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A proposal for the algebra of a novel noncommutative spacetime

Markus B. Fröb, Albert Much, Kyriakos Papadopoulos

TL;DR

This work provides a mathematically controlled construction of a Lorentz-invariant noncommutative spacetime by building a Weyl algebra from Lorentz-invariant coordinate commutators and employing a Krein-space quantization with a Dereziński–Meissner–type state. The coordinate operators are realized in a GNS representation, enabling a Lorentzian distance functional $D(\chi_p,\chi_q)$ and a causal measure that remain well-defined at the quantum level and converge to classical Minkowski geometry in the appropriate limit. The framework reveals Planck-scale corrections to distances and a fuzzy, measurement-dependent causal structure that reduces to microcausality in the strict localization limit. It also links to the EPS program by showing how spacetime geometry can be recovered from algebraic data and highlights potential implications for spacetime topology and causal structure at the quantum level. Overall, the paper offers a concrete model for quantum spacetime where Lorentzian geometry emerges from noncommutative algebra, with quantifiable, localization-dependent quantum corrections accessible to phenomenology.

Abstract

We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field theory, we systematically construct a quantum spacetime algebra whose geometric and causal properties are derived from first principles. Using the Weyl algebra formalism and the Gelfand--Naimark--Segal (GNS) construction, we rigorously define operator-valued coordinates that respect Lorentz symmetry and encode quantum gravitational effects through nontrivial commutation relations. We show how the emergent quantum spacetime exhibits minimal length effects, which deliver both classical Minkowski distances and quantum corrections proportional to the Planck length squared. Furthermore, we establish that noncommutativity respects a fuzzy form of causality, where the quantum causal structure gives back the light cone in the classical limit, vanishing for spacelike separations and encoding a time orientation for timelike intervals.

A proposal for the algebra of a novel noncommutative spacetime

TL;DR

This work provides a mathematically controlled construction of a Lorentz-invariant noncommutative spacetime by building a Weyl algebra from Lorentz-invariant coordinate commutators and employing a Krein-space quantization with a Dereziński–Meissner–type state. The coordinate operators are realized in a GNS representation, enabling a Lorentzian distance functional and a causal measure that remain well-defined at the quantum level and converge to classical Minkowski geometry in the appropriate limit. The framework reveals Planck-scale corrections to distances and a fuzzy, measurement-dependent causal structure that reduces to microcausality in the strict localization limit. It also links to the EPS program by showing how spacetime geometry can be recovered from algebraic data and highlights potential implications for spacetime topology and causal structure at the quantum level. Overall, the paper offers a concrete model for quantum spacetime where Lorentzian geometry emerges from noncommutative algebra, with quantifiable, localization-dependent quantum corrections accessible to phenomenology.

Abstract

We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field theory, we systematically construct a quantum spacetime algebra whose geometric and causal properties are derived from first principles. Using the Weyl algebra formalism and the Gelfand--Naimark--Segal (GNS) construction, we rigorously define operator-valued coordinates that respect Lorentz symmetry and encode quantum gravitational effects through nontrivial commutation relations. We show how the emergent quantum spacetime exhibits minimal length effects, which deliver both classical Minkowski distances and quantum corrections proportional to the Planck length squared. Furthermore, we establish that noncommutativity respects a fuzzy form of causality, where the quantum causal structure gives back the light cone in the classical limit, vanishing for spacelike separations and encoding a time orientation for timelike intervals.
Paper Structure (7 sections, 4 theorems, 48 equations)

This paper contains 7 sections, 4 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Quantum theory of noncommutative coordinates
  3. The Weyl algebra
  4. Krein space
  5. Dereziński--Meissner state
  6. The noncommutative spacetime
  7. Distance functional

Key Result

Lemma 2.2

For any $f \neq 0$ for which $\sigma(f,g) \neq 0$ for at least one $g \in H$, we have ${\mathopen{}\mathclose{\left\lVert{ W(f) - \mathbbm{1} }\right\rVert}} = 2$.

Theorems & Definitions (14)

  • Definition 2.1: Weyl Algebra
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 4 more