In Hawking's paper "Particle Creation by Black Holes" he says the following:

The operator $\phi$ can be expressed as $$\phi=\sum_i f_i a_i+\bar{f}_ia_i^\dagger.$$

The solutions $\{f_i\}$ of the wave equation $f_{i;ab}g^{ab}=0$ can be chosen so that on past null infinity $\mathscr{I}^-$ they form a complete family satisfying the orthonormality condition (1.2) where the surface $S$ is $\mathscr{I}^-$ and so that they contain only positive frequencies with respect to the canonical affine parameter on $\mathscr{I}^-$. The operators $a_i$ and $a_i^\dagger$ have the natural interpretation as the annihilation and creation operators for ingoing particles i.e. for particles at past null infinity $\mathscr{I}^-$.

Now I'm quite probably missing something extremely basic here. But why the coefficients of the modes which are positive frequency with respect to the canonical affine parameter on $\mathscr{I}^-$ can be interpreted as creation and annihilation operators of particles on $\mathscr{I}^-$?

I do know that the basic point of QFT is indeed: (1) pick a set of modes which are complete in the KG inner product and positive/negative frequency with respect to some timelike Killing vector field and (2) expand the field in these modes, upon quantization, the coefficients become creation and annihilation operators in a Fock space giving a "particle" interpretation.

But here still. Here we have a few issues:

  1. The modes are not positive frequency with respect to a timelike Killing field, but rather with respect to a parameter which is actually a null coordinate. In that case, how does one justify that the coefficients become creation and annihilation operators upon quantizing?

  2. Still, I can't see why we can interpret the resulting creation and annihilation operators as creating and annihilating particles on $\mathscr{I}^-$? Why on $\mathscr{I}^-$? How do we justify this?


All that is a bit rough and modern views on this subject exist (see e.g. our book https://www.springer.com/it/book/9783319643427 for a recent book also on these ideas, there is a free version of the book in the archives).

However, I guess you are considering a spacetime containing a black hole. Close to past null infinity, spacetime is assumed to be similar to Minkowski spacetime (the black hole forms later). It is possible to fix the null parameter on that null surface such that it coincides with (extends) the standard Killing time in the considered spacetime far from the event horizon. In this picture, that is nothing but Minkowski time. Actually there is a conformal trasformation involved in the procedure that is singular at null infinity and everything goes right if dealing with massless particles (see the quoted book). So, you may assume to deal with the standard quantization procedure in Minkowski spacetime as soon as you stay close to the past null infinity.

Actually these modes cannot be complete and information must be added concerning the past event horizon for instance. This route leads in particular to the so-called Unruh state (are the quoted book).


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