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Orthogonal pairs of Euler elements II: Geometric Bisognano--Wichmann and Spin--Statistics Theorems

Vincenzo Morinelli, Karl-Hermann Neeb, Gestur Olafsson

Abstract

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element $h$, characterized by the property that $ad h$ is diagonalizable with eigenvalues in $\{-1, 0, 1\}$. These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements $(h, k)$ orthogonal if $e^{i π\operatorname{ad} h}(k) = -k.$ Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.

Orthogonal pairs of Euler elements II: Geometric Bisognano--Wichmann and Spin--Statistics Theorems

Abstract

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element , characterized by the property that is diagonalizable with eigenvalues in . These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements orthogonal if Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.

Paper Structure

This paper contains 24 sections, 23 theorems, 199 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abstract Euler wedges
  4. Orthogonal pairs of Euler elements
  5. Examples
  6. Space- and timelike orthogonal pairs
  7. Geometric Bisognano--Wichmann and Spin--Statistics
  8. Nets of standard subspaces on abstract wedges
  9. Results on nets of standard subspaces
  10. Nets of standard subspaces on causal spacetimes
  11. On 1+2-dimensional Minkowski space
  12. On two-dimensional conformal nets
  13. Applications to nets of von Neumann algebras
  14. Results on nets of von Neumann algebras
  15. Nets of von Neumann algebras on causal spacetimes
  16. ...and 9 more sections

Key Result

Theorem 2.5

(MNO25)($Z_3$-Theorem) Let $G$ be a connected Lie group with Lie algebra ${\mathfrak g}$ and $h \in {\mathfrak g}$ a symmetric Euler element for which the Lie algebra involution $\tau_h^{\mathfrak g}$ integrates to $G$. Then $Z_3(G)$ is generated by finitely many elements of the form $\zeta_{h,k} :=

Figures (4)

  • Figure 1: The Minkowski space $M_0$ in $\widetilde{M}^{1,1}={\mathbb R}^2$. The dark grey region is the Rindler wedge $W_R$, and the light grey region is the left wedge $W_L$.
  • Figure 2: The light grey region is the the forward light cone $V_+$ and the dark grey region is the strip $S_+$. Both wedge regions are contained in $M_0$.
  • Figure 3: $M^{1,1}$ is obtained by identifying the dotted and the dashed edges, respectively, of $M_0\subset\widetilde{M^{1,1}}$. In the first picture, the dark gray region is $\mathcal{D}_{(-k_0,-k_0)}$ and the light grey region is $\mathcal{D}_{(k_0,k_0)}$. In the second picture the dark gray region is $\mathcal{D}_{(-k_0,k_0)}$ and the light grey region is $\mathcal{D}_{(k_0,-k_0)}$.
  • Figure 4: The cylinder is obtained by identifying the dotted lines in $\widetilde{M}^{1,1}$. The dark and the light grey wedge regions are identified because they are transformed into each other by $r_s(2\pi)$; see \ref{['eq:rs']}.

Theorems & Definitions (55)

  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • Example 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • Definition 3.1
  • ...and 45 more