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Explaining the Born rule in the intuitionistic interpretation of quantum mechanics

Arkady Bolotin

Published Oct 6, 2016• 1 citation
OpenAlex

Abstract

This paper presents a novel explanation of the cause of quantum probabilities and the Born rule based on the intuitionistic interpretation of quantum mechanics where propositions obey constructive (intuitionistic) logic. The use of constructive logic makes it possible (through a replacement of the concept of truth with the concept of constructive probability) to abandon the law of excluded middle in the intuitionistic interpretation.

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Oxford University Press eBooks•Jul 20, 2017•0 citations
This paper presents a novel explanation of the cause of quantum probabilities and the Born rule based on the intuitionistic interpretation of quantum mechanics where propositions obey constructive (intuitionistic) logic. The use of constructive logic makes it possible (through a replacement of the concept of truth with the concept of constructive probability) to abandon the law of excluded middle in the intuitionistic interpretation.
In the present paper, the decision problem of the Schrödinger equation (asking whether or not a given Hamiltonian operator has the nonempty solution set) is represented as a logical statement. As it is shown in the paper, the law of excluded middle would be applicable to the introduced statement if and only if quantum fundamentalism (asserting that everything in the universe is ultimately describable in quantum-mechanical terms) held. But, since the decision problem of the Schrödinger equation is in general undecidable, such a statement is allowed to be other than true or false, explicitly, it may fail to have truth values at all. This makes possible to abandon the law of excluded middle together with quantum fundamentalism in the proposed intuitionistic interpretation of quantum mechanics.
Abstract Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. Assuming that our usual inference procedure is working rationally for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from epistemological viewpoints, we can regard quantum mechanics as a natural extension of classical probability theory.
Within the many versions of the Copenhagen interpretation numerous hints can be found about how to reason about quantum mechanics. However, strict guidelines are often missing. The standard quantum logic is sometimes opted as providing such a guideline but, in itself, it is again puzzling due to the failure of the law of distributivity. In this paper I take an instrumentalist approach inspired by the Copenhagen interpretation towards the interpretation of propositions in a physical theory. Starting with so-called elementary propositions for a physical theory I introduce logical connectives in accordance with this approach and relate the thereby expanding set of propositions to mathematical objects within the theory in a consistent way. This is first done for classical mechanics and then for quantum mechanics. In both cases this approach leads to the development of intuitionistic logics rather than classical ones. It also follows that the obtained intuitionistic quantum logic is a proper generalization of the obtained intuitionist logic for classical mechanics. Finally, some hints are provided in how these logics are to be used in practical situations.
In order to solve the problem of which is the real logic of Quantum Mechanics (QM), we suggest to study the mutual relationships between the physical theory (as a whole, that is both its experimental part and its theoretical part), its mathematics and its logic. That results to be a more general problem than what is considered by traditional studies on Quantum Logic (QL). These studies focus attention on the problem of the relationship between the experimental part of the physical theory and only its mathematics, from which logic is then derived. The new approach, that we suggest, generalises the interpretation that one of us (A. D.) successfully tested on the foundations of classical physical theories. The foundations of a theory are interpreted in terms of two options on the kind of mathematics, either classical one or constructive one, and the kind of logic, either classical logic or intuitionistic logic. It is remarkable that each theory making use of intuitionistic logic – i. e. L. Carnot’s classical mechanics, S. Carnot’s thermodynamics, Lobacevskij’s non-Euclidean geometry is organised in order to solve an universal problem (expressed by a double negated sentence), rather than in an entirely deductive way; hence, a non-classical logic does not belong exclusively to QM. From an historical and foundational analysis about QM, we conclude that in QM it is the law of double negation and not modular law (introduced by Birkhoff and von Neumann without giving physical evidence to it) which fails, i. e. which represents the true difference between Classical Logic (CL) and QL. We have looked for previous works on QL supporting our suggestion; in the past Destouches-Fevrier (1948), Fine (1974), Bell-Hallett (1981), Gauthier (1983), Adelman-Corbett (1995) already defended this thesis on the relevance of double negation law for QL. Under this light, QL results to have the same logic of a classical theory organised in a problematic manner. Hence, QM has to be referred to intuitionistic logic, in the same way every theory which is developed around a fundamental problem, expressed by a double negated sentence, does. In particular, by following the founder of intuitionism, Brouwer, we reject the identification of intuitionistic logic with an entirely deductive system; rather, the logical system of QL, as a theory, is organised in such a way to put a crucial problem to be solved. In agreement with this kind of organisation, among the several QM formulations we choose Heisenberg’s formulation, which was ignored by previous authors on QL. This formulation is aimed to solve the problem of quantum measurement, that originates the uncertainty methodological principle. Moreover, it is remarkable that it makes use of an operational mathematics (discrete matrices only). We interpret its characteristic arguing about its principal problem through double negated sentences; it results to be a moving from arguments belonging to non-classical logic expressed by a double negated sentence to the recognition of a suitable mathematics for QM formulation. In sum, QL shows how it is possible to state the relationship between QM and mathematics. This point has been almost ignored by the most traditional approaches to QL, which rather drew it from an a priori mathematics of QM (commonly, Hilbert space). As a verification of our approach, we show that even both Lobachevskij’s geometry and S. Carnot’s thermodynamics argued in order to achieve mathematical formulas expressing the content of the basic double negated sentences of respective theory. In the same way the recent book on QM by T. F. Jordan formalised his matrix mechanics. 1. Logic in classical physics and in Quantum Mechanics An historical analysis of Quantum Mechanics (QM) shows that there were at the origin of the theory two distinct formulations: Matrix Mechanics (Heisenberg, 1925) and Wave Mechanics (Schroedinger, 1926). Although Dirac subsumed these two formulations in a comprehensive mathematical framework (Hilbert space), we think that the organisation of the principles is different in Matrix Mechanics from that in Wave Mechanics, and this difference implies different logics. So we think that in order to characterise the logic of QM it is necessary to distinguish between this two formulations. In Matrix Mechanics, Heisenberg’s basic idea was to explain atomic spectra not in terms of atomic orbits (as Old Quantum Theory tried to make), but only by means of the study of experimental data; i. e., frequencies and intensities of light both emitted and absorbed by matter. Through a
We can use quantum theory to explain an enormous variety of phenomena by showing why they were to be expected and what they depend on. These explanations of probabilistic phenomena involve applications of the Born rule: to accept quantum theory is to let relevant Born probabilities guide one’s credences about presently inaccessible events. We use quantum theory to explain a probabilistic phenomenon by showing how its probabilities follow from a correct application of the Born rule, thereby exhibiting the phenomenon’s dependence on the quantum state to be assigned in circumstances of that type. This is not a causal explanation since a probabilistic phenomenon is not constituted by events that may manifest it: but each of those events does depend causally on events that actually occur in those circumstances. Born probabilities are objective and sui generis, but not all Born probabilities are chances.