 So, we now continue with the path integral formalism and what we are trying to achieve is to create a formalism to calculate correlation functions in quantum field theory ok. So, there are several steps that we go through to get to that point and I will explain that in a few minutes. Let us begin with winding up some interesting points about the path integral formula. One is that one is that it automatically gives time ordered products ok. So, the point is suppose we think of e raise to i s we might be dropping h cross very soon. So, but let me put it this time. We can think of e raise to i s i s over h cross as a weight function and define some kind of average of some some function that depends on q any function dependent on q. So, it would be actually a functional well actually q and t q and p. We will see in what sense this is an average value and yet not really not to your ordinary average value, but certainly a very important quantity. So, in fact, this is how mostly path integral is used for all practical purposes. The main purpose of the path integral is not to calculate a transition amplitude from initial to final state nobody does that. What one does is actually use it to compute certain kinds of expectation values which we eventually will call Green's functions ok. So, so we can note here more generally correlation functions. So, that is the end of that side comment. So, the point is that therefore, so consider this property. So, suppose you are suppose you have two operators at two different times which you are trying to calculate. So, by this I mean o at q's of t 1 p of t 1 and this is o 1 o of q of t 2 p of t 2 right. So, suppose you have two different operators unfortunately I do not mean to match this one with one. So, we could call it something as call it one like this and two ok. It is not that this has to do with this, but there is a time coordinate we are using for the first operator and another time coordinate you are using for the other operator. Then this will become integral d p d q of and now as per our instruction we put o 1 and I am dropping all the detail of what this is o 1 o 2 e raised to i s over h cross right. This is what it asks us to do. This is the definition of that product, averaged product ok. But now we see what happens if we implement the detail of this we get a diagram like this that I have q t and we are suppressing p because we do not have that many axis without getting confused. So, this is t i and t f and then let us say I draw t 1 t 2 something like this. Now I change my notation because I unfortunately used ok. So, I have time slices, but let us say t 1 is over here and t 2 is over here. So, the instruction says that you will take starting with t i q i to q f at t f you are supposed to do this right t 2 is here right. So, wherever you like some one path is like this. Now, because to take account of this t 1 and t 2 what we will do is we can introduce additional slice exactly at that point ok. So, of course, then you can get something like go up to here then here and then you go there and then there and then there and then at t 2 you know of course, you have to reach the same final point. So, you could add additional slices and then it is just a matter of the detail of setting up the time slicing. Ultimately you are supposed to take the limit of the all the slices coming very close to each other infinite number of time slices. So, automatically the t 1 and t 2 are going to get covered in that, but we are just illustrating that also strategically in this sliced version before having got to the limit make sure that t 1 and t 2 are included in the list of slices, but once we do this we see that if t 1 is. So, the point is the slices are ordered. So, if t 1 is less than t 2 then t 1 slice will appear before t 2 in above example, t 1 is less than t 2. Therefore, in the averaging the operator O 1 will automatically appear first when you begin to average it will get averaged first and similarly vice versa ok. So, in quantum mechanics we are concerned about ordering of operators what we will find is that automatically if t 1 is less than t 2 then t 1 will appear actually to this side because you are slicing like this. So, it will go like this and then if it is t 2 then 2 will appear later. So, in the time ordering prescription what we say. So, what we do is that we write a compact formula for this one more important comment to make. Note that the path integral is a very interesting formula because it actually involves no operators. The path integral formulation of quantum mechanics does not talk about any operators. It does talk about initial state and final state between which the transition amplitude is calculated. So, the states are there, but we do not have to introduce any operators because all the operators are just classical. It is just that they are going to be integrated functionally by this complicated time slice measure ok. So, note that one more comment formulation does not require promoting dynamical variables to operators. We automatically get average values or expectation values by doing the time slice functional integral. So, here particularly, but it does maintain and by time ordered we mean if t 1 is. So, whichever is before gets integrated first and our quantum mechanics notation is that we go in state to out state left to right. So, if t 1 is less than t 2 then it has the meaning of being O 2 t 2 O 1 t 1 and if t 1 is greater than t 2 then t 1 will be on this side. Thus we find that this two point function as you might call it or this product operator product average value will be automatically ordered in the path integral. So, this is still continuing with comment number 1. Thus this integral d p d q of O 1 t 1 by that we mean actually this whole thing. Because it is always a functional of p and q. We could write just not to get completely lost is q 1 p 1 right O 2 of q 2 p 2 e raise to i S over h cross will automatically calculate the expectation value of the time ordered product of it will calculate automatically the time ordered average value. As you would have done I mean in quantum mechanics the order becomes important. So, this will automatically give this whether you like it or not, but amazingly enough that is exactly what you like as you know from your quantum 3 course you write. Now, the second comment is about the classical limit. I write these things out in long detail and not always very accurate, but it is a marker for you to remember what was said because if you write out formally without the remarks then you will probably not understand what was the motivation for what. So, with the understanding that within the expressions O 1 and O 2 p's will need to go to the left of the right of the q's q's you could have of course, defined your whole path integral with some other ordering prescription. You could have said q's to the right p's to the left it is matter of slicing remember how we started inserting the complete set of states you could have started it from this end instead of that end. So, whatever prescription you followed there will automatically apply here please will go need to the left of the q's in our notation in our convention or whichever convention was used to derive the basic formula very good. So, you have to take care of that part and which is where effectively you will be constrained by what quantum interpretation you are giving to your or what is your proposed quantum operator is implicitly there in the formula, but you do not have to I mean in terms of formalism when you use it you do not