The canonical formulation of higher-order theories of gravity can only be accomplished by introducing additional degrees of freedom, namely, the extrinsic curvature tensor $K_{ij}$. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric $h_{ij}$ must also be fixed at the boundary. While in the Ostrogradsky, Dirac, and Horowitz formalisms the extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz formalism fixes the Ricci scalar $R$ instead. It has been taken for granted that the Hamiltonian structures corresponding to all of the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that this indeed is true, but only for a class of higher-order theories. However, for more general higher-order theories—e.g., dilatonic coupled Gauss-Bonnet gravity in the presence of a curvature-squared term—the Hamiltonian obtained following the 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 only the modified Horowitz’ formalism can produce a viable quantum description of the theory, since it only admits a classical analogue under an appropriate semiclassical approximation. Thus, fixing the Ricci scalar $R$ at the boundary appears to be a fundamental issue for a canonical formulation of higher-order theories of gravity.

• Received 8 May 2017

DOI:https://doi.org/10.1103/PhysRevD.96.084025