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Causal quantum-mechanical localization observables in lattices of real projections

Gandalf Lechner, Ivan Romualdo de Oliveira

TL;DR

The work shows that relocating quantum localization from the projection lattice $\mathcal{P}(\mathcal{H})$ to the real-projection lattice $\mathcal{P}_{\mathbb{R}}(\mathcal{H})$ with symplectic complementation enables causal, Poincaré-covariant localization observables, avoiding classic No-Go theorems. It constructs a spacetime localization observable via the Brunetti-Guido-Longo (BGL) map for positive-energy Poincaré representations, and proves uniqueness under natural axioms; it also establishes a Gleason-type result in the symplectic setting showing the absence of universal probability measures, while demonstrating a cluster-like approximate additivity at macroscopic scales. The framework ties causality to modular localization and Lorentz invariance, producing a structure that mirrors core quantum-field-theoretic features, including wedges and standard subspaces. Practically, the theory provides an essentially unique mechanism for localization in a relativistic setting and clarifies how Newton-Wigner-type localization emerges only approximately for large separation, with implications for interpreting localization in relativistic quantum systems. $${}$$

Abstract

Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the lattice of real linear projections with symplectic complementation, the well-known No-Go theorems of Hegerfeldt and Malament no longer apply: Causal and Poincaré covariant localization observables exist. In this setting, several features of quantum field theory, such as Lorentz symmetry and modular localization, emerge automatically. In the case of a particle described by a massive positive energy representation of the Poincaré group, the Brunetti-Guido-Longo map defines a spacetime localization observable that is unique under some natural further assumptions. Regarding possible probabilistic interpretations of such a structure, a Gleason theorem and a cluster theorem for symplectic complements are established. These imply that evaluating such localization observables in states yields a fuzzy probability measure that fails to be a measure because it is not additive. However, for separation scales that are large in comparison to the Compton wavelength, the emerging modular localization picture is essentially additive and approximates the one of Newton-Wigner.

Causal quantum-mechanical localization observables in lattices of real projections

TL;DR

The work shows that relocating quantum localization from the projection lattice to the real-projection lattice with symplectic complementation enables causal, Poincaré-covariant localization observables, avoiding classic No-Go theorems. It constructs a spacetime localization observable via the Brunetti-Guido-Longo (BGL) map for positive-energy Poincaré representations, and proves uniqueness under natural axioms; it also establishes a Gleason-type result in the symplectic setting showing the absence of universal probability measures, while demonstrating a cluster-like approximate additivity at macroscopic scales. The framework ties causality to modular localization and Lorentz invariance, producing a structure that mirrors core quantum-field-theoretic features, including wedges and standard subspaces. Practically, the theory provides an essentially unique mechanism for localization in a relativistic setting and clarifies how Newton-Wigner-type localization emerges only approximately for large separation, with implications for interpreting localization in relativistic quantum systems.

Abstract

Quantum-mechanical observables for spatial and spacetime localization are considered from a lattice-theoretic perspective. It is shown that when replacing the lattice of all complex orthogonal projections underlying the Born rule by the lattice of real linear projections with symplectic complementation, the well-known No-Go theorems of Hegerfeldt and Malament no longer apply: Causal and Poincaré covariant localization observables exist. In this setting, several features of quantum field theory, such as Lorentz symmetry and modular localization, emerge automatically. In the case of a particle described by a massive positive energy representation of the Poincaré group, the Brunetti-Guido-Longo map defines a spacetime localization observable that is unique under some natural further assumptions. Regarding possible probabilistic interpretations of such a structure, a Gleason theorem and a cluster theorem for symplectic complements are established. These imply that evaluating such localization observables in states yields a fuzzy probability measure that fails to be a measure because it is not additive. However, for separation scales that are large in comparison to the Compton wavelength, the emerging modular localization picture is essentially additive and approximates the one of Newton-Wigner.
Paper Structure (20 sections, 23 theorems, 80 equations, 5 figures)

This paper contains 20 sections, 23 theorems, 80 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Involution lattices and logics
  3. Basic definitions
  4. Space- and spacetime logics
  5. Logics and involution lattices on a Hilbert space
  6. Effects and spectral order
  7. The involution lattice of von Neumann algebras
  8. Localization Observables
  9. General observables on involution lattices
  10. Covariant spatial and spacetime localization observables
  11. Translationally covariant causal localization observables
  12. No localization observables in $\mathcal{P}(\mathcal{H})$
  13. Constraints on causal localization observables in $\mathcal{P}_{\mathbb{R}}(\mathcal{H})$
  14. Lorentz symmetry and wedges
  15. Spacetime localization observables in $\mathcal{P}_{\mathbb{R}}(\mathcal{H})$
  16. ...and 5 more sections

Key Result

Lemma 2.7

The causal completion map is an injective $\sigma$-additive logic homomorphism, making $\operatorname{c}(\EuScript{B}(\mathbb{R}^d))$ a Boolean sublogic of $\mathcal{C}(\mathbb{R}^{d+1})$. It is covariant w.r.t. the actions of the Euclidean group $\mathbf{E}(d)$ on $\EuScript{B}(\mathbb{R}^d)$ and $\mathcal{C}(\mathbb{R}^{d+

Figures (5)

  • Figure 1: a) $\mathcal{O}_1$ and $\mathcal{O}_2$ are disjoint but not separated. b),c) Failure of distributivity: $\mathcal{O}_1\wedge\mathcal{O}_2=\emptyset=\mathcal{O}_1\wedge\mathcal{O}_3$ (b), but $\mathcal{O}_1\wedge(\mathcal{O}_2\vee\mathcal{O}_3)\neq\emptyset$ (c). In these figures, the time axis is vertical and the spatial axis horizontal as usual.
  • Figure 2: The causal completion map $\operatorname{c}(A)=(\{0\}\times A)"$.
  • Figure 3: As the spatial regions $A+|t|{\mathbb B}$ and $B$ remain separated for small times, $\tau_t(f(A))$ remains separated from $f(B)$ for a causal spatial localization observable $f$.
  • Figure 4: Illustration of the configurations in \ref{['eq:Ivstfat']} and \ref{['eq:geometry']}.
  • Figure 5: The right wedge $W=W(e_1)$, the orbits of its boost group $\Lambda_{e_1}$, and the reflection $j_{e_1}$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5: Spatial localization regions, the Boolean logic of a Cauchy surface
  • Example 2.6: Spacetime localization regions, the logic of Minkowski spacetime
  • Lemma 2.7
  • proof
  • Example 2.8: Orthogonal projections, the logic of quantum mechanics
  • Example 2.9: Real orthogonal projections
  • ...and 50 more