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So good morning, welcome back. Today we are going to discuss spin or rather spinner fields on curved spaces but also on curved space times.

And spin on curved spaces, curved space times is a topic that cannot be made precise without the full formalism of principal bundles, associated bundles and so on.

So this is another honest application of what we did, so that's 6.2 spinors or spinner fields on curved spaces and space times.

And in order to develop the definition of what a spinner field is, well it will be a section of an appropriate bundle and that appropriate bundle, that spin bundle, will be the associated bundle of some suitable principal bundle.

It will be the spin frame bundle, that principal bundle. But in order to write down all of this we need some preparation and the first thing we need to look at is the spin group.

So spin groups. What is a spin group? A spin group is a double cover, I'll explain that in a second in this context, that's a more general topological notion but in this context of Lie groups it's quite simple,

a spin group is a double cover of the special orthogonal group S O N.

So that would be the group of all the linear maps from R N to R N such that an inner product is kept constant but because if we're just on a vector space R N we can always use Sylvester's theorem and if it's a positive definite inner product we could equally well take the standard inner product.

So that is if Rx, Ry equals xy is kept constant and this is the standard inner product on R N so xy is defined as the sum i equals 1 to N xi yi.

So a spin group is a double cover of the special orthogonal group. What does that mean? That means we can construct a Lie group homomorphism from that spin group and that spin group will be called spin of N.

That is its official name spin N and we can construct a Lie group homomorphism so of course in the context of Lie groups the twiddle doesn't mean linear, there is no linear structure, it means the compatibility with this structure.

I could write as a Lie group into S O N and note I just asked for a homomorphism that means it's structure preserving map, well let's give it a name, let's call it map row.

I don't ask for it to be a Lie group isomorphism so it's not one to one. Indeed it's the very definition of the spin group that the kernel of this map, the kernel of row, well you know the kernel from linear algebra where it's all those elements

in the domain which are mapped to 0, well 0 is the neutral element of the group operation on a vector space and that's the addition, well here the group operation is a different one.

In any case we want all those in the domain that are mapped to the neutral element in the target but in the group of course the neutral element is the identity.

So where the kernel, that's just the definition of the kernel, where the kernel and that's now the definition of spin where the kernel is isomorphic to the group Z 2 and you know Z 2 is the group with just two elements.

It certainly needs the identity element whatever it is and it needs a second element and whatever the group operation is E with E must give you the identity again.

Okay that's Z 2. Okay so again a spin group is a double cover and this double cover in the Lie group sense here simply means, well in this context of SON it means that you have a Lie group homomorphism which has a kernel with two elements.

In other words this map is 2 to 1, yeah it's 2 to 1 so there are two here that map to the same over here and not only two there, always pairs that map to the same.

So that's the map that's 2 to 1. In other words the map is 2 to 1. Okay so that's the spin group and we will construct the spin group explicitly and we'll have need to construct it explicitly in cases that interest us.

I give you the result first for various dimensions and we will concentrate on three and four dimensions on three dimensional space and four dimensional space time which is still what seems to be the case out there.

So let's look at the dimension N and let's write down what is spin N.

And we'll also look at spin, well I'll do that in a second. So in one, two, three dimensions what's this spin N group? Well what you all need to know is that in three dimensions the spin three group, spin three is SU2.

Okay one can check this so that if N is three then SU2 does this trick and we'll study that in detail.

In two dimensions it's U1 I think and in one dimension it's O1. Okay and that goes on. Now if I say that goes on one calls these here coincidental isomorphisms.

So we have spin N is as a group isomorphic to.

But in some dimensions there is simply no other classical simple classical group where this fits in.

At least not in this simple way.

So this seems like a complicated definition but as soon as we understood this we can of course work with this as usual.

Okay so if people say oh SU2 is the spin so if you see the spin three group you know it's SU2.

If somebody talks about U1 you would know it's spin two. Maybe they use it in another context but that's the case.

Now I said here that we're looking at SON. However we could also look at so extension of this idea.

Look at a row that takes goes from SOPQ sorry that defines spin PQ which is a double cover of SOP,Q and by P,Q I mean that I no longer have a positive definite inner product.

But I have an inner product with P pluses and Q minuses in the Sylvester form. Okay and obviously this is of interest in relativity.

Because there we have the Lorentz group SO13 and there we need that.

So we can also ask about the spin P,Q groups and I just give a few examples of interest.

And the examples of interest to us are well if any it's certainly the 1,3 example because that's a Minkowski inner product.

So we talk about the Lorentz group that goes with it. The question is what is the double cover?

So this remains. So the idea is again that the kernel of row still ought to be isomorphic to Z2.

So the question is what do we find for spin 1,3? What are the spinners in three dimensions?

And it turns out it's SL2,C. Well it's isomorphic to it so we could we can just take SL2,C.

You can't see this. We look at this. Okay so this needs to be proven. That's why I said result. So don't be puzzled.

Then it's maybe interesting to look at 1,2. That's like you take a thin film or something and you look at electrons in there.

Well if the electrons are effectively constrained to this in this thin film you of course have the Lorentz group SO1,2.

And there the spin group is given by SL2,R. It just so happens. SL is special linear.

So it's the general linear group but only those elements that have determinant 1.

Of course this was incomplete. What I wrote down here is the orthogonal group. I have to restrict this to determinant plus 1.

So this S here is this extra condition that the determinant be 1. I'm sorry I suppressed that.

And I think spinners in 1,1 so in 1 plus one dimensional space-time I think it's GL1,R.