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Ghost fields in quantum field theory have been a long-standing problem.Specifically, theories with higher derivatives involve ghosts that appear inthe Hamiltonian in the form of linear momenta term, which is commonly known asthe Ostrogradski ghost. Higher derivative theories may involve both types ofconstraints i.e. first class and second class. Interestingly, these higherderivative theories may have non-Hermitian Hamiltonian respecting$\mathcal{PT}-$symmetries. In this paper, we have considered the$\mathcal{PT}-$symmetric nature of the extended Maxwell-Chern-Simon's theoryand employed the second class constraints to remove the linear momenta termscausing the instabilities. We found that the removal is not complete ratherconditions arise among the coefficients of the operator ${Q}$.

Ghosts in higher derivative Maxwell-Chern-Simon's theory and $\mathcal{PT}-$symmetry