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American Journal of Physics

Quantum logic

Carl G. Adler, James F. Wirth

Published May 1, 1983• 4 citations
Publisher

Abstract

A discussion of quantum logic is presented. An example, drawn from elementary quantum mechanics, is used to illustrate the failure of the classical distributive law which is quantum logic’s most singular feature. Furthermore, an elaboration and explanation of an existing analog suitable for classroom demonstration is also presented. Finally, some remarks are made on the possible impact of quantum logic.

DOI
10.1119/1.13227

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A discussion of quantum logic is presented. An example, drawn from elementary quantum mechanics, is used to illustrate the failure of the classical distributive law which is quantum logic’s most singular feature. Furthermore, an elaboration and explanation of an existing analog suitable for classroom demonstration is also presented. Finally, some remarks are made on the possible impact of quantum logic.
This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.
What do we expect of a quantum logical interpretation of quantum mechanics worthy of the name? In what sense should it resolve the paradoxes? What scope should it have? Should quantum logic, if it is successful in the micro world, replace classical logic in the macro world and in mathematics? We distinguish between two kinds of quantum logical interpretations of quantum mechanics. An example of the first kind, the activist kind, will do much more than simply rewrite the assertions that quantum mechanics makes about the quantum world in an unfamiliar notation - in the elementary language of quantum mechanics. It will offer a solution to the major puzzles of quantum mechanics. It will show that the paradoxes of quantum mechanics can be resolved if we do away with classical logic and replace it with quantum logic, at least in our descriptions of quantum phenomena. It may defend a form of quantum-mechanical realism, a sort of quantum logical particle view of quantum systems, protecting it from the paradoxes that classical logic would impose on it. The second kind of quantum logical interpretation, the passive or quietist kind, is less ambitious. A quietist interpretation offers a solution to the paradoxes only in the following sense. It will assert that quantum mechanics, when it describes quantum phenomena in the elementary language, need not generate paradox, that the so-called paradoxes are not even formulable in quantum logic.
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
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.
The book investigates the ontology and logic of quantum physics. The first part discusses the relationship of theory and observation and different views on the ontological status of scientific theories. It introduces the fundamentals of quantum mechanics and some of its interpretations and their compatibility with various ontological positions. In the second part, implications of quantum mechanics on classical logic, especially on the distributive law and bivalence, as discussed by Garrett Birkhoff & John von Neumann (1936) and Hilary Putnam (1968), and their counterarguments are reconstructed and discussed. It is concluded that classical logic is sufficient for dealing with quantum mechanical propositions.