A long goodbye to mathematics (a secret hello to physics?) /

A sad day's post

Grothendieck did not derive his inspiration from physics and its mathematical problems. Not that his mind was incapable of grasping this area—he had thought about it secretly before 1967—but the moral principles that he adhered to relegate physics to the outer darkness, especially after Hiroshima. It is surprising that some of Grothendieck’s most fertile ideas regarding the nature of space and symmetries have become naturally wed to the new directions in modern physics. It is this unexpected marriage—and its occasionally comical aspects—that I would like to talk about here. “A mad day’s work”, as you know, is the subtitle given to The Marriage of Figaro by Beaumarchais. From a certain distance there is less cause for astonishment; the concepts of space and symmetry are so fundamental that they are necessarily central to any serious scientific reflection. Mathematicians as influential as Bernhard Riemann or Hermann Weyl, to name only a few, have undertaken to analyze these concepts on the dual levels of mathematics and physics...

Grothendieck’s broken dream was to develop a theory of motives, which would in particular unify Galois theory and topology. At the moment we have only odd bits of this theory, but I would like to conclude with a magnificent, quite unexpected development, in which physics and mathematics come together again...

Drinfeld has introduced a group GRT1 called the (graded) Grothendieck{Teichmuller group. It is a scheme of groups over the field Q, and it therefore has a Lie algebra, denoted grt1. To describe this Lie algebra would require me to give precise information on the Knizhnik/Zamolodchikov equations, which play a fundamental role in the theory of conformal elds. It is conjectured that the Lie algebra grt1 is a free Lie algebra with generators ψ₃, ψ₅, ψ₇ ... corresponding in a natural way to the numbers ζ (3), ζ (5), ζ (7) ... Moreover, GRT1 plays the role of the Galois group for transcendental numbers of the form ζ (k₁... k_r), since it acts (conjecturally) on the algebra A by automorphisms.  

At almost the same time, at the institute, Connes and Kontsevich had just discovered a natural occurrence of the group GRT1 in fundamental problems of physics:

1. Connes and Kreimer [3] discovered how to make the Lie algebra grt1 (and other similar Lie algebras) act on the algebra corresponding to Feynman diagrams. It represents a new type of symmetry, not acting on any particular model of eld theory, but sweeping away a whole class of possible Lagrangians.

2. Kontsevich [9] has recently solved the problem of quantization by deformation for Poisson manifolds. The set of possible quantizations has a symmetry group, and Kontsevich conjectures that it is isomorphic to GRT .

In both problems the numbers ζ (k₁... k_r) arise as the values of certain integrals...

A MAD DAY'S WORK: FROM GROTHENDIECK TO CONNES AND KONTSEVICH, THE EVOLUTION OF CONCEPTS OF SPACE AND SYMMETRY

PIERRE CARTIER

Article electronically published on July 12, 2001

(Version française du texte de Cartier disponible ici)

Last words to Alexander Grothendieck himself

... séduit par le prestige soudain de la physique atomique, c’est pourtant pour des études de physique que je me suis d’abord inscrit à l’ Université de Montpellier, avec l’idée de m’initier aux mystères de la structure de la matière et de la nature de l’énergie. Mais j’ai vite compris que si je voulais m’initier à des mystères, ce n’était pas en suivant les cours de la Fac que j’y arriverais, mais en travaillant par mes propres moyens, seul, avec ou sans livres. Comme je n’avais pas le flair, ni l’appareillage, pour apprendre la physique de cette façon-là ; j’ai renvoyé la chose à des temps plus propices, Je me suis alors mis à faire des maths, tout en suivant "de loin" quelques cours, dont aucun ne pouvait me satisfaire, ni m’apporter rien au delà de ce que je pouvais trouver dans les manuels courants.

Récoltes et semailles , p. 495

Le petit enfant découvre le monde comme il respire - le flux et le reflux de sa respiration lui font accueillir le monde en son être délicat, et le font se projeter dans le monde qui l’accueille. L’adulte aussi découvre, en ces rares instants où il a oublié ses peurs et son savoir, quand il regarde les choses ou lui-même avec des yeux grands ouverts, avides de connaître, des yeux neufs - des yeux d’enfant.

Récoltes et semailles , p. 128 (version Yves Pocchiola?)

//last edit 6 june 2016

Poser un problème mathématique et le résoudre à la physicienne

Le mathématicien adulte et l'enfant physicien?

To demonstrate the cardinal difference between the ways problems are posed and solved by physicists and by mathematicians, Arnold provides the following problem for children: “On a bookshelf there are two volumes of Pushkin’s poetry. The thickness of the pages of each volume is 2 cm and that of each cover 2 mm. A worm holes through from the first page of the first volume to the last page of the second, along the normal direction to the pages. What distance did it cover?” Usually kids have no problems to find the unexpected correct answer, 4 mm, in contrast to adults. For example, the editors of the highly respectable physics journal initially corrected the text of the problem itself into: “from the last page of first volume to the first page of the second” to “match” the answer given by Arnold [1, 17]. The secret of kids lies in the experimental method used by them: they simple go to the shelf and see how the first page of the first volume and the last page of the second are situated with respect to each other...

