Equivalent and inequivalent canonical structures of higher order theories of gravity

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.