Equivalent and inequivalent canonical structures of higher order theories of gravity

TL;DR

This paper investigates the canonical structures of higher order gravity theories, showing that different formalisms are equivalent only for certain classes, and that for more complex theories, their Hamiltonians are not canonically related, affecting quantum dynamics.

Contribution

It demonstrates the non-equivalence of Hamiltonian structures in various formalisms for complex higher order gravity theories, highlighting the limitations of existing assumptions.

Findings

Hamiltonian structures are equivalent only for a specific class of theories.

For complex theories like dilatonic Gauss-Bonnet gravity, Hamiltonians differ across formalisms.

Quantum descriptions are viable but dynamics differ, preventing a unique formalism choice.

Abstract

Canonical formulation of higher order theory of gravity can only be accomplished associating additional degrees of freedom, which are extrinsic curvature tensor. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric, must also be fixed at the boundary. While, in all the three, viz. Ostrogradski's, Dirac's and Horowitz' formalisms, extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz' formalism fixes Ricci scalar, instead. It has been taken as granted that the Hamiltonian structure corresponding to all the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that indeed it is true, but only for a class of higher order theory. However, for more general higher order theories, e.g. dilatonic coupled Gauss-Bonnet gravity in the presence of curvature squared term, the Hamiltonian obtained following modified Horowitz' formalism is found to be different from the others, and is not related under canonical transformation. Further, it has also been demonstrated that although all the formalisms produce viable quantum description, the dynamics is different and not canonically related to modified Horowitz' formalism. Therefore it is not possible to choose the correct formalism which leads to degeneracy in Hamiltonian.