84% go to infinitely negative energies. 01:17:45.650 --> 01:17:50.250 align:middle line:84% And then your system will be in big problem. 01:17:50.250 --> 01:17:52.140 align:middle line:90% OK. 01:17:52.140 --> 01:17:53.970 align:middle line:90% So these are the two-- 01:17:53.970 --> 01:18:01.000 align:middle line:90% 01:18:01.000 --> 01:18:05.560 align:middle line:90% are the most prominent problems. 01:18:05.560 --> 01:18:09.170 align:middle line:84% And then people tried many different ways 01:18:09.170 --> 01:18:12.010 align:middle line:84% to try to avoid them, et cetera, including some very 01:18:12.010 --> 01:18:13.480 align:middle line:90% ingenious solutions, et cetera. 01:18:13.480 --> 01:18:16.153 align:middle line:84% Though, we will not go into them. 01:18:16.153 --> 01:18:17.820 align:middle line:84% But let me just mention there's actually 01:18:17.820 --> 01:18:21.670 align:middle line:84% even-- if you can address those problems, 01:18:21.670 --> 01:18:24.340 align:middle line:84% still, the relativistic quantum mechanics 01:18:24.340 --> 01:18:28.280 align:middle line:84% will not make sense for a very fundamental reason. 01:18:28.280 --> 01:18:30.580 align:middle line:84% So these are more like superficial reasons 01:18:30.580 --> 01:18:32.960 align:middle line:84% why that equation does not quite work. 01:18:32.960 --> 01:18:33.460 align:middle line:90% OK? 01:18:33.460 --> 01:18:37.270 align:middle line:84% But there's actually a more fundamental reason 01:18:37.270 --> 01:18:43.810 align:middle line:84% why relativistic quantum mechanics even as a concept 01:18:43.810 --> 01:18:44.720 align:middle line:90% does not make sense. 01:18:44.720 --> 01:18:45.220 align:middle line:90% OK? 01:18:45.220 --> 01:18:51.030 align:middle line:90% 01:18:51.030 --> 01:18:57.240 align:middle line:84% So by definition, if you want to interpret this kind of thing 01:18:57.240 --> 01:18:58.990 align:middle line:90% as a wave function-- 01:18:58.990 --> 01:18:59.490 align:middle line:90% OK? 01:18:59.490 --> 01:19:01.990 align:middle line:84% So what's an interpretation of the wave function, 01:19:01.990 --> 01:19:03.390 align:middle line:90% which we already said? 01:19:03.390 --> 01:19:07.830 align:middle line:84% So this describes the amplitude for a single particle 01:19:07.830 --> 01:19:10.870 align:middle line:90% at some point, OK-- 01:19:10.870 --> 01:19:14.760 align:middle line:90% at some point x at time t. 01:19:14.760 --> 01:19:18.420 align:middle line:84% But now, if you have two particles, what do you do? 01:19:18.420 --> 01:19:24.240 align:middle line:84% You introduce the location for particle 1, and the particle 2, 01:19:24.240 --> 01:19:25.810 align:middle line:90% and the t. 01:19:25.810 --> 01:19:26.310 align:middle line:90% OK? 01:19:26.310 --> 01:19:28.047 align:middle line:90% So this describes two particles. 01:19:28.047 --> 01:19:29.880 align:middle line:84% And if you want to describe three particles, 01:19:29.880 --> 01:19:31.297 align:middle line:84% then you have to introduce more x. 01:19:31.297 --> 01:19:31.797 align:middle line:90% OK? 01:19:31.797 --> 01:19:34.230 align:middle line:84% Remember, that's what you did in non-relativistic quantum 01:19:34.230 --> 01:19:35.190 align:middle line:90% mechanics. 01:19:35.190 --> 01:19:37.050 align:middle line:84% So in non-relativistic quantum mechanics, 01:19:37.050 --> 01:19:40.140 align:middle line:84% this makes sense, OK, and just because there's 01:19:40.140 --> 01:19:44.460 align:middle line:84% no mixing between the different branches-- 01:19:44.460 --> 01:19:45.960 align:middle line:84% say, single particle, two particles. 