WEBVTT 00:00:00.000 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:04.320 align:middle line:90% [RUSTLING] 00:00:04.320 --> 00:00:06.240 align:middle line:90% [CLICKING] 00:00:06.240 --> 00:00:11.360 align:middle line:90% 00:00:11.360 --> 00:00:13.610 align:middle line:84% PROFESSOR: So, last time, we talked 00:00:13.610 --> 00:00:19.200 align:middle line:84% about path integral formulation of quantum mechanics. 00:00:19.200 --> 00:00:21.845 align:middle line:84% So let me just quickly remind you the main idea. 00:00:21.845 --> 00:00:39.100 align:middle line:90% 00:00:39.100 --> 00:00:49.120 align:middle line:84% OK, so classically, if we specify the initial location-- 00:00:49.120 --> 00:00:52.420 align:middle line:84% say, if you add t prime, x prime, 00:00:52.420 --> 00:00:59.680 align:middle line:84% and then the final location, t and x so at time t, you add x, 00:00:59.680 --> 00:01:03.700 align:middle line:84% and then there's a unique trajectory, OK? 00:01:03.700 --> 00:01:06.310 align:middle line:90% And there's a unique trajectory. 00:01:06.310 --> 00:01:13.010 align:middle line:84% So, quantum mechanically, you ask a different question. 00:01:13.010 --> 00:01:18.560 align:middle line:84% You ask, what is the transition amplitude 00:01:18.560 --> 00:01:23.600 align:middle line:84% if you say, at t prime, you add location x prime? 00:01:23.600 --> 00:01:26.300 align:middle line:84% And what's the transition amplitude at time 00:01:26.300 --> 00:01:27.875 align:middle line:90% t you add the location x? 00:01:27.875 --> 00:01:30.542 align:middle line:90% 00:01:30.542 --> 00:01:36.510 align:middle line:84% And then the answer for this question, now, is equal to-- 00:01:36.510 --> 00:01:44.700 align:middle line:84% you just sum of all possible paths between t prime and x 00:01:44.700 --> 00:01:47.580 align:middle line:90% prime and t x, OK? 00:01:47.580 --> 00:01:50.460 align:middle line:84% So you just sum of all possible paths. 00:01:50.460 --> 00:01:53.220 align:middle line:90% And we sum weights, OK? 00:01:53.220 --> 00:01:59.460 align:middle line:84% So, more explicitly-- and with the weight, 00:01:59.460 --> 00:02:02.190 align:middle line:90% you sum of all paths. 00:02:02.190 --> 00:02:10.050 align:middle line:84% And the weight is given by exponential i h S. 00:02:10.050 --> 00:02:12.900 align:middle line:84% So, for each path, you have an action. 00:02:12.900 --> 00:02:16.050 align:middle line:84% And then you just evaluate, for each path, the action. 00:02:16.050 --> 00:02:20.040 align:middle line:84% And then, here, also, the explicit h bar 00:02:20.040 --> 00:02:25.620 align:middle line:84% here and because the action has the unit of h bar. 00:02:25.620 --> 00:02:31.350 align:middle line:84% Yeah, so this would be the quantum mechanical answer, OK? 00:02:31.350 --> 00:02:35.580 align:middle line:84% So this is a formulation of quantum mechanics 00:02:35.580 --> 00:02:38.230 align:middle line:90% using path integral. 00:02:38.230 --> 00:02:44.960 align:middle line:84% And so we can write this more explicitly, 00:02:44.960 --> 00:02:48.290 align:middle line:84% mathematical notation for the summing over paths 00:02:48.290 --> 00:02:49.650 align:middle line:90% is the following. 00:02:49.650 --> 00:02:53.930 align:middle line:84% So we introduce a notation like this, which you integrate over 00:02:53.930 --> 00:02:57.087 align:middle line:84% all possible trajectories between x 00:02:57.087 --> 00:03:02.270 align:middle line:90% t prime equal to x and x t-- 00:03:02.270 --> 00:03:05.075 align:middle line:90% x prime and x t equal to x. 00:03:05.075 --> 00:03:08.170 align:middle line:90% 00:03:08.170 --> 00:03:12.340 align:middle line:84% And then you-- with weight S x t. 00:03:12.340 --> 00:03:17.400 align:middle line:90% 00:03:17.400 --> 00:03:23.880 align:middle line:84% And the x t is a function of x t which you 00:03:23.880 --> 00:03:27.990 align:middle line:90% integrate from t prime to t. 00:03:27.990 --> 00:03:30.940 align:middle line:90% You have t double prime. 00:03:30.940 --> 00:03:36.660 align:middle line:84% So 1/2 m x dot squared minus V x, OK? 00:03:36.660 --> 00:03:39.630 align:middle line:84% So this is for the one-dimensional particle. 00:03:39.630 --> 00:03:44.040 align:middle line:90% 00:03:44.040 --> 00:03:50.070 align:middle line:84% And, also, this-- so all the trajectories-- this fixed 00:03:50.070 --> 00:03:53.700 align:middle line:84% endpoint at t prime should be equal to x prime, and the time 00:03:53.700 --> 00:03:56.130 align:middle line:90% t should be equal to x. 00:03:56.130 --> 00:04:01.880 align:middle line:84% And, also, this integration is a shorthand notation for limit. 00:04:01.880 --> 00:04:05.930 align:middle line:84% So this DX(t) should be understood 00:04:05.930 --> 00:04:13.950 align:middle line:84% as a limit, when you take angle to infinity, this factor, 00:04:13.950 --> 00:04:18.450 align:middle line:90% and divide it by 2 pi i delta t. 00:04:18.450 --> 00:04:25.440 align:middle line:84% So, essentially, you divide your paths by intervals of delta t. 00:04:25.440 --> 00:04:28.605 align:middle line:84% And then you integrate over all possible. 00:04:28.605 --> 00:04:33.430 align:middle line:90% 00:04:33.430 --> 00:04:34.090 align:middle line:90% Yeah. 00:04:34.090 --> 00:04:38.740 align:middle line:84% OK, so you separate the paths from t, 00:04:38.740 --> 00:04:45.090 align:middle line:90% t prime to t by n segments. 00:04:45.090 --> 00:04:47.840 align:middle line:90% So this is the t i. 00:04:47.840 --> 00:04:52.340 align:middle line:84% And then the location will be x i. 00:04:52.340 --> 00:04:55.940 align:middle line:84% So this is t i, and location-- and then you integrate 00:04:55.940 --> 00:04:59.090 align:middle line:90% the x i over that, OK? 00:04:59.090 --> 00:05:04.160 align:middle line:84% So this is the formulation of path integral 00:05:04.160 --> 00:05:05.990 align:middle line:90% for quantum mechanics. 00:05:05.990 --> 00:05:08.960 align:middle line:90% Any questions on this? 00:05:08.960 --> 00:05:10.035 align:middle line:90% Yes? 00:05:10.035 --> 00:05:10.900 align:middle line:84% AUDIENCE: Can you do this computation 00:05:10.900 --> 00:05:12.817 align:middle line:84% in some arbitrary basis that's not necessarily 00:05:12.817 --> 00:05:13.533 align:middle line:90% a position basis? 00:05:13.533 --> 00:05:14.200 align:middle line:90% PROFESSOR: Yeah. 00:05:14.200 --> 00:05:16.840 align:middle line:84% Yeah, in principle, you can also do this in momentum basis-- 00:05:16.840 --> 00:05:19.810 align:middle line:90% yeah, momentum space. 00:05:19.810 --> 00:05:22.810 align:middle line:84% And, also, you can actually also generalize-- 00:05:22.810 --> 00:05:28.160 align:middle line:90% 00:05:28.160 --> 00:05:34.700 align:middle line:84% yeah, we will talk about how you generalize this to other basis. 00:05:34.700 --> 00:05:37.640 align:middle line:84% But this is the fundamental formulation. 00:05:37.640 --> 00:05:40.550 align:middle line:84% And when you write it in other basis, 00:05:40.550 --> 00:05:42.950 align:middle line:84% essentially, you reduce it to this basis 00:05:42.950 --> 00:05:49.540 align:middle line:84% because this is the most intuitive way to formulate it. 00:05:49.540 --> 00:05:50.710 align:middle line:90% Any other questions? 00:05:50.710 --> 00:05:53.970 align:middle line:90% 00:05:53.970 --> 00:05:55.740 align:middle line:90% OK, good. 00:05:55.740 --> 00:06:00.750 align:middle line:84% So a simple example is just a free particle. 00:06:00.750 --> 00:06:03.670 align:middle line:90% 00:06:03.670 --> 00:06:12.020 align:middle line:84% When you have a free particle, then the S 00:06:12.020 --> 00:06:13.710 align:middle line:90% is particularly simple. 00:06:13.710 --> 00:06:21.400 align:middle line:84% You just-- integration of 1/2 m x squared, OK? 00:06:21.400 --> 00:06:26.360 align:middle line:84% So we described that when S is equal to this-- 00:06:26.360 --> 00:06:29.450 align:middle line:84% and then this is like a Gaussian integral, OK? 00:06:29.450 --> 00:06:32.800 align:middle line:84% Because this is quadratic in x, so this 00:06:32.800 --> 00:06:34.630 align:middle line:90% is like a Gaussian integral. 00:06:34.630 --> 00:06:36.640 align:middle line:90% And so you can reduce-- 00:06:36.640 --> 00:06:38.875 align:middle line:84% so, in this case, you have a Gaussian integral. 00:06:38.875 --> 00:06:44.800 align:middle line:90% 00:06:44.800 --> 00:06:47.710 align:middle line:84% Then, schematically, you have something like this. 00:06:47.710 --> 00:07:00.980 align:middle line:90% 00:07:00.980 --> 00:07:07.710 align:middle line:84% You can-- when x is of this form, 00:07:07.710 --> 00:07:10.330 align:middle line:84% you can write it in this general form. 00:07:10.330 --> 00:07:12.810 align:middle line:84% You can write the path integral in this general form. 00:07:12.810 --> 00:07:16.140 align:middle line:84% And the k, in this case, will be some delta function. 00:07:16.140 --> 00:07:21.030 align:middle line:84% Yeah, I wrote down k last time explicitly, 00:07:21.030 --> 00:07:23.250 align:middle line:84% and then this is like a Gaussian integral. 00:07:23.250 --> 00:07:29.920 align:middle line:90% 00:07:29.920 --> 00:07:33.850 align:middle line:84% And in the Gaussian integral, we can just directly write down 00:07:33.850 --> 00:07:39.640 align:middle line:84% its answer under some constant divided by det of k. 00:07:39.640 --> 00:07:43.870 align:middle line:84% So this determinant is defined in the spatial functions 00:07:43.870 --> 00:07:45.850 align:middle line:90% and C is some constant. 00:07:45.850 --> 00:07:46.890 align:middle line:90% OK. 00:07:46.890 --> 00:07:53.430 align:middle line:84% So you may also view this just as a lateral-- 00:07:53.430 --> 00:07:55.995 align:middle line:84% this also just follow from this continuum limit, OK? 00:07:55.995 --> 00:07:58.290 align:middle line:84% It follows from this discrete case, which 00:07:58.290 --> 00:08:00.330 align:middle line:90% you take a continuum limit. 00:08:00.330 --> 00:08:00.870 align:middle line:90% Yes? 00:08:00.870 --> 00:08:03.120 align:middle line:84% AUDIENCE: So when you evaluate that Gaussian integral, 00:08:03.120 --> 00:08:05.670 align:middle line:84% there's an i in front in the exponent. 00:08:05.670 --> 00:08:06.570 align:middle line:90% PROFESSOR: Right. 00:08:06.570 --> 00:08:08.773 align:middle line:84% AUDIENCE: How does this converge? 00:08:08.773 --> 00:08:09.690 align:middle line:90% PROFESSOR: Yeah, yeah. 00:08:09.690 --> 00:08:12.600 align:middle line:84% This is the same as you do the standard Gaussian integral 00:08:12.600 --> 00:08:13.710 align:middle line:90% with the i. 00:08:13.710 --> 00:08:17.130 align:middle line:84% Yeah, that integral, you can also do. 00:08:17.130 --> 00:08:19.290 align:middle line:84% AUDIENCE: But shouldn't this diverge if it's just-- 00:08:19.290 --> 00:08:22.170 align:middle line:90% PROFESSOR: No, it's a-- 00:08:22.170 --> 00:08:25.800 align:middle line:90% yeah, just with a single-- 00:08:25.800 --> 00:08:31.630 align:middle line:84% so this integral, we can define, right? 00:08:31.630 --> 00:08:34.179 align:middle line:84% Yeah, this integral-- we know how to do this integral. 00:08:34.179 --> 00:08:36.120 align:middle line:90% It's the same as this one. 00:08:36.120 --> 00:08:36.620 align:middle line:90% Yeah. 00:08:36.620 --> 00:08:40.760 align:middle line:90% 00:08:40.760 --> 00:08:44.550 align:middle line:84% So you have to do a little bit mathematical trick 00:08:44.550 --> 00:08:46.650 align:middle line:90% to do this integral. 00:08:46.650 --> 00:08:48.232 align:middle line:90% But this integral is defined. 00:08:48.232 --> 00:08:49.065 align:middle line:90% We can give a value. 00:08:49.065 --> 00:08:53.190 align:middle line:90% 00:08:53.190 --> 00:08:54.150 align:middle line:90% Other questions? 00:08:54.150 --> 00:08:59.920 align:middle line:90% 00:08:59.920 --> 00:09:01.700 align:middle line:90% OK, good. 00:09:01.700 --> 00:09:08.380 align:middle line:84% So this is the path integral for quantum mechanics 00:09:08.380 --> 00:09:11.000 align:middle line:90% with one degree of freedom. 00:09:11.000 --> 00:09:15.100 align:middle line:84% So, now, we can just immediately generalize this 00:09:15.100 --> 00:09:18.620 align:middle line:90% to more degrees of freedom, OK? 00:09:18.620 --> 00:09:21.110 align:middle line:84% And there's nothing really change. 00:09:21.110 --> 00:09:23.240 align:middle line:84% When you have more than one degree of freedom, 00:09:23.240 --> 00:09:24.650 align:middle line:90% you just integrate over-- 00:09:24.650 --> 00:09:29.480 align:middle line:84% say, if you have, say, three particles, 00:09:29.480 --> 00:09:32.180 align:middle line:84% then you just integrate the D X 1, dx2, 00:09:32.180 --> 00:09:35.660 align:middle line:84% D X 3 and then use the same action, OK? 00:09:35.660 --> 00:09:40.010 align:middle line:84% So it's just straightforward, generalize. 00:09:40.010 --> 00:09:42.196 align:middle line:84% And then we can generalize to field theory 00:09:42.196 --> 00:09:44.571 align:middle line:84% because field theory, we just take the numbers of degrees 00:09:44.571 --> 00:09:47.780 align:middle line:90% of freedom to infinity, OK? 00:09:47.780 --> 00:09:48.815 align:middle line:90% So, we can now-- 00:09:48.815 --> 00:09:54.230 align:middle line:90% 00:09:54.230 --> 00:09:54.730 align:middle line:90% yeah. 00:09:54.730 --> 00:09:57.280 align:middle line:90% 00:09:57.280 --> 00:10:01.375 align:middle line:84% Let me just-- so to field theory-- 00:10:01.375 --> 00:10:15.900 align:middle line:90% 00:10:15.900 --> 00:10:21.080 align:middle line:84% So, in this case, we just need to replace 00:10:21.080 --> 00:10:23.540 align:middle line:90% in the different places by-- 00:10:23.540 --> 00:10:26.660 align:middle line:84% so, remember, so the dynamical variable 00:10:26.660 --> 00:10:28.970 align:middle line:90% for a single particle-- 00:10:28.970 --> 00:10:30.800 align:middle line:90% you have this operator. 