Equivalence Principle, Higher Dimensional Möbius Group and the Hidden Antisymmetric Tensor of Quantum Mechanics

We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one–dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Möbius invariance which in turn univocally fixes the quantum version of the Hamilton–Jacobi equation. This holds also in the relativistic case, so that we obtain both the time–dependent Schrödinger equation and the Klein–Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric two–tensor which underlies Quantum Mechanics and sheds new light on the nature of the Quantum Hamilton–Jacobi equation.