WEBVTT 00:00:00.000 --> 00:00:06.776 align:middle line:90% [SQUEAKING][RUSTLING][CLICKING] 00:00:06.776 --> 00:00:11.140 align:middle line:90% 00:00:11.140 --> 00:00:12.520 align:middle line:90% HONG LIU: OK. 00:00:12.520 --> 00:00:14.240 align:middle line:90% Let us start. 00:00:14.240 --> 00:00:19.600 align:middle line:84% So last time, we started talking about interacting theories. 00:00:19.600 --> 00:00:24.790 align:middle line:84% And here is the simplest interacting theory 00:00:24.790 --> 00:00:28.600 align:middle line:84% with the Lagrangian density of the form. 00:00:28.600 --> 00:00:38.360 align:middle line:90% 00:00:38.360 --> 00:00:38.860 align:middle line:90% OK. 00:00:38.860 --> 00:00:41.560 align:middle line:84% So this is just the simplest one. 00:00:41.560 --> 00:00:44.590 align:middle line:84% And we will use this as our example 00:00:44.590 --> 00:00:50.080 align:middle line:84% to illustrate how you treat interacting theories, OK-- 00:00:50.080 --> 00:00:53.770 align:middle line:84% how you treat the particles with interactions. 00:00:53.770 --> 00:00:56.170 align:middle line:84% Once we understand how to do one theory, 00:00:56.170 --> 00:01:01.820 align:middle line:84% essentially you will know how to do all of them. 00:01:01.820 --> 00:01:06.370 align:middle line:90% And we also said that the-- 00:01:06.370 --> 00:01:13.180 align:middle line:84% a key quantity when talking about interactions-- 00:01:13.180 --> 00:01:14.620 align:middle line:90% a key observable-- 00:01:14.620 --> 00:01:16.960 align:middle line:90% is called S-matrix. 00:01:16.960 --> 00:01:23.630 align:middle line:84% So S-matrix is defined as the following. 00:01:23.630 --> 00:01:26.650 align:middle line:90% So S beta alpha beta-- 00:01:26.650 --> 00:01:35.740 align:middle line:84% beta alpha equal to beta U plus infinity minus infinity alpha. 00:01:35.740 --> 00:01:39.570 align:middle line:84% So you start with some alpha initial state, 00:01:39.570 --> 00:01:41.360 align:middle line:84% which are widely separated, which 00:01:41.360 --> 00:01:45.850 align:middle line:84% you can treat as a collections of free particles. 00:01:45.850 --> 00:01:50.530 align:middle line:84% And then say as T goes to minus infinity, 00:01:50.530 --> 00:01:53.350 align:middle line:90% they are very widely separated. 00:01:53.350 --> 00:01:56.080 align:middle line:84% But then as time evolves, they will come together, 00:01:56.080 --> 00:01:57.790 align:middle line:90% the scatter. 00:01:57.790 --> 00:02:01.430 align:middle line:84% Then they will separate into far future. 00:02:01.430 --> 00:02:04.460 align:middle line:84% And so that's our final state beta. 00:02:04.460 --> 00:02:07.480 align:middle line:84% And so S alpha beta is the transition amplitude 00:02:07.480 --> 00:02:08.440 align:middle line:90% between the two. 00:02:08.440 --> 00:02:08.940 align:middle line:90% OK? 00:02:08.940 --> 00:02:12.130 align:middle line:84% So this is the written in terms of the Schrodinger picture. 00:02:12.130 --> 00:02:18.350 align:middle line:84% You can also written in terms of the Heisenberg picture, 00:02:18.350 --> 00:02:21.110 align:middle line:84% which is normally written in this form. 00:02:21.110 --> 00:02:21.610 align:middle line:90% OK. 00:02:21.610 --> 00:02:26.300 align:middle line:90% 00:02:26.300 --> 00:02:27.860 align:middle line:90% So-- and we often-- 00:02:27.860 --> 00:02:30.590 align:middle line:90% 00:02:30.590 --> 00:02:34.070 align:middle line:84% so it's often convenient to write 00:02:34.070 --> 00:02:38.750 align:middle line:84% this S alpha beta separated into identity part delta 00:02:38.750 --> 00:02:43.310 align:middle line:84% beta alpha and then plus i T beta alpha. 00:02:43.310 --> 00:02:46.920 align:middle line:84% So this will capture the effect of the interactions. 00:02:46.920 --> 00:02:49.260 align:middle line:84% We'll capture the effect of the interactions. 00:02:49.260 --> 00:02:54.470 align:middle line:90% And so-- and since the-- 00:02:54.470 --> 00:02:57.350 align:middle line:84% such as-- yeah, we always consider 00:02:57.350 --> 00:03:02.670 align:middle line:84% theories which are both spatial and the time translation 00:03:02.670 --> 00:03:03.170 align:middle line:90% invariant. 00:03:03.170 --> 00:03:06.440 align:middle line:84% So you have energy momentum conservation. 00:03:06.440 --> 00:03:10.550 align:middle line:90% And so i T alpha-- 00:03:10.550 --> 00:03:16.840 align:middle line:84% i T beta alpha will always contain a piece proportional-- 00:03:16.840 --> 00:03:19.060 align:middle line:90% we also put i here-- 00:03:19.060 --> 00:03:25.810 align:middle line:90% proportional to the moment-- 00:03:25.810 --> 00:03:27.532 align:middle line:90% the momentum conservation. 00:03:27.532 --> 00:03:34.420 align:middle line:90% 00:03:34.420 --> 00:03:35.080 align:middle line:90% OK. 00:03:35.080 --> 00:03:38.620 align:middle line:84% So p alpha-- and the p alpha is the total initial momentum. 00:03:38.620 --> 00:03:46.100 align:middle line:90% 00:03:46.100 --> 00:03:47.860 align:middle line:90% So here is full momentum, OK? 00:03:47.860 --> 00:03:49.570 align:middle line:90% So I suppress the indices. 00:03:49.570 --> 00:03:54.790 align:middle line:84% And the p beta is the total final momentum. 00:03:54.790 --> 00:04:00.770 align:middle line:90% 00:04:00.770 --> 00:04:03.250 align:middle line:84% So this part is purely kinematic. 00:04:03.250 --> 00:04:06.290 align:middle line:84% So the M alpha beta is the physical part. 00:04:06.290 --> 00:04:06.790 align:middle line:90% OK. 00:04:06.790 --> 00:04:09.640 align:middle line:84% So this is normally called the scattering amplitude. 00:04:09.640 --> 00:04:20.380 align:middle line:90% 00:04:20.380 --> 00:04:22.210 align:middle line:90% OK. 00:04:22.210 --> 00:04:25.900 align:middle line:84% So our goal is to be able to calculate the scattering 00:04:25.900 --> 00:04:27.290 align:middle line:90% amplitude. 00:04:27.290 --> 00:04:27.790 align:middle line:90% OK. 00:04:27.790 --> 00:04:31.550 align:middle line:84% So this is something-- some version 00:04:31.550 --> 00:04:33.050 align:middle line:84% of the scattering amplitude that can 00:04:33.050 --> 00:04:34.650 align:middle line:90% be measured in the experiments. 00:04:34.650 --> 00:04:37.310 align:middle line:84% So theoretically, we want to be able to calculate 00:04:37.310 --> 00:04:38.700 align:middle line:90% this quantity. 00:04:38.700 --> 00:04:42.320 align:middle line:84% So they'll be able to compare with the experiments. 00:04:42.320 --> 00:04:45.225 align:middle line:90% So we also mentioned-- 00:04:45.225 --> 00:04:47.240 align:middle line:84% I will not repeat here-- we also mentioned 00:04:47.240 --> 00:04:50.330 align:middle line:84% that this S alpha beta, S beta alpha 00:04:50.330 --> 00:04:54.770 align:middle line:84% also have various properties, which 00:04:54.770 --> 00:05:02.310 align:middle line:84% you should go back to your notes of last class to review them. 00:05:02.310 --> 00:05:08.570 align:middle line:84% So one of the key points is the LSZ theorem, 00:05:08.570 --> 00:05:11.620 align:middle line:84% which said, which I just mentioned, 00:05:11.620 --> 00:05:24.360 align:middle line:84% that M alpha beta can be obtained from a correlation 00:05:24.360 --> 00:05:25.950 align:middle line:90% function, like this. 00:05:25.950 --> 00:05:34.250 align:middle line:84% Say, for example, for that theory, OK? 00:05:34.250 --> 00:05:36.770 align:middle line:84% So the n would correspond to the total number 00:05:36.770 --> 00:05:40.070 align:middle line:84% of particles of the initial plus final states. 00:05:40.070 --> 00:05:42.470 align:middle line:84% So suppose you have 2 to 2 scattering, 00:05:42.470 --> 00:05:45.260 align:middle line:90% and then n would be equal to 4. 00:05:45.260 --> 00:05:47.720 align:middle line:84% So if you start from 2 particles, 00:05:47.720 --> 00:05:50.450 align:middle line:84% you scattering to 3 particles, then n will be 5. 00:05:50.450 --> 00:05:53.240 align:middle line:90% OK, just the total number, yeah. 00:05:53.240 --> 00:05:56.360 align:middle line:84% So later, we will discuss, in more detail, how you 00:05:56.360 --> 00:05:59.240 align:middle line:90% obtain M alpha beta from here. 00:05:59.240 --> 00:06:01.850 align:middle line:84% But here, at the moment, it's enough for us 00:06:01.850 --> 00:06:05.300 align:middle line:84% to know that the scattering amplitude can be 00:06:05.300 --> 00:06:07.410 align:middle line:90% extracted from this quantity. 00:06:07.410 --> 00:06:10.682 align:middle line:90% Oh, sorry, I should-- 00:06:10.682 --> 00:06:12.140 align:middle line:84% So here, it should be time ordered. 00:06:12.140 --> 00:06:19.700 align:middle line:90% 00:06:19.700 --> 00:06:23.250 align:middle line:84% And so let me just explain the notation here. 00:06:23.250 --> 00:06:30.500 align:middle line:84% So the omega is the vacuum of your interacting theory. 00:06:30.500 --> 00:06:32.870 align:middle line:90% And then the T is time ordering. 00:06:32.870 --> 00:06:41.180 align:middle line:90% 00:06:41.180 --> 00:06:45.020 align:middle line:84% It means that, whoever with the larger time, 00:06:45.020 --> 00:06:48.350 align:middle line:90% it will appear earlier. 00:06:48.350 --> 00:06:51.330 align:middle line:84% Whoever with a larger time will appear earlier. 00:06:51.330 --> 00:06:53.780 align:middle line:90% And so any questions on this? 00:06:53.780 --> 00:07:00.530 align:middle line:90% 00:07:00.530 --> 00:07:04.960 align:middle line:84% So our goal would be to tell you how to calculate this quantity. 00:07:04.960 --> 00:07:07.010 align:middle line:84% So once you know how to calculate this quantity, 00:07:07.010 --> 00:07:09.010 align:middle line:84% then we will be able to calculate the scattering 00:07:09.010 --> 00:07:09.700 align:middle line:90% amplitude. 00:07:09.700 --> 00:07:13.840 align:middle line:84% And then you will be able to compare with experiments. 00:07:13.840 --> 00:07:18.680 align:middle line:90% And good. 00:07:18.680 --> 00:07:22.940 align:middle line:84% So in principle, we already know how to calculate this quantity 00:07:22.940 --> 00:07:25.550 align:middle line:90% from what you learned in 8.06. 00:07:25.550 --> 00:07:30.170 align:middle line:84% So for people who have taken 8.06, here, you learned. 00:07:30.170 --> 00:07:33.470 align:middle line:84% Yeah, so we mentioned last time, this theory cannot be solved, 00:07:33.470 --> 00:07:34.340 align:middle line:90% exactly. 00:07:34.340 --> 00:07:36.230 align:middle line:84% And the only way we can treat this theory 00:07:36.230 --> 00:07:42.140 align:middle line:84% is to take lambda small and then treat this as a perturbation, 00:07:42.140 --> 00:07:43.910 align:middle line:84% treat this as a small perturbation, 00:07:43.910 --> 00:07:46.640 align:middle line:84% and then expand all your physical quantities 00:07:46.640 --> 00:07:48.080 align:middle line:90% in power series of lambda. 00:07:48.080 --> 00:07:50.940 align:middle line:90% 00:07:50.940 --> 00:07:53.780 align:middle line:84% So this procedure is called perturbation theory. 00:07:53.780 --> 00:07:58.190 align:middle line:84% And in your quantum mechanics class, 00:07:58.190 --> 00:08:03.140 align:middle line:84% you should have already learned how to do perturbation theory. 00:08:03.140 --> 00:08:07.790 align:middle line:84% For example, in 8.06, that's a big part of the 8.06 class. 00:08:07.790 --> 00:08:12.440 align:middle line:84% Anyway, so just with 8.06, in principle, you already 00:08:12.440 --> 00:08:13.410 align:middle line:90% know how to do this. 00:08:13.410 --> 00:08:15.086 align:middle line:84% Let me just outline the procedure-- 00:08:15.086 --> 00:08:22.350 align:middle line:90% 00:08:22.350 --> 00:08:23.440 align:middle line:90% a computation. 00:08:23.440 --> 00:08:28.980 align:middle line:90% 00:08:28.980 --> 00:08:31.470 align:middle line:84% So that is a correlation function. 00:08:31.470 --> 00:08:35.045 align:middle line:84% So it's called a time ordered correlation function. 00:08:35.045 --> 00:08:51.190 align:middle line:90% 00:08:51.190 --> 00:08:55.880 align:middle line:84% So what I will call this is a naive perturbation theory. 00:08:55.880 --> 00:08:58.380 align:middle line:84% So this is the perturbation theory 00:08:58.380 --> 00:09:03.530 align:middle line:84% you would do after you learned 8.06, OK, 00:09:03.530 --> 00:09:05.147 align:middle line:84% after you learned 8.06, and then you 00:09:05.147 --> 00:09:06.980 align:middle line:84% will be able to do this perturbation theory. 00:09:06.980 --> 00:09:12.350 align:middle line:90% 00:09:12.350 --> 00:09:16.640 align:middle line:84% But we will do something more clever, OK? 00:09:16.640 --> 00:09:19.460 align:middle line:84% We will do something more clever. 00:09:19.460 --> 00:09:22.610 align:middle line:84% But let me just remind you, equipped 00:09:22.610 --> 00:09:24.980 align:middle line:84% with your previous quantum mechanical knowledge, 00:09:24.980 --> 00:09:27.890 align:middle line:84% how would you treat this problem? 00:09:27.890 --> 00:09:34.030 align:middle line:84% So now let's consider, again, that theory as example. 00:09:34.030 --> 00:09:38.410 align:middle line:84% So the equation of motion gives you 00:09:38.410 --> 00:09:44.290 align:middle line:84% partial square, minus m square phi equal to, say, lambda 00:09:44.290 --> 00:09:45.580 align:middle line:90% divided by 6 phi cubed. 00:09:45.580 --> 00:09:48.690 align:middle line:90% 00:09:48.690 --> 00:09:57.060 align:middle line:84% And so we can solve this equation even though we cannot 00:09:57.060 --> 00:09:59.430 align:middle line:90% solve this equation exactly. 00:09:59.430 --> 00:10:01.620 align:middle line:84% We can solve it for perturbatively in lambda. 00:10:01.620 --> 00:10:09.960 align:middle line:90% 00:10:09.960 --> 00:10:12.370 align:middle line:84% So the way we do it is the following. 00:10:12.370 --> 00:10:19.230 align:middle line:84% We write phi equal to phi 0, and the lambda phi 1 plus lambda 00:10:19.230 --> 00:10:22.260 align:middle line:90% squared phi 2, et cetera. 00:10:22.260 --> 00:10:24.030 align:middle line:90% So lambda is a small number. 