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

TL;DR

This paper investigates Noether currents in higher derivative theories with differential field transformations, revealing how surface terms can affect gauge invariance and clarifying ambiguities in the Noether procedure for extended gravity models.

Contribution

It provides a simple representation for Noether integrals in higher derivative theories and addresses gauge invariance issues in extended gravity models derived via DFTA.

Findings

Surface integrals can violate gauge invariance in some higher derivative theories.

The ambiguity in the Noether procedure is resolved in a general setting.

Results are applied to extended gravity theories like mimetic, disformal, and Regge-Teitelboim models.

Abstract

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 vanishes 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.