WEBVTT 00:00:00.000 --> 00:00:02.390 align:middle line:90% [SQUEAKING] 00:00:02.390 --> 00:00:03.346 align:middle line:90% [RUSTLING] 00:00:03.346 --> 00:00:05.258 align:middle line:90% [CLICKING] 00:00:05.258 --> 00:00:10.038 align:middle line:90% 00:00:10.038 --> 00:00:11.160 align:middle line:90% [SIDE CONVERSATIONS] 00:00:11.160 --> 00:00:12.345 align:middle line:90% PROFESSOR: OK, let us start. 00:00:12.345 --> 00:00:15.510 align:middle line:90% 00:00:15.510 --> 00:00:24.090 align:middle line:84% So, last time, we discussed how to calculate such a correlation 00:00:24.090 --> 00:00:47.990 align:middle line:84% function, say Gn, in a single particle theory, 00:00:47.990 --> 00:00:50.540 align:middle line:90% so using path integral. 00:00:50.540 --> 00:00:57.110 align:middle line:84% So the goal is to calculate this time-ordered product 00:00:57.110 --> 00:00:59.360 align:middle line:84% correlation function, the vacuum correlation function, 00:00:59.360 --> 00:01:03.110 align:middle line:84% of this time-ordered product in this theory. 00:01:03.110 --> 00:01:08.510 align:middle line:84% So, last time, we described how to do this using path integral. 00:01:08.510 --> 00:01:12.570 align:middle line:84% And we derive the beautiful formula. 00:01:12.570 --> 00:01:15.620 align:middle line:84% So the formula is given by the following. 00:01:15.620 --> 00:01:26.760 align:middle line:84% It's a Gn is given by the ratio of two path integrals DX(t). 00:01:26.760 --> 00:01:29.370 align:middle line:90% 00:01:29.370 --> 00:01:31.425 align:middle line:84% So if we call this thing to be x. 00:01:31.425 --> 00:01:34.220 align:middle line:90% 00:01:34.220 --> 00:01:39.747 align:middle line:84% So the x and the exponential i S xt. 00:01:39.747 --> 00:01:43.950 align:middle line:90% 00:01:43.950 --> 00:01:49.050 align:middle line:84% And then, the DX(t), just the pure path integral. 00:01:49.050 --> 00:01:52.920 align:middle line:90% 00:01:52.920 --> 00:01:56.860 align:middle line:84% So, here, I didn't write to the upper limit and lower limit. 00:01:56.860 --> 00:01:58.980 align:middle line:84% So it should be understood that the boundary 00:01:58.980 --> 00:02:03.210 align:middle line:84% condition for both path integral is that the x-- 00:02:03.210 --> 00:02:05.850 align:middle line:84% So it should be from minus infinity-- 00:02:05.850 --> 00:02:08.039 align:middle line:84% the time range should be from minus infinity 00:02:08.039 --> 00:02:09.350 align:middle line:90% to plus infinity. 00:02:09.350 --> 00:02:14.700 align:middle line:84% And then we can choose the value of x at both ends just 00:02:14.700 --> 00:02:17.220 align:middle line:90% to be zero. 00:02:17.220 --> 00:02:21.390 align:middle line:84% And, yeah, and the slight subtlety 00:02:21.390 --> 00:02:25.020 align:middle line:84% is that, when you evaluate S, so there 00:02:25.020 --> 00:02:29.760 align:middle line:84% should be the epsilon parameter because we-- 00:02:29.760 --> 00:02:32.040 align:middle line:84% in deriving the path integral we have, 00:02:32.040 --> 00:02:37.170 align:middle line:84% say we need to give the Hamiltonian a slightly 00:02:37.170 --> 00:02:38.970 align:middle line:90% say epsilon part. 00:02:38.970 --> 00:02:41.320 align:middle line:84% So that will also affect your action. 00:02:41.320 --> 00:02:45.180 align:middle line:84% So your action will also have a small epsilon dependence. 00:02:45.180 --> 00:02:49.080 align:middle line:84% And so, at the end of the day, after you 00:02:49.080 --> 00:02:51.030 align:middle line:84% have done this calculation, and then 00:02:51.030 --> 00:02:53.280 align:middle line:90% you said epsilon goes to 0. 00:02:53.280 --> 00:03:00.000 align:middle line:84% So the epsilon goes to 0 at the end of here. 00:03:00.000 --> 00:03:01.680 align:middle line:84% So when we write the epsilon, it's 00:03:01.680 --> 00:03:05.970 align:middle line:84% always should be assumed that this is a positive number. 00:03:05.970 --> 00:03:08.850 align:middle line:84% This is infinitesimal positive number. 00:03:08.850 --> 00:03:12.080 align:middle line:90% 00:03:12.080 --> 00:03:12.630 align:middle line:90% Good. 00:03:12.630 --> 00:03:13.595 align:middle line:90% Any questions on this? 00:03:13.595 --> 00:03:26.150 align:middle line:90% 00:03:26.150 --> 00:03:29.360 align:middle line:84% So, now, in principle, with this formula, 00:03:29.360 --> 00:03:33.500 align:middle line:84% then we can now calculate this quantity. 00:03:33.500 --> 00:03:36.090 align:middle line:90% We can calculate this quantity. 00:03:36.090 --> 00:03:39.740 align:middle line:84% And we can calculate this n-point function. 00:03:39.740 --> 00:03:43.700 align:middle line:84% So, but, in practice, it's actually more convenient, 00:03:43.700 --> 00:03:46.190 align:middle line:84% rather than to calculate the Gn, because we often 00:03:46.190 --> 00:03:49.580 align:middle line:84% need to know the such correlation 00:03:49.580 --> 00:03:51.650 align:middle line:90% function for different n. 00:03:51.650 --> 00:03:53.338 align:middle line:84% We often need to know such correlation 00:03:53.338 --> 00:03:54.380 align:middle line:90% function for different n. 00:03:54.380 --> 00:03:58.730 align:middle line:84% So not only say sometimes we are interested in n equal to 2, 00:03:58.730 --> 00:04:02.720 align:middle line:84% sometimes interested in n equal to 3, 4, et cetera. 00:04:02.720 --> 00:04:07.580 align:middle line:84% And then, there is a nice trick to-- 00:04:07.580 --> 00:04:10.700 align:middle line:84% you can try to calculate, then say, 00:04:10.700 --> 00:04:15.770 align:middle line:84% in general, is to use this technique called generating 00:04:15.770 --> 00:04:23.710 align:middle line:84% functional, which we started talking 00:04:23.710 --> 00:04:28.240 align:middle line:84% about at the end of last lecture. 00:04:28.240 --> 00:04:31.830 align:middle line:84% And so, the basic idea of the generating functional 00:04:31.830 --> 00:04:34.890 align:middle line:84% can be easily understood by consider just 00:04:34.890 --> 00:04:37.800 align:middle line:90% this one-dimensional example. 00:04:37.800 --> 00:04:45.830 align:middle line:84% Say, if you are interested in doing an integral like this, 00:04:45.830 --> 00:04:48.090 align:middle line:90% xn. 00:04:48.090 --> 00:04:50.010 align:middle line:84% And, if you are interested in this integral 00:04:50.010 --> 00:04:52.840 align:middle line:84% for different value of integer n, 00:04:52.840 --> 00:05:00.970 align:middle line:84% then it's more convenient to consider such an integral, Za 00:05:00.970 --> 00:05:02.225 align:middle line:90% given by-- 00:05:02.225 --> 00:05:08.640 align:middle line:90% 00:05:08.640 --> 00:05:11.220 align:middle line:84% So the reason I put the i here is just, 00:05:11.220 --> 00:05:14.130 align:middle line:84% yeah, I didn't specify the range of x. 00:05:14.130 --> 00:05:17.430 align:middle line:84% And if x is from minus infinity to plus infinity, 00:05:17.430 --> 00:05:18.900 align:middle line:90% it's not easy to-- yeah. 00:05:18.900 --> 00:05:24.457 align:middle line:84% Yeah, put i here, just so that this integral can be defined. 00:05:24.457 --> 00:05:26.790 align:middle line:84% Depending on circumstances, you don't have to put the i. 00:05:26.790 --> 00:05:31.020 align:middle line:84% Say if x is from 0 to infinity, then 00:05:31.020 --> 00:05:34.170 align:middle line:84% I can just put, say, minus lambda a, with lambda 00:05:34.170 --> 00:05:36.790 align:middle line:90% to be a positive number. 00:05:36.790 --> 00:05:37.860 align:middle line:90% And then, that's fine. 00:05:37.860 --> 00:05:41.520 align:middle line:90% 00:05:41.520 --> 00:05:47.370 align:middle line:84% So the benefit of considering this integral is that-- oh, no, 00:05:47.370 --> 00:05:50.670 align:middle line:84% no, no, not the-- sorry, I should-- 00:05:50.670 --> 00:05:52.570 align:middle line:90% x, yeah, xa. 00:05:52.570 --> 00:05:55.580 align:middle line:90% 00:05:55.580 --> 00:05:56.210 align:middle line:90% Yeah. 00:05:56.210 --> 00:05:58.753 align:middle line:84% So the benefit of doing this integral 00:05:58.753 --> 00:06:00.170 align:middle line:84% is that if you notice, say, if you 00:06:00.170 --> 00:06:02.435 align:middle line:84% take a derivative with respect to a, 00:06:02.435 --> 00:06:05.060 align:middle line:84% and then, that will bring down a factor of x. 00:06:05.060 --> 00:06:12.305 align:middle line:84% So it's a-- so if you take a derivative with a, 00:06:12.305 --> 00:06:14.630 align:middle line:84% you bring down a factor of x, and you 00:06:14.630 --> 00:06:17.290 align:middle line:84% take the derivative twice with a, 00:06:17.290 --> 00:06:19.830 align:middle line:84% then you bring down a factor of x square. 00:06:19.830 --> 00:06:22.610 align:middle line:84% And if you do it n times, and then you bring down 00:06:22.610 --> 00:06:25.910 align:middle line:90% a factor of x to the power n. 00:06:25.910 --> 00:06:29.720 align:middle line:84% So, essentially, this Zn then can 00:06:29.720 --> 00:06:37.500 align:middle line:84% be written as partial n Za, then partial a, you derivative n 00:06:37.500 --> 00:06:38.280 align:middle line:90% times. 00:06:38.280 --> 00:06:42.150 align:middle line:84% And then you set a equal to 0 because, in the end, we want-- 00:06:42.150 --> 00:06:45.600 align:middle line:84% in this integral, there's no exponential piece. 00:06:45.600 --> 00:06:47.070 align:middle line:84% And you set that equal to 0, then 00:06:47.070 --> 00:06:48.612 align:middle line:84% you get rid of the exponential piece. 00:06:48.612 --> 00:06:50.250 align:middle line:90% Then you have factor of xn. 00:06:50.250 --> 00:06:51.690 align:middle line:90% So there's still-- yeah. 00:06:51.690 --> 00:06:56.520 align:middle line:90% So I still need to put i n here. 00:06:56.520 --> 00:07:00.390 align:middle line:84% So if you know how to compute the Za, and then 00:07:00.390 --> 00:07:04.050 align:middle line:84% you only need to take derivatives to do Zn. 00:07:04.050 --> 00:07:09.840 align:middle line:84% So taking derivatives is much easier than doing integrals. 00:07:09.840 --> 00:07:12.840 align:middle line:84% So, in other words, we can also write it, 00:07:12.840 --> 00:07:17.570 align:middle line:84% expand Za in terms of power series in a. 00:07:17.570 --> 00:07:24.270 align:middle line:84% And then, the Zn would be the coefficient. 00:07:24.270 --> 00:07:27.940 align:middle line:90% 00:07:27.940 --> 00:07:32.350 align:middle line:84% So Zn would be the coefficient for a to the power n. 00:07:32.350 --> 00:07:37.500 align:middle line:90% 00:07:37.500 --> 00:07:38.780 align:middle line:90% Yeah. 00:07:38.780 --> 00:07:39.470 align:middle line:90% Good? 00:07:39.470 --> 00:07:43.145 align:middle line:84% Yeah, so we call this Za the generating function. 00:07:43.145 --> 00:07:51.030 align:middle line:90% 00:07:51.030 --> 00:07:52.700 align:middle line:84% So now we can use the similar idea 00:07:52.700 --> 00:07:55.870 align:middle line:90% to generalize to this case. 00:07:55.870 --> 00:07:58.180 align:middle line:84% And, now, here, this is-- here it's just 00:07:58.180 --> 00:07:59.920 align:middle line:90% one-dimensional integral. 00:07:59.920 --> 00:08:01.820 align:middle line:84% Here we have a functional integral. 00:08:01.820 --> 00:08:03.860 align:middle line:90% So now this is the function. 00:08:03.860 --> 00:08:06.640 align:middle line:84% So, essentially, you just generalize this generating 00:08:06.640 --> 00:08:09.970 align:middle line:84% function into a generating functional. 00:08:09.970 --> 00:08:13.700 align:middle line:84% And so, we can consider the following object. 00:08:13.700 --> 00:08:16.510 align:middle line:84% So in order to compute this, then 00:08:16.510 --> 00:08:25.330 align:middle line:84% we can consider the analog of this 00:08:25.330 --> 00:08:31.330 align:middle line:84% a is we consider object called J. J now will depend on t. 00:08:31.330 --> 00:08:32.830 align:middle line:90% Yeah, J is a function. 00:08:32.830 --> 00:08:39.250 align:middle line:84% J is a function because x now becomes a function. 00:08:39.250 --> 00:08:44.480 align:middle line:84% And so, introduce a path integral like this. 00:08:44.480 --> 00:08:46.480 align:middle line:90% So, from now on, I will not-- 00:08:46.480 --> 00:08:49.660 align:middle line:84% when I don't write the range, then it's always-- 00:08:49.660 --> 00:08:51.490 align:middle line:84% you should always keep this in mind-- 00:08:51.490 --> 00:08:53.800 align:middle line:84% the range will always from minus infinity 00:08:53.800 --> 00:08:56.890 align:middle line:84% to plus infinity with the boundary value to be 0. 00:08:56.890 --> 00:09:00.200 align:middle line:90% 00:09:00.200 --> 00:09:07.370 align:middle line:84% And then, we can just say, I have standard xt. 00:09:07.370 --> 00:09:11.820 align:middle line:84% Then I can add the analog of this. 00:09:11.820 --> 00:09:16.880 align:middle line:84% So, remember, again, you think t is just an index. 00:09:16.880 --> 00:09:21.230 align:middle line:84% And then you just, essentially, imagine if you have multiple x. 00:09:21.230 --> 00:09:23.660 align:middle line:90% And then you just sum over them. 00:09:23.660 --> 00:09:26.240 align:middle line:84% So sum over them, in this case, just corresponding 00:09:26.240 --> 00:09:27.720 align:middle line:90% to an integral. 00:09:27.720 --> 00:09:31.130 align:middle line:84% And so, we just have a piece like this. 00:09:31.130 --> 00:09:33.760 align:middle line:90% 00:09:33.760 --> 00:09:35.410 align:middle line:84% So this would be the generalization 00:09:35.410 --> 00:09:36.800 align:middle line:90% of that equation. 00:09:36.800 --> 00:09:42.190 align:middle line:84% And so, integration over t just can be imagined as a sum. 00:09:42.190 --> 00:09:45.280 align:middle line:84% Imagine if you have multiple x and multiple a, 00:09:45.280 --> 00:09:47.260 align:middle line:84% and then you need to sum over them. 00:09:47.260 --> 00:09:50.770 align:middle line:84% And then, yeah, so this is just a generalization of that. 00:09:50.770 --> 00:09:52.285 align:middle line:84% Do you have any questions on this? 00:09:52.285 --> 00:09:57.010 align:middle line:90% 00:09:57.010 --> 00:10:03.350 align:middle line:84% So now, similarly, now if we take a functional derivative 00:10:03.350 --> 00:10:09.740 align:middle line:84% with respect to Jt, then we can bring down a factor of xt. 00:10:09.740 --> 00:10:12.950 align:middle line:84% So, more explicitly, so let me just remind you, 00:10:12.950 --> 00:10:16.730 align:middle line:84% that the rule of doing functional derivative. 00:10:16.730 --> 00:10:25.310 align:middle line:84% So if you have a function, delta Jt prime, with respect 00:10:25.310 --> 00:10:27.920 align:middle line:84% to the derivative of Jt, and then 00:10:27.920 --> 00:10:30.410 align:middle line:84% that just gives you a delta function. 