‘Über die Grundlagen der Quantenmechanik.’ Offprint from Mathematische Annalen, Band 98, Heft 1, pp. 1-30.
Berlin: Julius Springer, 1928. First edition, very rare offprint, of the paper that is at once the first published attempt at an axiomatic foundation of quantum mechanics, David Hilbert’s last publication in physics, and John von Neumann’s first publication on the subject that was to occupy the remainder of his scientific life. The present copy carries, in pencil on the front wrapper, the ownership signature of Paul Peter Ewald (1888–1985): the physicist whom Hilbert had personally brought to Göttingen in 1912 to be his ‘tutor in physics.’ That Hilbert’s own physics tutor should have retained the offprint of Hilbert’s valedictory physics paper, whose protagonist was the young Hungarian who within five years would supersede it, closes a circuit of intellectual patronage across the most consequential fifteen years of twentieth-century theoretical physics. Ewald had come to Göttingen as a student of mathematics in 1906, and by 1907 was one of Hilbert’s Ausarbeiter—students entrusted with taking down Hilbert’s lectures and preparing fair copies for the mathematical reading room. He moved to Munich later in 1907 to work under Arnold Sommerfeld; his 1912 doctoral dissertation, on the propagation of electromagnetic waves through periodic arrays of oscillators, supplied the theoretical framework for what was about to become the most celebrated physical experiment of its decade. On a January 1912 walk through Munich’s Englischer Garten, Ewald described the dissertation’s results to Max von Laue, who at that moment was struggling to devise an experiment to prove that X-rays were electromagnetic waves; within weeks, von Laue, Walter Friedrich, and Paul Knipping had performed the diffraction experiment that founded modern X-ray crystallography and that won Laue the 1914 Nobel Prize in Physics. Ewald’s own subsequent contributions—the Ewald sphere in reciprocal space, the Ewald summation for Coulomb lattice sums, the Ewald–Oseen extinction theorem in crystal optics—are standard fixtures of condensed-matter physics and remain in daily use a century later. In the spring of 1912, with Ewald newly returned from Munich, Hilbert wrote to Sommerfeld requesting a special assistant for physics; Sommerfeld sent Ewald back, and he was received in Göttingen as Hilbert’s ‘physics tutor,’ the designation under which the appointment would be informally remembered for the rest of his life. Hilbert’s expectations of the arrangement were characteristically ambitious: Ewald recalled, in later life, Hilbert summarising his research programme to him in a single sentence—mathematics had been reformed; physics was next; chemistry would follow. Ewald returned to Munich as Sommerfeld’s assistant in 1913, took the chair of theoretical physics at the Technische Hochschule Stuttgart in 1921, and held it until 1937, when the Nazi legislation against Jewish academics drove him into exile—first to Queens’ College Cambridge, then, from 1949, to the Polytechnic Institute of Brooklyn; he served as the founding editor-in-chief of Acta Crystallographica from 1948 to 1959 and as president of the International Union of Crystallography from 1960 to 1963, and his 1962 volume Fifty Years of X-Ray Diffraction, edited for the IUCr, remains the standard first-person history of the field he had seeded on the Englischer Garten walk half a century earlier. That Ewald retained the present offprint in his working library is itself significant: the paper was not only an artefact of Hilbert’s final physics course but a document in which one of Ewald’s own lifelong concerns—the extension of mathematical rigour to physical theory, the programme Hilbert had first articulated to him in 1912—was being carried out by Hilbert’s latest generation of students. It is an offprint that, in a sense Hilbert would have recognised, circles back to its own intellectual origin. By the time of the paper here offered, Hilbert was at the close of a forty-year career that had reshaped every branch of mathematics he had touched. He had presented his programme of axiomatic foundations at the second International Congress in Paris in 1900, in a lecture whose twenty-three open problems set the agenda of much of twentieth-century mathematics; he had given the Grundlagen der Geometrie in 1899; and he had developed integral-equation theory and the spectral theory of bounded self-adjoint operators — in the work that, in von Neumann’s extension to unbounded operators, would furnish exactly the apparatus called for in the paper here offered — in a celebrated cycle of papers between 1904 and 1910. He had taken the chair at Göttingen in 1895 and was to retire from it in 1930. The Göttingen mathematical institute he and Felix Klein had built had become, by the late 1920s, the foremost mathematical centre in the world, and the years 1925–1933 represent its short final flourishing. Within five years of the present offprint, the National Socialist legislation of April 1933 had driven from Göttingen most of those who had built and inhabited it: Courant, Born, Bernays, Noether, Landau, Hellinger, Weyl (who departed for the new Institute for Advanced Study in Princeton), and von Neumann himself, who had emigrated to Princeton in 1930. Hilbert, then sixty-eight and in increasingly fragile health, remained in Göttingen until his death in February 1943; he is reported to have remarked to the new Reich Minister of Education that mathematics in Göttingen no longer existed. The paper emerged from a lecture course, ‘Mathematische Methoden der Quantentheorie,’ that Hilbert delivered at Göttingen