 So, now that we remember why SU 2 is like S 3, but here you say there is no S 3. So, what Jackie when Rossi read Rabi propose is to look at R 3 and choose maps u of x because we are in R 3 such that u approaches minus 1 as mod x goes to infinity. You choose only those u of x such that you would reach the. So, u is a map right from R 3 at every point you have to erect a u of x, but what you do is that when you reach this we sometimes call it S 2 infinity the sphere at infinity we make sure that the u of x here is minus 1. So, long as you choose a map like this you have done a it is a 1 to 1 map, but you have chosen entire S 2 infinity to be mapped into one point of SU 2. You can take a simple example if you like you take a Gaussian plus infinity and minus infinity are both mapped to 0 right. So, you can take any function that is localized in R 3 asymptotically just goes to one point of the real axis. So, these are maps of R 3 into real axis where regardless of where you start all the infinities mapped into one point. We are doing the same thing except that instead of real valued map it is a group valued map and in the group space we will all map it into one point. Effectively therefore, the R 3 itself is compactified R 3 itself is converted to S 3 by this kind of argument. The outermost sphere is mapped to one point. So, it is a S 3 becomes an S 3. So, we will come back to it again. How do you do this? How do you choose such a map? Well it is very easy once it is explained. Example choose u of x to be equal to x square minus a squared divided by x squared plus a squared times identity plus i times 2 times x times a over x square plus a square times x cap dot tau. We tied up we mixed up space and group space. This is a map of its domain is the values x in R 3 and its range is SU 2 group. We can easily check that and note that the what we were writing as theta cap and I was showing you here is literally chosen to be x cap the direction in space you are going out. So, it is exactly x cap of the real space. Now this looks mysterious, but one can check easily that this is an SU 2 element because well you could do u dagger u and you will find that it is equal to 1 and, but it is a little easier if you notice the clever form of it is given. You can think of consider cos theta by 4 to be equal to x over square root of x square plus a square and sin theta by 4 to be a over square root of x square plus a square right. Then the puzzle is solved because this is cos square minus sin square that is what this is because this gets squared that will become cos of theta by 2 and this is 2 times sin theta cos theta. So, it is exactly that form cos theta by 2 sin theta by 2. So, you have to choose the angle theta of x such that it is cosine inverse of 4 times x over square root this. Now one can see what this map does all it does this is just one example it is a very easy to it is a nice example, but one did not have to do this. Note that the map is such that at x equal to 0 it just becomes equal to at x equal to 0 this is equal to minus 1 this becomes minus 1 and x equal to 0 this term is not there. So, it is a map which is actually opposite it is u x equal to 0 is minus 1 and u equal to infinity x equal to infinity it will be plus 1 and there this will reduce to 0 because it will go as 1 over x. So, it will go to 0. So, actually the Jackie Rabi choice is like this you could put a minus sign in front of it if you do not like it it does not matter. So, that is the particular choice in this solution, but note that what this means is that we can try to plot x as a function of and this is of course, modulus x right this x is same as that is what we mean x square meaning is clear it is a vector squared. So, thus note that we constructed a theta which at x equal to 0 better to look at sin function because we know sin theta becomes theta. So, as theta goes to 0 as x goes to 0 this becomes equal to 1 right and that means, that it is equal to pi by theta by 4 is equal to pi by 2 that means, that theta is equal to or rather we can just plot the theta by 4 function because that is all that enters into the picture. So, this is equal to pi by 2 at the origin and asymptotically this reduces to 0 and this reduces to 1 this becomes 1. So, asymptotically it is going to 0 as a function of mod x, but this is all you need you did not have to choose the very nice form which is whatever you like to call it Laurentian or something you could choose anything that you like so, long as it is monotonic and so, long as it starts with pi by 2 and goes to 0 it is fine. I can construct for you a reverse example we know take a tan hyperbolic function which starts with 0 and goes to 1. So, all you do is you multiply you take pi by 2 well. So, I will give this as a you think of other functions that can still be written as some kind of transcendental functions and which have that same property. Then you can stuff it in here it will look more complicated, but it will it will not be very different the 2 will be related by a simple gauge trans local gauge transformation. So, now what we so, at least this one you can see why it is firstly a nothing, but an SU 2 element is just that we have chosen those values for the theta except that it is a r valued radius valued map and every at every point of r it is an SU 2 element and it maps the r 3 into the SU 2 both of which are converted to S 3 because in r 3 we chose S infinity 2 to also all be mapped into one point of U. Now, the interesting fact about this particular map is that we cannot shrink it to 0 although there are no obstructions in the space. So, remember in the case of Abelian case the gauge group was a 1 valued space lambda was one thing that parameterized