Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras - Beygi, Alireza; Klevansky, Sandra Pamela; Bender, Carl M. (Arxiv, 2018)
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 $η$, which are quadratically nilpotent ($η^2=0$), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: $ηη^{PT}+η^{PT}η=-1$, where $η^{PT}$ is the PT adjoint of $η$, and $ηη^{CPT}+η^{CPT}η=1$, 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.