• Nikolay L. Chuprikov posted an update in the group Group logo of 2018 Workshop on Wigner’s Friend2018 Workshop on Wigner’s Friend 4 years, 7 months ago

    I want to present here my personal understanding of this issue, which is based on the ensemble interpretation of quantum mechanics. To begin with, a recent extension of Wigner’s friend thought experiment and its original variant only make sense if the Schrödinger cat paradox is interpreted as a measurement problem. However, in my opinion, such an interpretation of the paradox is erroneous. This paradox is the result of the presence in the quantum theory of an internal problem which is associated with the generally accepted (but incorrect in the general case) formulation of the superposition principle for closed quantum systems, and which must be solved without abandoning the concept of “closed systems”. In this sense, modern quantum mechanics is an unfinished theory. As a consequence, none of the known interpretations of quantum mechanics can also be considered as logically completed. This also applies to the ensemble interpretation which, in my opinion, most consistently reflects the statistical nature of quantum mechanics and does not introduce elements alien to the quantum theory.

    In my opinion, the following provisions of this interpretation correctly reflect the essence of quantum theory and cannot be revised when solving the paradoxes of Schrödinger’s cat and friend Wigner (using the example of one-particle quantum dynamics): (1) all concepts and provisions of quantum theory relate to quantum one-particle ensembles (the behavior of a particle in an infinite set of identical experiments with one particle is deterministic and predictable) whose properties are determined by an external physical context (that is, by an external field that enters into the Hamiltonian, as well as by the boundary conditions or by initial conditions) under which the ensembles move; (2) The behavior of a particle in a single one-particle experiment is random and is outside the scope of quantum theory.

    In particular, the wave function (state vector) describes the state of a quantum one-particle ensemble; the square of the wave function module in the coordinate and momentum representations describe the real properties of the ensemble, which are uniquely determined by the external physical context. In a separate one-particle experiment, the measurement of the coordinate (or momentum) of a particle gives one of the eigenvalues of this observable. The measurement affects the particle, without affecting the physical context that determines the properties of the corresponding one-particle ensemble. That is, there can be no talk of any collapse of the wave function. Figuratively speaking, the role of experiment in quantum mechanics is not to measure the coordinate or momentum of a particle in one single-particle experiment, but to “measure” the distribution of particles over coordinates and momenta with the help of an infinite number of identical single-particle experiments.

    All the above refers to the Heisenberg uncertainty relation for the coordinate and momentum of a particle. This inequality characterizes one of the most important properties of quantum ensembles, imposing a restriction on the standard deviations for these two random variables, and not on the measurement errors of theirs. All the talk that the coordinate and momentum of a particle cannot be measured in a single one-particle experiment is purely speculative, as it has nothing to do with Heisenberg’s uncertainty principle.

    So, from the point of view of the ensemble interpretation, the Schrödinger cat paradox is not a measurement problem. In this regard, the proponents of this interpretation argue that the paradox is thus disappears. However, it is not. The real problem that this paradox highlights is that the principle of superposition contradicts the principles of macroscopic realism. And now it is time to move to that provision of the ensemble interpretation, which is erroneous (and which is common to all interpretations). According to this provision, all quantum states of a particle are divided into two classes: (a) pure states — all quantum states that are specified by the state vector (wave function); (b) mixed states – all states that are specified by the density operator ( ).

    Note that item (a) is based on the modern formulation of the superposition principle, according to which any superposition of pure states of a quantum particle is also its pure state. In other words, according to this formulation, the Schrödinger representation is irreducible for any quantum single-particle process. However, this is not the case if we are dealing with a quantum process in which the energy spectrum of a particle is continuous. By the example of scattering a particle on a one-dimensional potential barrier, we showed that in the distant past and the distant future of this process (when the state of a particle coincides with the corresponding in- or out-asymptote), the space of its states represents the sum of two orthogonal subspaces that are invariant with respect to the actions of the coordinate and momentum operators.

    We found that here there is a superselection rule according to which vector states fall into two classes: pure vector states and mixed vector states. For example, the initial state of a particle (which, say, falling on a barrier from the left) belongs to one superselection sector and, therefore, is a pure vector state. At the same time, the final state of the particle, which is a superposition of the transmitted and reflected wave packets, is a mixed vector state. Thus, in the course of the scattering process, Schrödinger quantum dynamics transforms a pure state of a particle into a mixed one. It is important to remind that we deal with a closed quantum system. All observables (and characteristic times) can be defined only for pure vector states, that is, separately for that part of the initial ensemble, the final state of which is described by the transmitted wave packet, as well as for that part of the initial ensemble, the final state of which is described by the reflected wave packet. The modern quantum model of this process does not imply such a detailed description and, therefore, should be revised. It can be shown that the new model should be nonlinear — the initial and final states of each of these two subprocesses are “stitched” at the center of the symmetric barrier with the help of nonlinear boundary conditions: at the “stitching” point, the probability density and probability current density for each subprocess must be continuous (the reflection subprocess is nonzero only to the left of this point).

    This approach implies the revision of all modern models of quantum single-particle processes with a continuous energy spectrum. In particular, this concerns the modern model of the double-slit experiment and the alpha decay model of the radioactive nucleus, where the decay occurs via tunneling the alpha-particle through the potential barrier created by the parent nucleus.

    I am realizing that this approach goes against all known approaches (for none of them calls into question the principle of superposition). However, the protracted history of solving the Schrödinger cat paradox within the framework of well-known approaches, which shows that paradoxes only multiply, gives me the right to hope that alternative approaches are also worthy of attention and discussion.