PT-Symmetric Representations of Fermionic Algebras

A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum mechanics from bosonic systems (systems for which $T^2=1$) to fermionic systems (systems for which $T^2=-1$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT-symmetric matrix representations for operator algebras of the form ${\eta }^2=0$, ${\bar{\eta }}^2=0$, $\eta \bar{\eta }+\bar{\eta }=\alpha 1$, where $\overline{eta}={\eta }^{PT}=PT\eta T^{-1}{P}^{-1}$. It is easy to construct matrix representations for the Grassmann algebra ($\alpha =0$). However, one can only construct matrix representations for the fermionic operator algebra ($\alpha \mathrm{\ne }0$) if $\alpha =-1$; a matrix representation does not exist for the conventional value $\alpha =1$.