Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1107.1874 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Imaginary-scaling versus indefinite-metric quantization of the Pais-Uhlenbeck oscillator

Ali Mostafazadeh

Published Nov 14, 2011• 23 citations
arXivHTMLPDF

Abstract

Using the Pais-Uhlenbeck oscillator as a toy model, we outline a consistent alternative to the indefinite-metric quantization scheme that does not violate unitarity. We describe the basic mathematical structure of this method by giving an explicit construction of the Hilbert space of state vectors and the corresponding creation and annihilation operators. The latter satisfy the usual bosonic commutation relation and differ from those of the indefinite-metric theories by a sign in the definition of the creation operator. This change of sign achieves a definitization of the indefinite metric that gives life to the ghost states without changing their contribution to the energy spectrum.

DOI
10.1103/physrevd.84.105018

Related papers

A generalized non-Hermitian Pais-Uhlenbeck quantum Hamiltonian, its Hermitian equivalence and position-dependent mass correspondence

Bijan Kumar Bagchi, A. Ghose Choudhury, Partha Guha

We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.

arXiv:1212.2092•Dec 10, 2012•0 citations

A toy model of bosonic non-canonical quantum field

Marek Czachor, Monika Syty

A harmonic oscillator is an indefinite-frequency one if the parameter $ω$ is replaced by an operator. An ensemble of $N$ such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and $N$ can be finite. Due to the operator character of $ω$ the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large $N$ one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail.

arXiv:hep-th/0112011•Dec 2, 2001•2 citations

On the construction of a pseudo-Hermitian quantum system with a pre-determined metric in the Hilbert space

Pijush K Ghosh

A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.

arXiv:0908.1321•Journal of Physics A Mathematical and Theoretical•Mar 4, 2010•13 citations

Spectrum-generating algebra and intertwiners of the resonant Pais-Uhlenbeck oscillator

Andreas Fring, Ian Marquette, Takano Taira

We study the quantum Pais-Uhlenbeck oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden $su(2)$ Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant Pais-Uhlenbeck oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.

quant-pharXiv:2601.20752•Journal of Physics A Mathematical and Theoretical•Jan 29, 2026•0 citations

Spectrum-generating algebra and intertwiners of the resonant Pais–Uhlenbeck oscillator

Andreas Fring, Ian Marquette, Takano Taira

We study the quantum Pais–Uhlenbeck (PU) oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden (Formula presented) (Formula presented) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant PU oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.

•Jul 24, 2026•0 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.

arXiv:quant-ph/0508195•Journal of Physics A Mathematical and General•Jul 26, 2006•103 citations
Using the Pais-Uhlenbeck oscillator as a toy model, we outline a consistent alternative to the indefinite-metric quantization scheme that does not violate unitarity. We describe the basic mathematical structure of this method by giving an explicit construction of the Hilbert space of state vectors and the corresponding creation and annihilation operators. The latter satisfy the usual bosonic commutation relation and differ from those of the indefinite-metric theories by a sign in the definition of the creation operator. This change of sign achieves a definitization of the indefinite metric that gives life to the ghost states without changing their contribution to the energy spectrum.
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.
A harmonic oscillator is an indefinite-frequency one if the parameter ωωω is replaced by an operator. An ensemble of NNN such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and NNN can be finite. Due to the operator character of ωωω the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large NNN one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail.
A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.
We study the quantum Pais-Uhlenbeck oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden su(2)su(2)su(2) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant Pais-Uhlenbeck oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.
We study the quantum Pais–Uhlenbeck (PU) oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden (Formula presented) (Formula presented) Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant PU oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.
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.