00:10:30.800 --> 00:10:35.810 align:middle line:84% And then when you go to field theory, the counterpart of this 00:10:35.810 --> 00:10:45.700 align:middle line:84% is the phi, is your field variable. 00:10:45.700 --> 00:10:49.240 align:middle line:90% And then-- so this quantity-- 00:10:49.240 --> 00:10:51.770 align:middle line:90% 00:10:51.770 --> 00:10:56.780 align:middle line:84% when you go to field theory, the corresponding quantity 00:10:56.780 --> 00:11:03.860 align:middle line:90% is phi x at some t and phi x-- 00:11:03.860 --> 00:11:11.525 align:middle line:84% say, phi prime x at time t prime. 00:11:11.525 --> 00:11:12.025 align:middle line:90% OK? 00:11:12.025 --> 00:11:17.460 align:middle line:90% 00:11:17.460 --> 00:11:19.020 align:middle line:84% Again, the way to think about this 00:11:19.020 --> 00:11:22.080 align:middle line:84% is just you should view this x as label. 00:11:22.080 --> 00:11:26.460 align:middle line:84% And then this is just as if we have infinite number 00:11:26.460 --> 00:11:29.490 align:middle line:90% of positions, OK? 00:11:29.490 --> 00:11:31.830 align:middle line:84% You just have infinite number of phi's, which 00:11:31.830 --> 00:11:35.550 align:middle line:84% are eigenvalues of this operator at time t, at t prime, 00:11:35.550 --> 00:11:37.095 align:middle line:90% and, similarly, here, OK? 00:11:37.095 --> 00:11:41.280 align:middle line:90% 00:11:41.280 --> 00:11:43.980 align:middle line:84% So, again, this one-- you can write it 00:11:43.980 --> 00:11:48.390 align:middle line:84% in the form in terms of the position 00:11:48.390 --> 00:11:50.610 align:middle line:84% eigenstate-- so this is the position 00:11:50.610 --> 00:11:51.855 align:middle line:90% eigenstate in field theory. 00:11:51.855 --> 00:12:01.160 align:middle line:90% 00:12:01.160 --> 00:12:03.710 align:middle line:84% OK, so this is the position of eigenstate in field theory. 00:12:03.710 --> 00:12:05.570 align:middle line:84% And then we can use the same technique 00:12:05.570 --> 00:12:07.310 align:middle line:90% we discussed last time. 00:12:07.310 --> 00:12:11.540 align:middle line:84% You just split this into many, many infinitesimal pieces 00:12:11.540 --> 00:12:14.420 align:middle line:84% by splitting the time interval from t prime 00:12:14.420 --> 00:12:18.050 align:middle line:84% to t and split into many pieces and then 00:12:18.050 --> 00:12:21.080 align:middle line:84% just do it over and over and insert a complete set of state 00:12:21.080 --> 00:12:25.630 align:middle line:90% of phi, OK? 00:12:25.630 --> 00:12:27.160 align:middle line:90% And, now, you just-- 00:12:27.160 --> 00:12:34.250 align:middle line:84% so H is now, of course, is the field counterpart of it 00:12:34.250 --> 00:12:35.535 align:middle line:90% because H-- 00:12:35.535 --> 00:12:36.160 align:middle line:90% you could, too. 00:12:36.160 --> 00:12:38.080 align:middle line:84% But the details of this does not matter. 00:12:38.080 --> 00:12:46.090 align:middle line:90% 00:12:46.090 --> 00:12:49.270 align:middle line:90% Say, this is the Hamiltonian. 00:12:49.270 --> 00:12:51.340 align:middle line:84% You can just insert this into here, 00:12:51.340 --> 00:12:53.710 align:middle line:84% and then you just repeat the same procedure, OK? 00:12:53.710 --> 00:12:56.860 align:middle line:84% Just keep in mind, now, you just have infinite number 00:12:56.860 --> 00:12:58.840 align:middle line:90% of degrees of freedom, OK? 00:12:58.840 --> 00:13:00.160 align:middle line:90% That's the only difference. 00:13:00.160 --> 00:13:02.890 align:middle line:84% Your rotation becomes a little bit more complicated. 00:13:02.890 --> 00:13:07.110 align:middle line:84% But if you keep a very clear mind, 00:13:07.110 --> 00:13:11.340 align:middle line:84% with the correspondence with the single particle case, 00:13:11.340 --> 00:13:14.100 align:middle line:90% then everything is the same, OK? 00:13:14.100 --> 00:13:17.670 align:middle line:84% If you understand how to translate the rotation here 00:13:17.670 --> 00:13:19.410 align:middle line:84% to the rotation here, then everything 00:13:19.410 --> 00:13:21.180 align:middle line:90% is just exactly the same, OK? 00:13:21.180 --> 00:13:24.250 align:middle line:90% Just everything goes through. 00:13:24.250 --> 00:13:27.220 align:middle line:84% So I will not repeat that procedure. 00:13:27.220 --> 00:13:30.160 align:middle line:84% And, now, you just find this quantity. 00:13:30.160 --> 00:13:33.070 align:middle line:84% Then, again, we can write this in terms of path integral. 00:13:33.070 --> 00:13:36.750 align:middle line:84% So, now, this can be written as in terms 00:13:36.750 --> 00:13:37.830 align:middle line:90% of path integral of phi. 00:13:37.830 --> 00:13:44.830 align:middle line:84% Now, you integrate over all possible configurations of phi 00:13:44.830 --> 00:13:47.580 align:middle line:84% with is the boundary condition phi 00:13:47.580 --> 00:13:53.640 align:middle line:90% t prime x equal to phi prime x. 00:13:53.640 --> 00:13:57.210 align:middle line:90% And the phi t x-- 00:13:57.210 --> 00:14:00.160 align:middle line:90% equal to phi x. 00:14:00.160 --> 00:14:06.400 align:middle line:84% And, again, you just have exponential i S phi. 00:14:06.400 --> 00:14:11.000 align:middle line:90% And S is the action for phi. 00:14:11.000 --> 00:14:12.305 align:middle line:90% So, now, the S-- 00:14:12.305 --> 00:14:19.730 align:middle line:90% 00:14:19.730 --> 00:14:27.920 align:middle line:90% is now-- so, in this case-- 00:14:27.920 --> 00:14:30.110 align:middle line:84% yeah, it's an integration of the Lagrangian density. 00:14:30.110 --> 00:14:34.310 align:middle line:90% And L is 1/2. 00:14:34.310 --> 00:14:36.755 align:middle line:84% Yeah, it's just what we wrote down before. 00:14:36.755 --> 00:14:45.720 align:middle line:90% 00:14:45.720 --> 00:14:46.910 align:middle line:90% OK, some potential phi-- 00:14:46.910 --> 00:14:47.410 align:middle line:90% OK. 00:14:47.410 --> 00:14:49.870 align:middle line:90% 00:14:49.870 --> 00:14:54.060 align:middle line:84% So for any scalar field theory, these just work identically, 00:14:54.060 --> 00:14:54.560 align:middle line:90% OK? 00:14:54.560 --> 00:14:56.810 align:middle line:84% And, again, you just get the integration 00:14:56.810 --> 00:14:59.720 align:middle line:84% over the Lagrangian, which is become 00:14:59.720 --> 00:15:03.080 align:middle line:84% four integral of the Lagrangian density, OK? 00:15:03.080 --> 00:15:04.100 align:middle line:90% Any questions on this? 00:15:04.100 --> 00:15:16.480 align:middle line:90% 00:15:16.480 --> 00:15:18.390 align:middle line:90% Good. 00:15:18.390 --> 00:15:27.680 align:middle line:84% So whenever you get confused about path integral in field 00:15:27.680 --> 00:15:34.430 align:middle line:84% theory, then try to translate into the language 00:15:34.430 --> 00:15:39.050 align:middle line:84% of a single particle by doing this kind of replacement. 00:15:39.050 --> 00:15:42.500 align:middle line:84% And then you will be able to settle your problem. 00:15:42.500 --> 00:15:45.200 align:middle line:84% And then whenever you get confused about the path 00:15:45.200 --> 00:15:47.580 align:middle line:84% integral in this quantum mechanics, 00:15:47.580 --> 00:15:53.690 align:middle line:84% then just reduce it by a finite dimensional integral, 00:15:53.690 --> 00:15:56.480 align:middle line:84% and then you should be able to understand, OK? 00:15:56.480 --> 00:16:00.770 align:middle line:84% Just always reduce it to the simple case. 00:16:00.770 --> 00:16:02.870 align:middle line:84% And, often, your confusion can be understood 00:16:02.870 --> 00:16:05.730 align:middle line:90% in that simpler case, OK? 00:16:05.730 --> 00:16:06.230 align:middle line:90% Yeah. 00:16:06.230 --> 00:16:11.070 align:middle line:90% 00:16:11.070 --> 00:16:11.680 align:middle line:90% Good. 00:16:11.680 --> 00:16:16.630 align:middle line:84% So, now, let me explain a little bit what this notation means, 00:16:16.630 --> 00:16:17.200 align:middle line:90% OK? 00:16:17.200 --> 00:16:20.950 align:middle line:84% So this said in words means you integrate over 00:16:20.950 --> 00:16:27.640 align:middle line:84% all possible configurations of phi between t and t prime. 00:16:27.640 --> 00:16:34.510 align:middle line:84% And the final configuration-- the initial configuration to be 00:16:34.510 --> 00:16:37.187 align:middle line:84% phi prime, and the final configuration to be phi-- 00:16:37.187 --> 00:16:38.380 align:middle line:90% phi x-- 00:16:38.380 --> 00:16:42.340 align:middle line:84% and the initial to be phi prime x, OK? 00:16:42.340 --> 00:16:44.895 align:middle line:84% So let me just say, previously, DX(t)-- 00:16:44.895 --> 00:16:48.090 align:middle line:90% 00:16:48.090 --> 00:16:51.450 align:middle line:84% if you think about it in terms of discrete case corresponding 00:16:51.450 --> 00:16:52.740 align:middle line:90% to-- you just sum-- 00:16:52.740 --> 00:16:59.310 align:middle line:84% take the product of all the o at different time d x i, OK? 00:16:59.310 --> 00:17:02.460 align:middle line:84% So in the continuum limit, it's like you just 00:17:02.460 --> 00:17:05.130 align:middle line:84% take the product of all possible value of t 00:17:05.130 --> 00:17:08.280 align:middle line:84% and then integrate over the value of x 00:17:08.280 --> 00:17:10.669 align:middle line:90% at that particular value of t. 00:17:10.669 --> 00:17:13.319 align:middle line:84% So that's the meaning of this DX(t), 00:17:13.319 --> 00:17:18.490 align:middle line:84% which is up to a prefactor corresponding to that. 00:17:18.490 --> 00:17:25.710 align:middle line:84% So, similarly, phi is the same thing. 00:17:25.710 --> 00:17:30.560 align:middle line:84% So, now, remember, phi x has now become the dynamical variable. 00:17:30.560 --> 00:17:35.100 align:middle line:90% So this now becomes sum over-- 00:17:35.100 --> 00:17:36.545 align:middle line:90% take product over t. 00:17:36.545 --> 00:17:40.510 align:middle line:90% 00:17:40.510 --> 00:17:41.010 align:middle line:90% Oh, yeah. 00:17:41.010 --> 00:17:41.740 align:middle line:90% No, no, no. 00:17:41.740 --> 00:17:46.080 align:middle line:84% First, you think of take product over x. 00:17:46.080 --> 00:17:48.600 align:middle line:90% So this is a label of x. 00:17:48.600 --> 00:17:51.150 align:middle line:84% So x are the labels of phi, and you 00:17:51.150 --> 00:17:53.130 align:middle line:90% take all possible value of x. 00:17:53.130 --> 00:17:56.340 align:middle line:84% And then each of them is just like a-- 00:17:56.340 --> 00:18:00.090 align:middle line:90% 00:18:00.090 --> 00:18:00.870 align:middle line:90% OK? 00:18:00.870 --> 00:18:06.510 align:middle line:84% So just like you have so many different variables, 00:18:06.510 --> 00:18:10.530 align:middle line:84% each x label one degree of freedom, OK? 00:18:10.530 --> 00:18:11.940 align:middle line:90% And then you have phi t. 00:18:11.940 --> 00:18:15.150 align:middle line:90% And, now, then you use this one. 00:18:15.150 --> 00:18:20.520 align:middle line:90% Then you have t x d phi. 00:18:20.520 --> 00:18:26.400 align:middle line:90% 00:18:26.400 --> 00:18:28.240 align:middle line:84% Is it clear what this equation means? 00:18:28.240 --> 00:18:34.940 align:middle line:90% 00:18:34.940 --> 00:18:38.770 align:middle line:84% So this step tells you that you have 00:18:38.770 --> 00:18:41.240 align:middle line:84% many, many different degrees of freedom. 00:18:41.240 --> 00:18:45.700 align:middle line:84% So this step is just enumerate all possible degrees of freedom 00:18:45.700 --> 00:18:47.627 align:middle line:90% which are labeled by x. 00:18:47.627 --> 00:18:49.210 align:middle line:84% It's just like your standard integral. 00:18:49.210 --> 00:18:52.660 align:middle line:84% If you have five-dimensional integral, and each-- 00:18:52.660 --> 00:18:55.930 align:middle line:84% and then you just have a product of five different variables, 00:18:55.930 --> 00:18:56.650 align:middle line:90% OK? 00:18:56.650 --> 00:19:01.150 align:middle line:84% Here, we have all different possible value of x variables. 00:19:01.150 --> 00:19:03.880 align:middle line:84% And that variable is a function of t. 00:19:03.880 --> 00:19:06.160 align:middle line:84% And then, now, we use this equation. 