 Now, what we are driving towards is quantization of the gauge fields, a quantization of the young mill system. And the canonical structure becomes very important because you remember that we started our course by defining the path integral using the canonical formalism. We had the canonical variables and we had the fundamental relation q p equal to we traded these two things right. q p equal to i h cross is same as saying that the q p overlap is equal to e raise to i q p over h cross and square root 2 pi h cross. So, canonical structure is very important to us for quantizing and that is why we have to be careful about what the canonical structure of these equations is. And so, towards quantization quite a lot of subtleties arise. Now, historically it turned out that this was very early days and nobody thought that young mills fields will become important. They were proposed in 1956 and throughout the 1960s people were mostly concerned about spontaneous symmetry breaking. They were trying to get pions as goals you know superconductivity of the strong force and things like that. But Feynman had begun to worry about this and there was famous quantum gravity person called DeWitt. Bryce DeWitt is a name not known much now to the to the ignorant, but to the those who know it is a big name because Bryce DeWitt gave his full force to quantize quantum gravity along the canonical method, but actually more adopting swingers he was a Schwinger student. So, more adopting Schwingers Schwinger had his own way of saying how you vary the action. So, it is a functional approach, but not path integral one and then in later life Bryce was a professor at Texas Austin and I have taken courses from him. So, Bryce DeWitt wrote a trilogy. So, he has a book called Dynamical Theory of Groups and Fields in which he meant to attack this problem of systems that have this kind of redundancy and as we know general theory of relativity has the huge redundancy due to reparameterization invariance and the covariant derivative of differential geometry. So, Bryce like a juggernaut just cleaned out the whole field of doing what was required. So, Newton was somehow interested in this in the Young and Mills proposal and he observed that from a very physical point of view that the gauge conditions shows rather that the superfluous degrees of freedom can be co-variantly removed by introducing Gauss. Gauss diagram is one that has a scalar loop it could have anything outside, but it has a scalar, but the diagram has overall minus sign. So, it has to be subtracted. So, if you draw the diagram you do not get any minus sign, but you say. So, you say that diagrams like this have to be subtracted. So, you can interpret it by saying that well you know in Wicks theorem that when you have closed loops of fermions you get a minus sign because of the anti commuting property. So, you would have had psi psi from this vertex. So, you know how the Wicks theorem will go is that you have a mu psi bar gamma mu psi normal ordered and a mu psi well you would have a mu psi bar gamma nu psi and you have to link this psi to this psi bar and this psi bar to this psi, but in doing. So, that would give you this diagram. So, one a mu coming in reducing psi and psi bar you know one arrow going this way one and another one like this and then you have to. So, this is one vertex this is the other vertex this is the product of the two. So, you join up these lines that joining up in Wicks prescription is coupling this psi bar with this psi and this psi with this psi bar, but that will require going taking this psi across one psi bar. So, sorry this is normal ordered. So, there is no question of going across that, but it entails a negative sign from having to take the psi across to this side ok. So, then it becomes a propagator. So, that minus sign normally comes if you have the fermion statistics there, but here it is difficult to see how minus sign would come. So, people said well treat, but and it is required to be scalar there are no indices on it. So, people said well all you do is that you claim that it is a scalar, but a fermion, but it never appears in the asymptotic states. So, it is not going to violate our spin statistics theorem. Spin statistics theorem says that scalar cannot be a fermion, scalar has to be a boson all integer spin and 0 spin have to be bosons, but this violates normal that condition. However, it is not going to be seen outside it is only a trick in the diagrammatic calculation. So, that is what Feynman observed and he probably did not say it was fermionic loop he just said these need to be subtracted. Bryce Dawitt in his own juggernaut way just came to the point and say ok you need to put this and you need to put this and he never realized that he had done the same thing. But out in Lebedev Institute Russia two Russians were worrying about this and they had read Feynman's paper. So, Fadyev and Popov this name is 2 D's and 2 E's, but there is an apostrophe here. They gave a completely field theoretic interpretation, because they had a colleague called Berezin who for his own mysterious reasons because he was a Russian had already worked out how to deal with field theory of or quantum mechanics of Grassmann variables. Well I know why he had got it out. So, Berezin's observation was that if you are going to do path integral quantization of fermions, fermions have anticommuting property. So, classical limit also they must be anticommuting, but if it is so, then they cannot be ordinary numbers. So, they have to be what are called Grassmann numbers which anticommute their product is not even I mean you cannot express them as numbers and those variables have to be anticommuting. So, just around 1962-1963 he worked this out Feynman is also around 1961-62 and then Fadyev and Popov realized that what they need to say is that they use Berezin's method, but they violate spin statistics theorem then they can actually get Feynman's ghost diagrams in a systematic way from the path integral. So, that is what we are going to do next yeah. Oh I just wanted to tell the story that in America nobody was worrying about this because they were not looking at Yang-Mills field theory, but in Russia it was stronger nobody was supposed to look at field theory at all because Landau had