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Physical Review A

Extended PT - and CPT -symmetric representations of fermionic algebras

Omar Cherbal, D. A. Trifonov

Published May 23, 2012• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.85.052123

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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 ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). 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 ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, 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 $\ensuremath{\alpha}=1$.
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We describe a novel class of quantum mechanical particle oscillations in both relativistic and nonrelativistic systems based on P T symmetry and T 2 = − 1 (relevant for fermions), where P is parity and T is time reversal. The Hamiltonians are chosen at the outset to be self-adjoint with respect to a P T inner product. The quantum mechanical time evolution is based on a modified C P T inner product constructed in terms of a suitable C operator. The resulting quantum mechanical evolution is shown to be unitary and probability is conserved by the oscillations. Published by the American Physical Society 2024
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |T2=1T^{2}=1T2=1|⁠. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |T2=−1T^{2}=-1T2=−1|⁠. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |HHH| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |HHH| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |hhh|⁠, so that the generalized Kramers degeneracy of |HHH| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.
In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian is PT -symmetric. Here P is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo-Hermiticity or, what is the same, self-adjoint operators in Kre˘ın spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019).