From [formula omitted] algebra to κ-Lorentz quasigroup. A deformation of relativistic symmetry

Abstract

We consider an algebraic ansatz for the class of nonlinear D=4 Poincaré algebras and show that it contains only the quantum κ- Poincar e ́ (real Hopf) algebras, obtained recently by the contraction of U q (O(3,2)). We derive the explicit formulae for the finite κ-Lorentz transformations generated by the realizations of the κ- Poincar e ́ algebra in D=4 momentum space. These finite κ-Lorentz transformations form a quasigroup, with generalized composition law of the boost parameters (rapidities). We consider further the (2 s+1)-component field realizations with arbitrary spin s and their finite κ-Lorentz transformations. For s= 1 2 we obtain the κ-covariant Dirac equation, derived from the finite κ-Lorentz boost formula. After the coupling of the κ-deformed Dirac equation to the electromagnetic potential we show that in the lowest order (linear) in el/ κ the κ-corrections to the hydrogen atom energy levels vanish but the value g=2 of the electron's magnetic moment is modified ( g=2→ g=2[1+( m el / κ)]. Finally the space-time description of κ-relativistic fields is briefly discussed.