declared in 1960 that field theory is dead QFT is dead in a major conference he said this based on his analysis of quantum electrodynamics. So, there are two problems first of all you would like to know the good news that all the perturbation theory that you have been using is wrong from mathematics point of view because it can be proved that the perturbation theory perturbation expansion you make is only asymptotically convergent or they are actually called asymptotically divergent series. So, first thing is that perturbation series is an asymptotic series they just call it asymptotic series. This means that the terms grow smaller I just draw a diagram like this. So, series goes like this keeps growing smaller, but then it begins to diverge again the terms begin to diverge again ok. Now, this somehow was already noticed by Pankare much earlier and so, he had warned people that you should cut off your calculation here while it is still converging cut it off and interpret that as the answer do not go to the bitter end of the series. So, that was one problem that anyway perturbation series people had doubts about, but Landau further showed for QED that there is Landau ghost. Now, this is a different ghost not that ghost which is an essential singularity at order 1 over alpha. So, if you do QED to 137th order then there is a essential problem. So, he said that it is not even benign like what Pankare said. So, this is Pankare's prescription, but that there is an essential singularity and therefore, this does not make sense. I am quoting this because it is important to remember in history of science that great men can be when they are wrong they are very badly wrong. So, there was of course, no reason to stop studying quantum field theory, but because Landau declared it nobody in Russia was working in quantum field theory. So, Fadiyev recalls many years later about 5 years later he has given some set of lectures I think in TIFR actually there on the video that it was like censorship. So, they secretly brought out a preprint, but did not show it to anyone and then the preprint migrated to Europe secretly in somebody's suitcase and when it reached the west everybody was enlightened and then said hey because by then it was late 60s 1967 or so that, but I think so in 4 to 7 they first wrote this in 1964 the preprint migrated out they never published it, but by late 60s young males theories as applicable to weak interaction had come back in vogue. So, it was Ben Lee I think B. W. Lee one of the pioneers of perturbation theory for spontaneously broken electromagnetic theory he brought it out as a Fermilab preprint. He reprint he translated and produced it as a Fermilab preprint and after that by 1967 they actually published it as a physics letters paper in the typical Russian style where each page is like 50 pages of calculation. So, they just wrote up the note their whole lecture notes they wrote in 4 pages and that appeared as physics letters been 1967 and I think David's book up is actually this books date is also 1967, but he had already a set of papers in physical review 3 papers. So, it is called the trilogy. To all the older quantum gravity people up to year 2000 David's trilogy was the Bible of why quantum gravity does not work or if you had to do it how you would do it and it is very useful document because if you actually do calculate he has done all the calculation which you cannot imagine doing. Now therefore, this Padya and Popov came out with a very elegant way of doing things they were of course, in this mold of Dirac. So, they said that corresponding to a gauge constraint you also put gauge conditions on the phase space and we write like this because this is a this is a formal equality it is not a not really a function it is some expression on the space of q's and p's which is set to 0. And so, introduce sorry I am sorry that is the second line. So, constraint the first line is constraints that is our these things ok. So, we already wrote C for it. So, we have constraints C. So, we introduce gauge conditions this nomenclature also has lot of disputes, but it has stayed. So, I will not bore you with that. So, we introduce gauge conditions gamma a such that they are complementary to this C they choose a particular point out of this. So, the problem with this C is that if you draw some n dimensional phase space ok. So, let us say q q i and then p i lots of dimensions and you have a constraint surface right in this big space you have you are supposed to be restricted to that and the point is that if you actually try to do path integral and kept cutting it at lot of places you are doing redundant integration. If you are doing integration within this then you are doing redundant integration because they are they are all equivalent you should cut it only once. So, the gauge condition gamma are supposed to be something that cut them in a judicious way. So, gamma should fix a particular trajectory out of it for the timing I will draw I mean I do not know how to draw it, but so it will choose a particular point out of these. So, that you are not taking care you are not integrating over superfluous degree sorry I am sorry one moment. So, the fact that C equal. So, superfluous integration would happen perpendicular to it because C equal to 0 is the physical surface. So, you would be integrating within that and going outside of it is actually not required. So, you want to fix it so that you make a particular choice of unique variables out of the constrained ones and that is done by erecting a complementary surface gamma. Only thing is I do not want any overlap at all I just want one point out of it right. So, the correct thing to show would be to draw a stack of these 0 1 I mean any number any fixed number would be fine. Any one of those is a valid surface and you want to select one out of it that is the correct thing to show which is a little hard to draw ok. So, it is equal to some alpha i at some constants. So, any one of them is fine because if you set it equal to some constant it is ok. So, you put this gauge conditions which make a choice out of this stack of surfaces and an example is we know take Maxwell we know the equation 0 or some constant or some external current . This reads box of a nu right d mu d mu a nu