Negative heat capacity for a Klein-Gordon oscillator in non-commutative complex phase space

We obtain exact solutions to the two-dimensional Klein-Gordon oscillator in a non-commutative complex phase space to first order in the non-commutativity parameter. We derive the exact non-commutative energy levels and show that the energy levels split to $2m$ levels. We find that the non-commutativity plays the role of a magnetic field interacting automatically with the spin of a particle induced by the non-commutativity of complex phase space. The effect of the non-commutativity parameter on the thermal properties is discussed. It is found that the dependence of the heat capacity $C_V$ on the non-commutative parameter gives rise to a negative quantity. Phenomenologically, this effectively confirms the presence of the effects of self-gravitation induced by the non-commutativity of complex phase space.