Reflection Positivity and Conformal Symmetry

TL;DR

This paper explores reflection positivity in the context of Lie groups and conformal symmetry, establishing a mathematical framework and applying it to the conformal group of the sphere and half-space models.

Contribution

It develops an abstract theory of reflection positivity using distributions and applies it specifically to the conformal group of the sphere and half-space models.

Findings

Established a framework for reflection positive distributions on Lie groups.

Applied the theory to the conformal group of the sphere.

Connected reflection positivity with conformal symmetry in mathematical physics.

Abstract

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If $\g$ is a Lie algebra with an involutive automorphism $\tau$. Decompose $\g = \fh \oplus \fq = \ker(\tau - \1) \oplus \ker(\tau + \1)$ into $\tau$-eigenspaces and let $\g^c := \fh \oplus i \fq$. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the level of unitary representations. The idea is simple on the Lie algebra level: Let $(\pi,\cH^0)$ be a representation of $\g$ where $\pi$ acts by skew-symmetric operators. Assume that there exists a unitary operator $J$ of order two such that $J \pi J = \pi\circ \tau$ and a $\g$-invariant subspace $\cK^0$ which is {\it $J$-positive}. Then complex linear extension leads to a representation of $\g^c$ on $\cK^0$ by operators which are skew-symmetric with respect to $h_J$, so that we obtain a "unitary" representation of $\g^c$ on the pre-Hilbert space $\cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}$. The aim of this article is twofold. First we discuss reflection positivity in an abstract setting using {\it reflection positive distributions} on the Lie group $G_\tau =G\rtimes {1,\tau}$ and {\it reflection positive distribution vectors} of a unitary representation of $G_\tau$. Then we apply these ideas to the conformal group $\OO_{1,n+1}^+(\R)$ of the sphere $\bS^n$ as well as the the half-space picture mostly used in physics.