Nonlinear fermions and coherent states

Abstract

Nonlinear fermions of degree n (n-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA† + A†nAn = 1. The (n + 1)th-order nilpotency of these operators follows from the existence of unique A-vacuum. Supposing appropriate (n + 1)th-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) × (n + 1) matrix realization of the related para-Grassmann algebra is provided. General (n + 1)th-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 are briefly discussed.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.