00:10:30.410 --> 00:10:31.910 align:middle line:84% That just give you a delta function. 00:10:31.910 --> 00:10:34.830 align:middle line:90% 00:10:34.830 --> 00:10:37.050 align:middle line:84% So now let's look at-- with this rule, 00:10:37.050 --> 00:10:42.930 align:middle line:84% then let's just look at the delta, delta Jt on the ZJ. 00:10:42.930 --> 00:10:49.290 align:middle line:90% 00:10:49.290 --> 00:10:51.180 align:middle line:90% So and then, yeah. 00:10:51.180 --> 00:10:53.250 align:middle line:84% So, now, since this is Jt, let's just, 00:10:53.250 --> 00:10:55.950 align:middle line:84% for convenience, let's just put this to be t prime. 00:10:55.950 --> 00:10:58.510 align:middle line:90% 00:10:58.510 --> 00:11:00.242 align:middle line:90% Let's put it to t prime. 00:11:00.242 --> 00:11:02.200 align:middle line:84% And then, when you take derivative with respect 00:11:02.200 --> 00:11:07.085 align:middle line:84% to Jt, and then you directly take derivative with this guy. 00:11:07.085 --> 00:11:08.710 align:middle line:84% And then, when you take the derivative, 00:11:08.710 --> 00:11:10.960 align:middle line:84% the delta function will get rid of this integral. 00:11:10.960 --> 00:11:13.060 align:middle line:90% And then you just have xt. 00:11:13.060 --> 00:11:21.520 align:middle line:84% And then you will just get i DX(t) exponential i S. 00:11:21.520 --> 00:11:25.400 align:middle line:84% And, yeah, sorry, you will get DX(t). 00:11:25.400 --> 00:11:27.790 align:middle line:84% And then you have a factor of xt. 00:11:27.790 --> 00:11:29.920 align:middle line:90% So let me just write Dx-- 00:11:29.920 --> 00:11:32.860 align:middle line:90% again, to save, you just add Dx. 00:11:32.860 --> 00:11:37.900 align:middle line:84% And you have xt from taking derivative of Jt. 00:11:37.900 --> 00:11:43.150 align:middle line:84% And then exponential i S plus i Jx. 00:11:43.150 --> 00:11:48.370 align:middle line:90% So let me just save some effort. 00:11:48.370 --> 00:11:53.170 align:middle line:84% Yeah, you should keep in mind that this Dt is, actually, I'm 00:11:53.170 --> 00:11:54.910 align:middle line:84% just using a simplified notation. 00:11:54.910 --> 00:11:58.130 align:middle line:90% 00:11:58.130 --> 00:11:58.630 align:middle line:90% Good. 00:11:58.630 --> 00:11:59.547 align:middle line:90% Any questions on this? 00:11:59.547 --> 00:12:04.060 align:middle line:90% 00:12:04.060 --> 00:12:14.480 align:middle line:84% So, now, if I also introduce Z0, defined to be ZJ equal to 0. 00:12:14.480 --> 00:12:17.520 align:middle line:84% So this is just the original integral. 00:12:17.520 --> 00:12:20.930 align:middle line:84% So Z0 is essentially the downstairs. 00:12:20.930 --> 00:12:31.730 align:middle line:84% So Z0 is just the Dx exponential i S. So when J equal to 0, 00:12:31.730 --> 00:12:32.960 align:middle line:90% and then we just have this. 00:12:32.960 --> 00:12:35.600 align:middle line:84% So, essentially, this is the downstairs here. 00:12:35.600 --> 00:12:38.870 align:middle line:84% So we also-- let's also introduce this definition. 00:12:38.870 --> 00:12:41.510 align:middle line:90% 00:12:41.510 --> 00:12:44.450 align:middle line:90% And then we then find-- 00:12:44.450 --> 00:12:47.150 align:middle line:84% so, by comparing the two, then we 00:12:47.150 --> 00:12:53.120 align:middle line:84% immediately conclude that the one point function of x hat t0 00:12:53.120 --> 00:12:57.950 align:middle line:84% is just given by 1 over i, because this i there, 00:12:57.950 --> 00:13:02.290 align:middle line:90% and 1 over Z0, then delta J-- 00:13:02.290 --> 00:13:07.630 align:middle line:90% delta ZJ, delta Jt. 00:13:07.630 --> 00:13:11.820 align:middle line:90% And then you set Jt equal to 0. 00:13:11.820 --> 00:13:15.020 align:middle line:84% So, similarly, here, if you set a equal to 0, so you take 00:13:15.020 --> 00:13:20.000 align:middle line:84% derivative, and then you bring down a factor of xt. 00:13:20.000 --> 00:13:22.010 align:middle line:90% And then you set J equal to 0. 00:13:22.010 --> 00:13:24.740 align:middle line:84% And then you just have that integral with xt there. 00:13:24.740 --> 00:13:28.500 align:middle line:84% And then you get this one point function. 00:13:28.500 --> 00:13:32.440 align:middle line:84% So, now, you can just immediately generalize. 00:13:32.440 --> 00:13:40.750 align:middle line:84% So this n-point function, Gn then can be written as, you 00:13:40.750 --> 00:13:41.800 align:middle line:90% just take one-- 00:13:41.800 --> 00:13:43.850 align:middle line:90% now you have take n derivatives. 00:13:43.850 --> 00:13:47.710 align:middle line:84% Now you have a factor of 1 over i to the power n. 00:13:47.710 --> 00:13:50.500 align:middle line:84% Again, you divide it by Z0 because you always 00:13:50.500 --> 00:13:53.710 align:middle line:90% need to divide by this piece. 00:13:53.710 --> 00:14:00.700 align:middle line:84% And then you just take Z derivative n times. 00:14:00.700 --> 00:14:05.400 align:middle line:90% So you take Jt1, Jtn. 00:14:05.400 --> 00:14:08.820 align:middle line:84% So this time variable should match with the time variable 00:14:08.820 --> 00:14:11.700 align:middle line:90% in the original definition. 00:14:11.700 --> 00:14:14.920 align:middle line:84% And then, after you take the derivative, 00:14:14.920 --> 00:14:18.280 align:middle line:90% you set J equal to 0. 00:14:18.280 --> 00:14:25.540 align:middle line:84% So, and then, this gives you the n-point function. 00:14:25.540 --> 00:14:28.790 align:middle line:84% If you know how to compute this ZJ, and, again, 00:14:28.790 --> 00:14:30.550 align:middle line:90% you only need to do derivatives. 00:14:30.550 --> 00:14:32.080 align:middle line:90% And then it's much simpler. 00:14:32.080 --> 00:14:34.570 align:middle line:84% Then you only need to do the path integral once, 00:14:34.570 --> 00:14:38.460 align:middle line:84% and then you just doing the derivatives. 00:14:38.460 --> 00:14:39.435 align:middle line:90% Any questions on this? 00:14:39.435 --> 00:14:53.960 align:middle line:90% 00:14:53.960 --> 00:14:57.690 align:middle line:84% So, now, if you keep in mind of this, 00:14:57.690 --> 00:15:03.110 align:middle line:84% we can also rewrite this expression as the following. 00:15:03.110 --> 00:15:05.240 align:middle line:90% We can also rewrite-- 00:15:05.240 --> 00:15:13.580 align:middle line:84% alternatively, we can also write ZJ divided by Z0 as following 0 00:15:13.580 --> 00:15:25.560 align:middle line:84% and the time ordered exponential i, dt, Jt, xt, 0. 00:15:25.560 --> 00:15:28.320 align:middle line:90% 00:15:28.320 --> 00:15:33.180 align:middle line:84% So, as I mentioned before, that in that formula, in principle, 00:15:33.180 --> 00:15:36.090 align:middle line:90% x can be anything. 00:15:36.090 --> 00:15:38.290 align:middle line:90% x can be anything. 00:15:38.290 --> 00:15:41.940 align:middle line:84% So, now, in this path integral, you 00:15:41.940 --> 00:15:45.042 align:middle line:84% can just imagine you separate this term-- 00:15:45.042 --> 00:15:46.500 align:middle line:84% the sum of the two exponential, you 00:15:46.500 --> 00:15:48.214 align:middle line:90% can just write it as a product. 00:15:48.214 --> 00:15:49.672 align:middle line:84% And then you have the exponential i 00:15:49.672 --> 00:15:52.410 align:middle line:84% S, and the exponential i, this piece. 00:15:52.410 --> 00:15:55.380 align:middle line:84% And then we treat that piece to be x. 00:15:55.380 --> 00:15:59.250 align:middle line:84% And then, that gives you this formula. 00:15:59.250 --> 00:16:02.400 align:middle line:84% If you treat that piece to be x, it give you this formula. 00:16:02.400 --> 00:16:04.200 align:middle line:84% Then you ask, what is the meaning 00:16:04.200 --> 00:16:07.760 align:middle line:84% when we have a time ordering of this exponential? 00:16:07.760 --> 00:16:10.330 align:middle line:84% So the meaning is that, imagine you 00:16:10.330 --> 00:16:13.570 align:middle line:90% expand this in power series. 00:16:13.570 --> 00:16:19.780 align:middle line:84% Imagine expand in power series of x. 00:16:19.780 --> 00:16:25.210 align:middle line:84% And then you just order each term in the power series. 00:16:25.210 --> 00:16:27.790 align:middle line:84% You can pick, because now each term is a polynomial, 00:16:27.790 --> 00:16:29.950 align:middle line:90% you can time-order them. 00:16:29.950 --> 00:16:30.970 align:middle line:90% You can time-order them. 00:16:30.970 --> 00:16:33.550 align:middle line:90% 00:16:33.550 --> 00:16:35.320 align:middle line:90% So, now, this object-- 00:16:35.320 --> 00:16:37.460 align:middle line:90% so I can also write this object. 00:16:37.460 --> 00:16:40.780 align:middle line:84% So when you expand this object, and then you-- 00:16:40.780 --> 00:16:42.700 align:middle line:90% the first time is just 1. 00:16:42.700 --> 00:16:49.027 align:middle line:84% And then, the next term just Dt, Jt, then 00:16:49.027 --> 00:16:49.985 align:middle line:90% the one-point function. 00:16:49.985 --> 00:16:54.570 align:middle line:90% 00:16:54.570 --> 00:17:03.270 align:middle line:84% And then, under the n'th term, yeah, et cetera. 00:17:03.270 --> 00:17:05.099 align:middle line:84% So you can, when you expand it, you 00:17:05.099 --> 00:17:09.329 align:middle line:84% can just write from n equal to infinity, 0 to infinity, 00:17:09.329 --> 00:17:13.589 align:middle line:90% i to the power n, n factorial. 00:17:13.589 --> 00:17:18.589 align:middle line:84% Now you have n integrals, t1, tn. 00:17:18.589 --> 00:17:30.015 align:middle line:84% Then you have times, you have Gn, t1, tn, and the Jt1, Jtn. 00:17:30.015 --> 00:17:32.860 align:middle line:90% 00:17:32.860 --> 00:17:34.150 align:middle line:90% So this is the typical-- 00:17:34.150 --> 00:17:36.140 align:middle line:90% the n'th term is like that. 00:17:36.140 --> 00:17:39.730 align:middle line:84% So when you expand this to n'th power, you 00:17:39.730 --> 00:17:41.530 align:middle line:90% group all the x together. 00:17:41.530 --> 00:17:43.990 align:middle line:90% Because J just C number-- 00:17:43.990 --> 00:17:46.150 align:middle line:84% integration and the J are C numbers. 00:17:46.150 --> 00:17:47.470 align:middle line:90% You can take them out. 00:17:47.470 --> 00:17:52.480 align:middle line:84% And then this part is just the x between the zeros 00:17:52.480 --> 00:17:58.250 align:middle line:84% and then you have this factor of J. Yes. 00:17:58.250 --> 00:18:00.200 align:middle line:84% AUDIENCE: Where did the n factorial come from? 00:18:00.200 --> 00:18:02.030 align:middle line:84% PROFESSOR: Oh, just when you expand the exponential, 00:18:02.030 --> 00:18:03.080 align:middle line:90% there's always n factorial. 00:18:03.080 --> 00:18:04.205 align:middle line:90% AUDIENCE: Oh, right, sorry. 00:18:04.205 --> 00:18:05.560 align:middle line:90% PROFESSOR: Yeah. 00:18:05.560 --> 00:18:06.060 align:middle line:90% Good. 00:18:06.060 --> 00:18:06.977 align:middle line:90% Any questions on this? 00:18:06.977 --> 00:18:14.920 align:middle line:90% 00:18:14.920 --> 00:18:15.830 align:middle line:90% Yes. 00:18:15.830 --> 00:18:17.830 align:middle line:84% AUDIENCE: When you separate out the exponentials 00:18:17.830 --> 00:18:21.150 align:middle line:84% into the e to the i S and then the i integral. 00:18:21.150 --> 00:18:22.010 align:middle line:90% PROFESSOR: Yeah. 00:18:22.010 --> 00:18:25.191 align:middle line:84% AUDIENCE: Is that S, and the other term, like operator. 00:18:25.191 --> 00:18:26.941 align:middle line:84% So you'd have to have another contribution 00:18:26.941 --> 00:18:29.600 align:middle line:90% from the commutators? 00:18:29.600 --> 00:18:31.200 align:middle line:90% PROFESSOR: Sorry, say it again? 00:18:31.200 --> 00:18:35.150 align:middle line:84% AUDIENCE: So, S, in that case, it functions of an an operator. 00:18:35.150 --> 00:18:37.794 align:middle line:84% And then the other one is also like an operator. 00:18:37.794 --> 00:18:40.433 align:middle line:84% So wouldn't that introduce commentators? 00:18:40.433 --> 00:18:41.350 align:middle line:90% PROFESSOR: No, no, no. 00:18:41.350 --> 00:18:42.970 align:middle line:84% Because, in the path integral, they're 00:18:42.970 --> 00:18:44.067 align:middle line:90% just ordinary functions. 00:18:44.067 --> 00:18:44.567 align:middle line:90% Right? 00:18:44.567 --> 00:18:44.905 align:middle line:90% AUDIENCE: Oh. 00:18:44.905 --> 00:18:46.280 align:middle line:84% PROFESSOR: In the path integrals, 00:18:46.280 --> 00:18:47.450 align:middle line:90% they're ordinary functions. 00:18:47.450 --> 00:18:48.880 align:middle line:84% So, in the path integral, they're 00:18:48.880 --> 00:18:50.770 align:middle line:90% always just ordinary functions. 00:18:50.770 --> 00:18:53.890 align:middle line:84% But then, when we rewrite them in terms of the operator 00:18:53.890 --> 00:18:57.580 align:middle line:90% form and then so-- 00:18:57.580 --> 00:19:00.340 align:middle line:84% the left-hand side, So the x, they're 00:19:00.340 --> 00:19:01.880 align:middle line:90% just ordinary functions. 00:19:01.880 --> 00:19:04.090 align:middle line:84% But in the right-hand side, I'm now writing it 00:19:04.090 --> 00:19:05.650 align:middle line:90% in terms of the operator form. 00:19:05.650 --> 00:19:09.480 align:middle line:84% And now, indeed, now the ordinary matters, 00:19:09.480 --> 00:19:10.890 align:middle line:90% Now the ordering matters. 00:19:10.890 --> 00:19:11.700 align:middle line:90% Yeah. 00:19:11.700 --> 00:19:15.240 align:middle line:84% So I'm just using this form-- yeah, so, in this formula, 00:19:15.240 --> 00:19:18.000 align:middle line:84% the right-hand side is just the ordinary functions. 00:19:18.000 --> 00:19:20.610 align:middle line:84% But the left-hand side involving some operator 00:19:20.610 --> 00:19:22.245 align:middle line:84% sandwiched between the ground state. 00:19:22.245 --> 00:19:22.745 align:middle line:90% Yeah. 00:19:22.745 --> 00:19:27.120 align:middle line:90% 00:19:27.120 --> 00:19:30.430 align:middle line:90% Other questions? 00:19:30.430 --> 00:19:31.150 align:middle line:90% Yes. 00:19:31.150 --> 00:19:33.430 align:middle line:84% AUDIENCE: So in that formula in the middle, 00:19:33.430 --> 00:19:36.753 align:middle line:84% where Gn is expressed as the function derivative of Z-- 00:19:36.753 --> 00:19:37.420 align:middle line:90% PROFESSOR: Yeah. 00:19:37.420 --> 00:19:39.400 align:middle line:84% AUDIENCE: Does it matter what order I take the derivatives? 00:19:39.400 --> 00:19:41.050 align:middle line:84% PROFESSOR: No, that doesn't matter. 00:19:41.050 --> 00:19:45.130 align:middle line:84% Because, again, J is ordinary functions. 00:19:45.130 --> 00:19:45.640 align:middle line:90% Yes. 00:19:45.640 --> 00:19:47.515 align:middle line:84% Yeah, because this is just the path integral. 00:19:47.515 --> 00:19:50.680 align:middle line:90% 00:19:50.680 --> 00:19:52.390 align:middle line:90% J is just ordinary functions. 