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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.
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As pointed out in Refs. [ 29 , 7 ] because PU-oscillator does not involve nonlinear interactions, strictly speaking the fact that the spectrum of the Hamiltonian is not bounded below does not cause any serious problem. However the inclusion of generic nonlinear interactions results in the emergence of an instability
Cited in the paper.
Throughout this article we use the term “correspondence principle” to mean a clear rule that assigns to each operator describing an observable of a quantum system a particular classical observable that is by definiton a real-valued function of the phase space of the undelying classical system. It is this correspondence that gives physical meaning to the quantum observables
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Smooth functions having a compact support belong to the domain of A A . Because these functions constitute a dense subset of L 2 ( ℝ ) L^{2}(\mathbb{R}) , A A is densely defined. It is also easy to see that A − 1 = 𝒫 A A^{-1}={\cal P}A . Therefore, as operators acting in L 2 ( ℝ ) L^{2}(\mathbb{R}) , the domains of A A and A − 1 A^{-1} coincide
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S. Kuzhel, private communication
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A 2 A^{2} is the restriction of the parity operator 𝒫 \mathcal{P} on the domain of A 2 A^{2} . The parity operator is defined by ( 𝒫 ψ ) ( x ) := ψ ( − x ) (\mathcal{P}\psi)(x):=\psi(-x) for every function ψ : ℝ → ℂ \psi:\mathbb{R}\to\mathbb{C} defined on ℝ \mathbb{R}
Cited in the paper.