One of the most fascinating gedankenexperiment (thought experiments) of physics was formulated by Einstein and two of his collaborators, Podolsky and Rosen (EPR), in the 1930s. A lot of ink followed, but basically most physicists today believe that the argument of EPR has been superseded by the early experiments of Aspect on pairs of entangled photons, and by other similar experiments that followed, which proved, through the violation of the famous Bell’s inequalities, that entanglement is a reality of the physical world with which we have to deal.
On the other hand, EPR, in their reasoning, though not consciously, had pointed their finger at something deeper about quantum physics, which had nothing to do with the experimental discovery of quantum entanglement. The question is very subtle, and this explains why it took almost fifty years from the article of EPR (which was published in 1935) to understand all the consequences of their reasoning, which in fact was a reasoning ex absurdum.
The understanding came thanks to the work of the Belgian physicist Diederik Aerts, who in his research doctorate (we are in the early eighties) tried to understand, at a fundamental level, which mathematical structure was possessed by a theory capable of describing composite systems, formed by multiple entities, such as bipartite systems consisting of two physical entities. In doing so, he obviously began to study the simplest situations, and the simplest of all was obviously that of a bipartite system whose parts were separated, in the sense that the measurement of physical quantities carried out on one of the two parts could not influence in whatsoever way the measurement of physical quantities carried out on the other part. This is obviously a common situation in our daily reality, composed of macroscopic bodies with spatial properties.
But Aerts’ big surprise was that he was able to prove that the quantum formalism was incapable of describing this simple situation of two separate entities, and more generally the situation of separate measurement processes, that is, measurements that do not influence each other. In other words, EPR, which in their reasoning had implicitly assumed that quantum theory was instead able to describe situations of separation, thus arriving at their paradox, had put their finger on a type of incompleteness that at that time, understandably, they did not at all hypothesized.
The elements of reality that quantum theory was unable to describe (and for this reason was to be considered incomplete) did not have to do with the position and momentum of the elementary entities, but with the possibility of describing separate processes of measurement.
In my opinion, it is surprising that today the scientific community has not been able to appreciate the solution of the EPR paradox proposed by Aerts. There are various reasons I think for this, for example the fact that the mathematical approach that Aerts used, and which originated in the works of the school of quantum physics in Geneva, and more particularly in those of Constantin Piron, is different from that taught in traditional physics manuals.
One of the differences lies in the fact that the axiomatic construction of the theory does not start by postulating the Hilbert space, as the space of the states of a physical system. The primary focus of the Geneva approach, or rather, of the Geneva-Brussels approach, are the experimental tests and the properties that they allow to define in an operational way. The space of states is something that must be reconstructed afterwards, and only appropriate axioms on the lattice of the properties of a system are able to give life to a Hilbertian structure, which is usually taken for granted.
Now, this more fundamental approach is exactly what enabled Aerts to realize the incompleteness of quantum mechanics. Not by an absurd reasoning, like the one formulated by EPR, but by an explicit highlighting of the missing structure in the mathematical formalism. In essence, a number of axioms were needed to reconstruct the Hilbertian quantum state space. However, two of them were flagrantly violated by separate systems. Therefore, quantum mechanics could not describe such experimental situations.
In the short article that I have linked below, I wanted to bring to the attention of the community of physicists this neglected but important result of Aerts, explaining in a simple way its content and the implications. Probably a good part of the article can also be read by non-physicists.