Briefly:

In particular, quantum theory missed asymptotic superselection rules which restrict the action of the superposition principle in some scattering problems for closed systems with asymptotically free dynamics.

In details:

As is known, when discussing Bell inequalities, it is very important to identify all (explicit and implicit) assumptions that are important for deriving these inequalities. Our goal is to draw attention to one assumption that is used in the analysis of these inequalities, but from our point of view it is erroneous. The fact is that at the present time it is customary to believe that modern quantum mechanics is a complete, internally consistent theory. In particular, it is considered as an indisputable truth its provision according to which all vector states of particles, including superpositions of single-particle states localized in the disjoint (macroscopically distinguishable) spatial regions, as well as vector states of the EPR-pair (the pair of electrons separated by a space-like interval) are always pure quantum states.

Our idea is that modern quantum theory does not possess this property (and the formulation of the problem of its interpretation is premature because the formulation of this problem is reasonable only when the mathematical apparatus of QM is completed and internally consistent). Namely, there is reason to believe that the question of the legality of the unlimited use of the superposition principle in some single-particle quantum processes, as well as in the theory of EPR pairs, is not properly well-regulated in modern quantum theory. More concretely, we believe that the issue of “purity” of vector states in some cases should be reconsidered due to super-selection rules that restrict the principle of superposition; ultimately, this makes the concept of collapse of the wave function (vector state) unnecessary in such cases.

Using the example of a completed scattering of a particle on a one-dimensional symmetric potential barrier [1] (arXiv: 1805.03952v5; ), we showed that the quantum dynamics of a particle in this process is governed by the asymptotic superselection rule, which restricts the action of the superposition principle in this one-particle process. According to this rule, the wave function describing the (vector) state of a scattering particle – the scattering state – is a mixed vector state. But, unlike ‘ordinary’ mixed states described by the density operator (whose square is not equal to the operator itself), no observable can be introduced for this vector state (that is, for the whole quantum ensemble of scattering particles). This can be done only for the sub-ensembles of transmitted and reflected particles, the individual dynamics of which at all stages of scattering are reconstructed in [1] according to the in- and out-asymptotes of a scattering state (their dynamics can be experimentally investigated only indirectly).

The theory [1] of mixed vector states arising in the one-dimensional completed scattering process contains some features of well-known approaches to fundamental problems of quantum mechanics, but does not coincide with any of them.

Examples:

(1) The well-known asymptotic superselection rules destinated to convert pure quantum states into mixed ones, were obtained for open quantum systems. Whereas, the asymptotic superselection rule [1] was obtained for a closed quantum system.

(2) In the theory of the collapse of the wave function, non-unitarity plays a key role. In [1], the dynamics of the transmission subprocess is non-unitary (the dynamics of the process itself is unitary, but the superselection rule forbids introducing observables for it).

(3) In the GRW theory, nonlinear effects play a key role in the transformation of pure quantum states into mixed ones. In [1], although the dynamics of the process itself is linear, the dynamics of its transmission subprocesses is nonlinear. The point is that the incoming and outgoing waves, in the stationary state to describe this subprocess, are “stitched” together in the center of the finite spatial region (where a symmetric barrier is nonzero) with making use of nonlinear conditions of continuity — they assume the continuity of this subprocess’ wave function and the continuity of the corresponding probability flow density (but do not imply the continuity of the first derivative of this wave function at this point).

We note that in case of the “uncompleted scattering process” of a particle on a one-dimensional potential barrier, the asymptotic superselection rule does not appear.

Summary: the proposed approach takes all the “good” of the known approaches to the fundamental problems of quantum mechanics, but respects the very quantum mechanics with its base concept of ‘closed systems’ and linear unitary dynamics; it shows that the notion of mixed vector states is that tool which can reconcile the linear unitary quantum theory of closed systems with classical physics.