Sums of generalized harmonic series for kids from five to fifteen

Z.K. Silagadze

(Submitted on 16 Mar 2010)

Le lecteur est évidemment invité à découvrir dans l'article en question ce que cette parabolle peut illustrer. Dans la même veine ...

Expansion de la physique et consolidation des mathématiques (et réciproquement)

“The trouble with physics” is the title of an interesting and well-informed polemic by Lee Smolin against String Theory and present main stream physics at large. He notices a stagnation in physics, so much promise, so little fulfillment [Sm06, p. 313], a predominance of anti-foundational spirit and contempt for visions, partly related to the mathematization paradigm of the 1970s, according to Smolin: Shut up and calculate. Basically, Smolin may be right. Børge Jessen, the Copenhagen mathematician and close collaborator of Harald Bohr once suggested to distinguish in sciences and mathematics between periods of expansion and periods of consolidation. Clearly physics had a consolidation period in the first half of the 20th century with relativity and quantum mechanics... while, to me, the mathematics of that period is characterized by an almost chaotic expansion in thousands of directions. Following that way of looking, mathematics of the second half of the 20th century is characterized by an enormous consolidation, combining so disparate fields like partial differential equations and topology in index theory, integral geometry and probability in point processes, number theory, statistical mechanics and cryptography, etc. A true period of consolidation for mathematics, while - at least from the outside - one can have the impression that physics ... of the second half of the 20th century were characterized merely by expansion, new measurements, new effects - and almost total absence of consolidation or, at least failures and vanity of all trials in that direction. Indeed, there have been impressive successes in recent physics, in spite of the absence of substantial theoretical progress in physics: perhaps the most spectacular and for applications most important discovery has been the High Temperature Superconducting property of various ceramic materials by Bednorz and Muller - seemingly without mathematical or theoretical efforts but only by systematic combinatorial variation of experiments - in the tradition of the old alchemists, [BeMu87].

The remarkable advances in fluid dynamics, weather prediction, oceanography, climatic modelling are mainly related to new observations and advances in computer power while the equations have been studied long before. Nevertheless, I noticed a turn to theory among young experimental physicists in recent years, partly related to investigating the energy landscapes in material sciences, partly to the re-discovery of the interpretational difficulties of quantum mechanics in recent quantum optics.

Bernhelm Booss-Bavnbek,
Steps towards Quantum Gravity and the practice of science: will the merger of mathematics and physics work? 23 novembre 2009

La gravitation quantique : (péril physique ou) promesse mathématique (?)

When we write... of “unprecedented challenges, where the achievements of spacetime physics and quantum field theory are called into question” we are aware that large segments of the physics community actually are questioning the promised unified quantum gravity. We shall not repeat the physicists’ skepticism which was skillfully gathered and elaborated, e.g., by Lee Smolin in [92]. Here we shall only add a skeptical mathematical voice, i.e., a remark made by Yuri Manin in a different context [76], elaborated in [77], and then try to draw a promising perspective out of Manin’s remark. The Closing round table of the International Congress of Mathematicians (Madrid, August 22–30, 2006) was devoted to the topic "Are pure and applied mathematics drifting apart?" As panelist, Manin subdivided the mathematization, i.e., the way mathematics can tell us something about the external world, into three modes of functioning (similarly Bohle, Booß and Jensen 1983, [10], see also [13]):

• (i) An (ad-hoc, empirically based) mathematical model “describes a certain range of phenomena, qualitatively or quantitatively, but feels uneasy pretending to be something more”. Manin gives two examples for the predictive power of such models, Ptolemy’s model of epicycles describing planetary motions of about 150 BCE, and the standard model of around 1960 describing the interaction of elementary particles, besides legions of ad-hoc models which hide lack of understanding behind a more or less elaborated mathematical formalism of organizing available data. 

• (ii) A mathematically formulated theory is distinguished from an ad-hoc model primarily by its “higher aspirations. A theory, so to speak, is an aristocratic model.” Theoretically substantiated models, such as Newton’s mechanics, are not necessarily more precise than ad-hoc models; the coding of experience in the form of a theory, however, allows a more flexible use of the model, since its embedding in a theory universe permits a theoretical check of at least some of its assumptions. A theoretical assessment of the precision and of possible deviations of the model can be based on the underlying theory. 

• (iii) A mathematical metaphor postulates that “some complex range of phenomena might be compared to a mathematical construction”. As an example, Manin mentions artificial intelligence with its “very complex systems which are processing information because we have constructed them, and we are trying to compare them with the human brain, which we do not understand very well – we do not understand almost at all. So at the moment it is a very interesting mathematical metaphor, and what it allows us to do mostly is to sort of cut out our wrong assumptions. If we start comparing them with some very well-known reality, it turns out that they would not work.”

