PT-Symmetric Representations of Fermionic Algebras

Carl M. Bender, S. P. Klevansky

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 \(\bar{eta}=\eta^{PT} =PT \eta T^{-1}P^{-1}\). 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 (\(\alpha \neq 0\)) if \(\alpha= -1\); a matrix representation does not exist for the conventional value \(\alpha=1\).
Subjects: High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

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