# Asymptotic quantization and scattering theory

There is one idea presented on some papers by Ashtekar called "asymptotic quantization". This is reviewed in the case of gravity on this paper. Moreover, this seems to be the basis of the recent work on the IR sector reviewed in this paper.

I'm trying to understand the high-level conceptual idea behind this asymptotic quantization and to connect it with what I already now. Please correct me if I got any misconceptions in what I say.

## Asymptotic Quantization

In summary, what Ashtekar does is:

1. He considers asymptotically flat spacetimes at null infinity;

2. He supposes the theory is specified by a Lagrangian density $$\mathcal{L}$$ and then uses the Wald-Zoupas formalism to construct a sympletic form $$\Omega$$ out of integration over $$\mathcal{I}^\pm$$.

3. With this he constructs a phase space of radiative modes with an associated Poisson bracket.

4. He quantizes the theory by constructing a Hilbert space out of the phase space by taking solutions of positive frequency with respect to retarted/advanced time depending on whether we have $$\mathcal{I}^+$$ or $$\mathcal{I}^-$$ and taking its associated Fock space.

## Scattering Theory

In standard scattering theory we have a system described by a Hilbert space $$\mathscr{H}$$ with complicated evolution. We want to approximate this complicated dynamics by the free one on the asymptotic regions.

Then we seek out Hilbert spaces $$\mathscr{H}_{\text{in/out}}$$ describing the asymptotic theory, and the so-called Moller operators $$\Omega_\mp : \mathscr{H}_{\text{in/out}}\to \mathscr{H}$$. The image of the Moller operators are then the states for which such approximation is fine, the scattering states.

In particular the $$\mathcal{S}$$-matrix is $$\Omega_+^\dagger \Omega_-$$ which maps the in to the out space directly, thus allowing transition amplitudes of the scattering states to be found through these descriptor states.

Now finding the right asymptotic theory seems non-trivial in QFT. In fact, in QED if one assumes the asymptotic theory to be free one ends up with IR divergences setting $$\mathcal{S}$$ to zero. Correcting the asymptotic theory seems the starting point of the Faddeev-Kulish dressed states approach to remove IR divergences.

## Is asymptotic quantization a method to construct $$\mathscr{H}_{\text{in/out}}$$ or $$\mathscr{H}$$?

Now, I have one intuition that Ashtekar's asymptotic quantization is meant to construct the spaces $$\mathscr{H}_{\text{in/out}}$$. I just can't find the way to phrase this more precisely. I get this impression because it is constructed out of asymptotic data.

So is this the idea in the end? Is asymptotic quantization all about constructing $$\mathscr{H}_{\text{in/out}}$$ that will serve as a starting point for a scattering theory?

Or I'm getting it all wrong, and asymptotic quantization is actually a means to construct $$\mathscr{H}$$, the full Hilbert space, thereby only really working for the free theories?

• Why don't you send him a mail about your queries? I did a few weeks back and he replied. – Sounak Sinha Oct 8 at 2:16