00:19:06.160 --> 00:19:10.090 align:middle line:84% And each of them is like here, OK? 00:19:10.090 --> 00:19:12.340 align:middle line:84% And then, now, you have the product 00:19:12.340 --> 00:19:14.560 align:middle line:90% of all possible value of t. 00:19:14.560 --> 00:19:17.950 align:middle line:84% And then you integrate, then, all possible values of phi, 00:19:17.950 --> 00:19:21.340 align:middle line:90% say, at point x and t, OK? 00:19:21.340 --> 00:19:26.970 align:middle line:84% But, remember, t always is between t prime and t, OK? 00:19:26.970 --> 00:19:27.873 align:middle line:90% Yes? 00:19:27.873 --> 00:19:30.290 align:middle line:84% AUDIENCE: If I'm understanding your notation for the first 00:19:30.290 --> 00:19:33.860 align:middle line:84% equation, the second equality-- what is plugging in-- 00:19:33.860 --> 00:19:35.540 align:middle line:90% PROFESSOR: You mean here? 00:19:35.540 --> 00:19:36.705 align:middle line:90% AUDIENCE: The one above it. 00:19:36.705 --> 00:19:37.580 align:middle line:90% No, the one above it. 00:19:37.580 --> 00:19:38.120 align:middle line:90% PROFESSOR: Yeah, yeah. 00:19:38.120 --> 00:19:39.537 align:middle line:84% AUDIENCE: So that second quality-- 00:19:39.537 --> 00:19:41.373 align:middle line:84% so when you relabel i to the t, that's-- 00:19:41.373 --> 00:19:42.290 align:middle line:90% PROFESSOR: Yeah, yeah. 00:19:42.290 --> 00:19:44.910 align:middle line:84% Yeah, here, I just write it in the continuum form. 00:19:44.910 --> 00:19:48.170 align:middle line:90% So, here, I labeled it by t i. 00:19:48.170 --> 00:19:53.340 align:middle line:84% So each location t i, I have d x i in that form. 00:19:53.340 --> 00:19:55.700 align:middle line:84% But if I go to continuum limit and go to infinite limit, 00:19:55.700 --> 00:20:01.970 align:middle line:84% essentially, at each point t, I have integration of x. 00:20:01.970 --> 00:20:08.720 align:middle line:84% And so that's roughly the continuum form of that. 00:20:08.720 --> 00:20:10.325 align:middle line:90% And, here, it's similar. 00:20:10.325 --> 00:20:13.230 align:middle line:90% 00:20:13.230 --> 00:20:15.930 align:middle line:90% OK? 00:20:15.930 --> 00:20:17.300 align:middle line:90% Good. 00:20:17.300 --> 00:20:19.550 align:middle line:84% So once you have learned this trick, 00:20:19.550 --> 00:20:21.890 align:middle line:84% do the reduction, and I think you 00:20:21.890 --> 00:20:23.690 align:middle line:84% will be able to settle all your confusions 00:20:23.690 --> 00:20:25.460 align:middle line:90% about these definitions, OK? 00:20:25.460 --> 00:20:28.880 align:middle line:90% 00:20:28.880 --> 00:20:29.380 align:middle line:90% OK. 00:20:29.380 --> 00:20:33.260 align:middle line:90% 00:20:33.260 --> 00:20:39.365 align:middle line:84% So, other than that, other than this additional label x-- 00:20:39.365 --> 00:20:50.390 align:middle line:90% so other than additional label-- 00:20:50.390 --> 00:20:55.010 align:middle line:90% 00:20:55.010 --> 00:20:56.990 align:middle line:90% I should call it here label x. 00:20:56.990 --> 00:21:11.440 align:middle line:84% So the field theory path integral for a scalar field 00:21:11.440 --> 00:21:24.320 align:middle line:84% is essentially identical to that in quantum mechanics, OK? 00:21:24.320 --> 00:21:28.630 align:middle line:90% 00:21:28.630 --> 00:21:30.100 align:middle line:90% So just to emphasize this point. 00:21:30.100 --> 00:21:34.080 align:middle line:90% 00:21:34.080 --> 00:21:36.230 align:middle line:84% So, here, actually, you can define this general. 00:21:36.230 --> 00:21:38.420 align:middle line:84% It doesn't matter what your V is, OK? 00:21:38.420 --> 00:21:41.210 align:middle line:84% You can choose whatever V you want. 00:21:41.210 --> 00:21:43.790 align:middle line:90% The story does not change. 00:21:43.790 --> 00:21:47.430 align:middle line:84% Just like when we do the single particle case, 00:21:47.430 --> 00:21:49.580 align:middle line:84% it doesn't matter what this V is, OK? 00:21:49.580 --> 00:21:52.350 align:middle line:84% You can choose arbitrary V, if you want. 00:21:52.350 --> 00:21:55.010 align:middle line:90% But if we choose V to be-- 00:21:55.010 --> 00:22:03.270 align:middle line:84% when v is for the free theory, then-- 00:22:03.270 --> 00:22:05.940 align:middle line:84% in this case, of course, a particularly simple. 00:22:05.940 --> 00:22:23.127 align:middle line:84% And then, in this case, the path integral is again Gaussian, OK? 00:22:23.127 --> 00:22:24.710 align:middle line:84% The path integral is, again, Gaussian. 00:22:24.710 --> 00:22:26.810 align:middle line:90% I will not write it again. 00:22:26.810 --> 00:22:28.070 align:middle line:90% So, again, that formula-- 00:22:28.070 --> 00:22:33.290 align:middle line:84% and this formula applies just this case, 00:22:33.290 --> 00:22:35.180 align:middle line:90% in a more complicated space. 00:22:35.180 --> 00:22:38.030 align:middle line:84% K is the operator in a more complicated space, OK? 00:22:38.030 --> 00:22:40.640 align:middle line:84% So, now, you not only have the space of t, 00:22:40.640 --> 00:22:42.950 align:middle line:84% but you also have the space of x. 00:22:42.950 --> 00:22:46.880 align:middle line:84% But once you generalize to some space of functions, 00:22:46.880 --> 00:22:47.630 align:middle line:90% it doesn't matter. 00:22:47.630 --> 00:22:49.670 align:middle line:84% This function becomes more complicated, OK? 00:22:49.670 --> 00:22:52.698 align:middle line:84% And so, conceptually, they're the same. 00:22:52.698 --> 00:22:53.990 align:middle line:90% Conceptually, they're the same. 00:22:53.990 --> 00:23:02.510 align:middle line:90% 00:23:02.510 --> 00:23:04.160 align:middle line:90% Good? 00:23:04.160 --> 00:23:10.090 align:middle line:84% So, now, for interacting theory, then, essentially, then 00:23:10.090 --> 00:23:12.290 align:middle line:84% the quantum field theory then reduce 00:23:12.290 --> 00:23:15.170 align:middle line:90% to doing this path integral, OK? 00:23:15.170 --> 00:23:17.090 align:middle line:84% If you know how to do this path integral, 00:23:17.090 --> 00:23:21.140 align:middle line:84% and then, essentially, you know how to solve the theory, OK? 00:23:21.140 --> 00:23:24.540 align:middle line:84% You know how to solve the theory. 00:23:24.540 --> 00:23:27.450 align:middle line:84% And so, as I said, in the free theory case, 00:23:27.450 --> 00:23:29.450 align:middle line:84% it's essentially reduced to a Gaussian integral. 00:23:29.450 --> 00:23:32.270 align:middle line:84% Then everything is simple, and we 00:23:32.270 --> 00:23:37.230 align:middle line:84% will go back to discuss that in a little bit more detail later. 00:23:37.230 --> 00:23:42.200 align:middle line:84% And, now, let's just think a little bit 00:23:42.200 --> 00:23:46.470 align:middle line:84% how to treat the interacting case. 00:23:46.470 --> 00:23:57.440 align:middle line:84% So, recall, our goal is to compute this object-- 00:23:57.440 --> 00:24:00.530 align:middle line:84% the vacuum expectation value of time-ordered correlation 00:24:00.530 --> 00:24:06.180 align:middle line:90% functions, OK? 00:24:06.180 --> 00:24:09.840 align:middle line:84% We want to compute this object, OK? 00:24:09.840 --> 00:24:12.150 align:middle line:84% So, now, we will discuss how we use path integral 00:24:12.150 --> 00:24:13.950 align:middle line:90% to compute this object, OK? 00:24:13.950 --> 00:24:17.170 align:middle line:90% 00:24:17.170 --> 00:24:22.495 align:middle line:84% So before doing that, let's first try to do this in-- 00:24:22.495 --> 00:24:25.980 align:middle line:90% 00:24:25.980 --> 00:24:29.010 align:middle line:84% so before doing that, let's first 00:24:29.010 --> 00:24:32.820 align:middle line:84% discuss how to do this in quantum mechanics, 00:24:32.820 --> 00:24:34.480 align:middle line:84% before doing this in field theory. 00:24:34.480 --> 00:24:36.780 align:middle line:84% And we can just first understand how 00:24:36.780 --> 00:24:40.020 align:middle line:84% we compute the similar object in quantum mechanics. 00:24:40.020 --> 00:24:41.942 align:middle line:84% And once we understand how to do that, 00:24:41.942 --> 00:24:43.650 align:middle line:84% again, the generalization to field theory 00:24:43.650 --> 00:24:45.780 align:middle line:90% will be straightforward. 00:24:45.780 --> 00:24:48.210 align:middle line:90% OK. 00:24:48.210 --> 00:24:49.550 align:middle line:90% Good? 00:24:49.550 --> 00:24:56.860 align:middle line:84% So, first, we talk about time-ordered correlation 00:24:56.860 --> 00:25:07.770 align:middle line:84% function in quantum mechanics, OK? 00:25:07.770 --> 00:25:12.240 align:middle line:84% So we will be introducing a number of tricks, 00:25:12.240 --> 00:25:17.510 align:middle line:84% and those tricks will then take over to field theory, OK? 00:25:17.510 --> 00:25:20.910 align:middle line:84% They will be carried over to field theory. 00:25:20.910 --> 00:25:25.130 align:middle line:84% So, again, let's just consider this system, OK? 00:25:25.130 --> 00:25:27.580 align:middle line:84% Let's consider a system like this-- 00:25:27.580 --> 00:25:31.780 align:middle line:84% just with the Lagrangian of this form, just one particle theory, 00:25:31.780 --> 00:25:32.710 align:middle line:90% OK? 00:25:32.710 --> 00:25:40.070 align:middle line:84% And then the analog of this object is this object-- 00:25:40.070 --> 00:25:42.010 align:middle line:90% let's call it G n-- 00:25:42.010 --> 00:25:47.680 align:middle line:84% would be, say-- so let's call the 0 is the vacuum 00:25:47.680 --> 00:25:49.120 align:middle line:90% of the field theory-- 00:25:49.120 --> 00:25:50.890 align:middle line:84% of this quantum mechanical system. 00:25:50.890 --> 00:25:54.040 align:middle line:84% And then you have time-ordered, and then you 00:25:54.040 --> 00:25:59.560 align:middle line:84% have x t 1 because the dynamical variable here for quantum 00:25:59.560 --> 00:26:05.650 align:middle line:84% mechanics is just x and x t n at different time and then 00:26:05.650 --> 00:26:06.520 align:middle line:90% here, OK? 00:26:06.520 --> 00:26:11.920 align:middle line:84% So we want to compute such object in quantum mechanics. 00:26:11.920 --> 00:26:17.700 align:middle line:90% 00:26:17.700 --> 00:26:19.720 align:middle line:90% OK? 00:26:19.720 --> 00:26:23.260 align:middle line:84% So our goal is to develop techniques-- yeah, 00:26:23.260 --> 00:26:26.200 align:middle line:84% so you should imagine all these are operators, OK? 00:26:26.200 --> 00:26:29.150 align:middle line:90% All these are operators. 00:26:29.150 --> 00:26:32.485 align:middle line:84% So our goal is to develop techniques to calculate this. 00:26:32.485 --> 00:26:37.250 align:middle line:90% 00:26:37.250 --> 00:26:44.240 align:middle line:84% OK, so before doing that, let's first 00:26:44.240 --> 00:26:49.970 align:middle line:84% understand how to do this time-ordering using 00:26:49.970 --> 00:26:51.350 align:middle line:90% path integral, OK? 00:26:51.350 --> 00:26:54.300 align:middle line:90% 00:26:54.300 --> 00:26:56.850 align:middle line:84% So before I do that, do you have any questions? 00:26:56.850 --> 00:27:03.555 align:middle line:90% 00:27:03.555 --> 00:27:04.055 align:middle line:90% OK. 00:27:04.055 --> 00:27:08.550 align:middle line:90% 00:27:08.550 --> 00:27:13.710 align:middle line:84% So it turns out, actually, path integral 00:27:13.710 --> 00:27:18.513 align:middle line:84% is some of the natural, the most natural objects 00:27:18.513 --> 00:27:20.680 align:middle line:84% to think about such kind of time-ordered correlation 00:27:20.680 --> 00:27:22.390 align:middle line:90% functions, OK? 00:27:22.390 --> 00:27:25.630 align:middle line:84% And we will see, in a minute, the path integral 00:27:25.630 --> 00:27:28.450 align:middle line:84% is actually the most natural thing to-- 00:27:28.450 --> 00:27:30.130 align:middle line:84% the most natural framework to think 00:27:30.130 --> 00:27:32.960 align:middle line:84% about this kind of time-ordered correlation functions. 00:27:32.960 --> 00:27:37.120 align:middle line:84% So before doing that, let's consider one simple example. 00:27:37.120 --> 00:27:47.100 align:middle line:84% So, firstly, consider-- so let's consider some other time, t 1, 00:27:47.100 --> 00:27:50.190 align:middle line:90% between t prime and t, OK? 00:27:50.190 --> 00:27:53.835 align:middle line:84% So we consider again, we go back to this problem, OK? 00:27:53.835 --> 00:27:54.960 align:middle line:90% We go back to this problem. 00:27:54.960 --> 00:27:58.680 align:middle line:84% But we can see there's some time in between t prime and t, OK? 00:27:58.680 --> 00:28:02.200 align:middle line:84% So, now, let's consider this object. 00:28:02.200 --> 00:28:05.200 align:middle line:84% So the simplest case of this one-- 00:28:05.200 --> 00:28:06.670 align:middle line:90% you just take n equal to 1, OK? 00:28:06.670 --> 00:28:08.050 align:middle line:90% You just have one of them. 00:28:08.050 --> 00:28:11.420 align:middle line:90% 00:28:11.420 --> 00:28:12.795 align:middle line:84% Let's consider the simplest case. 00:28:12.795 --> 00:28:21.180 align:middle line:90% 00:28:21.180 --> 00:28:24.310 align:middle line:90% So let's consider this object. 00:28:24.310 --> 00:28:27.190 align:middle line:90% So this is easy to do. 00:28:27.190 --> 00:28:30.880 align:middle line:84% We can just use the same technique we used before. 00:28:30.880 --> 00:28:39.270 align:middle line:84% So, essentially, most of the tricks in quantum mechanics 00:28:39.270 --> 00:28:42.480 align:middle line:90% just reduce to one trick. 00:28:42.480 --> 00:28:44.700 align:middle line:84% Yeah, I say most of the case, OK? 00:28:44.700 --> 00:28:46.950 align:middle line:84% Most of the case, tricks in quantum mechanics 00:28:46.950 --> 00:28:49.500 align:middle line:90% reduce to one trick. 00:28:49.500 --> 00:28:53.740 align:middle line:84% What is that trick, which we already used over and over, 00:28:53.740 --> 00:28:56.295 align:middle line:84% say, in driving the path integral? 00:28:56.295 --> 00:28:57.170 align:middle line:90% AUDIENCE: Identities? 00:28:57.170 --> 00:28:59.400 align:middle line:84% PROFESSOR: Yeah, insert identities. 00:28:59.400 --> 00:29:02.300 align:middle line:84% So if you know how to insert identities 00:29:02.300 --> 00:29:04.310 align:middle line:84% at the right location at the right time, 00:29:04.310 --> 00:29:08.600 align:middle line:90% and, essentially, you can-- 00:29:08.600 --> 00:29:12.570 align:middle line:84% yeah, you know all the tricks in quantum mechanics. 