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arXiv:quant-ph/0612009

Comment on "Dirac Quantization of Pais-Uhlenbeck Fourth Order Oscillator"

Katarzyna Bolonek-Lasoń, Kosinski, Piotr

Published Dec 1, 2006• 3 citations
arXivHTMLPDF

Abstract

The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the limit is defined, the limiting theory can be hardly viewed as satisfactory.

DOI
10.48550/arxiv.quant-ph/0612009

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The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the limit is defined, the limiting theory can be hardly viewed as satisfactory.
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.
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 discuss the quantum dynamics of the PU oscillator, i.e. the system with the LagrangianWhen Ω 1 = Ω 2 , the free PU oscillator has a pure point spectrum that is dense everywhere.When Ω 1 = Ω 2 , the spectrum is continuous, E ∈ {-∞, ∞}.The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary.Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved.We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates.The free nonstandard PU oscillator is unitary at Ω 1 = Ω 2 , but unitarity is broken in the equal frequencies limit.
Li and Miao [Phys. Rev. A 85, 042110 (2012)] proposed a non-Hermitian Hamiltonian that is neither Hermitian nor PTPTPT symmetric but exhibits real eigenvalues for some values of the model parameters. In order to explain this fact, they resorted to PTPTPT-pseudo Hermiticity and to a so-called permutation symmetry. Here we show that the spectrum of this Hamiltonian can be easily analyzed in the usual way in terms of exact or broken antiunitary symmetries that appear to be more relevant than the permutation symmetry. In addition, we show why the authors' Hamiltonian and the well-known Pais-Uhlenbeck oscillator lead to the same fourth-order differential equation for the coordinates.