00:19:52.390 --> 00:19:53.860 align:middle line:84% This is just some functionals of J. 00:19:53.860 --> 00:19:55.720 align:middle line:84% You can just take arbitrary derivatives. 00:19:55.720 --> 00:20:01.246 align:middle line:84% Also, you notice, on the left-hand side, 00:20:01.246 --> 00:20:06.830 align:middle line:84% so Gn is a function of t1 and tn. 00:20:06.830 --> 00:20:09.200 align:middle line:84% So Gn, as a function of t1 and tn 00:20:09.200 --> 00:20:12.500 align:middle line:84% is actually completely symmetric. 00:20:12.500 --> 00:20:14.030 align:middle line:84% Because, under this time ordering, 00:20:14.030 --> 00:20:16.322 align:middle line:84% it doesn't matter how you order them. 00:20:16.322 --> 00:20:18.530 align:middle line:84% Because they're just ordered by time ordering anyway. 00:20:18.530 --> 00:20:23.270 align:middle line:84% So it doesn't matter how I write the ordering here. 00:20:23.270 --> 00:20:26.870 align:middle line:84% So Gn is a symmetric function of t1 and tn. 00:20:26.870 --> 00:20:29.490 align:middle line:90% So you can see it here. 00:20:29.490 --> 00:20:33.320 align:middle line:84% So, here, because all the derivative commute, and then 00:20:33.320 --> 00:20:35.636 align:middle line:84% this is a symmetric function of t1 and tn. 00:20:35.636 --> 00:20:36.136 align:middle line:90% Yeah. 00:20:36.136 --> 00:20:41.570 align:middle line:90% 00:20:41.570 --> 00:20:44.890 align:middle line:90% Other questions? 00:20:44.890 --> 00:20:45.400 align:middle line:90% Good? 00:20:45.400 --> 00:20:47.980 align:middle line:90% OK. 00:20:47.980 --> 00:20:51.790 align:middle line:84% So, in the future, we often just computing this object. 00:20:51.790 --> 00:20:54.620 align:middle line:84% We just often computing this object. 00:20:54.620 --> 00:20:57.920 align:middle line:90% And then, that will tell us-- 00:20:57.920 --> 00:21:00.110 align:middle line:84% then that will give us the generating functional 00:21:00.110 --> 00:21:01.610 align:middle line:90% of the correlation functions. 00:21:01.610 --> 00:21:04.580 align:middle line:84% Then we can just obtain correlation functions 00:21:04.580 --> 00:21:05.930 align:middle line:90% by taking derivatives. 00:21:05.930 --> 00:21:08.930 align:middle line:90% 00:21:08.930 --> 00:21:12.380 align:middle line:84% So now let's look at the explicit example 00:21:12.380 --> 00:21:16.990 align:middle line:90% to illustrate how this works. 00:21:16.990 --> 00:21:25.130 align:middle line:84% So let's just consider simple example, a harmonic oscillator. 00:21:25.130 --> 00:21:28.640 align:middle line:84% So, almost always, a harmonic oscillator is a good example. 00:21:28.640 --> 00:21:31.660 align:middle line:90% 00:21:31.660 --> 00:21:32.230 align:middle line:90% Good? 00:21:32.230 --> 00:21:37.900 align:middle line:84% So, in this case, so, again, we look at this object. 00:21:37.900 --> 00:21:43.620 align:middle line:90% But now S is-- 00:21:43.620 --> 00:21:47.470 align:middle line:84% so we always now take from minus infinity to plus infinity. 00:21:47.470 --> 00:21:49.050 align:middle line:90% We have dt. 00:21:49.050 --> 00:21:50.610 align:middle line:90% Then you have 1/2. 00:21:50.610 --> 00:21:52.860 align:middle line:84% So let's take this to be my Lagrangian. 00:21:52.860 --> 00:21:56.880 align:middle line:90% 00:21:56.880 --> 00:21:59.910 align:middle line:90% So let me call this omega 0. 00:21:59.910 --> 00:22:04.870 align:middle line:84% So this is considered the essential harmonic oscillator. 00:22:04.870 --> 00:22:06.660 align:middle line:90% So I take the m equal to 1. 00:22:06.660 --> 00:22:10.300 align:middle line:90% 00:22:10.300 --> 00:22:13.340 align:middle line:90% And I consider this Jt. 00:22:13.340 --> 00:22:14.740 align:middle line:90% Yeah. 00:22:14.740 --> 00:22:18.730 align:middle line:84% We are interested in computing this object. 00:22:18.730 --> 00:22:28.230 align:middle line:84% So, also, I should mention, in practice, 00:22:28.230 --> 00:22:31.590 align:middle line:90% so you always interested in xt. 00:22:31.590 --> 00:22:34.380 align:middle line:84% Say, when you calculate n-point functions, 00:22:34.380 --> 00:22:38.200 align:middle line:90% even though the value of t-- 00:22:38.200 --> 00:22:41.710 align:middle line:84% say, suppose you want to calculate the Gn. 00:22:41.710 --> 00:22:45.190 align:middle line:90% So Gn, you have n values of t. 00:22:45.190 --> 00:22:49.480 align:middle line:84% And so, outside that n values of t, 00:22:49.480 --> 00:22:51.530 align:middle line:90% you can just take the J to be-- 00:22:51.530 --> 00:22:52.030 align:middle line:90% yeah. 00:22:52.030 --> 00:22:57.130 align:middle line:84% So we can always take J to go to 0 at the plus minus infinity. 00:22:57.130 --> 00:23:00.880 align:middle line:84% Yeah, so that helps your integral to converge. 00:23:00.880 --> 00:23:02.425 align:middle line:84% Yeah, this is just a side remark. 00:23:02.425 --> 00:23:06.220 align:middle line:90% 00:23:06.220 --> 00:23:07.210 align:middle line:90% Good? 00:23:07.210 --> 00:23:12.230 align:middle line:84% So, now, first, to compute this object or this object, 00:23:12.230 --> 00:23:16.990 align:middle line:84% we need to first understand what is this S epsilon. 00:23:16.990 --> 00:23:19.900 align:middle line:84% We need to understand what is this s epsilon. 00:23:19.900 --> 00:23:25.240 align:middle line:84% So, remember, so, for the harmonic oscillator, 00:23:25.240 --> 00:23:31.510 align:middle line:84% the Hamiltonian is p squared divided by 2m plus 1/2 omega 0 00:23:31.510 --> 00:23:32.770 align:middle line:90% squared x squared. 00:23:32.770 --> 00:23:36.010 align:middle line:90% 00:23:36.010 --> 00:23:40.510 align:middle line:84% And so, here is the Lagrangian, and this is the Hamiltonian. 00:23:40.510 --> 00:23:42.910 align:middle line:84% They are related by the Legendre transform. 00:23:42.910 --> 00:23:46.410 align:middle line:90% 00:23:46.410 --> 00:23:51.690 align:middle line:84% So now, in order to do this H, go to H minus i epsilon. 00:23:51.690 --> 00:23:58.230 align:middle line:84% And then, now this become p squared divided by 2m 1 00:23:58.230 --> 00:24:02.940 align:middle line:84% minus i epsilon, then plus 1/2 omega 0 squared, 00:24:02.940 --> 00:24:06.030 align:middle line:90% x squared, 1 minus i epsilon. 00:24:06.030 --> 00:24:10.050 align:middle line:84% So you multiply the both term by 1 minus i epsilon. 00:24:10.050 --> 00:24:15.600 align:middle line:84% And then, to obtain what is the corresponding S for this, 00:24:15.600 --> 00:24:17.820 align:middle line:84% you just do a lot of Legendre transform back 00:24:17.820 --> 00:24:20.480 align:middle line:90% into a Lagrangian. 00:24:20.480 --> 00:24:25.040 align:middle line:84% So then you find that the L epsilon, 00:24:25.040 --> 00:24:28.340 align:middle line:84% which you do corresponding to the Legendre 00:24:28.340 --> 00:24:30.620 align:middle line:90% transform of this-- 00:24:30.620 --> 00:24:34.080 align:middle line:84% yeah, so this is a trivial exercise, which we can do. 00:24:34.080 --> 00:24:38.060 align:middle line:84% Then you find that this give, gets 1/2 x squared 1 00:24:38.060 --> 00:24:41.040 align:middle line:90% plus i epsilon. 00:24:41.040 --> 00:24:45.300 align:middle line:84% And then, yeah, so, this part, essentially, does not change, 00:24:45.300 --> 00:24:49.530 align:middle line:84% 1/2 omega 0 squared x squared 1 minus i epsilon. 00:24:49.530 --> 00:24:52.020 align:middle line:90% So that's what you get. 00:24:52.020 --> 00:24:54.060 align:middle line:84% Essentially, when you invert it, when 00:24:54.060 --> 00:24:58.410 align:middle line:84% you do the Legendre transform to go to p, to x, somehow 00:24:58.410 --> 00:25:00.420 align:middle line:84% this becomes from 1 minus i epsilon 00:25:00.420 --> 00:25:01.455 align:middle line:90% become 1 plus i epsilon. 00:25:01.455 --> 00:25:03.600 align:middle line:84% Yeah, you can easily check yourself 00:25:03.600 --> 00:25:07.150 align:middle line:90% because you did do an inversion. 00:25:07.150 --> 00:25:13.995 align:middle line:84% And so, now let's write this into a more convenient form. 00:25:13.995 --> 00:25:18.570 align:middle line:90% 00:25:18.570 --> 00:25:21.600 align:middle line:84% So to write it in a more convenient form, we always-- 00:25:21.600 --> 00:25:28.170 align:middle line:84% remember, we always treated the x as a two x's sandwiched 00:25:28.170 --> 00:25:29.820 align:middle line:90% by some differential operator. 00:25:29.820 --> 00:25:32.620 align:middle line:84% So we can do integration by parts. 00:25:32.620 --> 00:25:34.590 align:middle line:84% So we can do integration by part to write it 00:25:34.590 --> 00:25:42.450 align:middle line:84% as minus 1/2 x partial t squared plus omega 0 squared. 00:25:42.450 --> 00:25:48.180 align:middle line:84% Then minus i epsilon omega squared plus i epsilon 00:25:48.180 --> 00:25:51.290 align:middle line:90% partial t square x. 00:25:51.290 --> 00:25:53.461 align:middle line:90% And then plus total derivative. 00:25:53.461 --> 00:26:03.460 align:middle line:90% 00:26:03.460 --> 00:26:05.470 align:middle line:84% So the total derivative always vanished 00:26:05.470 --> 00:26:09.190 align:middle line:90% because we always-- yeah. 00:26:09.190 --> 00:26:11.090 align:middle line:84% Just, we always impose boundary conditions. 00:26:11.090 --> 00:26:15.190 align:middle line:84% So that at t equal to plus minus infinity, they go to 0. 00:26:15.190 --> 00:26:17.980 align:middle line:90% 00:26:17.980 --> 00:26:20.410 align:middle line:84% And so, now let's look at this object. 00:26:20.410 --> 00:26:24.920 align:middle line:90% 00:26:24.920 --> 00:26:29.870 align:middle line:84% So let's look at this object, the epsilon dependence. 00:26:29.870 --> 00:26:33.110 align:middle line:84% So omega squared is just a positive number. 00:26:33.110 --> 00:26:36.500 align:middle line:84% Multiply epsilon, it's still a positive number. 00:26:36.500 --> 00:26:37.730 align:middle line:90% And epsilon infinitesimal. 00:26:37.730 --> 00:26:41.730 align:middle line:84% So we can just still call it i epsilon. 00:26:41.730 --> 00:26:45.750 align:middle line:84% So, now, the partial t square acting on epsilon. 00:26:45.750 --> 00:26:50.280 align:middle line:84% So partial t square is a negative definite operator. 00:26:50.280 --> 00:26:53.420 align:middle line:90% 00:26:53.420 --> 00:26:56.870 align:middle line:84% Because, remember, whenever you do a Fourier transform 00:26:56.870 --> 00:26:58.850 align:middle line:90% on x, so partial t-- 00:26:58.850 --> 00:27:01.118 align:middle line:84% a single factor gives you i omega. 00:27:01.118 --> 00:27:02.660 align:middle line:84% Then, if you have a partial t square, 00:27:02.660 --> 00:27:05.330 align:middle line:84% then give you minus omega squared. 00:27:05.330 --> 00:27:11.810 align:middle line:84% So partial t square is a negative definite operator. 00:27:11.810 --> 00:27:14.360 align:middle line:84% So that means that this is also a negative number 00:27:14.360 --> 00:27:16.340 align:middle line:90% times i epsilon. 00:27:16.340 --> 00:27:19.040 align:middle line:84% And then, that means we can just write the whole thing just 00:27:19.040 --> 00:27:24.950 align:middle line:84% as 1/2 x partial t square plus omega 0 square 00:27:24.950 --> 00:27:28.910 align:middle line:90% then minus i epsilon x. 00:27:28.910 --> 00:27:35.000 align:middle line:90% 00:27:35.000 --> 00:27:35.500 align:middle line:90% Good? 00:27:35.500 --> 00:27:38.130 align:middle line:90% Is this clear? 00:27:38.130 --> 00:27:42.130 align:middle line:84% Yeah, because just anything, any positives 00:27:42.130 --> 00:27:47.105 align:middle line:84% in multiplying epsilon still give you epsilon. 00:27:47.105 --> 00:27:48.480 align:middle line:84% Because it's just a small number. 00:27:48.480 --> 00:27:49.230 align:middle line:90% It doesn't matter. 00:27:49.230 --> 00:27:55.100 align:middle line:90% 00:27:55.100 --> 00:27:57.740 align:middle line:84% So now we can write this S xt, now 00:27:57.740 --> 00:28:04.970 align:middle line:84% we can write the S epsilon xt then 00:28:04.970 --> 00:28:09.420 align:middle line:84% in the following form in the minus 1/2. 00:28:09.420 --> 00:28:13.320 align:middle line:84% Let me write as dt, dt prime as we wrote before. 00:28:13.320 --> 00:28:21.330 align:middle line:84% Then xt Kt, t prime and xt prime. 00:28:21.330 --> 00:28:34.300 align:middle line:84% And then, Kt, t prime is just given by delta t minus t prime, 00:28:34.300 --> 00:28:41.350 align:middle line:84% say partial t prime squared minus omega 0 squared 00:28:41.350 --> 00:28:48.290 align:middle line:90% minus epsilon, or plus, yeah. 00:28:48.290 --> 00:28:53.870 align:middle line:84% So, again, we just introduce a lot of t prime. 00:28:53.870 --> 00:28:57.360 align:middle line:84% And then I introduce a delta function. 00:28:57.360 --> 00:29:00.120 align:middle line:90% And now we have a matrix form. 00:29:00.120 --> 00:29:02.510 align:middle line:84% Now, again, this action has a matrix structure. 00:29:02.510 --> 00:29:12.712 align:middle line:90% 00:29:12.712 --> 00:29:16.350 align:middle line:84% So now, this S depend on epsilon. 00:29:16.350 --> 00:29:17.490 align:middle line:90% Now S depend on epsilon. 00:29:17.490 --> 00:29:21.770 align:middle line:90% 00:29:21.770 --> 00:29:22.270 align:middle line:90% Good? 00:29:22.270 --> 00:29:24.850 align:middle line:90% 00:29:24.850 --> 00:29:25.350 align:middle line:90% OK? 00:29:25.350 --> 00:29:26.280 align:middle line:90% Good? 00:29:26.280 --> 00:29:29.370 align:middle line:84% So now we can evaluate-- now we can now ready 00:29:29.370 --> 00:29:33.300 align:middle line:90% to evaluate this path integral. 00:29:33.300 --> 00:29:36.777 align:middle line:84% Now we are ready to evaluate the path integral. 00:29:36.777 --> 00:29:37.860 align:middle line:90% So let's first look at Z0. 00:29:37.860 --> 00:29:41.475 align:middle line:90% 00:29:41.475 --> 00:29:42.600 align:middle line:90% And let's first look at Z0. 00:29:42.600 --> 00:29:49.660 align:middle line:84% Z0 is just the Gaussian integral we already said before. 00:29:49.660 --> 00:29:57.070 align:middle line:84% So the Z0 is the Gaussian integral. 00:29:57.070 --> 00:29:59.670 align:middle line:90% So I will be schematic. 00:29:59.670 --> 00:30:05.630 align:middle line:90% 00:30:05.630 --> 00:30:09.575 align:middle line:90% Yeah, this x dot K dot x. 00:30:09.575 --> 00:30:13.961 align:middle line:90% 00:30:13.961 --> 00:30:20.400 align:middle line:84% So this is a shorthand notation to denote this two integral. 