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Okay, alright, so as I was saying, in this lecture we'll construct representations

and we'll do it for some of the concrete groups, the special unitary groups SU2, SU3.

I'll show you, we'll see the so-called Heisenberg group, we'll see the Poincare group,

the symmetry group of spacetime, and that'll be about it.

Now, one key ingredient which also appeared in the last lecture was that of a spectral measure.

So let me just make that right away, that definition, so definition,

this plays a role in this lecture and also in the previous lecture.

So we'll define what's called a projection-valued measure.

So as the word says, it's something which behaves like a measure,

but the values are not numbers, but projections on orthogonal, orthogonal projections on closed subspaces.

So say on the Borel-Sigma algebra of some space M.

So we have a subological space M, we have the Sigma algebra, we have the collection of all Borel sets,

this is a nice subalgebra of the space of all sets, and we have an assignment E,

maybe better call it B goes to P of B, this should be an orthogonal projection

in some Hilbert space H, a Hilbert space H, and for each Borel set, we have an orthogonal projection.

And this should satisfy that if we take intersection of two sets,

and you take the measure that is the projection corresponding to the intersection,

then it should be the product of the two orthogonal projections.

So you see in particular it means that they commute, because if you take V intersection U,

then they should give you the same thing over here, so they have to commute.

And then it should have the important property for measures that it's what we call sigma additive,

so what we call sigma additive.

That is if you take a countable union of sets, if it's a disjoint union,

so all the sets U1, U2, U3 and so on are all Borel sets,

so that is they are in the Sigma algebra generated by the open and close sets,

so the very nice subalgebra, then this should simply be the sum.

And the convergence of this one should be meant in what we call the strong operator topology,

so sum convergent, what we call strongly IE on each.

Every time we take a vector in the space, then I can apply the projection to that vector,

I can apply the individual terms in this sum at the vector,

and then this sum should be convergent in the Hilbert space,

and it should converge to the P to the projection value measure

applied to the disjoint union of these Borel sets.

So these are open or closed sets, or they are whatever sets you can get out

by doing countable operations on open and closed sets, unions, intersections, take difference and so on.

We also want to normalize the measure, so it should be normalized,

so that P of the whole space is the identity.

This we call a projection value measure, and such things occur in the theory a lot,

so we have already seen that if we have a self-adjoint operator,

then this one has associated to it what we call a spectral measure,

what we sometimes call E, this is the one that entered this funny quantum mechanical interpretation,

that if you apply this one to a Borel measure, you get an orthogonal projection,

and then if you take a state V, and you take the, okay, so we saw this quantity for state V,

of course this is a projection, so you can also take the square,

this is also the same as the square, and it's self-adjoint, so it's also the same as E, B, V, E, B, V,

so it's also the same as just taking the projection applied to your state,

so this is the one that arose in that interpretation, if V is a unit vector,

then this is the probability that the observable H in that state will be measured to have a value in the Borel set B,

so this was a spectral measure, this is something which happens on the real line,