00:10:24.030 --> 00:10:29.460 align:middle line:84% You imagine you expand phi in power series of lambda. 00:10:29.460 --> 00:10:31.680 align:middle line:84% And then you just plug in this power series 00:10:31.680 --> 00:10:36.210 align:middle line:84% into this equation, and then you equate two side 00:10:36.210 --> 00:10:38.640 align:middle line:90% with the same power of lambda. 00:10:38.640 --> 00:10:39.930 align:middle line:90% With the same power of lambda? 00:10:39.930 --> 00:10:46.410 align:middle line:84% So the 0th order, you just get our previous equation, 00:10:46.410 --> 00:10:47.790 align:middle line:90% phi 0 equal to 0. 00:10:47.790 --> 00:10:52.030 align:middle line:84% So phi 0 is just like our previous free field equation. 00:10:52.030 --> 00:10:55.360 align:middle line:90% 00:10:55.360 --> 00:11:01.750 align:middle line:84% And then at 1st order, then you see phi 1, 00:11:01.750 --> 00:11:07.540 align:middle line:84% and now you have 1 over 6 phi 0 to the cube, and then 00:11:07.540 --> 00:11:08.050 align:middle line:90% et cetera. 00:11:08.050 --> 00:11:11.880 align:middle line:90% 00:11:11.880 --> 00:11:14.850 align:middle line:84% So this equation, we know how to solve. 00:11:14.850 --> 00:11:18.130 align:middle line:84% And once you know how to solve this equation, 00:11:18.130 --> 00:11:20.070 align:middle line:84% you find a solution for phi 0, then 00:11:20.070 --> 00:11:21.810 align:middle line:90% you plug it in this equation. 00:11:21.810 --> 00:11:23.820 align:middle line:84% And now this just becomes a linear equation. 00:11:23.820 --> 00:11:25.290 align:middle line:90% Now you can solve phi 1. 00:11:25.290 --> 00:11:27.870 align:middle line:84% And similarly, you can solve phi 2, et cetera. 00:11:27.870 --> 00:11:30.270 align:middle line:84% So order by order, you can solve all of them. 00:11:30.270 --> 00:11:33.780 align:middle line:84% And all of them become linear equations. 00:11:33.780 --> 00:11:40.230 align:middle line:90% 00:11:40.230 --> 00:11:50.670 align:middle line:84% So phi 1 -- all higher orders can be solved in terms of phi 00:11:50.670 --> 00:11:51.750 align:middle line:90% 0. 00:11:51.750 --> 00:11:56.640 align:middle line:84% So once you find the phi 0, and then phi 0 00:11:56.640 --> 00:11:58.200 align:middle line:90% can be solved in terms of phi 0. 00:11:58.200 --> 00:12:05.650 align:middle line:90% 00:12:05.650 --> 00:12:09.290 align:middle line:84% So classically, this gives you a solution. 00:12:09.290 --> 00:12:12.550 align:middle line:84% And as we said, quantum mechanically, we just 00:12:12.550 --> 00:12:16.630 align:middle line:84% treat this, treat all of this as operator equations. 00:12:16.630 --> 00:12:20.350 align:middle line:84% And then, essentially, you can just 00:12:20.350 --> 00:12:24.130 align:middle line:84% solve the phi, perturbatively, as the operator equation-- 00:12:24.130 --> 00:12:26.140 align:middle line:90% as a quantum operator. 00:12:26.140 --> 00:12:29.800 align:middle line:84% And then you solve this quantum operator equation, 00:12:29.800 --> 00:12:31.090 align:middle line:90% perturbatively. 00:12:31.090 --> 00:12:45.840 align:middle line:90% 00:12:45.840 --> 00:12:58.270 align:middle line:84% So similarly, we can solve this omega 00:12:58.270 --> 00:13:00.410 align:middle line:90% using perturbation theory. 00:13:00.410 --> 00:13:02.050 align:middle line:90% So first, we expand again. 00:13:02.050 --> 00:13:05.600 align:middle line:90% We expand this omega in power. 00:13:05.600 --> 00:13:08.170 align:middle line:84% So this is the vacuum of the interacting theory. 00:13:08.170 --> 00:13:12.685 align:middle line:84% And we expand it into a perturbative series. 00:13:12.685 --> 00:13:26.230 align:middle line:90% 00:13:26.230 --> 00:13:31.610 align:middle line:84% So the 0th order, you just get the free theory vacuum. 00:13:31.610 --> 00:13:35.600 align:middle line:84% And then you will have corrections from the lambda, 00:13:35.600 --> 00:13:38.780 align:middle line:84% here, due to you have interactions 00:13:38.780 --> 00:13:41.360 align:middle line:84% and because your Hamiltonian change. 00:13:41.360 --> 00:13:43.620 align:middle line:84% Because of the lambda, your Hamiltonian change. 00:13:43.620 --> 00:13:45.990 align:middle line:84% And so your ground state will also change. 00:13:45.990 --> 00:13:49.340 align:middle line:84% And so, now, you can use your 8.06 00:13:49.340 --> 00:13:54.680 align:middle line:84% to work out those higher order corrections. 00:13:54.680 --> 00:14:01.150 align:middle line:84% Yeah, so now when you worked out the phi, 00:14:01.150 --> 00:14:03.940 align:middle line:84% when you worked out the omega, now, 00:14:03.940 --> 00:14:06.140 align:middle line:90% you can work out this object. 00:14:06.140 --> 00:14:09.580 align:middle line:90% So let me just call this G n. 00:14:09.580 --> 00:14:23.000 align:middle line:84% So the G n then can be reduced to a calculation free theory. 00:14:23.000 --> 00:14:33.060 align:middle line:90% 00:14:33.060 --> 00:14:35.190 align:middle line:84% And you need perturbation theory, 00:14:35.190 --> 00:14:38.340 align:middle line:84% because everything can be expressed in terms of phi 0. 00:14:38.340 --> 00:14:41.250 align:middle line:84% And here, again, you expand in terms of phi 0. 00:14:41.250 --> 00:14:45.780 align:middle line:84% And the corrections can be obtained by, say, phi 0 00:14:45.780 --> 00:14:48.550 align:middle line:84% acting on a vacuum state, et cetera. 00:14:48.550 --> 00:14:51.240 align:middle line:84% So in the end, if you do this procedure, 00:14:51.240 --> 00:14:55.080 align:middle line:84% then you should be able to calculate this G n. 00:14:55.080 --> 00:14:57.750 align:middle line:84% Conceptually, it's not very difficult to calculate. 00:14:57.750 --> 00:15:01.450 align:middle line:90% 00:15:01.450 --> 00:15:03.850 align:middle line:84% But this procedure is actually very complicated. 00:15:03.850 --> 00:15:08.590 align:middle line:90% 00:15:08.590 --> 00:15:10.730 align:middle line:84% Yeah, I don't urge you to try it yourself. 00:15:10.730 --> 00:15:13.270 align:middle line:84% But I think you should try just to get 00:15:13.270 --> 00:15:18.665 align:middle line:84% the sense that this procedure is actually not very easy to do. 00:15:18.665 --> 00:15:20.290 align:middle line:84% So that's why, in quantum field theory, 00:15:20.290 --> 00:15:24.370 align:middle line:84% we actually need to develop a more sophisticated technique 00:15:24.370 --> 00:15:25.810 align:middle line:90% to treat this. 00:15:25.810 --> 00:15:29.770 align:middle line:84% Because that's what people did in the early days. 00:15:29.770 --> 00:15:34.268 align:middle line:84% In the early days, before systematic perturbation 00:15:34.268 --> 00:15:36.310 align:middle line:84% technique for quantum field theory was developed, 00:15:36.310 --> 00:15:38.470 align:middle line:84% that's how people did it in the past. 00:15:38.470 --> 00:15:41.590 align:middle line:90% 00:15:41.590 --> 00:15:42.790 align:middle line:90% And that was not easy. 00:15:42.790 --> 00:15:45.070 align:middle line:90% But people did it. 00:15:45.070 --> 00:15:47.650 align:middle line:90% 00:15:47.650 --> 00:15:49.600 align:middle line:84% So if you want to calculate, the 1st order, 00:15:49.600 --> 00:15:50.590 align:middle line:90% it's actually doable. 00:15:50.590 --> 00:15:55.600 align:middle line:84% But the higher orders become increasingly complicated. 00:15:55.600 --> 00:16:04.312 align:middle line:84% And quickly becomes very complicated 00:16:04.312 --> 00:16:05.520 align:middle line:90% when you go to higher orders. 00:16:05.520 --> 00:16:21.910 align:middle line:90% 00:16:21.910 --> 00:16:27.610 align:middle line:84% So eventually, people developed something more clever. 00:16:27.610 --> 00:16:32.040 align:middle line:84% So eventually, people developed something more clever. 00:16:32.040 --> 00:16:35.240 align:middle line:90% So there are two approaches. 00:16:35.240 --> 00:16:40.730 align:middle line:84% There are two more sophisticated approaches, which 00:16:40.730 --> 00:16:44.490 align:middle line:90% are, in the end, equivalent. 00:16:44.490 --> 00:16:48.320 align:middle line:84% OK, in the end, they're equivalent, 00:16:48.320 --> 00:16:51.680 align:middle line:84% but their derivation is different. 00:16:51.680 --> 00:16:55.940 align:middle line:84% So the first is you use so-called interaction picture. 00:16:55.940 --> 00:17:05.430 align:middle line:90% 00:17:05.430 --> 00:17:08.359 align:middle line:84% You use so-called interaction picture. 00:17:08.359 --> 00:17:10.682 align:middle line:84% So what do you do is the following. 00:17:10.682 --> 00:17:12.849 align:middle line:84% So you separate your Hamiltonian, total Hamiltonian, 00:17:12.849 --> 00:17:15.970 align:middle line:84% in terms of a free theory Hamiltonian and the interacting 00:17:15.970 --> 00:17:17.752 align:middle line:90% part. 00:17:17.752 --> 00:17:19.210 align:middle line:84% So the free theory Hamiltonian just 00:17:19.210 --> 00:17:30.500 align:middle line:84% comes from, say, if I write this as L0 minus that plus L I-- 00:17:30.500 --> 00:17:36.220 align:middle line:84% so this term is L I, so this part is L0. 00:17:36.220 --> 00:17:40.840 align:middle line:84% And so similarly, you can write your Hamiltonian-- also 00:17:40.840 --> 00:17:44.140 align:middle line:84% separate into the free theory Hamiltonian and the interacting 00:17:44.140 --> 00:17:45.910 align:middle line:90% part. 00:17:45.910 --> 00:17:49.630 align:middle line:84% And then the interacting picture-- 00:17:49.630 --> 00:17:53.440 align:middle line:84% it introduces the state into interacting picture, 00:17:53.440 --> 00:17:58.090 align:middle line:84% and you introduce the operator in the interacting picture. 00:17:58.090 --> 00:18:00.760 align:middle line:84% So the state, in the interacting picture, 00:18:00.760 --> 00:18:09.290 align:middle line:84% are related by the standard state in the Schrodinger 00:18:09.290 --> 00:18:09.790 align:middle line:90% picture. 00:18:09.790 --> 00:18:18.960 align:middle line:90% 00:18:18.960 --> 00:18:21.420 align:middle line:90% So let me just put S here. 00:18:21.420 --> 00:18:24.150 align:middle line:84% So this denotes the states in the Schrodinger picture, 00:18:24.150 --> 00:18:26.400 align:middle line:90% through this factor of H0-- 00:18:26.400 --> 00:18:29.550 align:middle line:90% basically, i H0 t. 00:18:29.550 --> 00:18:34.050 align:middle line:84% And the operators in the interacting picture 00:18:34.050 --> 00:18:41.490 align:middle line:84% are related to the operator in the Schrodinger picture, 00:18:41.490 --> 00:18:43.980 align:middle line:84% also, through this factor of the exponential of H0. 00:18:43.980 --> 00:18:51.323 align:middle line:90% 00:18:51.323 --> 00:18:52.740 align:middle line:84% Already in your quantum mechanics, 00:18:52.740 --> 00:18:54.990 align:middle line:84% you may have discussed the interaction picture. 00:18:54.990 --> 00:18:57.120 align:middle line:84% Have you seen the interaction picture before? 00:18:57.120 --> 00:18:57.720 align:middle line:90% Yeah? 00:18:57.720 --> 00:19:00.480 align:middle line:90% Yeah. 00:19:00.480 --> 00:19:04.200 align:middle line:84% So remember this evolves with the full Hamiltonian. 00:19:04.200 --> 00:19:06.600 align:middle line:84% In the Schrodinger picture, this evolves 00:19:06.600 --> 00:19:08.460 align:middle line:90% with the full Hamiltonian. 00:19:08.460 --> 00:19:13.350 align:middle line:84% And so, here, the interaction picture 00:19:13.350 --> 00:19:16.680 align:middle line:84% related this by evolution operator-- conjugate 00:19:16.680 --> 00:19:20.040 align:middle line:84% of the evolution operator for the free Hamiltonian. 00:19:20.040 --> 00:19:23.790 align:middle line:84% So this Psi I is still evolved in a complicated way. 00:19:23.790 --> 00:19:26.070 align:middle line:84% So in the interacting picture, the states still 00:19:26.070 --> 00:19:27.810 align:middle line:90% evolves in a complicated way. 00:19:27.810 --> 00:19:29.910 align:middle line:84% But the operator evolve very simply, 00:19:29.910 --> 00:19:32.790 align:middle line:84% because, in the Schrodinger picture, 00:19:32.790 --> 00:19:34.530 align:middle line:90% the operator don't evolve. 00:19:34.530 --> 00:19:37.020 align:middle line:84% And here, the evolution just controlled 00:19:37.020 --> 00:19:39.900 align:middle line:90% by the free Hamiltonian. 00:19:39.900 --> 00:19:42.210 align:middle line:84% So in the interacting picture, the states 00:19:42.210 --> 00:19:45.750 align:middle line:84% still evolve, complicatedly, but the operator 00:19:45.750 --> 00:19:47.790 align:middle line:90% evolves in a simple way. 00:19:47.790 --> 00:19:53.100 align:middle line:84% Anyway, with some manipulation, by introducing this interacting 00:19:53.100 --> 00:20:00.420 align:middle line:84% picture, you can design a more clever way to do this, 00:20:00.420 --> 00:20:03.330 align:middle line:90% to do the perturbation theory. 00:20:03.330 --> 00:20:05.550 align:middle line:84% And so I will not go through it, here. 00:20:05.550 --> 00:20:15.690 align:middle line:84% You should read the book, Peskin and Schroeder, section 4.2. 00:20:15.690 --> 00:20:18.900 align:middle line:84% And there they give a detailed discussion 00:20:18.900 --> 00:20:25.530 align:middle line:84% how you do perturbation theory using this interacting picture. 00:20:25.530 --> 00:20:29.940 align:middle line:84% So a second approach is to use the path integral. 00:20:29.940 --> 00:20:37.620 align:middle line:90% 00:20:37.620 --> 00:20:39.600 align:middle line:90% And just use path integral. 00:20:39.600 --> 00:20:43.080 align:middle line:84% So path integrals, in my own opinion, 00:20:43.080 --> 00:20:48.790 align:middle line:84% have lots of advantage over this interaction picture. 00:20:48.790 --> 00:20:50.310 align:middle line:84% So when you do interaction picture, 00:20:50.310 --> 00:20:52.200 align:middle line:84% you get the feeling, somehow, you're 00:20:52.200 --> 00:20:54.720 align:middle line:90% doing some very clever tricks. 