it. So, if you had a simply connected space you could shrink it to 0, but the gauge group space here is itself 3 dimensional. So, to shrink it to 0 you need a trivial 3 dimensional space a 3 dimensional space that is exactly r 3 you can shrink any U to 0, but a 3 dimensional space which has effectively secretly become an S 3 you cannot shrink it to 0. So, you cannot everywhere press this down and iron it out. So, because r 3 is rendered effectively S 3 thus this U cannot be made trivial trivial map and what trivial map can you have U 0 equal to 0 or U 0 equal to constant. Well you cannot have arbitrary constants you can only 0 after all U dagger U has to be 1. So, you can choose some possible constant values of U you cannot these two differ by topological for topological reasons they are not modifiable into each other by a continuous change ok. Well if you make non-trivial non-continuous changes then you will have defect defects in the derivative, but derivatives would turn on gauge fields. So, it would not be very nice to then you would have to turn on you would have to feed some energy. In fact, one can show that the energy would be very large. So, it cannot be done I am sorry it can be turned if you turn on if you feed some energy into the system then you can go from one to the other, but otherwise you cannot. This map also has a very interesting property that its integer powers are also gauge transformations denote the above U of x by U 1 of x i.e it is called winding in memory of if you take a circle if you take one circle and another circle, circle number 2 and if you go through 2 pi here and suppose here you go through 4 pi ok. So, this is called number of windings. So, map is characterized by around the circle around the S 1 2. So, that term winding number is a standard in topology borrowed from the one dimensional example. So, this one winds once because it maps R 3 into SU 2 1s. So, it is of course, 1 to 1, but it is not just that calculus requirement that is important it is that as topologically it covers the space once ok. But now consider U to the power n. So, let us call it U n I think better to write a square label because we have too many kind of indices. So, U n of x to be equal to U 1 and we take it to power n, where n is an integer plus or minus 1 plus positive or negative n belongs to z, but without 0. Then we know what happens if I have a unitary matrix its nth power is also unitary. So, all you have to do is U U dagger equal to 1, but if you take U to the power n U which is U U U U n times the U dagger to the power n is also U dagger U dagger n times you multiply the two all the U U daggers keep giving one you get answer 1. So, U n is also unitary matrix, but now you see what will happen what will happen to the n the n we can see over here this is nothing, but x i theta by 2 theta cap dot tau right this is what it is if you raise this to power n what will happen n will just go and sit multiply the theta right. So, here the theta was a function of x. So, now instead of starting with pi by 2 it will start times n times pi by 2 and cross through n minus 1 pi by 2 n minus 3 you know up to 0. So, it will go from n times pi by 2 and then go to 0 because the theta just got multiplied by n that is all that happens. So, you have to take here it will look complicated it is no longer simple, but it is cosine of n theta by 4 that will become that. So, the theta just becomes and theta got scaled up now sorry not that yes. So, the theta becomes n times whatever it was before, but it is still an SU 2 element, but we can now see that what this does is take the s 1 example if I wind around this once and this winds once that is one map. So, the trivial map is I wind around the whole circle I stay put at origin that is the trivial map then I wind around once I wind around once, but suppose as I reach pi I already reach 2 pi here then as I go from pi to 2 pi I wind once again that is a 2 double winding map then I reach one third of the circle I cover the whole circle here then I cover next two thirds I cover the whole circle last one third I cover the whole that is the winding number 3 map. That is exactly what happens here you will start from this minus 1 and hit plus 1 at not one third now, but somewhere and then. So, if you are raised to power 3 and then you will start and hit minus 1 again you will hit plus 1 again and finally, you will reach plus 1 after having crossed plus 1 2 times in between and then the third time at the infinity. So, it is a triple winding map that 3 windings cannot be shrunk to 2 winding and 2 winding cannot be shrunk to 1 winding. So, there are maps indexed by the integer z ok. So, they map R 3 or the compactified R 3. So, let us just say S 3 of space into SU 2 and number of times. The meaning of minus 1 is just winding the opposite way. So, in the case of circle as you go from 0 to 2 pi here you start in the opposite direction that is all it means. In the present case it will mean starting from minus pi by 2 and going there and etcetera. So, they are anti winding maps and they are also independent because the direction this direction cannot be converted into that direction ok. So, they map also n times and so, each one of these is an independent is a gauge transformation is a set. So, this set of gauge transformations indexed by n cannot be mapped into each other cannot be smoothly deformed into each other and I now include all z because the n equal to 0 case would be our u 0 this belongs to n equal to 0. So, such maps indexed by n can not be smoothly deformed into each other. We have a peculiar situation that the situation with no electromagnetic gauge fields is what we call vacuum ok. So, now observe or we say. So, by vacuum