Nets of standard subspaces on Lie groups

DOI10.1016/J.AIM.2021.107715zbMATH Open1487.22015arXiv2006.09832OpenAlexW3154092389WikidataQ115361987 ScholiaQ115361987MaRDI QIDQ2237351

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Publication date: 27 October 2021

Published in: (Search for Journal in Brave)

Abstract: Let G be a Lie group with Lie algebra

m a t h f r a k g

,

h i n f r a k g

an element for which the derivation ad(h) defines a 3-grading of

m a t h f r a k g

and

a u_{G}

an involutive automorphism of G inducing on

m a t h f r a k g

the involution

e^{p i i a d (h)}

. We consider antiunitary representations

U

of the Lie group

G_{a} u = G t i m e s e, a u_{G}

for which the positive cone

C_{U} = x i n m a t h f r a k g : - i p a r t i a l U (x) g e q 0

and

h

span

m a t h f r a k g

. To a real subspace E of distribution vectors invariant under

e x p (m a t h b b R h)

and an open subset

O s u b s e t e q G

, we associate the real subspace

H_{E} (O) s u b s e t e q H

, generated by the subspaces

U (v a r p h i) E

, where

v a r p h i i n C^{i} n f t y_{c} (O, m a t h b b R)

is a real-valued test function on

O

. Then

H_{E} (O)

is dense in

H_{E} (G)

for every non-empty open subset

O s u b s e t e q G

(Reeh--Schlider property). For the real standard subspace

V s u b s e t e q H

, for which

J_{V} = U (a u_{G})

is the modular conjugation and

D e l t a_{V}^{- i t / 2 p i} = U (e x p t h)

is the modular group, we obtain sufficient conditions to be of the form

H_{E} (S)

for an open subsemigroup

S s u b s e t e q G

. If

m a t h f r a k g

is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspacs

H_{E} (O)

,

O s u b s e t e q 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.

Full work available at URL: https://arxiv.org/abs/2006.09832

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