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Completing the quantum formalism in a contextually objective framework

Philippe Grangier 1
1 Laboratoire Charles Fabry / Optique Quantique
LCF - Laboratoire Charles Fabry
Abstract : In 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.
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Submitted on : Monday, November 29, 2021 - 4:14:06 PM
Last modification on : Tuesday, January 4, 2022 - 6:13:34 AM

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Philippe Grangier. Completing the quantum formalism in a contextually objective framework. Foundations of Physics, Springer Verlag, 2021, 51 (4), pp.76. ⟨10.1007/s10701-021-00424-1⟩. ⟨hal-03034473⟩

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