Pais-Uhlenbeck Oscillators Research Articles

Dynamical symmetries of supersymmetric oscillators

In this paper, we describe the dynamical symmetries of classical supersymmetric oscillators in one and two spatial (bosonic) dimensions. Our main ingredient is a generalized Poisson bracket which is defined as a suitable classical counterpart to commutators and anticommutators. In one dimension, i.e., in the presence of one bosonic and one fermionic coordinate, the Hamiltonian admits a U(1, 1) symmetry for which we explicitly compute the first integrals. It is found that suitable forms of the supercharges emerge in a natural way as fermionic conserved quantities. Following this, we describe classical supercharge operators based on the generalized Poisson bracket and subsequently define supersymmetry transformations. We perform a straightforward generalization to two spatial dimensions where the Hamiltonian has an overall U(2, 2) symmetry. We comment on plausible supersymmetric generalizations of the Pais-Uhlenbeck and isotonic oscillators, and also present the possibility of defining a generalized Nambu bracket within the classical formalism.

Oscillators from nonlinear realizations

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 construction, we discuss the so(2, 3) and G2(2) algebras.

Entanglement of higher-derivative oscillators in holographic systems

We study the quantum entanglement of coupled Pais–Uhlenbeck oscillators using the formalism of thermo-field dynamics. The entanglement entropy is computed for the specific cases of two and a ring of N coupled Pais–Uhlenbeck oscillators of fourth order. It is shown that the entanglement entropy depends on the temperatures, frequencies and coupling parameters of the different degrees of freedom corresponding to harmonic oscillators. We also make remarks on the appearance of instabilities of higher-derivative oscillators in the context of AdS/CFT correspondence. Finally, we advert to the information geometry theory by calculating the Fisher information metric for the considered system of coupled oscillators.

Minimal realization of ℓ-conformal Galilei algebra, Pais-Uhlenbeck oscillators and their deformation

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. Schr\"odinger) 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.

Myers–Pospelov model as an ensemble of Pais–Uhlenbeck oscillators: unitarity and Lorentz invariance violation

We study a generalization of a Pais-Uhlenbeck oscillator for fermionic variables. Next, we consider an ensemble of these oscillators and we identify a particular case of the Myers-Pospelov model which is relevant for effective theories of quantum gravity. Finally, by taking the advantage of this connection, we analyze, for this model, the unitarity at one loop order in the low energy regime where no ghost states can be created on-shell. This energy regime is the relevant one when we consider the Myers-Pospelov model as a true effective theory coming from new space-time structure.

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler–Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais–Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.

Remarks on quantization of Pais–Uhlenbeck oscillators

This work is concerned with a quantization of the Pais-Uhlenbeck oscillators from the point of view of their multi-Hamiltonian structures. It is shown that the 2n-th order oscillator with a simple spectrum is equivalent to the usual anisotropic n - dimensional oscillator.