00:29:12.570 --> 00:29:14.970 align:middle line:84% And so, here, we do the same thing. 00:29:14.970 --> 00:29:18.650 align:middle line:84% So since this, at t 1, so we just 00:29:18.650 --> 00:29:22.760 align:middle line:84% insert the complete set of position eigenstate at t 1. 00:29:22.760 --> 00:29:26.400 align:middle line:90% So let's just do this. 00:29:26.400 --> 00:29:44.750 align:middle line:84% So we integrate x 1, t 1 and the x 1, t 1, x hat, t 1, x prime, 00:29:44.750 --> 00:29:47.090 align:middle line:90% t prime, OK? 00:29:47.090 --> 00:29:49.600 align:middle line:90% 00:29:49.600 --> 00:29:51.220 align:middle line:90% So these are the-- 00:29:51.220 --> 00:29:54.930 align:middle line:90% so this integrates to 1, OK? 00:29:54.930 --> 00:29:57.470 align:middle line:90% This integrates to 1. 00:29:57.470 --> 00:30:04.820 align:middle line:84% So since this is eigenstate at t1, so this act on this will-- 00:30:04.820 --> 00:30:07.640 align:middle line:90% you just get your eigenvalue. 00:30:07.640 --> 00:30:13.040 align:middle line:90% So we just get d x1, x1. 00:30:13.040 --> 00:30:15.850 align:middle line:90% So you just get eigenvalue. 00:30:15.850 --> 00:30:25.435 align:middle line:84% And then you had xt, x1 t1, x1 t1, x prime, t prime, OK? 00:30:25.435 --> 00:30:30.130 align:middle line:90% 00:30:30.130 --> 00:30:34.260 align:middle line:84% But, now, we know how to do both of them. 00:30:34.260 --> 00:30:37.710 align:middle line:84% We just plug in our path integral formula 00:30:37.710 --> 00:30:41.740 align:middle line:90% to both of them, OK? 00:30:41.740 --> 00:30:45.870 align:middle line:90% 00:30:45.870 --> 00:30:47.450 align:middle line:90% So you can do this explicitly-- 00:30:47.450 --> 00:30:51.880 align:middle line:90% 00:30:51.880 --> 00:30:57.040 align:middle line:84% plug in in this expression there, OK? 00:30:57.040 --> 00:30:58.375 align:middle line:90% Plug in this expression there. 00:30:58.375 --> 00:31:01.250 align:middle line:90% 00:31:01.250 --> 00:31:02.780 align:middle line:90% But it's also very intuitive. 00:31:02.780 --> 00:31:06.720 align:middle line:84% You know what this looks like, OK? 00:31:06.720 --> 00:31:09.370 align:middle line:90% So this is like the following. 00:31:09.370 --> 00:31:15.820 align:middle line:84% So, previously, we just do all the paths between these two, 00:31:15.820 --> 00:31:16.320 align:middle line:90% OK? 00:31:16.320 --> 00:31:22.780 align:middle line:90% So, now, we are-- 00:31:22.780 --> 00:31:27.880 align:middle line:84% now you have this x t, and now we have t1. 00:31:27.880 --> 00:31:30.400 align:middle line:90% So suppose this is the time, t1. 00:31:30.400 --> 00:31:33.340 align:middle line:84% And suppose t goes up, and so this is t prime. 00:31:33.340 --> 00:31:35.930 align:middle line:90% 00:31:35.930 --> 00:31:40.190 align:middle line:84% So suppose this is the time t1, and this is t at prime, x 00:31:40.190 --> 00:31:42.930 align:middle line:90% prime, OK? 00:31:42.930 --> 00:31:45.046 align:middle line:90% So this corresponding to you-- 00:31:45.046 --> 00:31:49.140 align:middle line:84% you first do the path integral to some location t1-- 00:31:49.140 --> 00:31:51.900 align:middle line:90% integrate over all paths here. 00:31:51.900 --> 00:31:56.420 align:middle line:84% And then you multiply x1, and then you 00:31:56.420 --> 00:32:00.140 align:middle line:84% do all the path integral here, OK? 00:32:00.140 --> 00:32:03.210 align:middle line:84% And then you iterate over x1, OK? 00:32:03.210 --> 00:32:07.660 align:middle line:90% You integrate x1, OK? 00:32:07.660 --> 00:32:11.090 align:middle line:84% So, now, if you-- without this x1-- 00:32:11.090 --> 00:32:13.290 align:middle line:84% if we are without x1, then, of course, 00:32:13.290 --> 00:32:16.370 align:middle line:84% this is trivially equal to the previous one 00:32:16.370 --> 00:32:20.150 align:middle line:84% because you do all possible paths to t1 00:32:20.150 --> 00:32:22.190 align:middle line:84% and all possible paths from t1 to t, 00:32:22.190 --> 00:32:24.830 align:middle line:84% and then you iterate over all possible location here. 00:32:24.830 --> 00:32:27.740 align:middle line:84% Then it's the same as you integrate from here to there-- 00:32:27.740 --> 00:32:29.420 align:middle line:90% the arbitrary path, OK? 00:32:29.420 --> 00:32:34.310 align:middle line:84% So the only difference is, now, we multiply by x1 00:32:34.310 --> 00:32:37.130 align:middle line:90% and the value at t1. 00:32:37.130 --> 00:32:38.830 align:middle line:84% And, now, we can immediately write down, 00:32:38.830 --> 00:32:40.920 align:middle line:84% using path integral, what is this object? 00:32:40.920 --> 00:32:43.760 align:middle line:90% So this object is, essentially-- 00:32:43.760 --> 00:32:47.310 align:middle line:90% OK, it's just equal to-- 00:32:47.310 --> 00:32:51.205 align:middle line:84% so, now, I will use a simplified notation 00:32:51.205 --> 00:33:01.120 align:middle line:84% and say, here, write x prime, t prime, x t to be the limit, OK? 00:33:01.120 --> 00:33:03.730 align:middle line:84% And then, again, we integrate all possible paths 00:33:03.730 --> 00:33:04.540 align:middle line:90% between them. 00:33:04.540 --> 00:33:07.300 align:middle line:90% 00:33:07.300 --> 00:33:11.545 align:middle line:84% But, then, we just have x t1 here. 00:33:11.545 --> 00:33:16.580 align:middle line:90% 00:33:16.580 --> 00:33:19.370 align:middle line:84% You integrate over all possible paths between them. 00:33:19.370 --> 00:33:21.350 align:middle line:90% Just add, this time, t1. 00:33:21.350 --> 00:33:25.190 align:middle line:84% You multiply in the integrand the value of x at t1. 00:33:25.190 --> 00:33:30.260 align:middle line:84% Yeah, so this is essentially just the x1 here, OK? 00:33:30.260 --> 00:33:32.540 align:middle line:90% Is this clear? 00:33:32.540 --> 00:33:35.930 align:middle line:84% You can do this explicitly by plugging in those formulas 00:33:35.930 --> 00:33:37.010 align:middle line:90% and then manipulating. 00:33:37.010 --> 00:33:38.270 align:middle line:90% You will get this. 00:33:38.270 --> 00:33:44.310 align:middle line:84% But it's much easier to understand it heuristically 00:33:44.310 --> 00:33:44.985 align:middle line:90% using a diagram. 00:33:44.985 --> 00:33:50.700 align:middle line:90% 00:33:50.700 --> 00:33:52.880 align:middle line:90% Good? 00:33:52.880 --> 00:33:58.200 align:middle line:84% So this is very, very suggestive. 00:33:58.200 --> 00:34:04.020 align:middle line:84% It tells you, when we insert an operator here, 00:34:04.020 --> 00:34:07.320 align:middle line:84% what we do is just we translate the eigenvalue 00:34:07.320 --> 00:34:11.969 align:middle line:84% of this operator, plug it in the integrand. 00:34:11.969 --> 00:34:16.409 align:middle line:90% 00:34:16.409 --> 00:34:19.409 align:middle line:84% So, now, you can do the same thing. 00:34:19.409 --> 00:34:25.100 align:middle line:84% Suppose you have two operators, OK? 00:34:25.100 --> 00:34:28.400 align:middle line:90% So, now, let's look at-- 00:34:28.400 --> 00:34:29.750 align:middle line:90% you have two operators. 00:34:29.750 --> 00:34:36.865 align:middle line:90% 00:34:36.865 --> 00:34:37.365 align:middle line:90% OK? 00:34:37.365 --> 00:34:45.000 align:middle line:90% 00:34:45.000 --> 00:34:48.989 align:middle line:84% So you can almost immediately write down-- 00:34:48.989 --> 00:34:52.080 align:middle line:84% if you try to generalize that, what would you 00:34:52.080 --> 00:34:56.420 align:middle line:90% write down this, the answer? 00:34:56.420 --> 00:34:57.139 align:middle line:90% Yes? 00:34:57.139 --> 00:35:00.920 align:middle line:84% AUDIENCE: State the x and t1 and the x and t2 as an integral. 00:35:00.920 --> 00:35:02.600 align:middle line:90% PROFESSOR: Good. 00:35:02.600 --> 00:35:03.275 align:middle line:90% So you just-- 00:35:03.275 --> 00:35:08.040 align:middle line:90% 00:35:08.040 --> 00:35:08.540 align:middle line:90% DX(t). 00:35:08.540 --> 00:35:19.989 align:middle line:90% 00:35:19.989 --> 00:35:23.880 align:middle line:90% But this is almost correct-- 00:35:23.880 --> 00:35:29.590 align:middle line:84% not completely correct-- for one reason, which, actually, one 00:35:29.590 --> 00:35:30.880 align:middle line:90% of you asked before. 00:35:30.880 --> 00:35:33.120 align:middle line:90% Yes? 00:35:33.120 --> 00:35:35.120 align:middle line:84% AUDIENCE: Just because you haven't ordered them. 00:35:35.120 --> 00:35:36.050 align:middle line:90% PROFESSOR: Exactly. 00:35:36.050 --> 00:35:48.400 align:middle line:84% This is only equal to that only for when t1 is greater than t2. 00:35:48.400 --> 00:35:51.460 align:middle line:84% So, remember, all these paths-- they 00:35:51.460 --> 00:35:53.650 align:middle line:90% don't come back in time, OK? 00:35:53.650 --> 00:35:55.150 align:middle line:84% The paths-- they go forward in time. 00:35:55.150 --> 00:35:58.390 align:middle line:90% They don't come back. 00:35:58.390 --> 00:36:03.100 align:middle line:84% So, here, it's like you have to insert the two-- 00:36:03.100 --> 00:36:05.650 align:middle line:84% here, you just insert the two sets of complete states. 00:36:05.650 --> 00:36:06.530 align:middle line:90% One is here. 00:36:06.530 --> 00:36:07.540 align:middle line:90% One is here. 00:36:07.540 --> 00:36:10.040 align:middle line:84% But the path integral cannot come back, OK? 00:36:10.040 --> 00:36:10.540 align:middle line:90% So 00:36:10.540 --> 00:36:13.090 align:middle line:84% The order here have to be the same as the ordering 00:36:13.090 --> 00:36:15.690 align:middle line:90% of the path integral, OK? 00:36:15.690 --> 00:36:18.150 align:middle line:84% It has to be the same ordering of the path integral. 00:36:18.150 --> 00:36:26.035 align:middle line:84% So that means this is equal only for t1 greater than t2, OK? 00:36:26.035 --> 00:36:31.140 align:middle line:90% 00:36:31.140 --> 00:36:40.750 align:middle line:84% But the same thing happens but for t2 greater than t1. 00:36:40.750 --> 00:36:45.310 align:middle line:84% So suppose t2 is greater than t1. 00:36:45.310 --> 00:36:47.470 align:middle line:84% If you start this expression, then we can ask, 00:36:47.470 --> 00:36:50.282 align:middle line:84% what the corresponding operator form? 00:36:50.282 --> 00:36:52.240 align:middle line:84% So what do you think will be the operator form? 00:36:52.240 --> 00:36:58.780 align:middle line:90% 00:36:58.780 --> 00:37:03.100 align:middle line:84% Yeah, so for t1 greater than t2, we have this equal to that, OK? 00:37:03.100 --> 00:37:04.660 align:middle line:90% But, now, let's ask-- 00:37:04.660 --> 00:37:08.680 align:middle line:84% suppose it's opposite-- t2 greater than t1. 00:37:08.680 --> 00:37:11.080 align:middle line:84% But, on the right-hand side, we still have this one. 00:37:11.080 --> 00:37:13.838 align:middle line:84% But what should be the left-hand side? 00:37:13.838 --> 00:37:15.230 align:middle line:90% AUDIENCE: x t2 and x t1. 00:37:15.230 --> 00:37:16.022 align:middle line:90% PROFESSOR: Exactly. 00:37:16.022 --> 00:37:19.850 align:middle line:84% We just exchange x2 and t1 because path integral always 00:37:19.850 --> 00:37:21.330 align:middle line:90% follows the time order. 00:37:21.330 --> 00:37:24.800 align:middle line:84% So that means the order here has always to be time-ordered. 00:37:24.800 --> 00:37:27.120 align:middle line:84% So that means that the correct formula, 00:37:27.120 --> 00:37:31.290 align:middle line:90% which applies for all t1, t2-- 00:37:31.290 --> 00:37:32.730 align:middle line:90% it's just you time-order them. 00:37:32.730 --> 00:37:38.370 align:middle line:90% 00:37:38.370 --> 00:37:39.600 align:middle line:90% You just time-order them. 00:37:39.600 --> 00:37:44.320 align:middle line:90% 00:37:44.320 --> 00:37:48.820 align:middle line:84% So no matter for what the value of t1 and t2, 00:37:48.820 --> 00:37:52.810 align:middle line:84% if you time-order them so that the x which is the larger time 00:37:52.810 --> 00:37:57.810 align:middle line:84% always sit in the front, then it's always equal to this one, 00:37:57.810 --> 00:37:58.980 align:middle line:90% OK? 00:37:58.980 --> 00:38:05.120 align:middle line:84% So this, now, I'll just immediately generalize, OK? 00:38:05.120 --> 00:38:26.170 align:middle line:84% So this tells you, because of the path integral-- in the path 00:38:26.170 --> 00:38:28.420 align:middle line:84% integral, the time only goes forward. 00:38:28.420 --> 00:38:36.930 align:middle line:90% 00:38:36.930 --> 00:38:40.230 align:middle line:90% The time only goes forward. 00:38:40.230 --> 00:38:56.530 align:middle line:84% Then the time ordering naturally arises in path integrals, OK? 00:38:56.530 --> 00:39:00.430 align:middle line:90% 00:39:00.430 --> 00:39:03.710 align:middle line:84% So you can immediately generalize this. 00:39:03.710 --> 00:39:04.760 align:middle line:90% So for any t-- 00:39:04.760 --> 00:39:08.140 align:middle line:90% 00:39:08.140 --> 00:39:17.690 align:middle line:84% for any t1, tn between t prime and t, 00:39:17.690 --> 00:39:31.630 align:middle line:84% then you always have x t time-ordered, x hat t1, x tn, 00:39:31.630 --> 00:39:32.350 align:middle line:90% 0. 00:39:32.350 --> 00:39:48.040 align:middle line:84% And this is equal to x prime, t prime, x t, DX(t), and x t1, x 00:39:48.040 --> 00:39:54.210 align:middle line:90% tn, exponential i S x t, OK? 00:39:54.210 --> 00:39:59.200 align:middle line:90% 00:39:59.200 --> 00:40:09.100 align:middle line:84% So that's why we say, earlier, that, using path integral, 00:40:09.100 --> 00:40:10.780 align:middle line:84% our goal is to compute this object. 