we mean no electric magnetic fields it should be 0, but you can have pure gauge transformations many states in equivalent under gauge transformations of about type. Such gauge transformations are called large gauge transformations to distinguish them from small modifications of the u u 0. So, all modifications that are topologically equivalent to the trivial one we call them small like you perturbed around u equal to 0, but where you took the trouble of going all the way to infinity it is a lot of work to go to infinity and then soldered everything correctly then we call it a large gauge transformation. But now this causes a problem to quantum theory because what do we mean by vacuum in quantum theory because although these are these are all corresponding to H mu nu equal to 0 you could have done a large gauge transformation and the state may not remain the same it may change differ by a phase because there is no way for you to without turning on electromagnetic fields compare those two states you do not know what is the relative phase between them ok. So, what does the vacuum of the quantum theory look like and actually Jackie when Rabi do a careful like this is a very beautiful physical review letters paper from which year I forget, but if you just look for Jackie Rabi physical and this what do they say vacuum structure of gauge fields or theories or something like that 1973 or no 76 maybe. There is another thing you might think that this is all specific to SU 2, but SU 2. So, but it is certainly true of all SU n groups because if you have an SU n group all you do is take an SU 2 subgroup of it SU n groups are homogeneous spaces they are simply connected and homogeneous. So, you just take a shrink to an SU 2. So, all you have to do is keep mapping from your R 3 into some subgroup of that SU n you get the same answer. So, the answer does not change. In fact, I think there is a theorem of bot Raul bot map from any map from R 3 into any now this is the part I do not remember SU n or any connectedly group, but the weaker thing is certainly true which is a subgroup of the SU n. By the way this counting that we did with winding number and all that is also called the homotopy group of the maps. So, this set of maps. So, this is end of the Raul bot statement note that the set of maps indexed by n form a group by providing a multiplication rule. We will not go into detail, but you can guess what that rule is you just continuous continue into the next one you compress the first one. So, if you have done it 3 times then you say oh well I have to do 2 times more. So, you go to 2 more's you know you have 3 winding then you want to do a 2 winding well do 2 more windings to it. So, you can define such a rule that would be called multiplication. So, by providing a multiplication rule by and it is known as the homotopy group by 3 of SU 2 3 because we are mapping from S 3 mapping S 3 is into the space ok. Now, I wanted to talk about the vacuum structure let me see if I can do it in 10 minutes. So, let us see the effect of the vacuum of quantum field theory. So, let us so remember that small gauge transformations are all allowed, but the sector would remain the same all gauge transformations such that the U asymptotically approaches U 0 or constant. Such gauge transformations will leave you in that sector winding sector n it will not change the winding number at infinity. So, all small gauge transformations do not matter, but let us denote possible ground states well there is certainly ground states with so corresponding to winding number n as psi n. So, within which treated all treated equivalent unchanged under small gauge transformations. We might then consider the possibility that the true vacuum should be a linear superposition of all of these ok. So, there is one thing here we can check that and you can see the Jackiw-Rabbi paper that we can turn on F mu nu fields and change over from one such sector to another. Now, why is this important? It is important because it shows that the various sectors of this vacuum are not like completely independent. You can actually turn on some energetic activity and when the activity dies down you may find yourself in a different winding sector without having to do something violent dangerously violent ok. So, often it is shown like this that I have here the grand class of this is the functional axis all possible gauge field configurations if you like I will put this as well. And here we draw energy total energy of the system you calculate the Young Mills energy E square plus B square. Then we already know the sector U 0 this is the sector 0 and we know that there is a sector 1 and there is a sector 2 and so on. And of course, there is a sector minus 1 so on on the other side. What we do know is that when you get there you have 0 value of energy. What Jackiw and Rabbi show is that in fact, there is a finite barrier, but this barrier height cannot be defined ok. You have to turn on some electromagnetic fields to go from one to the other, but there is no lower limit on the how much electromagnetic energy you have to turn on. The reason is that the gauge theory has no scale in it is one of the thing. The classical theory at least you know that it is just there is only that dimensionless charge g. There is nowhere there is a dimensionless number in that whole Lagrangian. So, due to the so I should probably start putting some dashes here. So, we can check that this is true. The barrier can be demonstrated to exist to be finite, but with no lower bound. Any one map that you will construct will have a finite energy, but there is nothing intrinsic in the theory that says if you do it of this scale then it can be made. So, there is nothing to