Parametrization of the directions of the dynamic spin-quantization axis for rho, Ksup(ω) and Δsup(++) resonances by an arbitrary pencil of Lobachevsky's straight lines

The best of the published world's data on the t-dependence of the density matrix elements for rho, Ksup(ω), Δ++ -resonances at 2.8/17 GeV have been analyzed in the patterns of the Lobachevsky velocity space, which plays the role of the invariant phase space in high energy physics. A natural from the point of view of the velocity space parametrization of the invariant Donohue-Hogaasen axis Z', known as the dynamic spin quantization axis, by pencils of straight lines on the non-Euclidean plane of resonance production has been suggested by one of the authors. It is realised by an arbitrary pencil in the range of momentum transfers √-t=0/1 (0/2) GeV/c. The straight lines of this pencil are transformed one into another by a concrete subgroup of motions in this space which is equivalent to the corresponding subgroup of Lorentz transformations. This parametrization permits one to use the eigenvalues of the density matrix (Donohue-Hogaasen parameters α, β, UPSILON) for invariant description of polarization states of resonances. Their t-dependence has a simple and dynamically definite structure. The results, obtained by invariant spin quantization, significantly differ from those obtained by non-invariant spin quantization in the Jackson direction and in the reaction CMS direction