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

Taha Koohrokhi, S. Kartal, Ali Mohammadi

Published Aug 29, 2021• 0 citations
OpenAlex

Abstract

A quantum mechanical system with unbroken super- and parity-time ($\mathcal{PT}$)-symmetry is derived and analyzed. Here, we propose a new formalism to construct the complex $\mathcal{PT}$-symmetric superpartners by extending the additive shape invariant potentials to the complex domain. The probabilistic interpretation of a $\mathcal{PT}$-symmetric quantum theory is correlated with the calculation of a new linear operator called the $\mathcal{C}$ operator, instead of complex conjugation in conventional quantum mechanics. At the present work, we introduce an anti-$\mathcal{PT}$ ($\mathcal{APT}$) conjugation to redefine a new version of the inner product without any additional considerations. This $\mathcal{PT}$-supersymmetric quantum mechanics, satisfies essential requirements such as completeness, orthonormality as well as probabilistic interpretation.

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Anti-PT Transformations and Complex PT-Symmetric Superpartners

S. Kartal, Taha Koohrokhi, Ali Mohammadi

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.

•Sep 7, 2021•0 citations

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

Taha Koohrokhi, S. Kartal, Mohammadi, Ali

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.

arXiv:2108.12834•Aug 29, 2021•0 citations

The broken and unbroken phases of PT-symmetry and supersymmetry in quantum mechanics

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We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.

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A C*-Algebraic Perspective on Probabilistic Structures in the Unbroken and Broken PT-Symmetric Regimes

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PT-symmetry, invariance under the combined operations of spatial reflection and time reversal, arises naturally in non-Hermitian physics and has been attracting an increasing amount of attention in the literature. Here, we examine probabilistic structures within the unbroken and broken PT-symmetric regimes from a C*-algebraic perspective. We show that PT-unbroken probability is directly compatible with the operator-algebraic formulation of standard QM. However, though PT-broken Hamiltonians may generate well-defined non-unitary dynamics, basic structural properties required for quantum probability typically fail. We analyze this failure via relation to the GNS construction, ultimately demonstrating that an extension to a C*-algebraic probability theory on a Hermitian supersystem is required to define a meaningful probability calculus within the PT-broken system itself. Thereby, consistent probability assignment must be extrinsic to the PT-broken phase, and must be inherited from a dilation to a Hermitian system. We offer a novel interpretation of the broken PT-symmetric regime as a non-ordered, non-probabilistic subalgebra of a larger Hermitian, fully quantum system. Additionally, our work clarifies the structural importance of positivity in standard quantum mechanics.

Princeton Journal of Pre-Collegiate Research•Jan 1, 2026•0 citations

Generalized PT-, C-, and CPT-Symmetries, Position Operators, and Localized States of Klein-Gordon Fields

Mostafazadeh, A

We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space ${cal H}$ defined by this inner product is Hermitian. The quantum system having ${cal H}$ as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space $L^2(R^3)oplus L^2(R^3)$. We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.

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Features of Complex Supersymmetric Quantum Mechanics

Aishwarya Parthasarathi

The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.

•Sep 24, 2021•0 citations
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.
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.
We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.
PT-symmetry, invariance under the combined operations of spatial reflection and time reversal, arises naturally in non-Hermitian physics and has been attracting an increasing amount of attention in the literature. Here, we examine probabilistic structures within the unbroken and broken PT-symmetric regimes from a C*-algebraic perspective. We show that PT-unbroken probability is directly compatible with the operator-algebraic formulation of standard QM. However, though PT-broken Hamiltonians may generate well-defined non-unitary dynamics, basic structural properties required for quantum probability typically fail. We analyze this failure via relation to the GNS construction, ultimately demonstrating that an extension to a C*-algebraic probability theory on a Hermitian supersystem is required to define a meaningful probability calculus within the PT-broken system itself. Thereby, consistent probability assignment must be extrinsic to the PT-broken phase, and must be inherited from a dilation to a Hermitian system. We offer a novel interpretation of the broken PT-symmetric regime as a non-ordered, non-probabilistic subalgebra of a larger Hermitian, fully quantum system. Additionally, our work clarifies the structural importance of positivity in standard quantum mechanics.
We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space calH{cal H}calH defined by this inner product is Hermitian. The quantum system having calH{cal H}calH as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space L2(R3)oplusL2(R3)L^2(R^3)oplus L^2(R^3)L2(R3)oplusL2(R3). We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.
The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.