Relativistic crystallographic point groups in two dimensions

Abstract

Abstract Two-dimensional relativistic point groups are investigated and various equivalence classes of them are explicitly given. It is shown that every two-dimensional relativistic point group can always be generated by at most three elements (the total inversion, a proper Lorentz transformation of infinite order, and a mirror, i.e an improper Lorentz transformation). There are 7 abstract point groups (C1, C2, C∞, D2, D∞, C∞ × C2 and D∞ × C2) divided into an infinite number of geometric (or R- equivalent) and arithmetic (or Z-equivalent) crystal classes, which are indicated. The relativistic geometric crystal classes (conjugated crystallographic subgroups of the Lorentz group) are also given. All elements of GL(2,Z) that can be interpreted as crystallographic transformations of the Minkowskian metric space are discussed. The role of point group symmetry in physical systems having a crystallographic structure in space and time is considered and the importance of point groups for recognizing such systems in nature is underlined.