00:20:54.720 --> 00:20:58.050 align:middle line:84% And you don't know why you want to do those tricks. 00:20:58.050 --> 00:21:01.948 align:middle line:90% And you get some nice formulas. 00:21:01.948 --> 00:21:03.990 align:middle line:84% But somehow, you feel you're doing a little bit-- 00:21:03.990 --> 00:21:06.870 align:middle line:90% 00:21:06.870 --> 00:21:09.300 align:middle line:84% yeah, you're doing some kind of magic. 00:21:09.300 --> 00:21:12.780 align:middle line:84% But the path integral treatment is 00:21:12.780 --> 00:21:15.030 align:middle line:84% automatic in the sense it's straightforward. 00:21:15.030 --> 00:21:16.680 align:middle line:90% And you don't have to think. 00:21:16.680 --> 00:21:19.020 align:middle line:84% And it just follows from the path integral. 00:21:19.020 --> 00:21:21.270 align:middle line:84% And it's much easier to generalize. 00:21:21.270 --> 00:21:23.160 align:middle line:90% And it's physically intuitive. 00:21:23.160 --> 00:21:32.830 align:middle line:84% And so that's why we will describe this approach. 00:21:32.830 --> 00:21:35.350 align:middle line:84% Because it's more physically intuitive. 00:21:35.350 --> 00:21:37.870 align:middle line:84% Sometimes, when you do too much fancy mathematics, 00:21:37.870 --> 00:21:41.320 align:middle line:84% you feel you obscure the physical picture. 00:21:41.320 --> 00:21:44.672 align:middle line:84% And yeah, this interaction picture 00:21:44.672 --> 00:21:46.630 align:middle line:84% gives you a little bit of that kind of feeling. 00:21:46.630 --> 00:21:50.620 align:middle line:90% 00:21:50.620 --> 00:21:52.810 align:middle line:84% But it's important for you to know it. 00:21:52.810 --> 00:21:54.700 align:middle line:84% We don't have time to cover both. 00:21:54.700 --> 00:21:58.720 align:middle line:84% And so we choose just to do this in class. 00:21:58.720 --> 00:21:59.908 align:middle line:90% Yes, you have a question? 00:21:59.908 --> 00:22:01.450 align:middle line:84% AUDIENCE: In the interaction picture, 00:22:01.450 --> 00:22:03.740 align:middle line:84% why do both state and the operator 00:22:03.740 --> 00:22:05.450 align:middle line:90% evolve according to H sub 0? 00:22:05.450 --> 00:22:07.108 align:middle line:84% Shouldn't the state evolve according 00:22:07.108 --> 00:22:08.400 align:middle line:90% to the interaction Hamiltonian? 00:22:08.400 --> 00:22:09.025 align:middle line:90% HONG LIU: Yeah. 00:22:09.025 --> 00:22:11.640 align:middle line:90% Yeah, just by design. 00:22:11.640 --> 00:22:12.450 align:middle line:90% AUDIENCE: Oh, OK. 00:22:12.450 --> 00:22:14.550 align:middle line:84% HONG LIU: I just defined them this way. 00:22:14.550 --> 00:22:15.258 align:middle line:90% AUDIENCE: Oh, OK. 00:22:15.258 --> 00:22:16.883 align:middle line:84% HONG LIU: I just defined them this way. 00:22:16.883 --> 00:22:19.170 align:middle line:84% Turns out the quantity defined this way is actually 00:22:19.170 --> 00:22:23.370 align:middle line:84% convenient for doing perturbation theory. 00:22:23.370 --> 00:22:26.070 align:middle line:84% So that's why I say there's a little bit of magic here. 00:22:26.070 --> 00:22:29.640 align:middle line:84% And you have to introduce this kind of unintuitive quantities, 00:22:29.640 --> 00:22:33.360 align:middle line:84% which turns out actually to be quite simple when 00:22:33.360 --> 00:22:35.020 align:middle line:90% you do perturbation theory. 00:22:35.020 --> 00:22:35.520 align:middle line:90% Yeah. 00:22:35.520 --> 00:22:39.470 align:middle line:90% 00:22:39.470 --> 00:22:41.010 align:middle line:90% OK, good. 00:22:41.010 --> 00:22:42.230 align:middle line:90% Any questions on this? 00:22:42.230 --> 00:22:45.170 align:middle line:90% 00:22:45.170 --> 00:22:48.778 align:middle line:84% Now, let's just go plug into the path integral. 00:22:48.778 --> 00:22:50.570 align:middle line:84% And now let's just plug into path integral. 00:22:50.570 --> 00:23:06.620 align:middle line:90% 00:23:06.620 --> 00:23:09.860 align:middle line:84% So path integral is an alternative way 00:23:09.860 --> 00:23:12.350 align:middle line:90% to formulate quantum mechanics. 00:23:12.350 --> 00:23:15.770 align:middle line:84% It's equivalent, say, to the Schrodinger equation 00:23:15.770 --> 00:23:17.370 align:middle line:90% or Heisenberg equation. 00:23:17.370 --> 00:23:19.370 align:middle line:84% It's equivalent to your standard way of thinking 00:23:19.370 --> 00:23:21.980 align:middle line:90% about quantum mechanics. 00:23:21.980 --> 00:23:25.880 align:middle line:84% So path integral will give you a better conceptual way 00:23:25.880 --> 00:23:29.510 align:middle line:84% for non-relativistic quantum mechanics. 00:23:29.510 --> 00:23:32.930 align:middle line:84% Path integral, most of the time, they 00:23:32.930 --> 00:23:38.880 align:middle line:84% give you a better conceptual way to think 00:23:38.880 --> 00:23:40.900 align:middle line:90% about quantum mechanics. 00:23:40.900 --> 00:23:44.860 align:middle line:84% But, in terms of calculations, it often 00:23:44.860 --> 00:23:47.200 align:middle line:90% does not offer any advantage. 00:23:47.200 --> 00:23:52.180 align:middle line:84% So that's why the textbooks, for undergraduate quantum 00:23:52.180 --> 00:23:55.372 align:middle line:84% mechanics, they don't just do path integral. 00:23:55.372 --> 00:23:57.580 align:middle line:84% Because to do the calculation, it's still much easier 00:23:57.580 --> 00:23:58.830 align:middle line:90% to solve Schrodinger equation. 00:23:58.830 --> 00:24:02.530 align:middle line:90% 00:24:02.530 --> 00:24:06.342 align:middle line:84% But conceptually, it gives you a better picture. 00:24:06.342 --> 00:24:08.050 align:middle line:84% And for quantum field theory, it actually 00:24:08.050 --> 00:24:09.115 align:middle line:90% become much more useful. 00:24:09.115 --> 00:24:13.103 align:middle line:90% 00:24:13.103 --> 00:24:14.395 align:middle line:90% Is there any questions on this? 00:24:14.395 --> 00:24:17.640 align:middle line:90% 00:24:17.640 --> 00:24:18.140 align:middle line:90% OK, good. 00:24:18.140 --> 00:24:18.640 align:middle line:90% OK. 00:24:18.640 --> 00:24:22.690 align:middle line:90% 00:24:22.690 --> 00:24:26.970 align:middle line:84% OK, so first, we are going just to review 00:24:26.970 --> 00:24:28.720 align:middle line:84% the story for the non-relativistic quantum 00:24:28.720 --> 00:24:31.900 align:middle line:90% mechanics. 00:24:31.900 --> 00:24:35.950 align:middle line:84% And once we understand this and the immediate generalization-- 00:24:35.950 --> 00:24:37.910 align:middle line:90% the field theory is immediate. 00:24:37.910 --> 00:24:40.450 align:middle line:84% Generalization to field theory is immediate. 00:24:40.450 --> 00:24:48.640 align:middle line:84% So we will just use the example of just one particle, just 00:24:48.640 --> 00:24:51.100 align:middle line:84% a particle in one dimension, just 00:24:51.100 --> 00:24:54.250 align:middle line:90% with the most familiar one. 00:24:54.250 --> 00:24:57.340 align:middle line:90% So use this as an example. 00:24:57.340 --> 00:25:01.030 align:middle line:84% Once we understand this example, then you essentially 00:25:01.030 --> 00:25:02.495 align:middle line:90% understand everything. 00:25:02.495 --> 00:25:07.310 align:middle line:90% 00:25:07.310 --> 00:25:09.350 align:middle line:90% Good. 00:25:09.350 --> 00:25:12.530 align:middle line:84% So now the key point about the path integral-- 00:25:12.530 --> 00:25:16.070 align:middle line:90% 00:25:16.070 --> 00:25:18.030 align:middle line:84% previously, in quantum mechanics, 00:25:18.030 --> 00:25:21.530 align:middle line:84% you start with the Schrodinger equation. 00:25:21.530 --> 00:25:24.020 align:middle line:84% But the path integral starts with a different object. 00:25:24.020 --> 00:25:27.630 align:middle line:90% 00:25:27.630 --> 00:25:29.420 align:middle line:84% So the object which the path integral 00:25:29.420 --> 00:25:32.180 align:middle line:84% is most convenient to treat is the propagator 00:25:32.180 --> 00:25:33.035 align:middle line:90% we discussed before. 00:25:33.035 --> 00:25:38.580 align:middle line:90% 00:25:38.580 --> 00:25:45.440 align:middle line:84% So this is the object which we defined, in the Heisenberg 00:25:45.440 --> 00:25:49.050 align:middle line:90% picture, to be this object. 00:25:49.050 --> 00:25:50.930 align:middle line:84% So this is the transition amplitude. 00:25:50.930 --> 00:25:53.040 align:middle line:90% If you have a particle-- 00:25:53.040 --> 00:25:56.960 align:middle line:84% if you have an eigenstate, with x prime and t prime 00:25:56.960 --> 00:26:01.730 align:middle line:84% and go into a position eigenstate at the location, x, 00:26:01.730 --> 00:26:03.440 align:middle line:90% at time t. 00:26:03.440 --> 00:26:06.803 align:middle line:84% So we mentioned before, if you load this object, 00:26:06.803 --> 00:26:07.970 align:middle line:90% essentially it's equivalent. 00:26:07.970 --> 00:26:09.890 align:middle line:84% You have solved the Schrodinger equation. 00:26:09.890 --> 00:26:12.680 align:middle line:84% Because from the wave equation at t prime, 00:26:12.680 --> 00:26:17.420 align:middle line:90% then by integrated with this-- 00:26:17.420 --> 00:26:20.720 align:middle line:84% convoluted with this object, then 00:26:20.720 --> 00:26:22.420 align:middle line:84% you can get the wave function at time t. 00:26:22.420 --> 00:26:24.980 align:middle line:90% 00:26:24.980 --> 00:26:31.280 align:middle line:84% And so this can also be written in terms of Schrodinger picture 00:26:31.280 --> 00:26:36.530 align:middle line:90% as this. 00:26:36.530 --> 00:26:40.550 align:middle line:84% Just start with the position eigenstate 00:26:40.550 --> 00:26:44.300 align:middle line:84% at x prime, then you evolve for t minus t prime, 00:26:44.300 --> 00:26:49.685 align:middle line:84% and then you ask the overlap with the x. 00:26:49.685 --> 00:26:53.796 align:middle line:90% 00:26:53.796 --> 00:26:56.410 align:middle line:90% Any questions on this? 00:26:56.410 --> 00:27:00.460 align:middle line:90% So let me call this object k. 00:27:00.460 --> 00:27:06.490 align:middle line:84% This is k x t, x prime, t prime, OK? 00:27:06.490 --> 00:27:10.730 align:middle line:84% If you know the wave function, so the wave function at t 00:27:10.730 --> 00:27:16.277 align:middle line:84% x can be written as a wave function at the t x prime, k t 00:27:16.277 --> 00:27:20.350 align:middle line:90% x, t prime, x prime. 00:27:20.350 --> 00:27:24.060 align:middle line:84% Yeah, I think I changed the order. 00:27:24.060 --> 00:27:27.680 align:middle line:84% So just to be consistent with the order I wrote there. 00:27:27.680 --> 00:27:31.940 align:middle line:90% t prime, psi, t prime, x prime. 00:27:31.940 --> 00:27:34.520 align:middle line:84% So if you know the wave function at t prime, 00:27:34.520 --> 00:27:39.750 align:middle line:84% and then, if I integrate this, you know the wave function. 00:27:39.750 --> 00:27:44.690 align:middle line:84% So knowing this object is equivalent to solving 00:27:44.690 --> 00:27:45.780 align:middle line:90% the system. 00:27:45.780 --> 00:27:50.120 align:middle line:84% So now we will describe a way to compute this object. 00:27:50.120 --> 00:27:53.270 align:middle line:84% We will describe a way to compute this object. 00:27:53.270 --> 00:27:56.870 align:middle line:90% 00:27:56.870 --> 00:28:00.710 align:middle line:84% So do you have any questions on this? 00:28:00.710 --> 00:28:01.210 align:middle line:90% Good. 00:28:01.210 --> 00:28:03.850 align:middle line:90% 00:28:03.850 --> 00:28:09.230 align:middle line:84% OK, so we will do this by using a trick. 00:28:09.230 --> 00:28:12.180 align:middle line:90% 00:28:12.180 --> 00:28:14.830 align:middle line:84% So imagine this is your time axis. 00:28:14.830 --> 00:28:17.020 align:middle line:90% So we start with t prime. 00:28:17.020 --> 00:28:22.330 align:middle line:84% And then we end at t, So this is the time axis. 00:28:22.330 --> 00:28:24.660 align:middle line:90% End at t. 00:28:24.660 --> 00:28:28.590 align:middle line:84% So what we will do is we divide it into segments. 00:28:28.590 --> 00:28:34.320 align:middle line:84% Divide this interval by infinite number of segments, 00:28:34.320 --> 00:28:37.410 align:middle line:84% and each segment is infinitesimal, of course. 00:28:37.410 --> 00:28:43.920 align:middle line:84% So we label this point by 0, 1, 2, etcetera. 00:28:43.920 --> 00:28:50.960 align:middle line:90% So t0 would be just be t prime. 00:28:50.960 --> 00:28:58.430 align:middle line:84% And the last point will be n, and the tn would be t. 00:28:58.430 --> 00:29:01.910 align:middle line:84% So we divided this by n intervals. 00:29:01.910 --> 00:29:05.030 align:middle line:90% 00:29:05.030 --> 00:29:07.920 align:middle line:84% So the width of the interval is delta t 00:29:07.920 --> 00:29:11.840 align:middle line:84% equal to t minus t prime divided by n. 00:29:11.840 --> 00:29:13.202 align:middle line:90% So here is delta t. 00:29:13.202 --> 00:29:16.320 align:middle line:90% 00:29:16.320 --> 00:29:19.920 align:middle line:84% And if we take n go to infinity, and then this integral 00:29:19.920 --> 00:29:20.700 align:middle line:90% goes to 0. 00:29:20.700 --> 00:29:27.880 align:middle line:90% 00:29:27.880 --> 00:29:32.060 align:middle line:84% So the reason for considering this? 00:29:32.060 --> 00:29:34.400 align:middle line:84% Just now, we can rewrite the expression, 00:29:34.400 --> 00:29:49.130 align:middle line:84% minus i H t prime in terms of many i H delta t, 00:29:49.130 --> 00:29:50.165 align:middle line:90% and then n times. 00:29:50.165 --> 00:29:52.523 align:middle line:90% 00:29:52.523 --> 00:29:53.690 align:middle line:90% Essentially, I have n times. 00:29:53.690 --> 00:30:00.510 align:middle line:90% 00:30:00.510 --> 00:30:01.875 align:middle line:90% So now what I'm going to do? 00:30:01.875 --> 00:30:05.190 align:middle line:90% 00:30:05.190 --> 00:30:12.570 align:middle line:84% Then we have this x, say, exponential minus i H delta t. 00:30:12.570 --> 00:30:16.470 align:middle line:84% So now let's write this n times, separately-- 00:30:16.470 --> 00:30:20.340 align:middle line:90% 00:30:20.340 --> 00:30:20.865 align:middle line:90% x prime. 