PT-Symmetric Representations of Fermionic Algebras

Abstract

A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum mechanics from bosonic systems (systems for which $T^2=1$) to fermionic systems (systems for which $T^2=-1$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT-symmetric matrix representations for operator algebras of the form ${\eta }^2=0$, ${\bar{\eta }}^2=0$, $\eta \bar{\eta }+\bar{\eta }=\alpha 1$, where $\overline{eta}={\eta }^{PT}=PT\eta T^{-1}{P}^{-1}$. It is easy to construct matrix representations for the Grassmann algebra ($\alpha =0$). However, one can only construct matrix representations for the fermionic operator algebra ($\alpha \ne 0$) if $\alpha =-1$; a matrix representation does not exist for the conventional value $\alpha =1$.

5 pages, 2 figures

References in corpus (6)

• Making Sense of Non-Hermitian Hamiltonians
• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• PT-symmetric laser-absorber
• Faster than Hermitian Quantum Mechanics
• The ODE/IM Correspondence
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• Nonlinear Topological Edge States in a non-Hermitian Array of Optical Waveguides Embedded in an Atomic Gas
• Model pseudofermionic systems: connections with exceptional points
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Spectral singularities of odd-$PT$-symmetric potential
• Acceleration toward polarization singularity inspired by relativistic ExB drift
• Damping and Pseudo-fermions
• Generation of families of spectra in PT-symmetric quantum mechanics and scalar bosonic field theory
• Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras
• Instantaneous modulations in time-varying complex optical potentials
• Non-Hermitian CP-Symmetric Dirac Hamiltonians with Real Energy Eigenvalues
• Pseudo-Hermitian systems with PT-symmetry: Degeneracy and Krein space
• Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with anti-$\mathcal{P}\mathcal{T}$ symmetry