The equivalence postulate of quantum mechanics

The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently form ulated equivalence principle (EP). The latter, stating that all physical sy stems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the quantum stationary HJ equation (Q SHJE). This is a third order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories de pend on the Planck length through hidden variables which arise as initial c onditions. The formulation has manifest p - q duality, a consequence of the involutive nature of the Legendre transformation and of its recently obser ved relation with second order linear differential equations. This reflects in an intrinsic psi(D) - psi duality between linearly independent solution s of the Schrodinger equation. Unlike Bohm's theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic pro perty of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence o f the quantum potential which differs from the Bohmian one and plays the ro le of particle's self-energy. Furthermore, the QSHJE is defined only if the ratio psi(D)/psi is a local homeomorphism of the extended real line into i tself. This is an important feature as the L-2(R) condition, which in the C openhagen formulation is a consequence of the axiomatic interpretation of t he wave function, directly follows as a basic theorem which only uses the g eometrical gluing conditions of psi(D)/psi at q = +/-infinity as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunneling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non stationa ry higher dimensional quantum HJ equation and the relativistic extension ar e derived.