Generalized Grassmannian Coherent States For Pseudo-Hermitian $n$ Level Systems

TL;DR

This paper extends fermionic coherent states to n-level pseudo-Hermitian systems using generalized Grassmann variables, exploring their mathematical properties and constructing specific states for a deformed group.

Contribution

It introduces a generalized framework for Grassmannian coherent states in pseudo-Hermitian n-level systems, including bi-overcompleteness and resolution of identity.

Findings

Derived bi-overcompleteness condition for these states

Constructed Grassmannian coherent and squeezed states for deformed group $SU_q(2)$

Analyzed resolution of identity with appropriate Grassmann weight functions

Abstract

The purpose of this paper is to generalize fermionic coherent states for two-level systems described by pseudo-Hermitian Hamiltonian \cite{Trifonov}, to n-level systems. Central to this task is the expression of the coherent states in terms of generalized Grassmann variables. These kind of Grassmann coherent states satisfy bi-overcompleteness condition instead of over-completeness one, as it is reasonably expected because of the biorthonormality of the system. Choosing an appropriate Grassmann weight function resolution of identity is examined. Moreover Grassmannian coherent and squeezed states of deformed group $SU_{q}(2)$ for three level pseudo-Hermitian system are presented.