Nets of standard subspaces on Lie groups

TL;DR

This paper constructs nets of standard subspaces on Lie groups with specific symmetries, leading to models of free quantum fields on causal homogeneous spaces, and verifies their properties in certain semisimple cases.

Contribution

It introduces a new framework for associating nets of standard subspaces to Lie groups with 3-gradings, extending quantum field theory models to new geometric settings.

Findings

Established the Reeh–Schlider property for the constructed nets.

Derived conditions for nets to be of the form H_E(S) on open subsemigroups.

Verified criteria for nets satisfying the Bisognano–Wichman property in semisimple Lie groups.

Abstract

Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $\tau_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{\pi i ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_\tau = G \rtimes \{e,\tau_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspace $H_E(O) \subseteq H$, generated by the subspaces $U(\varphi)E$, where $\varphi \in C^\infty_c(O,\mathbb{R})$ is a real-valued test function on $O$. Then $H_E(O)$ is dense in $H_E(G)$ for every non-empty open subset $O \subseteq G$ (Reeh--Schlider property). For the real standard subspace $V \subseteq H$, for which $J_V = U(\tau_G)$ is the modular conjugation and $\Delta_V^{-it/2\pi} = U(\exp th)$ is the modular group, we obtain sufficient conditions to be of the form $H_E(S)$ for an open subsemigroup $S \subseteq G$. If $\mathfrak{g}$ is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspacs $H_E(O)$, $O \subseteq G$, 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.