 So, now we go on to the more formal aspects and luckily this book of Ramon has a nice bridge to going from single particle quantum mechanics to the field theory. So, towards transition to the quantum theory and let me also make some pre remarks and then repeat them again because these are very important remarks without which we will we will all get lost in formalism. So, some remarks can be repeated. So, we can set towards QFT Green's functions. So, in QFT we find that on this set ok, this set with all possible values of n captures all the physical effects of the theory. So, the purpose of quantum field theory or any theory that you have any quantum theory that you have is to be able to calculate endpoint functions. What they mean and how we use them will come out later, but you can just think about the fact that if you had some very general function if you know all its moments in sense of statistics then you can always recover the original function. So, the contents are all in all the endpoint functions and that is what we are going to move towards and towards that let me also give away the other main trick. So, the way we calculate this is to define a generating function. Generating function will mean that we introduce an auxiliary variable actually auxiliary function some f of t ok. We have Q of t P of t, but there is some forcing function f of t and then we define in the presence of f right are all brackets here here. To be equal to integral d P d Q o e raise to i times s and now on we can say farewell to h cross no more h cross ok, i s and plus i s of P Q and we add to this i times. So, as you remember as integral d t of P Q dot minus h. So, here we need to add an integral d t of f of t times the Q of t and whatever t i to t f yes. Also actually actually we are going to get rid of the momentum. So, we should we can do two things we can add some variable for plus i times. So, we can add auxiliary functions f of t and g of t which keep track of each of these and this whole thing became too big, but you know what is going on in fact. So, it is all in the exponent, but to not get carried away with too much generality we will in practice be using only the position dependent path integral. So, the more important thing the denominator ok. So, but divided by d P d Q of e raise to i s of P Q plus i integral f Q plus i integral g P. So, you have to divide out by the thing without the insertion of O. Now, we shall be using the path integral only in the Q formalism, where a, b are the degrees of freedom. Then the path integral can be written only in terms of Q. This was exercise one last time which I am sure all of you worked out the same evening. So, up to an overall normalization n we write n is O and we would have the same n aside from some product some multiplicative factors of measure 0 because of the slices where that enters, but as we said. So, we define it like this do not struggle over the fact that the front factor is going to look the same. So, that is what it is. If this is how we define it then and so, this was in presence of f then what we can see is that this is actually equal to the same as yeah. So, the denominator and times to get the O of Q down all I have to do is differentiate this expression with respect to f enough number of times and I will backtrack a little bit. I will say that we will define this with without the denominator. So, we for simplicity. So, this is technically how it is defined I mean in real life to calculate averages, but for the time being we define also begin by considering just Q. So, the main trick I am about to talk about is that the expectation value of Q of t then is simply equal to variation with respect to f of t of this object. So, let us write Q Q dot S of Q Q dot and plus it remains and this is in presence of f. If you want to calculate and some factor I I have to remove. So, since there is an I over here. So, it is actually variation which is I this is clear right because if you vary with respect to f then a Q will come down and it will automatically reproduce this where O is Q and then the exponential will remain as it is. So, and this is standard step metric right. So, that is all we are using. So, and likewise let us say Q t 1 Q t 2 would be equal to d square of I delta f of t 1 I delta f of t 2 times the functional integral ok. So, now, what we will do is to introduce the harmonic oscillator in this language as a warm up to the real thing. We will find that there is a nice analogy of non relativistic harmonic oscillator to the relativistic propagator because the relativistic propagator is of the form p 0 square minus p vector squared that acts like omega squared plus p squared. The harmonic oscillator Hamiltonian is quadratic in Q and P. So, it matches exactly. So, now, consider taking the harmonic oscillator and as you good students remember there is d below them, but we have stopped writing it because it just becomes cumbersome right. So, this transition amplitude in this case is aside from the n which we accumulate due to the momentum space integration and we put that Q t f t in. So, for convenience we said m equal to 1 and we also introduce a means well m equal to 1, but most important thing is introduce a damping factor e raise to minus epsilon and we put it proportional to Q squared. I should not call it factor, but it becomes factor in the exponential. So, then this is equal to one half of Q dot squared and then minus omega squared, but there is also minus epsilon Q squared. So, this minus sign I write as I times I, but that is with a plus sign. So, omega squared minus I epsilon times Q squared and then plus f Q. So, this whole thing d t. So, note that this epsilon is such that as if you extend the T i to T f to very large values and it picks up a large contribution to this then it will make the path integral go to 0. It will make it converge in the limit of T i and T f going to infinity, but we put it also not because we are worried about our mathematician friends and convergence or anything it is actually a very clever device which comes useful, but it shows that the. So, to tell you because you already know from quantum theory in the Feynman propagator you have to have the I epsilon prescription p square minus n square plus I epsilon that I epsilon comes out exactly correct with the correct sign. If you pretend to tell people look I am regulating this by putting a damping factor. So, that is what is nice about the path integral. So, now next thing we need to do is rewrite this in terms of energy integrals. So, since we are run coming closer to end of the class time what I will do is I will write the what we are going towards and we will prove the formulae next time. So, what we can show using this is that limit of times going to infinity in the presence of f is equal to q f infinity q i minus infinity without anything times n e raise to. So, f equal to 0 times e raise to minus i over 2 integral of d e f tilde e and f tilde of minus e this you would be familiar with Fourier transform you get opposite signs when there is product of things divided by e square minus omega square plus I epsilon where the f tilde are just Fourier transform. So, the point is that this object that was to be computed if you just did the without any f then the answer for O cannot depend on any q of t right because q of t got integrated out there is a big integral over d q of t. So, the meaning of what we are calculating becomes clear if you insert this f as a temporary device and then let then drop the f in the end after you have calculated. So, here I should have said one more step was here these are still in the presence of f, but to get it in the in the end without any f what you have to do is this and then evaluated at f equal to 0 that is the prescription. So, the f is a sorry right now we can see that this expression in the presence of f has no q of t in it and it is a function of in fact, just the f it is Fourier transform, but it is function only of f there is no sign of any q in it, but we can now compute the expectation value of q t 1 q t 2 etcetera by just varying with respect to f t 1 f t 2 of course, they are Fourier transforms of this variable, but you can extract the two point functions out of this. So, thus we have a of f of t right. So, this is the meaning of this is what we call generating functional or generating function from which by varying the auxiliary variable we can extract the end point averages. So, it contains all the information that you need to calculate all the physical quantities of interest ok. So, from here on I am actually just using Ramon's book for next couple of turns. So, you can read it and the comments that I have been making about momentum space the canonical path integral the phase space path integral as being the correcting is also commented by him. He derives it using more like the more intuitive Feynman approach and assuming that it is given by Lagrangian, but then before switching on to these topics there is a short discussion of why actually you have to use the p if you do not then you get some spurious factors which usually do not bother you ever, but for keeping things straight and then you do not have to put any n at all you get the exact answer.