00:30:20.865 --> 00:30:24.120 align:middle line:90% 00:30:24.120 --> 00:30:27.207 align:middle line:90% So now let's insert. 00:30:27.207 --> 00:30:29.290 align:middle line:84% So the mathematical tool that we are going to use? 00:30:29.290 --> 00:30:34.650 align:middle line:84% Insert, between each such factor, 00:30:34.650 --> 00:30:38.010 align:middle line:84% a complete set of states, a complete set 00:30:38.010 --> 00:30:39.930 align:middle line:90% of position eigenstates. 00:30:39.930 --> 00:30:49.140 align:middle line:90% So we insert at t i. 00:30:49.140 --> 00:30:55.410 align:middle line:84% Yeah because here, it's t0, then going to here 00:30:55.410 --> 00:30:56.970 align:middle line:90% will be t1, et cetera. 00:30:56.970 --> 00:31:01.940 align:middle line:84% So at t i, you insert a complete set of eigenstate. 00:31:01.940 --> 00:31:06.420 align:middle line:90% 00:31:06.420 --> 00:31:11.750 align:middle line:84% You just insert one position eigenstate. 00:31:11.750 --> 00:31:15.900 align:middle line:90% 00:31:15.900 --> 00:31:27.780 align:middle line:84% So now this object, k, then can be written as then d x 1, 00:31:27.780 --> 00:31:29.430 align:middle line:90% d x n minus 1. 00:31:29.430 --> 00:31:32.490 align:middle line:90% So at each here, you-- 00:31:32.490 --> 00:31:33.360 align:middle line:90% OK? 00:31:33.360 --> 00:31:42.810 align:middle line:84% And then you have x exponential minus i H delta t x n minus 1. 00:31:42.810 --> 00:31:47.980 align:middle line:90% 00:31:47.980 --> 00:31:51.940 align:middle line:84% And then all the way, here, into x1 exponential 00:31:51.940 --> 00:31:56.080 align:middle line:90% minus i H delta t, x prime. 00:31:56.080 --> 00:32:01.670 align:middle line:90% 00:32:01.670 --> 00:32:06.560 align:middle line:84% So the purpose for doing so is that now we 00:32:06.560 --> 00:32:12.230 align:middle line:84% can calculate each of such things, explicitly. 00:32:12.230 --> 00:32:14.910 align:middle line:90% And now we will see. 00:32:14.910 --> 00:32:17.925 align:middle line:84% So now let's consider just one of such factor. 00:32:17.925 --> 00:32:23.350 align:middle line:90% 00:32:23.350 --> 00:32:30.180 align:middle line:84% So factor, here, have the form x i plus 1 exponential minus i 00:32:30.180 --> 00:32:33.390 align:middle line:90% h delta t x i. 00:32:33.390 --> 00:32:38.040 align:middle line:84% So any factor, here, have the form for some value of i. 00:32:38.040 --> 00:32:40.830 align:middle line:90% 00:32:40.830 --> 00:32:44.610 align:middle line:84% So i start from 0 all the way to n minus 1. 00:32:44.610 --> 00:32:47.710 align:middle line:90% 00:32:47.710 --> 00:32:51.520 align:middle line:84% So now, let's compute this quantity. 00:32:51.520 --> 00:32:54.740 align:middle line:90% 00:32:54.740 --> 00:32:56.650 align:middle line:84% And now this quantity is actually computable, 00:32:56.650 --> 00:33:02.350 align:middle line:84% even though this object is very hard to compute, 00:33:02.350 --> 00:33:06.710 align:middle line:84% even though this object is very hard to compute. 00:33:06.710 --> 00:33:08.320 align:middle line:84% This object is very hard to compute. 00:33:08.320 --> 00:33:12.950 align:middle line:84% Because you have the potential there, et cetera. 00:33:12.950 --> 00:33:16.160 align:middle line:84% In general, we don't know how to compute this exponential. 00:33:16.160 --> 00:33:18.560 align:middle line:84% But now the trick is that, once you separate them 00:33:18.560 --> 00:33:21.890 align:middle line:84% into infinitesimal steps, then we can actually calculate 00:33:21.890 --> 00:33:23.903 align:middle line:90% the infinitesimal step. 00:33:23.903 --> 00:33:25.820 align:middle line:84% And once you can calculate infinitesimal step, 00:33:25.820 --> 00:33:29.630 align:middle line:84% you just add them together, then become the whole thing. 00:33:29.630 --> 00:33:30.785 align:middle line:90% So that's the basic idea. 00:33:30.785 --> 00:33:33.490 align:middle line:90% 00:33:33.490 --> 00:33:35.755 align:middle line:84% So to do this contain two elements. 00:33:35.755 --> 00:33:43.120 align:middle line:90% 00:33:43.120 --> 00:33:46.940 align:middle line:84% So the first element is to recognize-- 00:33:46.940 --> 00:33:50.660 align:middle line:84% now, remember, h, here, is the operator. 00:33:50.660 --> 00:33:52.820 align:middle line:84% So let me just to emphasize here, 00:33:52.820 --> 00:33:55.160 align:middle line:84% now, let me put a hat, here, just to emphasize here 00:33:55.160 --> 00:33:56.810 align:middle line:90% is the operator. 00:33:56.810 --> 00:33:59.720 align:middle line:84% And so if you write this explicitly, 00:33:59.720 --> 00:34:07.030 align:middle line:90% this is minus i delta t. 00:34:07.030 --> 00:34:10.360 align:middle line:84% And then you have p squared divided by 2m. 00:34:10.360 --> 00:34:15.020 align:middle line:90% Let me just also put the-- 00:34:15.020 --> 00:34:17.680 align:middle line:84% yeah, to emphasize the operators, and the V hat. 00:34:17.680 --> 00:34:25.320 align:middle line:90% 00:34:25.320 --> 00:34:29.475 align:middle line:84% So now, because the delta t is small, 00:34:29.475 --> 00:34:34.880 align:middle line:84% it goes to 0, then to leading order in delta t, 00:34:34.880 --> 00:34:36.739 align:middle line:90% so this is the first element. 00:34:36.739 --> 00:34:38.000 align:middle line:90% The first element is a point. 00:34:38.000 --> 00:34:41.060 align:middle line:84% Then to leading order in delta t, 00:34:41.060 --> 00:34:43.550 align:middle line:84% we can actually just separate the exponential. 00:34:43.550 --> 00:34:48.409 align:middle line:84% You can say this is approximately equal 00:34:48.409 --> 00:34:57.350 align:middle line:84% to exponential i delta t, p hat squared divided by 2m, 00:34:57.350 --> 00:35:03.340 align:middle line:84% and the exponential i delta t, V hat x. 00:35:03.340 --> 00:35:07.390 align:middle line:84% And the only corrections are order delta t squared. 00:35:07.390 --> 00:35:12.940 align:middle line:90% 00:35:12.940 --> 00:35:19.160 align:middle line:84% So this, you can easily see from the Campbell House-- 00:35:19.160 --> 00:35:23.050 align:middle line:90% Campbell or Baker? 00:35:23.050 --> 00:35:26.890 align:middle line:90% Yeah, anyway BCH formula. 00:35:26.890 --> 00:35:29.590 align:middle line:90% So from the BCH formula-- 00:35:29.590 --> 00:35:31.330 align:middle line:90% let me write it here. 00:35:31.330 --> 00:35:37.975 align:middle line:84% Exponential A plus B is equal to exponential A exponential B. 00:35:37.975 --> 00:35:44.240 align:middle line:84% And then exponential minus 1/2 A commutator B, then 00:35:44.240 --> 00:35:47.000 align:middle line:90% plus with more commutators. 00:35:47.000 --> 00:35:52.472 align:middle line:84% And now each A and B are proportional to delta t. 00:35:52.472 --> 00:35:54.680 align:middle line:84% So here is proportional to delta t, here proportional 00:35:54.680 --> 00:35:55.220 align:middle line:90% to delta t. 00:35:55.220 --> 00:36:00.620 align:middle line:84% And here, it involves at least the product of A and B. 00:36:00.620 --> 00:36:04.640 align:middle line:84% So here, it's all higher order in delta t squared. 00:36:04.640 --> 00:36:08.750 align:middle line:84% So this is one of the key reason we want to do this. 00:36:08.750 --> 00:36:12.050 align:middle line:84% We want to separate into individual steps. 00:36:12.050 --> 00:36:13.820 align:middle line:90% Because we can factorize them. 00:36:13.820 --> 00:36:16.190 align:middle line:84% Because, in general, we cannot factorize this guy. 00:36:16.190 --> 00:36:20.180 align:middle line:84% This guy is very complicated, because p and x don't commute. 00:36:20.180 --> 00:36:21.770 align:middle line:84% In general, we cannot factorize them. 00:36:21.770 --> 00:36:22.760 align:middle line:90% It's very complicated. 00:36:22.760 --> 00:36:25.550 align:middle line:84% For a finite t, here, we cannot factorize them. 00:36:25.550 --> 00:36:27.590 align:middle line:84% But for delta t, we can factorize them. 00:36:27.590 --> 00:36:30.140 align:middle line:90% We get this structure. 00:36:30.140 --> 00:36:32.740 align:middle line:84% So once you have this structure, and then this thing 00:36:32.740 --> 00:36:33.740 align:middle line:90% becomes easy to compute. 00:36:33.740 --> 00:36:37.930 align:middle line:90% 00:36:37.930 --> 00:36:39.285 align:middle line:90% The second thing you can show. 00:36:39.285 --> 00:37:12.780 align:middle line:90% 00:37:12.780 --> 00:37:14.340 align:middle line:84% So the second thing you can show? 00:37:14.340 --> 00:37:18.500 align:middle line:84% Now, you can show that x i plus 1-- 00:37:18.500 --> 00:37:22.740 align:middle line:84% so now you plug in this factorized form 00:37:22.740 --> 00:37:26.340 align:middle line:90% into this expression. 00:37:26.340 --> 00:37:29.890 align:middle line:90% And now you can compute this. 00:37:29.890 --> 00:37:36.510 align:middle line:84% So you have minus i delta t, p hat squared divided by 2m, 00:37:36.510 --> 00:37:42.600 align:middle line:84% minus i delta t, V x hat and x i. 00:37:42.600 --> 00:37:44.410 align:middle line:90% So those are just eigenvalues. 00:37:44.410 --> 00:37:45.300 align:middle line:90% So there's no hat. 00:37:45.300 --> 00:37:46.440 align:middle line:90% But here, there's a hat. 00:37:46.440 --> 00:37:49.860 align:middle line:90% They are operators. 00:37:49.860 --> 00:37:53.430 align:middle line:84% So here, you can now compute this object. 00:37:53.430 --> 00:37:56.550 align:middle line:84% So this becomes an elementary exercise in quantum mechanics. 00:37:56.550 --> 00:37:59.060 align:middle line:90% 00:37:59.060 --> 00:38:01.040 align:middle line:90% I will not. 00:38:01.040 --> 00:38:04.400 align:middle line:84% There's like two line calculation here. 00:38:04.400 --> 00:38:07.250 align:middle line:84% I will leave it to your homework, 00:38:07.250 --> 00:38:10.250 align:middle line:84% so you can calculate this object. 00:38:10.250 --> 00:38:12.920 align:middle line:84% Let me just write down the result. 00:38:12.920 --> 00:38:16.430 align:middle line:84% So this becomes the following object-- 00:38:16.430 --> 00:38:22.250 align:middle line:84% m divided by 2pi i delta t exponential-- 00:38:22.250 --> 00:38:25.940 align:middle line:84% so you can evaluate this thing, explicitly-- 00:38:25.940 --> 00:38:39.650 align:middle line:84% 1/2 i m, x i plus 1 minus x i delta t squared delta t, 00:38:39.650 --> 00:38:45.020 align:middle line:90% minus i delta t V x. 00:38:45.020 --> 00:38:45.965 align:middle line:90% So you just get this. 00:38:45.965 --> 00:38:48.480 align:middle line:90% 00:38:48.480 --> 00:38:50.170 align:middle line:90% So this is the answer. 00:38:50.170 --> 00:38:53.520 align:middle line:90% So this is the second element. 00:38:53.520 --> 00:38:57.540 align:middle line:84% Now this is readily computable using the standard technique. 00:38:57.540 --> 00:39:02.610 align:middle line:84% So I will remember to put it in your homework. 00:39:02.610 --> 00:39:10.190 align:middle line:90% 00:39:10.190 --> 00:39:16.840 align:middle line:84% So now we have to use a little bit of imagination. 00:39:16.840 --> 00:39:21.420 align:middle line:84% So now we have to imagine-- use a little bit of imagination. 00:39:21.420 --> 00:39:24.395 align:middle line:84% I should have drawn this line longer or bigger. 00:39:24.395 --> 00:39:26.145 align:middle line:84% Let me just draw this a little bit bigger. 00:39:26.145 --> 00:39:29.580 align:middle line:90% 00:39:29.580 --> 00:39:31.995 align:middle line:84% Sorry, I drew this line too short. 00:39:31.995 --> 00:39:34.680 align:middle line:90% 00:39:34.680 --> 00:39:38.560 align:middle line:90% So let me just draw this longer. 00:39:38.560 --> 00:39:41.840 align:middle line:90% So here is t prime. 00:39:41.840 --> 00:39:43.910 align:middle line:90% Here is t. 00:39:43.910 --> 00:39:46.340 align:middle line:84% Here, you have different locations. 00:39:46.340 --> 00:39:50.890 align:middle line:84% So I also draw the delta t to be big. 00:39:50.890 --> 00:39:54.210 align:middle line:90% So at each t i, you have x i. 00:39:54.210 --> 00:39:56.760 align:middle line:90% We insert an x i, OK? 00:39:56.760 --> 00:39:58.290 align:middle line:90% So this is t i. 00:39:58.290 --> 00:40:00.750 align:middle line:90% We insert an x i. 00:40:00.750 --> 00:40:05.160 align:middle line:84% So at t1, we insert x1, et cetera. 00:40:05.160 --> 00:40:09.420 align:middle line:84% So now you imagine, for each choice, 00:40:09.420 --> 00:40:12.300 align:middle line:84% for each value of x, in the end, we 00:40:12.300 --> 00:40:14.130 align:middle line:84% need to integrate over all possible x. 00:40:14.130 --> 00:40:17.100 align:middle line:90% 00:40:17.100 --> 00:40:23.940 align:middle line:84% So for each particular value of x1, x2, x3, x n, 00:40:23.940 --> 00:40:31.175 align:middle line:84% we can view it as a function of t evaluated at that point. 00:40:31.175 --> 00:40:35.230 align:middle line:90% 00:40:35.230 --> 00:40:43.330 align:middle line:84% Now, let's consider, you have a function, x t, 00:40:43.330 --> 00:40:49.495 align:middle line:84% so that when you evaluate it at t i, the value is x i plus x i. 00:40:49.495 --> 00:40:55.190 align:middle line:90% 00:40:55.190 --> 00:41:02.220 align:middle line:84% So if you think this way, now this thing in the exponential 00:41:02.220 --> 00:41:06.660 align:middle line:84% will become a very familiar object. 00:41:06.660 --> 00:41:10.020 align:middle line:90% So what is this object? 00:41:10.020 --> 00:41:10.520 align:middle line:90% Yes? 00:41:10.520 --> 00:41:11.930 align:middle line:90% A Lagrangian? 00:41:11.930 --> 00:41:14.120 align:middle line:84% Exactly, just become the Lagrangian. 00:41:14.120 --> 00:41:17.010 align:middle line:84% Because this is just like the time derivative, 00:41:17.010 --> 00:41:19.550 align:middle line:84% so this is just like x dot square. 