Quantum superposition principle and geometry

Abstract

Within the Geometrical Formulation of Quantum Mechanics, quantum states are rays in the standard Hilbert space of the theory. The resulting formulation is very elegant from the geometrical viewpoint, since it allows to cast the main postulates of the theory in terms of two geometric structures, namely a symplectic structure and a Riemannian metric. However, the superposition principle of quantum mechanics is not naturally incorporated, since the quantum state space is non-linear. In this note we offer some steps to incorporate the superposition principle within the geometric description. In this respect, we argue that it is necessary to make the distinction between a projective superposition principle and a decomposition principle that should replace the standard superposition principle. We illustrate our proposal with two very well known examples, namely the spin 1/2 system and the two slit experiment, where the distinction is clear.