Nonlinear Fermions and Coherent States

TL;DR

This paper introduces nonlinear fermions of degree n with unique algebraic properties, constructs associated coherent states, and explores their mathematical structure and eigenstates.

Contribution

It develops a new framework for nonlinear fermions, including their algebra, coherent states, and matrix representations, expanding the understanding of finite-dimensional quantum systems.

Findings

Defined nonlinear fermions with degree n and specific anticommutation relations.

Constructed coherent states and displacement-operator-like states for these fermions.

Provided matrix representations and analyzed eigenstates of ladder operators.

Abstract

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.