Two-and four-dimensional representations of the PT- and CPT-symmetricfermionic 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., 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.