Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras

TL;DR

This work builds a PT-symmetric fermionic framework by redefining inner products and adjoints ($\mathcal{PT}$ and $\mathcal{CPT}$) for a nilpotent fermionic operator $\eta$ ($\eta^2=0$). It delivers explicit 2D and 4D matrix representations, analyzes ground and excited states, and shows how the choice of adjoint governs the resulting fermionic algebra (with $-\mathds{1}$ for $\mathcal{PT}$ and $+\mathds{1}$ for $\mathcal{CPT}$). A solvable model of fermions interacting with bosons is solved exactly, yielding an explicit spectrum $E_N=Nm+M-g^2/m$ and revealing mass renormalization $M- g^2/m$ in the PT-symmetric setting. The results provide a structured approach to second-quantized PT-symmetric theories, with potential applications in many-body physics and beyond.

Abstract

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.

Figures (1)

• Figure 1: The fermion (solid line) can emit or absorb bosons (wavy gray lines). These are the only possible interactions, so the fermion number is conserved.