In the Atiyah problem on configurations of points, one defines smooth complex-valued functions $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ on the configuration space of $n$ distinct points in $\mathbb{R}^3$ (more precisely, $\mathbf{x}_i \in \mathbb{R}^3$ for $i = 1, \ldots, n$ and the $\mathbf{x}_i$ are distinct). The $n$-point function $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ is defined as the normalized determinant of a complex $n$ by $n$ matrix containing the coefficients of $n$ complex polynomials of degree at most $n - 1$ constructed using the pairwise directions between the $\mathbb{x}_i$ and the Hopf map. Please refer, for example, to the article by Atiyah and Sutcliffe entitled "The geometry of point particles" for more detail.

This suggests the following question. Could these $n$-point functions $D$ be the vacuum expectation values of a Euclidean QFT? I had asked a related question on MO, and this post is my attempt to make my thoughts more precise. Here is a link to the previous post:

Is there, mathematically speaking, a QFT with the following properties?

I can check that all the Osterwalder-Schrader (OS) axioms E0-E4 are satisfied with the exceptions of E2, reflection positivity. In the case of E2, I am not sure if the functions $D$ satisfy E2 or not, and I am kind of stuck on how to check if it holds or not. Is it enough to consider finitely many test functions which are delta functions, perhaps?

I know that $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ gets complex conjugated when the $\mathbf{x}_i$ all get reflected with respect to some plane in $\mathbb{R}^3$.

Your help is kindly appreciated.

In the Atiyah problem on configurations of points, one defines smooth complex-valued functions $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ on the configuration space of $n$ distinct points in $\mathbb{R}^3$ (more precisely, $\mathbf{x}_i \in \mathbb{R}^3$ for $i = 1, \ldots, n$ and the $\mathbf{x}_i$ are distinct). The $n$-point function $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ is defined as the normalized determinant of a complex $n$ by $n$ matrix containing the coefficients of $n$ complex polynomials of degree at most $n - 1$ constructed using the pairwise directions between the $\mathbb{x}_i$ and the Hopf map. Please refer, for example, to the article by Atiyah and Sutcliffe entitled "The geometry of point particles" for more detail.

This suggests the following question. Could these $n$-point functions $D$ be the vacuum expectation values of a Euclidean QFT? I had asked a related question on MO, and this post is my attempt to make my thoughts more precise, using the help I got there, particularly from @PedroLauridsenRibeiro. Here is a link to the previous post:

Is there, mathematically speaking, a QFT with the following properties?

I can check that all the Osterwalder-Schrader (OS) axioms E0-E4 are satisfied with the exceptions of E2, reflection positivity. In the case of E2, I am not sure if the functions $D$ satisfy E2 or not, and I am kind of stuck on how to check if it holds or not. Is it enough to consider finitely many test functions which are delta functions, perhaps?

I know that $D(\mathbf{x}_1, \ldots, \mathbf{x}_n)$ gets complex conjugated when the $\mathbf{x}_i$ all get reflected with respect to some plane in $\mathbb{R}^3$.

Your help is kindly appreciated.