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

So last time I described the geometry of Lorentzian manifolds,

particularly concentrate on globally hyperbolic space times,

which are characterized by the fact that they have a Cauchy surface.

And the main use of the existence of Cauchy surface for us is that we have a well-posed initial value problem,

the Cauchy problem is well-posed, so we have globally hyperbolic Lorentzian manifolds.

And we consider on this manifold differential equation that prototype is the wave equation,

so the equations we consider are normally hyperbolic differential equation.

And in coordinates they are typically, they are of the form, so we have a differential operator,

and this operator is of the form g mu nu d mu d nu plus a mu d mu plus b,

and differential operator of this form we call normally hyperbolic.

So the connection with the structure of the Lorentzian manifold is that here the inverse metric tensor appears.

Now these operators have nice properties on this kind of Lorentzian manifolds,

namely they can characterize a solution uniquely in terms of initial values on a given Cauchy surface,

and what we mainly use is the existence of green functions, so what is a green function?

So if we look at operators g from say the smooth functions with compact support to the smooth functions,

there is a property that p times g is equal to g times p is equal to the identity,

where I mean by identity the embedding of the smooth functions with compact support into the space of smooth functions.

So this is the general definition of a green function, we often use the description of green functions in terms of the integral kernel,

so we write g f of y in the form of an integral, here we take the volume element of our space time,

which is characterized in terms of the Lorentzian metric, and g, we use here the same symbol as tradition in physics,

so this is now, looks like a function of two variables, but is actually a distribution,

and so this is another way to write down the action of green functions.

Now we have, there are many green functions on the Lorentzian manifolds,

because two green functions differ by a solution of this equation, and all solutions of the wave equations are in the kernel of the operator p,

and so we could add them to this operator g, and this would not change these relations.

Now there are two special green functions, which can be characterized in terms of the support properties, namely there is the retarded and the advance green function.

Now the retarded green function is defined by the support properties of this function g applied to f, f is a function with compact support,

and the support of the function g f should be contained in the future of the support of f, so the future is just defined,

J plus of some set R is a set of all x and m, such that x can be reached by a causal curve,

from the region R, and if necessary I take the closure here, so it should be a closed set.

So this is the condition on the retarded function, and in particular we assume that the support of f is compact,

so in the past this function f in the past of this support is of course zero, and then although the solution should be zero,

but since the initial value problem is well posed, you can have each Cauchy surface in the past, the support of f, you have initial value zero, so the solution would be zero.

So the retarded green function is unique, and the same holds for the advanced green function,

which I characterize by an index a, and that we have to replace future by past, everything else is the same.

So this is a very important and useful structure, there are of course a lot of other green functions, some of them will also play a role,

but in general there is no other uniquely fixed green function, so to fix other green functions one has to introduce some additional structure.

A particular green function which one needs in quantum field theory is the Feynman propagator,

but the Feynman propagator can be defined only if one has some notion of positive and negative frequencies, and this is not meaningful on a generic space time.

But these retarded and advanced propagators exist in every space time, moreover they have the nice feature that they depend only on the local geometry,

so if you have some sub region which in itself is globally hyperbolic and you compute in the sub region the retarded and the advanced solution,

you get exactly the restriction of these operators, just because of the uniqueness, otherwise there would be several possibilities.

So this is a structure which is canonically associated to the space time.

Now we have to look more carefully at the singularities of these integral kernels, so these are not functions, these are distributions,

and we have to understand the singularities of these green functions.

Now the method by which one can analyze this is the method of mycolocal analysis, and in particular we need the concept of a wave front set.

Wave front set of a distribution.

So traditionally in physics one would just take the Fourier transform of the distribution and try to understand it better,

that's what one usually does with the propagators in field theory, but the problem of course on a curved space time is that there's no generic Fourier transform,