Spacetime Quasicrystals

TL;DR

The paper extends the theory of self-similar Euclidean quasicrystals to Minkowski spacetime by constructing Lorentzian quasicrystals via a spacetime cut-and-project framework using Lorentzian lattices. It develops both a Euclidean foundation (lattices, Coxeter theory, and algebraic number theory) and a Lorentzian generalization, revealing that spacetime quasicrystals exhibit infinite orientational symmetry, lack local scale invariance in the window approach, and can display self-duality or global scale invariance through weighting schemes. Two explicit spacetime examples are constructed: a (1+1)-dimensional quasicrystal from the odd self-dual lattice ${I}_{3,1}$ and a (3+1)-dimensional family from the even self-dual lattice ${ m II}_{9,1}$, highlighting rich structure and potential cosmological implications. The authors speculate on physical relevance, proposing a picture in which our 3+1D universe densely fills a symmetric $T^{9,1}$ torus and may shed light on hierarchy relations such as $M_{ m Pl}M_{ m vac} oughly\= M_{ m EW}^{2}$, connecting discretized spacetime, scale invariance, and quantum gravity concepts.”

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 $M_{\rm Pl}M_{\rm vac}\approx M_{\rm EW}^{2}$ between the Planck, vacuum energy, and electroweak scales ($M_{\rm Pl}$, $M_{\rm vac}$, $M_{\rm EW}$).

Figures (14)

• Figure 1: Left, a patch of the Penrose tiling penrose1974role given by rhombs in purple, underlaid by its "inflation" in pink. Right top, the thin (red) and thick (blue) Penrose rhombs, along with their edge-matching rules and inflation rules. Right bottom, Conway's version of the tiles: "darts" (red) and "kites" (blue) are given with their inflation rules. The two versions are equivalent since tilings built from kites and darts are mutually locally derivable from tilings by rhombs. For an introduction to the Penrose tiling, see Ref. gardner1977extraordinary.
• Figure 2: The Coxeter-Dynkin diagrams for the irreducible Coxeter groups of finite/spherical type (left panel) and affine/planar type (right panel). The non-crystallographic cases ($H_{4}$, $H_{3}$ and $I_{2}^{n}$) are grouped in the box at lower left. Each affine diagram (at right) may be obtained by adding a single "extending root" to the corresponding finite crystallographic diagram (at left); or, alternatively, by removing a single extending node (any of the black nodes) from any affine diagram (at right) we obtain the corresponding finite crystallographic diagram (at left).
• Figure 3: The "Coxeter complex" showing mirrors (geodesic curves) and images of the fundamental domain (triangular regions) for various Coxeter groups: https://commons.wikimedia.org/wiki/File:Icosahedral_reflection_domains.png: the spherical (finite) $[5,3]$ (icosahedral) group; https://commons.wikimedia.org/wiki/File:Tiling_Dual_Semiregular_V4-6-12_Bisected_Hexagonal.svg: the affine (planar) $[6,3]$ group; https://en.wikipedia.org/wiki/File:Hyperbolic_domains_642.png: the hyperbolic $[6,4]$ group.
• Figure 4: Obtaining non-crystallographic subgroups of finite Coxeter groups by folding. Top row: the foldings yielding the $H_{4}$ and $I_{2}^{30}$ subgroups of $E_{8}$. Middle row: the foldings yielding the $H_{3}$ and $I_{2}^{10}$ subgroups of $D_{6}$. Bottom row: the foldings yielding the $I_{2}^{5}$, $I_{2}^{8}$ and $I_{2}^{12}$ subgroups of $A_{4}$, $B_{4}$ and $F_4$ respectively.
• Figure 5: An illustration of the C&P scheme explained in Sec. \ref{['subsec:CNP']}. Left panel: constructing the quasicrystal $\Lambda_{{\rm ph}}^{{\bf t},W}$. For each point ${\bf x}\in\Lambda^{{\bf t}}$, if the "internal" point ${\bf x}_{\rm in}=\Pi_{\rm in}{\bf x}$ lies inside the window ${\cal W}$, we include the "physical" point ${\bf x}_{\rm ph}=\Pi_{\rm ph}{\bf x}$ in the quasicrystal $\Lambda_{{\rm ph}}^{{\bf t},{\cal W}}$ (see Sec. Sec. \ref{['subsec:CNPconstruction']} for details, included the weighted variant of the C&P scheme). Right panel: The corresponding Fourier transform $\widehat{\Lambda}_{{\rm ph}}^{{\bf t},W}$ is produced by a closely analogous C&P scheme: for each point ${\bf p}\in\Lambda_{\ast}$ (the dual lattice), the Fourier transform includes a delta function located at the "physical" momentum ${\bf p}_{\rm ph}=\Pi_{\rm ph}{\bf p}$, with a coefficient given by $\widehat{W}(-{\bf p}_{in})$, where $\widehat{W}$ is the Fourier transform of the "top-hat" or "indicator" function in the shape of the window ${\cal W}$, which is evaluated at the "internal" momentum ${\bf p}_{\rm in}=\Pi_{\rm in}{\bf p}$. As a display convention, one can represent the "height" of the delta function at ${\bf p}_{\rm ph}$ by a circular disk centered at ${\bf p}_{\rm ph}$, with area proportional to the coefficient $\widehat{W}(-{\bf p}_{\rm in})$, as indicated by the red circles in the right panel.