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So, I want to add some remarks to the lecture of yesterday.

So, just to remind you, we had the following structure.

We had spacetimes, we had embeddings of one spacetime into another spacetime, which satisfies certain conditions, which I mentioned explicitly.

Now, we can associate to the spacetimes other objects, and what we then did was, we, for instance, looked at the C infinity functions,

and they are interpreted as possible field configurations, so we associate to every spacetime its space of C infinity functions,

and then this association has properties of a functor, so we have to associate to chi, map between these two spaces, and there we use the pullback.

So, just a composition of 5s chi, and this gives a contravariant functor, so the composition log goes in the opposite direction.

We have another functor, which is a functor of test function spaces,

but here we can embed a test function, chi was assumed to be injective, so we can embed the test function in the smaller region as a test function in the larger region,

so here we can use the pull forward, so this is a covariant functor,

and we have the functor of maps from the space of configurations to the real numbers,

so this was, I call them these maps, F of m, F of n, so these are maps from the space of configurations to the numbers, real or complex,

okay let me take complex here, and here again this is a functor, now again a covariant functor,

I call this embedding alpha chi and alpha chi, F as a field configuration phi is defined to be F of phi, composed with chi,

so this goes again in the opposite direction to this functor here, so this is again a covariant functor.

And the error of the pullback is in the wrong direction?

Yes, the pullback goes in the other direction, you are right, yes, you should note the error in this direction.

So this is the basic structure from which we start, now the next step is to introduce the Lagrangian,

and the Lagrangian was defined to be a natural transformation between these two functors,

and this means that there exists mappings Lm from D from m to F of m,

and we have mappings Ln from D of n to F of n, and if we have this embedding from m into n,

we have here the corresponding morphism d chi which was the push forward,

and here we have this map alpha chi, and the natural condition says that this diagram is commutative.

Now we added one further condition, so natural condition would be that this map is even linear,

but this turns out to be too strong requirement for quantum field theory,

so we use the somewhat weaker condition, I mean we required additivity,

which says that these maps satisfies this identity if the support of F intersected with the support of H is empty.

And the nice fact is that this implies that these functionals Lm of F themselves are already local functionals,

so local functionals were defined in terms of the configurations F local F of phi plus psi plus chi F of phi plus psi

minus F of psi plus F of psi plus chi phi psi chi in C infinity of m,

and support of phi intersected with support of chi is empty.

So this is this additivity property, and the nice fact is that this additivity property together with certain smoothness condition on the functionals

implies that these Fs are local functionals in the traditional sense,

so they are given by functions which depend only locally on the field itself and its derivatives.

So it's given by a function on the jet space and you integrate this function then over the space time m.

So here the consequences was so, this were consequences Lm of F is local,

and another consequence is that this direct consequence of this commutativity of this diagram is that

this Lagrangian is automatically invariant under all isometries of the space time,

so it's unnatural to use Lagrangians which are not symmetric under all isometries of the space time.

And moreover it makes sense to speak of the same Lagrangian for different space time,

so you can just say what the phi to the 4th theory means on an arbitrary background.

Now we added more structures, namely we had these equivalence classes of Lagrangians

defined by the fact that the two Lagrangians differ only at points where the function F is not constant.

And then we defined the class of a Lagrangian as an action.

So we then could define the Euler-Lagrange derivatives.

These are just called S' so the Euler-Lagrange equation would be that S,

so this was defined as the functional derivative of Lm of F with respect to phi if on,

okay let me write it in this way, so this is understood as a distribution and the F should be equal to 1 on the support of H.

And so this then again gives a natural transformation which is defined in a coherent way on all space times at the same time.