Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1104.4156 · Physical Review A

PT -symmetric representations of fermionic algebras

Carl M. Bender, S. P. Klevansky

Published Aug 25, 2011• 23 citations
arXivHTMLPDF

Abstract

A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics from bosonic systems (systems for which ${\mathcal{T}}^{2}=\mathbf{1}$) to fermionic systems (systems for which ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct $\mathcal{P}\mathcal{T}$-symmetric matrix representations for operator algebras of the form ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, where $\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}$. It is easy to construct matrix representations for the Grassmann algebra ($\ensuremath{\alpha}=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\ensuremath{\alpha}\ensuremath{\ne}0$) if $\ensuremath{\alpha}=\ensuremath{-}1$; a matrix representation does not exist for the conventional value $\ensuremath{\alpha}=1$.

DOI
10.1103/physreva.84.024102

Related papers

Extended PT - and CPT -symmetric representations of fermionic algebras

Omar Cherbal, D. A. Trifonov

In a recent paper, Bender and Klevansky [Phys. Rev. A 84, 024102 (2011)] considered $\mathcal{PT}$-symmetric matrix representations for fermionic operator algebras of the form ${\ensuremath{\xi}}^{2}={\overline{\ensuremath{\xi}}}^{2}=0$, $\ensuremath{\xi}\overline{\ensuremath{\xi}}+\overline{\ensuremath{\xi}}\ensuremath{\xi}=\ensuremath{\varepsilon}1$, where $\overline{\ensuremath{\xi}}$ is the $\mathcal{PT}$ transform of $\ensuremath{\xi}$. They constructed such algebras for $\ensuremath{\varepsilon}=\ensuremath{-}1$ and established that it is not possible to construct a matrix representation for the standard fermionic algebra ($\ensuremath{\varepsilon}=1$). Bender and Klevansky used the formalism developed by Jones-Smith and Mathur [Phys. Rev. A 82, 042101 (2010)] which extends $\mathcal{PT}$-symmetric quantum mechanics to the case of odd time-reversal symmetry (fermionic case). By using the same formalism, we show that $\mathcal{PT}$-symmetric matrix representations exist for both standard ($\ensuremath{\varepsilon}=1)$ and abnormal ($\ensuremath{\varepsilon}=\ensuremath{-}1)$ fermionic algebras if one takes $\overline{\ensuremath{\xi}}$ as adjoint $\ensuremath{\xi}$ with respect to the $\mathcal{CPT}$ and $\mathcal{PT}$ inner products, respectively. This general result is illustrated for the example of a typical quaternionic four-level model by an explicit construction of the fermionic creation and annihilation operators which satisfy all the criteria of $\mathcal{PT}$ quantum mechanics for the odd time-reversal symmetry.

Physical Review A•May 23, 2012•12 citations

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

Alireza Beygi, S. P. Klevansky, Carl M. Bender

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that ${\mathcal{T}}^{2}=\ensuremath{-}1$ for fermionic systems. In $\mathcal{PT}$-symmetric quantum mechanics an operator has both $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints. Fermionic operators $\ensuremath{\eta}$, which are quadratically nilpotent (${\ensuremath{\eta}}^{2}=0$), and algebras with $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints can be constructed. These algebras obey different anticommutation relations: $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{PT}}+{\ensuremath{\eta}}^{\mathcal{PT}}\ensuremath{\eta}=\ensuremath{-}\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{PT}}$ is the $\mathcal{PT}$ adjoint of $\ensuremath{\eta}$, and $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{CPT}}+{\ensuremath{\eta}}^{\mathcal{CPT}}\ensuremath{\eta}=\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{CPT}}$ is the $\mathcal{CPT}$ adjoint of $\ensuremath{\eta}$. This paper presents matrix representations for the operator $\ensuremath{\eta}$ and its $\mathcal{PT}$ and $\mathcal{CPT}$ adjoints in two and four dimensions. A $\mathcal{PT}$-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.

arXiv:1803.10034•Physical Review A•Mar 28, 2018•6 citations

ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

Francisco Marcelo Fernández

We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.

arXiv:1301.7639•Jan 29, 2013•1 citation

ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

Francisco Marcelo Fernández

We discuss the construction of real matrix representations of PT-symmetric operators. We show the limitation of a general recipe presented some time ago for non-Hermitian Hamiltonians with antiunitary symmetry and propose a way to overcome it. Our results agree with earlier ones for a particular case.

Acta Polytechnica•Apr 30, 2014•3 citations

Fermionic functionals without Grassmann numbers

Hrvoje Nikolić

Since any fermionic operator ψcan be written as ψ=q+ip, where q and p are hermitian operators, we use the eigenvalues of q and p to construct a functional formalism for calculating matrix elements that involve fermionic fields. The formalism is similar to that for bosonic fields and does not involve Grassmann numbers. This makes possible to perform numerical fermionic lattice computations that are much faster than not only other algorithms for fermions, but also algorithms for bosons.

arXiv:hep-th/0210307•Oct 31, 2002•0 citations

PT-symmetry and PT-frame

Zhengli Chen

In recent years,some elementary points of the PT-symmetry and its mathematical or physical descriptions have been studied.This paper gives a discussion of PT-symmetric quantum theory by applying the methods of operator algebra and matrix.Some important results and properties are provided.

