Reply to `Comment on Gravity and the Poincare' Group'

TL;DR

This paper clarifies that the problematic constraint algebra in a Poincaré gauge theory model is not caused by the gauge formulation itself, and demonstrates how fixing an extra symmetry resolves the issue without altering the equations of motion.

Contribution

It shows that the non-linear symmetry is not the cause of the constraint algebra problem and that fixing this symmetry restores the proper Poincaré algebra.

Findings

The constraint algebra can be made to satisfy the Poincaré algebra after fixing the extra symmetry.

Fixing the additional symmetry does not change the equations of motion.

The objections raised by Strobl are addressed and shown to be immaterial.

Abstract

In the first order form, the model considered by Strobl presents, besides local Lorentz and diffeomorphism invariances, an additional local non-linear symmetry. When the model is realized as a Poincar\'e gauge theory according to the procedure outlined in Refs.[1,2], the generators of the non-linear symmetry are responsible for the ``nasty constraint algebra''. We show that not only the Poincar\'e gauge theoretic formulation of the model is not the cause of the emerging of the undesirable constraint algebra, but actually allows to overcome the problem. In fact one can fix the additional symmetry without breaking the Poincar\'e gauge symmetry and the diffeomorphisms, so that, after a preliminary Dirac procedure, the remaining constraints uniquely satisfy the Poincar\'e algebra. After the additional symmetry is fixed, the equations of motion are unaltered. The objections to our method raised by Strobl in Ref.[3] are then immaterial. Some minor points put forward in Ref.[3] are also discussed.