Lorentzian Quasicrystals and the Irrationality of Spacetime

Abstract

Ordered structures that tile the plane in an aperiodic fashion - thus lacking translational symmetry - have long been considered in the mathematical literature. A general method for the construction of quasicrystals is known as *cut-and-project* ($\mathsf{CNP}$ for short), where an irrational slice ''cuts'' a higher-dimensional space endowed with a lattice and suitably chosen lattice points are further ''projected'' down onto the slice to form the vertices of the quasicrystal. However, all of the known examples of $\mathsf{CNP}$ quasicrystals are Euclidean. In this talk, after presenting the main ingredients of the Euclidean prescription, we will extend it to Lorentzian spacetimes and develop Spacetime $\mathsf{CNP}$. This will allow us to discuss the first-ever examples of spacetime quasicrystals, one in (1+1)- and another in (1+3)-dimensional spacetime. Finally, we will argue why the latter construction might be relevant for *our Lorentzian spacetime*. In particular, we shall appreciate how the picture of a quasi-crystalline spacetime could provide a potentially new string-compactification scheme that can naturally accommodate for the hierarchy problem and the smallness of our cosmological constant. Lastly, we will briefly comment on its relevance to quantum gravity; first, as a conformal Lorentzian structure of no intrinsic scale, and second through the connection of quasicrystals to quantum error-correcting codes.