00:41:19.550 --> 00:41:23.210 align:middle line:84% And this is just like V times delta t. 00:41:23.210 --> 00:41:26.700 align:middle line:90% 00:41:26.700 --> 00:41:27.975 align:middle line:90% So this is equal to-- 00:41:27.975 --> 00:41:30.800 align:middle line:90% 00:41:30.800 --> 00:41:33.130 align:middle line:90% so go from here to here. 00:41:33.130 --> 00:41:35.470 align:middle line:90% Now, this is just equal to-- 00:41:35.470 --> 00:41:47.920 align:middle line:84% with the same prefactor, just become exponential i delta t 00:41:47.920 --> 00:41:51.410 align:middle line:90% times your Lagrangian density. 00:41:51.410 --> 00:41:53.030 align:middle line:90% Yeah, just a Lagrangian, here. 00:41:53.030 --> 00:41:57.260 align:middle line:84% Just the Lagrangian x, x dot evaluated at t i. 00:41:57.260 --> 00:42:02.120 align:middle line:90% 00:42:02.120 --> 00:42:04.740 align:middle line:90% So you have the Lagrangian. 00:42:04.740 --> 00:42:09.750 align:middle line:84% So remember, the Lagrangian is just 1/2 00:42:09.750 --> 00:42:11.840 align:middle line:90% m x dot squared minus V x. 00:42:11.840 --> 00:42:15.960 align:middle line:90% 00:42:15.960 --> 00:42:20.430 align:middle line:84% So if you take all this delta t factor and this i factor, 00:42:20.430 --> 00:42:23.460 align:middle line:84% and then this just becomes 1/2 m x dot square. 00:42:23.460 --> 00:42:26.910 align:middle line:84% And this just becomes V all evaluated at x i. 00:42:26.910 --> 00:42:29.070 align:middle line:90% And x i is the value at t i. 00:42:29.070 --> 00:42:32.630 align:middle line:90% So all evaluated at t i. 00:42:32.630 --> 00:42:35.250 align:middle line:84% So now you recognize, you just get your Lagrangian there. 00:42:35.250 --> 00:42:39.660 align:middle line:90% 00:42:39.660 --> 00:42:48.200 align:middle line:84% So now, we can write down the full t, full K. 00:42:48.200 --> 00:42:57.250 align:middle line:84% So now the full K become limit n go to infinity, 00:42:57.250 --> 00:42:59.940 align:middle line:90% because we take delta t go to 0. 00:42:59.940 --> 00:43:03.200 align:middle line:84% So we need to take the n go to infinity. 00:43:03.200 --> 00:43:10.540 align:middle line:84% So m 2 pi i delta t to the power m divided by 2. 00:43:10.540 --> 00:43:13.570 align:middle line:84% So each one gives you square root. 00:43:13.570 --> 00:43:19.360 align:middle line:84% And then you have d x1, d xn minus 1. 00:43:19.360 --> 00:43:28.010 align:middle line:84% But now the integrand becomes i delta t. 00:43:28.010 --> 00:43:35.540 align:middle line:84% Now you just sum over i from 0 to n minus 1, L, 00:43:35.540 --> 00:43:40.760 align:middle line:84% your Lagrangian, x, x prime evaluated at t i. 00:43:40.760 --> 00:43:43.310 align:middle line:90% 00:43:43.310 --> 00:43:46.340 align:middle line:84% Because now, the product in the exponential 00:43:46.340 --> 00:43:47.330 align:middle line:90% just become the sum. 00:43:47.330 --> 00:43:53.440 align:middle line:90% 00:43:53.440 --> 00:43:57.800 align:middle line:84% So now you recognize this essentially 00:43:57.800 --> 00:44:05.240 align:middle line:84% is the discrete version of the integral of L with dt. 00:44:05.240 --> 00:44:08.270 align:middle line:90% It's just over time. 00:44:08.270 --> 00:44:09.305 align:middle line:90% So now we can write. 00:44:09.305 --> 00:44:11.970 align:middle line:90% 00:44:11.970 --> 00:44:14.400 align:middle line:84% So the exponential, we can essentially-- 00:44:14.400 --> 00:44:16.120 align:middle line:84% so let me keep this integral here. 00:44:16.120 --> 00:44:18.540 align:middle line:84% So the exponential part, we can write it 00:44:18.540 --> 00:44:27.760 align:middle line:84% as exponential times i from t prime to t, 00:44:27.760 --> 00:44:32.890 align:middle line:84% say dt double prime, L, x, x dot. 00:44:32.890 --> 00:44:34.840 align:middle line:90% So this just becomes that. 00:44:34.840 --> 00:44:38.360 align:middle line:84% The integrand just becomes like that. 00:44:38.360 --> 00:44:42.520 align:middle line:84% But we still have this whole bunch of complicated stuff. 00:44:42.520 --> 00:44:44.230 align:middle line:84% Take n go to infinity-- this thing. 00:44:44.230 --> 00:44:49.150 align:middle line:90% 00:44:49.150 --> 00:44:53.230 align:middle line:84% So normally, when you have something complicated, 00:44:53.230 --> 00:44:58.310 align:middle line:84% again, there's an unfailing trick in physics to do it. 00:44:58.310 --> 00:44:59.420 align:middle line:90% What is that trick? 00:44:59.420 --> 00:45:00.520 align:middle line:90% Do you know? 00:45:00.520 --> 00:45:03.950 align:middle line:90% 00:45:03.950 --> 00:45:05.090 align:middle line:90% Give it a name? 00:45:05.090 --> 00:45:09.290 align:middle line:84% Exactly, we just rename it into something simple. 00:45:09.290 --> 00:45:11.750 align:middle line:84% So we just call the whole thing D X(t). 00:45:11.750 --> 00:45:14.630 align:middle line:90% 00:45:14.630 --> 00:45:16.610 align:middle line:84% This is just supposed to represent this limit. 00:45:16.610 --> 00:45:19.460 align:middle line:90% 00:45:19.460 --> 00:45:22.505 align:middle line:84% And so this is the condition that x 00:45:22.505 --> 00:45:28.410 align:middle line:84% t prime equal to x prime, and x t equal to t-- 00:45:28.410 --> 00:45:32.000 align:middle line:90% or x t equal to x. 00:45:32.000 --> 00:45:42.730 align:middle line:84% And the meaning of this D X(t), if you 00:45:42.730 --> 00:45:48.460 align:middle line:84% think about it, as I mentioned, each particular choice of x1 00:45:48.460 --> 00:45:55.710 align:middle line:84% and to x n minus 1, you can think of some function 00:45:55.710 --> 00:45:59.490 align:middle line:90% of x evaluated at t i. 00:45:59.490 --> 00:46:02.505 align:middle line:84% So now, you integrate over all possible such values. 00:46:02.505 --> 00:46:05.280 align:middle line:90% 00:46:05.280 --> 00:46:07.860 align:middle line:84% You integrate over all possible such functions. 00:46:07.860 --> 00:46:11.260 align:middle line:90% 00:46:11.260 --> 00:46:18.990 align:middle line:84% So this just corresponding to integrate 00:46:18.990 --> 00:46:32.420 align:middle line:84% all possible functions x t satisfying-- 00:46:32.420 --> 00:46:35.980 align:middle line:84% so mathematically, this means satisfying 00:46:35.980 --> 00:46:41.390 align:middle line:84% that the x t prime equal to x prime, and x t equal to x. 00:46:41.390 --> 00:46:46.210 align:middle line:84% So because we always fix the two end points. 00:46:46.210 --> 00:46:48.005 align:middle line:84% You only integrate in the middle. 00:46:48.005 --> 00:46:49.255 align:middle line:90% You only integrate the middle. 00:46:49.255 --> 00:46:51.980 align:middle line:90% 00:46:51.980 --> 00:46:55.960 align:middle line:84% So now this has a very simple physical interpretation. 00:46:55.960 --> 00:47:01.070 align:middle line:90% 00:47:01.070 --> 00:47:03.730 align:middle line:84% So now this has a very simple physical interpretation. 00:47:03.730 --> 00:47:18.140 align:middle line:90% 00:47:18.140 --> 00:47:21.520 align:middle line:84% So integrate over all possible functions x, 00:47:21.520 --> 00:47:25.585 align:middle line:84% between t and t prime, with the end point fixed. 00:47:25.585 --> 00:47:28.150 align:middle line:90% 00:47:28.150 --> 00:47:33.190 align:middle line:84% Physically, this just means, suppose-- 00:47:33.190 --> 00:47:35.920 align:middle line:90% 00:47:35.920 --> 00:47:37.210 align:middle line:90% so this is the t. 00:47:37.210 --> 00:47:39.730 align:middle line:90% This is the x-axis. 00:47:39.730 --> 00:47:43.630 align:middle line:84% So suppose that the t prime is here. 00:47:43.630 --> 00:47:50.140 align:middle line:84% So here is the, say, x prime at t prime. 00:47:50.140 --> 00:47:54.070 align:middle line:90% And so this is the x t. 00:47:54.070 --> 00:47:57.820 align:middle line:90% So this is the x prime, t prime. 00:47:57.820 --> 00:48:00.700 align:middle line:90% And this point is x and t. 00:48:00.700 --> 00:48:03.610 align:middle line:84% So you integrate all possible functions between them just 00:48:03.610 --> 00:48:05.650 align:middle line:90% corresponding to-- 00:48:05.650 --> 00:48:09.070 align:middle line:84% you integrate over all paths between them. 00:48:09.070 --> 00:48:14.220 align:middle line:90% 00:48:14.220 --> 00:48:16.470 align:middle line:84% You integrate over all possible paths between them. 00:48:16.470 --> 00:48:21.150 align:middle line:90% 00:48:21.150 --> 00:48:35.780 align:middle line:84% So this is the same as integrate all paths between them. 00:48:35.780 --> 00:48:37.055 align:middle line:90% And the thing is weighted. 00:48:37.055 --> 00:48:39.950 align:middle line:90% 00:48:39.950 --> 00:48:48.330 align:middle line:84% And each path is weighted with the integrand. 00:48:48.330 --> 00:48:52.160 align:middle line:90% 00:48:52.160 --> 00:49:02.130 align:middle line:84% It's weighted by exponential i S. 00:49:02.130 --> 00:49:04.680 align:middle line:90% And the S is just your action. 00:49:04.680 --> 00:49:08.570 align:middle line:90% 00:49:08.570 --> 00:49:14.160 align:middle line:84% So this is, again, from t prime to t. 00:49:14.160 --> 00:49:20.512 align:middle line:90% 00:49:20.512 --> 00:49:21.720 align:middle line:90% Yeah, let me just write down. 00:49:21.720 --> 00:49:25.110 align:middle line:90% 00:49:25.110 --> 00:49:26.295 align:middle line:90% By this factor, OK? 00:49:26.295 --> 00:49:29.250 align:middle line:90% 00:49:29.250 --> 00:49:31.560 align:middle line:84% So the so the integration, with the Lagrangian, 00:49:31.560 --> 00:49:34.020 align:middle line:90% is just your action. 00:49:34.020 --> 00:49:37.380 align:middle line:84% And then you find that, now, you can 00:49:37.380 --> 00:49:43.010 align:middle line:84% write this propagate this transition amplitude in terms 00:49:43.010 --> 00:49:44.850 align:middle line:84% of the expression corresponding to integrate 00:49:44.850 --> 00:49:48.630 align:middle line:84% all possible paths, sum over all possible paths, 00:49:48.630 --> 00:49:52.020 align:middle line:84% and with the weight by the action, exponential action. 00:49:52.020 --> 00:49:56.440 align:middle line:90% 00:49:56.440 --> 00:50:03.640 align:middle line:84% So this started as a technical trick, 00:50:03.640 --> 00:50:07.150 align:middle line:84% because computing this thing is complicated. 00:50:07.150 --> 00:50:10.630 align:middle line:84% And then we can make mathematical progress 00:50:10.630 --> 00:50:14.830 align:middle line:84% by dividing them into small intervals. 00:50:14.830 --> 00:50:17.020 align:middle line:90% But once you do that-- 00:50:17.020 --> 00:50:19.720 align:middle line:84% but now we obtain a different way. 00:50:19.720 --> 00:50:24.220 align:middle line:84% But now, we actually obtain something conceptually new. 00:50:24.220 --> 00:50:26.640 align:middle line:84% You just say that quantum mechanics. 00:50:26.640 --> 00:50:32.400 align:middle line:84% So classically, remember, there is 00:50:32.400 --> 00:50:35.610 align:middle line:84% a fixed path from x t prime to t, 00:50:35.610 --> 00:50:37.590 align:middle line:84% just follow the equation of motion. 00:50:37.590 --> 00:50:39.690 align:middle line:84% But now we say quantum mechanically, we just 00:50:39.690 --> 00:50:41.460 align:middle line:90% sum over all possible paths. 00:50:41.460 --> 00:50:43.830 align:middle line:84% So that's the quantum uncertainty compared 00:50:43.830 --> 00:50:46.710 align:middle line:90% with the classical mechanics. 00:50:46.710 --> 00:50:48.870 align:middle line:84% So starting with that mathematical trick, 00:50:48.870 --> 00:50:52.170 align:middle line:84% but now we actually obtain something conceptually brand 00:50:52.170 --> 00:50:53.240 align:middle line:90% new. 00:50:53.240 --> 00:50:55.670 align:middle line:84% Yeah, because quantum mechanically, the difference 00:50:55.670 --> 00:50:57.740 align:middle line:84% from classical mechanics, just now 00:50:57.740 --> 00:51:00.282 align:middle line:84% we just sum over all possible paths, which 00:51:00.282 --> 00:51:01.490 align:middle line:90% is physically very intuitive. 00:51:01.490 --> 00:51:04.050 align:middle line:84% Because quantum mechanically, you can just go anywhere-- 00:51:04.050 --> 00:51:05.840 align:middle line:90% OK, this uncertainty principle. 00:51:05.840 --> 00:51:10.240 align:middle line:90% 00:51:10.240 --> 00:51:10.740 align:middle line:90% Good. 00:51:10.740 --> 00:51:11.900 align:middle line:90% Any questions on this? 00:51:11.900 --> 00:51:13.170 align:middle line:90% Yes? 00:51:13.170 --> 00:51:16.790 align:middle line:84% Are there constraints on what path is really possible? 00:51:16.790 --> 00:51:19.280 align:middle line:84% Does it have to be differentiable? 00:51:19.280 --> 00:51:20.697 align:middle line:90% Yeah, this is a good question. 00:51:20.697 --> 00:51:23.410 align:middle line:90% 00:51:23.410 --> 00:51:29.960 align:middle line:84% So to make sense, rigorously, this quantity is actually-- 00:51:29.960 --> 00:51:34.770 align:middle line:84% yeah, to make rigorous, this quantity is not easy to do. 00:51:34.770 --> 00:51:36.650 align:middle line:90% So we will not go into there. 00:51:36.650 --> 00:51:40.220 align:middle line:84% That needs to go into lots of mathematics. 00:51:40.220 --> 00:51:42.230 align:middle line:84% So normally, we don't need to worry 00:51:42.230 --> 00:51:46.220 align:middle line:84% about the precise mathematics, how to define that path. 00:51:46.220 --> 00:51:48.800 align:middle line:90% 00:51:48.800 --> 00:51:52.070 align:middle line:84% Mostly, we think this from the conceptual picture. 00:51:52.070 --> 00:51:57.950 align:middle line:84% And also, when we do technical manipulations, as we will see, 00:51:57.950 --> 00:52:00.530 align:middle line:84% often, you don't need to know the details, how you 00:52:00.530 --> 00:52:02.210 align:middle line:90% define this measure, precisely. 00:52:02.210 --> 00:52:02.810 align:middle line:90% Yeah. 00:52:02.810 --> 00:52:04.130 align:middle line:90% Yes? 00:52:04.130 --> 00:52:06.590 align:middle line:90% Sir, a question. 00:52:06.590 --> 00:52:08.330 align:middle line:90% There's no coming back in time. 00:52:08.330 --> 00:52:10.820 align:middle line:90% I mean, that's right? 