Semiscalar representations of the Lorentz group and propagation equations

We construct semiscalar linear representations of the inhomogeneous Lorentz group by considering the invariance of linear propagation equations. There is only one semiscalar representation, and the most general linear propagation equation that admits this Lorentz representation is a telegrapher/Maxwell-Cattaneo type equation, whose elementary solutions propagate at the speed of light. Under a Lorentz boost along the x¹ axis, the propagated field variable transforms as U'=U exp q( gamma ( upsilon )( upsilon )x¹-c²t). If one imposes U'=U, then the Lorentz boost of the propagation equation acquires a velocity-dependent convection-type term. In the Newtonian limit c to infinity , the equation reduces to the Fourier heat equation, and previous results on semiscalar representations of the Galilean group are regained.