Journal of Xi'an University of Arts and Science•Jan 1, 2013•0 citations
A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics from bosonic systems (systems for which T2=1{\mathcal{T}}^{2}=\mathbf{1}T2=1) to fermionic systems (systems for which T2=\ensuremath−1{\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}T2=\ensuremath−1). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT\mathcal{P}\mathcal{T}PT-symmetric matrix representations for operator algebras of the form \ensuremathη2=0{\ensuremath{\eta}}^{2}=0\ensuremathη2=0, \ensuremathη‾2=0{\overline{\ensuremath{\eta}}}^{2}=0\ensuremathη​2=0, \ensuremathη\ensuremathη‾+\ensuremathη‾\ensuremathη=\ensuremathα1\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}\ensuremathη\ensuremathη​+\ensuremathη​\ensuremathη=\ensuremathα1, where \ensuremathη‾=\ensuremathηPT=PT\ensuremathηT\ensuremath−1P\ensuremath−1\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}\ensuremathη​=\ensuremathηPT=PT\ensuremathηT\ensuremath−1P\ensuremath−1. It is easy to construct matrix representations for the Grassmann algebra (\ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0). However, one can only construct matrix representations for the fermionic operator algebra (\ensuremathα\ensuremath≠0\ensuremath{\alpha}\ensuremath{\ne}0\ensuremathα\ensuremath=0) if \ensuremathα=\ensuremath−1\ensuremath{\alpha}=\ensuremath{-}1\ensuremathα=\ensuremath−1; a matrix representation does not exist for the conventional value \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1.
In a recent paper, Bender and Klevansky [Phys. Rev. A 84, 024102 (2011)] considered PT\mathcal{PT}PT-symmetric matrix representations for fermionic operator algebras of the form ${\ensuremath{\xi}}^{2}={\overline{\ensuremath{\xi}}}^{2}=0$, $\ensuremath{\xi}\overline{\ensuremath{\xi}}+\overline{\ensuremath{\xi}}\ensuremath{\xi}=\ensuremath{\varepsilon}1$, where $\overline{\ensuremath{\xi}}$ is the PT\mathcal{PT}PT transform of $\ensuremath{\xi}$. They constructed such algebras for $\ensuremath{\varepsilon}=\ensuremath{-}1$ and established that it is not possible to construct a matrix representation for the standard fermionic algebra ($\ensuremath{\varepsilon}=1$). Bender and Klevansky used the formalism developed by Jones-Smith and Mathur [Phys. Rev. A 82, 042101 (2010)] which extends PT\mathcal{PT}PT-symmetric quantum mechanics to the case of odd time-reversal symmetry (fermionic case). By using the same formalism, we show that PT\mathcal{PT}PT-symmetric matrix representations exist for both standard ($\ensuremath{\varepsilon}=1)$ and abnormal ($\ensuremath{\varepsilon}=\ensuremath{-}1)$ fermionic algebras if one takes $\overline{\ensuremath{\xi}}$ as adjoint $\ensuremath{\xi}$ with respect to the CPT\mathcal{CPT}CPT and PT\mathcal{PT}PT inner products, respectively. This general result is illustrated for the example of a typical quaternionic four-level model by an explicit construction of the fermionic creation and annihilation operators which satisfy all the criteria of PT\mathcal{PT}PT quantum mechanics for the odd time-reversal symmetry.
Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that ${\mathcal{T}}^{2}=\ensuremath{-}1$ for fermionic systems. In PT\mathcal{PT}PT-symmetric quantum mechanics an operator has both PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints. Fermionic operators $\ensuremath{\eta}$, which are quadratically nilpotent (${\ensuremath{\eta}}^{2}=0$), and algebras with PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints can be constructed. These algebras obey different anticommutation relations: $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{PT}}+{\ensuremath{\eta}}^{\mathcal{PT}}\ensuremath{\eta}=\ensuremath{-}\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{PT}}$ is the PT\mathcal{PT}PT adjoint of $\ensuremath{\eta}$, and $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{CPT}}+{\ensuremath{\eta}}^{\mathcal{CPT}}\ensuremath{\eta}=\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{CPT}}$ is the CPT\mathcal{CPT}CPT adjoint of $\ensuremath{\eta}$. This paper presents matrix representations for the operator $\ensuremath{\eta}$ and its PT\mathcal{PT}PT and CPT\mathcal{CPT}CPT adjoints in two and four dimensions. A PT\mathcal{PT}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.