 Now, one of the main consequences of this gauge invariance is that all the charges you have in the system have to have only integer multiples of q because when lambda shifts by 2 pi by q 2 n pi by q you have to get back the same answer ok. So, to electromagnetism given 2 charges now both of those will leave this combination derivative this combination derivative is called covariant derivative I will come to stating that also, but the point is that if in psi 1 transformation if lambda equal to 2 pi by q 1 then there is no transformation psi 1 is go to psi 1. In psi 2 we will have e raise to i q 2 over q 1 times 2 pi. So, if the system is single valued under such transformations then we need that q 1 q 2 over q 1 must be an integer ok. I guess I have to supplement it by some way in which you can compare the psi with its original and so on, but the upshot is that if the system is actually unchanged in one sector then in the other sector should not be stuck with some phase. So, if q 1 is the smaller number then q 2 over q 1 should be integer or vice versa whichever is smaller the other one has to be an integer multiple of that. So, an interesting actual thing that happened was that landau and lift sheets were working out a possible low energy theory for superconductivity. So, in superconductivity one deals with a charged condensate or a charged effective field which is a scalar. So, this is the so called condensate. So, but it is charged. So, one had to write for it something like that what do we have plus or minus I have forgotten now, but whatever and in good old condensate or notation something like this and this is a phi. So, lift sheets recalls in his tribute to landau that they had the possibility of. So, what should this q be and landau said just put electron charge because if it is not compatible with electron charge then gauge invariance will be a problem. So, he just said let it be equal to E it turns out it is twice E because the condensate is actually psi psi by electron pair condensate. So, landau's choice and in fact, q equal to twice E and mine yeah twice E for superconductivity. Now, that in hindsight and retrospect sounds a very simple outcome, but it is a very deep one because if you are a theorist trying to propose something to experimentalist to look for and if you got confused at that point you would say well it can come out any charge then experimentalist will say well. So, what charge should we look for and so on well it will look exactly like ordinary electromagnetism just turned out to be twice the charge instead of just q itself it could have come out 4, but it would be a integer multiple of the charge due to for compatibility. At least otherwise under close loops you would not recover single valued systems. There is a beautiful application where actually the system can come to 2 and pi times its original value when there are magnetic flux tubes threading the system ok. So, the gauge function lambda is actually not single valued if you have obstructions where you cannot shrink loops, but we will discuss that separately. So, we can complete the discussion of u 1 gauge invariance at this point. Now, the second part is so now we go on to the more advanced topic of non-abillion gauge invariance. So, now this gauge invariance as I was mentioning here was proposed by while that had a relation to something else that was going on in general theory of relativity. So, we can comment a little bit on this whole idea of gauge invariance or symmetry and in the literature there is actually a difference of opinion how should one refer to this. So, redundancy. So, if you think as if it is this profound invariance that you can introduce magically some lambda which traverses through all the equations, but never appears in the final physics answers you might think that this is some kind of an invariance and often called a symmetry, but the other opinion is that you just have extra degrees of freedom that you put in so that the formalism looks covariant. For example, electromagnetism the propagating electromagnetic wave is essentially a plane wave and if you have some preferred directions epsilon 1 and epsilon 2 yeah these are unit vectors then a typical electric field will have specific projections along this and the direction of propagation is normal to this plane k right. So, I do not want to put anything more in this, but epsilon 1 epsilon 2 are perpendicular to the direction of propagation. So, electromagnetic field is transverse in a propagating wave it can do strange things within this plane it can be rotating or doing anything, but it remains perpendicular. Therefore, there are only 2 degrees of freedom and the B field is determined by E field completely by from the Faraday's law. The true degrees of freedom true number of degrees of freedom in propagating electromagnetic wave are only 2. So, what are 4 MU's doing? Now the MU has a great advantage because it is a covariant expression Lorentz covariant field. So, all the description all the physics you want to write you have a test to check whether the equations are physically relevant by checking that they remain gauge covariant they remain Lorentz covariant. So, it is good to have a Lorentz covariant notation, but if you do this then in the internal space in the space of possible field values you have some redundancy to the extent of an entire space time dependent scalar field ok. So, thus Lorentz covariant description entails an ambiguity in the field space or rather redundancy not ambiguity. It is ambiguous to the extent that you can add or subtract, but you can therefore, say that there is a redundancy of the description. But most of the time in most textbooks and most discussion this thing is emphasized for it being some kind of a symmetry ok. The answer is that so, if you think that this is some kind of a paradox or contradiction well answer is that both points of you are correct and ultimately the equations remain equations this is what is good about physics or science. You can put what words you like, but that is the covariant derivative you will have to use. So, the answers will all remain the same. But by thinking that it is some kind of a symmetry a lot of satisfaction was drawn the most important practical implication that it results in is this uniqueness of the charge that you have to introduce and this is what we will see when we got to non-Abelian case. So, I will just say the real. So, what this means we will see later, but idea is what we discussed there it is only one charge value and it is integer value. So, the that reduces the number of independent couplings to be read from nature right. If you have a metal then its elastic constant changes as you go from one metal to the other. Would not it be very nice if all the elastic constants were integer multiples of each other. It would be boring world, but and maybe in the nano world something like that happens. But that kind of universality reduces the need to do fresh experiments or writing different descriptions for different substances they all have a common description. The idea of non-Abelian gauge invariance there. So, this is called local symmetry. So, gauge invariance is also sometimes called we also have the usual QM ambiguity is called global symmetry. In fact, most physicists now like to think that the QM ambiguity is something superfluous, but in fact, the global symmetries lead