twice weekly through the winter semester of 1926–1927. By that date Hilbert was sixty-four, convalescing from the near-fatal onset of pernicious anaemia the previous year—he owed his recovery, he would later acknowledge, to the Minot–Murphy–Whipple liver therapy whose discoverers would receive the 1934 Nobel Prize in Medicine—and he had not published a physics paper since the second instalment of his ‘Grundlagen der Physik’ in 1917. But he had followed the new quantum mechanics from its inception with close and proprietary attention. He had invited Werner Heisenberg to present matrix mechanics in the Göttingen mathematical seminar in November 1925, within four months of its publication; he had followed the parallel rise of Schrödinger’s wave mechanics through the spring of 1926; and by the autumn of that year he had judged, correctly, that the new physics demanded a mathematical foundation that did not yet exist. His two assistants for the course were Lothar Nordheim, a former student of Max Born whom Hilbert had recently taken on as his physics assistant, and a twenty-two-year-old Hungarian named Janos (German Johann, later John) von Neumann, newly arrived from Budapest and Zürich on an International Education Board fellowship that Hilbert himself had secured—over initial Board reservations about the candidate’s youth—on Richard Courant’s recommendation. The division of labour among the three men, as Nordheim later described it, was unambiguous. Hilbert, ill and several degrees removed from the concrete calculations of the new theory, supplied the conceptual programme and delivered the lectures; their preparation fell to Nordheim, who, in his own candid later recollection, had to teach Hilbert quantum mechanics and wave mechanics from week to week. Nordheim wrote the expository prose. Von Neumann, who attended the course and rapidly emerged as its best mathematician, contributed the technical elaboration. The paper was received by Mathematische Annalen on 6 April 1927 and appeared as the lead article of Band 98, Heft 1, early the following year. Its introduction set out the attribution with characteristic Göttingen exactness: the memoir had emerged from Hilbert’s lecture course; Nordheim had assisted ‘materially’; ‘important parts of the mathematical elaboration’ had been contributed by von Neumann; and Nordheim had prepared the final version. Reid, writing half a century later, is more candid: ‘Hilbert did almost none of the work, the spirit was quite definitely his.’ The programme of the paper was one Hilbert had pursued continuously since the early 1890s, and which had already yielded, in the Grundlagen der Geometrie of 1899, the most influential axiomatic treatment of any branch of mathematics since Euclid. In geometry, Hilbert had shown, the primitive notions—point, line, plane—and their interrelations can be so stringently specified by a system of axioms that a purely analytical apparatus, linear algebra, emerges as their unique realisation and can then be turned back on the geometry to generate theorems unavailable to synthetic reasoning. The proposal of the 1927 paper was to carry through the analogous programme for quantum mechanics: to specify, by a sufficiently stringent set of physical postulates on probability amplitudes, an analytical apparatus that would uniquely satisfy those postulates and that could then be used, in the reverse direction, to generate new physical predictions. The authors laid down six axioms, extending and recasting the four postulates that Pascual Jordan had given in his ‘Neue Begründung der Quantenmechanik’ a few months earlier. Axiom I defined the probability amplitude φ(x, y; F₁, F₂) for two dynamical quantities F₁ and F₂; Axiom III specified the condition under which a dynamical quantity takes a sharp value; Axiom IV laid down the addition and multiplication theorems for amplitudes; Axiom VI demanded that probabilities be independent of the choice of coordinate system. The analytical apparatus they extracted from the axioms was an operator calculus in which every dynamical variable corresponds to an operator and every probability amplitude to a quantity built out of inner products. The paper is at once historically decisive and mathematically broken, and the authors themselves were unusually frank about its brokenness. The transformation theory of Dirac and Jordan, on which the formalism rested, was built on the manipulation of ‘improper functions’—above all the Dirac δ-function, which behaves as a function under integration but is not, in any rigorous sense, a function at all. (Laurent Schwartz’s theory of distributions, which retroactively legitimised the δ-function by reconstructing it as a distribution on a suitable test space, belongs to the late 1940s.) Hilbert, Nordheim, and von Neumann acknowledged in the closing paragraph of the paper that they had presented the theory in a ‘mathematically still imperfect version.’ Nordheim, looking back in a later interview, was blunter: the paper was elegant but ‘absolutely inacceptable mathematically.’ Crucially, the authors announced in a footnote that a new paper by von Neumann, rigorously replacing the improper-function machinery, was to appear shortly in the Göttinger Nachrichten. That footnoted paper—‘Mathematische Begründung der Quantenmechanik,’ received by the Nachrichten on 20 May 1927, barely six weeks after the paper here offered—is the pivot of the subsequent history. In it, and in the two further papers that followed within twelve months, von Neumann discarded the δ-function entirely, replaced it with a rigorous theory of (generally unbounded) self-adjoint operators on a separable Hilbert space, proved the spectral theorem in the form needed for the physics, and in so doing gave quantum mechanics the mathematical foundation that, with inessential modifications, it still has. The 1932 Springer monograph Mathematische Grundlagen der Quantenmechanik, the book that defined the subject for three generations, is the systematic exposition of the programme begun in those 1927 Nachrichten papers. It is therefore not an overstatement to say that the paper here offered is the document out of which modern quantum mechanics, in its mathematically rigorous form, was driven into existence by dissatisfaction with its own inadequacies. Von Neumann himself always dated his entry into quantum theory from the collaboration on it. The reception of the paper here offered, and of von Neumann’s subsequent Nachrichten papers of 1927, was at first largely confined to the small circle of mathematical physicists who could follow the abstract operator-theoretic apparatus. Its broader influence dates from the 1932 publication of von Neumann’s monograph Mathematische Grundlagen der Quantenmechanik (Berlin: Springer), the systematic treatise that formalised every result of the 1927 papers and added the no-hidden-variables theorem and the measurement-projection postulate that would dominate subsequent foundational discussion. Garrett Birkhoff and von Neumann’s 1936 paper ‘The Logic of Quantum Mechanics’ (Annals of Mathematics 37) extended the operator framework to a non-classical propositional calculus, founding the field of quantum logic that George Mackey, Josef Jauch, Constantin Piron, David Foulis, and Charles Randall would develop through the 1960s and 1970s. Andrew Gleason’s theorem of 1957, which establishes that every probability measure on the projection lattice of a Hilbert space of dimension three or higher derives from a density operator, gave the operator formalism its canonical statistical foundation; Simon Kochen and E. P. Specker’s 1967 paper, which uses Gleason’s techniques to rule out a broad class of non-contextual hidden-variable theories, established the formalism’s contemporary status as the unique mathematical home for non-classical probability. The C*-algebra reformulation of quantum mechanics, and its generalisation to quantum field theory in the Haag–Kastler axiomatic framework of the 1960s, descend in a direct line from the operator-algebra programme that begins in the paper here offered. In every subsequent foundations debate — from the Bell theorem of 1964 through the recent quantum-information rephrasing of the formalism — the operator-on-Hilbert-space machinery has provided the common technical vocabulary. Offprints of Mathematische Annalen papers of this period were distributed in small numbers—typically twenty-five to fifty copies—to the authors’ personal allocation. They are almost invariably encountered today only in the working libraries of contemporaries, and are seldom met with separately on the market. We locate no copy of the present offprint on OCLC and no auction record on RBH. The paper itself is read today either in the bound journal volume or in the third volume of Hilbert’s Gesammelte Abhandlungen (1935), but the separate Springer offprint is the form in which Hilbert’s own circle read and cited it in 1927 and 1928, at the moment of its composition and of the immediate repudiation of its formalism by its own junior co-author. The paper is the first systematic mathematical treatment of quantum mechanics as a theory of operators on a Hilbert space, prepared by Hilbert’s guest John von Neumann and his assistant Lothar Nordheim from Hilbert’s Göttingen lectures of the preceding winter. It is the opening document of the programme that von Neumann would extend in his own three papers of 1927–1930 and complete in his 1932 Mathematische Grundlagen der Quantenmechanik — the textbook that has determined the mathematical presentation of quantum mechanics ever since, and the source of the modern abstract formalism (the C*-algebra, the density operator, the projection-valued measure). Dirac’s 1930 Principles of Quantum Mechanics is its working-physicist’s complement; where Dirac gave physicists the bra-ket notation and the heuristic δ-function, von Neumann gave mathematicians the rigorous Hilbert-space apparatus to which the formalism would later be anchored. Ewald’s copy — Paul Peter Ewald, founder of dynamical X-ray diffraction theory, was one of the central German theorists of the period and an intimate of the Göttingen circle — is the working-scientist’s copy of the founding text. References: Mehra & Rechenberg, The Historical Development of Quantum Theory, Vol. 6, Part I (2000), pp. 401–411; Reid, Hilbert (1970; rev. ed. Hilbert–Courant, 1986), pp. 129, 183; Jammer, The Conceptual Development of Quantum Mechanics (1966), pp. 309–312; Lacki, ‘The early axiomatizations of quantum mechanics: Jordan, von Neumann, and the continuation of Hilbert’s program,’ Archive for History of Exact Sciences 54 (2000), pp. 279–318; Duncan & Janssen, ‘(Never) mind your p’s and q’s: Von Neumann versus Jordan on the foundations of quantum theory,’ European Physical Journal H 38 (2013), pp. 175–259. 8vo (233 x 157 mm), pp. 1-30. Original printed cream wrappers, a trifle toned, with a small early paper repair to the lower outer corner of the front wrapper; a fresh and clean copy, the text unopened in places. Front wrapper inscribed in pencil “Ewald” (underlined) to the upper right corner, with the inventory number “I/662” to the upper left. Housed in a custom folding case with green cloth spine and cream paper-covered boards, printed paper label to the front board.
Item #6686
Price: $3,500.00
