# Ghost free boson-fermion co-existence system

## TL;DR

This paper investigates a system combining bosonic and fermionic fields, establishing ghost-free conditions and ensuring unique evolution by analyzing Hamiltonian and Lagrangian formulations.

## Contribution

It derives conditions for ghost-free coexistence of bosons and fermions, linking Hamiltonian and Lagrangian criteria, and addresses the uniqueness of time evolution in such systems.

## Key findings

- Ghost-free condition matches first-order fermion equations
- Additional condition ensures unique time evolution
- Hamiltonian and Lagrangian formalisms are consistent

## Abstract

We study co-existence system of both bosonic and fermionic degrees of freedom. For such system with up to first derivatives in Lagrangian, we find Ostrogradsky-type ghost-free condition in Hamiltonian analysis, which is found to be the same with requiring that the equations of motion of fermions are first-order in Lagrangian formulation. When fermionic degrees of freedom are present, 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 system.