2020/06/16 by Neeb, Karl-Hermann, Olafsson, Gestur · 1 citation
#22E45 #81R05 #81T05 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2006.09832
Let G be a Lie group with Lie algebra \mathfrakg, h ∈ \frakg an element for which the derivation ad(h) defines a 3-grading of \mathfrakg and τG an involutive automorphism of G inducing on \mathfrakg the involution eπi ad(h). We consider antiunitary representations U of the Lie group Gτ= G \rtimes \e,τG\ for which the positive cone CU = \ x ∈ \mathfrakg : -i ∂ U(x) ≥ 0\ and h span \mathfrakg. To a real subspace E of distribution vectors invariant under exp(ℝ h) and an open subset O ⊆ G, we associate the real subspace HE(O) ⊆ H, generated by the subspaces U(φ)E, where φ∈ C^∞c(O,ℝ) is a real-valued test function on O. Then HE(O) is dense in HE(G) for every non-empty open subset O ⊆ G (Reeh--Schlider property). For the real standard subspace V ⊆ H, for which JV = U(τG) is the modular conjugation and ΔV-it/2π = U(exp th) is the modular group, we obtain sufficient conditions to be of the form HE(S) for an open subsemigroup S ⊆ G. If \mathfrakg is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspacs HE(O), O ⊆ G, satisfying the Bisognano--Wichman property for some domains O. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag--Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.