00:52:10.820 --> 00:52:13.940 align:middle line:90% In principle. 00:52:13.940 --> 00:52:18.030 align:middle line:84% Yeah, there's no coming back in time, just from here to there. 00:52:18.030 --> 00:52:18.530 align:middle line:90% Yeah. 00:52:18.530 --> 00:52:19.940 align:middle line:90% There's no coming back in time. 00:52:19.940 --> 00:52:24.670 align:middle line:84% But the path can wind around as you want. 00:52:24.670 --> 00:52:29.170 align:middle line:84% Yeah, but time, you cannot come back. 00:52:29.170 --> 00:52:32.110 align:middle line:90% Other questions? 00:52:32.110 --> 00:52:34.260 align:middle line:90% OK, good. 00:52:34.260 --> 00:52:39.450 align:middle line:84% So there's also another form of this path integral. 00:52:39.450 --> 00:52:40.850 align:middle line:90% So I will not do it here. 00:52:40.850 --> 00:52:44.510 align:middle line:90% It's a slight variance of this. 00:52:44.510 --> 00:52:46.040 align:middle line:90% Let me just write it down. 00:52:46.040 --> 00:52:49.350 align:middle line:84% Again, I can put it in your homework. 00:52:49.350 --> 00:52:54.950 align:middle line:84% So there's also a Hamiltonian form, which is much less used. 00:52:54.950 --> 00:52:57.380 align:middle line:84% But sometimes this is also useful. 00:52:57.380 --> 00:53:00.950 align:middle line:90% It's a Hamiltonian form. 00:53:00.950 --> 00:53:06.230 align:middle line:84% So you can also write the k, same object as the following, 00:53:06.230 --> 00:53:08.300 align:middle line:90% as x. 00:53:08.300 --> 00:53:09.935 align:middle line:84% Again, the same boundary condition, x 00:53:09.935 --> 00:53:13.610 align:middle line:84% t prime equal to x prime and x t. 00:53:13.610 --> 00:53:17.570 align:middle line:84% So you integrate over all possible paths, as here. 00:53:17.570 --> 00:53:23.620 align:middle line:84% But you also integrate over all possible momentum, 00:53:23.620 --> 00:53:25.240 align:middle line:90% within this time range. 00:53:25.240 --> 00:53:31.880 align:middle line:84% And then the integrand becomes the Hamiltonian. 00:53:31.880 --> 00:53:33.830 align:middle line:84% You write it in terms of the Hamiltonian. 00:53:33.830 --> 00:53:43.430 align:middle line:84% So t prime, t dt double prime, then p x dot minus H. 00:53:43.430 --> 00:53:46.190 align:middle line:90% So H is your Hamiltonian. 00:53:46.190 --> 00:53:49.040 align:middle line:84% And the p, now, is just some function, some arbitrary 00:53:49.040 --> 00:53:49.800 align:middle line:90% function. 00:53:49.800 --> 00:53:54.170 align:middle line:84% And so for any choice of x and t p, 00:53:54.170 --> 00:53:57.620 align:middle line:90% you can evaluate this quantity. 00:53:57.620 --> 00:53:59.870 align:middle line:84% And then you sum all of them together. 00:53:59.870 --> 00:54:04.280 align:middle line:84% And so this alternative form, which can be easily-- 00:54:04.280 --> 00:54:08.660 align:middle line:84% again, a couple of lines, which you can 00:54:08.660 --> 00:54:10.160 align:middle line:84% show to be equivalent to this form. 00:54:10.160 --> 00:54:17.240 align:middle line:90% 00:54:17.240 --> 00:54:20.150 align:middle line:84% To show this is related to the second step, 00:54:20.150 --> 00:54:23.180 align:middle line:84% here, the intermediate step of doing this one 00:54:23.180 --> 00:54:24.440 align:middle line:90% will lead to this one. 00:54:24.440 --> 00:54:33.425 align:middle line:90% 00:54:33.425 --> 00:54:33.925 align:middle line:90% Good. 00:54:33.925 --> 00:54:36.600 align:middle line:90% 00:54:36.600 --> 00:54:39.490 align:middle line:90% Any questions? 00:54:39.490 --> 00:54:47.160 align:middle line:90% 00:54:47.160 --> 00:54:51.135 align:middle line:84% So now let me just talk about the example. 00:54:51.135 --> 00:54:58.700 align:middle line:90% 00:54:58.700 --> 00:55:03.470 align:middle line:84% So how you can use this method to calculate, say, 00:55:03.470 --> 00:55:06.875 align:middle line:90% some systems, simple systems? 00:55:06.875 --> 00:55:11.000 align:middle line:90% 00:55:11.000 --> 00:55:13.160 align:middle line:84% So as I said, this method is not very 00:55:13.160 --> 00:55:16.740 align:middle line:84% efficient for non relativistic quantum mechanics-- 00:55:16.740 --> 00:55:22.340 align:middle line:84% and so even of a different conceptual picture. 00:55:22.340 --> 00:55:24.920 align:middle line:84% And you can see from those examples. 00:55:24.920 --> 00:55:27.990 align:middle line:84% So now let's consider a simple case. 00:55:27.990 --> 00:55:30.620 align:middle line:84% So we just consider a free particle. 00:55:30.620 --> 00:55:35.870 align:middle line:84% So there's no V. So just consider a free particle. 00:55:35.870 --> 00:55:43.530 align:middle line:90% 00:55:43.530 --> 00:55:47.640 align:middle line:84% So here, S just 1/2 m x dot square. 00:55:47.640 --> 00:55:52.960 align:middle line:90% 00:55:52.960 --> 00:55:55.240 align:middle line:84% Let me just use q, since my notes use q. 00:55:55.240 --> 00:55:57.910 align:middle line:90% It doesn't matter. 00:55:57.910 --> 00:56:03.475 align:middle line:84% So let's try to compute, in this theory, the amplitude. 00:56:03.475 --> 00:56:04.100 align:middle line:90% Let's call Z t. 00:56:04.100 --> 00:56:06.990 align:middle line:90% 00:56:06.990 --> 00:56:12.880 align:middle line:84% Say at time 0, you are at q equal to 0. 00:56:12.880 --> 00:56:18.190 align:middle line:84% And then at time t, again, you add q equal to 0. 00:56:18.190 --> 00:56:21.610 align:middle line:84% So you just come back to the same point. 00:56:21.610 --> 00:56:23.090 align:middle line:90% Come back to the same point. 00:56:23.090 --> 00:56:26.080 align:middle line:84% So corresponding to this situation, 00:56:26.080 --> 00:56:29.260 align:middle line:84% if you call this direction to be t and this direction 00:56:29.260 --> 00:56:35.620 align:middle line:84% to be q, so, at some initial point, you are here. 00:56:35.620 --> 00:56:38.830 align:middle line:84% And your final point, at some t, the value of t, 00:56:38.830 --> 00:56:41.230 align:middle line:84% again, you go back to the same point in q. 00:56:41.230 --> 00:56:44.020 align:middle line:84% And then we just sum over all paths here-- 00:56:44.020 --> 00:56:48.490 align:middle line:90% 00:56:48.490 --> 00:56:49.330 align:middle line:90% but free particle. 00:56:49.330 --> 00:56:53.820 align:middle line:90% 00:56:53.820 --> 00:56:57.780 align:middle line:84% So in order to compute this guy, now, we have to work with this. 00:56:57.780 --> 00:57:01.170 align:middle line:84% Because this is only a formal notation. 00:57:01.170 --> 00:57:04.260 align:middle line:84% Remember, this is a formal notation for this object. 00:57:04.260 --> 00:57:06.340 align:middle line:84% OK, if you really want to compute this-- 00:57:06.340 --> 00:57:08.370 align:middle line:84% So this is called the path integral. 00:57:08.370 --> 00:57:09.870 align:middle line:84% So this is called the path integral. 00:57:09.870 --> 00:57:12.690 align:middle line:84% So if you want to really calculate this path integral, 00:57:12.690 --> 00:57:16.690 align:middle line:84% explicitly, then you have to go back to this form. 00:57:16.690 --> 00:57:20.220 align:middle line:84% You have to go back to this form. 00:57:20.220 --> 00:57:24.240 align:middle line:90% So the z t is equal to-- 00:57:24.240 --> 00:57:27.210 align:middle line:84% in this case, limit n go to infinity. 00:57:27.210 --> 00:57:33.180 align:middle line:90% 00:57:33.180 --> 00:57:35.550 align:middle line:90% Yeah, let me just not copy. 00:57:35.550 --> 00:57:40.530 align:middle line:84% Yeah, anyway, you can write down. 00:57:40.530 --> 00:57:42.090 align:middle line:90% You just copy this down. 00:57:42.090 --> 00:57:45.750 align:middle line:90% Just change this. 00:57:45.750 --> 00:57:50.820 align:middle line:84% Just to save time, change this x i to q. 00:57:50.820 --> 00:57:54.600 align:middle line:84% And now the integrand, you only have 1/2 m q squared. 00:57:54.600 --> 00:57:56.550 align:middle line:90% There's no return. 00:57:56.550 --> 00:58:00.340 align:middle line:84% You can just plug that in, into here. 00:58:00.340 --> 00:58:03.510 align:middle line:84% So now, in principle, you can calculate that integral. 00:58:03.510 --> 00:58:05.190 align:middle line:84% And I will leave it to your exercise 00:58:05.190 --> 00:58:06.960 align:middle line:90% to do in your homework. 00:58:06.960 --> 00:58:10.360 align:middle line:84% And you find that it will recover the standard answer. 00:58:10.360 --> 00:58:12.780 align:middle line:84% So at the end of the day, when you do this integral, 00:58:12.780 --> 00:58:15.690 align:middle line:84% this mini-integral, then you find that this gives you-- 00:58:15.690 --> 00:58:21.510 align:middle line:90% 00:58:21.510 --> 00:58:24.990 align:middle line:84% actually, I didn't write down the final answer. 00:58:24.990 --> 00:58:25.545 align:middle line:90% Let me see. 00:58:25.545 --> 00:58:31.280 align:middle line:90% 00:58:31.280 --> 00:58:33.890 align:middle line:84% So you find the final answer is a very simple one, 00:58:33.890 --> 00:58:39.770 align:middle line:90% just 2 pi i t square root. 00:58:39.770 --> 00:58:44.960 align:middle line:84% So you can do this, an infinite number of integrals, 00:58:44.960 --> 00:58:47.010 align:middle line:90% and then you get that answer. 00:58:47.010 --> 00:58:49.010 align:middle line:84% Of course, you can calculate that thing, easily, 00:58:49.010 --> 00:58:52.610 align:middle line:84% by solving the Schrodinger equation 00:58:52.610 --> 00:58:54.690 align:middle line:90% using your standard method. 00:58:54.690 --> 00:58:57.050 align:middle line:84% But this is a good exercise to do. 00:58:57.050 --> 00:59:02.420 align:middle line:90% 00:59:02.420 --> 00:59:05.270 align:middle line:84% So this is one method to calculate it. 00:59:05.270 --> 00:59:07.700 align:middle line:84% But this method is not very efficient. 00:59:07.700 --> 00:59:09.800 align:middle line:84% Because each time, if we have to go through this 00:59:09.800 --> 00:59:11.870 align:middle line:84% limit, and it will be very annoying. 00:59:11.870 --> 00:59:14.960 align:middle line:90% 00:59:14.960 --> 00:59:18.980 align:middle line:84% So now there's a more abstract method. 00:59:18.980 --> 00:59:22.610 align:middle line:84% Now, there's a more abstract method but much less 00:59:22.610 --> 00:59:25.310 align:middle line:90% mathematically rigorous. 00:59:25.310 --> 00:59:27.080 align:middle line:84% You can make it mathematically rigorous, 00:59:27.080 --> 00:59:30.830 align:middle line:84% but I'm not going to make it mathematically rigorous. 00:59:30.830 --> 00:59:36.233 align:middle line:84% But when you get used to it, it's much easier to manipulate. 00:59:36.233 --> 00:59:38.150 align:middle line:84% So now I'm going to talk about a second method 00:59:38.150 --> 00:59:40.470 align:middle line:90% to calculate this. 00:59:40.470 --> 00:59:41.945 align:middle line:90% So this is the first method. 00:59:41.945 --> 00:59:46.740 align:middle line:90% 00:59:46.740 --> 00:59:48.435 align:middle line:90% You just do honest integral. 00:59:48.435 --> 00:59:51.670 align:middle line:90% 00:59:51.670 --> 00:59:52.785 align:middle line:90% So the second method? 00:59:52.785 --> 01:00:01.354 align:middle line:90% 01:00:01.354 --> 01:00:05.140 align:middle line:84% Let me call this one for the first method. 01:00:05.140 --> 01:00:06.340 align:middle line:90% And the second method-- 01:00:06.340 --> 01:00:35.140 align:middle line:90% 01:00:35.140 --> 01:00:37.260 align:middle line:84% So in the second method, let's first look at this. 01:00:37.260 --> 01:00:40.290 align:middle line:84% Oh, yeah, sorry, this should be integrated over dt. 01:00:40.290 --> 01:00:44.070 align:middle line:84% So this is a m and q's creates the Lagrangian. 01:00:44.070 --> 01:00:46.030 align:middle line:84% Here, we're talking about the action. 01:00:46.030 --> 01:00:49.050 align:middle line:84% So here we look at the action which you appear in the path 01:00:49.050 --> 01:00:52.080 align:middle line:84% integral in that formal form then 01:00:52.080 --> 01:00:56.340 align:middle line:84% you have s is equal to 1/2 m, then 01:00:56.340 --> 01:01:00.240 align:middle line:84% you have dt from t equal to 0 to capital T. 01:01:00.240 --> 01:01:03.390 align:middle line:84% From t equal to 0 to capital T. And then we 01:01:03.390 --> 01:01:06.030 align:middle line:90% have 1/2 m q dot square. 01:01:06.030 --> 01:01:11.510 align:middle line:90% 01:01:11.510 --> 01:01:13.890 align:middle line:90% So we have that. 01:01:13.890 --> 01:01:18.510 align:middle line:84% So now I'm going to slightly rewrite this expression. 01:01:18.510 --> 01:01:24.320 align:middle line:84% So I'm going to write as following, m, from 0 to T, dt-- 01:01:24.320 --> 01:01:25.910 align:middle line:90% I will do integration by parts. 01:01:25.910 --> 01:01:30.070 align:middle line:90% 01:01:30.070 --> 01:01:30.900 align:middle line:90% Oh, sorry. 01:01:30.900 --> 01:01:32.490 align:middle line:90% So I'm already out. 01:01:32.490 --> 01:01:36.300 align:middle line:84% Then I have q minus delta t squared, q. 01:01:36.300 --> 01:01:39.170 align:middle line:90% 01:01:39.170 --> 01:01:41.520 align:middle line:84% So integration by parts, so the derivative will only 01:01:41.520 --> 01:01:42.750 align:middle line:90% act on one of the q. 01:01:42.750 --> 01:01:44.580 align:middle line:90% So I get a minus sign. 01:01:44.580 --> 01:01:48.660 align:middle line:84% And the boundary term is all 0, because the q, at the initial 01:01:48.660 --> 01:01:50.522 align:middle line:90% and the final value, are 0. 01:01:50.522 --> 01:01:51.855 align:middle line:90% Yeah, so you can check yourself. 01:01:51.855 --> 01:01:56.830 align:middle line:90% 01:01:56.830 --> 01:01:58.930 align:middle line:84% So I'm going to write this expression a little bit 01:01:58.930 --> 01:01:59.430 align:middle line:90% further. 01:01:59.430 --> 01:02:04.160 align:middle line:90% 01:02:04.160 --> 01:02:10.030 align:middle line:84% I'm going to introduce two t, t and t prime. 