The equivalence postulate of quantum mechanics.

*(English)* Zbl 0978.81002
Summary: The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently formulated equivalence principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, uniquely leads to the quantum stationary HJ equation (QSHJE). This is a third-order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest $p-q$ duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic ${\psi}^{D}$-$\psi $ duality between linearly independent solutions of the Schrödinger equation. Unlike the case for Bohm’s theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of a 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 itself. This is an important feature as the ${L}^{2}\left(R\right)$ condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ${\psi}^{D}/\psi $ at $q=\pm \infty $ 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-stationary higher-dimensional quantum HJ equation and the relativistic extension are derived.

##### MSC:

81P05 | General and philosophical topics in quantum theory |