On the canonical equivalence between Jordan and Einstein frames

A longstanding issue is the classical equivalence between the Jordan and the Einstein frames, which is considered just a field redefinition of the metric tensor and the scalar field. In this work, based on the previous result that the Hamiltonian transformations from the Jordan to the Einstein frame are not canonical on the extended phase space, we study the possibility of the existence of canonical transformations. We show that on the reduced phase space -- defined by suitable gauge fixing of the lapse and shifts functions -- these transformations are Hamiltonian canonical. Poisson brackets are replaced by Dirac's brackets following the Bergman-Dirac's procedure. The Hamiltonian canonical transformations map solutions of the equations of motion in the Jordan frame into solutions of the equations of motion in the Einstein frame.