 So, another way of looking at this is to think of this. So, the point is these overall things are not that important what the all the material is here. So, we define we say that W of f well it is f of t. So, W of f is just e raise to i times d of t 1 minus t 2 full form of f d f. So, we call this expression the W of f, but now if you are astute and observant you notice that ultimately the path integral was supposed to give you this factor pulls out of everything else which is exactly what had happened when you did that stationary phase right. When we did stationary phase you said if it is stationary at the point where the f hits an extremum. So, e raise to i f 0 over lambda f of x 0 over lambda just came out and then the remaining term became Gaussian integral. This is actually like that what comes out is essentially what is the classical path and so, we think of this as essentially the effective classical action ok. So, we define this to be equal to so, i times z of f. So, this z of f is actually what we call the effective effective action, but in terms of the current instead of in terms of the variables. For the time being just remember that this is how we write this is the concepts we introduce we call W of f to be this in field theory it will make more sense. And however, this is effectively saying that I am also defining log of W as some new quantity, but its logic is that it is whatever is in the exponent as if it had it was the result after you had integrated out all the dynamics and that is the dominant thing that is what you would see in the classical limit. So, it is some it is called effective action actually you have seen this in stat mech. In statistical mechanics you have say if you have the phase space thing d q d p e raise to minus beta h and time some operator. So, what you do there is you have some extra you put minus mu n let us say right. Then you get after you do the integral you get the potential there the thermodynamic potential appears in the exponent e to the minus beta say some free energy or something. So, things like this are used in other context as well I have rubbed it out because I am not quite sure what it is, but I am trying to give you the idea that this side has a. So, this actually came out of doing the d q the big d q, but after all the dust settles we just identify the leading term as the new effective action. So, it contains actually the information of all the quantum mechanics that is going on right because by varying it enough times you can calculate all the correlation functions that is the point. At least for the harmonic oscillator case the d can is in fact, this this is equal to d because look at it this is the answer in front of you. So, if you varied with respect to f 1 2 remarks 1 is that 2 this is the average I should have used a bar or something. So, not quantum mechanical, but this is in fact, equal to because what happens you vary this with respect to f 1 d 1 to f 2 comes down. So, we do sequentially. So, I d by d f t 2 of this exponent f d f will be equal to the f d f in the denominator, but with 1 f removed. So, it will become equal to this i times this minus i. So, plus a half integral d t d t 2 this is varying with respect to t 2. So, d t 1 will remain f of t 1 times d t 1 minus t 2 right and times the exponent itself right half has to go away because it is t 1 and t 2, but they are both arbitrary integration variables. So, thanks. So, half goes away here and we vary second time with respect to now t 1 we should have used different variables where it is ok. So, I then gives then this f gets plucked off. So, we get d of t 1 minus t 2 and then we are supposed to evaluate it at f equal to 0 that is good. So, I forgot to say that here this, but with at f equal to 0 the averaging prescription says that you vary and then set the auxiliary variable to 0. So, then you will get that yeah some details need to be checked, but I think this is correct. So, this is one thing the second thing is that this d t is very nicely an analog of the Feynman propagator. So, d t is equal to theta of t raise to minus i omega t under the 1 over 2 i omega. This expression you should memorize by heart although I have not done it. The thing is that the you know the proof of this right quantum 3 you have done this many times. So, you know this you have when t is greater than 0 you can close in the upper half plane when t is less than 0 you can close in the. So, if theta of t then your you this minus sign is inherited from what wherever the t yeah that minus sign will be inherited from this minus sign when t is positive you close in the upper half plane when t is negative you close in the lower half plane. So, you get this. So, this is very nicely what the Stuckelberg Feynman propagator looks like what is the word for it pre sages. So, Feynman tried to apply the path integral to electrodynamics, but the boundary condition was still a problem and if you are any nice guy you will say well you would put same boundary conditions as on the electrodynamic Green's function. So, you must have done Jackson chapter 13. So, where he computes the Green's function for classical electromagnetic field it is just quadratic it is just box equal to 0. So, that box becomes 1 over p square, but if you just want classical electrodynamics then you want influences to propagate only forward in time. So, the prescription you put is not I epsilon, but simply put both the poles above the axis. So, that you get contribution from both the poles everything goes forward in time nothing goes backward in time that is what Feynman tried and he was not getting the answer. So, then he looked up this paper by. So, Feynman's where he does the calculation correctly he cites this paper by Stuckelberg which was published in Helvetica Physica Acta I forget the spelling some again Physica Acta this was Helvetica as you know Switzerland. So, Stuckelberg was a slightly crazy genius who had mental problems. So, half of the year he would be in the asylum and then other half will come out and write quantum field theory papers. My advisor ECG Sudarshan was proud to note that he shared the initials with him which were also ECG. So, it is ECG Stuckelberg good. So, now we go to field theory. So, one last comment is that this particular. So, it is sub comment because I said only or we can take it as third remark is that this particular fact that the poles are above and below like this on the omega axis on the complex E plane. So, this is my notation for the complex plane of the variable. This means that this path of integration for E could actually as well have been rotated to this path and instead of integrating like