00:40:10.780 --> 00:40:13.390 align:middle line:84% So that's why, actually, using path integral to go compute it 00:40:13.390 --> 00:40:17.290 align:middle line:84% is very natural because the time-ordering-- it's just 00:40:17.290 --> 00:40:20.980 align:middle line:84% very natural from the point of view of path integral. 00:40:20.980 --> 00:40:21.982 align:middle line:90% Yes? 00:40:21.982 --> 00:40:23.458 align:middle line:84% AUDIENCE: Shouldn't the x be x' t'? 00:40:23.458 --> 00:40:27.073 align:middle line:90% 00:40:27.073 --> 00:40:28.240 align:middle line:90% PROFESSOR: Oh, sorry, sorry. 00:40:28.240 --> 00:40:31.150 align:middle line:90% Yeah, yeah, yeah. 00:40:31.150 --> 00:40:31.680 align:middle line:90% Right. 00:40:31.680 --> 00:40:32.180 align:middle line:90% Good. 00:40:32.180 --> 00:40:39.100 align:middle line:90% 00:40:39.100 --> 00:40:41.180 align:middle line:90% Any questions on this? 00:40:41.180 --> 00:40:42.355 align:middle line:90% So this is a key formula. 00:40:42.355 --> 00:40:45.513 align:middle line:90% 00:40:45.513 --> 00:40:46.805 align:middle line:90% OK, so this is the key formula. 00:40:46.805 --> 00:40:51.020 align:middle line:90% 00:40:51.020 --> 00:40:51.520 align:middle line:90% OK? 00:40:51.520 --> 00:40:55.850 align:middle line:90% 00:40:55.850 --> 00:40:56.350 align:middle line:90% Good. 00:40:56.350 --> 00:40:57.267 align:middle line:90% Any questions on this? 00:40:57.267 --> 00:40:59.780 align:middle line:90% 00:40:59.780 --> 00:41:03.270 align:middle line:84% So, now, we have got the time-ordering. 00:41:03.270 --> 00:41:05.190 align:middle line:90% OK, now, we want to compute. 00:41:05.190 --> 00:41:08.790 align:middle line:84% But here is the vacuum correlation function, OK? 00:41:08.790 --> 00:41:13.110 align:middle line:84% So we have to actually go to the vacuum. 00:41:13.110 --> 00:41:19.740 align:middle line:84% Here, it's in such between the position eigenstate, OK? 00:41:19.740 --> 00:41:22.920 align:middle line:84% So, now, we have to see how to do this for the vacuum. 00:41:22.920 --> 00:41:26.670 align:middle line:84% So, now, let me just introduce a simplified rotation. 00:41:26.670 --> 00:41:30.330 align:middle line:84% So let me call this whole thing X-- 00:41:30.330 --> 00:41:33.590 align:middle line:90% capital X, OK? 00:41:33.590 --> 00:41:38.450 align:middle line:84% I call this whole thing capital X. 00:41:38.450 --> 00:41:44.670 align:middle line:84% So, now, so we want to consider the vacuum correlation 00:41:44.670 --> 00:41:45.170 align:middle line:90% function. 00:41:45.170 --> 00:41:54.710 align:middle line:90% 00:41:54.710 --> 00:41:58.455 align:middle line:90% So the idea is very simple, OK? 00:41:58.455 --> 00:42:19.520 align:middle line:90% 00:42:19.520 --> 00:42:20.030 align:middle line:90% Right. 00:42:20.030 --> 00:42:21.020 align:middle line:90% OK. 00:42:21.020 --> 00:42:24.050 align:middle line:84% So we want to consider a vacuum function. 00:42:24.050 --> 00:42:37.240 align:middle line:84% We want to interested in the G n for arbitrary t1, 00:42:37.240 --> 00:42:42.400 align:middle line:84% tn belongs to minus infinity to plus infinity, OK? 00:42:42.400 --> 00:42:44.950 align:middle line:90% 00:42:44.950 --> 00:42:46.195 align:middle line:90% So you want to compute this. 00:42:46.195 --> 00:42:52.780 align:middle line:90% 00:42:52.780 --> 00:42:56.140 align:middle line:84% OK, and the G n, using our notation, 00:42:56.140 --> 00:43:00.100 align:middle line:90% we know equal to 0, X, 0, OK? 00:43:00.100 --> 00:43:05.950 align:middle line:90% 00:43:05.950 --> 00:43:09.720 align:middle line:90% So I denote 0 for that. 00:43:09.720 --> 00:43:15.340 align:middle line:84% So now, again, we insert a complete set of states, OK? 00:43:15.340 --> 00:43:17.860 align:middle line:84% Well, again, we insert a complete set of states. 00:43:17.860 --> 00:43:28.340 align:middle line:84% We insert 1 equal to dx, x t, which 00:43:28.340 --> 00:43:35.270 align:middle line:84% is also equal to dx prime, x prime, t prime, x prime, t 00:43:35.270 --> 00:43:37.010 align:middle line:90% prime, OK? 00:43:37.010 --> 00:43:39.820 align:middle line:90% 00:43:39.820 --> 00:43:45.670 align:middle line:84% This t goes to plus infinity, and t prime 00:43:45.670 --> 00:43:50.040 align:middle line:84% goes to minus infinity into here, OK? 00:43:50.040 --> 00:43:52.525 align:middle line:90% So insert that into here. 00:43:52.525 --> 00:43:53.025 align:middle line:90% OK? 00:43:53.025 --> 00:43:55.550 align:middle line:90% 00:43:55.550 --> 00:43:56.720 align:middle line:90% And then we find-- 00:43:56.720 --> 00:44:00.470 align:middle line:90% 00:44:00.470 --> 00:44:01.550 align:middle line:90% yeah, so let me-- 00:44:01.550 --> 00:44:06.060 align:middle line:84% I need the bigger space, so let me just write it down. 00:44:06.060 --> 00:44:07.040 align:middle line:90% So if you do that-- 00:44:07.040 --> 00:44:11.330 align:middle line:90% 00:44:11.330 --> 00:44:12.970 align:middle line:90% so this trick is general. 00:44:12.970 --> 00:44:16.450 align:middle line:84% It does not restrict to the vacuum. 00:44:16.450 --> 00:44:19.900 align:middle line:84% So, in any case, evaluating any states, 00:44:19.900 --> 00:44:24.570 align:middle line:84% you can reduce it to the path integral 00:44:24.570 --> 00:44:29.990 align:middle line:84% by doing this trick of inserting these guys. 00:44:29.990 --> 00:44:34.550 align:middle line:84% So then we have Gn, now, just equal to-- 00:44:34.550 --> 00:44:38.450 align:middle line:84% now we have limit t goes to plus infinity, 00:44:38.450 --> 00:44:41.930 align:middle line:90% t prime goes to minus infinity. 00:44:41.930 --> 00:44:45.020 align:middle line:90% Then we have dx, dx prime. 00:44:45.020 --> 00:44:50.870 align:middle line:84% So we insert one here and the one here, in both places. 00:44:50.870 --> 00:44:52.820 align:middle line:90% Here, we insert the upper line. 00:44:52.820 --> 00:44:55.550 align:middle line:84% Here, we insert the lower line, OK? 00:44:55.550 --> 00:45:07.870 align:middle line:84% Then we have 0, x, t, then x t, x, x prime, 00:45:07.870 --> 00:45:14.045 align:middle line:84% t prime, and then x prime, t prime, then 0-- 00:45:14.045 --> 00:45:17.150 align:middle line:90% 00:45:17.150 --> 00:45:19.720 align:middle line:90% times. 00:45:19.720 --> 00:45:20.220 align:middle line:90% OK? 00:45:20.220 --> 00:45:25.690 align:middle line:90% 00:45:25.690 --> 00:45:32.760 align:middle line:90% So, now, we just have-- 00:45:32.760 --> 00:45:36.280 align:middle line:90% yeah, so let me just save some-- 00:45:36.280 --> 00:45:41.840 align:middle line:84% so this then become dx, dx prime. 00:45:41.840 --> 00:45:43.015 align:middle line:90% So these two are simple. 00:45:43.015 --> 00:45:45.650 align:middle line:90% 00:45:45.650 --> 00:45:49.055 align:middle line:84% So, essentially, they just become the vacuum wave 00:45:49.055 --> 00:45:51.330 align:middle line:90% function, OK? 00:45:51.330 --> 00:46:01.092 align:middle line:84% So we have psi 0 star x and psi 0 x prime. 00:46:01.092 --> 00:46:03.300 align:middle line:84% And then we have this path integral, which we already 00:46:03.300 --> 00:46:06.480 align:middle line:90% know how to do, OK? 00:46:06.480 --> 00:46:08.847 align:middle line:84% Then we have this path integral, OK? 00:46:08.847 --> 00:46:10.180 align:middle line:90% Then we have that path integral. 00:46:10.180 --> 00:46:15.870 align:middle line:90% 00:46:15.870 --> 00:46:18.090 align:middle line:90% So you just insert-- 00:46:18.090 --> 00:46:22.080 align:middle line:84% so when we call star, you just insert the star here, OK? 00:46:22.080 --> 00:46:25.930 align:middle line:90% 00:46:25.930 --> 00:46:30.010 align:middle line:84% And, now, with t and t prime to infinity-- 00:46:30.010 --> 00:46:33.470 align:middle line:90% 00:46:33.470 --> 00:46:36.260 align:middle line:84% and, yeah, the reason we take t prime and t go to infinity 00:46:36.260 --> 00:46:39.210 align:middle line:84% is obvious because we want to include-- 00:46:39.210 --> 00:46:42.440 align:middle line:84% so, here, the t has to go between the t prime and the t. 00:46:42.440 --> 00:46:44.780 align:middle line:84% So if we want to have arbitrary t-- 00:46:44.780 --> 00:46:49.320 align:middle line:84% and then we want the t prime to go to minus infinity 00:46:49.320 --> 00:46:53.150 align:middle line:90% and t go to plus infinity. 00:46:53.150 --> 00:46:53.960 align:middle line:90% OK? 00:46:53.960 --> 00:47:03.110 align:middle line:84% So we have used that-- the x prime, t prime 0 is just 00:47:03.110 --> 00:47:08.500 align:middle line:84% equal to psi 0 x prime, just the ground state wave function, OK? 00:47:08.500 --> 00:47:21.060 align:middle line:90% 00:47:21.060 --> 00:47:22.830 align:middle line:90% So that means that-- 00:47:22.830 --> 00:47:27.870 align:middle line:84% so this trick actually works for other states, too, OK? 00:47:27.870 --> 00:47:30.390 align:middle line:90% You can do the similar trick. 00:47:30.390 --> 00:47:35.190 align:middle line:84% And you just essentially get two more integrals to do. 00:47:35.190 --> 00:47:38.820 align:middle line:84% And you take that path integral, and then 00:47:38.820 --> 00:47:42.120 align:middle line:84% you just integrate it over the initial and the final wave 00:47:42.120 --> 00:47:43.640 align:middle line:90% functions, OK? 00:47:43.640 --> 00:47:47.800 align:middle line:90% 00:47:47.800 --> 00:47:50.700 align:middle line:84% OK, so the trick applies to any state-- 00:47:50.700 --> 00:48:03.220 align:middle line:90% 00:48:03.220 --> 00:48:05.830 align:middle line:84% can you see-- not just the vacuum correlation functions. 00:48:05.830 --> 00:48:09.350 align:middle line:90% 00:48:09.350 --> 00:48:09.998 align:middle line:90% Yes? 00:48:09.998 --> 00:48:12.290 align:middle line:84% AUDIENCE: So does this still count as-- because this is 00:48:12.290 --> 00:48:14.458 align:middle line:84% still quantum mechanics what we're doing, 00:48:14.458 --> 00:48:15.250 align:middle line:90% your ground state-- 00:48:15.250 --> 00:48:18.310 align:middle line:84% I mean, this is an energy eigenket, 00:48:18.310 --> 00:48:20.260 align:middle line:90% and it's the ground state, or-- 00:48:20.260 --> 00:48:21.140 align:middle line:90% PROFESSOR: Yeah. 00:48:21.140 --> 00:48:21.640 align:middle line:90% Sorry. 00:48:21.640 --> 00:48:22.270 align:middle line:90% Say it again. 00:48:22.270 --> 00:48:25.060 align:middle line:84% AUDIENCE: So since this is still quantum mechanics, I'm saying, 00:48:25.060 --> 00:48:27.220 align:middle line:84% this 0 here is like an energy eigenket, right? 00:48:27.220 --> 00:48:28.177 align:middle line:90% PROFESSOR: Yeah, yeah. 00:48:28.177 --> 00:48:29.260 align:middle line:90% Yeah, it's a ground state. 00:48:29.260 --> 00:48:32.620 align:middle line:84% It's the it's the lowest-energy eigenstate, yeah. 00:48:32.620 --> 00:48:33.470 align:middle line:90% It doesn't matter. 00:48:33.470 --> 00:48:37.630 align:middle line:84% You can also take other states, too. 00:48:37.630 --> 00:48:40.150 align:middle line:84% It just will be some wave function there. 00:48:40.150 --> 00:48:42.010 align:middle line:84% It will be some wave function there. 00:48:42.010 --> 00:48:43.600 align:middle line:90% Yeah. 00:48:43.600 --> 00:48:45.010 align:middle line:90% Yes? 00:48:45.010 --> 00:48:46.970 align:middle line:84% AUDIENCE: So [INAUDIBLE] someone else said, 00:48:46.970 --> 00:48:51.420 align:middle line:84% which evolves in time, then moves [INAUDIBLE].. 00:48:51.420 --> 00:48:52.750 align:middle line:90% PROFESSOR: Sorry? 00:48:52.750 --> 00:48:53.250 align:middle line:90% Oh, yeah. 00:48:53.250 --> 00:48:57.040 align:middle line:84% Yeah, if you not take-- yeah, that's a very good question. 00:48:57.040 --> 00:48:59.360 align:middle line:90% So if you don't take-- 00:48:59.360 --> 00:49:03.760 align:middle line:84% and so if you take other states, and then, in general, there 00:49:03.760 --> 00:49:05.773 align:middle line:84% may be some dependence on t prime 00:49:05.773 --> 00:49:06.940 align:middle line:90% and t in this wave function. 00:49:06.940 --> 00:49:10.060 align:middle line:90% Yeah, indeed, yeah. 00:49:10.060 --> 00:49:14.270 align:middle line:84% So you have to take the limit, also, in the wave function. 00:49:14.270 --> 00:49:14.770 align:middle line:90% Yeah. 00:49:14.770 --> 00:49:17.110 align:middle line:84% But, here, we are using the simplification 00:49:17.110 --> 00:49:20.530 align:middle line:84% that the ground state is actually time-independent. 00:49:20.530 --> 00:49:21.170 align:middle line:90% Good. 00:49:21.170 --> 00:49:24.870 align:middle line:90% Yeah, that's a good question. 00:49:24.870 --> 00:49:26.060 align:middle line:90% Other questions? 00:49:26.060 --> 00:49:28.970 align:middle line:90% 00:49:28.970 --> 00:49:31.580 align:middle line:90% OK, good. 00:49:31.580 --> 00:49:37.730 align:middle line:84% So this works, essentially, for any state. 00:49:37.730 --> 00:49:39.740 align:middle line:84% But for the ground state, actually, there's 00:49:39.740 --> 00:49:44.420 align:middle line:84% another trick which can simplify the problem, which 00:49:44.420 --> 00:49:48.950 align:middle line:84% you don't even have to do this two additional integral, OK? 00:49:48.950 --> 00:49:51.830 align:middle line:90% You can actually directly-- 00:49:51.830 --> 00:50:01.830 align:middle line:84% so, for the ground state, for the vacuum, 00:50:01.830 --> 00:50:09.775 align:middle line:84% there's actually another trick to get rid of-- 00:50:09.775 --> 00:50:15.560 align:middle line:90% 00:50:15.560 --> 00:50:22.960 align:middle line:84% to make the-- these two additional integrals 00:50:22.960 --> 00:50:30.840 align:middle line:90% unnecessary, OK? 00:50:30.840 --> 00:50:33.355 align:middle line:84% So, now, let me tell you how you do this trick. 