Conservation Laws and Stability of Field Theories of Derived Type

TL;DR

This paper explores the relationship between symmetries and conserved quantities in higher-derivative relativistic field theories, demonstrating the existence of multiple energy-momentum tensors and conditions for stability.

Contribution

It introduces a framework for constructing multiple conserved tensors in derived-type theories and links them to symmetries using the Lagrange anchor, extending understanding of stability in such models.

Findings

Multiple conserved energy-momentum tensors exist in derived systems.

Stability is achieved if the wave operator has simple, real roots.

Explicit examples include Pais-Uhlenbeck oscillator and higher-derivative scalar fields.

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.