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arXiv:1805.08583 · Annals of Physics

Separation of conditions as a prerequisite for quantum theory

Hans A. De Raedt, Mikhail I. Katsnelson, Dennis Willsch, Kristel F. L. Michielsen

Published Feb 10, 2019• 8 citations
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Abstract

We introduce the notion of “separation of conditions” meaning that a description of statistical data obtained from experiments, performed under a set of different conditions, allows for a decomposition such that each partial description depends on mutually exclusive subsets of these conditions. Descriptions that allow a separation of conditions are shown to entail the basic mathematical framework of quantum theory. The Stern–Gerlach and the Einstein–Podolsky–Rosen–Bohm experiment with three, respectively nine possible outcomes are used to illustrate how the separation of conditions can be used to construct their quantum theoretical descriptions. It is shown that the mathematical structure of separated descriptions implies that, under certain restrictions, the time evolution of the data can be described by the von Neumann/Schrödinger equation.

DOI
10.1016/j.aop.2019.01.012

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We introduce the notion of “separation of conditions” meaning that a description of statistical data obtained from experiments, performed under a set of different conditions, allows for a decomposition such that each partial description depends on mutually exclusive subsets of these conditions. Descriptions that allow a separation of conditions are shown to entail the basic mathematical framework of quantum theory. The Stern–Gerlach and the Einstein–Podolsky–Rosen–Bohm experiment with three, respectively nine possible outcomes are used to illustrate how the separation of conditions can be used to construct their quantum theoretical descriptions. It is shown that the mathematical structure of separated descriptions implies that, under certain restrictions, the time evolution of the data can be described by the von Neumann/Schrödinger equation.
It is shown that the quantum theoretical description of statistical data resulting from experiments with a finite number of different outcomes emerges by organizing the data such that the descriptions of the preparation and measurement stage are separated as much as possible. The quantum theoretical description that derives from this elementary principle of separation is void of the usual postulates/interpretations regarding wave functions, observables, quantization rules, Born's rule, and the like. The separation principle is illustrated by application to the Stern-Gerlach and the Einstein-Podolsky-Rosen-Bohm experiment and generalizes in a trivial manner. The von Neunman equation and therefore also the Schr\odinger equation are shown to follow directly from the mathematical structure that renders possible separated descriptions.
We propose and develop the thesis that the quantum theoretical description of experiments emerges from the desire to organize experimental data such that the description of the system under scrutiny and the one used to acquire the data are separated as much as possible. Application to the Stern-Gerlach and Einstein-Podolsky-Rosen-Bohm experiments are shown to support this thesis. General principles of logical inference which have been shown to lead to the Schrödinger and Pauli equation and the probabilistic descriptions of the Stern-Gerlach and Einstein-Podolsky-Rosen-Bohm experiments, are used to demonstrate that the condition for the separation procedure to yield the quantum theoretical description is intimately related to the assumptions that the observed events are independent and that the data generated by these experiments is robust with respect to small changes of the conditions under which the experiment is carried out.
A theory of quantum measurement was introduced some time ago that was based on the notion of the so-called separation status. This separation status had a spatial, local character, so that the theory worked only in special cases. Nevertheless, it enabled a description of state reduction process that was specific in where, when and under which objective conditions the process occurs and that preserved the unitary transformation symmetry of quantum mechanics. Now, in an accompanying paper (arxiv:1411.5524), a completely general mathematical definition of the status is given and analyzed. The present paper reformulates the theory of state reduction accordingly. A general mathematical form of the process is postulated and illustrated by examples of Stern-Gerlach experiment and of screening.
A general theory based on six postulates is introduced. The basic notions are theoretical variables that are associated with an observer or with a group of communicating observers. These variables may be accessible or inaccessible. From these postulates, the ordinary formalism of quantum theory is derived. The mathematical derivations are not given in this article, but I refer to the recent articles. Three possible applications of the general theory can be given as follows: (1) the variables may be decision variables connected to the decisions of a person or a group of persons, (2) the variables may be statistical parameters or future data, and (3) most importantly, the variables are physical variables in some context. The last application gives a completely new foundation of quantum mechanics, a foundation which in my opinion is much easier to understand than ordinary formalism. So-called paradoxes like that of Schrödinger’s cat can be clarified under the theory. Explanations of the outcomes of David Bohm’s version of the EPR (Einstein–Podolsky–Rosen) experiment and the Bell experiment are provided. Finally, references to links toward relativity theory and quantum field theory are given. The concluding remarks point to further possible developments.
We review recent work that employs the framework of logical inference to establish a bridge between data gathered through experiments and their objective description in terms of human-made concepts. It is shown that logical inference applied to experiments for which the observed events are independent and for which the frequency distribution of these events is robust with respect to small changes of the conditions under which the experiments are carried out yields, without introducing any concept of quantum theory, the quantum theoretical description in terms of the Schrödinger or the Pauli equation, the Stern-Gerlach or Einstein-Podolsky-Rosen-Bohm experiments. The extraordinary descriptive power of quantum theory then follows from the fact that it is plausible reasoning, that is common sense, applied to reproducible and robust experimental data.
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.