PT-symmetric representations of fermionic algebras

2011/04/21 by Carl M. Bender, S. P. Klevansky · 2 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Combinatorics #Formalism (music) #Mathematics #Nonlinear Photonic Systems #Physics #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1103/physreva.84.024102

5 pages, 2 figures

arxiv created 2011/04/21 · openalex publication_date 2011/08/25 · arxiv updated 2015/03/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends PT-symmetric quantum mechanics from bosonic systems (systems for which T²=1) to fermionic systems (systems for which T²=\ensuremath-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 \ensuremathη²=0, \ensuremathη²=0, \ensuremathη\ensuremathη+\ensuremathη\ensuremathη=\ensuremathα1, where \ensuremathη=\ensuremathη^PT=PT\ensuremathηT^\ensuremath-1P^\ensuremath-1. It is easy to construct matrix representations for the Grassmann algebra (\ensuremathα=0). However, one can only construct matrix representations for the fermionic operator algebra (\ensuremathα\ensuremath≠0) if \ensuremathα=\ensuremath-1; a matrix representation does not exist for the conventional value \ensuremathα=1.

Citations

• Must a Hamiltonian be Hermitian?
• Making sense of non-Hermitian Hamiltonians

Cited by

• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• Model pseudofermionic systems: Connections with exceptional points