The following content has been provided by the University of Erlangen-Nürnberg.

I have shown you some field theoretic constructions of conformal field theories,

notably these non-abelian current algebras and stress energy tensors,

in the way the physicists would do it by constructing free fields on a Fox space

and then doing some standard constructions like WIC products and play with that.

In this lecture I want to do basically the same thing again,

but in a more mathematical language, in the language of operator algebras.

And we have notably C-star algebras.

C-star algebra is an algebra which, well roughly speaking,

I don't want to give the formal definition,

but it has an algebra that has all the properties like a star subalgebra on a Hilbert space

should have a closed, topologically closed, a norm closed star subalgebra on a Hilbert space should have.

So whenever you have the intention of representing an algebra on a Hilbert space,

then you should start from a C-star algebra to begin with.

But the C-star algebra itself may be something more abstract,

and it may have different representations on different Hilbert spaces, which may be inequivalent,

but it consists of bounded operators that have a conjugate, an adjoint,

and it has a norm, and the norm and the adjoint satisfy the same relations

as they would do for bounded operators on a Hilbert space.

And yeah, okay, so this is the language I will use now.

And in particular, it means that we will not look at fields anymore,

and field operators, which are necessarily unbounded operators, with a single exception.

But rather things, in order to make the contact with the fields,

you should think of the objects that I am constructing as exponentials of self-adjoint unbounded operators,

turning them into unitary operators.

And we have already seen the Weyl algebra,

which is defined as one starts with a linear real symplectic space of functions,

could just be the Schwarz functions on the real line or something like this.

And so one has a symplectic form, which is an excursion metric

and a real valued skew symmetric form on this space of functions.

And then you define these relations, plus saying that each of these W is unitary,

which by virtue of these equations implies that the adjoint of this is W of minus F.

And then one can show that these relations define a unique C star algebra.

The star operation is manifestly here.

And the non-trivial question is, is there a norm on this algebra?

And it turns out, yes, there is a unique norm.

And then one may close the algebra generated by these unitaries,

close this algebra under this norm, and the result is a C star algebra,

which carries the name of CCR.

This stands for canonical commutation relations.

And the only input was sigma.

Yeah, your symplectic vector space with a symplectic form.

So this is a C star algebra.

And the next thing is, how do I get representations of a C star algebra on a Hilbert space?

And here the basic tool is, this is unspecific, C, well, A, any C star algebra.

And then the basic tool is the GNS construction, a Gelfand-Neumann-Neumark-Sieger construction.

And it is the following.

If I have an omega, a state on A.

State means it's a linear functional.

It is real valued on the self-adjoint elements.