The explicit form of the unitary representation of the Poincar\'e group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators

Abstract

We geometrically derive the explicit form of the Unitary representation of the Poincare group and use it to apply speed-of-light boosts to simple polarization basis to end up with Hawton-Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while Pryce connection is metric semi-symmetric, the flat Hawton-Baylis connection is not semi-symmetric. Finally we discuss localizability of photon states localized on closed loops and show that photon states on the circle, both unnormalized improper states and finite norm wave packet smeared over washer-like regions are strictly localized with respect to Hawton-Baylis operators with commuting components and also with respect to the noncommutative Jauch-Piron-Amrein POV measure.