The quantum and Klein-Gordon oscillators in a non-commutative complex space and the thermodynamic functions

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In this work we study the quantum and Klein-Gordon oscillators in non-commutative complex space. We show that the quantum oscillator in non-commutative complex space obeys an equation similar to the equation of motion of an electron with spin in a commutative space in an external uniform magnetic field. Therefore the wave function ψ(z,z) takes values in C⁴, spin up, spin down, particle, antiparticle, a result which is obtained by the Dirac theory. We derive the thermodynamic functions associated to the partition function. We show that the non-commutativity affects energy at the high temperature limit. The Klein Gordon oscillator in non-commutative complex space also has a similar equation of motion to that a fermions with spin 1/2 in a commutative space in a constant magnetic field. The energy levels could be obtained by exact solution.