Noether Currents and Maxwell-Type Equations of Motion in Higher Derivative Gravity Theories

Taichiro Kugo

doi:10.3390/sym13081408

Abstract

In general coordinate invariant gravity theories whose Lagrangians contain arbitrarily high order derivative fields, the Noether currents for the global translation and for the Nakanishi’s IOSp(8|8) choral symmetry containing the BRS symmetry as its member are constructed. We generally show that for each of these Noether currents, a suitable linear combination of equations of motion can be brought into the form of a Maxwell-type field equation possessing the Noether current as its source term.

Highlights

The equation of motion for the Yang-Mills field Aμa in the covariant gauge is given in the formCitation: Kugo, T
Where D ν F aμν is the covariant divergence of the field strength F aμν ; jμa is the color current from the matter field; and B a, c a and ca are Nakanishi-Lautrup (NL), Faddeev-Popov (FP)
Ghost, and anti-ghost fields, respectively. This equation was first noted by Ojima [1] to be rewritten into the form of the Maxwell-type equation of motion: Theories

Summary

Introduction

The equation of motion for the Yang-Mills field Aμa in the covariant gauge is given in the form. We will consider a general gravity theory which is invariant under the general coordinate (GC) transformation and contains arbitrarily high order derivatives of gravity and matter fields, and we derive a concrete form of the Noether current for the rigid translation, i.e., energy momentum tensor; derive the Maxwell-type gravity equation of motion in a gauge-unfixed, i.e., classical system; derive the Maxwell-type equation analogous to Equation (2) in a gauge-fixed quantum system in the de Donder–Nakanishi gauge; Symmetry 2021, 13, 1408. We derive an expression for the energy-momentum tensor for such a general system as the Noether current for the translation invariance and show that the gravity field equation of motion can be cast into the form of the Maxwell-type equation. In Appendix B, to obtain some familiarity with the OSp-symmetry, we briefly study the simplest model, a OSp(2|2)-invariant scalar field system on flat Minkowski background; the OSp(2|2) Noether current is derived, and the IOSp(2|2) algebra is confirmed from the canonical (anti-)commutation relations

Gravity Equation of Motion in a Generic Higher Derivative System

Quantum Theory with de Donder Gauge

Conclusions