Higher time derivatives in the microcanonical ensemble describe dynamics of flux-coupled classical and quantum oscillators

Abstract

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 (hyper)surface. We provide as concrete example of these statements the Pais--Uhlenbeck oscillator and show that it can describe a system that makes sense in the microcanonical ensemble. It describes two oscillators that are coupled by imposing a fixed phase difference, that thereby describes a non--local interaction between them. The consistent quantum dynamics can straightforwardly be expressed using two pairs of creation and annihilation operators, with the phase difference describing a flux, that describes the interaction. The properties of the action imply that particular solutions, that would describe independent oscillators, are, in general, not admissible.The reason is that the coordinate transformation, that would decouple the oscillators isn't a symmetry of the action--unless a "BPS bound" is saturated. Only then do they decouple. But, in these cases, the action does describe one, not two, oscillators, anyway and the higher derivative term is a coordinate artifact.