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arXiv:2108.12834

Anti-$\mathcal{PT}$ Transformations And Complex Non-Hermitian $\mathcal{PT}$-Symmetric Superpartners

Taha Koohrokhi, S. Kartal, Mohammadi, Ali

Published Aug 29, 2021• 0 citations
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Abstract

We propose a new algebraic formalism for constructing complex non-Hermitian $\mathcal{PT}$-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time ($\mathcal{PT}$)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the $\mathcal{CPT}$-inner product is defined in $\mathcal{PT}$-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-$\mathcal{PT}$ ($\mathcal{APT}$)-inner product, $\langle A|B\rangle\equiv |A\rangle^{\mathcal{APT}}.|B\rangle$, which replaces the previous versions without any additional considerations. This $\mathcal{PT}$-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the $\rm ^{3}H(d,n)^{4}He$ reaction.

DOI
10.48550/arxiv.2108.12834

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Anti-$\mathcal{PT}$ Transformations and Complex $\mathcal{PT}$-Symmetric Superpartners

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$\mathcal{PT}$-Supersymmetric Square Well and Barrier

Taha Koohrokhi

The Parity-Time ($\mathcal{PT}$) symmetric potentials are derived by non-Hermitian supersymmetric quantum mechanics for square well and barrier. These $\mathcal{PT}$-supersymmetric square well and barrier. The partners have complex partners. The partners are isospectral with real energies. $\mathcal{PT}$-symmetry is only unbroken for the bound states.

arXiv:2001.09418•Jan 26, 2020•1 citation
Anti-PT\mathcal{PT}PT Transformations And Complex Non-Hermitian PT\mathcal{PT}PT-Symmetric Superpartners
We propose a new algebraic formalism for constructing complex non-Hermitian PT\mathcal{PT}PT-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time (PT\mathcal{PT}PT)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the CPT\mathcal{CPT}CPT-inner product is defined in PT\mathcal{PT}PT-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-PT\mathcal{PT}PT (APT\mathcal{APT}APT)-inner product, ⟨A∣B⟩≡∣A⟩APT.∣B⟩\langle A|B\rangle\equiv |A\rangle^{\mathcal{APT}}.|B\rangle⟨A∣B⟩≡∣A⟩APT.∣B⟩, which replaces the previous versions without any additional considerations. This PT\mathcal{PT}PT-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the 3H(d,n)4He\rm ^{3}H(d,n)^{4}He3H(d,n)4He reaction.
Anti-PT\mathcal{PT}PT Transformations and Complex PT\mathcal{PT}PT-Symmetric Superpartners
A quantum mechanical system with unbroken super- and parity-time (PT\mathcal{PT}PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT\mathcal{PT}PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT\mathcal{PT}PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C\mathcal{C}C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT\mathcal{PT}PT (APT\mathcal{APT}APT) conjugation to redefine a new version of the inner product without any additional considerations. This PT\mathcal{PT}PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.
Abstract A quantum mechanical system with unbroken super-and parity-time (PT)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex PT-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a PT-symmetric quantum theory is correlated with the calculation of a new linear operator called the C operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-PT (A PT) conjugation to redefine a new version of the inner product without any additional considerations. This PT-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.
With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.
Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.
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.
PT\mathcal{PT}PT-Supersymmetric Square Well and Barrier
The Parity-Time (PT\mathcal{PT}PT) symmetric potentials are derived by non-Hermitian supersymmetric quantum mechanics for square well and barrier. These PT\mathcal{PT}PT-supersymmetric square well and barrier. The partners have complex partners. The partners are isospectral with real energies. PT\mathcal{PT}PT-symmetry is only unbroken for the bound states.