00:50:33.355 --> 00:50:34.980 align:middle line:84% And this trick is very important, also, 00:50:34.980 --> 00:50:37.980 align:middle line:84% in quantum field theory because, if you 00:50:37.980 --> 00:50:40.185 align:middle line:84% do a harmonic oscillator, of course, 00:50:40.185 --> 00:50:42.990 align:middle line:90% you know the wave function. 00:50:42.990 --> 00:50:45.660 align:middle line:84% But if I give you a aharmonic oscillator, which 00:50:45.660 --> 00:50:49.140 align:middle line:84% we don't know how to solve the ground state wave function, 00:50:49.140 --> 00:50:52.445 align:middle line:84% and then this will be a nightmare because then 00:50:52.445 --> 00:50:53.820 align:middle line:84% you don't know the wave function. 00:50:53.820 --> 00:50:56.445 align:middle line:84% And for quantum field theory, in particular, in the interacting 00:50:56.445 --> 00:51:00.510 align:middle line:84% case, we don't know the wave function, OK? 00:51:00.510 --> 00:51:03.390 align:middle line:84% So even though this formally gives you the answer, 00:51:03.390 --> 00:51:07.000 align:middle line:84% but, in practice, it's actually often not convenient to use, 00:51:07.000 --> 00:51:07.500 align:middle line:90% OK? 00:51:07.500 --> 00:51:09.330 align:middle line:84% But, fortunately, for the ground state, 00:51:09.330 --> 00:51:11.610 align:middle line:84% there is additional trick which, actually, 00:51:11.610 --> 00:51:16.880 align:middle line:84% you don't need to use the ground state wave function at all, OK? 00:51:16.880 --> 00:51:18.630 align:middle line:84% So, now, I tell you this additional trick. 00:51:18.630 --> 00:51:21.150 align:middle line:84% But this trick is only specific to the vacuum, OK? 00:51:21.150 --> 00:51:23.490 align:middle line:84% You cannot apply it to other states, OK? 00:51:23.490 --> 00:51:24.990 align:middle line:90% So this is specific to vacuum. 00:51:24.990 --> 00:51:28.220 align:middle line:90% 00:51:28.220 --> 00:51:31.350 align:middle line:84% So, now, let's forget about this thing, OK? 00:51:31.350 --> 00:51:34.350 align:middle line:84% Now, let's forget about this thing. 00:51:34.350 --> 00:51:38.010 align:middle line:90% Start coming back from here, OK? 00:51:38.010 --> 00:51:39.720 align:middle line:84% So, now, let's coming back from here. 00:51:39.720 --> 00:51:41.250 align:middle line:84% Now, let's coming back from here. 00:51:41.250 --> 00:51:53.200 align:middle line:84% Just consider-- again, I take limit t goes to plus infinity, 00:51:53.200 --> 00:51:56.600 align:middle line:84% and t prime goes to minus infinity. 00:51:56.600 --> 00:52:03.255 align:middle line:84% Let's look at this object x t, x, x prime, t prime, OK? 00:52:03.255 --> 00:52:09.630 align:middle line:90% 00:52:09.630 --> 00:52:11.380 align:middle line:90% OK? 00:52:11.380 --> 00:52:18.160 align:middle line:84% So, now, again, we are going to insert identity, OK? 00:52:18.160 --> 00:52:20.450 align:middle line:84% Now, we are, again, going to insert the identity. 00:52:20.450 --> 00:52:22.760 align:middle line:84% And, now, we insert the identity in a different way. 00:52:22.760 --> 00:52:25.610 align:middle line:84% Again, we insert the identity here 00:52:25.610 --> 00:52:28.890 align:middle line:90% and insert the identity here. 00:52:28.890 --> 00:52:32.880 align:middle line:84% But, now, we insert identity expressed in the complete set 00:52:32.880 --> 00:52:35.660 align:middle line:90% of energy eigenstates, OK? 00:52:35.660 --> 00:52:40.690 align:middle line:84% So, now, we insert the complete set of energy eigenstates. 00:52:40.690 --> 00:52:44.700 align:middle line:90% I just formally label it by m. 00:52:44.700 --> 00:52:47.130 align:middle line:90% OK? 00:52:47.130 --> 00:52:52.020 align:middle line:84% So, now, let's just insert in the two places, OK? 00:52:52.020 --> 00:52:52.520 align:middle line:90% Yeah. 00:52:52.520 --> 00:52:55.370 align:middle line:90% 00:52:55.370 --> 00:52:58.850 align:middle line:84% OK, so, now, we have two identities. 00:52:58.850 --> 00:52:59.960 align:middle line:90% Then we have n and m. 00:52:59.960 --> 00:53:02.970 align:middle line:90% 00:53:02.970 --> 00:53:04.410 align:middle line:90% We have n and m. 00:53:04.410 --> 00:53:06.690 align:middle line:90% So, now, I will-- 00:53:06.690 --> 00:53:07.990 align:middle line:90% so you keep this in mind. 00:53:07.990 --> 00:53:11.010 align:middle line:84% I will not copy this over and over, OK? 00:53:11.010 --> 00:53:14.010 align:middle line:84% So keep this limit in mind-- always, t prime. 00:53:14.010 --> 00:53:17.500 align:middle line:90% It goes to minus infinity. 00:53:17.500 --> 00:53:18.900 align:middle line:90% So, now, we have-- 00:53:18.900 --> 00:53:26.790 align:middle line:84% and then we have x t with m, and m x with n, 00:53:26.790 --> 00:53:31.410 align:middle line:90% and n with x prime, t prime, OK? 00:53:31.410 --> 00:53:44.920 align:middle line:90% 00:53:44.920 --> 00:53:46.555 align:middle line:84% So, now, if we look at this sum-- 00:53:46.555 --> 00:53:54.550 align:middle line:90% 00:53:54.550 --> 00:54:01.590 align:middle line:84% if n and m equal to 0, this is the object we want, OK? 00:54:01.590 --> 00:54:03.240 align:middle line:90% This is the object we want. 00:54:03.240 --> 00:54:07.090 align:middle line:84% But this also contains many, many other stuff. 00:54:07.090 --> 00:54:11.100 align:middle line:84% So, now, we will use the trick to isolate n equal to 0 00:54:11.100 --> 00:54:13.650 align:middle line:90% and m equal to 0 piece, OK? 00:54:13.650 --> 00:54:14.655 align:middle line:90% You isolate that piece. 00:54:14.655 --> 00:54:17.322 align:middle line:90% 00:54:17.322 --> 00:54:18.780 align:middle line:84% That's what we are going to do now. 00:54:18.780 --> 00:54:27.780 align:middle line:90% 00:54:27.780 --> 00:54:35.160 align:middle line:84% So this is the commonly-used technique in quantum field 00:54:35.160 --> 00:54:36.510 align:middle line:90% theory-- 00:54:36.510 --> 00:54:38.370 align:middle line:84% not only in quantum field theory, actually-- 00:54:38.370 --> 00:54:41.145 align:middle line:90% in many areas of physics. 00:54:41.145 --> 00:54:45.480 align:middle line:90% 00:54:45.480 --> 00:54:47.400 align:middle line:90% So, now, let's first-- 00:54:47.400 --> 00:54:49.410 align:middle line:84% to explain the trick, let's look at this object. 00:54:49.410 --> 00:54:52.180 align:middle line:90% 00:54:52.180 --> 00:54:53.910 align:middle line:84% So let's look at this object, OK? 00:54:53.910 --> 00:54:56.630 align:middle line:90% 00:54:56.630 --> 00:55:02.600 align:middle line:84% So this object is limit t prime goes to minus infinity, 00:55:02.600 --> 00:55:07.640 align:middle line:84% some energy eigenstate n and x prime, t prime. 00:55:07.640 --> 00:55:10.620 align:middle line:84% So, again, this is written in the Heisenberg picture. 00:55:10.620 --> 00:55:15.050 align:middle line:84% So we need to-- so translate to your more familiar language 00:55:15.050 --> 00:55:16.830 align:middle line:90% of the Schrodinger picture. 00:55:16.830 --> 00:55:25.050 align:middle line:84% So we need to write it as n, and then you have expansion i H t 00:55:25.050 --> 00:55:28.170 align:middle line:90% prime and then x prime, OK? 00:55:28.170 --> 00:55:33.080 align:middle line:84% So, now, it is the standard Schrodinger picture state. 00:55:33.080 --> 00:55:35.690 align:middle line:90% 00:55:35.690 --> 00:55:37.720 align:middle line:84% So, now, since n is the energy eigenstate, 00:55:37.720 --> 00:55:40.450 align:middle line:90% we can just act this one, OK? 00:55:40.450 --> 00:55:45.320 align:middle line:84% So this is just equal to limit t prime goes 00:55:45.320 --> 00:55:58.450 align:middle line:84% to minus infinity exponential i En t prime and x prime, OK? 00:55:58.450 --> 00:56:03.580 align:middle line:90% 00:56:03.580 --> 00:56:11.340 align:middle line:84% So, now, we will try to select the ground state. 00:56:11.340 --> 00:56:15.390 align:middle line:84% And we do that by doing the following. 00:56:15.390 --> 00:56:24.990 align:middle line:84% So, now, imagine giving En a small imaginary part, OK? 00:56:24.990 --> 00:56:30.500 align:middle line:90% 00:56:30.500 --> 00:56:39.265 align:middle line:84% OK, imagine-- yeah, take En goes to En 1 minus epsilon. 00:56:39.265 --> 00:56:44.500 align:middle line:90% 00:56:44.500 --> 00:56:50.250 align:middle line:84% So epsilon is a small positive number, 00:56:50.250 --> 00:56:53.860 align:middle line:90% which is equivalent to taking-- 00:56:53.860 --> 00:56:56.580 align:middle line:84% or, you just take H, your Hamiltonian, 00:56:56.580 --> 00:56:59.830 align:middle line:90% take into 1 minus epsilon, OK? 00:56:59.830 --> 00:57:04.080 align:middle line:84% Give your Hamiltonian a slightly imaginary part, OK? 00:57:04.080 --> 00:57:07.100 align:middle line:90% It's equivalent, OK? 00:57:07.100 --> 00:57:12.490 align:middle line:90% So, now, with this-- 00:57:12.490 --> 00:57:15.940 align:middle line:90% and now we have-- 00:57:15.940 --> 00:57:22.120 align:middle line:84% so limit t prime goes to minus infinity. 00:57:22.120 --> 00:57:32.920 align:middle line:84% So, now, we have exponential i En, 1 minus i epsilon t prime, 00:57:32.920 --> 00:57:36.330 align:middle line:90% and n x prime, OK? 00:57:36.330 --> 00:57:39.050 align:middle line:90% 00:57:39.050 --> 00:57:40.310 align:middle line:90% So this is just equal to that. 00:57:40.310 --> 00:57:48.930 align:middle line:90% 00:57:48.930 --> 00:57:51.920 align:middle line:90% So, now, you see-- 00:57:51.920 --> 00:57:56.600 align:middle line:84% suppose we normalize our state so that-- 00:57:56.600 --> 00:58:06.210 align:middle line:84% yeah, so, by definition, En are greater than E0 for n 00:58:06.210 --> 00:58:11.770 align:middle line:90% greater than 0, OK? 00:58:11.770 --> 00:58:18.400 align:middle line:84% So the effect of putting epsilon here, when you multiply it, 00:58:18.400 --> 00:58:21.850 align:middle line:84% it still give you a factor like this-- 00:58:21.850 --> 00:58:25.090 align:middle line:90% epsilon En times t prime, OK? 00:58:25.090 --> 00:58:27.835 align:middle line:84% So give you a real factor like this. 00:58:27.835 --> 00:58:35.700 align:middle line:90% 00:58:35.700 --> 00:58:37.190 align:middle line:90% OK? 00:58:37.190 --> 00:58:42.270 align:middle line:84% So, now, t prime goes to minus infinity. 00:58:42.270 --> 00:58:45.230 align:middle line:90% And the En is greater than E0. 00:58:45.230 --> 00:58:50.320 align:middle line:84% So for any state which is not ground state, 00:58:50.320 --> 00:58:53.080 align:middle line:84% this factor will be exponentially small-- 00:58:53.080 --> 00:58:53.860 align:middle line:90% goes to 0. 00:58:53.860 --> 00:58:56.890 align:middle line:84% So the ratio between this factor and the ground state 00:58:56.890 --> 00:59:02.170 align:middle line:84% goes to 0 for any excited states, OK? 00:59:02.170 --> 00:59:05.170 align:middle line:84% Because of the En minus E0 greater than 0. 00:59:05.170 --> 00:59:07.840 align:middle line:84% And this goes to minus infinity, and the epsilon 00:59:07.840 --> 00:59:08.980 align:middle line:90% is positive, OK? 00:59:08.980 --> 00:59:12.800 align:middle line:90% 00:59:12.800 --> 00:59:19.450 align:middle line:84% So this implies-- so we continue. 00:59:19.450 --> 00:59:19.950 align:middle line:90% Sorry. 00:59:19.950 --> 00:59:24.310 align:middle line:84% It's just-- so let's continue over here. 00:59:24.310 --> 00:59:31.960 align:middle line:84% So this means we get, essentially, 0 for n 00:59:31.960 --> 00:59:36.131 align:middle line:84% not equal to 0 and the exponential i 00:59:36.131 --> 00:59:45.825 align:middle line:84% E0 1 minus epsilon t prime for n equal to 0, OK? 00:59:45.825 --> 00:59:49.250 align:middle line:90% 00:59:49.250 --> 00:59:49.940 align:middle line:90% Yes? 00:59:49.940 --> 00:59:53.830 align:middle line:84% AUDIENCE: If you want to formally show this-- 00:59:53.830 --> 00:59:58.755 align:middle line:84% that all energy levels that are not in a ground state 00:59:58.755 --> 01:00:02.620 align:middle line:84% get exponentially suppressed, do you divide by the ground state 01:00:02.620 --> 01:00:05.740 align:middle line:84% energy into the [INAUDIBLE] energy, or how do you do it? 01:00:05.740 --> 01:00:09.558 align:middle line:84% PROFESSOR: Oh, you just take out the overall factor 01:00:09.558 --> 01:00:10.975 align:middle line:84% corresponding to the ground state. 01:00:10.975 --> 01:00:14.240 align:middle line:90% 01:00:14.240 --> 01:00:18.670 align:middle line:84% Yeah, then all the other coefficients will go to 0. 01:00:18.670 --> 01:00:19.840 align:middle line:90% Yes? 01:00:19.840 --> 01:00:21.430 align:middle line:84% AUDIENCE: What is the justification 01:00:21.430 --> 01:00:24.400 align:middle line:84% or why are we allowed to add a small imaginary part 01:00:24.400 --> 01:00:25.570 align:middle line:90% to the energy? 01:00:25.570 --> 01:00:27.670 align:middle line:84% PROFESSOR: It's a mathematical trick. 01:00:27.670 --> 01:00:28.340 align:middle line:90% AUDIENCE: OK. 01:00:28.340 --> 01:00:31.537 align:middle line:84% PROFESSOR: At the end of the day, we set epsilon go to 0. 