Coherent states, vacuum structure and infinite component relativistic wave equations

It is commonly claimed in the recent literature that certain solutions to wave equations of positive energy of Dirac-type with internal variables are characterized by a non-thermal spectrum. As part of that statement, it was said that the transformations and symmetries involved in equations of such type corresponded to a particular representation of the Lorentz group. In this paper, we give the general solution to this problem emphasizing the interplay between the group structure, the corresponding algebra and the physical spectrum. This analysis is completed with a strong discussion and proving that: (i) the physical states are represented by coherent states; (ii) the solutions in [Yu. P. Stepanovsky, Nucl. Phys. B (Proc. Suppl.) 102 (2001) 407–411; 103 (2001) 407–411] are not general, (iii) the symmetries of the considered physical system in [Yu. P. Stepanovsky, Nucl. Phys. B (Proc. Suppl.) 102 (2001) 407–411; 103 (2001) 407–411] (equations and geometry) do not correspond to the Lorentz group but to the fourth covering: the Metaplectic group [Formula: see text].