Noether currents in theories with higher derivatives and theories with differential field transformation in action (DFTA)

The article is devoted to the investigation of the Noether currents and integrals of motion in the special subclass of theories with higher field derivatives - theories under differential field transformations in action (DFTA). Under fairly general assertions, we derive a simple representation for the integrals as sums of old integrals of motion, terms that vanish on the old equations of motion and surface integrals. We show that for some cosmological theories of that kind with high-order derivatives the last contribution can violate gauge invariance, specifically, diffeomorphisms. Then we investigate ambiguity of the Noether procedure for higher derivative theories and fix it in the way the problem resolves in quite general setting. The obtained results are discussed for the several extensions of the General Relativity, which are obtained by DFTA, in particular, for mimetic, disformal and Regge-Teitelboim theories of gravity. In particular, we calculate Iyer-Wald black hole entropy from the corrected currents and also discuss the derivation of the integral constraints on Sachs-Wolfe effect.

Automated summary

This article investigates the Noether currents and integrals of motion in a specific subclass of theories with higher field derivatives, known as theories under differential field transformations in action (DFTA). The article derives a simple representation for the integrals, which consist of old integrals of motion, terms that vanish on the old equations of motion, and surface integrals. In certain cosmological theories with high-order derivatives, it is found that the surface integrals can violate gauge invariance, specifically diffeomorphisms. The article also addresses the ambiguity of the Noether procedure in higher derivative theories and resolves it in a general setting. The obtained results are discussed in the context of several extensions of General Relativity, including mimetic, disformal, and Regge-Teitelboim theories of gravity. The article concludes by calculating the Iyer-Wald black hole entropy from the corrected currents and discussing the derivation of integral constraints on the Sachs-Wolfe effect.