to conserved charges. So, which are not space time dependent global means not space time dependent local means space time dependent. These result in conserved charges by Noether's theorem Noether's theorem whereas, local symmetries enforce that the Lagrangian can only be the covariant derivative. So, you are not allowed to put only the ordinary derivative d mu psi you have to put it in this combination only. So, it restricts the kind of interactions you can have and as we also saw this coupling or charge that is an it is a numerical factor that number is also fixed and at best you can have integer multiples of that. So, that is the restriction put by local symmetries. And so, sometimes Weinberg calls these dynamical symmetry symmetry of dynamics or interactions. So, symmetry of interactions or dynamical symmetries and these are real symmetries because they give you charges conserved charges. These things give you no new conserved charges, but they enforce a kind of the kind of interactions you have. So, that is the distinction between the two kinds of symmetries. Now, there is also a geometric picture associated with gauge transformations which we can use to extend this idea to the non-ibelian case. So, the geometric origins go back to what is called Kalutza Klein theories, well actually Kalutza theory. In Einstein's theory, general relativity we have gravities described by a symmetric second rank tensor g mu nu with 1, 2, 3 and 4, 0, 1, 2, 3 I am sorry. Now, Kalutza proposed that suppose you use 5 dimensions, so suppose we put g A B and put 0, 1, 2, 3 and well actually 5 or if you want we will leave it 4. So, and suppose that this part of the matrix is the usual g mu nu, then you will be left with the components here which because it is symmetric they are the same and an extra component here. And then Kalutza proposed that since we do not see the 5th dimension may be the dimension is very small it is called up it is compactified. So, although it is supposed to be a 5 dimensional metric, so we assume the 5th dimension to be 5th dimension which is 4. So, we do not see it well if we do not see it then how will it manifest? Well the point is these so it is some kind of a circle. So, we have space. So, this is the 4 dimensional space and then there would be an internal direction at each point right. So, this is the 5th dimension at every point in space you will have this internal direction which we cannot probe which because it is too tiny. Trying to probe would be equivalent to exciting these ok, but think about it there are they basically will behave like a mu field because they will they are 4 components. So, the internal direction you cannot you cannot rotate 3 into 4 because this is too small to rotate this into that, but any excitation here to the 4 dimensional observer will look like a 4 vector and that will be an additional thing which would be a new scalar not the one we have been writing before. There would be a scalar. So, in Kaluza's theory the presence of this compactified dimension will appear as ordinary Einstein relativity plus an electromagnetic vector potential and some new other scalar degree of freedom ok. And the scale of this will determine the charge. So, the physical scale of. So, in the we have been writing actually q times lambda right this combination. So, what we now do is that we put a scale l on this. So, ideally lambda should have been lambda would be equal to some distance d divided by an intrinsic scale l and then this would be equal to times l times q if you do then together it will look l times another unit. So, q is dimensionless the ordinary charge is dimensionless and lambda is dimensionless, but what I am saying is that what you call. So, this lambda was like an angle it is an angular variable, but if this circumference is equal to l then the angle would be some distance traversed on this divided by that l and then this charge q could as well be thought of as determined by this value l in the units of 1 over l ok. It counts a units of 1 over l. So, this is the way because when d reaches 2 pi l is when the system has to come back to itself. So, the charge would be determined by the size of the internal dimensions and it would look like in 4-D description this looks like like viles gauge invariant EM fields gauge covariant potentials. Now, I jump to that conclusion, but this is the proof that Kalooja gave. So, what Kalooja showed was that if that if you transform it like a space time metric a 5 dimensional space time metric, but restrict yourself to compactified transformations on this then effectively for this pieces of the metric the look as if you are doing viles gauge transformation ok. So, this is the proof of Kalooja. So, you start with full Einstein general covariance. So, thus this is like saying we got gauge invariance out of general covariance. Now, I could also lecture you a little bit on what is meant by general covariance. There are two parts to general covariance. So, general covariance with lower phase G and C is same as reparameterization invariance. This is not a principle of physics all it says is geometrically you can choose coordinate patches as you like that is what reparameterization invariance means. It has no physical content whatsoever. Lot of people seem to think that sending x mu to x mu prime correct thing would to write would be x prime mu equal to some functions f mu of the old x mu. This is there because it is a choice of coordinate system. It has nothing physical about it. However, the general covariance of general relativity is that there is a a metric tensor G mu has been G mu nu has been singled out such that that when you make the transformation this is the transformation the G mu nu obeys. How many people have done general relativity? Everyone does that good. So, you know the transformation rule for G. So, the existence of a selected out metric along with reparameterization together that is what constitutes the physical law that the selected metric has to transform co-variantly under the reparameterization invariance that together constitutes general covariance and results in requires or further require that all derivatives to be co-variant derivatives where the G gamma is derived from that this G. So, it is of course, this requirement that this is compatible with this is what is called Riemannian geometry, but that this whole package together is what constitutes general covariance ok. So, this at least is a clarification of how Weinberg's book on general relativity uses these words. You will find that at some point he switches from this to this and that is the difference between the two. So, whatever saying was that it is not so much that you have reparameterization invariance in 5 dimensions that is important, but that actually exists physical fields A mu which will exactly transform the while way as observed in 4 dimensions under the general covariance of 5 dimensions. So, in general in 5 dimensions you will also introduce its co-variant derivative you will have a selected metric tensor and that. So, in fact, when you propose this curved up you already proposed a specific geometry. So, you proposed a particular metric in those 5 dimensions and that is what and how that transforms is what is the physical principle of gauge invariance ok. So, the gauge invariance can be derived from general covariance in this sense ok. So, we will stop here today.