  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.
