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Operator calculus: the lost formulation of quantum mechanics

A mathematical reconstruction

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Abstract

Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in historical accounts of quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions.

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Notes

  1. See Jordan (1927).

  2. Werner Heisenberg (1901–1976), Pascual Jordan (1902–1980).

  3. See, e.g., Mehra and Rechenberg (1982), Beller (1999), Lacki (2000).

  4. See, e.g., Masani (1989), Im (1996), Boffi (2008), Darrigol (2009), Duncan and Janssen (2009).

  5. See. e.g., Styer et al. (2002), Bonolis (2004).

  6. See. e.g., Peters (2004).

  7. It is not the goal of this paper to establish direct connections between the works mentioned and the one analysed here; this is instead suggested as a possible line of research.

  8. In what follows, AHQP will stand for the Archive for the History of Quantum Physics.

  9. Wiener (1926), received 20th April 1925.

  10. A result of this interest is the previous joint paper by Born and Jordan (1925b); the connection of this paper with Born’s studies on collisions has been considered in Im (1996).

  11. Da wird es wohl ruhiger zugehen Wiener, den Sie ja nach Göttingen eingeladen haben, gefällt mir sehr gut. (Letter from Born to Hilbert 28th November 1925. Niedersächsische Staats- und Universitätsbibliothek Göttingen (SUB), MS_D.HILBERT_40_A).

  12. It is known, nevertheless, that during Born’s sudden departure from Germany, some of his letters were lost, so the lack of complementary information does not necessarily mean that the discussion did not occur.

  13. Nevertheless, Norbert Wiener did not entirely dissociate himself from quantum mechanics, as has already been remarked in Mehra and Rechenberg (1982).

  14. It is interesting to note that Eckart’s work Eckart (1926) also includes the influences of Kornel Lanczos. Despite its value, the paper by Eckart has not been explored much in historiography.

  15. In the following, NWP will stand for the following reference to Wiener Papers: Norbert Wiener Papers, MC 22, box 2. Massachusetts Institute of Technology, Archives and Special Collections, Cambridge, Massachusetts.

  16. As we will see later, despite the fact that the authors try to generalise matrices to general operators, the differential operator plays a leading role in the argumentation.

  17. Schrödinger was actually in Zurich at that time.

  18. The quantum condition in the form given in the text is generally attributed to Born and is considered a postulate. It is at the core of many characteristics of quantum mechanics, such as discreteness and quantum jumps.

  19. Specifically, three years later, Dirac faced the same problem with the assumption of a \(\beta \)-particle leaving the nucleus of an atom and interacting with the surrounding electrons (Dirac 1927, p. 624).

  20. Wiener (1926).

  21. We do not make a distinction between Fourier transforms and Fourier integrals. Wiener almost always used the second expression in Wiener (1926).

  22. See, for example, Steen (1973), Lützen (1982), Birkhoff and Kreyszig (1984).

  23. The works we will discuss in Sect. 13 are certainly mentioned in Lützen (1982), although we develop an independent point of view, with the specific goal of connecting the works to Born–Wiener’s paper.

  24. Van der Waerden attributes the mentioned section directly to Born (van der Waerden 1967, p. 56).

  25. To enable a comparison with the original paper, we will refer to the Born–Wiener enumeration of formulas as BW(nn).

  26. In the case of linear uniform motion, the object defined will still be an operator, but one must take care when trying to refer it to the definition made here by the authors.

  27. Although this paper is the joint work of the whole team, some contributions of the second author must be singled out due to their significance in the dynamics of our research. This is the case in the present section. References to this note will be added for the same purpose in other parts of this paper.

  28. [...] diese nennen wir die zum Energiewert W gehörige Spaltensumme der ,,Matrix“, die dem Operator q entspricht. (Born and Wiener’s emphasis).

  29. A more detailed explanation of this claim is not difficult but quite long.

  30. Wiener used this technique extensively in his previous paper on Operational Calculus (Wiener 1926).

  31. In the present context, a Leibniz result is a generalisation of the “Leibniz rule” for the derivation of a product of functions \((fg)' = f'g +fg'\).

  32. An operator (resp. matrix) A on a Hilbert space H with inner product \( < \cdot \, , \cdot > \) is said to be Hermitian if \( \forall x, y \in H \Rightarrow<{A}x,y> = <x,{A} y>\).

  33. In the Born–Wiener development of the transpose, there are some sign errors that eventually cancel each other but that somehow affect their final proposal for the Hermite condition. The key formula in Born and Wiener’s reasoning is the factorisation of \({q}_1\) in such a way that

    $$\begin{aligned} {q}_1 (V,W)= F\left( \dfrac{2 \pi i}{h}W \right) \cdot U \left( \dfrac{2 \pi }{h}(W-V) \right) , \end{aligned}$$
    (10)

    where U is defined as the Fourier coefficient of u(t) by means of

    $$\begin{aligned} U(\alpha ) = \lim _{T \rightarrow \infty }\dfrac{1}{2T} \int _{-T}^{T}{e}^{i \alpha t}u(t) \text {d}t. \end{aligned}$$

    In this way, “transposing” W and V by simply interchanging them in (10) also introduces the conjugation when \( u(t) = {e}^{i \omega _0 t}\), as in the case of the harmonic oscillator.

  34. Lieber Kollege Born! Die Korrektur ist nach Ihrer Anzeige verbessert worden, und Jordan erhält sofort Nachricht davon. Die Fehler beschädigen in keiner Weise die wirkliche Validität unseres Argumentes. (NWP, Wiener to Born 14th February 1926). By the date the letter was written, the paper had already been received but not yet published.

