Abstract
Authoritative appraisals have qualified this book as an “axiomatic” theory. However, given that its essential content is no more than an analogy, its theoretical organization cannot be axiomatic. Indeed, in the first edition Dirac declares that he had avoided an axiomatic presentation. Moreover, I show that the text aims to solve a basic problem (How quantum mechanics is similar to classical mechanics?). A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors non-axiomatically. Four characteristic features of a new model of organizing a theory were recognized. A careful examination of Dirac’s text shows that it actually applied this kind of organization of a theory, confirming formally what Kronz and Lupher suggested through intuitive categories (pragmatism and rigour), i.e. Dirac’s formulation of Quantum mechanics represents a distinct theoretical approach from von Neumann’s axiomatic approach. However, since the second edition Dirac has changed his approach: although relying again on analogy, his theory refers to the axiomatic method. Some considerations on the odd paths which led to the present formulation of QM are added. They suggest that a new foundation of this theory needs to be found.
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Notes
Robert Oppenheimer also commented on its abstract character (quoted by Kragh 1990, p. 79 and Kragh 2013, p. 250); see some more appraisals on this page and the following one of Kragh’s later paper. Remarkably, a review of this book ended with the following words: “here is a physicist who is approaching the highest department of modern physics with such a completely abstract point of view that he is regarded as a pure mathematician by many of his colleagues” (Koopman 1935, p. 474).
The page number in round brackets refers to Dirac’s book; the Roman capital number preceding it refers to the edition number.
Poincaré (1903, Chap. “Optique et Electricité”) and Poincaré (1905, Chap. VII) had already distinguished two kinds of physical theory. Yet, his characterization was largely ignored. The same occurred for Einstein’s characterization of two organizations of a scientific theory, i.e. a “principle theory” (e.g. phenomenological thermodynamics) and “constructive theory” (e.g. statistical mechanics) (Frisch 2006). They correspond to respectively PO and AO unlike the intuitive meanings of Einstein’s words. Unfortunately, Einstein’s appraisal of Dirac’s book—“we owe to Dirac the most logically perfect presentation of this theory [QM]” (quoted in Kragh 2013, p. 251), corresponds to both kinds of organization, depending on the two ways of understanding the adverb “logically” according to classical logic or intuitionist logic.
Notice that the titles of the sections and their sequence are not very informative. The title of Sect. 3 ought to include the word “state”; Sect. 4 includes indeterminacy, a subject already treated in the previous section; Sect. 5 adds a “further discussion on photon”, which actually introduces a qualification that “we shall find from [the new] mathematical theory”. Similar remarks apply to the sections and their sequence of the following chapters. Hence, his synthetic presentation of the book obliges the reader to discover by himself the real structure of the theory.
Let us recall that not even the mathematicians opposing Hilbert’s axiomatic organization—i.e. Brouwer and partly Weyl and Heyting—ever characterized how a theory can be organized other than axiomatically and perhaps did not even imagine that this new organization exists.
He does not hide his appeal to analogy. In the first edition of the book he makes use of this word in the title of Sect. 33 and in several pages, e.g. pp. 11 (four times), 84, 86, 88, 93, 94, 95, 96, 98. 106.
However Dirac’s book never mentions Bohr’s principle of correspondence (Mehra and Rechenberg 1986; vol. VI, p. 331). See also (Darrigol 1992, pp. 326–327).
It is not by chance that the mathematician Garrett Birkhoff’s appraisal of the book stresses the typical requirement of a traditional scientist: “He impresses me as being at least comparatively deficient in appreciation of quantitative principles, logical consistency and completeness, and possibilities of systematic exposition and extension of a central theory” (quoted in Kragh 1990, p. 280).
Pauli (1932). Both Schroedinger and Slater were two physicists that rejected Weyl’s mathematical approach. Weyl himself echoed the widespread criticisms that his book had received, when in the Preface of the second German edition (1931) he wrote: “It has been rumoured that the ‘group pest’ is gradually being cut out of quantum physics” (p. x). See also Wigner (1959, p. v), that denounces a past, “great reluctance among physicists toward accepting group theoretical arguments and the group theoretical point of view”.
