Group Representations of Lorentz Transformations Extended to Superluminal Observers

TL;DR

This work extends the proper orthochronous Lorentz group to include superluminal boosts, forming the extended Lorentz group $\mathcal{L}_{\text{ext}}$ via a Klein four extension. The extended Poincaré group $\mathcal{P}_{\text{ext}}$ is then classified using Mackey induction, revealing a unified treatment of non-lightlike orbits and a doubled, interlinked structure for lightlike (massless) orbits through $SO(2,1)$-type little groups. Differentiating the extended representations yields wave equations for all sectors, including tachyonic and novel massless representations, derived from the Casimir operators. The paper further argues that parity-asymmetric phenomena in electroweak decays can be understood as kinematical consequences of the extended group, with $SO(2,1)$ covariance playing the central role. This framework offers a coherent, representation-theoretic lens on tachyonic physics and extended spacetime symmetries with potential implications for fundamental symmetries and particle phenomenology.

Abstract

We construct an extension of the proper orthochronous Lorentz group that includes space-time transformations for observers moving with superluminal relative velocities in arbitrary direction. This extension is generated by a realization of the Klein four group depending on polar and azimuthal angles identifying a spatial direction and is obtained with matrices representing infinite velocity limits of superluminal Lorentz boosts. The resulting group has the same identity component of the whole Lorentz group O(3,1) but involutive operators corresponding to an infinite speed boost and its negative in place of parity and time reversal. Different spatial directions in the definition of Klein group realization give rise to equivalent group extensions. We then define the extended Poincare group including translations and classify its unitary irreducible representations (UIRs). The resulting UIRs are induced from Wigner's UIRs of standard Poincare group and depend on the action of the extended Lorentz group defined on momentum space. UIRs corresponding to non lightlike orbits restrict to the ordinary Poincare subgroup as a multiplicity one direct sum of a massive forward, a massive backward and a tachyonic Wigner UIR while for lightlike orbits as two inequivalent direct sum representations combiningh linearly a forward and backward massless Wigner UIR. We then derive wave equations corresponding to solutions of the Casimir eigenvalue problem of Poincare algebra obtained differentiating the above representations. This set of equations contains all the wave equations known to date in quantum field theory together with new wave equations describing tachyonic behaviour and a new class of massless representations. We finally show that tachyonic wave functions provide a relevant representation theoretic tool for interpretation of parity violation phenomena in quantum field theory