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: $etaeta^{PT}+eta^{PT}eta=-1$, where $eta^{PT}$ is the PT adjoint of $eta$, and $etaeta^{CPT}+eta^{CPT}eta=1$, 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.