Nets of standard subspaces on Lie groups

Document Type


Publication Date



Let G be a Lie group with Lie algebra g, h∈g an element for which the derivation ad h defines a 3-grading of g and τG an involutive automorphism of G inducing on g the involution eπiadh. We consider antiunitary representations (U,H) of the Lie group Gτ=G⋊{idG,τG} for which the positive cone CU={x∈g:−i∂U(x)≥0} and h span g. To a real subspace E⊆H−∞ of distribution vectors invariant under exp⁡(Rh) and an open subset O⊆G, we associate the real subspace HE(O)⊆H, generated by the subspaces U−∞(φ)E, where φ∈Cc∞(O,R) is a real-valued test function on O. Then HE(O) generates the complex Hilbert space HE(G):=HE(G)+iHE(G)‾ for every non-empty open subset O⊆G (Reeh–Schlieder 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 g is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspaces HE(O), O⊆G, satisfying the Bisognano–Wichmann property in a suitable sense. 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.

Publication Source (Journal or Book title)

Advances in Mathematics

This document is currently not available here.