  35. Hieraus folgt ohne weiteres der Energiesatz und die Frequenz-bedingung, wobei nur statt der Matrizenformel \(\dot{\mathbf{q }}= \frac{2 \pi i}{h}(\mathbf{W} {} \mathbf{q} -\mathbf{q} {} \mathbf{W} )\) die Operatorformel (12) \(\dot{{q}}= {Dq} - {qD}\) zu gebrauchen ist. (Born and Wiener 1926a, p. 182).

  36. See Sect.10— Inertial Motion.

  37. Note 27 also applies to this section.

  38. In our paper we talk about an operator space as the space in which the solutions for the equations of motion are given. It is not really a vector space. In fact, Wiener substitutes the constants of the field by series of differential operators, so it would be more accurate to talk about a module over a ring.

  39. The authors’ paper was released before the rise of the statistical interpretation of quantum mechanics and the subsequent questioning of determinism by Heisenberg’s work (Heisenberg 1927).

  40. In other words: substitution of equation (14) by the alternative

    $$\begin{aligned} {q} =\varphi _1({D}) {e}^{i \omega _0 t} + \varphi _2({D}){e}^{- i \omega _0 t}, \end{aligned}$$
    (BW(32))

    or by any other combination leads to the same results as those obtained by the authors. We omit here the details of the explanation, which are not difficult and are based mainly on the symmetry of the equations involved.

  41. In this instance, we consider \(\varphi _1({D})\) and \(\varphi _2({D})\) to be parameters.

  42. Strictly speaking, the paragraph would have established \(W_0\) as the first point of an open set in \(\mathbb {R}\), which is meaningless. Notice that the existence of discrete energy levels can be concluded from the Fourier analysis and that the quantum condition is used just to establish the recurrence between them.

  43. \(\varphi = \sum _{k=0}a_k {D}^k \).

  44. Ignoring the constants and using \(u=2 \pi i W /h \) one has \(\varphi _1(u) = \sqrt{\dfrac{hu}{\pi i \mu }}\).

  45. We emphasise this because, as we have seen, the operators were supposed to have a specific series development. See also footnote 38.

  46. The issues with “non-commutativity” are treated in more detail in this section.

  47. von Neumann suggests the solution pointed out here but qualifies it as an “inexact way” of approaching the problem because of his rejection of the \(\delta \)-function.

  48. \(\frac{2 \pi }{h}(V-W) = \xi \), and writing \(\hat{q}(\xi )\) instead of q(V, W).

  49. It is the first term of Wiener’s function \(\varPsi \) in (33) of Sect. 13 that gives \(\varPsi \) its delta-function character.

  50. See Sect. 13—Annex: Mathematical precedents. Wiener uses this technique in (36).

  51. See footnote 23.

  52. See Courant and Hilbert (1924, p. 309); we have used the simpler notation found in Courant and Hilbert (1970, p. 309).

  53. Note 27 also applies to this section.

  54. In other words, the solution would be the weighted average of the elongations obtained by the singular forces, where the forces act as relative weights.

  55. K is nevertheless continuous at \(\xi \) and continuously differentiable everywhere else.

  56. A note about terminology follows. It is common to use the term “distribution” applied also to the function g entering in the definition of G (for example, when shaping a Gaussian “distribution”). According to the statisticians’ use of the term distribution, one should distinguish between the density g and its distribution G, the latter being a non-decreasing function with limit 1, as in the case of the Heaviside function.

  57. See Peters (2004) for other approaches to the \(\delta \)-function, including the one of Courant presented here; this work also analyses the lack of acceptance of the \(\delta \)-function at that time.

  58. The condition is intended to exclude functions that could be increasing towards infinity at some point. The asymptotic behaviour towards zero as x goes to infinity is guaranteed by the integrability of f.

  59. \(L^2\) convergence.

  60. This is more clearly seen in one of the intermediate expressions of Wiener’s paper.

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Authors and Affiliations

  1. Centre for the History of Science, Research Module C, Carrer de Can Magrans s/n, 08193, Bellaterra, Barcelona, Spain

    Gonzalo Gimeno

  2. Centre for the History of Science, Universitat Autònoma de Barcelona, Barcelona, Spain

    Mercedes Xipell & Marià Baig

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  1. Gonzalo Gimeno
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Correspondence to Gonzalo Gimeno.

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Communicated by Tilman Sauer.

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We are particularly grateful to Arianna Borrelli, whose previous (unpublished) work on the topic has inspired several sections of this paper; we also thank the members of the department of History of Science of the Technische Universität Berlin for their encouragement and suggestions. We also thank Anthony Duncan for sharing his points of view with us. This paper has additionally benefitted from the debates in the framework of the Fourth Conference on History of Quantum Physics (HQ-4), 2015, in Spain. We thank Pablo Soler and Miguel Gimeno for their careful review and comments. We thank Jaume Navarro and two anonymous referees for their suggestions, which have decidedly contributed to the readability and understandability of this work. We acknowledge the support of the Spanish Ministry of Science and Innovation (PID2019-105131GB-I00)

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Gimeno, G., Xipell, M. & Baig, M. Operator calculus: the lost formulation of quantum mechanics. Arch. Hist. Exact Sci. 75, 283–322 (2021). https://doi.org/10.1007/s00407-020-00262-z

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