Abstract
Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that T²=-1 for fermionic systems. In PT-symmetric quantum mechanics an operator has both PT and CPT adjoints. Fermionic operators η, which are quadratically nilpotent (η²=0), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: ηη^PT+η^PTη=-1, where η^PT is the PT adjoint of η, and ηη^CPT+η^CPTη=1, where η^CPT is the CPT adjoint of η. This paper presents matrix representations for the operator η 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.