Nonlinear Fermions and Coherent States

Nonlinear fermions of degree \(n\) (\(n\)-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation \(AA^\dagger + {A^\dagger}^n A^n = 1\). The (\(n+1\))-order nilpotency of these operators follows from the existence of unique \(A\)-vacuum. Supposing appropreate (\(n+1\))-order nilpotent para-Grassmann variables and integration rules the sets of \(n\)-fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The \((n+1)\times(n+1)\) matrix realization of the related para-Grassmann algebra is provided. General \((n+1)\)-order nilpotent ladder operators of finite dimensional systems are expressed as polynomials in terms of \(n\)-fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties briefly discussed.