 Today, we begin with the topic of Yang-Mills Fields because these are become the corner stone of all of modern physics. And in other words, this topic is called non-ibelian gauge symmetry. What is interesting about this concept is that all the particle physics we know is very elegantly described in terms of this principle. What it does is that it makes all couplings universal the value of like the electromagnetic charge, the electron charge value of the electron charge everything else is a multiple of that. So, similarly and therefore, all electromagnetic interactions are determined by one charge. Similarly, the weak force and the strong force. So, electromagnetism all the three are described by the same quote symmetry principle. And so, I should put quote marks on this I will explain what that means. And what we find is that these two are sort of intertwined ok. So, together they form what we now call electroweak theory. So, there is an underlying gauge principle based on the group 1 SU 3 C cross SU 2 L cross U 1 Y where C is color this is strong force and these two together give the weak force plus U 1 electromagnetism. So, this is an advertisement or a trailer if you like of what this topic is all about ok. So, thus all the known forces at terrestrial level are completely described by this particular framework. So, let us begin this by seeing what is the usual idea of gauge invariance or the electromagnetism. So, before we get to become very sophisticated let us try to track back what is how one deduces this and how possibly the first people who made this observation about this kind of symmetry saw this. And that starts with the Lorentz force in Lagrangian framework or Hamiltonian framework. So, first of all it turns out. So, if you see Goldstein's book which is the only one we used to use when we were young. Nowadays the book has a co-author and it is somewhat rewritten, but if you see Goldstein's book this occurs fairly early and he shows that that book contains the proof that and it would probably be also there in Landau and Lipschitz classical mechanics you know that to get. So, V is x dot to get this right you need a Lagrangian which is of the form t minus q phi. So, what we do is we propose that this point that e is derivable from a scalar potential right we know that this can always be done Faraday's law and Ampere's law together will allow you to do this and Coulomb's law and so and Ampere's law along with divergence B equal to 0 will allow you to write B in terms of a vector potential. In that case using this we can rewrite the Lorentz force in Lagrangian form as the kinetic energy minus q times phi q is the charge plus q over C A dot V and therefore, if you find the canonical momentum P x equal to we should write variational derivative then that will be equal to M x dot and plus q over C A x. So, the canonical momentum of electromagnetism already has the potentials in it not just the velocities. So, you can then construct the Hamiltonian out of this Hamiltonian has to be written out by doing P x dot you know and x dot has to be written in terms of P minus q over C A etcetera writing P x dot minus l. So, you can do the remainder, but the point is the momentum canonical momentum now looks like this. So, in quantum mechanics we know that P goes to minus i gradient yes and eventually we have to get to a covariant form and there is some mismatch between the way we write here and the covariant notation. So, for example, the gradient operator. So, if you have to write this as a in index notation then the gradient operator is automatically a covariant vector people know co and contravariant right contravariants are written with upper index and covid down index mainly to remember whether you are referring what whether you are referring to the original frame of reference or its dual frame of reference. So, because of this now we write that P in electromagnetism what we need is that P i is equal to old P i minus q over C lower A i and as far as I remember there is a mixed notation here it turns out that one of these indices is up instead of down ok. So, if you take x to be the contravariant vector that is right this A x is covariant and so, there is an opposite sign, but this is the prescription that is that correctly works. So, wherever you see minus i gradient you replace it by A lower i, but that is same as minus i yes times. So, what happens is that at this point we actually have because of this mixed notation it is like this and then what one does is to write to covariant version. So, this is sort of the physics notation and additionally what happens is that the Hamiltonian gets from the minus l it gets a plus q phi. So, this also means h goes to old h plus q times A 0 and therefore, i i d by d t has to be replaced by no for A 0 there will not be any sign change. So, yeah so, that becomes equal to i d by d t no sorry. So, the because of the minus l the q phi appears here with plus sign. So, this is actually minus and right and so, I want here eventually plus i q A 0 and modulo whatever sign confusion there is it can be worked out I have checked it once in detail. What we will adopt is that together and this is what we will be using and I think if forced to argue I will say that yeah. So, this is upper i and that is because or when you do this substitution you are supposed to think that it is upper, but you have to put a gradient and therefore, there is a relative minus sign, but if you do this you can begin by saying this is your prescription. The only problem is that you have to check that it reproduces Lorentz force correctly if you started with the idea of describing physics correctly and it does match. So, if this is so, what well the interesting point is that this is the way to couple electromagnetism to charges the coupling between charges and the electromagnetic fields is through this prescription only wherever there is a canonical momentum you replace it by canonical momentum plus this contribution from the gauge fields. So, in the in quantum mechanics where we have Schrodinger equation where the Hamiltonian begins with 1 over 2 m grad square psi it becomes so, plus v psi etcetera, but we are not going to use this too long, but so, we have to then replace it by so, this goes over 2 and there is a minus h cross square by 2 m. There is h cross somewhere here as well yeah. So, there will actually be a 1 over h cross here because actually this p goes to h cross gradient. So, there is a q c over h cross. So, we can write it once here there is an h cross c, but