If one takes seriously the postulate of quantum mechanics in which physicalstates are rays in the standard Hilbert space of the theory, one is naturallylead to a geometric formulation of the theory. Within this formulation ofquantum mechanics, the resulting description is very elegant from thegeometrical viewpoint, since it allows to cast the main postulates of thetheory in terms of two geometric structures, namely a symplectic structure anda Riemannian metric. However, the usual superposition principle of quantummechanics is not naturally incorporated, since the quantum state space isnon-linear. In this note we offer some steps to incorporate the superpositionprinciple within the geometric description. In this respect, we argue that itis necessary to make the distinction between a 'projective superpositionprinciple' and a 'decomposition principle' that extend the standardsuperposition principle. We illustrate our proposal with two very well knownexamples, namely the spin 1/2 system and the two slit experiment, where thedistinction is clear from the physical perspective. We show that the twoprinciples have also a different mathematical origin within the geometricalformulation of the theory.
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