and minus d nu of so, exchanging order of these. So, Lorentz not Lorentz observes that if you set d mu a mu to 0 then we get only box a nu equal to 0 which are just the allumbarsian independent equations for each variable. So, this it is no longer tied up with derivatives of the other components and they become all free ok. So, this is the gauge this is our gauge condition gamma or gamma a ok the condition is this of course, that also does not fix everything because as we know there are two different things. So, in this if you now set a 0 to be trivially 0 then you will get that divergence a would have to be 0 which is that a has to be purely a curl like field, but this is the meaning of putting this. Now, in the canonical language we can then think of the gammas as some kind of conjugates to the gauge conditions. So, there is a ancient book called Hansen, Raji and Tytelbein. So, I am going to explain a few things out of that book. This is a very arcane book, but somebody has brought out a new version now we were taught by Tytelbein. So, and Tytelbein has now a new name he is called Boonster. Tytelbein is a very clever Chilean physicist. He was working on quantum gravity according to Bryce David program, but using functional methods path integral methods and John Wheeler thought that he was going to crack the problem of quantum gravity. Anyway Tytelbein went back from Texas to Chile and started a new institute and then later he changed his name to Boonster. Whatever that is coming back to this the philosophy can be stated like this that we yes here. So, the philosophy is that identify CA with the with new configuration space variables, new coordinates, some subset right. So, from Q i you switch to C 1, C 2, C j and then Q star 1, 2, 3 the starred one will be free. So, you somehow do this transformation on the phase space and then you try to so, yeah then you try to treat the gammas. Gammas themselves will not be conjugate to the C, but you arrange it so that you find conjugate. So, then solve for P i in. So, now you claim that so, your gamma a were on the phase space. So, they were functions of Q i and P i. This can be written as gamma a of Q i of you know the Q tilde. So, C Q star P i of sorry C Q star and P P star. So, you make a this is all formal nobody is telling you that you can actually do it. The claim is that if you can first you adopt some coordinate transformation such that the ones that are unencumbered are grouped as Q star and the ones that have any problem with them you just put the C 1 to j whatever constraints you have treat them as coordinates on this phase space now. So, it is a non-trivial statement. So, actually not so, I should say coordinates on the phase space. Then formally propose that P 1, P 2, P n similarly becomes that I should have written. So, where can we write it? Now, this is a purely formal statement for the time being you do not know what the hell the capital P's are, but you do know the conjugates to the unencumbered ones. So, you list them and put here some symbols P. Now, you substitute this over here of course, what to substitute you do not know. So, you are going to just write them as functions of Q to begin with, but let me just write out the formal procedure then you can think about what is actually doable and what is a merely a statement of existence of such transformations, but the point is it conceptually clarifies to us what is going on. So, if you do this wherever in the old phase space you replace all the small Q and P by the condition C and the unencumbered Q stars some formal P's and P stars and similarly old P's like this and that said that is equal to 0 there will be as many of them as there are this fictitious P's. So, you solve these to find P's in terms of gamma. So, you invert this to determine the capital P in terms of the old Q and P. So, you solve for capital P in terms of Q star P star C that solve that is really because you are going to invert it and C's are anyway equal to 0. So, but you will have some formal expression for the capital P in terms of Q star P star and the C's and now you. So, in order for this inversion to be possible it is necessary that the gamma and the old P are conjugate to each and the P's are conjugate to each other the new P's. So, we need that. So, we are coming to somewhat crucial condition that comes out of all this rather formal mumbo jumbo which is that very. So, the formal statement is that this set of equations should not be equal to 0, but this we can now cleverly state as right we need this because we are going to to find this solution you need to there has to be independence of the variables gamma and there have to be as many gammas as P's and they cannot be coinciding with each other. Otherwise you will have a 0 in the if there is any 0 in the determinant then you will not be able to invert. So, you require this, but then you look at it and that is note that this is nothing, but the Poisson bracket of gamma A with the C's because there are as many capital P's as gamma. So, temporarily use that index, but that is same as saying that the determinant of gamma A C B should not be equal to 0. This is the real requirement on gammas. So, now you see that you do not have to worry about this fictitious P's after all, but this is how one thinks to arrive at that answer and we can see that this C and this gamma will have a nonzero commutator the Poisson bracket. So, you can do it, it is actually visible trivially, but anyway you can do it. So, because there is no A 0 in this and the only dot there is A 0, other things are all canonical, sorry I am going to reverse one minute. So, what we mean is the A's and pi's of course, pi i with A i will have non-trivial Poisson bracket delta 3 of xx prime. They are of course, functions of space, but that you do not x prime and at a fixed time we do this here. So, you will find that there are non-trivial Poisson brackets left over and then it is not collapsing to 0. So, that selects for you the physical phase space. Now, next time we will complete the Fadiya Popov trick where we will see that this determinant is what is going to be an important feature, not that determinant itself after one more reinterpretation of that determinant it will enter in the final answer. So, we will stop today.