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
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 put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference. The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever. However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73]. Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law. This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure. What went wrong? The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Originally published in 1981, this book forms volume 15 of the Encyclopedia of Mathematics and its Applications. The text provides a clear and thorough treatment of its subject, adhering to a clean exposition of the mathematical content of serious formulations of rational physical alternatives of quantum theory as elaborated in the influential works of the period, to which the authors made a significant contribution. The treatment falls into three distinct, logical parts: in the first part, the modern version of accumulated wisdom is presented, avoiding as far as possible the traditional language of classical physics for its interpretational character; in the second part, the individual structural elements for the logical content of the theory are laid out; in part three, the results of section two are used to reconstruct the usual Hilbert space formulation of quantum mechanics in a novel way.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].