# How to properly make sense of the $\mathcal{S}$-matrix as a correlator on a sphere?

In the book "Lectures on the Infrared Structure of Gravity and Gauge Theories" by Andrew Strominger, the author discusses in Chapter 3 the idea of "The $$\mathcal{S}$$-matrix as a Celestial Correlator".

He says:

The alternate description of scattering is as a type of correlation function on a sphere, depicted in figure 16. In the massless case, "in" and "out" massless particles are labelled by operators denoted $$\mathcal{O}_k(z,\bar{z}).\tag{3.0.1}$$ Here, $$z = \frac{x^1+ i x^2}{r+x^3}\tag{3.0.2}$$ denotes the point on the sphere at past or future null infinity where the particle of type $$k$$ enters or exits spacetime.

After a brief discussion about the massive case, which doesn't affect my doubt, the author continues:

It is then natural to express the $$n$$-particle scattering amplitudes in the form of a celestial correlator on $$\mathcal{CS}^2$$: $$\langle \text{out}|\mathcal{S}|\text{in}\rangle \to \langle \mathcal{O}_1(z_1,\bar{z}_1),\dots, \mathcal{O}_n(z_n,\bar{z}_n)\tag{3.0.3}\rangle$$ Note that no assumptions are being made here, we are simply rewriting the $$\mathcal{S}$$-matrix in a different notation.

Now my question is on making sense of Eq. (3.0.3) precisely.

What really are the $$\mathcal{O}_k$$ operators? My intuition tells me that they are just the creation/annihilation operators $$a(\mathbf{p}),a^\dagger(\mathbf{p})$$ expressed in terms of the asymptotic fields at $$\mathcal{I}^\pm$$ and with $$\mathbf{p}$$ written in terms of $$(z,\bar{z})$$. Is that the case?

Also what is the state with respect to which the mean value in Eq. (3.0.3) is taken? Is it the same vacuum on the asymptotic Fock space?

In summary, what is the precise way to translate the LHS of Eq. (3.0.3) to the RHS?

Update: This update is meant to make the point clearer. The issue here is not with the motivation. In fact the motivation seems clear from the context of Strominger's work in the context of flat space holography.

The simple relabeling is also clear for me. In fact, a massless particle has a 4-momentum of the form $$p^\mu = (E,E\hat{\mathbf{p}})$$ where $$\hat{\mathbf{p}}\in S^2$$ is a unit vector. Using stereographic projection and complex coordinates we can parameterize $$\hat{\mathbf{p}} = \hat{\mathbf{p}}(z,\bar{z})$$ , hence $$p^\mu = p^\mu(E,z,\bar{z})$$. It is the clear that a matrix element $$\mathcal{A}(p_1',\dots, p_m', p_1,\dots, p_n)=\langle p_1',\dots, p_m' | \mathcal{S} | p_1,\dots, p_n\rangle$$

is a function $$\mathcal{A}(p_1',\dots, p_m', p_1,\dots, p_n) = \mathcal{A}((E_1',z_1',\bar{z}_1'),\dots, (E_m',z_m',\bar{z}_m'), (E_1,z_1,\bar{z}_1),\dots, (E_n,z_n,\bar{z}_n))$$

So, that $$\mathcal{S}$$-matrix elements are functions of the tuples $$(E,z,\bar{z})$$ for the several particles is clear and seems indeed trivial. What is not clear (and seems non-trivial to me) is why these functions mean values of products of operators $$\mathcal{O}_k(z,\bar{z})$$

The following answer is based on my interpretation of the article you cited, and on some of its references.

### "Motivation" for the LHS $$\rightarrow$$ RHS step

What's the point of re-writing $$\mathcal{S}$$ into $$\mathcal{O}_k$$? It's probably just to break down a potentially complex $$\mathcal{S}$$ into "manageable" single-particle/excitation operators $$\mathcal{O}_k$$. Rendering the calculation easier. And more importantly disentangling an $$n-$$body problem into $$n$$ $$1-$$body problems.

This is similar to working out the mathematical expression of perturbative amplitudes in terms of simple(r) step-by-step Feynman rules for each line & vertex.

### What are the $$\mathcal{O}_k$$ operators?

For this paper (one of the references in yours actually) I found the following half-decent definition:

The $$n$$-particle scattering amplitudes $$\mathcal{A}_n$$ of any four-dimensional quantum field thoery (QFT$$_4$$) can be described as a collection of $$n$$-point correlation functions on the two-sphere ($$S^2$$) with coordinates $$(z,\bar{z})$$: $$\mathcal{A}_n = \langle O_1(E_1, z_1, \bar{z}_1) \dots O_n(E_n, z_n, \bar{z}_n) \rangle,$$ where $$O_k$$ creates (if $$E_k<0$$) or annihilates (if $$E_k>0$$) an asymptotic particle with energy $$|E_k|$$ at the point $$(z_k, > \bar{z}_k)$$ where the particle crosses the asymptotic $$S^2$$ at null infinity ($$\mathcal{I}$$).

Based on the use of the word "asymptotic", it seems to me that your suggestion might be right. You are ignoring the specifics of the interactions, and just looking at $$t\rightarrow -\infty$$ when the particle was created ($$\mathcal{O}_1(z_1, \bar{z}_1)$$) and at $$t\rightarrow \infty$$ when the particle was destroyed ($$\mathcal{O}_1(z_2, \bar{z}_2)$$). I used the same subscript $$1$$ for the same praticle, though the creation and destruction occur at distinct angles $$(z, \bar{z})$$.

### What is the state wrt which the mean is taken?

Because they are talking about "asymptotic" particles, and because they want to make $$n$$ $$1-$$body problems out of an $$n-$$body one, the state is the vacuum.

I.e., you start with nothing and create your single particle $$1$$ from it. Then you destroy so it returns to nothing.

You can use the non-interacting vacuum $$|0\rangle$$, but since it talks about non-Abelian stuff immediately afterwards you might want to use the interacting vacuum $$|\Omega\rangle$$.