https://hal-iogs.archives-ouvertes.fr/hal-03034473Grangier, PhilippePhilippeGrangierLaboratoire Charles Fabry / Optique Quantique - LCF - Laboratoire Charles Fabry - IOGS - Institut d'Optique Graduate School - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueCompleting the quantum formalism in a contextually objective frameworkHAL CCSD2021[PHYS.PHYS.PHYS-OPTICS] Physics [physics]/Physics [physics]/Optics [physics.optics]Edwards, Marie-Laure2021-11-29 16:14:062023-03-24 14:53:242021-12-17 14:15:15enJournal articleshttps://hal-iogs.archives-ouvertes.fr/hal-03034473/document10.1007/s10701-021-00424-1application/pdf1In standard quantum mechanics (QM), a state vector $| \psi \rangle$ may belong to infinitely many different orthogonal bases, as soon as the dimension $N$ of the Hilbert space is at least three. On the other hand, a complete physical observable $A$ (with no degeneracy left) is associated with a $N$-dimensional orthogonal basis of eigenvectors. In an idealized case, measuring $A$ again and again will give repeatedly the same result, with the same eigenvalue. Let us call this repeatable result a modality $\mu$, and the corresponding eigenstate $| \psi \rangle$. A question is then: does $| \psi \rangle$ give a complete description of $\mu$ ? The answer is obviously no, since $| \psi \rangle$ does not specify the full observable $A$ that allowed us to obtain $\mu$; hence the physical description given by $| \psi \rangle$ is incomplete, as claimed by Einstein, Podolsky and Rosen in their famous article in 1935. Here we want to spell out this provocative statement, and in particular to answer the questions: if $| \psi \rangle$ is an incomplete description of $\mu$, what does it describe ? is it possible to obtain a complete description, maybe algebraic ? Our conclusion is that the incompleteness of standard QM is due to its attempt to describe systems without contexts, whereas both are always required; they can be separated only outside the measurement periods.