The aim of this work is to prove analytically the existence of symmetric
periodic solutions of the family of Hamiltonian systems with Hamiltonian
function H(q_1,q_2,p_1,p_2)=
1/2(q_1^2+p_1^2)+1/2(q_2^2+p_2^2)+ a q_1^4+b
q_1^2q_2^2+c \q_2^4 with three real
parameters a, b and c. Moreover, we characterize the stability of these
periodic solutions as function of the parameters. Also, we find a
first-order analytical approach of these symmetric periodic solutions.
We emphasize that these families of periodic solutions are different
from those that exist in the literature.
The main goal of this work is to study the Dirac oscillator as a quantum field using the canonical formalism of quantum field theory and to develop the canonical quantization procedure for this system in (1 + 1) and (3 + 1) dimensions. This is possible because the Dirac oscillator is characterized by the absence of the Klein paradox and by the completeness of its eigenfunctions.We show that the Dirac oscillator field can be seen as constituted by infinite degrees of freedom which are identified as decoupled quantum linear harmonic oscillators. We observe that while for the free Dirac field the energy quanta of the infinite harmonic oscillators are the relativistic energies of free particles, for the Dirac oscillator field the quanta are the energies of relativistic linear harmonic oscillators.
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