Lorentzian Quasicrystals and the Irrationality of Spacetime

          @misc{ pirsa_PIRSA:25060084,
            doi = {10.48660/25060084},
            url = {https://pirsa.org/25060084},
            author = {Mygdalas, Sotirios},
            keywords = {},
            language = {en},
            title = {Lorentzian Quasicrystals and the Irrationality of Spacetime},
            publisher = {Perimeter Institute},
            year = {2025},
            month = {jun},
            note = {PIRSA:25060084 see, \url{https://pirsa.org}}
          }
          

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 subspace to form the vertices of the quasicrystal. However, most of the known examples of $\mathsf{CNP}$ quasi-tilings are Euclidean. In this talk, after presenting the main ingredients of the Euclidean prescription, we will extend it to Lorentzian spacetimes and develop Lorentzian $\mathsf{CNP}$. This will allow us to discuss the first ever examples of Lorentzian 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 comment on its relevance to quantum geometry and quantum gravity; first, as a conformal Lorentzian structure of no intrinsic scale, and second through the connection of quasicrystals to quantum error-correcting codes.