Cryptoreality of nonanticommutative Hamiltonians

Ivanov, E.; Smilga, Andrei

doi:10.1088/1126-6708/2007/07/036

Cited by 18 publications

(29 citation statements)

References 28 publications

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“…With respect to the new inner product appropriate for the PT-symmetric phase of the Lee model, the Hamiltonian becomes self-adjoint and its ghost state is reinterpreted as an ordinary quantum state with positive PT norm. This same procedure has been applied to other problematic models [10].…”

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confidence: 99%

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

2008

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This is the published version of the paper.This version of the publication may differ from the final published version. A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed. This realization possesses no states of negative norm and has a real energy spectrum that is bounded below. The key to this construction is the recognition that in this realization the Hamiltonian is not Dirac Hermitian. However, the Hamiltonian is symmetric under combined space reflection P and time reversal T. The Hilbert space that is appropriate for this PT-symmetric Hamiltonian is identified and it is found to have a positive-definite inner product. Furthermore, the time-evolution operator is unitary. Permanent

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confidence: 99%

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

2008

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“…The periods (14,15) are all inter-related under rotations in the complex E plane (the monodromies) 5 . For example,…”

Section: Complex Cubic Potential and Elliptic Dynamicsmentioning

confidence: 99%

Exceptional points in quantum and classical dynamics

Smilga

2009

J. Phys. A: Math. Theor.

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We note that when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits singular behaviour associated with the restructuring of classical trajectories. A system with the crypto-Hermitian Hamiltonian H = (p 2 + z 2 )/2−igz 5 and hyper-elliptic classical dynamics is studied in detail. Analogies with supersymmetric Yang-Mills dynamics are elucidated.

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“…Let us start by reviewing the example of a modified harmonic oscillator subject to a complex canonical transformation [24,25]. This has also been studied in a form that exploits the PT symmetry in the framework of the non-Hermitian Hamiltonian approach [23,32,33,34]. The strategy is based on the observation that the complex Lagrangian…”

Section: Reality Conditions and The Complex Oscillatormentioning

confidence: 99%

An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

Dector

Morales‐Técotl²,

Urrutia³

et al. 2009

SIGMA

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Abstract. Our purpose in this paper is to analyze the Pais-Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais-Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level.

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Cryptoreality of nonanticommutative Hamiltonians

Cited by 18 publications

References 28 publications

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

Exceptional points in quantum and classical dynamics

An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator

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