Résumé:
In our thesis, we presented the basic tools of the construction of the formal deformation
theory in the general case. Then we illustrated how we can solve some problems of
deformed relativistic quantum mechanics, through the path integral supersymmetric
formalism, where we considered two approaches in the deformed case.
The first approach is related to the presence of a minimal length and its effect on spin
relativistic systems. Hence we suggest the dynamic study of the Dirac oscillator in one
dimension in the momentum by the Feynman standard techniques and Kleinert as well. In
the same context, we also studied the dynamics of a variable particle mass that has halfspin,
subject to the interaction of linear potential, following two different techniques,
namely the Dirac equation in one dimension and in the configurations space and the path
integrals formalism in momentum space. By the direct method, we solved the equation
using the approximation technique of the ordinary quantum mechanics; we obtained the
shift of relativistic energy levels. Concerning the second method, the Green function was
constructed by Feynman approach. In both methods, we found the same energetic levels
quantity at the first order of deformation parameter.
Now, we deal with the second approach which is the introduction of non-commutative
geometry in phase space. We treated the relativistic oscillator case (Klein-Gordon and
Dirac) in the presence of a magnetic field. In addition, we also applied the same formalism
to the Fashbach-Villars equation for spin zero case. When we introduced the Foldy-
Wouthuysen (F-W) transformation to diagonalise the Hamiltonian.
In each application, the propagators are calculated; the wave function and corresponding
energy spectra are derived.
