# Reflection positivity on spheres

## TL;DR

This paper constructs a reflection positive Hilbert space on spheres, leading to irreducible unitary spherical representations of the Lorentz group, bridging Euclidean and relativistic quantum field theories via complex domains.

## Contribution

It specializes a reflection positivity construction to spheres and characterizes the resulting unitary representations of the Lorentz group, connecting geometric domains with quantum field theory.

## Key findings

- Constructed a reflection positive Hilbert space on spheres.
- Derived irreducible unitary spherical representations of the Lorentz group.
- Linked geometric domains with quantum field theoretical structures.

## Abstract

In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $\mathbb{S}^n$. We determine the resulting Osterwalder--Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group $G^c = \mathrm{O}_{1,n}(\mathbb{R})^{\uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $\Xi$, known as the crown of the hyperboloid, containing a half-sphere $\mathbb{S}^n_+$ and the hyperboloid $\mathbb{H}^n$ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of $G^c$ in spaces of holomorphic functions on $\Xi$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.