Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that $T^2=-1$ for fermionic systems. In PT-symmetric quantum mechanics an operator has both PT and CPT adjoints. Fermionic operators $\eta$, which are quadratically nilpotent ($\eta^2=0$), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: $\eta\eta^{PT}+\eta^{PT}\eta=-1$, where $\eta^{PT}$ is the PT adjoint of $\eta$, and $\eta\eta^{CPT}+\eta^{CPT}\eta=1$, where $\eta^{CPT}$ is the CPT adjoint of $\eta$. This paper presents matrix representations for the operator $\eta$ and its PT and CPT adjoints in two and four dimensions. A PT-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.