01:19:45.960 --> 01:19:49.140 align:middle line:84% Because a particle cannot be created and destroyed 01:19:49.140 --> 01:19:53.130 align:middle line:90% in a non-relativistic system. 01:19:53.130 --> 01:19:55.590 align:middle line:84% But in a relativistic system, you 01:19:55.590 --> 01:19:57.760 align:middle line:90% can always create particles. 01:19:57.760 --> 01:19:58.260 align:middle line:90% OK? 01:19:58.260 --> 01:20:02.085 align:middle line:84% If you have enough energy, you can create new particles. 01:20:02.085 --> 01:20:04.210 align:middle line:90% A pair create electrons. 01:20:04.210 --> 01:20:04.710 align:middle line:90% OK? 01:20:04.710 --> 01:20:06.600 align:middle line:84% It happens in the accelerator all the time. 01:20:06.600 --> 01:20:10.150 align:middle line:84% Or electrons-- they can annihilate into photons. 01:20:10.150 --> 01:20:10.650 align:middle line:90% OK? 01:20:10.650 --> 01:20:15.940 align:middle line:84% So the particles are not conserved, OK, 01:20:15.940 --> 01:20:16.690 align:middle line:90% are not conserved. 01:20:16.690 --> 01:20:20.650 align:middle line:84% So this kind of wave function concept don't even make sense. 01:20:20.650 --> 01:20:24.750 align:middle line:84% So if you want to describe-- so that means, whenever you have 01:20:24.750 --> 01:20:28.260 align:middle line:84% special relativity and quantum mechanics together, 01:20:28.260 --> 01:20:32.910 align:middle line:84% you must have a framework which can describe arbitrary number 01:20:32.910 --> 01:20:35.540 align:middle line:90% of particles at the same time. 01:20:35.540 --> 01:20:36.040 align:middle line:90% OK? 01:20:36.040 --> 01:20:38.110 align:middle line:84% Because particle numbers can change all the time, 01:20:38.110 --> 01:20:41.760 align:middle line:84% OK, because of annihilation and the creation effect. 01:20:41.760 --> 01:20:47.220 align:middle line:84% And that cannot be achieved by this kind of concept. 01:20:47.220 --> 01:20:49.290 align:middle line:84% It cannot be achieved by wave function. 01:20:49.290 --> 01:20:51.810 align:middle line:84% It turns out miraculously that it can 01:20:51.810 --> 01:20:53.700 align:middle line:90% be achieved by field theory. 01:20:53.700 --> 01:20:57.750 align:middle line:84% It turns out that field theory, once you quantize it, 01:20:57.750 --> 01:21:00.750 align:middle line:84% automatically gives you a framework 01:21:00.750 --> 01:21:04.240 align:middle line:84% to describe arbitrary number of particles, 01:21:04.240 --> 01:21:08.880 align:middle line:84% OK, in the unified manner, OK-- single particle, two particles, 01:21:08.880 --> 01:21:10.170 align:middle line:90% arbitrary number of particles. 01:21:10.170 --> 01:21:12.780 align:middle line:84% And so this is one of the magic of field theory. 01:21:12.780 --> 01:21:15.870 align:middle line:84% So that's why the field theory plays such an important role 01:21:15.870 --> 01:21:21.960 align:middle line:84% in particle physics, is because the excitations of fields 01:21:21.960 --> 01:21:23.760 align:middle line:84% automatically provides the mechanism 01:21:23.760 --> 01:21:25.870 align:middle line:84% to describe arbitrary number of particles. 01:21:25.870 --> 01:21:26.370 align:middle line:90% OK. 01:21:26.370 --> 01:21:28.760 align:middle line:90% So we will stop here today. 01:21:28.760 --> 01:21:32.000 align:middle line:90%