Lorentz-boost eigenmodes

Plane waves and cylindrical or spherical vortex modes are important sets of solutions of quantum and classical wave equations. These are eigenmodes of the energy-momentum and angular-momentum operators, i.e., generators of spacetime translations and spatial rotations, respectively. Here we describe another set of wave modes: eigenmodes of the ``boost momentum'' operator, i.e., a generator of Lorentz boosts (spatiotemporal rotations). Akin to the angular momentum, only one (say, $z)$ component of the boost momentum can have a well-defined quantum number. The boost eigenmodes exhibit invariance with respect to the Lorentz transformations along the z axis, leading to scale-invariant wave forms and steplike singularities moving with the speed of light. We describe basic properties of the Lorentz-boost eigenmodes and argue that these can serve as a convenient basis for problems involving causal propagation of signals.