01:00:31.537 --> 01:00:33.620 align:middle line:84% And then you go back to your original Hamiltonian. 01:00:33.620 --> 01:00:35.390 align:middle line:90% So this is just a trick. 01:00:35.390 --> 01:00:37.520 align:middle line:84% If you're doing this order properly, 01:00:37.520 --> 01:00:41.490 align:middle line:84% it just naturally selects the ground state for you. 01:00:41.490 --> 01:00:42.060 align:middle line:90% Yeah. 01:00:42.060 --> 01:00:43.680 align:middle line:84% Yeah, just a pure, mathematical trick. 01:00:43.680 --> 01:00:44.250 align:middle line:90% Yes? 01:00:44.250 --> 01:00:46.420 align:middle line:84% AUDIENCE: Epsilon has to have a certain sign, right? 01:00:46.420 --> 01:00:49.300 align:middle line:84% PROFESSOR: Yeah, epsilon-- when we write this epsilon, 01:00:49.300 --> 01:00:53.500 align:middle line:84% we always assume epsilon is greater than 0. 01:00:53.500 --> 01:00:56.260 align:middle line:90% So, yeah. 01:00:56.260 --> 01:00:58.352 align:middle line:90% You always assume-- yeah. 01:00:58.352 --> 01:01:00.504 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:01:00.504 --> 01:01:00.825 align:middle line:90% 01:01:00.825 --> 01:01:01.450 align:middle line:90% PROFESSOR: Hmm? 01:01:01.450 --> 01:01:04.908 align:middle line:84% AUDIENCE: I don't see why you can do that [INAUDIBLE].. 01:01:04.908 --> 01:01:05.825 align:middle line:90% PROFESSOR: Oh, I can-- 01:01:05.825 --> 01:01:08.630 align:middle line:90% 01:01:08.630 --> 01:01:11.000 align:middle line:90% I have a path integral. 01:01:11.000 --> 01:01:15.050 align:middle line:84% And, essentially, I can design the rule 01:01:15.050 --> 01:01:19.070 align:middle line:84% I want, as far as, at the end of the day, 01:01:19.070 --> 01:01:24.210 align:middle line:90% I get the desired quantity. 01:01:24.210 --> 01:01:24.960 align:middle line:90% Yeah. 01:01:24.960 --> 01:01:27.340 align:middle line:84% But the order of limit is very important. 01:01:27.340 --> 01:01:29.110 align:middle line:90% You have to be very careful. 01:01:29.110 --> 01:01:30.940 align:middle line:90% Yeah. 01:01:30.940 --> 01:01:32.115 align:middle line:90% Yes? 01:01:32.115 --> 01:01:33.490 align:middle line:84% AUDIENCE: Can you get around this 01:01:33.490 --> 01:01:37.290 align:middle line:84% with some sort of a stationary phase or saddle point 01:01:37.290 --> 01:01:37.890 align:middle line:90% expansion? 01:01:37.890 --> 01:01:39.265 align:middle line:84% Because, in the end, you're going 01:01:39.265 --> 01:01:40.860 align:middle line:84% to be integrating exponential factors. 01:01:40.860 --> 01:01:43.620 align:middle line:84% And so the term that dominates is-- 01:01:43.620 --> 01:01:45.360 align:middle line:90% PROFESSOR: No, here, it's not-- 01:01:45.360 --> 01:01:49.080 align:middle line:84% no, here, you cannot do a stationary phase because if you 01:01:49.080 --> 01:01:50.760 align:middle line:90% just-- yeah, yeah. 01:01:50.760 --> 01:01:55.370 align:middle line:84% You really need to suppress the other contribution-- 01:01:55.370 --> 01:01:56.480 align:middle line:90% not only suppress them. 01:01:56.480 --> 01:01:58.470 align:middle line:90% Just, they have to go to 0. 01:01:58.470 --> 01:02:03.270 align:middle line:90% Yeah-- genuinely go to 0. 01:02:03.270 --> 01:02:09.670 align:middle line:84% OK, so, similarly, with the same trick, just the same thing, 01:02:09.670 --> 01:02:14.100 align:middle line:84% in one shot, it selects n equal to 0. 01:02:14.100 --> 01:02:17.340 align:middle line:84% But you see, actually, it also selects m equal to 0 01:02:17.340 --> 01:02:22.530 align:middle line:84% because, you see, with the same sign, 01:02:22.530 --> 01:02:27.660 align:middle line:84% you can easily check yourself when t goes to plus infinity. 01:02:27.660 --> 01:02:31.660 align:middle line:90% 01:02:31.660 --> 01:02:38.740 align:middle line:84% So these now become limit t go to plus infinity, 01:02:38.740 --> 01:02:47.610 align:middle line:84% say, exponential minus i En, 1 minus epsilon t times x n. 01:02:47.610 --> 01:02:59.910 align:middle line:84% And then, again, this become 0 for m not equal to 0. 01:02:59.910 --> 01:03:01.635 align:middle line:90% And then the ground state-- 01:03:01.635 --> 01:03:07.536 align:middle line:90% 01:03:07.536 --> 01:03:08.700 align:middle line:90% for n equal to-- 01:03:08.700 --> 01:03:13.137 align:middle line:90% 01:03:13.137 --> 01:03:18.750 align:middle line:90% for m equal to 0, OK? 01:03:18.750 --> 01:03:27.420 align:middle line:84% So this simple trick beautifully selects the m equal to 0 01:03:27.420 --> 01:03:29.040 align:middle line:90% and n equal to 0 piece, OK? 01:03:29.040 --> 01:03:30.000 align:middle line:90% So, now, we can write-- 01:03:30.000 --> 01:03:34.650 align:middle line:90% 01:03:34.650 --> 01:03:37.710 align:middle line:84% so this is just the ground state wave function. 01:03:37.710 --> 01:03:40.620 align:middle line:84% And this is just the ground state wave function, OK? 01:03:40.620 --> 01:03:46.660 align:middle line:84% So, now, we have limit t prime goes to minus infinity, 01:03:46.660 --> 01:03:55.800 align:middle line:84% and t goes to plus infinity, x t, x, x prime, t prime. 01:03:55.800 --> 01:03:58.820 align:middle line:84% So, now, you go to-- so you have the ground state wave function 01:03:58.820 --> 01:04:10.770 align:middle line:90% star, x prime, x, now 0, x, 0. 01:04:10.770 --> 01:04:12.420 align:middle line:84% And then you have this phase factor. 01:04:12.420 --> 01:04:20.510 align:middle line:90% 01:04:20.510 --> 01:04:22.775 align:middle line:84% And with everything else, goes to 0, OK? 01:04:22.775 --> 01:04:31.560 align:middle line:90% 01:04:31.560 --> 01:04:37.300 align:middle line:90% So we get that, OK? 01:04:37.300 --> 01:04:42.820 align:middle line:84% And so this is the-- so, now, after those factors, 01:04:42.820 --> 01:04:45.730 align:middle line:90% we get what we want, OK? 01:04:45.730 --> 01:04:46.690 align:middle line:90% We get what we want. 01:04:46.690 --> 01:04:53.156 align:middle line:90% 01:04:53.156 --> 01:04:56.860 align:middle line:84% So, but, still, you have lots of lowering factors. 01:04:56.860 --> 01:05:00.220 align:middle line:90% But they can be easily got off. 01:05:00.220 --> 01:05:03.910 align:middle line:84% They can be easily get rid of using the following. 01:05:03.910 --> 01:05:16.700 align:middle line:90% 01:05:16.700 --> 01:05:18.580 align:middle line:90% So, now, you also-- 01:05:18.580 --> 01:05:23.620 align:middle line:84% similarly, the exact same reason, whether-- 01:05:23.620 --> 01:05:26.260 align:middle line:84% so in this discussion, whether you have x there, 01:05:26.260 --> 01:05:28.420 align:middle line:90% or what's the form of x? 01:05:28.420 --> 01:05:29.590 align:middle line:90% Doesn't matter. 01:05:29.590 --> 01:05:32.030 align:middle line:90% x can be anything. 01:05:32.030 --> 01:05:34.180 align:middle line:90% So x can be anything. 01:05:34.180 --> 01:05:38.740 align:middle line:84% So you can just take x equal to 1, OK? 01:05:38.740 --> 01:05:44.560 align:middle line:84% And then you also have limit t goes to plus infinity. 01:05:44.560 --> 01:05:47.200 align:middle line:90% t prime goes to minus infinity. 01:05:47.200 --> 01:05:51.160 align:middle line:90% The x t, x prime, t prime-- 01:05:51.160 --> 01:05:59.110 align:middle line:84% so this is also equal to psi 0 x, psi 0 star, x prime. 01:05:59.110 --> 01:06:03.220 align:middle line:84% And, now, it's the 0 with 0 because the you've 01:06:03.220 --> 01:06:05.290 align:middle line:90% set x equal to 1, OK? 01:06:05.290 --> 01:06:06.920 align:middle line:90% And then you just ground state. 01:06:06.920 --> 01:06:09.220 align:middle line:90% And so this is just equal to 1. 01:06:09.220 --> 01:06:18.590 align:middle line:84% And then you have the same factor, OK? 01:06:18.590 --> 01:06:27.760 align:middle line:90% 01:06:27.760 --> 01:06:34.740 align:middle line:84% So, now, we can just find G n, OK? 01:06:34.740 --> 01:06:36.050 align:middle line:90% So this object is Gn. 01:06:36.050 --> 01:06:38.560 align:middle line:90% 01:06:38.560 --> 01:06:41.305 align:middle line:84% So Gn, then, can be defined as the ratio between the two. 01:06:41.305 --> 01:06:46.030 align:middle line:90% 01:06:46.030 --> 01:06:47.020 align:middle line:90% We take the limit-- 01:06:47.020 --> 01:06:48.550 align:middle line:90% t plus infinity. 01:06:48.550 --> 01:06:52.600 align:middle line:90% t prime goes to minus infinity. 01:06:52.600 --> 01:07:02.920 align:middle line:84% Have x t, x prime, t prime, and x t, and x, x t, OK? 01:07:02.920 --> 01:07:06.770 align:middle line:84% You can just find that as a ratio between the two. 01:07:06.770 --> 01:07:12.620 align:middle line:84% So this gives you the expression for calculating 01:07:12.620 --> 01:07:14.720 align:middle line:90% this correlation function. 01:07:14.720 --> 01:07:19.840 align:middle line:84% And the x is any time-ordered product, OK? 01:07:19.840 --> 01:07:23.130 align:middle line:84% And so, now, for each of them, we know the path integral. 01:07:23.130 --> 01:07:25.630 align:middle line:84% And then you can just write down the path integral for each. 01:07:25.630 --> 01:07:29.930 align:middle line:90% 01:07:29.930 --> 01:07:33.800 align:middle line:84% So, now, you see this G n is, in principle, 01:07:33.800 --> 01:07:36.500 align:middle line:84% a physical observable that's actually 01:07:36.500 --> 01:07:39.720 align:middle line:84% involving the ratio of two path integrals-- 01:07:39.720 --> 01:07:44.130 align:middle line:84% one without any x and one with some x, 01:07:44.130 --> 01:07:47.260 align:middle line:84% with your desired correlation functions. 01:07:47.260 --> 01:07:50.550 align:middle line:84% So I mentioned before that when you evaluate the path 01:07:50.550 --> 01:07:52.740 align:middle line:84% integrals, you will get that kind C 01:07:52.740 --> 01:07:54.120 align:middle line:90% determinant or those things. 01:07:54.120 --> 01:07:56.340 align:middle line:84% They just cancel between upstairs and downstairs. 01:07:56.340 --> 01:08:00.050 align:middle line:84% Later, we will see explicitly, OK? 01:08:00.050 --> 01:08:02.855 align:middle line:84% Yeah, so this is a beautiful formula, OK? 01:08:02.855 --> 01:08:06.530 align:middle line:90% 01:08:06.530 --> 01:08:08.690 align:middle line:90% Yeah. 01:08:08.690 --> 01:08:10.310 align:middle line:90% And then you don't have to-- 01:08:10.310 --> 01:08:14.630 align:middle line:90% 01:08:14.630 --> 01:08:15.160 align:middle line:90% yeah. 01:08:15.160 --> 01:08:16.970 align:middle line:84% Yeah, so all those factors canceled. 01:08:16.970 --> 01:08:19.250 align:middle line:84% You don't need to know the ground state wave function. 01:08:19.250 --> 01:08:20.917 align:middle line:84% You can just perform that path integral. 01:08:20.917 --> 01:08:25.450 align:middle line:90% 01:08:25.450 --> 01:08:36.080 align:middle line:84% And here, in principle, you can also take arbitrary x and x 01:08:36.080 --> 01:08:38.300 align:middle line:90% prime, OK? 01:08:38.300 --> 01:08:39.979 align:middle line:84% You can take arbitrary x and x prime. 01:08:39.979 --> 01:08:41.689 align:middle line:90% It doesn't matter. 01:08:41.689 --> 01:08:44.479 align:middle line:84% So, conventionally, we just take x equal to x prime equal 01:08:44.479 --> 01:08:49.899 align:middle line:90% to 0, just for simplicity, OK? 01:08:49.899 --> 01:08:51.270 align:middle line:90% We can just take them to 0. 01:08:51.270 --> 01:09:06.540 align:middle line:90% 01:09:06.540 --> 01:09:07.040 align:middle line:90% Good. 01:09:07.040 --> 01:09:10.270 align:middle line:90% Any questions on this? 01:09:10.270 --> 01:09:11.170 align:middle line:90% Yes. 01:09:11.170 --> 01:09:14.199 align:middle line:84% AUDIENCE: Why can you specify both points 01:09:14.199 --> 01:09:17.080 align:middle line:90% in your [INAUDIBLE]? 01:09:17.080 --> 01:09:20.609 align:middle line:84% PROFESSOR: Yeah, I just-- the only thing matters is there-- 01:09:20.609 --> 01:09:24.270 align:middle line:84% so they only come into this prefactor, 01:09:24.270 --> 01:09:25.979 align:middle line:84% and then cancelled between the two. 01:09:25.979 --> 01:09:27.750 align:middle line:90% It doesn't matter. 01:09:27.750 --> 01:09:31.710 align:middle line:84% So all the factors there and these factors, they 01:09:31.710 --> 01:09:32.640 align:middle line:90% just cancel. 01:09:32.640 --> 01:09:38.250 align:middle line:84% And you're just left with this x0 left, so it doesn't matter. 01:09:38.250 --> 01:09:40.290 align:middle line:84% They cancel anyway, so it doesn't matter 01:09:40.290 --> 01:09:43.340 align:middle line:90% where you put x and x prime. 01:09:43.340 --> 01:09:43.973 align:middle line:90% Yes? 01:09:43.973 --> 01:09:45.890 align:middle line:84% AUDIENCE: So we didn't actually take the limit 01:09:45.890 --> 01:09:47.760 align:middle line:84% as epsilon moves to 0, or anything, 01:09:47.760 --> 01:09:49.580 align:middle line:90% so it doesn't actually matter. 01:09:49.580 --> 01:09:52.080 align:middle line:84% PROFESSOR: So, normally, when you calculate that at the end, 01:09:52.080 --> 01:09:53.939 align:middle line:84% and then you take epsilon goes to 0. 01:09:53.939 --> 01:09:58.370 align:middle line:84% So we just, normally, leave that implicit. 01:09:58.370 --> 01:09:59.870 align:middle line:84% Yeah, when you do the path integral, 01:09:59.870 --> 01:10:02.330 align:middle line:90% you keep a small epsilon. 01:10:02.330 --> 01:10:08.630 align:middle line:84% So after you finish, in the end, you take the epsilon goes to 0. 