01:02:10.030 --> 01:02:13.370 align:middle line:90% I write it as q. 01:02:13.370 --> 01:02:16.430 align:middle line:84% Yeah, let me, now, just write it as 1/2. 01:02:16.430 --> 01:02:19.025 align:middle line:90% Let me just write it as 1/2 q t. 01:02:19.025 --> 01:02:21.680 align:middle line:90% 01:02:21.680 --> 01:02:31.400 align:middle line:84% And I write here minus m delta t minus t prime, partial t 01:02:31.400 --> 01:02:34.520 align:middle line:90% prime squared, q t prime. 01:02:34.520 --> 01:02:39.520 align:middle line:90% 01:02:39.520 --> 01:02:42.410 align:middle line:90% So these two are the same. 01:02:42.410 --> 01:02:45.280 align:middle line:84% So I introduce additional t prime, 01:02:45.280 --> 01:02:47.360 align:middle line:84% but, also, I introduce a delta function. 01:02:47.360 --> 01:02:50.380 align:middle line:84% So when you evaluate that delta function with t prime integral, 01:02:50.380 --> 01:02:53.470 align:middle line:84% you just get back to the expression above. 01:02:53.470 --> 01:02:57.220 align:middle line:90% 01:02:57.220 --> 01:02:59.355 align:middle line:84% So now I'm going to pull this into the k. 01:02:59.355 --> 01:03:06.460 align:middle line:90% 01:03:06.460 --> 01:03:09.820 align:middle line:84% Yeah, actually, sorry, I want to put this minus sign out, just 01:03:09.820 --> 01:03:11.590 align:middle line:90% for convenience. 01:03:11.590 --> 01:03:13.540 align:middle line:90% So let me just call this in k. 01:03:13.540 --> 01:03:15.543 align:middle line:90% So what I get is the minus. 01:03:15.543 --> 01:03:16.960 align:middle line:84% Yeah, it doesn't matter, actually. 01:03:16.960 --> 01:03:18.160 align:middle line:90% So sorry. 01:03:18.160 --> 01:03:19.370 align:middle line:90% Let me put it here. 01:03:19.370 --> 01:03:22.270 align:middle line:90% Yeah, it doesn't matter. 01:03:22.270 --> 01:03:32.310 align:middle line:84% So here, we can just write 1/2 dt dt prime, q t, k, t, 01:03:32.310 --> 01:03:34.555 align:middle line:90% t prime, q t prime. 01:03:34.555 --> 01:03:42.507 align:middle line:90% 01:03:42.507 --> 01:03:43.340 align:middle line:90% [AUDIENCE] Question. 01:03:43.340 --> 01:03:45.470 align:middle line:90% Yeah? 01:03:45.470 --> 01:03:48.470 align:middle line:84% [AUDIENCE] What's the integral of dt prime? 01:03:48.470 --> 01:03:49.100 align:middle line:90% Yeah. 01:03:49.100 --> 01:03:50.750 align:middle line:84% The integral of dt prime, you can just 01:03:50.750 --> 01:03:53.960 align:middle line:84% evaluate using this delta function and then this go away. 01:03:53.960 --> 01:03:56.090 align:middle line:84% And then this here, this all become t, 01:03:56.090 --> 01:03:57.533 align:middle line:90% then you just reduce to that. 01:03:57.533 --> 01:03:59.450 align:middle line:84% [AUDIENCE] So is it like an all time integral? 01:03:59.450 --> 01:03:59.900 align:middle line:90% Yeah. 01:03:59.900 --> 01:04:00.400 align:middle line:90% Yeah. 01:04:00.400 --> 01:04:01.070 align:middle line:90% No, no, no. 01:04:01.070 --> 01:04:04.220 align:middle line:90% It's the same from 0 to T. Yeah. 01:04:04.220 --> 01:04:04.940 align:middle line:90% Yeah. 01:04:04.940 --> 01:04:05.870 align:middle line:90% The same range. 01:04:05.870 --> 01:04:08.180 align:middle line:90% OK. 01:04:08.180 --> 01:04:09.850 align:middle line:84% So now I just call this into the k. 01:04:09.850 --> 01:04:14.530 align:middle line:90% 01:04:14.530 --> 01:04:19.630 align:middle line:84% So now, I think we have done this before. 01:04:19.630 --> 01:04:25.750 align:middle line:84% And now I want you to view this t and q as an index. 01:04:25.750 --> 01:04:29.800 align:middle line:84% Imagine this is just a continuous index. 01:04:29.800 --> 01:04:32.830 align:middle line:84% And then q is just like a vector-- 01:04:32.830 --> 01:04:34.690 align:middle line:90% q t prime is like a vector. 01:04:34.690 --> 01:04:35.960 align:middle line:90% q t is like a vector. 01:04:35.960 --> 01:04:37.730 align:middle line:90% And this is the same vector. 01:04:37.730 --> 01:04:41.200 align:middle line:84% And now this is just a matrix between the two vector. 01:04:41.200 --> 01:04:44.950 align:middle line:90% 01:04:44.950 --> 01:04:59.555 align:middle line:84% So now, we can view q t as a vector, with t as the index. 01:04:59.555 --> 01:05:04.740 align:middle line:90% 01:05:04.740 --> 01:05:10.340 align:middle line:90% And then the k is the matrix-- 01:05:10.340 --> 01:05:15.100 align:middle line:90% 01:05:15.100 --> 01:05:25.130 align:middle line:84% it's a matrix in the space of q t. 01:05:25.130 --> 01:05:38.470 align:middle line:90% 01:05:38.470 --> 01:05:44.740 align:middle line:84% So when I write this form, then this integral, dq t, 01:05:44.740 --> 01:05:50.410 align:middle line:84% so then we have this Z T equal to dq t. 01:05:50.410 --> 01:05:52.390 align:middle line:84% So for simplicity, let me just suppress 01:05:52.390 --> 01:05:55.930 align:middle line:84% the initial and the final just for simplicity. 01:05:55.930 --> 01:06:07.020 align:middle line:84% And then you just have i, then 1/2 i dt dt prime. 01:06:07.020 --> 01:06:09.930 align:middle line:90% Also let me just suppress the-- 01:06:09.930 --> 01:06:13.710 align:middle line:90% then you have q dot k dot q. 01:06:13.710 --> 01:06:18.900 align:middle line:84% Let me just use this simplified notation. 01:06:18.900 --> 01:06:21.930 align:middle line:90% This is the same as that guy. 01:06:21.930 --> 01:06:22.710 align:middle line:90% Yes? 01:06:22.710 --> 01:06:24.570 align:middle line:84% [AUDIENCE] Is this like ad hoc that you 01:06:24.570 --> 01:06:26.820 align:middle line:84% left the first q as a function of t 01:06:26.820 --> 01:06:30.840 align:middle line:84% and not made it a function of t prime instead? 01:06:30.840 --> 01:06:32.200 align:middle line:90% It doesn't matter. 01:06:32.200 --> 01:06:33.838 align:middle line:90% You can do that it way, too. 01:06:33.838 --> 01:06:34.755 align:middle line:90% Yeah, it won't change. 01:06:34.755 --> 01:06:40.440 align:middle line:90% 01:06:40.440 --> 01:06:42.210 align:middle line:90% So now you compare this form. 01:06:42.210 --> 01:06:45.390 align:middle line:90% 01:06:45.390 --> 01:06:49.380 align:middle line:84% So now, this look like a Gaussian integral. 01:06:49.380 --> 01:06:51.340 align:middle line:84% Now this look like a Gaussian integral. 01:06:51.340 --> 01:06:57.270 align:middle line:84% So now, let's recall your Gaussian integral, 01:06:57.270 --> 01:07:00.660 align:middle line:90% which you'll be familiar with. 01:07:00.660 --> 01:07:07.540 align:middle line:84% When you have the x exponential minus a divided by 2x square, 01:07:07.540 --> 01:07:10.885 align:middle line:84% then that gives you 2 pi divided by a. 01:07:10.885 --> 01:07:15.580 align:middle line:90% 01:07:15.580 --> 01:07:17.320 align:middle line:84% So this is a one-dimensional integral. 01:07:17.320 --> 01:07:19.320 align:middle line:84% But you can also have an n-dimensional integral. 01:07:19.320 --> 01:07:33.990 align:middle line:90% 01:07:33.990 --> 01:07:36.390 align:middle line:90% So here, the index is a sum. 01:07:36.390 --> 01:07:40.170 align:middle line:84% So this is equal to 2 pi to the power 01:07:40.170 --> 01:07:51.660 align:middle line:84% n divided by 2, square root delta A, of this matrix A. 01:07:51.660 --> 01:07:55.140 align:middle line:84% So now this is just a generalization 01:07:55.140 --> 01:08:02.250 align:middle line:84% of this integral, except the index becomes continuous. 01:08:02.250 --> 01:08:06.860 align:middle line:84% And here, the k have two indices. 01:08:06.860 --> 01:08:08.120 align:middle line:90% It's just like m, n. 01:08:08.120 --> 01:08:10.400 align:middle line:90% So here is like q m and q n. 01:08:10.400 --> 01:08:11.915 align:middle line:90% And this is just like m n. 01:08:11.915 --> 01:08:15.530 align:middle line:90% 01:08:15.530 --> 01:08:17.479 align:middle line:84% So this is just like an ordinary-- 01:08:17.479 --> 01:08:19.189 align:middle line:84% this is just like a Gaussian integral 01:08:19.189 --> 01:08:24.950 align:middle line:84% but in the space of infinite dimensional vectors. 01:08:24.950 --> 01:08:26.569 align:middle line:84% Yeah, of course, not just infinite. 01:08:26.569 --> 01:08:28.402 align:middle line:84% It's uncountable infinite dimension, 01:08:28.402 --> 01:08:29.735 align:middle line:90% because it's a continuous index. 01:08:29.735 --> 01:08:32.600 align:middle line:90% 01:08:32.600 --> 01:08:43.479 align:middle line:84% So now with this understanding, with this realization, 01:08:43.479 --> 01:08:45.430 align:middle line:84% we can just directly write down the answer. 01:08:45.430 --> 01:08:49.000 align:middle line:90% 01:08:49.000 --> 01:09:01.160 align:middle line:84% We say then the Z T must be just given by some constant divided 01:09:01.160 --> 01:09:03.859 align:middle line:90% by delta k, determinant k. 01:09:03.859 --> 01:09:10.310 align:middle line:90% 01:09:10.310 --> 01:09:12.319 align:middle line:84% Yeah, you just generalize this formula. 01:09:12.319 --> 01:09:13.639 align:middle line:90% You generalize this formula. 01:09:13.639 --> 01:09:16.630 align:middle line:90% 01:09:16.630 --> 01:09:19.090 align:middle line:84% So just now, k is the determinant 01:09:19.090 --> 01:09:23.350 align:middle line:84% in the more complicated space, in the space of functions. 01:09:23.350 --> 01:09:26.670 align:middle line:90% 01:09:26.670 --> 01:09:31.770 align:middle line:84% So the only unfortunate thing is that both C and the delta k 01:09:31.770 --> 01:09:34.260 align:middle line:90% actually are divergent. 01:09:34.260 --> 01:09:37.970 align:middle line:90% 01:09:37.970 --> 01:09:42.760 align:middle line:90% So C is our constant. 01:09:42.760 --> 01:09:47.350 align:middle line:84% And the delta K is the determinant 01:09:47.350 --> 01:09:57.280 align:middle line:90% of K in the space of functions. 01:09:57.280 --> 01:10:04.480 align:middle line:90% 01:10:04.480 --> 01:10:06.640 align:middle line:84% So this isn't like the K is the matrix. 01:10:06.640 --> 01:10:09.940 align:middle line:84% It's a matrix defined in the space of q t. 01:10:09.940 --> 01:10:11.860 align:middle line:90% q t is the space of functions. 01:10:11.860 --> 01:10:15.450 align:middle line:90% 01:10:15.450 --> 01:10:18.630 align:middle line:84% So this content can be divergent. 01:10:18.630 --> 01:10:21.570 align:middle line:90% 01:10:21.570 --> 01:10:24.090 align:middle line:84% But this doesn't matter, as we will later see, 01:10:24.090 --> 01:10:29.070 align:middle line:84% such a constant don't matter very much for your physics. 01:10:29.070 --> 01:10:32.740 align:middle line:84% And yeah, I warned you before, in quantum field theory, 01:10:32.740 --> 01:10:34.810 align:middle line:84% we will see divergences everywhere. 01:10:34.810 --> 01:10:37.490 align:middle line:84% So the key is to recognize which divergences are important 01:10:37.490 --> 01:10:40.500 align:middle line:84% and which divergence are not important. 01:10:40.500 --> 01:10:44.730 align:middle line:84% So unimportant divergence, you just forget them. 01:10:44.730 --> 01:10:49.110 align:middle line:84% Yeah, so we will later see that such a C will always 01:10:49.110 --> 01:10:51.900 align:middle line:84% cancel when you can see the physical quantities. 01:10:51.900 --> 01:10:53.460 align:middle line:90% And this will never matter. 01:10:53.460 --> 01:10:57.450 align:middle line:90% 01:10:57.450 --> 01:11:02.600 align:middle line:84% But still, we need to talk about how 01:11:02.600 --> 01:11:05.360 align:middle line:90% to think about the determinant. 01:11:05.360 --> 01:11:10.170 align:middle line:84% What do we mean by determinant here? 01:11:10.170 --> 01:11:12.230 align:middle line:84% So we do it in the same way as you 01:11:12.230 --> 01:11:13.805 align:middle line:90% would do in the matrix space. 01:11:13.805 --> 01:11:21.900 align:middle line:90% 01:11:21.900 --> 01:11:26.680 align:middle line:84% So if you have a finite dimensional matrix, 01:11:26.680 --> 01:11:29.800 align:middle line:84% there's a way to calculate the determinant, yeah, 01:11:29.800 --> 01:11:32.230 align:middle line:84% the standard way, which you learned. 01:11:32.230 --> 01:11:35.110 align:middle line:90% 01:11:35.110 --> 01:11:38.830 align:middle line:84% Say a1, a1 1, a2 2, et cetera, there's a complicated formula, 01:11:38.830 --> 01:11:40.600 align:middle line:90% OK. 01:11:40.600 --> 01:11:46.570 align:middle line:84% But that way does not generalize to such a k, 01:11:46.570 --> 01:11:49.940 align:middle line:90% with continuous indices. 01:11:49.940 --> 01:11:52.450 align:middle line:84% So we must find a way to generalize the determinant, 01:11:52.450 --> 01:11:59.320 align:middle line:84% here, in a way, which can apply to such a k, which is defined 01:11:59.320 --> 01:12:02.060 align:middle line:90% in the space of functions. 01:12:02.060 --> 01:12:04.030 align:middle line:84% So there's a lot of way we can calculate 01:12:04.030 --> 01:12:12.010 align:middle line:84% the determinant, a way without using the standard formula. 01:12:12.010 --> 01:12:15.430 align:middle line:90% So you do this as follows. 01:12:15.430 --> 01:12:18.630 align:middle line:84% Remember, the determinant of A is also 01:12:18.630 --> 01:12:22.460 align:middle line:84% the product of all the eigenvalues of A 01:12:22.460 --> 01:12:25.670 align:middle line:84% So you can just find all the eigenvalues of A. 01:12:25.670 --> 01:12:27.380 align:middle line:90% You take the product of them. 01:12:27.380 --> 01:12:31.250 align:middle line:84% And then that is the determinant of A. 01:12:31.250 --> 01:12:33.560 align:middle line:84% So this way to define determinant 01:12:33.560 --> 01:12:35.360 align:middle line:90% actually generalize. 01:12:35.360 --> 01:12:38.240 align:middle line:84% It doesn't matter how many eigenvalues you have. 01:12:38.240 --> 01:12:44.700 align:middle line:84% So now we can just find all the eigenvalues of k. 01:12:44.700 --> 01:12:54.320 align:middle line:84% Just find all the eigenvalues of k. 01:12:54.320 --> 01:12:56.270 align:middle line:84% And then we just take the product of them. 01:12:56.270 --> 01:12:58.520 align:middle line:90% That gives us the determinant. 