00:30:20.400 --> 00:30:23.750 align:middle line:90% I think it's positive i. 00:30:23.750 --> 00:30:24.250 align:middle line:90% Oh, yeah. 00:30:24.250 --> 00:30:25.150 align:middle line:90% Yeah, minus sign. 00:30:25.150 --> 00:30:26.410 align:middle line:90% I have a minus sign here. 00:30:26.410 --> 00:30:27.210 align:middle line:90% So it's minus i. 00:30:27.210 --> 00:30:28.350 align:middle line:90% Yeah. 00:30:28.350 --> 00:30:29.280 align:middle line:90% Good? 00:30:29.280 --> 00:30:32.010 align:middle line:84% So, and this, as we said before, this is just 00:30:32.010 --> 00:30:35.250 align:middle line:84% given by some constant and determinant 00:30:35.250 --> 00:30:40.133 align:middle line:84% K. This is just some constant determinant K. Yes? 00:30:40.133 --> 00:30:40.800 align:middle line:90% AUDIENCE: Sorry. 00:30:40.800 --> 00:30:42.240 align:middle line:90% [INAUDIBLE] 00:30:42.240 --> 00:30:42.840 align:middle line:90% PROFESSOR: OK. 00:30:42.840 --> 00:30:45.552 align:middle line:84% AUDIENCE: Why did you have to go to the Hamiltonian 00:30:45.552 --> 00:30:49.300 align:middle line:90% to put that 1 minus i epsilon? 00:30:49.300 --> 00:30:50.110 align:middle line:90% PROFESSOR: Right. 00:30:50.110 --> 00:30:53.140 align:middle line:84% It's because that's our previous rule. 00:30:53.140 --> 00:30:57.220 align:middle line:84% Because we say, in order to derive this, 00:30:57.220 --> 00:31:00.495 align:middle line:84% we use the cheek to take the H, go to-- 00:31:00.495 --> 00:31:01.910 align:middle line:90% yeah, H 1 minus epsilon. 00:31:01.910 --> 00:31:02.410 align:middle line:90% Yeah. 00:31:02.410 --> 00:31:05.770 align:middle line:84% So we want to know how this translate into the behavior 00:31:05.770 --> 00:31:06.430 align:middle line:90% in the action. 00:31:06.430 --> 00:31:09.140 align:middle line:90% 00:31:09.140 --> 00:31:10.100 align:middle line:90% Other questions? 00:31:10.100 --> 00:31:15.730 align:middle line:90% 00:31:15.730 --> 00:31:17.510 align:middle line:90% Good? 00:31:17.510 --> 00:31:23.030 align:middle line:84% So this is just the same as we discussed before just 00:31:23.030 --> 00:31:25.850 align:middle line:84% given by some constant and determinant K. 00:31:25.850 --> 00:31:29.150 align:middle line:84% As we said before, that the C is typically divergent. 00:31:29.150 --> 00:31:31.580 align:middle line:84% Determinant the K, so typically divergent. 00:31:31.580 --> 00:31:33.740 align:middle line:84% But we will see, it doesn't matter. 00:31:33.740 --> 00:31:37.160 align:middle line:84% So now we will see, it doesn't-- so, previously, we said, 00:31:37.160 --> 00:31:38.810 align:middle line:90% this will not matter. 00:31:38.810 --> 00:31:42.270 align:middle line:84% But now we will see it explicitly. 00:31:42.270 --> 00:31:43.460 align:middle line:90% So now let's look at the ZJ. 00:31:43.460 --> 00:31:46.190 align:middle line:90% 00:31:46.190 --> 00:32:01.750 align:middle line:84% So ZJ is the same integral, x dot K dot x. 00:32:01.750 --> 00:32:05.110 align:middle line:84% But, now, with this additional term. 00:32:05.110 --> 00:32:08.130 align:middle line:84% So let me just write, again, in the simplified 00:32:08.130 --> 00:32:11.100 align:middle line:90% notation, as J dot x. 00:32:11.100 --> 00:32:16.490 align:middle line:84% So you view the integration as a huge sum of vector-- yeah, 00:32:16.490 --> 00:32:17.115 align:middle line:90% vector product. 00:32:17.115 --> 00:32:21.420 align:middle line:90% 00:32:21.420 --> 00:32:23.430 align:middle line:84% So if this is a finite dimensional integral, 00:32:23.430 --> 00:32:26.700 align:middle line:90% you say, I know how to do this. 00:32:26.700 --> 00:32:32.160 align:middle line:84% We know how to do this because this is just 00:32:32.160 --> 00:32:37.050 align:middle line:84% the Gaussian with a linear piece. 00:32:37.050 --> 00:32:41.650 align:middle line:84% So we can just write down the answer. 00:32:41.650 --> 00:32:46.650 align:middle line:84% So let's just-- yeah, the rule is that you just 00:32:46.650 --> 00:32:49.440 align:middle line:84% treat it as a finite dimensional integral 00:32:49.440 --> 00:32:51.840 align:middle line:84% and write the answer for the finite dimensional integral. 00:32:51.840 --> 00:32:55.530 align:middle line:84% And then you translate the language 00:32:55.530 --> 00:32:58.050 align:middle line:84% in terms of this functional case. 00:32:58.050 --> 00:33:00.400 align:middle line:90% So we can write it-- 00:33:00.400 --> 00:33:04.617 align:middle line:84% so, again, this would be C divided by delta K. Yeah. 00:33:04.617 --> 00:33:05.700 align:middle line:90% So let me just remind you. 00:33:05.700 --> 00:33:08.880 align:middle line:84% Maybe just let me just do a little bit slower. 00:33:08.880 --> 00:33:12.180 align:middle line:84% So let me just remind you the standard story 00:33:12.180 --> 00:33:14.040 align:middle line:90% for such a Gaussian integral. 00:33:14.040 --> 00:33:21.805 align:middle line:84% So if you have dx1, dxn, exponential minus 1/2 xi, 00:33:21.805 --> 00:33:27.450 align:middle line:90% Aij, xj plus Ji, xi. 00:33:27.450 --> 00:33:31.730 align:middle line:84% So if you have an integral like this, 00:33:31.730 --> 00:33:33.290 align:middle line:84% we know how to compute this integral. 00:33:33.290 --> 00:33:34.610 align:middle line:90% We can just compute-- 00:33:34.610 --> 00:33:39.560 align:middle line:84% include the xi into here by completing the square. 00:33:39.560 --> 00:33:42.615 align:middle line:84% And then, what you get is the following. 00:33:42.615 --> 00:33:44.240 align:middle line:84% After you complete the square, you just 00:33:44.240 --> 00:33:47.400 align:middle line:84% get the original Gaussian integral. 00:33:47.400 --> 00:33:53.765 align:middle line:84% And so, what you get is you get the 2 pi D over 2, 00:33:53.765 --> 00:33:56.760 align:middle line:90% or your previous case, delta a. 00:33:56.760 --> 00:33:59.810 align:middle line:84% And then, the results you get from the complete integral 00:33:59.810 --> 00:34:05.788 align:middle line:90% is Ji A minus 1 Ij, Jj. 00:34:05.788 --> 00:34:08.330 align:middle line:84% So when you complete the square, you get the additional term. 00:34:08.330 --> 00:34:09.267 align:middle line:90% That's what you get. 00:34:09.267 --> 00:34:11.434 align:middle line:84% And this is coming from doing the Gaussian integral. 00:34:11.434 --> 00:34:14.135 align:middle line:90% 00:34:14.135 --> 00:34:16.260 align:middle line:84% So now we just have an infinite dimensional version 00:34:16.260 --> 00:34:18.100 align:middle line:90% of this integral. 00:34:18.100 --> 00:34:22.500 align:middle line:84% And we can just write down the result immediately. 00:34:22.500 --> 00:34:24.929 align:middle line:84% We can just write down the result immediately. 00:34:24.929 --> 00:34:29.350 align:middle line:90% So we just copy that thing. 00:34:29.350 --> 00:34:33.960 align:middle line:84% So we have C. So this C will be the exact the same as that C, 00:34:33.960 --> 00:34:38.370 align:middle line:84% because this just comes from doing a Gaussian integral as 00:34:38.370 --> 00:34:41.540 align:middle line:90% if J is not there. 00:34:41.540 --> 00:34:46.120 align:middle line:84% So we have the same C. We have the same delta k. 00:34:46.120 --> 00:34:50.530 align:middle line:84% Then, according to the rule there, up to the i, 00:34:50.530 --> 00:34:54.850 align:middle line:84% which is you have to put in, then we get 1/2 i. 00:34:54.850 --> 00:34:59.320 align:middle line:84% Then you have-- then we should have J k minus 1 00:34:59.320 --> 00:35:03.220 align:middle line:90% J. So this is essentially that. 00:35:03.220 --> 00:35:08.060 align:middle line:84% You take the inverse of A. So here we get that. 00:35:08.060 --> 00:35:12.490 align:middle line:84% And this, if we translate back into this kind of function 00:35:12.490 --> 00:35:21.040 align:middle line:84% language, so this just gives you C delta K exponential i 00:35:21.040 --> 00:35:22.540 align:middle line:90% divided by 2. 00:35:22.540 --> 00:35:25.570 align:middle line:90% Then you have dt, dt prime. 00:35:25.570 --> 00:35:32.370 align:middle line:84% Then you have Jt, K minus 1, t, again, t prime. 00:35:32.370 --> 00:35:34.710 align:middle line:84% So K minus 1 should be understood as the inverse 00:35:34.710 --> 00:35:37.770 align:middle line:90% of this K and Jt prime. 00:35:37.770 --> 00:35:48.900 align:middle line:90% 00:35:48.900 --> 00:35:53.630 align:middle line:84% So the K minus 1 is defined as follows. 00:35:53.630 --> 00:35:56.290 align:middle line:90% So the K is-- 00:35:56.290 --> 00:35:58.750 align:middle line:84% so you just, again, is the function generalization 00:35:58.750 --> 00:36:01.420 align:middle line:90% of the matrix case. 00:36:01.420 --> 00:36:06.190 align:middle line:84% So you just have t prime, K, t, t prime, 00:36:06.190 --> 00:36:10.330 align:middle line:84% and K minus 1, t prime, t double prime 00:36:10.330 --> 00:36:14.230 align:middle line:84% should be equal to delta t minus t double prime. 00:36:14.230 --> 00:36:17.860 align:middle line:84% So that's how you define the K, K minus 1. 00:36:17.860 --> 00:36:19.720 align:middle line:84% And this is like a matrix product. 00:36:19.720 --> 00:36:23.020 align:middle line:90% Just now you treat the t prime-- 00:36:23.020 --> 00:36:26.920 align:middle line:84% yeah, t prime, you sum over that, and then, yeah. 00:36:26.920 --> 00:36:33.400 align:middle line:84% So this is just like you have kmn, k minus n, and k 00:36:33.400 --> 00:36:35.500 align:middle line:90% equal to delta mk. 00:36:35.500 --> 00:36:38.230 align:middle line:84% You just translate the n into the integral. 00:36:38.230 --> 00:36:41.230 align:middle line:90% And the t is corresponding to m. 00:36:41.230 --> 00:36:43.720 align:middle line:84% And the t double prime corresponding to K. 00:36:43.720 --> 00:36:45.640 align:middle line:84% And the delta function corresponding to that. 00:36:45.640 --> 00:36:48.210 align:middle line:90% 00:36:48.210 --> 00:36:50.700 align:middle line:84% So that's how we define the K minus 1. 00:36:50.700 --> 00:36:59.120 align:middle line:84% And so, this is the result for ZJ. 00:36:59.120 --> 00:37:04.680 align:middle line:90% 00:37:04.680 --> 00:37:07.950 align:middle line:84% So, now, the physical object is this object 00:37:07.950 --> 00:37:10.650 align:middle line:90% is the DJ divided by Z0. 00:37:10.650 --> 00:37:12.460 align:middle line:84% Because you get the expectation value, 00:37:12.460 --> 00:37:15.780 align:middle line:90% we always need to divide by Z0. 00:37:15.780 --> 00:37:21.000 align:middle line:84% So now, if we take the ratio, so now I can erase this. 00:37:21.000 --> 00:37:29.780 align:middle line:84% So if we take the ratio, ZJ divided by Z0, 00:37:29.780 --> 00:37:32.660 align:middle line:84% we find all these factor canceled. 00:37:32.660 --> 00:37:36.060 align:middle line:84% So this factor cancels with that factor. 00:37:36.060 --> 00:37:38.010 align:middle line:90% So it doesn't matter. 00:37:38.010 --> 00:37:39.620 align:middle line:90% So we just get exponential. 00:37:39.620 --> 00:37:46.105 align:middle line:84% So, let me, again, using this shorthand notation, i over 2, 00:37:46.105 --> 00:37:55.070 align:middle line:84% J K minus 1 J. So this is the physical quantity. 00:37:55.070 --> 00:37:58.590 align:middle line:90% 00:37:58.590 --> 00:38:02.010 align:middle line:84% And when we expand this in powers of J, 00:38:02.010 --> 00:38:04.920 align:middle line:84% then we get the correlation-- then the coefficient of J 00:38:04.920 --> 00:38:06.480 align:middle line:90% give you correlation functions. 00:38:06.480 --> 00:38:08.010 align:middle line:90% Or we can just take derivatives. 00:38:08.010 --> 00:38:08.910 align:middle line:90% Yes. 00:38:08.910 --> 00:38:12.688 align:middle line:84% AUDIENCE: Yeah, so last time I thought you said the C 00:38:12.688 --> 00:38:15.080 align:middle line:84% and the determinant of K can be infinite. 00:38:15.080 --> 00:38:18.654 align:middle line:84% So is it OK to divide infinity by infinity and just say it's 1 00:38:18.654 --> 00:38:20.100 align:middle line:90% in this case? 00:38:20.100 --> 00:38:23.280 align:middle line:84% PROFESSOR: No, it's not one, because they are actually-- 00:38:23.280 --> 00:38:27.300 align:middle line:84% yeah, as you do in your Pset, say, 00:38:27.300 --> 00:38:34.410 align:middle line:84% if you have a free particle, that ratio is actually-- 00:38:34.410 --> 00:38:36.270 align:middle line:84% you can calculate to be a finite number. 00:38:36.270 --> 00:38:37.170 align:middle line:90% AUDIENCE: Oh. 00:38:37.170 --> 00:38:39.240 align:middle line:84% PROFESSOR: Even though their ratio is actually, 00:38:39.240 --> 00:38:40.770 align:middle line:90% both are infinite. 00:38:40.770 --> 00:38:44.750 align:middle line:84% But the ratio is actually a finite number. 00:38:44.750 --> 00:38:45.250 align:middle line:90% Yeah. 00:38:45.250 --> 00:38:47.380 align:middle line:84% Yeah, same thing with the harmonic oscillator. 00:38:47.380 --> 00:38:48.198 align:middle line:90% Yeah. 00:38:48.198 --> 00:38:49.990 align:middle line:84% Yeah, but the key thing is that we actually 00:38:49.990 --> 00:38:51.350 align:middle line:90% don't need to worry about them. 00:38:51.350 --> 00:38:52.910 align:middle line:90% They just cancel. 00:38:52.910 --> 00:38:53.410 align:middle line:90% Yeah. 00:38:53.410 --> 00:38:57.830 align:middle line:90% 00:38:57.830 --> 00:38:58.790 align:middle line:90% Other questions? 00:38:58.790 --> 00:39:04.390 align:middle line:90% 00:39:04.390 --> 00:39:11.200 align:middle line:84% So now, this is our final result for the harmonic oscillator. 00:39:11.200 --> 00:39:17.630 align:middle line:84% And except we still have to invert this K. 00:39:17.630 --> 00:39:25.430 align:middle line:84% We still have to invert this K. But, in fact-- 00:39:25.430 --> 00:39:30.120 align:middle line:90% 00:39:30.120 --> 00:39:35.040 align:middle line:84% but, before we do that, first we can see what is this-- 00:39:35.040 --> 00:39:37.110 align:middle line:84% whether there's any physical interpretation 00:39:37.110 --> 00:39:38.400 align:middle line:90% for this K minus 1. 00:39:38.400 --> 00:39:41.220 align:middle line:90% 00:39:41.220 --> 00:39:44.980 align:middle line:84% So let's just consider the following situation. 00:39:44.980 --> 00:39:47.950 align:middle line:84% So let's consider a two-point function. 00:39:47.950 --> 00:39:50.490 align:middle line:84% So, first, from here, you can immediately see, 00:39:50.490 --> 00:39:55.780 align:middle line:84% the one-point function is given by what, the vacuum 00:39:55.780 --> 00:39:59.690 align:middle line:90% one-point function of x? 00:39:59.690 --> 00:40:04.940 align:middle line:84% So can you see what is the vacuum one-point function for x 00:40:04.940 --> 00:40:08.460 align:middle line:90% without doing calculation? 00:40:08.460 --> 00:40:09.820 align:middle line:90% AUDIENCE: 0? 00:40:09.820 --> 00:40:12.010 align:middle line:90% PROFESSOR: Yes, 0. 00:40:12.010 --> 00:40:14.260 align:middle line:84% So the reason it's 0, it says because if you 00:40:14.260 --> 00:40:17.590 align:middle line:84% get one-point function, you take one derivative with J. 00:40:17.590 --> 00:40:20.230 align:middle line:84% So when you take one derivative is J, because of here, 00:40:20.230 --> 00:40:21.880 align:middle line:90% it's J square. 