Comment on "Arnowitt--Deser--Misner representation and Hamiltonian analysis of covariant renormalizable gravity" by M. Chaichian, M. Oksanen, A. Tureanu

TL;DR

This paper corrects the constraint algebra in a previous analysis of covariant renormalizable gravity, showing that the original claims about gauge invariance under spatial diffeomorphisms were incorrect due to an algebraic mistake.

Contribution

The authors identify and correct the constraint algebra in the Hamiltonian analysis of the referenced gravity models, clarifying their gauge invariance properties.

Findings

The true algebra of constraints differs from the original claim.

The actions are invariant under spatial diffeomorphisms despite the algebra correction.

The corrected algebra impacts the gauge transformations of the fields.

Abstract

The partial Hamiltonian analysis of the actions presented in the paper by M. Chaichian, M. Oksanen, A. Tureanu (Eur. Phys. J. C 71, 1657 (2011)) is incorrect; the true algebra of constraints differs from what they claim for their choice of momentum constraint. Our blind acceptance of the correctness of their constraint algebra led us to conclude, wrongly, that a few of the models presented by the authors (sharing the same constraint algebra) are not invariant under spatial diffeomorphism. We "proved" this by using Noether's second theorem (see first version of the paper), but we then found a mistake in our calculations. The differential identity of spatial diffeomorphism is intact, therefore, their actions are invariant; but in this case, the spatial diffeomorphism gauge symmetry cannot be compatible with their algebra. We now explicitly demonstrate that the actual algebra of constraints is different, and briefly describe how it affects the generator and gauge transformations of the fields.