PT-symmetric representations of fermionic algebras

A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) 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, (eta) over bar (2) = 0, eta(eta) over bar + (eta) over bar eta = alpha 1, where (eta) over bar = eta(PT) = PT eta T-1P-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 not equal 0) if alpha = -1; a matrix representation does not exist for the conventional value alpha = 1.