00:40:21.880 --> 00:40:24.460 align:middle line:84% You will bring down a factor of J. 00:40:24.460 --> 00:40:30.100 align:middle line:84% Then, when you set the J equal to 0, and then that will be 0. 00:40:30.100 --> 00:40:32.410 align:middle line:84% So the one-point function automatically is 0. 00:40:32.410 --> 00:40:34.600 align:middle line:84% And that's consistent with our expectation. 00:40:34.600 --> 00:40:36.970 align:middle line:84% In the harmonic oscillator, the one-point function of x 00:40:36.970 --> 00:40:41.980 align:middle line:84% is always 0 because x involve a or a dagger. 00:40:41.980 --> 00:40:45.400 align:middle line:84% When you sandwiched between two zeros, it's just 0. 00:40:45.400 --> 00:40:47.800 align:middle line:84% But now, so notice, the non-vanishing one 00:40:47.800 --> 00:40:49.190 align:middle line:90% is the two-point function. 00:40:49.190 --> 00:40:51.650 align:middle line:84% So now let's consider the two-point function. 00:40:51.650 --> 00:40:54.100 align:middle line:84% So two-point function by definition 00:40:54.100 --> 00:40:59.030 align:middle line:84% should be the Feynman function because this 00:40:59.030 --> 00:41:02.120 align:middle line:90% is a time-ordered product. 00:41:02.120 --> 00:41:04.770 align:middle line:84% So the two-point function, by definition, 00:41:04.770 --> 00:41:05.970 align:middle line:90% is the Feynman function. 00:41:05.970 --> 00:41:06.820 align:middle line:90% It's the G2. 00:41:06.820 --> 00:41:09.710 align:middle line:90% 00:41:09.710 --> 00:41:13.880 align:middle line:84% And so, this is given by just this expand-- 00:41:13.880 --> 00:41:19.550 align:middle line:90% yeah, just 1 over Z0 i square. 00:41:19.550 --> 00:41:23.930 align:middle line:90% You take ZJ, two derivatives. 00:41:23.930 --> 00:41:26.600 align:middle line:90% 00:41:26.600 --> 00:41:32.165 align:middle line:84% delta Jt, delta Jt prime, and then you take J equal to 0. 00:41:32.165 --> 00:41:40.430 align:middle line:90% 00:41:40.430 --> 00:41:41.675 align:middle line:90% So the-- yeah. 00:41:41.675 --> 00:41:44.300 align:middle line:84% So the two-point function is a Feynman propagator [INAUDIBLE].. 00:41:44.300 --> 00:41:45.360 align:middle line:90% 00:41:45.360 --> 00:41:47.570 align:middle line:84% So, here, we can just see what we get from here. 00:41:47.570 --> 00:41:50.200 align:middle line:90% 00:41:50.200 --> 00:41:54.940 align:middle line:84% So when you take two derivatives on this, 00:41:54.940 --> 00:41:57.380 align:middle line:84% you take two derivative on J again. 00:41:57.380 --> 00:41:59.590 align:middle line:84% So the first derivative on J you bring down 00:41:59.590 --> 00:42:03.720 align:middle line:84% a factor of K times J. And your second derivative, 00:42:03.720 --> 00:42:05.020 align:middle line:90% we can do two things. 00:42:05.020 --> 00:42:06.900 align:middle line:84% You can act on the exponential again. 00:42:06.900 --> 00:42:09.900 align:middle line:84% And then you can act-- or you can act on the J factor 00:42:09.900 --> 00:42:11.400 align:middle line:84% which you bring down the first time. 00:42:11.400 --> 00:42:13.970 align:middle line:90% 00:42:13.970 --> 00:42:15.470 align:middle line:84% But you have to act on the factor 00:42:15.470 --> 00:42:18.770 align:middle line:84% of J you bring the first time, because you have any free J 00:42:18.770 --> 00:42:19.670 align:middle line:90% left. 00:42:19.670 --> 00:42:23.240 align:middle line:84% And when you set J equal to 0, there will be equal to 0. 00:42:23.240 --> 00:42:24.800 align:middle line:84% So the both derivative should act 00:42:24.800 --> 00:42:27.980 align:middle line:90% on this J, which come together. 00:42:27.980 --> 00:42:34.770 align:middle line:84% So, here, then you get minus i, essentially, K minus 1 t and t 00:42:34.770 --> 00:42:35.850 align:middle line:90% prime. 00:42:35.850 --> 00:42:42.780 align:middle line:84% Just take these two derivatives, then you get K minus 1. 00:42:42.780 --> 00:42:50.480 align:middle line:84% So now, we learned something nice is that this K minus 1, 00:42:50.480 --> 00:42:53.090 align:middle line:84% it's actually the Feynman propagator. 00:42:53.090 --> 00:42:55.790 align:middle line:84% We discussed before, the Feynman function, we discussed before. 00:42:55.790 --> 00:42:59.090 align:middle line:90% 00:42:59.090 --> 00:43:02.000 align:middle line:84% Yeah, the harmonic oscillator version of the Feynman 00:43:02.000 --> 00:43:03.170 align:middle line:90% function. 00:43:03.170 --> 00:43:06.240 align:middle line:84% So, previously, we defined for the field theory. 00:43:06.240 --> 00:43:08.540 align:middle line:84% So this is the harmonic oscillator version 00:43:08.540 --> 00:43:09.665 align:middle line:90% of the Feynman function. 00:43:09.665 --> 00:43:15.350 align:middle line:90% 00:43:15.350 --> 00:43:23.590 align:middle line:84% So we find that K minus 1 t, t prime is just equal to-- 00:43:23.590 --> 00:43:26.560 align:middle line:90% actually, i GF t, t prime. 00:43:26.560 --> 00:43:29.290 align:middle line:90% 00:43:29.290 --> 00:43:31.040 align:middle line:84% So you find that this is just given by GF. 00:43:31.040 --> 00:43:38.620 align:middle line:90% 00:43:38.620 --> 00:43:48.130 align:middle line:84% So, and then, we learned that the ZJ divided by Z0 00:43:48.130 --> 00:44:10.100 align:middle line:84% is just equal to exponential now minus 1/2 J GF J. 00:44:10.100 --> 00:44:11.480 align:middle line:90% So now, it's just GF. 00:44:11.480 --> 00:44:13.850 align:middle line:84% It's just everything determined by this GF. 00:44:13.850 --> 00:44:16.980 align:middle line:90% 00:44:16.980 --> 00:44:21.000 align:middle line:84% So now we have a consistency check. 00:44:21.000 --> 00:44:25.370 align:middle line:84% Now we have a consistency check because, previously, we 00:44:25.370 --> 00:44:29.300 align:middle line:84% have discussed that the GF should satisfy 00:44:29.300 --> 00:44:32.120 align:middle line:90% certain differential equation. 00:44:32.120 --> 00:44:36.470 align:middle line:84% And here we have a differential equation for K. 00:44:36.470 --> 00:44:40.010 align:middle line:84% So K minus 1 also satisfy a differential equation. 00:44:40.010 --> 00:44:44.120 align:middle line:90% It just satisfy this equation. 00:44:44.120 --> 00:44:50.780 align:middle line:84% And so, now, if you plug in there and into here, 00:44:50.780 --> 00:44:55.920 align:middle line:84% OK if by definition, so from this equation. 00:44:55.920 --> 00:45:03.770 align:middle line:84% So let me call this equation star, star, star, 00:45:03.770 --> 00:45:05.000 align:middle line:90% and this equation star. 00:45:05.000 --> 00:45:09.160 align:middle line:90% 00:45:09.160 --> 00:45:19.080 align:middle line:84% So from equation star and star, star, 00:45:19.080 --> 00:45:24.840 align:middle line:84% and we find that K minus 1 should satisfy 00:45:24.840 --> 00:45:34.390 align:middle line:84% the following equation of the partial t squared 00:45:34.390 --> 00:45:41.770 align:middle line:84% minus omega 0 squared plus omega 0 minus i epsilon GF. 00:45:41.770 --> 00:45:42.940 align:middle line:90% Yeah. 00:45:42.940 --> 00:45:53.070 align:middle line:84% K minus 1 should be equal to the delta t minus t prime. 00:45:53.070 --> 00:45:55.080 align:middle line:84% So let me just make sure I get the sign correct. 00:45:55.080 --> 00:45:58.810 align:middle line:90% 00:45:58.810 --> 00:46:00.500 align:middle line:90% Yeah. 00:46:00.500 --> 00:46:02.630 align:middle line:90% So, yeah. 00:46:02.630 --> 00:46:06.430 align:middle line:90% Yeah, I think this is right. 00:46:06.430 --> 00:46:10.390 align:middle line:84% And now, if you plug in this expression, 00:46:10.390 --> 00:46:13.060 align:middle line:84% and then you find that this equation is actually exactly 00:46:13.060 --> 00:46:16.180 align:middle line:84% our definition of the Feynman propagator before. 00:46:16.180 --> 00:46:18.250 align:middle line:84% Here we don't have spatial derivatives. 00:46:18.250 --> 00:46:22.690 align:middle line:84% But if you look at it, in particular this i epsilon is 00:46:22.690 --> 00:46:24.430 align:middle line:84% precisely the i epsilon previously we 00:46:24.430 --> 00:46:27.580 align:middle line:84% need to use to define the Feynman propagator. 00:46:27.580 --> 00:46:30.790 align:middle line:84% And now we find that the epsilon prescription, which 00:46:30.790 --> 00:46:35.440 align:middle line:84% we previously used as a trick to define the Feynman propagator 00:46:35.440 --> 00:46:38.860 align:middle line:84% is actually recovered by this procedure-- 00:46:38.860 --> 00:46:42.603 align:middle line:84% recovered by that procedure of H goes to, yeah, 1 00:46:42.603 --> 00:46:43.270 align:middle line:90% minus i epsilon. 00:46:43.270 --> 00:46:46.290 align:middle line:90% 00:46:46.290 --> 00:46:47.940 align:middle line:90% Just everything is consistent. 00:46:47.940 --> 00:46:50.700 align:middle line:84% Just now you precisely recover that procedure. 00:46:50.700 --> 00:46:53.780 align:middle line:90% 00:46:53.780 --> 00:46:55.100 align:middle line:90% Any questions on this? 00:46:55.100 --> 00:47:06.190 align:middle line:90% 00:47:06.190 --> 00:47:07.120 align:middle line:90% Yes. 00:47:07.120 --> 00:47:09.287 align:middle line:84% AUDIENCE: So I guess in that expression right there. 00:47:09.287 --> 00:47:10.420 align:middle line:90% PROFESSOR: Yeah. 00:47:10.420 --> 00:47:12.670 align:middle line:84% AUDIENCE: If J is a function, if you just integrate it 00:47:12.670 --> 00:47:14.560 align:middle line:90% over t and t prime? 00:47:14.560 --> 00:47:16.510 align:middle line:84% Exponential negative 1/2 J dot GF dot J? 00:47:16.510 --> 00:47:18.255 align:middle line:84% PROFESSOR: Oh, you mean this expansion? 00:47:18.255 --> 00:47:18.880 align:middle line:90% AUDIENCE: Yeah. 00:47:18.880 --> 00:47:19.840 align:middle line:90% PROFESSOR: Yeah. 00:47:19.840 --> 00:47:23.422 align:middle line:84% So if I write it explicitly, you just have, yeah. 00:47:23.422 --> 00:47:24.880 align:middle line:84% Let me write it explicit since this 00:47:24.880 --> 00:47:26.380 align:middle line:90% is a very important equation. 00:47:26.380 --> 00:47:35.620 align:middle line:84% Yeah, so this is just dt, dt prime, Jt, GF, t, t prime, 00:47:35.620 --> 00:47:37.510 align:middle line:90% and Jt prime. 00:47:37.510 --> 00:47:39.470 align:middle line:90% Yeah. 00:47:39.470 --> 00:47:39.970 align:middle line:90% Good. 00:47:39.970 --> 00:47:45.540 align:middle line:90% 00:47:45.540 --> 00:47:48.270 align:middle line:90% So this is all consistent. 00:47:48.270 --> 00:47:53.700 align:middle line:84% So this i epsilon prescription, which we did here, 00:47:53.700 --> 00:47:56.130 align:middle line:84% automatically recovers the i epsilon prescription 00:47:56.130 --> 00:47:59.950 align:middle line:84% in the definition of the Feynman function we defined before. 00:47:59.950 --> 00:48:04.100 align:middle line:90% So it's very nice. 00:48:04.100 --> 00:48:08.960 align:middle line:84% And, in particular, so in momentum space, if you 00:48:08.960 --> 00:48:11.780 align:middle line:84% find the momentum space, GF omega, 00:48:11.780 --> 00:48:14.480 align:middle line:84% if you go to the momentum space, and then you 00:48:14.480 --> 00:48:18.395 align:middle line:84% find just equal to i omega squared minus omega 00:48:18.395 --> 00:48:21.320 align:middle line:90% 0 squared plus i epsilon. 00:48:21.320 --> 00:48:23.660 align:middle line:84% So you see this is actually the previous. 00:48:23.660 --> 00:48:27.050 align:middle line:84% So if you compare with our previous expression 00:48:27.050 --> 00:48:31.580 align:middle line:84% for the Feynman function, when you set k square to 0 00:48:31.580 --> 00:48:35.120 align:middle line:84% and replace the omega 0 square by m square, 00:48:35.120 --> 00:48:37.940 align:middle line:84% then that's exactly the one we derived before. 00:48:37.940 --> 00:48:46.395 align:middle line:90% 00:48:46.395 --> 00:48:46.895 align:middle line:90% Good. 00:48:46.895 --> 00:48:50.040 align:middle line:90% 00:48:50.040 --> 00:48:54.670 align:middle line:84% So now we can try to find the n-point functions. 00:48:54.670 --> 00:48:57.000 align:middle line:84% So now we can work out the all n-point functions. 00:48:57.000 --> 00:49:04.030 align:middle line:90% 00:49:04.030 --> 00:49:08.980 align:middle line:84% So now you can immediately conclude 00:49:08.980 --> 00:49:17.650 align:middle line:84% from the way we do the one-point function that all 00:49:17.650 --> 00:49:26.830 align:middle line:84% n-point functions, so general n for general odd n Gn 00:49:26.830 --> 00:49:30.200 align:middle line:90% is always 0. 00:49:30.200 --> 00:49:32.770 align:middle line:90% Gn is always 0. 00:49:32.770 --> 00:49:34.510 align:middle line:90% So the reason is the following. 00:49:34.510 --> 00:49:37.150 align:middle line:90% 00:49:37.150 --> 00:49:44.490 align:middle line:84% When we carry out this procedure to take the derivatives, 00:49:44.490 --> 00:49:48.175 align:middle line:84% so because the J is always paired in this exponential. 00:49:48.175 --> 00:49:49.800 align:middle line:84% So when you take one derivative, you're 00:49:49.800 --> 00:49:53.490 align:middle line:84% bringing down another J. So because, in the end, 00:49:53.490 --> 00:49:54.500 align:middle line:90% we set J equal to 0. 00:49:54.500 --> 00:49:58.940 align:middle line:84% So you have to get rid of all these J which you bring down 00:49:58.940 --> 00:50:00.800 align:middle line:90% from the exponential. 00:50:00.800 --> 00:50:04.710 align:middle line:84% And then that means, n has to be even. 00:50:04.710 --> 00:50:07.660 align:middle line:84% So if n is odd, then there's always one J left. 00:50:07.660 --> 00:50:11.420 align:middle line:84% And when you set J equal to 0, and then will be 0. 00:50:11.420 --> 00:50:13.910 align:middle line:84% So this is also consistent with your experience 00:50:13.910 --> 00:50:16.940 align:middle line:84% from harmonic oscillator because, in the harmonic 00:50:16.940 --> 00:50:19.880 align:middle line:84% oscillator, if you have all the number of x, then 00:50:19.880 --> 00:50:22.970 align:middle line:84% you have all the number of a and a dagger. 00:50:22.970 --> 00:50:25.970 align:middle line:84% If you have all the number of a and a dagger together, 00:50:25.970 --> 00:50:28.370 align:middle line:84% there's no way when they're sandwiched between 0 00:50:28.370 --> 00:50:32.750 align:middle line:84% and they can annihilate each other, and you will-- yeah. 00:50:32.750 --> 00:50:35.390 align:middle line:90% So you will always get 0. 00:50:35.390 --> 00:50:39.080 align:middle line:90% And so, here, we get it. 00:50:39.080 --> 00:50:39.830 align:middle line:90% Yeah. 00:50:39.830 --> 00:50:47.105 align:middle line:84% So now, for the even n, there's also a simple answer. 