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So in this lecture I'll continue by being rather concrete and look at the Poincare group.

And I told you last time that the homogeneous Lorentz transformations can be thought of as

covered twice by the group SL2C simply by the action on symmetric on Hermitian

two-by-two matrices and then the translations. So this is acting on Minkowski space

which is R4 by a translation in the R4 variable and by Lorentz rotations in the

coming from the action of SL2C. And so you remember last time we set up the machinery

that in groups of this nature there's a theorem saying that we can get all unitary irreducible

representations by using the so-called Macchi machine or sometimes called the Macchi little group

method which tells us that we should should study so that this is actually a group of the type Hn.

I specified last time what that meant. N was a normal Abelian subgroup.

So the way this one is acting in Minkowski space it's easy to see translations form a normal subgroup

Abelian and H is this universal covering of the Lorentz group connected component.

So our task is to find so we we know we can find the unitary irreducible representations

by this orbit method that we know that everything in here is coming as an induced representation

to Hn. What we have to do we have to look at a certain subgroup that our last time called Hchi

and then we have to induce make unitary induction of certain direct product representation of this group here.

So Chi is a character of N so Chi is a character of N so this is a one dimensional unitary representation of N

and rho and rho has to be a unitary irreducible representation of the stabilizer group.

And you remember the setting we are imagining H acting so H acts on acting on N

and so since it acts on N by conjugation it also acts by contrarian conjugation on the characters on the character group

and so this is the action that we have on N hat.

Okay so the recipe was that you get all irreducible unitaries of G this way

and there was also another thing that that this depends only well on the orbit.

So let me just write this OH so this is the orbit of H acting in the dual space like that

and the choice of this rho.

Okay this means up to conjugation so if you take a different point on the orbit

there will be a different character but it's going to be the same orbit.

Of course the stabilizer group H Chi, H Chi the stabilizer group will of course be conjugate to this guy

and so the representation that you choose at this other point should be exactly conjugate to your rho.

So it's just choose a point on the orbit that's enough and then choose a rho

and then we get up to equivalence all unitary irreducible representations this way.

And so this example illustrates rather well so we get all this way.

And so let me just show you what's called the Wigner classification.

So this was actually done in the old days, this was back in the old days when physicists and mathematicians

had a lot more in common and today Eugene Wigner was as we also saw last time

crucial instrumental in understanding the role of Lie groups and representations of Lie groups in physics.

So in this case we're dealing with relativistic particles and so Wigner gave the classification.

And so all we have to do you see is just to look at the orbit so in this case of course N

well can be identified with N hat, it's just Minkowski space.

And let's write coordinates x1, x2, x3, x4 and so we have a picture like that

x2, x3 and then x4 in this direction and so what are the orbits?

What are the orbits you know? I mean these are the mass shells so here's a mass shell

corresponding to a negative square.

Okay so we have orbits.

This is a picture I guess I'm thinking of this one as being coordinates maybe I'll call coordinates over here

in the N, I'll call those for x's and maybe I'll call p's the coordinate over here.

So there will be a p1, p2, p3, p4 and you will have exactly the same orbits over in the p space.

And so we'll have orbits of this nature so there will be orbits like this we can call them M0 plus

which are the p's for which let's say minus p1 squared minus p2 squared minus p3 squared plus p4 squared