Group Representations of Lorentz Transformations Extended to Superluminal Observers

TL;DR

This paper extends the Lorentz group to include superluminal observers, classifies the resulting representations, and derives new wave equations, including tachyonic and massless types, with implications for quantum field theory.

Contribution

It introduces a novel extension of the Lorentz group for superluminal observers and classifies its unitary irreducible representations, leading to new wave equations in quantum field theory.

Findings

Extended Lorentz group includes superluminal transformations.

Classified unitary irreducible representations of the extended Poincare group.

Derived new wave equations for tachyonic and massless particles.

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