Ostrogradsky-Hamilton approach to geodetic brane gravity

TL;DR

This paper develops an Ostrogradsky-Hamilton formalism for geodetic brane gravity, revealing new insights into its constraints, gauge structure, and degrees of freedom by treating it as a second-order derivative theory with boundary terms.

Contribution

It introduces a novel Ostrogradsky-Hamiltonian approach to geodetic brane gravity, accounting for boundary terms and second-order derivatives, which was not considered in previous treatments.

Findings

Full set of phase space constraints identified

Gauge transformations generated by constraints determined

Physical degrees of freedom counted and analyzed

Abstract

We develop the Ostrogradsky-Hamilton formalism for geodetic brane gravity, described by the Regge-Teitelboim geometric model in higher codimension. We treat this gravity theory as a second-order derivative theory, based on the extrinsic geometric structure of the model. As opposed to previous treatments of geodetic brane gravity, our Lagrangian is linearly dependent on second-order time derivatives of the field variables, the embedding functions. The difference resides in a boundary term in the action, usually discarded. Certainly, this suggests applying an appropriate Ostrogradsky-Hamiltonian approach to this type of theories. The price to pay for this choice is the appearance of second class constraints. We determine the full set of phase space constraints, as well as the gauge transformations they generate in the reduced phase space. Additionally, we compute the algebra of constraints and explain its physical content. In the same spirit, we deduce the counting of the physical degrees of freedom. We comment briefly on the naive formal canonical quantization emerging from our development.