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arXiv:1803.10034 · Physical Review A

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

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

Published Mar 28, 2018• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.97.032128

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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.
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A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles 1,21,21,2 with a Hilbert space Cd1⊗Cd2\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}Cd1​⊗Cd2​, a UE exists when min⁡(d1,d2)≥3\min{(d_1,d_2)}\geq 3min(d1​,d2​)≥3 and (d1,d2)≠(3,3)(d_1,d_2)\neq (3,3)(d1​,d2​)=(3,3). It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd if and only if d≥3d\geq 3d≥3 (resp. d≥8d\geq 8d≥8). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.
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