Lorentzian Quasicrystals and the Irrationality of Spacetime

Speaker: Sotiris Mygdalas (Perimeter Institute)
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 (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 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 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.