We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in X implies minimal substitution in the form P → P −eA(X). We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not. PACS numbers: 03.65.Ca, 11.30.Er, 02.30.Mv 1We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in X implies minimal substitution in the form P → P −eA(X). We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not. PACS numbers: 03.65.Ca, 11.30.Er, 02.30.Mv 1

Jan 1, 20097 citations

Time evolution of non-Hermitian Hamiltonian systems

Carla Figueira de Morisson Faria, A Fring

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT -symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT -symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.

arXiv:quant-ph/0604014Journal of Physics A Mathematical and GeneralJul 5, 2006121 citations

Disappearing Q operator

Hugh F Jones, R. J. Rivers

In the Schr\"odinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\ensuremath{\eta}\ensuremath{\equiv}{e}^{\ensuremath{-}Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$ . These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.In the Schr\"odinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\ensuremath{\eta}\ensuremath{\equiv}{e}^{\ensuremath{-}Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.

arXiv:hep-th/0612093Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsJan 26, 200733 citations

Exact Metric Operators as the Ground State functions of the Hermitian Conjugates of a Class of Quasi-Hermitian Hamiltonians

Abouzeid M. Shalaby

We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if $ψ$ is an eigen function of $H$ , then $ϕ=ηψ$ is an eigen function of $H^{\dagger}$ with the same eigen value. Thus, knowing any pair $(ψ,ϕ)$ one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the $Q$ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if $ψ$ is an eigen function of $H$, then $ϕ=ηψ$ is an eigen function of $H^{\dagger}$ with the same eigen value. Thus, knowing any pair $(ψ,ϕ)$ one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the $Q$ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.

arXiv:0706.2509Jun 17, 20070 citations

Metric operator in pseudo-Hermitian quantum mechanics and the imaginary cubic potential

Ali Mostafazadeh

We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, , in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v ( x ) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v ( x ) = i x 3 . This reveals an infinite class of previously unknown - as well as non- -inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, , in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v ( x ) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v ( x ) = i x 3 . This reveals an infinite class of previously unknown - as well as non- -inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.

arXiv:quant-ph/0508195Journal of Physics A Mathematical and GeneralJul 26, 2006103 citations

On pseudo-Hermitian Hamiltonians and their Hermitian counterparts

H F Jones

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h , obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT -invariant Hamiltonian, with potential V = i gx 3 . In this case the corresponding h , which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P .In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h , obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT -invariant Hamiltonian, with potential V = i gx 3 . In this case the corresponding h , which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P .

arXiv:quant-ph/0411171Journal of Physics A Mathematical and GeneralFeb 10, 2005153 citations

Q

operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source

j(t)

. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.

ψ

is an eigen function of

H

, then

ϕ=ηψ

is an eigen function of

H^{\dagger}

with the same eigen value. Thus, knowing any pair

(ψ,ϕ)

one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the

Q

operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.