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So what I want to talk today is, I think it's number 6, yes, correct, yeah.

On quantization and the use of method of deformation quantization.

And I will also discuss perturbative renormalization.

Now, yesterday I described the framework which we use, so this is specialized to the case of a scalar field.

So we have the configuration space, just a smooth function on a globally hyperbolic space time.

Then we have our observables, which are just maps from the infinity from M into the real numbers.

And now takes complex numbers because for quantum theory this is more appropriate.

So this we call generically F of M, but of course we have to restrict the class of maps, but what the good restrictions are we have to find out.

Yesterday I mentioned one special class of maps, this was a local maps, local, so this I call F index local.

And then we have the regular ones, so those are the functional derivatives, smooth densities.

And in general I only admit functionals which are infinitely differentiable in the sense of the locally convex differential calculus.

Additive and smooth, additive and smooth. But we will see that we need further restrictions, but at the moment I don't want to discuss it, it will come out from the analysis.

Now then we had on this space time dynamics induced by Lagrangian, so the Lagrangian was defined to be the functor, natural transformation between two functors.

This is a functor of test function spaces, it's a functor of these function spaces. And we had an equivalence relation between Lagrangians,

which just means that L, Z, Z, if you compare two such Lagrangians evaluated on a test function F, and you look at the support,

then this has support only where the function F is not constant. And the equivalence class was an action, so action S is just a class of such Lagrangians.

Then we added one assumption which makes a contact to this space time structure, namely we assumed that the linearized Euler-Lagrange equation of this action is actually a normally hyperbolic operator.

So we could define the first derivative, which was just a field equation, and we had the second derivative, which is a differential operator acting on another configuration, psi,

and this would be the linearized field equation at the configuration phi. So around phi we expand the functional and then we find this linearized form.

Now, so the assumption was, assumption S, I should add here, index M, so this has to be understood on a given space time.

The assumption was that SM double prime of phi is a normally hyperbolic differential operator,

and we used the property that our space times are globally hyperbolic, and this it follows that this operator has unique retarded and advanced Green's functions.

Unique, advanced and retarded Green's functions, gr and ga, and we can look at the difference.

And this difference can be used to define a Poisson bracket, which is a Piles bracket.

So this is a Piles bracket.

So we have two such functionals in this space, and we define the Poisson bracket depending on this action S, but let me omit this here.

So this is defined to be the functional derivative of F, and here it takes the operator delta applied to the functional derivative of G.

And this map has all the properties of a Poisson bracket.

One has to be a little bit careful in order to guarantee that here, of course, you see you need that these functional derivatives exist,

as I always assumed that because the functions should be smooth, but this difference of Green functions here is singular.

So one needs that these functional derivatives are sufficiently well behaved.

This makes no problem if you compute the Poisson bracket of two such functionals, but you have to make sure that the results of Poisson bracket again is in the same class.

This requires some further restriction on the function of derivatives in terms of wave front sets, but I will discuss this in the context of quantum theory.

Okay, so this is the classical theory. This gives us just a classical field theory as a Poisson algebra.

Now we want to associate to this classical system a quantum system, and the idea is to use the formal deformation quantization.

So we define a new product, a star product, a so-called star product, which is...

This should then be in F of m, formal power series in h bar, and such that at h bar equals zero, this is just the classical product.

And the commutator with respect to the star product is just i times h bar times the Poisson bracket.

At h bar equals zero. Okay, so plus terms of higher order in h bar.

And this product should be associative. This is important. So this is an associative product.

Now up to now we are not able to do this directly for generic classical field theories. What we can do is we can do it for the free field.

So for the free field, what does it mean? Free field here means just that the Lagrangian is of second order in the field, which means that the third derivative of the action is zero.

In other words, if we look at the second derivative, this is this differential operator, this is then no longer dependent on phi.

And this is what makes... And then, so it follows that s double prime does not depend on phi. The same holds true also for delta now.

And then we can define the star product just by this formula. So I write here, so let me write it in this form.

So we have I h bar over one half. Here we have this operator delta. Here we take the functional derivative.

So this is at phi. Here we have functional f of phi one, g of phi two, and we evaluate it at the same field configuration.

So this is the formula and one cannot check that this defines associative product.

But as it stands, this is defined only for regular functionals because you have to iterate these derivatives and if you don't know whether the functional derivatives are smooth, it's not known whether you can pair them with these operator delta.