Kragh (2013, Sect. 6, p. 264) He suggested the following explanation: “Dirac was familiar with the new, mathematically abstract way of representing quantum theory, but he did not find it either more fundamental or very helpful. He preferred to treat group theory as part of quantum mechanics, which for him was the general science of non-commuting quantities” (here Kragh paraphrases a quotation whose original text is reported by Darrigol 1992, p. 302, fn. 29). In my opinion this explanation does not explain why Dirac emphasised, at the beginnings of the “Preface”, the theoretical novelty of “transformations” and moreover the basic role played by transformations in establishing the analogy. Moreover, Kragh’s and Darrigol’s quotation is from the year 1929, i.e. before the date of book’s first edition.
By the way it is not clear what Dirac’s above words “a complete account… of all general physical laws” mean, because no physical law is mentioned in previous chapters (apart from what he called the superposition principle). Even the subsequent proposition seems to deny his previous claim of having covered “all general physical laws”.
He seems to allude to a choice of PO when he recalls his previous education. “I think that if l had not had this engineering training, I should not have had any success with the kind of work that I did later on, because it was really necessary to get away from the point of view one should deal only with results which could be deduced logically from known exact laws which one accepted, in which one had implicit faith” (quotes in Kragh 1990, p. 280). “Again in 1965, he recommended that physicists… follow the more modest path of setting up ‘a theory with a reasonable practical standard of logic, rather like the way engineers work’” (quoted in Kragh 1990, p. 281). In the first quotation we see an implicit reference to his disbelief in an AO and in the second quotation his leaning to a PO.
About Dirac’s views on Copenhagen interpretation Kragh (1990, pp. 80–86) compares them with those of the most prominent physicists of his time. Dirac supported an independent, but similar, view of the mainstream.
According to Heisenberg: “Methodologically, his starting points were particular problems, not the wider relationship. When he described his approach, I often had the feeling that he looked upon scientific research much as some mountaineers look upon a tough climb. All that matters is to get over the next three yards. lf you do that long enough, you are bound to reach the top” (quoted in Kragh 1990, p. 281). This analogy seems to allude to a step-by-step development of a PO theory. which proceeds through local, inductive steps towards the discovery of a new method.
Here we have to overcome a deeply rooted prejudice according to which only primitive languages make use of double negations. For a long time Anglo-Saxon linguists ostracised the double negations, which explains why the importance of the DNPs was rarely noticed (Horn 2001, p. 79–82; Horn 2010, pp. 111–112).
In the following each negation of a DNP will be underlined in order to make it easier for the reader to recognize the negated words; whereas each modal word or a single word which is equivalent to a DNP (e.g. “only” = “nothing other than”) will be dotted underlined. Notice that the word “must” is a modal word, which, as all modal words, is equivalent (via the S4 model of modal logic) to a doubly negated proposition of intuitionist logic (a DNP); hence it is not equivalent to the corresponding affirmative proposition, owing to the failure of the double negation law of such a logic.
Here we have to overcome one more prejudice: all ad absurdum proofs can be reverted into direct proofs (Gardiès 1991). Instead this operation is possible only if one translates the conclusion, which is a DNP, into an affirmative proposition; hence, only if one makes use of the double negation law, which however belongs to classical logic and not to intuitionist logic.
Actually, the latter method is an ancient one. It characterizes Leibniz’s approach to theoretical physics (“our mind looks for invariants”). It was restated in 1783 by L. Carnot (“to look for the invariants of bodies collision”) in his original foundation of mechanics (Carnot 1783, pp. 18 and 43). Unfortunately, Dirac ignores these anticipations and moreover he recalls (I, p. vi) Weyl’s book (1928) not because it had introduced group theory into QM for the first time, but as an unqualified exception to the coordinates method. Dirac does not call it “group theory” owing to his underrating of it, preferring to call it the “symbolic” method of the invariants (I, p. vi). It is apparent from a general analysis of his use of group theory that Dirac emphasizes specific aspects of group theory, which will be considered merely particular features after the full introduction of group theory into theoretical physics.
