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From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment

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Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 (LT-XII/QTS-X 2017)

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Abstract

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares. In the words of André Weil [W] “as with most questions that ever attracted his attention, he never abandoned it”. Euler introduced on the way the alternating “phi-series”, the better converging companion of the zeta function, the first example of a polylogarithm at a root of unity. He realized - empirically! - that odd zeta values appear to be new (transcendental?) numbers. It is amazing to see how, a quarter of a millennium later, the numbers Euler played with, “however repugnant” this game might have seemed to his contemporary lovers of the “higher kind of calculus”, reappeared in the first analytic calculation (by Laporta and Remiddi) of \(g-2\) - the anomalous magnetic moment of the electron, the most precisely calculated and measured physical quantity [K]. Mathematicians, on the other hand, are reviving the dream of Galois of uncovering a group structure of the periods, including the same multiple zeta values - the mixed Tate motives, inspired by ideas of Grothendieck and appearing in a variety of subjects - from algebraic geometry to Feynman amplitudes.

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Notes

  1. 1.

    A first hand review of Euler’s work and an elementary introduction to multple zeta values is contained in Sects. 1–2 of Cartier’s Bourbaki lecture [24].

  2. 2.

    A definitive collection of Euler’s works, Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. By the time of the appearance of his first full scale biography [21], at the end of 2015 the edition is nearing completion with over 80 large volumes published. The Eneström index of Euler’s papers counts 866 entries. A concise (30-page) biography of Euler with color illustrations is contained in [34]; shorter biographical sketches can be found in [5, 62].

  3. 3.

    Introduced by Sommerfeld (1916): \(4\pi \epsilon _0\hbar c\alpha = e^2\); in modern particle physics texts the vacuum permittivity \(\epsilon _0\) is taken as unity.

  4. 4.

    Interviewed some 37 years later (in 1986) Norman Kroll said: “[The errors] were arithmetic...The thing that I learned from that is: in doing a complicated calculation, you have to take the same kinds of precautions that an experimenter takes to see that dirt doesn’t get in his apparatus. We had some internal checks but not nearly enough.” -[58].

  5. 5.

    Hans Dehmelt (1922–2017) shared the 1989 Nobel Prize in Physics “for the development of the ion trap technique”.

  6. 6.

    Kinoshita uses \(A_1^{(2n)}\) instead of \(a_n\).

  7. 7.

    Is it possible that the hero of these (and other) calculations, Stefano Laporta, never had a tenure in Bologna (as I learned from David Broadhurst, December, 2014)?

  8. 8.

    More generally one is dealing with integrands depending on external momenta and masses, again expressed as ratios of polynomials [16] (explained in Panzer’s thesis [47]).

  9. 9.

    “Mes principales méditations depuis quelque temps étaient dirigées sur l’application à l’analyse transcendante de la théorie de l’ambiguité.” - see [3].

  10. 10.

    Motivic periods were first associated with graph polynomials in [7]. An accessible review of formal (and motivic) double zeta values is contained in [25].

  11. 11.

    There are, unfortunately, two opposite conventions of writing this coaction, right and left, in [16, 17] and in [48], respectively. We adopt that of [48] which coincides with the one of our earlier work [59, 60].

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Acknowledgements

It is a pleasure to thank Pierre Cartier for his critical remarks. The author thanks IHES for hospitality during the final stage of this work. He acknowledges the help of Mikhail Stoilov and partial support by Bulgarian NSF Grant DN-18/1.

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  1. Institut des Hautes Etudes Scientifiques, 91440, Bures-sur-Yvette, France

    Ivan Todorov

  2. Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784, Sofia, Bulgaria

    Ivan Todorov

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  1. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Vladimir Dobrev

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Todorov, I. (2018). From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment. In: Dobrev, V. (eds) Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 . LT-XII/QTS-X 2017. Springer Proceedings in Mathematics & Statistics, vol 263. Springer, Singapore. https://doi.org/10.1007/978-981-13-2715-5_3

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