Remarks on quantization of Pais-Uhlenbeck oscillators

We consider a Hamiltonian formulation of the (2n+1)-order generalization of the Pais-Uhlenbeck oscillator with distinct frequencies of oscillation. This system is invariant under time translations. However, the corresponding Noether integral of motion is unbounded from below and can be presented as a direct sum of 2n one-dimensional harmonic oscillators with an alternating sign. If this integral of motion plays a role of a Hamiltonian, a quantum theory of the Pais-Uhlenbeck o...

We offer a new Hamiltonian formulation of the classical Pais-Uhlenbeck Oscillator and consider its canonical quantization. We show that for the non-degenerate case where the frequencies differ, the quantum Hamiltonian operator is a Hermitian operator with a positive spectrum, i.e., the quantum system is both stable and unitary. A consistent description of the degenerate case based on a Hamiltonian that is quadratic in momenta requires its analytic continuation into a complex ...

Ostrogradsky's method allows one to construct Hamiltonian formulation for a higher derivative system. An application of this approach to the Pais-Uhlenbeck oscillator yields the Hamiltonian which is unbounded from below. This leads to the ghost problem in quantum theory. In order to avoid this nasty feature, the technique previously developed in [Acta Phys. Polon. B 36 (2005) 2115] is used to construct an alternative Hamiltonian formulation for the multidimensional Pais-Uhlen...

We discuss the quantum dynamics of the Pais-Uhlenbeck oscillator. The Lagrangian of this higher-derivative model depends on two frequencies. When the frequencies are different, the free PU oscillator has a pure point spectrum that is dense everywhere. When the frequencies are equal, the spectrum is continuous. It is not bounded from below, running from minus to plus infinity, but this is not disastrous as the Hamiltonian is still Hermitian and the evolution operator is still ...

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 pos...

A system of two independent Bosonic Harmonic Oscillators is converted into the respective fourth-order derivative Pais-Uhlenbeck oscillator model. The conversion procedure displays transparently how the quantization of the fourth-order derivative Pais-Uhlenbeck oscillator has to be performed in order not to suffer from the divergence problems of the vacuum state and path integrals as conjectured most recently by P. D. Mannheim in his article ``Determining the normalization of...

We review the classical and quantum theory of the Pais-Uhlenbeck oscillator as the toy-model for quantizing f(R) gravity theories.

The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it becomes defective and can be written in canonical Jordan form. It is shown that any quadratic function of $K$ coordinates and $K$ momenta leads to a $2K$ differential equation for those dynamical variables. We illustrate all these features o...

We construct an N=2 supersymmetric extension of the Pais-Uhlenbeck oscillator for distinct frequencies of oscillation. A link to a set of decoupled N=2 supersymmetric harmonic oscillators with alternating sign in the Hamiltonian is introduced. Symmetries of the model are discussed in detail. The investigation of a quantum counterpart of the constructed model shows that the corresponding Fock space contains negative norm states and the energy spectrum of the system is unbounde...

We consider an N=2 supersymmetric odd-order Pais-Uhlenbeck oscillator with distinct frequencies of oscillation. The technique previously developed in [Acta Phys. Polon. B 36 (2005) 2115; Nucl. Phys. B 902 (2016) 95] is used to construct a family of Hamiltonian structures for this system.