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

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 $\eta$, 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.

8 pages, 1 figure

References in corpus (1)

• Making Sense of Non-Hermitian Hamiltonians

Cited by in corpus (6)

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• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• No-signaling principle and quantum brachistochrone problem in $PT$-symmetric fermionic two- and four-dimensional models
• $\mathcal{P}\mathcal{T}$-Symmetry in Hartree-Fock Theory
• Continuous quantum phase transition in the fermionic mass solutions of the Nambu-Jona-Lasinio model