01:10:08.630 --> 01:10:09.463 align:middle line:90% Yes. 01:10:09.463 --> 01:10:10.130 align:middle line:90% AUDIENCE: Sorry. 01:10:10.130 --> 01:10:13.400 align:middle line:84% Also, on why would you take x and x prime equals 0-- 01:10:13.400 --> 01:10:15.440 align:middle line:90% doesn't our path integral-- 01:10:15.440 --> 01:10:19.700 align:middle line:84% the bounds-- depend on x and x prime that you choose? 01:10:19.700 --> 01:10:22.550 align:middle line:84% PROFESSOR: No, the bound does not depend on it. 01:10:22.550 --> 01:10:27.290 align:middle line:84% The path integral is only limited by t prime and t. 01:10:27.290 --> 01:10:31.520 align:middle line:84% Yeah, initial value, they are the same-- 01:10:31.520 --> 01:10:34.040 align:middle line:84% of course, specific value of this path integral 01:10:34.040 --> 01:10:35.670 align:middle line:90% will depend on x and x prime. 01:10:35.670 --> 01:10:39.260 align:middle line:84% But they cancel between upstairs and downstairs. 01:10:39.260 --> 01:10:40.250 align:middle line:90% So they cancelled. 01:10:40.250 --> 01:10:40.850 align:middle line:90% Yeah. 01:10:40.850 --> 01:10:43.520 align:middle line:84% And so, in the end, this right-hand side 01:10:43.520 --> 01:10:45.140 align:middle line:84% does not depend on x and x prime. 01:10:45.140 --> 01:10:48.890 align:middle line:90% 01:10:48.890 --> 01:10:53.360 align:middle line:84% Yeah, also, you see here, if this formula makes any sense, 01:10:53.360 --> 01:10:57.140 align:middle line:84% the left-hand side does not depend on x and x prime. 01:10:57.140 --> 01:11:00.110 align:middle line:84% So the right-hand side must not depend on x and x prime, 01:11:00.110 --> 01:11:05.180 align:middle line:84% otherwise, this formula immediately wrong. 01:11:05.180 --> 01:11:06.920 align:middle line:90% Yes? 01:11:06.920 --> 01:11:15.335 align:middle line:84% AUDIENCE: [INAUDIBLE] have to keep the small epsilon back 01:11:15.335 --> 01:11:17.746 align:middle line:90% in [INAUDIBLE]. 01:11:17.746 --> 01:11:20.770 align:middle line:84% So, again, we take limit as epsilon goes to 0. 01:11:20.770 --> 01:11:22.020 align:middle line:90% PROFESSOR: Yeah, that's right. 01:11:22.020 --> 01:11:22.920 align:middle line:90% That's right. 01:11:22.920 --> 01:11:24.780 align:middle line:84% Yeah, just when you do the calculation, 01:11:24.780 --> 01:11:26.527 align:middle line:90% you keep a small epsilon. 01:11:26.527 --> 01:11:28.860 align:middle line:84% When you do the path integral, you keep a small epsilon. 01:11:28.860 --> 01:11:31.570 align:middle line:84% But when you finally calculate the final answer, 01:11:31.570 --> 01:11:33.880 align:middle line:90% you can set epsilon to 0. 01:11:33.880 --> 01:11:37.122 align:middle line:84% AUDIENCE: Do we do that before we take the t to infinity, 01:11:37.122 --> 01:11:39.590 align:middle line:90% also, or does it matter? 01:11:39.590 --> 01:11:42.710 align:middle line:84% PROFESSOR: No, you have to keep an epsilon. 01:11:42.710 --> 01:11:45.000 align:middle line:90% This, you have to take first. 01:11:45.000 --> 01:11:47.120 align:middle line:84% So this is part of your calculation. 01:11:47.120 --> 01:11:50.420 align:middle line:84% You have to always take this first 01:11:50.420 --> 01:11:52.250 align:middle line:84% because the final answer also does not 01:11:52.250 --> 01:11:54.050 align:middle line:90% depend on t and t prime. 01:11:54.050 --> 01:11:57.770 align:middle line:84% And because this is the vacuum, this correlation function 01:11:57.770 --> 01:12:00.200 align:middle line:84% does not depend on your t and t prime. 01:12:00.200 --> 01:12:03.980 align:middle line:90% So you always set the-- 01:12:03.980 --> 01:12:08.480 align:middle line:84% yeah, you keep a small epsilon, take this limit first, 01:12:08.480 --> 01:12:10.550 align:middle line:90% and do the calculation. 01:12:10.550 --> 01:12:11.960 align:middle line:90% Do the calculation. 01:12:11.960 --> 01:12:13.760 align:middle line:84% And then when you calculate everything, 01:12:13.760 --> 01:12:18.720 align:middle line:84% at the end of the day, then you set epsilon to 0. 01:12:18.720 --> 01:12:21.390 align:middle line:84% And then that guarantees you chose the vacuum. 01:12:21.390 --> 01:12:27.130 align:middle line:90% 01:12:27.130 --> 01:12:29.414 align:middle line:90% Good. 01:12:29.414 --> 01:12:35.190 align:middle line:84% There's one more trick we have to do, OK? 01:12:35.190 --> 01:12:35.690 align:middle line:90% Sorry. 01:12:35.690 --> 01:12:37.340 align:middle line:90% There are too many tricks. 01:12:37.340 --> 01:12:38.945 align:middle line:84% There's one more trick we have to do. 01:12:38.945 --> 01:12:41.810 align:middle line:90% 01:12:41.810 --> 01:12:44.000 align:middle line:84% So, now, we have told you how to calculate 01:12:44.000 --> 01:12:47.610 align:middle line:90% this guy for arbitrary x. 01:12:47.610 --> 01:12:55.220 align:middle line:90% But, often, it's actually not-- 01:12:55.220 --> 01:12:58.940 align:middle line:84% it's not the smartest way to directly calculate 01:12:58.940 --> 01:13:03.050 align:middle line:90% such an n-point function, OK? 01:13:03.050 --> 01:13:06.950 align:middle line:90% It's often not the smartest way. 01:13:06.950 --> 01:13:12.770 align:middle line:84% So there's something a little bit more clever. 01:13:12.770 --> 01:13:14.632 align:middle line:84% It's called generating functional. 01:13:14.632 --> 01:13:17.350 align:middle line:90% 01:13:17.350 --> 01:13:21.790 align:middle line:84% So some of you may have seen this, say, in your high school 01:13:21.790 --> 01:13:23.290 align:middle line:90% math competition. 01:13:23.290 --> 01:13:30.790 align:middle line:90% 01:13:30.790 --> 01:13:35.180 align:middle line:84% So suppose we want to do an integral like this, OK? 01:13:35.180 --> 01:13:39.270 align:middle line:84% So let's do-- again, let's go back to our ordinary integral. 01:13:39.270 --> 01:13:42.590 align:middle line:84% So let's just suppose we want to do integral like this. 01:13:42.590 --> 01:13:43.800 align:middle line:90% So do you have a parallel? 01:13:43.800 --> 01:13:46.310 align:middle line:90% Let's just say-- say this. 01:13:46.310 --> 01:13:49.190 align:middle line:84% We have some phase, like what you do in the path integral. 01:13:49.190 --> 01:13:52.350 align:middle line:84% And we want to calculate the x to the power m OK? 01:13:52.350 --> 01:13:55.100 align:middle line:90% n is some integer. 01:13:55.100 --> 01:13:58.010 align:middle line:84% Yeah, here, like you have an n-point function-- 01:13:58.010 --> 01:14:01.372 align:middle line:84% suppose we want to calculate this. 01:14:01.372 --> 01:14:02.830 align:middle line:84% So you can just calculate this guy. 01:14:02.830 --> 01:14:03.760 align:middle line:90% It's fine, OK? 01:14:03.760 --> 01:14:07.390 align:middle line:90% But there's a better way. 01:14:07.390 --> 01:14:09.790 align:middle line:90% Often, there's a better way. 01:14:09.790 --> 01:14:12.640 align:middle line:84% Instead of calculating this for individual n, 01:14:12.640 --> 01:14:21.540 align:middle line:84% you can actually consider the following quantity, Z a, 01:14:21.540 --> 01:14:30.830 align:middle line:84% which is defined to be dx exponential i lambda fx, then 01:14:30.830 --> 01:14:32.590 align:middle line:90% plus i-- 01:14:32.590 --> 01:14:37.300 align:middle line:90% say ax, OK? 01:14:37.300 --> 01:14:41.060 align:middle line:90% So you consider this object. 01:14:41.060 --> 01:14:44.110 align:middle line:84% so this object, when you expand this exponential-- 01:14:44.110 --> 01:14:49.710 align:middle line:90% 01:14:49.710 --> 01:14:52.720 align:middle line:90% yeah, so maybe I will-- 01:14:52.720 --> 01:14:57.640 align:middle line:84% I think I don't need this anymore. 01:14:57.640 --> 01:14:59.170 align:middle line:90% Yeah, let me just erase this. 01:14:59.170 --> 01:15:19.900 align:middle line:90% 01:15:19.900 --> 01:15:20.910 align:middle line:90% So, now, if you-- 01:15:20.910 --> 01:15:23.760 align:middle line:84% so why we are interested in this object? 01:15:23.760 --> 01:15:30.270 align:middle line:84% Because imagine you expand this exponential factor, and then, 01:15:30.270 --> 01:15:35.700 align:middle line:84% essentially, you get Z a is exponential n equal to 0 01:15:35.700 --> 01:15:43.700 align:middle line:84% to infinity, n factorial i a n, Zn, So? 01:15:43.700 --> 01:15:45.830 align:middle line:84% If you expand this thing in power series 01:15:45.830 --> 01:15:48.590 align:middle line:90% of a, a power series-- 01:15:48.590 --> 01:15:51.650 align:middle line:84% if you expand this and then just n-th term 01:15:51.650 --> 01:15:54.910 align:middle line:90% will give you an x power n. 01:15:54.910 --> 01:15:56.770 align:middle line:84% And then it's the same in that we just 01:15:56.770 --> 01:16:01.870 align:middle line:84% expand this Z a in power series, and then the Z n 01:16:01.870 --> 01:16:04.390 align:middle line:84% would be the n-th coefficient, OK? 01:16:04.390 --> 01:16:13.600 align:middle line:84% So if you compute Z a in one shot, and then the easy Z n, 01:16:13.600 --> 01:16:17.020 align:middle line:84% you can just do by a Taylor series. 01:16:17.020 --> 01:16:18.730 align:middle line:90% OK. 01:16:18.730 --> 01:16:26.710 align:middle line:84% So, sometimes, we also write Zn as, say, i to the power n. 01:16:26.710 --> 01:16:31.210 align:middle line:84% Then you take n-th derivative of Za over a, 01:16:31.210 --> 01:16:34.780 align:middle line:84% and then you set a equal to 0, OK? 01:16:34.780 --> 01:16:36.830 align:middle line:90% So this is the same thing. 01:16:36.830 --> 01:16:41.184 align:middle line:84% So when you take n derivatives of a, 01:16:41.184 --> 01:16:43.600 align:middle line:90% you get rid of the earlier-- 01:16:43.600 --> 01:16:45.940 align:middle line:90% the lower powers. 01:16:45.940 --> 01:16:47.680 align:middle line:84% And then the higher powers you get 01:16:47.680 --> 01:16:50.960 align:middle line:84% rid of by setting a equal to 0, OK? 01:16:50.960 --> 01:16:54.440 align:middle line:84% And then let's just pick this term, pick Zn term. 01:16:54.440 --> 01:16:58.730 align:middle line:90% 01:16:58.730 --> 01:17:00.530 align:middle line:84% So Za is called the generating function. 01:17:00.530 --> 01:17:08.520 align:middle line:90% 01:17:08.520 --> 01:17:09.487 align:middle line:90% OK? 01:17:09.487 --> 01:17:10.320 align:middle line:90% Generating function. 01:17:10.320 --> 01:17:16.760 align:middle line:90% 01:17:16.760 --> 01:17:18.320 align:middle line:90% Oh, OK. 01:17:18.320 --> 01:17:21.860 align:middle line:84% So, now, when we want to-- so one final remark. 01:17:21.860 --> 01:17:23.855 align:middle line:84% So, now, we want to compute this guy. 01:17:23.855 --> 01:17:26.760 align:middle line:90% 01:17:26.760 --> 01:17:37.320 align:middle line:84% So this Gn is equal to t, x1 to xn, so x t1 and x tn. 01:17:37.320 --> 01:17:41.400 align:middle line:90% 01:17:41.400 --> 01:17:45.260 align:middle line:84% Again, instead of computing each one of them, 01:17:45.260 --> 01:17:47.900 align:middle line:84% we can consider such a so-called generating function 01:17:47.900 --> 01:17:49.580 align:middle line:90% or generation of this idea. 01:17:49.580 --> 01:17:53.700 align:middle line:90% We can consider Z J t. 01:17:53.700 --> 01:17:56.880 align:middle line:90% 01:17:56.880 --> 01:17:58.290 align:middle line:90% We can consider this object-- 01:17:58.290 --> 01:18:06.340 align:middle line:84% DX(t) exponential i S, x t, and, now, 01:18:06.340 --> 01:18:17.280 align:middle line:84% add i dt from minus infinity to plus infinity, J t, x t, OK? 01:18:17.280 --> 01:18:21.430 align:middle line:84% So, remember, this is just the path integral. 01:18:21.430 --> 01:18:23.580 align:middle line:84% So let's just look at the upstairs, OK? 01:18:23.580 --> 01:18:26.110 align:middle line:90% 01:18:26.110 --> 01:18:34.280 align:middle line:84% Exponential x, exponential i S x(t). 01:18:34.280 --> 01:18:36.120 align:middle line:84% So, now, this starts the same thing. 01:18:36.120 --> 01:18:42.350 align:middle line:84% So, now, if you take this, imagine you expand this factor 01:18:42.350 --> 01:18:45.020 align:middle line:90% and then to n-th power. 01:18:45.020 --> 01:18:47.630 align:middle line:84% Then you will generate this term as a coefficient, OK? 01:18:47.630 --> 01:18:53.720 align:middle line:84% We will discuss that in more detail next time, OK? 01:18:53.720 --> 01:19:00.030 align:middle line:84% So, often, instead of computing this object directly, 01:19:00.030 --> 01:19:03.480 align:middle line:84% we compute a generating functional, so-called 01:19:03.480 --> 01:19:07.230 align:middle line:84% generating functional, which often 01:19:07.230 --> 01:19:09.820 align:middle line:90% makes things much easier, OK? 01:19:09.820 --> 01:19:13.420 align:middle line:90% 01:19:13.420 --> 01:19:16.720 align:middle line:84% Yeah, so with this preparation, and then 01:19:16.720 --> 01:19:22.510 align:middle line:84% we are ready to tackle how to treat this thing-- 01:19:22.510 --> 01:19:27.190 align:middle line:84% how to actually find such object in quantum mechanics. 01:19:27.190 --> 01:19:31.210 align:middle line:84% And once we have that, doing field theory will be automatic. 01:19:31.210 --> 01:19:34.450 align:middle line:84% And then you will, then, know how 01:19:34.450 --> 01:19:37.390 align:middle line:84% to actually calculate the scattering amplitude, which 01:19:37.390 --> 01:19:40.660 align:middle line:90% is a good achievement. 01:19:40.660 --> 01:19:43.500 align:middle line:90% OK, so let's stop here. 01:19:43.500 --> 01:19:48.000 align:middle line:90%