00:50:47.105 --> 00:50:50.180 align:middle line:90% 00:50:50.180 --> 00:50:53.980 align:middle line:90% If I have an n-point function-- 00:50:53.980 --> 00:51:00.190 align:middle line:84% so let me just write down Gn, say equal to 0 to t, 00:51:00.190 --> 00:51:06.850 align:middle line:90% say x hat t1, tn. 00:51:06.850 --> 00:51:14.850 align:middle line:84% So for even n, then you see that, again, 00:51:14.850 --> 00:51:17.160 align:middle line:84% because in order for this not to be 0, 00:51:17.160 --> 00:51:20.730 align:middle line:90% then all the x has to be paired. 00:51:20.730 --> 00:51:22.770 align:middle line:90% Then all the x has to be paired. 00:51:22.770 --> 00:51:25.020 align:middle line:90% So, in this case, we just sum-- 00:51:25.020 --> 00:51:27.810 align:middle line:90% 00:51:27.810 --> 00:51:36.380 align:middle line:84% so, in this case, the answer is just sum over all possible 00:51:36.380 --> 00:51:46.760 align:middle line:90% contractions between x ti's. 00:51:46.760 --> 00:51:53.960 align:middle line:84% So by contraction we mean, so if I have xti, 00:51:53.960 --> 00:51:58.980 align:middle line:84% xtj, we say there's a contraction between them. 00:51:58.980 --> 00:52:04.940 align:middle line:84% And so, that's just defined to be GF xi xj. 00:52:04.940 --> 00:52:07.980 align:middle line:90% 00:52:07.980 --> 00:52:09.060 align:middle line:90% Oh, no. ti, tj. 00:52:09.060 --> 00:52:13.410 align:middle line:90% 00:52:13.410 --> 00:52:16.570 align:middle line:90% So you just pair all of them. 00:52:16.570 --> 00:52:18.160 align:middle line:90% So each pair is a contraction. 00:52:18.160 --> 00:52:21.700 align:middle line:84% You just sum over all possible contractions, pair of them, 00:52:21.700 --> 00:52:25.930 align:middle line:84% and each pairing gives you a Feynman function. 00:52:25.930 --> 00:52:28.310 align:middle line:84% So each pairing gives you a Feynman function. 00:52:28.310 --> 00:52:34.600 align:middle line:84% And so, this is actually, in the early days of quantum field 00:52:34.600 --> 00:52:41.180 align:middle line:84% theory when you don't have a path integral, to show 00:52:41.180 --> 00:52:42.470 align:middle line:90% this is actually non-trivial. 00:52:42.470 --> 00:52:46.850 align:middle line:84% Because imagine if you do this time ordered product. 00:52:46.850 --> 00:52:49.053 align:middle line:90% There are many, many pieces. 00:52:49.053 --> 00:52:50.720 align:middle line:84% Because if you have an n-point function, 00:52:50.720 --> 00:52:52.803 align:middle line:84% then you have to write down all possible orderings 00:52:52.803 --> 00:52:54.380 align:middle line:90% between them. 00:52:54.380 --> 00:52:56.840 align:middle line:84% But, in the end, it's a very simple result. 00:52:56.840 --> 00:52:58.910 align:middle line:84% You just sum of all possible contractions. 00:52:58.910 --> 00:53:03.260 align:middle line:84% And each pair is the time ordered. 00:53:03.260 --> 00:53:06.740 align:middle line:84% And so, these the early days without path integral 00:53:06.740 --> 00:53:09.690 align:middle line:84% is actually a highly non-trivial result. 00:53:09.690 --> 00:53:12.600 align:middle line:84% And so, this was first proved by Wick. 00:53:12.600 --> 00:53:15.800 align:middle line:84% So this is called the Wick theorem. 00:53:15.800 --> 00:53:18.410 align:middle line:84% But now, we see, it's actually, if you know the path integral, 00:53:18.410 --> 00:53:21.030 align:middle line:84% then it's a trivial consequence of that. 00:53:21.030 --> 00:53:23.480 align:middle line:84% The path integral is actually quadratic in J. 00:53:23.480 --> 00:53:25.220 align:middle line:90% It's quadratic in J. 00:53:25.220 --> 00:53:26.960 align:middle line:84% So, yeah, so give you an example. 00:53:26.960 --> 00:53:30.180 align:middle line:90% 00:53:30.180 --> 00:53:32.000 align:middle line:84% So let's look at four-point functions. 00:53:32.000 --> 00:54:03.830 align:middle line:90% 00:54:03.830 --> 00:54:05.620 align:middle line:84% So if you look at four-point functions. 00:54:05.620 --> 00:54:13.273 align:middle line:90% 00:54:13.273 --> 00:54:15.190 align:middle line:84% So I don't even have to write that thing down. 00:54:15.190 --> 00:54:19.120 align:middle line:84% Let's just draw four dots below to the four points. 00:54:19.120 --> 00:54:22.180 align:middle line:90% 00:54:22.180 --> 00:54:25.240 align:middle line:84% And then, I just sum over all pairing between them. 00:54:25.240 --> 00:54:28.870 align:middle line:84% And each pairing will give me a Feynman propagator. 00:54:28.870 --> 00:54:34.990 align:middle line:84% So 1, I compare 1 and a 2 or 3 and 4. 00:54:34.990 --> 00:54:42.250 align:middle line:84% And I can also pair 1 and 3 and 2 and 4. 00:54:42.250 --> 00:54:50.400 align:middle line:84% And I can also pair 1 with 4 and the 2 with 3. 00:54:50.400 --> 00:54:56.935 align:middle line:90% 00:54:56.935 --> 00:55:00.500 align:middle line:90% Oh, 2-- 2 is here. 00:55:00.500 --> 00:55:01.420 align:middle line:90% So 2 with 3. 00:55:01.420 --> 00:55:04.270 align:middle line:90% I can also do that pairing. 00:55:04.270 --> 00:55:08.200 align:middle line:84% And so, if I write it in terms of the expressions, 00:55:08.200 --> 00:55:22.720 align:middle line:84% then I have GF t1, t2-- let me just, t1, t2, GF, t3, t4 00:55:22.720 --> 00:55:34.040 align:middle line:90% plus GF, t1, t3, GF, t2, t4. 00:55:34.040 --> 00:55:44.480 align:middle line:84% Then plus the GF, t1, t4 plus times GF, t2, t3. 00:55:44.480 --> 00:55:48.340 align:middle line:90% 00:55:48.340 --> 00:55:52.480 align:middle line:84% So you just sum over all such pairings. 00:55:52.480 --> 00:55:56.320 align:middle line:90% And each pairing gives you a GF. 00:55:56.320 --> 00:55:57.123 align:middle line:90% Yes. 00:55:57.123 --> 00:55:57.790 align:middle line:90% AUDIENCE: Sorry. 00:55:57.790 --> 00:55:59.540 align:middle line:84% I thought because of the time of ordering, 00:55:59.540 --> 00:56:03.180 align:middle line:90% you can't choose your pairing. 00:56:03.180 --> 00:56:04.893 align:middle line:90% There's only one way to pair. 00:56:04.893 --> 00:56:06.060 align:middle line:90% PROFESSOR: What do you mean? 00:56:06.060 --> 00:56:09.435 align:middle line:84% AUDIENCE: Why is it ok to do different pairings even though 00:56:09.435 --> 00:56:12.055 align:middle line:90% it's-- things are time ordered? 00:56:12.055 --> 00:56:14.980 align:middle line:84% PROFESSOR: Oh, what do you mean, you cannot do the pairing? 00:56:14.980 --> 00:56:16.483 align:middle line:84% AUDIENCE: The x's are time ordered. 00:56:16.483 --> 00:56:17.150 align:middle line:90% PROFESSOR: Yeah. 00:56:17.150 --> 00:56:17.820 align:middle line:90% AUDIENCE: The x's. 00:56:17.820 --> 00:56:18.487 align:middle line:90% PROFESSOR: Yeah. 00:56:18.487 --> 00:56:21.720 align:middle line:84% AUDIENCE: So wouldn't you have to pair them in that order? 00:56:21.720 --> 00:56:22.860 align:middle line:90% PROFESSOR: No. 00:56:22.860 --> 00:56:23.690 align:middle line:90% So that's the key. 00:56:23.690 --> 00:56:26.850 align:middle line:84% You just, somehow, this is the consequence of the theorem. 00:56:26.850 --> 00:56:29.257 align:middle line:84% If you want to just write down the orderings, 00:56:29.257 --> 00:56:30.840 align:middle line:84% then it's actually rather complicated. 00:56:30.840 --> 00:56:32.640 align:middle line:90% Yeah. 00:56:32.640 --> 00:56:35.940 align:middle line:84% But, somehow, the magic is that once you, say, write everything 00:56:35.940 --> 00:56:39.780 align:middle line:84% explicitly, you do all the everything, 00:56:39.780 --> 00:56:42.180 align:middle line:84% in the end you can group everything just in terms 00:56:42.180 --> 00:56:44.048 align:middle line:84% of product of the Feynman functions. 00:56:44.048 --> 00:56:44.548 align:middle line:90% Yeah. 00:56:44.548 --> 00:56:47.150 align:middle line:90% 00:56:47.150 --> 00:56:48.570 align:middle line:90% Yes. 00:56:48.570 --> 00:56:50.400 align:middle line:84% AUDIENCE: So for the harmonic oscillator, 00:56:50.400 --> 00:56:54.500 align:middle line:84% we know that at a given point in time, the position is Gaussian. 00:56:54.500 --> 00:56:58.750 align:middle line:84% And so, that would mean that the n-point function for all 00:56:58.750 --> 00:57:00.670 align:middle line:84% the t's equal to each other should 00:57:00.670 --> 00:57:04.720 align:middle line:84% be non-zero only for n equal to 1 or 2 00:57:04.720 --> 00:57:06.110 align:middle line:90% but not for n greater, right? 00:57:06.110 --> 00:57:08.380 align:middle line:84% Because only the first two moments of a Gaussian 00:57:08.380 --> 00:57:10.220 align:middle line:90% are non-zero. 00:57:10.220 --> 00:57:13.700 align:middle line:84% That seems inconsistent with the prescription over here. 00:57:13.700 --> 00:57:14.780 align:middle line:90% PROFESSOR: Sorry. 00:57:14.780 --> 00:57:17.000 align:middle line:90% Why are you saying that? 00:57:17.000 --> 00:57:18.770 align:middle line:90% AUDIENCE: At a given time, the-- 00:57:18.770 --> 00:57:21.140 align:middle line:84% PROFESSOR: No, but all these time are different. 00:57:21.140 --> 00:57:24.180 align:middle line:84% AUDIENCE: Right, but if I were to take the times to be equal. 00:57:24.180 --> 00:57:25.160 align:middle line:90% PROFESSOR: OK, yeah. 00:57:25.160 --> 00:57:29.320 align:middle line:84% AUDIENCE: Then so, for example, the variance of the particle 00:57:29.320 --> 00:57:32.898 align:middle line:84% position would be your two-point function evaluated at t and-- 00:57:32.898 --> 00:57:33.565 align:middle line:90% PROFESSOR: Yeah. 00:57:33.565 --> 00:57:34.440 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:57:34.440 --> 00:57:36.813 align:middle line:84% PROFESSOR: Yeah, this is just the x to the power n. 00:57:36.813 --> 00:57:37.480 align:middle line:90% AUDIENCE: Right. 00:57:37.480 --> 00:57:39.790 align:middle line:84% PROFESSOR: Yeah, the x to the power n 00:57:39.790 --> 00:57:42.850 align:middle line:90% is non-zero for even power. 00:57:42.850 --> 00:57:44.470 align:middle line:84% AUDIENCE: Yes, but then, so then, 00:57:44.470 --> 00:57:47.163 align:middle line:84% in this formula, that'd be GF t comma t. 00:57:47.163 --> 00:57:47.830 align:middle line:90% PROFESSOR: Yeah. 00:57:47.830 --> 00:57:48.880 align:middle line:90% AUDIENCE: Which is non-zero. 00:57:48.880 --> 00:57:49.547 align:middle line:90% PROFESSOR: Yeah. 00:57:49.547 --> 00:57:50.623 align:middle line:90% AUDIENCE: But-- 00:57:50.623 --> 00:57:52.290 align:middle line:84% PROFESSOR: Yeah, but the GF is non-zero. 00:57:52.290 --> 00:57:55.380 align:middle line:90% 00:57:55.380 --> 00:57:57.130 align:middle line:84% AUDIENCE: Right, but then, what I'm saying 00:57:57.130 --> 00:57:58.750 align:middle line:84% is, the four-point function then would also 00:57:58.750 --> 00:57:59.930 align:middle line:90% give you something non-zero. 00:57:59.930 --> 00:58:01.975 align:middle line:90% PROFESSOR: Yeah, it is non-zero. 00:58:01.975 --> 00:58:03.100 align:middle line:90% Yeah, it's very consistent. 00:58:03.100 --> 00:58:05.308 align:middle line:84% AUDIENCE: But the fourth moment of the Gaussian is 0. 00:58:05.308 --> 00:58:06.360 align:middle line:90% PROFESSOR: No. 00:58:06.360 --> 00:58:10.220 align:middle line:84% The fourth moment of Gaussian certainly is non-zero. 00:58:10.220 --> 00:58:14.000 align:middle line:84% Yeah, x4, yeah, Gaussian you have x4. 00:58:14.000 --> 00:58:16.088 align:middle line:90% That's certainly non-zero. 00:58:16.088 --> 00:58:19.469 align:middle line:84% AUDIENCE: But the means is 0, so all cumulants higher than 2 00:58:19.469 --> 00:58:21.550 align:middle line:90% are zero for a Gaussian? 00:58:21.550 --> 00:58:22.780 align:middle line:90% PROFESSOR: No, no, no. 00:58:22.780 --> 00:58:25.480 align:middle line:84% For the Gaussian, it is all how equivalent 00:58:25.480 --> 00:58:28.413 align:middle line:84% can be expressed in terms of the sigma 00:58:28.413 --> 00:58:29.705 align:middle line:90% and in terms of the two points. 00:58:29.705 --> 00:58:33.610 align:middle line:84% Yeah, here is just exactly what you show here. 00:58:33.610 --> 00:58:34.450 align:middle line:90% Yeah. 00:58:34.450 --> 00:58:36.310 align:middle line:84% Just everything can be expressed in terms 00:58:36.310 --> 00:58:38.900 align:middle line:90% of the two-point moment. 00:58:38.900 --> 00:58:39.400 align:middle line:90% Yeah. 00:58:39.400 --> 00:58:44.540 align:middle line:84% So if you set all the t to be 0, so this is GF is the same. 00:58:44.540 --> 00:58:45.310 align:middle line:90% So this is GF. 00:58:45.310 --> 00:58:47.510 align:middle line:84% Essentially, the only difference between them. 00:58:47.510 --> 00:58:48.207 align:middle line:90% So this is GF. 00:58:48.207 --> 00:58:50.290 align:middle line:84% Yeah, essentially you just add them together just, 00:58:50.290 --> 00:58:51.790 align:middle line:90% read GF squared. 00:58:51.790 --> 00:58:52.630 align:middle line:90% And, yeah. 00:58:52.630 --> 00:58:54.850 align:middle line:84% Yeah, just do this exercise yourself, 00:58:54.850 --> 00:58:56.350 align:middle line:90% and you will see it's the same. 00:58:56.350 --> 00:58:59.180 align:middle line:90% Yeah. 00:58:59.180 --> 00:58:59.680 align:middle line:90% Good. 00:58:59.680 --> 00:59:03.240 align:middle line:90% 00:59:03.240 --> 00:59:10.480 align:middle line:84% So now, with this preparation in quantum mechanics, 00:59:10.480 --> 00:59:13.890 align:middle line:84% now we can immediately just move in field theory. 00:59:13.890 --> 00:59:16.680 align:middle line:84% So now we can immediately move to field theory. 00:59:16.680 --> 00:59:30.938 align:middle line:84% So now you can say time ordered functions in field theory. 00:59:30.938 --> 00:59:37.030 align:middle line:90% 00:59:37.030 --> 00:59:39.780 align:middle line:84% So, before we do that, any other questions? 00:59:39.780 --> 00:59:46.660 align:middle line:90% 00:59:46.660 --> 00:59:48.580 align:middle line:90% Good. 00:59:48.580 --> 00:59:53.890 align:middle line:84% So, again, to go to field theory, 00:59:53.890 --> 00:59:59.760 align:middle line:84% now the only thing we need is just copy notations. 00:59:59.760 --> 01:00:01.630 align:middle line:84% You just need to change the notations. 01:00:01.630 --> 01:00:05.650 align:middle line:90% Just remember how you-- 01:00:05.650 --> 01:00:10.360 align:middle line:84% yeah, just replace the appropriate dynamic variable 01:00:10.360 --> 01:00:13.570 align:middle line:84% in quantum mechanics by the appropriate dynamic variables 01:00:13.570 --> 01:00:14.530 align:middle line:90% in field theory. 01:00:14.530 --> 01:00:17.620 align:middle line:90% And then that's it. 01:00:17.620 --> 01:00:19.880 align:middle line:90% So let's write down. 01:00:19.880 --> 01:00:21.970 align:middle line:84% So now we have, consider, in field theory, 01:00:21.970 --> 01:00:24.400 align:middle line:84% suppose we consider this n-point function. 