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The metric for non-Hermitian Hamiltonians: a case study

Dibwe Pierrot Musumbu

Published Dec 1, 2006• 0 citations

Abstract

We are studying a possible implementation of an appropriate framework for a proper non-Hermitian quantum theory.We present the case where for a non-Hermitian Hamiltonian with real eigenvalues, we define a new inner product on the Hilbert space with respect to which the non-Hermitian Hamiltonian is Quasi-Hermitian.The Quasi-hermiticity of the Hamiltonian introduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates, in which case the metric becomes a basis transformation.We use the non-Hermitian quadratic Hamiltonian to show that such a metric is not unique but can be uniquely defined by requiring to hermitize all elements of one of the irreducible sets defined on the set of all observables.We compare the constructed metric with specific known examples in the literature in which cases a unique choice is made.Opsomming Ons ondersoek die implementering van n gepaste raamwerk virn nie-Hermitiese kwantumteorie.Ons beskoun nie-Hermitiese Hamilton-operator met reele eiewaardes en definieer in gepaste binneproduk ten opsigte waarvan die operator kwasi-Hermitiese is.Die kwasi-Hermities aard van die Hamilton operator lei dan tot n stel bi-ortogonale toestande.Ons konstrueer n basistransformasie wat die linker en regter eietoestande van hierdie stel koppel.Hierdie transformasie word dan gebruik omn nuwe binneproduk op die Hilbert-ruimte te definieer.Die oorspronklike nie-HermitieseHamilton-operator is danHermitiesmet betrekking tot hierdie nuwe binneproduk.Ons gebruik die nie-Hermitiese kwadratieseHamilton-operator om te toon dat hierdie metriek nie uniek is nie, maar wel uniek bepaal kan word deur verder te vereis dat dit al die elemente van n onherleibare versameling operatoreHermitiseer. Ons vergelyk hierdie konstruksie met die bekende voorbeelde in die literatuur en toon dat die metriek in beide gevalle uniek bepaal kan word.

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We are studying a possible implementation of an appropriate framework for a proper non-Hermitian quantum theory.We present the case where for a non-Hermitian Hamiltonian with real eigenvalues, we define a new inner product on the Hilbert space with respect to which the non-Hermitian Hamiltonian is Quasi-Hermitian.The Quasi-hermiticity of the Hamiltonian introduces the bi-orthogonality between the left-hand eigenstates and the right-hand eigenstates, in which case the metric becomes a basis transformation.We use the non-Hermitian quadratic Hamiltonian to show that such a metric is not unique but can be uniquely defined by requiring to hermitize all elements of one of the irreducible sets defined on the set of all observables.We compare the constructed metric with specific known examples in the literature in which cases a unique choice is made.Opsomming Ons ondersoek die implementering van n gepaste raamwerk virn nie-Hermitiese kwantumteorie.Ons beskoun nie-Hermitiese Hamilton-operator met reele eiewaardes en definieer in gepaste binneproduk ten opsigte waarvan die operator kwasi-Hermitiese is.Die kwasi-Hermities aard van die Hamilton operator lei dan tot n stel bi-ortogonale toestande.Ons konstrueer n basistransformasie wat die linker en regter eietoestande van hierdie stel koppel.Hierdie transformasie word dan gebruik omn nuwe binneproduk op die Hilbert-ruimte te definieer.Die oorspronklike nie-HermitieseHamilton-operator is danHermitiesmet betrekking tot hierdie nuwe binneproduk.Ons gebruik die nie-Hermitiese kwadratieseHamilton-operator om te toon dat hierdie metriek nie uniek is nie, maar wel uniek bepaal kan word deur verder te vereis dat dit al die elemente van n onherleibare versameling operatoreHermitiseer. Ons vergelyk hierdie konstruksie met die bekende voorbeelde in die literatuur en toon dat die metriek in beide gevalle uniek bepaal kan word.

In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state

ψ

is controlled, in Hilbert space

{\cal L}

, by a Hamiltonian

\mathfrak{h}

which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces

\mathfrak{h}

by its isospectral but non-Hermitian avatar

H = Ω^{-1}\mathfrak{h}Ω

with

Ω^\daggerΩ= Θ\neq I

. Although acting in another, manifestly unphysical Hilbert space

{\cal H}

, the amended Hamiltonian

H \neq H^\dagger

can be perceived as self-adjoint with respect to the amended inner-product metric

Θ

. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of

H

we introduce and describe a specific new family of Hamiltonians

H

for which the metrics

Θ

become available in closed form.

Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA

{\mathcal{L}}

?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA

{\mathcal{H}}

?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.

Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.

Abstract In 1992, Scholtz et al. ( Ann. Phys., 213 (1992) 74) showed that a set of non-Hermitian operators can represent observables of a closed unitary quantum system, provided only that its elements are quasi-Hermitian ( i.e. , roughly speaking, Hermitian with respect to an ad hoc inner-product metric). We show that such a version of quantum mechanics admits a simultaneous closed-form representation of the metric and of the observables , in terms of auxiliary operators Z k with . At N = 2 the formalism degenerates to the well-known quantum mechanics using factorized metric , where is parity and where is charge.

A general strategy is provided to identify the most general metric for diagonalizable pseudo-Hermitian and anti-pseudo-Hermitian Hamilton operators represented by two-dimensional matrices. It is investigated how a permutation of the eigen-values of the Hamilton operator in the process of its diagonalization influences the metric and how this permutation equivalence affects energy eigen-values. We try to understand on one hand, how the metric depends on the normalization of the chosen left and right eigen-basis of the matrix representing the diagonalizable pseudo-Hermitian or anti-pseudo-Hermitian Hamilton operator, on the other hand, whether there has to exist a positive semi-definite metric required to set up a meaningful Quantum Theory even for non-Hermitian Hamilton operators of this type. Using our general strategy we determine the metric with respect to the two elements of the two-dimensional permutation group for various topical examples of matrices representing two-dimensional Hamilton operators found in the literature assuming on one hand pseudo-Hermiticity, on the other hand anti-pseudo-Hermiticity. The (unnecessary) constraint inferred by C. M. Bender and collegues that the

{\cal C}

-operator of

{\cal PT}

-symmetric Quantum Theory should be an involution (

{\cal C}^2=1

) is shown - in the unbroken phase of

{\cal PT}

-symmetry - to require the Hamilton operator to be symmetric. This inconvenient restriction had been already - with hesitation - noted by M. Znojil and H. B. Geyer in 2006 (arXiv:quant-ph/0607104). A Hamilton operator proposed by T. D. Lee and C. G. Wick is used to outline implications of the formalism to higher dimensional Hamilton operators.

Abstract The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.