About Dirac’s conception of mathematics, Kragh (1990, Chap. 14) discussed what may be called Dirac’s “principle of mathematical beauty”. Dirac certainly supported this idea to the extent of attributing a decisive role to it. But this principle is not a well-defined subject either in theoretical physics or in the philosophy of science. It is not surprising that Kragh’s discussion did not achieve certain results.
This function is defined equal to zero everywhere, except for the value 0, where s is equal to infinity, so that its integral on the entire range is 1. His first use of was in the paper (Dirac 1927). There he claimed: “Strictly, of course, δ(x) is not a proper function of x, but can be regarded only as a limit of a certain sequence of functions. All the same one can use δ(x) as though it were a proper function for practically all the purposes of quantum mechanics without getting incorrect results” (Dirac 1927, p. 625). However, in the first edition he felt the obligation to justify it. He devoted to it a section (I, § 22). A first weak justification (an analogy) was following one: “We get over the difficulty by allowing infinities of certain types to occur in the coefficients ap, which enables every ψ formally to be expressed in the required form. This is analogous to the device sometimes used in geometry, of avoiding the exception of parallel lines to the rule that two straight lines always meet in one point, by saying that parallel lines meet in a point at infinity” (I, p. 63). However, after having introduced this function he is more specific: “The introduction of the δ function into our analysis will not be in itself a source of lack of rigour in the theory, since any equation involving the δ function can be transcribed into an equivalent but usually more cumbersome form in which the δ function does not appear. The δ function is thus merely a convenient notation. The only lack of rigour in the theory arises from the fact that we perform operations on the abstract symbols, such as differentiation and integration with respect to parameters occurring in them, which are not rigorously defined. When these operations are permissible, the δ function may be used freely for dealing with the representatives of the abstract symbols, as though it were a continuous function, without leading to incorrect results. We can, in fact, even give a meaning to the δ function of an observable, provided it has a continuous range of eigenvalues, by means of the general definition of § 15” (I, p. 64). Actually, the first justification (a question of mere transcription) misinterprets the question. Which eventually is recognized in the use of “not rigorously defined” differential operations on this function. The question is circumvented by appealing to only the case “these operations are permessible”, without dealing with the contrary case.
Roberts also offered a detailed appraisal of Dirac’s mathematical presentation of QM: “Conclusions. The Dirac formalism may be regarded as being valid for a wide range of quantum systems provided we make a number of modifications, the most important of which are: (1) The bras are in 1–1 correspondence with a subset of the kets and not with all the kets. This is already implicit in Dirac's work, because he relaxes the requirement that the complete bracket expression should always be defined. (2) The observables used in representation theory should be continuous. (3) The term “commuting observable” is to be understood in the usual Hilbert space sense of commuting spectral resolutions. (4) A d-function normalization of a continuous spectrum is only possible when the corresponding spectral measure is absolutely continuous with respect to Lebesgue measure. (5) An eigenket of an observable is only of direct physical significance if it forms part of the integral eigendecomposition, associated with the spectral resolution of the corresponding self-adjoint operator on Hilbert space” (Roberts 1966, p. 1103).
Bokulich (2008, Chaps. 2–4) tried to classify the different kinds of organizations of a theory. Unfortunately she makes use of philosophical, loose distinctions. According to Bokulich, Heisenberg’s theory was “close”; this is a characteristic feature of an axiomatic theory, i.e. an AO; instead, the first formulation of QM clearly lacked of axioms and rather was based on a problem, hence a PO. In opposition, she characterizes Dirac’s theory as “open” with respect to subsequent developments, not through its structural features of present times.
To my knowledge no contemporary physicist noticed a change in Dirac’s approach to the kind of organization, nor his change of the basic philosophy in the fourth edition.
Notice that in the same years Emmy Noether changed algebra according to an axiomatic perspective so radically that it made H. Weyl feel he was an outdated mathematician (Weyl 1936, “Preface”).
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I am grateful to Professor David Braithwaite for having revised my insufficient English.
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Drago, A. Dirac’s Book The Principles of Quantum Mechanics as an Alternative Way of Organizing a Theory. Found Sci 28, 551–574 (2023). https://doi.org/10.1007/s10699-022-09835-3
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DOI: https://doi.org/10.1007/s10699-022-09835-3