Superselection rule in the problem of scattering a particle on a one-dimensional delta-potential

Recently, discussions of fundamental problems of (non-relativistic) quantum mechanics usually concern Bell’s inequalities. As for the discussions around the Schrödinger’s cat paradox, they faded into the background. At the same time, the discussion of Bell’s inequalities suggests that the Schrödinger’s cat paradox has already been resolved. Therefore, it is worth returning to the Schrödinger’s cat paradox to make sure that, apart from the well-known solution based on the concept of an “external observer,” no other logically possible solutions to this paradox (appealing to closed quantum systems) have been overlooked.

As it follows from this paradox, the quantum mechanical superposition principle is incompatible with the laws of classical mechanics on scales exceeding atomic scales. Consequently, there must be some mechanism that limits the validity of the superposition principle in quantum systems, where similar paradoxes arise, but which is not taken into account in the standard formalism of quantum mechanics. Here the question arises: “Does this mechanism already exist at the microlevel or does it arise ‘somewhere along the way from the microlevel to the macrolevel’?” It is now generally accepted that at the microlevel the validity of the superposition principle for closed one-particle systems is beyond doubt, and this mechanism (the decoherence mechanism) appears “somewhere along the way”.

However, there is reason to believe that for closed microsystems with a finite number of degrees of freedom, restrictions on the superposition principle already exist at the microlevel, and these restrictions are dictated by superselection rules. These rules should arise in quantum models of closed systems, in which spontaneous symmetry breaking occurs. A recent paper arXiv: 1805.03952v9 shows that this rule arises in the scattering state space, which describes the scattering of a particle by a one-dimensional delta potential barrier (no bound states).

In our opinion, the thought experiment with a cat, proposed by Schrödinger, played a dual role in discussions on the foundations of quantum mechanics. On the one hand, it (correctly) shows that it is the superposition principle that can cause the conflict between quantum mechanics and classical mechanics. On the other hand, due to the fact that a macroscopic object (cat) was included in the experiment, it involuntarily leads to the (erroneous) thought that the problem with the principle of superposition arises precisely at the macro level.

However, the superposition principle (in its current formulation) leads to paradoxes in some quantum problems precisely at the micro level. In other words, quantum mechanics with the superposition principle (in its current formulation) is itself internally inconsistent. And this can be shown by the example of the above mentioned one-dimensional scattering problem. The fact is that the modern quantum-mechanical description of this scattering process is based on two key provisions: (1) the states of a particle are scattering states, each of which has one in-asymptote and one out-asymptote — in the infinitely distant past and distant future, a particle moves as a free particle; (2) the Hamiltonian with the delta potential is self-adjoint — asymptotically free scattering states describe a unitary quantum dynamics.

But the specificity of this one-dimensional problem is that the out-asymptote, in the case of a one-sided incidence of a particle on the barrier, is a superposition of the left and right out-asymptotes. This means that this scattering process splits into two alternative subprocesses – transmission through the barrier (tunneling) and reflection from it. In other words, the provision about the existence of free dynamics, in this scattering process, in the infinitely distant future implies the division of this process into subprocesses. But this means that the quantum dynamics of a particle in this process is not unitary, which contradicts the second provision about the self-adjointness of the Hamiltonian in this problem. That is, the modern quantum model of the process of scattering a particle on a one-dimensional delta potential is contradictory – it (even without cat) contains a paradox! Its solution is suggested in the paper arXiv: 1805.03952v9.

Article written by