we will not henceforth write it. Now, the point is that and this was observation of Hermann Weyl that we have a gauge invariance in electron we have an invariance in electromagnetism. So, for the E m potentials we have the ambiguity that the vector potentials can be replaced by yeah again I am using a mixed notation because that gradient is actually covariant from the covariant notation whereas, I am writing upper index a's that that does not matter is just to observe this the overall sign will not matter. So, the point is if you change a by the gradient of a space time function lambda, then the curl of the nu a is same as curl of old a. So, this relation does not change and you can also set phi or a 0 and then d by dt probably with a 1 over c probably not d by dt of the same lambda and this works because if you look at the E field we can actually check about the 1 over c etcetera because if you take grad if you take d by dt of a it now gets a d by dt of grad lambda. So, we can check this quickly whether everything is right you can do it in your notebook on the side I am leaving some space here. So, this works out correctly if you put a 1 over c here right. So, because of that notational problem with up and down indices I think I need here a plus sign. So, we put a plus sign here in any case as we were saying it will not matter. So, this will become equal to minus grad a 0 minus 1 over c d by dt a and the remaining terms cancel because this is minus 1 over c grad d by dt of lambda with a minus sign and this is plus 1 over c d by dt of grad lambda. So, you need a plus sign there and a 1 over c in the a 0 transformation. Now, I have used the word ambiguity which is correct from the good old differential equations point of view because if you wanted to describe some physical fields a and b then you could have come up with some convenient a phi part the a 0 part and some convenient vector potentials that would you have to solve these equations right these are first order. So, given physical e and b you can always find phi and a as solutions of these differential equations because they are first order differential equations although they are coupled partial differential equations. The point then is that that solution quote will not be unique in fact, it will have a huge ambiguity. This if you have tried to solve PDE is first order PDE is by substituting into each other you know that unlike putting initial condition in ordinary differential equations here the initial conditions are functions of the variable which is not involved when your partial differential equations if you have only d by dx all you know is d by dx of f of x y is equal to x let us say. Then all you can tell is that therefore, f must be equal to half x squared plus any g of y right because you do not know what that part is. So, when you are solving them simultaneously the other conditions then help you determine g, but overall you may be left with a whole functional ambiguity in solutions of such partial differential equations and that is exactly what happens here you try to find these they are not uniquely determined by these two given conditions or these two equations. And so, there is a functional ambiguity and that anyway you observe enters through derivatives it is not directly that there is and there is only one scalar function although there are four components. There is only one overall space time the Lorentz scalar function that enters whose derivatives cause an ambiguity in identifying the gage potentials. So, now, one other thing I just would like you to remember is that a Lorentz scalar is a very strange object. A Lorentz scalar is something you have never seen all the scalars you know of will transform very strangely if you perform Lorentz transformations. One of our favorite scalar is length right it is invariant under rotation. So, we think it is a scalar, but length is not a Lorentz scalar. So, most things that you know are actually not going to qualify. In fact, I cannot think of anything that oh well the rest mass, but is a true scalar, but we already give it away because we say rest mass meaning mass in its own frame of reference which is which anywhere reduces to particular choice of frame. So, we do not know any fields space time fields that are scalars until very recently. It is all I am trying to say that somewhat counter intuitive what space time Lorentz scalar would be, but that is what one adds. Now, that ambiguity is transformed into magically into some kind of a symmetry and that was the observation of Hermann Weyl. So, Weyl said that therefore, the going back to the Schrodinger equation we know that the wave function has a overall phase problem. So, but now Hermann Weyl observes that if for this constant alpha if you instead put this same lambda then it will then you can arrange for this whole object to remain unchanged under this. So, and now we will stop writing q etcetera and let me write like this. So, what we do is that we simultaneously let thus if psi goes to e raised to i times q times lambda then d mu x d mu psi will go to i q d mu times psi d mu lambda plus times psi plus d mu psi times e raised to i q lambda x. So, and now then like we used to doing high school d mu yeah d mu plus i q a mu multiplied by psi will go over to. So, here you can multiply this on the left and right by e raised to i q lambda without any loss of generality because this is just an algebraic expression there is no derivative. Therefore, adding both the sides I will get that when I add these two the d mu psi here sorry the i q d mu lambda will exactly cancel with minus i q d mu lambda from the gauge field shift and I will simply get. Now, when you take this and put it here that e raised to i lambda will come here i q lambda is the charge unit has to be put in here. This again will act on it this combination, but this combination again will remain unchanged because a will have shifted and you can bring the overall phase all the way here and this psi also has the same phase. So, you can throw away that same phase the whole Hamiltonian has changed by only an overall phase the or you can just replace the psi by that nu psi with a space time dependent phase it will not change the Hamiltonian. In covariant notation it gets even better because it actually even cancels as we will see for scalar fields. Thus, with electromagnetic coupling and above changes in above transformations of psi of x as well as e raised to i q lambda x psi x describe the same system. And, so this is the statement of gauge invariance.