01:00:24.400 --> 01:00:30.610 align:middle line:84% And now, xn-- now xn denote the space time point now. 01:00:30.610 --> 01:00:33.250 align:middle line:90% So let me call the-- 01:00:33.250 --> 01:00:35.650 align:middle line:84% previously we call the vacuum for the field theory 01:00:35.650 --> 01:00:38.030 align:middle line:84% to be omega in the interacting theory. 01:00:38.030 --> 01:00:43.570 align:middle line:84% So now you can see that this quantity, phi x1 and the phi 01:00:43.570 --> 01:00:47.600 align:middle line:84% xn, let's look at this endpoint function between the vacuum. 01:00:47.600 --> 01:00:51.710 align:middle line:90% 01:00:51.710 --> 01:00:59.350 align:middle line:84% So now, again, if we call this thing to be X, 01:00:59.350 --> 01:01:01.240 align:middle line:84% So now we can immediately write down 01:01:01.240 --> 01:01:05.020 align:middle line:84% the answer in field theory, just the D phi. 01:01:05.020 --> 01:01:07.600 align:middle line:84% Now you just replace the DX by D phi. 01:01:07.600 --> 01:01:10.780 align:middle line:84% Then you'll, again, you just have this X, capital X. 01:01:10.780 --> 01:01:17.140 align:middle line:84% And then you have now i S phi then divided by D phi 01:01:17.140 --> 01:01:19.800 align:middle line:90% without i S phi. 01:01:19.800 --> 01:01:25.010 align:middle line:90% 01:01:25.010 --> 01:01:28.670 align:middle line:84% And another boundary condition is 01:01:28.670 --> 01:01:33.530 align:middle line:84% that the phi, when you do this integral, again, 01:01:33.530 --> 01:01:36.930 align:middle line:84% t goes to infinity to minus infinity to infinity. 01:01:36.930 --> 01:01:44.280 align:middle line:84% And now tx will-- should go to 0 for both t plus minus infinity. 01:01:44.280 --> 01:01:48.570 align:middle line:84% So that's the analog of a previous simple x equal to 0. 01:01:48.570 --> 01:01:53.970 align:middle line:84% And because it's here, remember, the x here is just the labels. 01:01:53.970 --> 01:01:56.580 align:middle line:84% And so, for each variable we require, 01:01:56.580 --> 01:02:00.390 align:middle line:84% it's go to 0 at t plus minus infinity. 01:02:00.390 --> 01:02:02.220 align:middle line:90% You could do 0. 01:02:02.220 --> 01:02:04.140 align:middle line:84% And also, normally, in field theory, 01:02:04.140 --> 01:02:08.670 align:middle line:84% we assume, yeah, in order for the integral 01:02:08.670 --> 01:02:11.080 align:middle line:84% to have well-defined behavior, et cetera. 01:02:11.080 --> 01:02:17.970 align:middle line:84% We often just assume also go to 0 in the spatial infinity. 01:02:17.970 --> 01:02:21.540 align:middle line:84% Yeah, this is just often for convenience. 01:02:21.540 --> 01:02:25.890 align:middle line:84% And physically, this also means that infinitely far away, 01:02:25.890 --> 01:02:29.970 align:middle line:84% and we assume that the field are not excited. 01:02:29.970 --> 01:02:34.200 align:middle line:84% Yeah, we are interested only in the, yeah, physical excitations 01:02:34.200 --> 01:02:35.040 align:middle line:90% in finite region. 01:02:35.040 --> 01:02:38.280 align:middle line:84% Anyway, so this is the condition we 01:02:38.280 --> 01:02:41.015 align:middle line:84% impose when doing the path integral in the field theory 01:02:41.015 --> 01:02:41.515 align:middle line:90% case. 01:02:41.515 --> 01:02:45.270 align:middle line:90% 01:02:45.270 --> 01:02:49.700 align:middle line:84% So now, again, you can introduce a generating functional. 01:02:49.700 --> 01:02:52.580 align:middle line:84% You can just, again, just copy notation, 01:02:52.580 --> 01:02:54.890 align:middle line:90% copy the previous formula. 01:02:54.890 --> 01:02:57.290 align:middle line:90% And by changing notation. 01:02:57.290 --> 01:03:23.600 align:middle line:90% 01:03:23.600 --> 01:03:28.300 align:middle line:84% So now the generating functional ZJ 01:03:28.300 --> 01:03:36.550 align:middle line:84% is defined to be D phi exponential i S phi. 01:03:36.550 --> 01:03:39.340 align:middle line:90% And then, now you just add-- 01:03:39.340 --> 01:03:44.560 align:middle line:84% now you integrate over all space time points and phi x. 01:03:44.560 --> 01:03:48.040 align:middle line:84% Again, we introduce a J. But now you integrate over 01:03:48.040 --> 01:03:49.090 align:middle line:90% all space time points. 01:03:49.090 --> 01:03:52.380 align:middle line:90% 01:03:52.380 --> 01:04:01.980 align:middle line:84% And then, similarly, the ZJ divided by Z0, 01:04:01.980 --> 01:04:03.720 align:middle line:90% so this we call-- 01:04:03.720 --> 01:04:09.780 align:middle line:84% yeah, the Z0, again, is just the integral without x, any x. 01:04:09.780 --> 01:04:18.550 align:middle line:84% So divided by Z0 is equal to now the omega 01:04:18.550 --> 01:04:24.310 align:middle line:84% and time-ordered product of the exponential i. 01:04:24.310 --> 01:04:39.420 align:middle line:90% 01:04:39.420 --> 01:04:40.870 align:middle line:90% So, again, just given by that. 01:04:40.870 --> 01:04:43.290 align:middle line:84% And, again, this time-ordered product, 01:04:43.290 --> 01:04:46.560 align:middle line:84% time-ordered exponential should be understood, 01:04:46.560 --> 01:04:49.620 align:middle line:84% as you expanded this in power series, 01:04:49.620 --> 01:04:51.340 align:middle line:84% when you expand it in power series, 01:04:51.340 --> 01:04:53.010 align:middle line:90% then you have powers of phi. 01:04:53.010 --> 01:04:56.640 align:middle line:84% Then you just order those phis in terms of time ordering. 01:04:56.640 --> 01:04:59.730 align:middle line:84% And then, again, the integration and J 01:04:59.730 --> 01:05:02.280 align:middle line:84% can be pulled outside this expectation value. 01:05:02.280 --> 01:05:07.380 align:middle line:90% 01:05:07.380 --> 01:05:08.550 align:middle line:90% Good? 01:05:08.550 --> 01:05:15.270 align:middle line:84% And the Z0 is just the same as ZJ equal to 0, 01:05:15.270 --> 01:05:26.670 align:middle line:90% so without any J. 01:05:26.670 --> 01:05:29.070 align:middle line:84% So this is, again, just immediately give you 01:05:29.070 --> 01:05:32.190 align:middle line:84% a general prescription for calculating 01:05:32.190 --> 01:05:36.660 align:middle line:84% n-point function in any theory, in any scalar theory. 01:05:36.660 --> 01:05:39.180 align:middle line:84% So, here, I don't even have to specify 01:05:39.180 --> 01:05:42.930 align:middle line:90% the precise form of the action. 01:05:42.930 --> 01:05:45.330 align:middle line:90% It just carry through. 01:05:45.330 --> 01:05:47.800 align:middle line:84% This also applies to interacting theory. 01:05:47.800 --> 01:05:50.680 align:middle line:84% So this formula also applies to interacting theory. 01:05:50.680 --> 01:05:56.920 align:middle line:90% And, yeah, it's very general. 01:05:56.920 --> 01:06:01.830 align:middle line:84% So this is the power, say, of this path integral formalism. 01:06:01.830 --> 01:06:04.440 align:middle line:84% So once you understand in the quantum mechanics case, 01:06:04.440 --> 01:06:07.530 align:middle line:84% go to quantum field theory is automatic. 01:06:07.530 --> 01:06:12.410 align:middle line:90% 01:06:12.410 --> 01:06:12.980 align:middle line:90% Good. 01:06:12.980 --> 01:06:15.230 align:middle line:84% So now let's look at how to treat-- 01:06:15.230 --> 01:06:17.270 align:middle line:90% how to do this thing-- 01:06:17.270 --> 01:06:21.000 align:middle line:84% calculate this thing in field theory. 01:06:21.000 --> 01:06:23.550 align:middle line:84% So, first, let's just look at the free field theory. 01:06:23.550 --> 01:06:26.520 align:middle line:90% 01:06:26.520 --> 01:06:29.792 align:middle line:84% And then, before we look at the interacting case, 01:06:29.792 --> 01:06:31.500 align:middle line:84% let's just look at the free field theory. 01:06:31.500 --> 01:06:34.850 align:middle line:90% 01:06:34.850 --> 01:06:38.870 align:middle line:84% So free field theory is almost identical, again, 01:06:38.870 --> 01:06:42.230 align:middle line:84% almost identical to the harmonic oscillator case 01:06:42.230 --> 01:06:44.310 align:middle line:84% because harmonic oscillator is also-- 01:06:44.310 --> 01:06:45.810 align:middle line:84% because the free theory will be also 01:06:45.810 --> 01:06:47.810 align:middle line:90% a quadratic Gaussian integral. 01:06:47.810 --> 01:06:50.390 align:middle line:84% So everything will be very similar to the harmonic 01:06:50.390 --> 01:06:51.450 align:middle line:90% oscillator case. 01:06:51.450 --> 01:06:55.220 align:middle line:84% We just need to, again, replace some notations. 01:06:55.220 --> 01:06:57.950 align:middle line:84% So now, let's consider the free field case. 01:06:57.950 --> 01:07:05.030 align:middle line:84% Consider this minus 1/2 partial mu phi, partial mu phi, 01:07:05.030 --> 01:07:10.280 align:middle line:84% minus 1/2 m square phi square without any nonlinear-- 01:07:10.280 --> 01:07:14.830 align:middle line:84% without any cubic or higher power term. 01:07:14.830 --> 01:07:18.840 align:middle line:84% And, again, the S can be written-- 01:07:18.840 --> 01:07:20.380 align:middle line:90% so S, in this case-- 01:07:20.380 --> 01:07:22.650 align:middle line:84% So, here, 0 means the free theory 01:07:22.650 --> 01:07:27.030 align:middle line:84% because later we will do interacting theory. 01:07:27.030 --> 01:07:30.990 align:middle line:84% So s, in this case, again, you can write it as-- 01:07:30.990 --> 01:07:34.230 align:middle line:90% you can integration by part. 01:07:34.230 --> 01:07:44.245 align:middle line:84% You can write it as d4x, d4x prime, then phi x, Kx, 01:07:44.245 --> 01:07:46.995 align:middle line:90% x prime, and then phi x prime. 01:07:46.995 --> 01:07:52.490 align:middle line:90% 01:07:52.490 --> 01:08:03.260 align:middle line:84% And now, the K is given by d squared 01:08:03.260 --> 01:08:06.080 align:middle line:90% plus m squared minus epsilon. 01:08:06.080 --> 01:08:08.540 align:middle line:84% Again, this epsilon comes from that thing. 01:08:08.540 --> 01:08:11.990 align:middle line:84% You just, if you work through, you just, yeah, 01:08:11.990 --> 01:08:13.670 align:middle line:90% it just minus epsilon. 01:08:13.670 --> 01:08:17.720 align:middle line:90% And then delta 4x minus x prime. 01:08:17.720 --> 01:08:22.149 align:middle line:90% 01:08:22.149 --> 01:08:23.500 align:middle line:90% And, yeah. 01:08:23.500 --> 01:08:26.180 align:middle line:90% 01:08:26.180 --> 01:08:26.680 align:middle line:90% Good. 01:08:26.680 --> 01:08:29.260 align:middle line:90% So, again, we can just-- 01:08:29.260 --> 01:08:35.590 align:middle line:84% shorthand notation, so this has a minus 1/2 phi dot K dot phi. 01:08:35.590 --> 01:08:37.990 align:middle line:84% Just now, the only difference is that you change it 01:08:37.990 --> 01:08:41.649 align:middle line:84% from integration of dt, integration of the full space 01:08:41.649 --> 01:08:43.060 align:middle line:90% time. 01:08:43.060 --> 01:08:46.240 align:middle line:84% Yeah, everything else is the same. 01:08:46.240 --> 01:08:46.740 align:middle line:90% And. 01:08:46.740 --> 01:08:47.729 align:middle line:90% Now, this is ZJ. 01:08:47.729 --> 01:08:52.990 align:middle line:90% 01:08:52.990 --> 01:09:02.930 align:middle line:84% ZJ just given by D phi exponential, again, 01:09:02.930 --> 01:09:04.449 align:middle line:84% just given by a Gaussian integral. 01:09:04.449 --> 01:09:09.620 align:middle line:90% 01:09:09.620 --> 01:09:16.890 align:middle line:84% So let me see whether I can squeeze in a Gaussian integral, 01:09:16.890 --> 01:09:22.830 align:middle line:84% phi dot K dot phi, then plus i J dot phi. 01:09:22.830 --> 01:09:28.090 align:middle line:84% So we can, again, write this in the simplified notation 01:09:28.090 --> 01:09:29.325 align:middle line:90% as i J dot phi. 01:09:29.325 --> 01:09:32.149 align:middle line:90% 01:09:32.149 --> 01:09:34.640 align:middle line:90% So, again, you just-- 01:09:34.640 --> 01:09:36.200 align:middle line:90% and then you find this-- 01:09:36.200 --> 01:09:41.540 align:middle line:84% you get some other C. Again, you get determinant K. 01:09:41.540 --> 01:09:49.670 align:middle line:84% And then you get exponential i over 2 J K1 J. 01:09:49.670 --> 01:09:52.290 align:middle line:84% So everything is exactly the same. 01:09:52.290 --> 01:09:56.390 align:middle line:84% And then, this is the same as Z0, this part. 01:09:56.390 --> 01:09:59.588 align:middle line:90% And then, again, they cancel. 01:09:59.588 --> 01:10:00.380 align:middle line:90% Again, they cancel. 01:10:00.380 --> 01:10:05.410 align:middle line:90% 01:10:05.410 --> 01:10:06.925 align:middle line:90% So, again, we find-- 01:10:06.925 --> 01:10:23.030 align:middle line:90% 01:10:23.030 --> 01:10:27.182 align:middle line:84% so, in this case, again, we find that K minus 1 x, 01:10:27.182 --> 01:10:33.230 align:middle line:84% x prime equal to i GF x, x prime. 01:10:33.230 --> 01:10:39.560 align:middle line:84% So this i epsilon prescription, so it's precisely, yeah, 01:10:39.560 --> 01:10:43.790 align:middle line:84% if you check the definition, so this i epsilon prescription m 01:10:43.790 --> 01:10:45.650 align:middle line:84% square goes to m squared minus epsilon, 01:10:45.650 --> 01:10:48.470 align:middle line:84% is also the precisely what we did before 01:10:48.470 --> 01:10:51.470 align:middle line:90% for the Feynman function. 01:10:51.470 --> 01:10:56.630 align:middle line:90% 01:10:56.630 --> 01:11:05.140 align:middle line:84% So, now, the final answer, the ZJ divide by Z0, 01:11:05.140 --> 01:11:08.260 align:middle line:84% yeah, I almost don't want to copy it. 01:11:08.260 --> 01:11:11.240 align:middle line:84% It's just exactly the same as this. 01:11:11.240 --> 01:11:17.410 align:middle line:84% You just replace the dt integral by d4x x integral. 01:11:17.410 --> 01:11:20.140 align:middle line:90% And the d4x integral. 01:11:20.140 --> 01:11:23.320 align:middle line:90% So, yeah, let me just write it. 01:11:23.320 --> 01:11:26.870 align:middle line:90% 01:11:26.870 --> 01:11:43.735 align:middle line:84% Exponential minus 1/2, then J GF J. Any questions on this? 01:11:43.735 --> 01:11:51.320 align:middle line:90% 01:11:51.320 --> 01:11:55.180 align:middle line:84% And, again, if you calculate n-point function, 01:11:55.180 --> 01:11:59.780 align:middle line:84% then you just get the identical structure as here. 01:11:59.780 --> 01:12:02.930 align:middle line:84% Just the identical structure here. 01:12:02.930 --> 01:12:08.740 align:middle line:84% The only difference is that you replace GF, t1, t2 by x1, x2, 01:12:08.740 --> 01:12:09.940 align:middle line:90% x3, x4, et cetera. 01:12:09.940 --> 01:12:12.430 align:middle line:90% Everything is just identical. 01:12:12.430 --> 01:12:15.490 align:middle line:84% So, to save time, I will not copy them again. 