Quantum relativistic oscillator. Modifying the Hamiltonian formalism of the relativistic string

A new internal position operator ${\xi }_{\mu }$=-$d_{\mu }$ and a new internal momentum operator ${\pi }_{\mu }$ are defined using an analog of constraint Hamiltonian mechanics. The new position and momentum do not fulfill the usual relativistic Heisenberg commutation relations and both have noncommuting components, but in the nonrelativistic limit they contract into the usual three-dimensional position and momentum. The new momentum ${\pi }_{\mu }$ is connected with an infinite-dimensional generalization ${\Gamma }_{\mu }$ of the Dirac matrices, which together with the intrinsic angular momentum $S_{\mu \nu }$ form an SO(3,2) algebra. A representation of this algebra provides the spectrum of hadron resonances, if hadrons are considered as collective vibrational and rotational excitations. A preliminary comparison between the predictions of this model and the experimental data leads to encouraging results.