2012/06/10 by Karl-Hermann Neeb, Neeb, Karl-Hermann, Gestur Olafsson +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT
paper · pdf · doi:10.48550/arxiv.1206.2039
arxiv created 2013/06/17 · arxiv updated 2013/06/18
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 τ. Decompose \g = \fh ⊕ \fq = ker(τ- \1) ⊕ ker(τ+ \1) into τ-eigenspaces and let \gc := \fh ⊕ 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 (π,\cH0) be a representation of \g where π acts by skew-symmetric operators. Assume that there exists a unitary operator J of order two such that J πJ = π∘ τ and a \g-invariant subspace \cK0 which is \it J-positive. Then complex linear extension leads to a representation of \gc on \cK0 by operators which are skew-symmetric with respect to hJ, so that we obtain a "unitary" representation of \gc on the pre-Hilbert space \cKJ0 := \cKJ/v \cK0 hJ(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 Gτ=G\rtimes 1,τ and \it reflection positive distribution vectors of a unitary representation of Gτ. Then we apply these ideas to the conformal group \OO1,n+1+(\R) of the sphere \bSn as well as the the half-space picture mostly used in physics.