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, even if they can be separated outside the measurement periods.
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Notes
In QM the assignment of values to measurement results is contextual (Kochen-Specker theorem), whereas the assignment of probabilities to measurement results is non-contextual (Gleason theorem, see also [7]). This creates endless confusions in the terminology, and extracontextuality is a good way out of them. For projectors eigenvalues 0 and 1 are contextual, whereas average values (probabilities) are non-contextual; this is another way to tell that there is no dispersion-free state
In [14] an incomplete direct product (i.e. a a sector in modern terminology) is defined as an equivalence class between vectors belonging to the infinite direct product. This idea can be extended in the algebraic framework, and an essential feature is that the asymptotic information about the sector can be encoded in a concise way, without specifying the full algebra (see e.g. Chi-Keung Ng, “On genuine infinite algebraic tensor products”, http://arxiv.org/abs/arXiv:1112.3128)
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The author thanks Alexia Auffèves, Nayla Farouki, Franck Laloë and Roger Balian for many useful discussions.
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Grangier, P. Completing the Quantum Formalism in a Contextually Objective Framework. Found Phys 51, 76 (2021). https://doi.org/10.1007/s10701-021-00424-1
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DOI: https://doi.org/10.1007/s10701-021-00424-1