ExtendedPT- andCPT-symmetric representations of fermionic algebras

Abstract

In a recent paper, Bender and Klevansky [Phys. Rev. A 84, 024102 (2011)] considered $\mathcal{PT}$-symmetric matrix representations for fermionic operator algebras of the form ${\ensuremath{\xi}}^{2}={\overline{\ensuremath{\xi}}}^{2}=0$, $\ensuremath{\xi}\overline{\ensuremath{\xi}}+\overline{\ensuremath{\xi}}\ensuremath{\xi}=\ensuremath{\varepsilon}1$, where $\overline{\ensuremath{\xi}}$ is the $\mathcal{PT}$ transform of $\ensuremath{\xi}$. They constructed such algebras for $\ensuremath{\varepsilon}=\ensuremath{-}1$ and established that it is not possible to construct a matrix representation for the standard fermionic algebra ($\ensuremath{\varepsilon}=1$). Bender and Klevansky used the formalism developed by Jones-Smith and Mathur [Phys. Rev. A 82, 042101 (2010)] which extends $\mathcal{PT}$-symmetric quantum mechanics to the case of odd time-reversal symmetry (fermionic case). By using the same formalism, we show that $\mathcal{PT}$-symmetric matrix representations exist for both standard ($\ensuremath{\varepsilon}=1)$ and abnormal ($\ensuremath{\varepsilon}=\ensuremath{-}1)$ fermionic algebras if one takes $\overline{\ensuremath{\xi}}$ as adjoint $\ensuremath{\xi}$ with respect to the $\mathcal{CPT}$ and $\mathcal{PT}$ inner products, respectively. This general result is illustrated for the example of a typical quaternionic four-level model by an explicit construction of the fermionic creation and annihilation operators which satisfy all the criteria of $\mathcal{PT}$ quantum mechanics for the odd time-reversal symmetry.