compare with. So, this barrier exists, but it has no lower bound, but it exists ok. But now what is more interesting is that the whatever the barrier is is only finite height. It is not large. It is not infinite or large. In fact, it can be arbitrarily small. We know in quantum mechanics that then systems tunnel into each other. If you leave the if you start by saying you can say oh forget all this junk I will sit at u equal to 0. How do you challenge me? Classically you will be fine because doing any going to anything else would require turning on the color electromagnetic fields. I say I will never turn them on in my world then you are safe. But in quantum mechanics you are not safe because the system may just tunnel. Therefore, you have to consider so the tunneling has to be taken into account. So, we propose a grand psi which is equal to sum over some C n times. So, psi n, but how do I determine this C n? Now, the point is that. So, what are the C n's? You may say they can be arbitrary which is true at first sight, but there is an interesting point. If we now slide this by one action of u 1 this would be equal to C n. Now you would get right C n times u acting on all these psi n, but we know that u 1 acting on psi n will make it psi n plus 1. Let me leave some space and up to some phase if I jump from one vacuum if I you know somehow went from this vacuum to the next vacuum I might accumulate some phase. Then it will become, but this is a bit awkward because physically this state is same as the original state. Heating by one large gauge transformation should not change physics that is by physics we mean the vacuum. So, the two things this and the original have to be the same up to one overall phase. They have to be same up to a phase. Now how do you see that this and that will have exactly same phase? Well, remember that this was summation from minus infinity to infinity right because the n is indexed by minus infinity to plus infinity. Therefore, the two things are provided theta n is actually just n times a constant theta. So, this is possible only if theta n is actually n times one constant theta. Then we can slide the summation by one unit take out one theta and then everything becomes the same. So, theta n would be n theta I take out e raise to i theta then I am just left with e raise to i theta n minus 1 n minus 1 times theta, but then I relabel n minus 1 in the summation then it just becomes the same old sum with the theta number theta and of course, remaining the same right n minus 1 times theta does not change, but I relabel the summation over the states. So, this happens provided theta n is n times a basic theta. Therefore, now we find that actually it is not. So, the state psi that is chosen can be. So, what value of theta? Well, the answer is it could be any theta any one possible value of course, 0 to 2 pi because now it has become phase, but 2 vacua that have different value of theta would now be different. So, what has happened is that we had a series of vacua oops series of vacua which was a discrete sum. We traded that discrete sum for a continuous indexing theta, but over a finite interval 0 to 2 pi. So, the physical states are listed by. So, this happens in any physical setting with any one preferred theta belonging to 0 to 2 pi right. Thus the q m vacua are labeled by a continuous parameter theta of this range instead of discrete index belonging to Z and that is where we can stop the story for the time being. But the point is that further you may say that who cares you know some theta is there it is actually unobservable, but if the theory is coupled to fermions which are massive then you can show that the theta actually becomes observable right for if it was like this you say oh there is I live in world with one value of theta and I will set it equal to 0. Nobody can argue against that like because you can choose the phase of your wave function, but when Young Mills theory is coupled to massive fermions this theta becomes unobservable. Not only that it appears as a term theta f f dual by some series of tricks you can transfer the, but then you are forced to have a f f dual term which you avoided at first. We argued that f f dual violates C p because the f dual is parity pseudo scale pseudo tensor, but now you are forced to have the pseudo tensor term you cannot avoid it and it actually becomes observable in induced moments. It becomes parity violating through electric dipole moments I wrote it in capitals because it is a very popular term of electron neutron electron proton etcetera. So, now the double mystery in nature no such theta is found in nature theta is found to be 0. So, far EDM has not been observed theta is bounded by 10 raise to minus 9 is really tiny if it exists it is really tiny. So, this is like the puzzle of Mr. Nobody I wish I wish he what is that limerick there is a limerick that Weinberg writes in his review on cosmological constant. So, you are in a cyclical trouble from all your clever arguments you found that there should be some theta that it should be observable, but in practice you find that its value in nature is exactly 0 or close to. So, now it poses you a problem why if you are allowed to have any possible vacuum in this range you have exactly 0 value in nature. So, this is called the strong C p puzzle of Q C d or the theta vacuum a puzzle of Q C d. So, it manifests itself in electric theory in a different way because there it spontaneously broken and then you do have a barrier height because the Higgs has a vacuum expectation value which sets its height, but in Q C d itself in pure Q C d because the quarks are massive theta should be non-zero, but it is found to be exactly 0 and we will stop with that.