Conservation Laws and Stability of Field Theories of Derived Type

Abstract

We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time gives rise to the series of symmetries of action functional. If the wave operator is given by n-th-order polynomial then this series includes n independent entries, which can be explicitly constructed. The Noether theorem is then used to construct an n-parameter set of second-rank conserved tensors. The canonical energy-momentum tensor is included in the series, while the other entries define independent integrals of motion. The Lagrange anchor concept is applied to connect the general conserved tensor in the series with the original space-time translation symmetry. This result is interpreted as existence of multiple energy-momentum tensors in the class of derived systems. To study stability we seek for bounded-conserved quantities that are connected with the time translations. We observe that the derived theory is stable if its wave operator is defined by a polynomial with simple and real roots. The general constructions are illustrated by the examples of the Pais-Uhlenbeck oscillator, higher-derivative scalar field, and extended Chern-Simons theory.

Nonlocal constants of motion in Lagrangian Dynamics of any order

“…If combined, K PU 1 and K PU 2 recover the two first integrals recently proposed by [15]. Another interesting situation generating first integrals arises when the Lagrangian, evaluated on a perturbed motion q λ (t), has constant derivative at λ = 0.…”

“…Approaching higher-order mechanics from a new nonlocal point of view provides new perspectives to identify novel first integrals without necessarily requiring invariance proprieties on the already difficult to investigate structure of higher-order Lagrangians. Hopefully Theorem 1.2 will provide a valuable tool to give a novel insight into stability proprieties of higher-order models and boundedness of related solutions as is done for second order equations in Kaparulin [16,15] and in the papers mentioned above.…”

Stability in the higher derivative Abelian gauge field theory

“…Now although the canonical energy is unbounded due to the nature of higher derivatives, the linear combination of these tensors (T k ) ν µ may give rise to bounded conserved charge which will stabilize the classical dynamics of derived model at free level, which also persists at quantum level. Moreover, as demonstrated at length in [40,41], when these conserved tensors are bounded in the free theory, the inclusion of consistent interactions will not spoil the stability of the coupling systems, at least at perturbative level.…”

“…especially, k = 0 corresponds to the spacetime translations invariance of the action of the derived model. Now the Noether's theorem tells us that each continuous symmetry of the action (2.9) determines a conserved quantity [40,41]…”

“…After that, a more general and systematic method was carried out as a guide to investigate the stabilities in a wide class of higher derivative systems named derived type theories [40][41][42][43][44]. Generally speaking, these derived theories are based on simpler free primary models whose equations of motion only involve first and second order differential operators without higher derivatives.…”