this you could as well integrate this. And according to complex analysis nothing would change because so long as there are no poles here this contours which are at infinity should not give anything and because so long as the signs are correct. So, even a tiny even if the value of t is tiny so long as there is a damping exponent you can always close it can close it. And so, this contour is equivalent to that contour and that is equivalent to just euclideanizing the time. Instead of t you are using a new variable tau which is equal to i times t is rotated by 90 degrees and just to tau times i t. So, comment number 3 is that the placing of poles rotating t rotating 2 let me write like this tau equal to i times t axis. This is a trick used quite a lot in quantum field theory and this is called weak rotation after GC weak. So, you can have some so less of a genuinely convergent Gaussian instead of having to have that fake Gaussian with an i stuck in it. And so, last time somebody was asking me this question what is all this trickery you know you go to imaginary axis you do this and everything is otherwise mathematically you will define. Well the point is that we really want to express the functional dependence on these f's this is all that matters because this functional dependence will always correctly reproduce what these averages are. And again whatever this expression averaging expression is what we are interested in is the dependence on t 1 and t 2. We want to eventually calculate a correlator or a endpoint function in which we are interested in the parametric dependence on t 1 t 2 t n and everything else is really a scaffolding. So, in it if there are these impressionistic flaws from the point of your mathematics the point is we are not actually trying to compute the value of it. We are trying to use this as a generating function for a formula this is the formula. This formula will correctly generate t 1 parametric dependence of this average value on t 1 and t 2 by doing this mumbo jumbo here ok. Whatever abracarabra do you do in the end the answer should come out right and that is so long as that part is right that is all you care. And so, these are ways of remembering how to generate it out of the full thing. So, if you think of it that way then you stop worrying about the mathematical precision of the formula or the of the process. The procedure looks quite shaky, but this is why you do not worry because in the end it is not that you are computing a value out of it, but you are computing a function out of it dependence of a function on its arguments. So, we can begin with how this appears in quantum field theory. Now, the curious point is that the non-relativistic harmonic oscillator serves as the motif for the free particle in relativistic field theory. So, here we will have action which is equal to integral d 4 x and then d mu phi d mu phi this is what we will this is the theory we will be dealing with m squared phi squared and then plus a j phi. So, we put half's in front of these real scalar field. So, this v of phi is not the actual potential energy of the system is this whole thing sorry. So, the kinetic energy is just half phi dot squared. So, the minus gradient phi square minus m square phi square and minus v phi together make up minus u t minus u just to change because unfortunately he has used v and I do not want to change that. So, if you think of Lagrangian s t minus u then here the u expression is integral d 4 x times grad phi squared plus m squared with half factors and plus v of phi. So, the so called potential energy part is this ok. Now, if we if we take this and then do the usual thing then the vacuum to vacuum amplitude can be written as and now we call this j the forcing function j to be equal to integral some normalization times integral d phi I will just write in more detail what d phi has to be e raised to i integral d 4 x times a half d mu phi d mu phi minus a half m square phi squared minus v of phi and then plus a j times phi. So, the i multiplies everything this is this and then plus j of x phi of x. So, this is how we define it and this is already in configuration space because of the reason that this is anyway quadratic in the velocity. So, does not matter m square phi square. Now, most of the steps go through as before and you can consider a Euclidean version by doing x 0 equal to x 0 bar equal to i, we had said tau equal to i t. So, you can always introduce Euclidean one if you do not like this one. We now say that this expression we refer to as w of j. Some warning if you read Isaacson and Riber, then the w and the z are exchanged Isaacson and Riber call this the z and the log of it w and I think Greiner's book also does that somehow Raman used the wrong notation, but once I am reading that book I cannot change the notation. So, and anyway it is a matter of symbols. So, this w of j can be shown to be equal to and this we can do next time or you can see most of it d 4 p exactly like we had for the f f. So, minus i over 2 where the p these p are 4 vectors and our definition notation for this is normalization 2 pi squared like the square root 2 pi square root 2 pi we had there this is the convention for the Fourier transform. So, then you get this answer you will not be too surprised because exactly like we got for the q dot square if you start with phi dot square and have d t then it will become omega times minus omega times phi tilde of omega phi tilde of minus omega it will become product like that. So, eventually you will just and then you will complete the squares etcetera. So, you will get this. So, you can try to do it for your own benefit. So, from here on starts the story of greens functions and we can as well stop here by just noting that we shall say that this w where this is really the average in other words. So, this is a functional if it is a functional and admits an analytic representation then you get oh this is not this is integrals over the right right. So, these are integrals over x 1, x 2, x n i to the n integral I should I do. So, that is what it is where this is j of x 1, j of x 2 etcetera. So, you can always write some kind of a power series for w as a functional of j and like multivariate calculus this is what you would do as the expansion. The coefficients is what we call greens functions or simply correlation function. So, we will see next time that this correlation function is a bit of an overkill and can also have redundant pieces in it. But if you take log of this w like that z that was defined then you get really the connected pieces right now this would not make sense, but let me just stop at that. So, we will stop with this.