01:12:58.520 --> 01:13:00.770 align:middle line:84% So this is the way which we can generalize. 01:13:00.770 --> 01:13:03.960 align:middle line:84% So how do we define the determinant eigenvalue? 01:13:03.960 --> 01:13:08.090 align:middle line:84% So when we define the eigenvalue of A m n, 01:13:08.090 --> 01:13:16.430 align:middle line:84% we find the eigenvector X will satisfy this behavior. 01:13:16.430 --> 01:13:23.300 align:middle line:84% So the n is summed, and then give you a this, right? 01:13:23.300 --> 01:13:28.250 align:middle line:84% So now the analog of the m n, here, is t and t prime. 01:13:28.250 --> 01:13:37.020 align:middle line:84% So the analog here is from 0 to T, t prime, k, t, t prime, 01:13:37.020 --> 01:13:38.490 align:middle line:90% some function, f n. 01:13:38.490 --> 01:13:41.460 align:middle line:90% 01:13:41.460 --> 01:13:44.250 align:middle line:84% So n, here, label different eigenvalues. 01:13:44.250 --> 01:13:45.300 align:middle line:90% So let me call it i. 01:13:45.300 --> 01:13:48.620 align:middle line:90% 01:13:48.620 --> 01:13:55.010 align:middle line:84% i t prime, you go to lambda i f t. 01:13:55.010 --> 01:13:57.890 align:middle line:90% 01:13:57.890 --> 01:14:01.220 align:middle line:84% So this would be the eigenvalue equation. 01:14:01.220 --> 01:14:06.380 align:middle line:84% Because now, the n index, here, is just replaced by t prime. 01:14:06.380 --> 01:14:10.290 align:middle line:84% The sum over n just replaced by integral. 01:14:10.290 --> 01:14:13.400 align:middle line:90% And then the rest is the same. 01:14:13.400 --> 01:14:17.870 align:middle line:84% So you just find all the eigenvalues of k the f t. 01:14:17.870 --> 01:14:23.240 align:middle line:84% And of course, it should satisfy the condition, here, 01:14:23.240 --> 01:14:28.212 align:middle line:90% the f 0 equal to f i-- 01:14:28.212 --> 01:14:32.090 align:middle line:84% yeah, f i T should satisfy the boundary condition. 01:14:32.090 --> 01:14:33.200 align:middle line:90% This is equal to 0. 01:14:33.200 --> 01:14:35.330 align:middle line:90% Because that's the integral. 01:14:35.330 --> 01:14:38.210 align:middle line:84% That's the condition of a path integral. 01:14:38.210 --> 01:14:40.760 align:middle line:84% The path integral, it's from 0 to 0. 01:14:40.760 --> 01:14:48.110 align:middle line:90% 01:14:48.110 --> 01:14:50.780 align:middle line:84% And you just solve this eigenvalue problem. 01:14:50.780 --> 01:14:53.270 align:middle line:84% This is a well-defined mathematical problem. 01:14:53.270 --> 01:14:57.010 align:middle line:84% Because k is just a differential operator. 01:14:57.010 --> 01:14:58.870 align:middle line:84% k is just some differential operator. 01:14:58.870 --> 01:15:00.950 align:middle line:84% And now, this becomes a well-defined mathematical 01:15:00.950 --> 01:15:01.450 align:middle line:90% problem. 01:15:01.450 --> 01:15:03.190 align:middle line:90% We can find these eigenvalues. 01:15:03.190 --> 01:15:06.220 align:middle line:84% And then we just take the product. 01:15:06.220 --> 01:15:10.070 align:middle line:84% So this way, we generalize that formula. 01:15:10.070 --> 01:15:13.420 align:middle line:84% So this is the exercise for you to do yourself. 01:15:13.420 --> 01:15:16.180 align:middle line:90% OK, k is just a quadratic. 01:15:16.180 --> 01:15:18.100 align:middle line:84% Differential is just delta t square. 01:15:18.100 --> 01:15:20.080 align:middle line:90% It's very easy. 01:15:20.080 --> 01:15:21.025 align:middle line:90% OK, it's very easy. 01:15:21.025 --> 01:15:24.870 align:middle line:90% 01:15:24.870 --> 01:15:29.950 align:middle line:90% So you find f i t. 01:15:29.950 --> 01:15:32.380 align:middle line:84% Yeah, actually, I think i is also 01:15:32.380 --> 01:15:35.830 align:middle line:90% not a very fortunate notation. 01:15:35.830 --> 01:15:39.310 align:middle line:90% Because it can be easily-- 01:15:39.310 --> 01:15:43.470 align:middle line:84% yeah, let me call it j, OK, so pi j t. 01:15:43.470 --> 01:15:48.340 align:middle line:90% 01:15:48.340 --> 01:15:51.520 align:middle line:84% So you can just solve this equation, easily, yourself. 01:15:51.520 --> 01:15:53.300 align:middle line:84% So let me just write down the answer. 01:15:53.300 --> 01:15:57.280 align:middle line:84% So the eigenvalue of pi j t divided 01:15:57.280 --> 01:16:02.230 align:middle line:84% by capital T. And with eigenvalues lambda j 01:16:02.230 --> 01:16:09.730 align:middle line:84% equal to m j square, pi squared, divided by t square. 01:16:09.730 --> 01:16:13.690 align:middle line:84% And the j equal to 1, 2, et cetera. 01:16:13.690 --> 01:16:17.420 align:middle line:90% 01:16:17.420 --> 01:16:20.320 align:middle line:90% So even though the k-- 01:16:20.320 --> 01:16:24.100 align:middle line:84% both index of k are continuous, but the eigenvalues 01:16:24.100 --> 01:16:27.400 align:middle line:84% are discrete, OK, a discrete infinite. 01:16:27.400 --> 01:16:29.620 align:middle line:84% The eigenvalue are discrete infinite, just labeled 01:16:29.620 --> 01:16:31.245 align:middle line:90% by integer j. 01:16:31.245 --> 01:16:32.620 align:middle line:84% You can easily convince yourself. 01:16:32.620 --> 01:16:35.140 align:middle line:90% Yeah, just solve this equation. 01:16:35.140 --> 01:16:36.190 align:middle line:90% It's easy to solve. 01:16:36.190 --> 01:16:40.810 align:middle line:90% 01:16:40.810 --> 01:16:43.065 align:middle line:84% So now we can find the determinant of k. 01:16:43.065 --> 01:16:49.350 align:middle line:90% 01:16:49.350 --> 01:16:51.720 align:middle line:84% Now we find the determinant of k. 01:16:51.720 --> 01:16:56.340 align:middle line:84% So this is given by product of n equal to 1i-- 01:16:56.340 --> 01:17:05.790 align:middle line:84% j equal to 1, to infinity, m j squared, 01:17:05.790 --> 01:17:07.346 align:middle line:90% pi squared divided by t square. 01:17:07.346 --> 01:17:14.180 align:middle line:90% 01:17:14.180 --> 01:17:17.510 align:middle line:90% So you say, what is this beast? 01:17:17.510 --> 01:17:20.870 align:middle line:84% So we have all these infinite things. 01:17:20.870 --> 01:17:22.100 align:middle line:90% We have these things. 01:17:22.100 --> 01:17:26.020 align:middle line:84% We have all these constants multiplied infinite times. 01:17:26.020 --> 01:17:29.320 align:middle line:90% And then we have this j square. 01:17:29.320 --> 01:17:31.330 align:middle line:90% And so now I claim-- 01:17:31.330 --> 01:17:34.330 align:middle line:90% 01:17:34.330 --> 01:17:37.810 align:middle line:84% again, I will not have time to do it here. 01:17:37.810 --> 01:17:44.560 align:middle line:84% I will claim this guy, once you do proper things, 01:17:44.560 --> 01:17:46.810 align:middle line:84% and then that gives you that answer. 01:17:46.810 --> 01:17:49.870 align:middle line:84% When you do that C over delta t, that can get you back 01:17:49.870 --> 01:17:52.210 align:middle line:90% to this answer. 01:17:52.210 --> 01:17:54.340 align:middle line:90% Get you back to that answer. 01:17:54.340 --> 01:17:58.135 align:middle line:84% So you will work through that in your homework problem. 01:17:58.135 --> 01:18:05.060 align:middle line:90% 01:18:05.060 --> 01:18:11.840 align:middle line:84% So this way, even though it looks hairy, 01:18:11.840 --> 01:18:14.660 align:middle line:84% but once you get used to it's much easier 01:18:14.660 --> 01:18:20.630 align:middle line:84% than doing those integrals and take the limit. 01:18:20.630 --> 01:18:24.020 align:middle line:84% It's much easier than doing this-- take the limit. 01:18:24.020 --> 01:18:26.852 align:middle line:84% And also, we will later see, most of the time, 01:18:26.852 --> 01:18:29.060 align:middle line:84% you actually don't need to calculate the determinant, 01:18:29.060 --> 01:18:31.190 align:middle line:90% explicitly. 01:18:31.190 --> 01:18:33.290 align:middle line:84% Later, we will see, most of the time, 01:18:33.290 --> 01:18:35.750 align:middle line:84% you don't need to calculate determinant, explicitly. 01:18:35.750 --> 01:18:38.150 align:middle line:84% You don't need to use those things. 01:18:38.150 --> 01:18:41.450 align:middle line:84% And just the path integral framework 01:18:41.450 --> 01:18:45.830 align:middle line:84% somehow provide a platform for you to do a lot of things. 01:18:45.830 --> 01:18:47.840 align:middle line:84% You don't actually need to evaluate it. 01:18:47.840 --> 01:18:50.060 align:middle line:84% It's actually provides a platform. 01:18:50.060 --> 01:18:52.490 align:middle line:84% That's the most important thing about the path integral. 01:18:52.490 --> 01:18:55.700 align:middle line:84% It's often not about how you evaluate it. 01:18:55.700 --> 01:18:58.290 align:middle line:90% 01:18:58.290 --> 01:19:00.770 align:middle line:90% Good. 01:19:00.770 --> 01:19:03.840 align:middle line:84% So before we conclude, let me just make some further remark. 01:19:03.840 --> 01:19:06.600 align:middle line:90% 01:19:06.600 --> 01:19:10.800 align:middle line:84% Because it's better if we just make a couple of remarks 01:19:10.800 --> 01:19:14.805 align:middle line:84% here so that we don't forget next time. 01:19:14.805 --> 01:19:23.370 align:middle line:90% 01:19:23.370 --> 01:19:26.725 align:middle line:84% yeah give me like two minutes to make some remark. 01:19:26.725 --> 01:19:45.380 align:middle line:90% 01:19:45.380 --> 01:19:49.190 align:middle line:84% So remember remarks I already said. 01:19:49.190 --> 01:19:51.980 align:middle line:84% We said that this is a new formulation of quantum 01:19:51.980 --> 01:19:56.150 align:middle line:84% mechanics, gives you a new conceptual way 01:19:56.150 --> 01:20:00.193 align:middle line:84% to think about quantum mechanics. 01:20:00.193 --> 01:20:01.610 align:middle line:84% I'll give you a new conceptual way 01:20:01.610 --> 01:20:05.240 align:middle line:84% to think about quantum mechanics. 01:20:05.240 --> 01:20:11.060 align:middle line:84% And you can show that these two formulations are equivalent. 01:20:11.060 --> 01:20:12.890 align:middle line:90% So this is equivalent. 01:20:12.890 --> 01:20:16.550 align:middle line:84% You can show that this is equivalent to the Schrodinger 01:20:16.550 --> 01:20:26.600 align:middle line:84% approach, to the Schrodinger formulation, 01:20:26.600 --> 01:20:30.965 align:middle line:84% by showing that the K, you calculate it this way. 01:20:30.965 --> 01:20:33.770 align:middle line:90% 01:20:33.770 --> 01:20:37.490 align:middle line:84% Using path integral, you find this K. 01:20:37.490 --> 01:20:40.005 align:middle line:84% You can show that the K actually satisfies the Schrodinger 01:20:40.005 --> 01:20:40.505 align:middle line:90% equation. 01:20:40.505 --> 01:20:52.350 align:middle line:90% 01:20:52.350 --> 01:20:54.662 align:middle line:90% So this will guarantee-- 01:20:54.662 --> 01:20:56.870 align:middle line:84% I already erased-- that guarantee your wave function, 01:20:56.870 --> 01:20:58.790 align:middle line:84% when you come over to this K, it will also 01:20:58.790 --> 01:21:01.190 align:middle line:84% satisfy the Schrodinger equation. 01:21:01.190 --> 01:21:03.050 align:middle line:84% So you can show that this actually satisfies 01:21:03.050 --> 01:21:04.092 align:middle line:90% the Schrodinger equation. 01:21:04.092 --> 01:21:09.250 align:middle line:90% 01:21:09.250 --> 01:21:11.220 align:middle line:84% So you can show that the K actually satisfies 01:21:11.220 --> 01:21:12.540 align:middle line:90% the Schrodinger equation. 01:21:12.540 --> 01:21:15.330 align:middle line:84% Again, I will leave it to your homework. 01:21:15.330 --> 01:21:21.180 align:middle line:90% 01:21:21.180 --> 01:21:21.930 align:middle line:90% Good. 01:21:21.930 --> 01:21:28.100 align:middle line:84% And the second point is just to highlight 01:21:28.100 --> 01:21:33.670 align:middle line:84% this contrast between the classical mechanics, which 01:21:33.670 --> 01:21:40.210 align:middle line:84% have a fixed path determined by equation of motion, 01:21:40.210 --> 01:21:45.790 align:middle line:84% and the quantum, you just sum over all possible paths 01:21:45.790 --> 01:21:48.550 align:middle line:90% weighted by your action. 01:21:48.550 --> 01:21:50.020 align:middle line:90% You just sum. 01:21:50.020 --> 01:21:52.270 align:middle line:84% Just a contrast, OK, just contrast. 01:21:52.270 --> 01:21:56.050 align:middle line:90% 01:21:56.050 --> 01:21:56.610 align:middle line:90% Yeah. 01:21:56.610 --> 01:21:58.630 align:middle line:90% Yeah, let's stop here. 01:21:58.630 --> 01:22:02.440 align:middle line:90% 01:22:02.440 --> 01:22:07.860 align:middle line:84% So the next time, then we are ready to go to field theory. 01:22:07.860 --> 01:22:11.880 align:middle line:84% So with this, if you are familiar with this, 01:22:11.880 --> 01:22:14.980 align:middle line:84% if you are familiar with this, get used to everything here, 01:22:14.980 --> 01:22:16.920 align:middle line:84% then going to field theory is just immediate. 01:22:16.920 --> 01:22:19.020 align:middle line:84% You're just changing the notation. 01:22:19.020 --> 01:22:20.790 align:middle line:84% Because, as we said, the field theory 01:22:20.790 --> 01:22:23.250 align:middle line:84% just goes on into quantum mechanics with infinite number 01:22:23.250 --> 01:22:24.180 align:middle line:90% of degrees of freedom. 01:22:24.180 --> 01:22:26.773 align:middle line:84% Once you understand 1 degree of freedom, 01:22:26.773 --> 01:22:28.440 align:middle line:84% then you can just generalize to infinite 01:22:28.440 --> 01:22:30.773 align:middle line:84% number of degrees of freedom just by changing notations. 01:22:30.773 --> 01:22:32.970 align:middle line:90% So this aspect will be the same. 01:22:32.970 --> 01:22:36.180 align:middle line:84% So we can immediately write down the path integral for quantum 01:22:36.180 --> 01:22:38.640 align:middle line:90% field theory next time. 01:22:38.640 --> 01:22:42.020 align:middle line:84% And then we can talk about the interactions. 01:22:42.020 --> 01:22:45.000 align:middle line:90%