Constructing ghost-free boson-fermion system

I first talk about the coexistence system of both bosonic and fermionic degrees of freedom in the context of point particle system. Even if a Lagrangian does not include higher derivatives, fermionic ghosts generally exist. For a Lagrangian with up to first derivatives, we find the fermionic ghost free condition in Hamiltonian analysis, which is found to be the same as requiring that the equations of motion of fermions be first order in Lagrangian formulation. When fermionic degrees of freedom are present, the uniqueness of time evolution is not guaranteed a priori because of the Grassmann property. We confirm that the additional condition, which is introduced to close Hamiltonian analysis, also ensures the uniqueness of the time evolution of the system. I also discuss a covariant extension of interactions between scalar fields and fermions in a flat space-time. I will also show that proposed derivative interaction terms between scalar fields and a Weyl fermion cannot be removed by field redefinitions.