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arXiv:0808.0139 · Symmetry Integrability and Geometry Methods and Applications

Comments on the Dynamics of the Pais-Uhlenbeck Oscillator

Andrei V. Smilga

Published Feb 12, 2009• 65 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3842/sigma.2009.017

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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.
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.
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.
We quantize a classically stable system of a harmonic oscillator polynomially coupled to a ghost with negative kinetic energy. We prove that due to an integral of motion with a positive discrete spectrum: i) the Hamiltonian has a pure point spectrum unbounded in both directions, ii) the evolution is manifestly unitary, iii) the vacuum is well-defined, iv) expectation values for squares of canonical variables are bounded. Numerical solutions of the Schrödinger equation confirm these results. We argue that the discrete spectrum of the integral of motion enforces stability for extended interactions.
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.
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.
Quantum dynamics of integrable systems is discussed. Localized wave packets generalizing the conventional coherent states of minimal uncertainty are constructed. The wave packet moves along a certain trajectory and does not change its shape for times of order 1 ¯h. In this Letter we suggest that the definition of a coherent state should be related to the Hamiltonian.The conventional coherent states [1] are defined as eigenstates of the annihilation operator α | z> = z | z> (1) The real and the imaginary parts of the complex variable z can be expressed by the mean values of the position ˆx and momentum ˆp operators in the coherent state. Under the oscillator Hamiltonian evolution Ut Ut|z> = |exp(−iωt)z> (2) where ω is the oscillator ′ √ s frequency. Then, because the state | z> is con-2¯h centrated around x = ω Rez it follows that Ut|z> has its support on the