Classical and quantum stability of higher-derivative dynamics

Abstract

We observe that a wide class of higher-derivative systems admits a bounded integral of motion that ensures the classical stability of dynamics, while the canonical energy is unbounded.We use the concept of a Lagrange anchor to demonstrate that the bounded integral of motion is connected with the time-translation invariance. A procedure is suggested for switching on interactions in free higher-derivative systems without breaking their stability. We also demonstrate the quantization technique that keeps the higher-derivative dynamics stable at quantum level.The general construction is illustrated by the examples of the Pais-Uhlenbeck oscillator, higherderivative scalar field model, and the Podolsky electrodynamics. For all these models, the positive integrals of motion are explicitly constructed and the interactions are included such that keep the system stable.

A conducting surface in Lee–Wick electrodynamics

“…where Z C [J ], defined in (22), is the Lee-Wick functional generator with boundary conditions (5) and Z S [0] is the functional generator (43) for the field S with no boundary conditions and no external source. As discussed previously, in this case Z S [0] has no physical effect and we can write from Eq.…”

“…Substituting (56) in (54) and integrating out on U and V it can be shown that the two field functional generator with the presence of conducting plate (54) is equal to the right hand side of (22), namely…”

“…Since its proposal, the theory has been standing out by its classical as well as its quantum aspects, as it exhibits many interesting peculiarities. We can mention, for instance, the fact that in this electrodynamics the self-energy of a point charge is finite in 3 + 1 dimensions [7][8][9][10][11][12], a Dirac string can produce a magnetic field [7], it stem a finite theory closely related to the Pauli-Villars regularization scheme [5,8,[13][14][15][16][17][18] where the divergences of the quantum electrodynamics are controllable [19][20][21] and it exhibits classical dynamical stability [22]. These features have made the model a widely studied subject in a variety of scenarios, mainly regarding its extension to the Standard Model [14,16,18,[23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern–Simons

“…In the work [19], it was noticed that the broad class of higher-derivative models are stable at classical level because they admit conserved tensors with bounded 00-component. The bounded conserved quantity turns out different from the canonical energy which can be unbounded for the same dynamics.…”

Casimir effect in Podolsky electrodynamics: A functional approach

“…10 The main idea of this branch of higher-order derivatives is very simple; we construct additional higher-order terms in standard Lagrangians in such a way that it preserves the original symmetries of the problem. This can be seen in the generalization of Utijama's work.…”