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 $η^2=0$, $\barη^2=0$, $η\barη+\bar η =α1$, where $\bar{eta}=η^{PT} =PT ηT^{-1}P^{-1}$. It is easy to construct matrix representations for the Grassmann algebra ($α=0$). However, one can only construct matrix representations for the fermionic operator algebra ($α\neq0$) if $α= -1$; a matrix representation does not exist for the conventional value $α=1$.

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