Clearly, Manin noted the deceptive formal similarity of the three ways of mathematization which are radically different with respect to their empirical foundation and scientific status. He expressed concern about the lack of distinction and how that may “influence our value systems”. In the words of [13, p. 73]: “Well founded applied mathematics generates prestige which is inappropriately generalized to support these quite different applications. The clarity and precision of the mathematical derivations here are in sharp contrast to the uncertainty of the underlying relations assumed. In fact, similarity of the mathematical formalism involved tends to mask the differences in the scientific extra-mathematical status, in the credibility of the conclusions and in appropriate ways of checking assumptions and results... Mathematization can – and therein lays its success – make existing rationality transparent; mathematization cannot introduce rationality to a system where it is absent ...or compensate for a deficit of knowledge.” 

Asked whether the last 30 years of mathematics’ consolidation raise the chance of consolidation also in phenomenologically and metaphorically expanding sciences, Manin hesitated to use such simplistic terms. He recalled the notion of Kolmogorov complexity of a piece of information, which is, roughly speaking, “the length of the shortest programme, which can be then used to generate this piece of information ...Classical laws of physics – such phantastic laws as Newton’s law of gravity and Einstein’s equations – are extremely short programmes to generate a lot of descriptions of real physical world situations. I am not at all sure that Kolmogorov’s complexity of data that were uncovered by, say, genetics in the human genome project, or even modern cosmology data ...is sufficiently small that they can be really grasped by the human mind.” In spite of our admiration of and sympathy with Manin’s thoughtfulness, the authors of this review shall reverse Manin’s argument and point to the astonishing shortness in the sense of Kolmogorov complexity of main achievements in one exemplary field of mathematics, in spectral geometry to encourage the new unification endeavor.

Some of the great unifications in physics were preceded by mature mathematical achievements (like John Bernoulli’s unification of light and particle movement after Leibniz’ and Newton’s infinitesimals and Einstein’s general relativity after Riemann’s and Minkowski’s geometries). Other great unifications in physics were antecedent to comprehensive mathematical theory (like Maxwell’s equations for electro- magnetism long before Hodge’s and de Rham’s vector analysis of differential forms). A few great unifications in physics paralleled mathematical break-throughs (like Newton’s unification of Kepler’s planetary movement with Galilei’s fall low paralleled calculus and Einstein’s 1905 heat explanation via diffusion paralleled the final mathematical understanding of the heat equation via Fourier analysis, Lebesgue integral and the emerging study of Brownian processes). In this section, we shall argue for our curiosity about the new unification, nourished by the remarkable shortness of basic achievements of spectral geometry and the surprisingly wide range of induced (inner-mathematical) explanations.

Bernhelm BOOSS-BAVNBEK, Giampiero ESPOSITO et Matthias LESCH,

Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry, 7 octobre 2007

Quand (est-ce que) le physicien passe la main au mathématicien (?)

Voici une réponse possible tirée d'une conférence d'un grand physicien américain :

[According to an idea from quantum chromodynamics*] the gluons are in fact massless, but we don't see them for the same reason that we don't see the quarks, which is what, as a result of the peculiar infrared properties of non-Abelian gauge theories, color is trapped; color particles like quarks and gluons can never be isolated. This has never been proved. There is now a million dollar prize offered by the Cray Foundation to anyone who succeeds in proving it rigorously, but since it is true [this is a matter settled by experiment*] I for one am happy to leave the proof to the mathematician. 

[Selon une argumentation tirée de la chromodynamique quantique*] les gluons sont en fait dépourvus de masse, mais nous ne pouvons pas les voir pour la même raison que nous ne pouvons voir les quarks,  à savoir que, en raison de certaines propriétés infrarouges particulières des théories de jauge non-abéliennes, la couleur est confinée, les particules avec une charge de couleur telles que les quarks et les gluons ne peuvent jamais être isolés. Or cela n'a jamais été démontré. Il y a d'ailleurs un prix d'un million de dollars offert par la Fondation Cray à toute personne qui réussira à prouver cela rigoureusement, mais puisque cette [idée*] est vraie [de par les preuves expérimentales*] je suis pour ma part heureux de laisser la démonstration au mathématicien. 

S. Weinberg, The making of the Standard Model 2003

* les textes entre crochets ont été ajoutés par moi pour faciliter la compréhension du texte, ils visent à expliciter au mieux la pensée de Weinberg mais ils dépendent naturellement de ma propre compréhension. J'invite le lecteur à se reporter à l'ensemble du texte original pour se faire une idée éventuellement plus juste.


Pour illustrer la différence entre la notion de preuve en physique et en mathématique, voici, sur le même sujet, un extrait tiré de la page 199 d'un récent livre d'Edward Shuryak "Quantum Many-Body Physics in a Nutshell" (livre qui a l'originalité à mon goût de discuter de façon très pédagogique la chromodynamique dans le cadre de la physique quantique à N-corps) :