have to promote, but I guess I agree with you. You will have to remember to order your operator that way and if your operator is different then you should have the additional pieces that come from that normal ordering which should be retained with the operator good. So, this is corrected by this, but I can tell you that just the way it is written an expression like this which although this is highly nontrivial in terms of real analysis or multivariate analysis and because of the oscillating there nothing mathematicians will faint several times over looking at this expression because it just makes no sense in any way at all. But anyway if you look at this then nowhere do you have to put any quantum variable and you get a answer for a transition amplitude between one quantum state and the other and so Wheeler John Wheeler who had the who always used to create jokes about this something without something. So, I used to say path integral is quantum mechanics without quantum mechanics Feynman was a student wrote the thesis with you. So, path integral is quantum mechanics without quantum mechanics ok not really as we have just learnt, but it looks it is good to tell people like that when they do not fully know the inequalities all right. So, the other comment was that the classical limit to understand this we need to understand what is called a stationary phase approximation in integration. So, suppose we have an integral j equal to and let me leave some gap here integral dx over the whole range e raise to i times some function of x ok. So, this is the basically the integral suppose I am integrating something that as e raise to i times some function of x which is what the form of our path integral is. To keep track of this h cross or to keep track of I mean some putting some scale in the problem we insert a book keeping constant lambda number lambda, but then consider this in the limit of lambda going to 0 or lambda small you do not have to take the technical limit, but what we mean is as lambda get smaller it is possible to arrive at an estimate of this integral. So, the point is that now we look at this it is integral dx suppose it was not any complicated f, but just e raise to i x then you know that it is sin x plus cos x plus i sin x. If you integrate over the whole range you are going to get nothing it is just going to keep canceling you will get some oscillatory answer depending on where you stop, but overall it is just it just gives you nothing definite as you go through several cycles they keep canceling each other. So, you do not get any very definite answer, but certainly not any divergent answer either ok. So, what you so if you then try to try to think of what happens to this f of x e raise to i f of x you can try to plot let us say the real part of it. So, suppose we plot real part of or that is to say cosine of f of x. Now, whatever f is cos can only oscillate between plus and minus 1. So, without worrying too much where the 0 of f is and so on we just I just draw some random curve it just keeps oscillating between minus 1 and plus 1, but now let us specialize on some points. So, such a thing might arise by let us say if f was monotonically progressing from here to here. So, over this it is just as if it is cosine x right it is happening rather regularly. So, you might say it is just going something straight like this and then it may turn over ok. So, suppose f is drawn here what you will find is that when f is doing something non-trivial this will keep oscillating, but precisely where f actually goes through a minimum it is stationary it f is not advancing then this course function will stay put there ok. So, what you say is that I will look at the points where f as extrema the points where f as extrema is where the progress of the argument of cosine is going to slow down and for some time it will stay near a constant value. So, we so, if x 0 is a minimum is an extremum and j receives a non-zero contribution. So, we just take not bothering to put and the point is as you make lambda smaller f will become much the exponent will become much larger. So, small changes in f will make f over lambda run over a long range. So, it will just make the oscillations much faster as lambda goes to 0 except at the point where it has actually hit a minimum. So, we get integral dx and we can write it as equal to e raised to i. So, we expand around x 0 the next term is not there because the derivative is 0 right. So, plus it would be f prime x 0 over times x minus x 0 over h cross over lambda right, but this is equal to 0 and we will just have the next term f double prime of x 0 times x minus x 0 over 2 squared over lambda dot dot dot. So, it is approximate I did forget the higher order terms and we proudly retain this term because we know how to do fake Gaussian integrals. So, all we have to do is that therefore, this becomes equal to e raised to i f x 0 over lambda comes out right it is a constant. What remains is our friend the ill defined Gaussian integral. So it is just equal to square root of your expert at doing this by now 2 lambda over i f prime, but now the consequence of this in our case is that the path integral receives non and so we can generalize to several instead of one extremum if it had several all it becomes the sum over each of them with contribution like this from each of them aside from the stationary phase there the fluctuation Gaussian fluctuations you get there. So, can be generalized to note that you should not have any guilt this is minus infinity to infinity. So, at any one point you did this now you go to another point you say oh, but when I am integrating there this enters no you do not have to worry do you put this expression for one and because all the nearby points is just going to give you infinite 0 because of oscillatory behavior. If you once you estimate it like this it will actually become just sum over all the extrema happily with integral minus infinity to infinity because everything else at each of them is just giving zeros. So, you can just generalize it by putting sum over n. So, can be generalized to several are extrema. So, in the path integral case we get maximum contribution or we get non zero contribution precisely where so with I have not put any below you can put with respect to x or with respect to p both, but wherever the variation of s is 0 and now s is a functional of the trajectory x t p t. So, precisely on that classical and this is the classical condition. So, at these points on the classical trajectory the path integral will give this answer some overall constant which is not of importance, but the point is that that is what will dominate the classical trajectories will dominate. And remember that s has to be measured in the units of h cross just as this f was measured in the units of lambda. So, if you are living in a world where s is much much bigger than h cross then any tiniest variation of the path from classical will make this f run so fast that it will wiggle so fast that it will contribute nothing to the integration. So, that is the answer. So, we can leave it at that. Thus the classical limit corresponds to quantum mechanics is this strange theory where you can superpose various states. So, if you have a cricket ball then its location here with momentum this can be superposed with its location there with momentum that and so on. But the fielder out there is managing to catch the ball where he is supposed to because any all the deviations that happened on the way are all cancelled out ok gets it exactly with the correct Q and P.