The One- and Two- components Bose charged gas are obtained in a large box. They constitute the neutral bosonic plasma. One obtains the non neutral bosonic plasma by confining identical charged particles of integer spin in Penning or Paul traps. By laser cooling one obtains ultracold plasma in the Paul trap. We studied the quantum degeneracy of non neutral plasma of spin zero identical particles, at high density and lows temperatures in harmonic Paul trap. We used the microscopic theory. The charged meson field was considered to construct the bosons operators, the space of quantum states, the fields operators of creation or annihilation of bosons of impulsion at the point . From these fields operators, we constructed the Hamiltonian operator of second quantization of the plasma and we determined the density matrix, the equation of motion, the Hartree-Fock equation, the elementary excitation specter, the ground state energy ...The density matrix leads to the macroscopic occupation of one quantum state of particle (BEC) in the trap.
