# How to understand Hawking's interpretation of the quantization of the field?

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?