[論文レビュー] Spacetime Quasicrystals

Latham Boyle, Sotirios Mygdalas|arXiv (Cornell University)|Jan 12, 2026

Quasicrystal Structures and Properties被引用数 0

ひとこと要約

The paper generalizes self-similar Euclidean quasicrystals to Minkowski spacetime, constructing the first Lorentzian quasicrystals and exploring their novel features and speculative physical relevance.

ABSTRACT

Self-similar quasicrystals (like the famous Penrose and Ammann-Beenker tilings) are exceptional geometric structures in which long-range order, quasiperiodicity, non-crystallographic orientational symmetry, and discrete scale invariance are tightly interwoven in a beautiful way. In this paper, we show how such structures may be generalized from Euclidean space to Minkowski spacetime. We construct the first examples of such Lorentzian quasicrystals (the spacetime analogues of the Penrose or Ammann-Beenker tilings), and point out key novel features of these structures (compared to their Euclidean cousins). We end with some (speculative) ideas about how such spacetime quasicrystals might relate to reality. This includes an intriguing scenario in which our infinite $(3+1)$D universe is embedded (like one of our spacetime quasicrystal examples) in a particularly symmetric $(9+1)$D torus $T^{9,1}$ (which was previously found to yield the most symmetric toroidal compactification of the superstring). We suggest how this picture might help explain the mysterious seesaw relationship $M_{ m Pl}M_{ m vac}\approx M_{ m EW}^{2}$ between the Planck, vacuum energy, and electroweak scales ($M_{ m Pl}$, $M_{ m vac}$, $M_{ m EW}$).

研究の動機と目的

• Generalize self-similar quasicrystals from Euclidean space to Minkowski spacetime.
• Develop spacetime cut-and-project schemes that yield non-crystallographic symmetry and discrete self-similarity.
• Provide concrete spacetime quasicrystal examples and analyze their unique Lorentzian properties.
• Discuss speculative physical implications and potential connections to fundamental scales and compactifications.

提案手法

• Review and extend Euclidean quasicrystal formalisms (cut-and-project and symmetric cut-and-project) to Lorentzian (spacetime) settings.
• Introduce Lorentzian lattices and their reflection symmetries to construct spacetime quasicrystals.
• Define a spacetime C&P scheme with appropriate windows/weighting to achieve non-crystallographic symmetry and scale features.
• Construct explicit examples in 1+1 dimensions from I_{3,1} and in 3+1 dimensions from II_{9,1}.
• Discuss implications of global versus local scale invariance and self-duality in spacetime quasicrystals.

実験結果

リサーチクエスチョン

• RQ1 How can quasicrystal concepts like quasiperiodicity, non-crystallographic symmetry, and discrete self-similarity be generalized to Minkowski spacetime?
• RQ2 What are the essential features and limitations of a spacetime cut-and-project construction?
• RQ3 Do spacetime quasicrystals admit Lorentzian symmetry enhancements and self-duality, and what are their implications?
• RQ4 What concrete spacetime quasicrystal examples can be realized from known self-dual Lorentzian lattices (e.g., I_{3,1}, II_{9,1})?
• RQ5 What speculative connections could spacetime quasicrystal structures have with fundamental scales and string theory compactifications?

主な発見

• First examples of Lorentzian quasicrystals (spacetime analogues of Penrose/Ammann-Beenker tilings) are constructed.
• Spacetime quasicrystals exhibit a dramatically larger (infinite) non-crystallographic symmetry group, impacting window/weighting choices.
• spacetime quasicrystals lack local scale invariance but can display global scale invariance or self-duality.
• Two explicit examples are given: a 1+1D spacetime quasicrystal from I_{3,1} and a 3+1D family from II_{9,1}.
• A speculative scenario embeds 3+1D spacetime in a symmetric 9+1D torus T^{9,1}, potentially relating to the M_Pl–M_vac–M_EW hierarchy and seesaw relations.