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So welcome to lecture number 12. Today we will talk about Stone's theorem and the construction of observables.

And indeed if you look at the axioms for the dynamics of quantum mechanics,

a question arises, namely how arbitrary is the stipulation that the unitary dynamics,

that is the dynamics in the absence of measurement, be controlled by this u of t.

So that was u of t was a one parameter family of real t of unitary maps given by x of minus i t h.

For some observable h, so for some self-adjoint operator h.

I mean this looks like a terribly strong assumption and the question is how arbitrary is that choice.

Of course if you say nothing else it's entirely arbitrary, but if you have ideas like for instance,

so this question is in the face of wanting something like if you develop a system for a time s

and then you develop further for a time period t, that should be something like you developed for a total time period t of s.

And if you develop for zero time then nothing should happen.

Those are reasonable demands on a time evolution and if you require this and you want this to be unitary

so that probabilities are being preserved, how arbitrary is this choice?

That's one question you can ask and another important question to ask is how does one practically construct observables?

Of course I can provide you with a definition of some operator and you can check whether it's self-adjoint or you can make it self-adjoint.

But how do we practically do that?

Well of course there are many levels to this question, but one level is how do you find a domain?

This is the question including the question of how to find a domain where the operator is at least essentially self-adjoint.

Because if it's essentially self-adjoint you consider its closure and you know there's a unique closure so you know you have thus defined a unique operator.

So both of these questions are answered by Stone's theorem. Both questions and of course more are answered by Stone's theorem which gives today's lecture its title.

In order to relieve you of the anxious anticipation how arbitrary is this?

So then Stone says, so this is Stone, not at all arbitrary, it's fixed.

And how does one practically construct observables? He says easily.

Okay so it's a really useful construction.

Now before we go there we need some terminology which is heavily used. So that's section one, one parameter groups.

And they're so-called generators.

So please recall that a group is a set G together with an operation diamond.

So where G is a set and diamond takes two elements of G and yields an element of G, but not in arbitrary fashion.

But you will certainly recall satisfying certainly associativity. So if you have a G you diamond it with H and then you diamond the result with a K.

And that's the same as G diamonded with H and K first. And this is true for all G, H, K in the set. I'm not writing that down, I said it.

The second is that there exists a neutral element, let's call it E in the group, and that is such that one for all, that for all G in G it is true that G diamonded with the E yields the G back.

And there's the fourth property that once you have a neutral element in the group you can formulate, no three, no three.

Once you have such a neutral element you can require that for all G in the group there exists another element in the group which gets the fancy name G inverse.

But it's just a name, it's just some other element that's guaranteed to be in the group such that G diamond G inverse or equally G inverse diamond G yields the neutral element.

So you need to neutral element law first. That's a group. You certainly know that.

Okay, that's general. So what's a one parameter group then?

So one parameter group is a group whose elements are continuously labeled.

If you want to be very fancy you can say it's a manifold, a one dimensional manifold that also carries a group operation.

And if you define that with certain continuity conditions and so on you get the concept of a Lie group which is at the intersection of algebra and differential geometry.

We don't need to go into these steps although what I'm going to write down is of course Lie group.

But we have a very simple technical definition. That's the following definition. A one parameter group.

Well let's call it G is a collection of elements because a group first and foremost is a set where these elements can all be labeled uniquely by a T in the real numbers.

So that's a one. Why should I say uniquely? No, they can be labeled by elements T in the real numbers.

And with whose set, I should say whose underlying set G is of this form and whose group operation I should add therefore can be written.

Well how can it be written? Look, you take a U of T that's an element of this group. You compose it with a U of S.

Let's stay general here. So whose composition with this diamond operation it will always look like this for a T and for an S.

And you know that the result must lie in the group again. That's what the diamond does. It takes two elements from the group and sends it to the group.

That means you need to be able to write the result as the U of a real number.

And well there will be some, I don't know how to call it, doesn't matter, delta that takes the T and that takes the S and combines them in suitable fashion to produce another real number.