Equivalent and inequivalent canonical structures of higher order theories of gravity

摘要

Canonical formulation of higher order theory of gravity can only beaccomplished associating additional degrees of freedom, which are extrinsiccurvature tensor. Consequently, to match Cauchy data with the boundary data,terms in addition to the three-space metric, must also be fixed at theboundary. While, in all the three, viz. Ostrogradski's, Dirac's and Horowitz'formalisms, extrinsic curvature tensor is kept fixed at the boundary, amodified Horowitz' formalism fixes Ricci scalar, instead. It has been taken asgranted that the Hamiltonian structure corresponding to all the formalisms withdifferent end-point data are either the same or are canonically equivalent. Inthe present study, we show that indeed it is true, but only for a class ofhigher order theory. However, for more general higher order theories, e.g.dilatonic coupled Gauss-Bonnet gravity in the presence of curvature squaredterm, the Hamiltonian obtained following modified Horowitz' formalism is foundto be different from the others, and is not related under canonicaltransformation. Further, it has also been demonstrated that no other than themodified Horowitz' formalism can produce a viable quantum description of thetheory, since it only admits classical analogue under appropriatesemi-classical approximation. Thus fixing Ricci scalar at the boundary appearsto be a fundamental issue for canonical formulation of higher order theory ofgravity.