Towards a group structure for superluminal velocity boosts

Canonical subluminal Lorentz boosts have a clear geometric interpretation. They can be neatly expressed as hyperbolic rotations, that leave both the family of $2$-sheet hyperboloids within the light cone, and the family $1$-sheet hyperboloids exterior to it, invariant. In this work, we construct a map between the two families of hypersurfaces and interpret the corresponding operators as superluminal velocity boosts. Though a physical observer cannot `jump' the light speed barrier, to pass from one regime to the other (at least not classically), the existence of superluminal motion does not, by itself, generate paradoxes. The implications of this construction for recent work on the `quantum principle of relativity', proposed by Dragan and Ekert, are discussed. The geometric picture reproduces their `superboost' operator in $(1+1)$ dimensions but generalises to $(1+3)$ dimensions in a very different way. This leaves open an important possibility, which appears to be closed to existing models, namely, the possibility of embedding the superluminal boosts within a group structure, without generating additional unwanted phenomenology, that contradicts existing experimental results. We prove that the set containing both subluminal and superluminal boosts forms a group, in $(1+1)$-dimensional spacetimes, and outline a program to extend these results to higher-dimensional geometries.