Nonlinear n-Pseudo Fermions

TL;DR

This paper introduces nonlinear n-pseudo-fermions with unique algebraic properties, constructs their coherent states, and explores their relation to pseudo-Hermitian systems, expanding the framework of quantum particles with nonlinear anticommutation relations.

Contribution

It presents the first formulation of nonlinear n-pseudo-fermions, including their algebra, coherent states, and connections to pseudo-Hermitian quantum systems.

Findings

Defined nonlinear n-pseudo-fermions with specific anticommutation relations.

Constructed n-pseudo-fermion number states and coherent states.

Explored the relation to finite-level pseudo-Hermitian systems.

Abstract

Nonlinear pseudo-fermions of degree n (n-pseudo-fermions) are introduced as (pseudo) particles with creation and annihilation operators $a$ and $b$, $b \neq a^\dagger$, obeying the simple nonlinear anticommutation relation $ab + b^n a^n = 1$. The (n+1)-order nilpotency of these operators follows from the existence of unique (up to a bi-normalization factor) $a$-vacuum. Supposing appropriate (n+1)-order nilpotent para-Grassmann variables and integration rules the sets of n-pseudo-fermion number states, and 'right' and 'left' ladder operator bi-overcomplete sets of coherent states are constructed. Explicit examples of n-pseudo-fermion ladder operators are provided, and the relation of pseudo-fermions to finite-level pseudo-Hermitian systems is briefly considered.