Quasi exactly solvable matrix models in sl(n)

We reconsider the quasi exactly solvable matrix models constructed recently by R. Zhdanov. The 2$\times$2 matrix operators representing the algebra sl(2) are generalized to matrices of arbitrary dimension and a similar construction is achieved for the algebra sl(n).Comment: LaTeX, 8 page

"Isochronic" dynamical systems and nullification of amplitudes

We construct the set of theories which share the property that the tree-level amplitudes nullify even if both initial and final states contain the same type of particles. The origin of this phenomenon lies in the fact that the reduced classical dynamics describes isochronic systems.Comment: 12 pages, 3 figure

Lorentz transformations, sideways shift and massless spinning particles

Recently (Phys. Rev. Lett. 114 (2015), 210402) the influence of the so called "Wigner translations" (more generally-Lorentz trans- formations) on circularly polarized Gaussian packets ( providing the solution to Maxwell equations in paraxial approximation) has been studied. It appears that, within this approximation, the Wigner trans- lations have an efect of shifting the wave packet trajectory parallel to itself by an amount proportional to the photon helicity. It has been suggested that this shift may result from specific properties of the algebra of Poincare generators for massless particles. In the present letter we describe the general relation between transformation proper- ties of electromagnetic field on quantum and classical levels. It allows for a straightforward derivation of the helicity-dependent transforma- tion rules. We present also an elementary derivation of the formula for sideways shift based on classical Maxwell theory. Some comments are made concerning the generalization to higher helicities and the relation to the coordinate operator defined long time ago by Pryce.Comment: 13 pages; no figures; section "Conclusions" added; missing normalization factor in eq.(20) inserted; final form to appear in Physics Letters