01:12:15.490 --> 01:12:16.565 align:middle line:90% So, do you have-- yeah. 01:12:16.565 --> 01:12:18.190 align:middle line:84% AUDIENCE: So in this particular theory, 01:12:18.190 --> 01:12:20.800 align:middle line:84% do we have the condition that phi goes to 0 01:12:20.800 --> 01:12:23.302 align:middle line:90% when t goes to infinity? 01:12:23.302 --> 01:12:24.010 align:middle line:90% PROFESSOR: Sorry? 01:12:24.010 --> 01:12:25.677 align:middle line:84% AUDIENCE: In this particular theory, Do. 01:12:25.677 --> 01:12:30.020 align:middle line:84% We have the condition that phi goes to 0? 01:12:30.020 --> 01:12:33.890 align:middle line:84% PROFESSOR: Yeah, so that ensures, 01:12:33.890 --> 01:12:36.470 align:middle line:84% when you do integration by parts, everything is 0. 01:12:36.470 --> 01:12:38.810 align:middle line:84% AUDIENCE: We have the solution for phi? 01:12:38.810 --> 01:12:40.070 align:middle line:90% PROFESSOR: Sorry? 01:12:40.070 --> 01:12:41.810 align:middle line:84% AUDIENCE: We know the solution for phi 01:12:41.810 --> 01:12:45.320 align:middle line:84% in terms of x and t right, and it's like a plane wave? 01:12:45.320 --> 01:12:48.305 align:middle line:90% 01:12:48.305 --> 01:12:50.180 align:middle line:84% PROFESSOR: No, this is the boundary condition 01:12:50.180 --> 01:12:51.930 align:middle line:90% in your path integral. 01:12:51.930 --> 01:12:53.430 align:middle line:84% Yeah, this is the boundary condition 01:12:53.430 --> 01:12:55.140 align:middle line:90% in your path integral. 01:12:55.140 --> 01:12:56.470 align:middle line:90% We're not talking about this. 01:12:56.470 --> 01:13:00.480 align:middle line:84% Yeah, so this is-- yeah, phi just the-- 01:13:00.480 --> 01:13:04.470 align:middle line:84% yeah, here, I didn't write down any explicit solution for phi. 01:13:04.470 --> 01:13:05.320 align:middle line:90% Sorry. 01:13:05.320 --> 01:13:08.802 align:middle line:84% What plane wave we are talking about? 01:13:08.802 --> 01:13:11.260 align:middle line:84% AUDIENCE: Oh, I was thinking about the solution for all phi 01:13:11.260 --> 01:13:14.480 align:middle line:84% field theory that we did a few lectures ago. 01:13:14.480 --> 01:13:15.530 align:middle line:90% PROFESSOR: Right. 01:13:15.530 --> 01:13:17.157 align:middle line:90% AUDIENCE: So that [INAUDIBLE]? 01:13:17.157 --> 01:13:17.990 align:middle line:90% PROFESSOR: Yeah, no. 01:13:17.990 --> 01:13:20.420 align:middle line:90% No, that will also go to 0. 01:13:20.420 --> 01:13:21.350 align:middle line:90% Yeah, no, no, no. 01:13:21.350 --> 01:13:23.060 align:middle line:90% That's an operator equation. 01:13:23.060 --> 01:13:24.050 align:middle line:90% That's an operator. 01:13:24.050 --> 01:13:26.880 align:middle line:84% Here, it's just the field in your path integral. 01:13:26.880 --> 01:13:30.510 align:middle line:84% So, here, you just integrate over all possible. 01:13:30.510 --> 01:13:35.000 align:middle line:84% There, when we write that, that's an operator equation 01:13:35.000 --> 01:13:36.690 align:middle line:90% with a and a dagger there. 01:13:36.690 --> 01:13:40.230 align:middle line:84% So here is just the ordinary function of space time, 01:13:40.230 --> 01:13:45.066 align:middle line:84% which we impose the boundary condition in the path integral. 01:13:45.066 --> 01:13:45.566 align:middle line:90% Yeah. 01:13:45.566 --> 01:13:52.380 align:middle line:90% 01:13:52.380 --> 01:13:58.130 align:middle line:84% And, similarly, in here, there's Wick theorem, just everything 01:13:58.130 --> 01:13:59.390 align:middle line:90% just goes through. 01:13:59.390 --> 01:14:00.620 align:middle line:90% Everything just goes through. 01:14:00.620 --> 01:14:03.330 align:middle line:90% 01:14:03.330 --> 01:14:03.830 align:middle line:90% Good. 01:14:03.830 --> 01:14:06.570 align:middle line:90% 01:14:06.570 --> 01:14:07.820 align:middle line:90% Any other questions? 01:14:07.820 --> 01:14:12.160 align:middle line:90% 01:14:12.160 --> 01:14:15.310 align:middle line:84% So, for the last few minutes, then we 01:14:15.310 --> 01:14:19.510 align:middle line:84% can venture a little bit into the interacting case. 01:14:19.510 --> 01:14:22.360 align:middle line:90% Was there any question? 01:14:22.360 --> 01:14:26.290 align:middle line:84% So now we can venture into interactions. 01:14:26.290 --> 01:14:29.180 align:middle line:84% So now we have our master formula. 01:14:29.180 --> 01:14:31.820 align:middle line:84% And now we can treat what happens in the interaction 01:14:31.820 --> 01:14:32.320 align:middle line:90% case. 01:14:32.320 --> 01:14:36.440 align:middle line:90% 01:14:36.440 --> 01:14:38.090 align:middle line:84% So now we can go to interacting theory. 01:14:38.090 --> 01:14:45.200 align:middle line:90% 01:14:45.200 --> 01:14:48.500 align:middle line:84% So in the interacting theory, let's just 01:14:48.500 --> 01:14:54.140 align:middle line:84% consider, say, the case which L equal to L0, 01:14:54.140 --> 01:14:57.560 align:middle line:90% then you pass some polynomial. 01:14:57.560 --> 01:14:59.780 align:middle line:84% For example, the simplest case is 01:14:59.780 --> 01:15:04.730 align:middle line:84% just lambda we discussed before to the phi 4. 01:15:04.730 --> 01:15:08.780 align:middle line:90% So plus some higher power term. 01:15:08.780 --> 01:15:11.150 align:middle line:84% So, for simplicity, I will just-- 01:15:11.150 --> 01:15:14.310 align:middle line:84% but what we will do, we will not depend on details form. 01:15:14.310 --> 01:15:15.980 align:middle line:90% Let me just write LI. 01:15:15.980 --> 01:15:19.340 align:middle line:84% Imagine you have some interacting terms. 01:15:19.340 --> 01:15:22.450 align:middle line:84% So, in this particular case, the LI is equal to that. 01:15:22.450 --> 01:15:24.200 align:middle line:84% But we can consider the more general case. 01:15:24.200 --> 01:15:29.240 align:middle line:84% Just you have some extra, yeah, something depend on phi. 01:15:29.240 --> 01:15:35.140 align:middle line:90% 01:15:35.140 --> 01:15:39.400 align:middle line:84% And, similarly, your Hamiltonian will also 01:15:39.400 --> 01:15:43.360 align:middle line:84% be the free theory Hamiltonian plus a interacting one. 01:15:43.360 --> 01:15:46.990 align:middle line:90% 01:15:46.990 --> 01:15:50.110 align:middle line:84% And the interacting one-- so let's, for simplicity, 01:15:50.110 --> 01:15:52.900 align:middle line:84% then this LI, as in this case, only depend on phi, 01:15:52.900 --> 01:15:57.580 align:middle line:84% does not depend on the, say, the time derivative of phi. 01:15:57.580 --> 01:16:00.440 align:middle line:84% So if it does not depend on the time derivative of phi, 01:16:00.440 --> 01:16:09.820 align:middle line:84% and then, essentially, LI, HI is essentially just minus d3x LI. 01:16:09.820 --> 01:16:12.310 align:middle line:84% When you do the Legendre transform 01:16:12.310 --> 01:16:16.780 align:middle line:84% to go from, say, L to H, don't change this term 01:16:16.780 --> 01:16:20.420 align:middle line:84% if it does not contain time derivative. 01:16:20.420 --> 01:16:21.760 align:middle line:90% So, yeah. 01:16:21.760 --> 01:16:23.770 align:middle line:84% So, and then, there's a very simple relation 01:16:23.770 --> 01:16:26.530 align:middle line:84% between this interacting term in the Lagrangian 01:16:26.530 --> 01:16:29.960 align:middle line:90% and also in the Hamiltonian. 01:16:29.960 --> 01:16:32.420 align:middle line:84% And this is a free theory Hamiltonian. 01:16:32.420 --> 01:16:36.800 align:middle line:84% And now we will write our total action 01:16:36.800 --> 01:16:42.185 align:middle line:84% in terms of the free theory part and the interacting part. 01:16:42.185 --> 01:16:46.880 align:middle line:90% 01:16:46.880 --> 01:16:55.610 align:middle line:84% And the interacting part is just given by d4 x LI. 01:16:55.610 --> 01:16:59.720 align:middle line:84% It's also the same as minus dt HI. 01:16:59.720 --> 01:17:11.110 align:middle line:90% 01:17:11.110 --> 01:17:11.920 align:middle line:90% Good? 01:17:11.920 --> 01:17:14.680 align:middle line:90% So just to set it up. 01:17:14.680 --> 01:17:16.390 align:middle line:90% And now we want to calculate. 01:17:16.390 --> 01:17:20.250 align:middle line:84% Again, we want to calculate this n-point function. 01:17:20.250 --> 01:17:23.100 align:middle line:90% Which did I erase it? 01:17:23.100 --> 01:17:26.483 align:middle line:84% Again, we want to calculate this n-point function. 01:17:26.483 --> 01:17:28.275 align:middle line:84% So this is the object we are interested in. 01:17:28.275 --> 01:17:32.270 align:middle line:90% 01:17:32.270 --> 01:17:37.070 align:middle line:84% And, again, we can just consider generating functional. 01:17:37.070 --> 01:17:40.340 align:middle line:84% And we can consider generating functional. 01:17:40.340 --> 01:17:45.650 align:middle line:90% 01:17:45.650 --> 01:17:46.820 align:middle line:90% Yeah. 01:17:46.820 --> 01:17:50.400 align:middle line:84% Before, actually, we do that, let's just consider-- 01:17:50.400 --> 01:17:52.400 align:middle line:84% yeah, let's just consider this n-point function. 01:17:52.400 --> 01:17:56.290 align:middle line:90% 01:17:56.290 --> 01:17:59.120 align:middle line:84% Yeah, we actually have one minute. 01:17:59.120 --> 01:18:02.000 align:middle line:90% We cannot do really much. 01:18:02.000 --> 01:18:05.360 align:middle line:84% Let me just tell you the basic idea. 01:18:05.360 --> 01:18:08.875 align:middle line:84% And then we will elaborate next time. 01:18:08.875 --> 01:18:11.770 align:middle line:90% 01:18:11.770 --> 01:18:15.790 align:middle line:84% So, now, let's imagine we want to compute this object. 01:18:15.790 --> 01:18:22.150 align:middle line:84% Again, we just use that formula, this Gn 01:18:22.150 --> 01:18:26.650 align:middle line:84% is equal to D phi X exponential i 01:18:26.650 --> 01:18:36.250 align:middle line:84% S divided by D phi exponential i S. So now, what I will do 01:18:36.250 --> 01:18:37.060 align:middle line:90% is, I will-- 01:18:37.060 --> 01:18:43.100 align:middle line:84% now, my S-- so the idea is the following. 01:18:43.100 --> 01:18:47.040 align:middle line:84% Now, this part, integrals are not doable. 01:18:47.040 --> 01:18:50.040 align:middle line:84% Because once you have these non-polynomial terms, 01:18:50.040 --> 01:18:56.240 align:middle line:84% or non-quadratic terms, we don't know how to do the integral. 01:18:56.240 --> 01:18:57.990 align:middle line:84% We don't know how to do this integral even 01:18:57.990 --> 01:19:03.550 align:middle line:90% for one-dimensional integral. 01:19:03.550 --> 01:19:06.100 align:middle line:84% So not to mention the path integral. 01:19:06.100 --> 01:19:08.160 align:middle line:84% We also don't know how to do such integral 01:19:08.160 --> 01:19:10.570 align:middle line:90% for a harmonic oscillator. 01:19:10.570 --> 01:19:12.200 align:middle line:90% But, yeah, same thing. 01:19:12.200 --> 01:19:14.350 align:middle line:84% We don't know how to do it for field theory. 01:19:14.350 --> 01:19:16.822 align:middle line:84% So, as we said before, even though we 01:19:16.822 --> 01:19:18.280 align:middle line:84% don't know how to do this integral, 01:19:18.280 --> 01:19:20.640 align:middle line:84% we can treat this perturbatively. 01:19:20.640 --> 01:19:22.980 align:middle line:90% So we treat this as a main term. 01:19:22.980 --> 01:19:24.510 align:middle line:84% And then we treat the lambda small, 01:19:24.510 --> 01:19:27.630 align:middle line:84% then we try to expand the power series of lambda. 01:19:27.630 --> 01:19:32.420 align:middle line:84% And now we can write the path integral as the following. 01:19:32.420 --> 01:19:40.600 align:middle line:84% D phi x, then we add S 0, then i SI, 01:19:40.600 --> 01:19:47.980 align:middle line:84% and then divided by D phi, exponential i S0 plus i SI. 01:19:47.980 --> 01:19:49.510 align:middle line:84% And, now, what we are going to do 01:19:49.510 --> 01:19:51.865 align:middle line:84% is, we just expand this term in power series. 01:19:51.865 --> 01:19:54.410 align:middle line:90% 01:19:54.410 --> 01:19:57.930 align:middle line:84% When we expanded this in power series, 01:19:57.930 --> 01:20:00.330 align:middle line:84% and then, essentially, we are reducing it 01:20:00.330 --> 01:20:03.640 align:middle line:84% to the path integral of the free theory. 01:20:03.640 --> 01:20:06.310 align:middle line:84% So, essentially, the upstairs and downstairs 01:20:06.310 --> 01:20:10.420 align:middle line:84% can be imagined as we are doing the free theory-- now 01:20:10.420 --> 01:20:11.695 align:middle line:90% this is a free theory vacuum. 01:20:11.695 --> 01:20:18.820 align:middle line:90% 01:20:18.820 --> 01:20:22.450 align:middle line:84% We can just view this as an integrand of exponential i0. 01:20:22.450 --> 01:20:25.360 align:middle line:90% So the upstairs just become-- 01:20:25.360 --> 01:20:29.460 align:middle line:90% so this just become the-- 01:20:29.460 --> 01:20:38.280 align:middle line:84% yeah, t x exponential i SI, now in the free theory. 01:20:38.280 --> 01:20:42.075 align:middle line:84% And then, downstairs also become 0 t exponential i 01:20:42.075 --> 01:20:44.100 align:middle line:90% SI in the free theory. 01:20:44.100 --> 01:20:47.220 align:middle line:90% 01:20:47.220 --> 01:20:50.010 align:middle line:84% And now we can just evaluate, in the free theory, 01:20:50.010 --> 01:20:52.560 align:middle line:84% such kind of correlation functions. 01:20:52.560 --> 01:20:55.860 align:middle line:84% And we evaluate them by expanding 01:20:55.860 --> 01:20:59.850 align:middle line:90% the SI in power series. 01:20:59.850 --> 01:21:04.080 align:middle line:84% And then just-- so everything becomes 01:21:04.080 --> 01:21:07.650 align:middle line:84% just doing some Taylor series expansion 01:21:07.650 --> 01:21:13.410 align:middle line:84% in the inside of the integral and then become very simple. 01:21:13.410 --> 01:21:15.600 align:middle line:90% You don't need any fancy stuff. 01:21:15.600 --> 01:21:18.930 align:middle line:84% First year your calculus you can do. 01:21:18.930 --> 01:21:21.780 align:middle line:90% First year calculus you can do. 01:21:21.780 --> 01:21:23.850 align:middle line:84% So when you expand that stuff, still you 01:21:23.850 --> 01:21:26.080 align:middle line:90% get something very complicated. 01:21:26.080 --> 01:21:28.380 align:middle line:84% And then you can use diagrammatic rules 01:21:28.380 --> 01:21:29.370 align:middle line:90% to simplify them. 01:21:29.370 --> 01:21:31.320 align:middle line:84% And that's called the Feynman diagrams. 01:21:31.320 --> 01:21:33.090 align:middle line:84% And now we can-- yeah, next time we 01:21:33.090 --> 01:21:37.050 align:middle line:84% will talk about Feynman diagrams to simplify such kind 01:21:37.050 --> 01:21:39.620 align:middle line:90% of expansion in power series. 01:21:39.620 --> 01:21:44.000 align:middle line:90%