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Noether’s Theorem and Symmetry
Open AccessArticle

Conservation Laws and Stability of Field Theories of Derived Type

Faculty of Physics, Tomsk State University, Tomsk 634050, Russia
Symmetry 2019, 11(5), 642;
Received: 4 April 2019 / Revised: 29 April 2019 / Accepted: 5 May 2019 / Published: 7 May 2019
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
PDF [256 KB, uploaded 7 May 2019]


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. View Full-Text
Keywords: Noether’s theorem; generalized symmetry; energy-momentum tensor; Lagrange anchor Noether’s theorem; generalized symmetry; energy-momentum tensor; Lagrange anchor
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Kaparulin, D.S. Conservation Laws and Stability of Field Theories of Derived Type. Symmetry 2019, 11, 642.

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