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.
The "Bargmann framework" allows us to prove the equivalence of the Pais-Uhlenbeck oscillator with a spin-zero particle in a circularly polarized periodic gravitational wave. Pushing forward the symmetries (which include Carroll symmetry) of the latter, we derive conserved charges for the Pais-Uhlenbeck oscillator.
We review the occurrence of negative energies in Pais-Uhlenbeck oscillator. We point out that in the absence of interactions negative energies are not problematic, neither in the classical nor in the quantized theory. However, in the presence of interactions that couple positive and negative energy degrees of freedom the system is unstable, unless the potential is bounded from bellow and above. We review some approaches in the literature that attempt to avoid the problem of negative energies in the Pais-Uhlenbeck oscillator.
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 tha…
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 Hamiltonian system possessing a generalized PT-symmetry (an involutive antilinear symmetry). We devise a real descriptio…
We have constructed coherent states for the higher derivative Pais-Uhlenbeck Oscillator. In the process we have suggested a novel way to construct coherent states for the oscillator having only negative energy levels. These coherent states have negative energies in general but their coordinate and momentum expectation values and dispersions behave in an identical manner as that of normal (positive energy) oscillator. The coherent states for the Pais-Uhlenbeck Oscillator have constant dispersions and a modified Heisenberg Uncertainty Relation. Moreover, under reasonable assumptions on parameter…
Two Lagrangian formulations for describing of the damped harmonic oscillator have been introduced by Bateman. For these models we construct higher derivative generalization which enjoys the l-conformal Newton-Hooke symmetry. The dynamics of generalized systems corresponds to the damped Pais-Uhlenbeck oscillator for a particular choice of its frequencies.
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-Uhlenbeck oscillator of arbitrary even order with distinct frequencies of oscillation. This construction is also generalized …
We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck oscillator, using Lie-Deprit series and Hamiltonian normal form theories. In particular, we explicitly describe the reduced phase space for this Hamiltonian system and give a proof for the existence of stable orbits for a certain class of self-interaction, found numerically in previous works, by using singular symplectic reduction.
We present the minimal realization of the $\ell$-conformal Galilei group in 2+1 dimensions on a single complex field. The simplest Lagrangians yield the complex Pais-Uhlenbeck oscillator equations. We introduce a minimal deformation of the $\ell$=1/2 conformal Galilei (a.k.a. Schrödinger) algebra and construct the corresponding invariant actions. Based on a new realization of the d=1 conformal group, we find a massive extension of the near-horizon Kerr-dS/AdS metric.
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.
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 unbounded from below.
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 oscillator faces a ghost problem. We construct an alternative canonical formulation for the system under study to avoid t…
It is demonstrated that the Pais-Uhlenbeck oscillator in arbitrary dimension enjoys the l-conformal Newton-Hooke symmetry provided frequencies of oscillation form the arithmetic sequence omega_k=(2k-1) omega_1, where k=1,...,n, and l is the half-integer (2n-1)/2. The model is shown to be maximally superintegrable. A link to n decoupled isotropic oscillators is discussed and an interplay between the l-conformal Newton-Hooke symmetry and symmetries characterizing each individual isotropic oscillator is analyzed.
The study of the symmetry of Pais-Uhlenbeck oscillator initiated in [Nucl. Phys. B 885 (2014) 150] is continued with special emphasis put on the Hamiltonian formalism. The symmetry generators within the original Pais and Uhlenbeck Hamiltonian approach as well as the canonical transformation to the Ostrogradski Hamiltonian framework are derived. The resulting algebra of generators appears to be the central extension of the one obtained on the Lagrangian level; in particular, in the case of odd frequencies one obtains the centrally extended l-conformal Newton-Hooke algebra. In this important cas…
Dynamical systems invariant under the action of the l-conformal Newton-Hooke algebras are constructed by the method of nonlinear realizations. The relevant first order Lagrangians together with the corresponding Hamiltonians are found. The relation to the Galajinsky and Masterov [Phys. Lett. B 723 (2013) 190] approach as well as the higher derivatives formulation is discussed. The generalized Niederer's transformation are presented which relate the systems under consideration to those invariant under the action of the l-conformal Galilei algebra [Nucl. Phys. B 876 (2013) 309]. As a nice applic…
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 unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preser…
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of unco…
We analyse the emergence of the Unruh effect within the context of a field Lagrangian theory associated to the Pais-Uhlenbeck fourth order oscillator model. To this end, we introduce a transformation that brings the Hamiltonian bounded from below and is consistent with $\mathcal{PT}$-symmetric quantum mechanics. We find that, as far as we consider different frequencies within the Pais-Uhlenbeck model, a particle together with an antiparticle of different masses are created as may be traced back to the Bogoliubov transformation associated to the interaction between the Unruh-DeWitt detector and…
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck \cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {\em ghosts} in the properties of the quantum mechanical solution. Bender and Mannheim \cite{Bender 08 a} were successful in persuading the corresponding quantum operator to `give up the ghost'. Their success had the advantage of making the model of Pais-Uhlenbeck acceptable to the physical community and in the process added further credit to the cause of ad…
By adding an imaginary interacting term proportional to ip_1p_2 to the Hamiltonian of a free anisotropic planar oscillator, we construct a new model which is described by the PT-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our PT-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positiv…
We construct the systems of the harmonic and Pais-Uhlenbeck oscillators, which are invariant with respect to arbitrary noncompact Lie algebras. The equations of motion of these systems can be obtained with the help of the formalism of nonlinear realizations. We prove that it is always possible to choose time and the fields within this formalism in such a way that the equations of motion become linear and, therefore, reduce to ones of ordinary harmonic and Pais-Uhlenbeck oscillators. The first-order actions, that produce these equations, can also be provided. As particular examples of this cons…
Motivated by the considerable success of alternative theories of gravity, we consider the toy model of a higher derivative Lagrangian theory, namely the Pais-Uhlenbeck oscillator studied in a recent paper by Hawking-Hertog. Its Euclidean Path Integral is studied with a certain detail and a pedagogical derivation of the propagator, which makes use of a Theorem due to Forman, is consequently proposed
We show that it is possible to consistently describe dynamical systems, whose equations of motion are of degree higher than two, in the microcanonical ensemble, even if the higher derivatives aren't coordinate artifacts. Higher time derivatives imply that there are more than one Hamiltonians, conserved quantities due to time translation invariance, and, if the volume in phase space, defined by their intersection, is compact, microcanonical averages can be defined and there isn't any instability, in the sense of Ostrogradsky, even though each Hamiltonian, individually, may define a non-compact …
