There is plenty of room in four dimensional spacetime

... for wandering through exotic smoothness

Progress in theoretical physics has often come as a result of questioning old assumptions, e.g., 

1. spacetime should be an absolute product, time × space, 

2. spacetime should be geometrically flat, 

3. spacetime should have trivial topology, and many others. 

Questioning these natural assumptions obviously has led to many rich discoveries. The Galilean structure of space and time in Newtonian physics was based on 1), which certainly seems “natural” from everyday experience. Of course, we now know from special relativity that such a product structure is not absolute but relative to the state of motion of the observer. Even granted such special relativistic insights, the geometric triviality of space, if not of spacetime, also seems to be an inevitable consequence of experience. The questioning of 2) however, led to the magnificent theory of general relativity. In hindsight, questioning of assumption 3) now seems to be part of a natural progression, and indeed, much work in modern theoretical physics calls on non-trivial topological models. In this questioning spirit then, it would seem to be well worthwhile to explore the recent discovery of exotic differentiable structures on topologically trivial spaces, especially R⁴. Almost all widely investigated physical theories make use of differential equations which of necessity require a manifold with such a structure. Of course, locally, all such structures are equivalent, so that the form of the equations and the local behavior of their solutions will be unchanged. Nevertheless, globally, the differentiable structures are not equivalent, so neither is the underlying physics. That is, such studies lead to fields that cannot be globally physically equivalent to any studied to date, and may offer a rich resource of new physical possibilities.

...

From the principle of general relativity as generally defined, we learn that two different smooth manifolds can represent the same physics, merely presented in different coordinate representation, if and only if they are diffeomorphic to each other. Until recently, this diffeomorphism class has been regarded by physicists as relatively trivial and the construction of “new” spacetime models seemed to require changes of the basic topology. From this review, however, it is apparent that this is not the case, that there are an infinity of physically inequivalent representations of spacetime all having the trivial topology of the first model, ℝ⁴.

It would seem very surprising, and contrary to much historical precedent, to have the sudden and unexpected discovery of the richness of mathematical models for four dimensional spacetime to be of no physical significance at all. 

Exotic Smoothness and Physics

Carl H. Brans

(Submitted on 4 May 1994)

 

The existence of... exotic structures is a strikingly counter-intuitive result. It means that although each of these manifolds is topologically equivalent to ℝ4, there is no local coordinate patch structure in which the global topological coordinates, ordered sets of four numbers, are everywhere smooth... The path to the discovery of such manifolds... is ... circuitous and mathematically involved ... 𝕊⁷. The bad news then is that following the argument in detail requires a great deal of mastery of many branches of mathematics. The good news, from our viewpoint, is that this wandering journey involves mathematical excursions touching on such strongly physics-based topics as Dirac spinors, moduli spaces of Yang-Mills instantons and even an intersection form, E8, identical to the Cartan form for the exceptional group recently studied in superstring theory...

Exotic Differentiable Structures and General Relativity

Carl H. Brans, Duane Randall

 [Submitted on 3 Dec 1992]

... contemplating quantum gravity speculations

In this article we will ... develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R⁴ to the exotic R⁴ is a quantization of geometry. 

... the model of a smooth manifold is not suitable to describe quantum gravity, but there is no sign for a discrete spacetime structure or higher dimensions in current experiments [41]. Therefore, we conjecture that the model of spacetime as a smooth 4-manifold can be used also in a quantum gravity regime, but then one has the problem to represent QFT by geometric methods (submanifolds for particles or fields etc.) as well to quantize GR. In particular, one must give meaning to the quantum state by geometric methods. Then one is able to construct the quantum theory without quantization. Here we implicitly assumed that the quantum state is real, i.e. the quantum state or the wave function has a real counterpart and is not a collection of future possibilities representing some observables. Experiments [75, 28, 83] supported this view. Then the wave function is not merely representing our limited knowledge of a system but it is in direct correspondence to reality! Then one has to go the reverse way: one has to show that the quantum state is produced by the quantization of a classical state. It is, however, not enough to have a geometric approach to quantum gravity (or the quantum field theory in general). What are the quantum fluctuations? What is the measurement process? What is decoherence and entanglement? In principle, all these questions have to be addressed too. Here, the exotic smoothness structure of 4-manifolds can help finding a way. A lot of work was done in the last decades to fulfill this goal. It starts with the work of Brans and Randall [32] and of Brans alone [29, 30, 31] where the special situation in exotic 4-manifolds (in particular the exotic R⁴) was explained. One main result of this time was the Brans conjecture: exotic smoothness can serve as an additional source of gravity. I will not present the whole history where I refer to Carl’s article.  

Here I will list only some key results which will be used in the following 

• Exotic smoothness is an extra source of gravity (Brans conjecture is true), see Asselmeyer [5] for compact manifolds and Sladkowski [86, 87] for the exotic R⁴. Therefore an exotic R⁴ is always curved and cannot be flat! 

• The exotic R⁴ cannot be a globally hyperbolic space (see [40] for instance), i.e. represented by M×R for some 3-manifold. Instead it admits complicated foliations [17]. Using non-commutative geometry, we are able to study these foliations (the leaf space) and get relations to QFT. For instance, the von Neumann algebra of a codimension one foliation of an exotic R⁴ must contain a factor of type III1 used in local algebraic QFT to describe the vacuum [11, 13, 19]. 

• The end of R⁴ (the part extending to infinity) is S³×R. If R⁴ is exotic then S³×R admits also an exotic smoothness structure. Clearly, there is always a topologically embedded 3-sphere but there is no smoothly embedded one. Let us assume the well known hyperbolic metric of the spacetime S³×R using the trivial foliation into leafs S³×{t} for all t ∈ R. Now we demand that S³×R carries an exotic smoothness structure at the same time. Then we will get only topologically embedded 3-spheres, the leafs S³×{t}. These topologically embedded 3-spheres are also known as wild 3-spheres. In [14], we presented a relation to quantum D-branes. Finally we proved in [16] that the deformation quantization of a tame embedding (the usual embedding) is a wild embedding. Furthermore we obtained a geometric interpretation of quantum states: wild embedded submanifolds are quantum states. Importantly, this construction depends essentially on the continuum, because wild embedded submanifolds admit always infinite triangulations. 

• For a special class of compact 4-manifolds we showed in [20] that exotic smoothness can generate fermions and gauge fields using the so-called knot surgery of Fintushel and Stern [51]. In the paper [10] we presented an approach using the exotic R⁴ where the matter can be generated (like in QFT). • The path integral in quantum gravity is dominated by the exotic smoothness contribution (see [6, 50, 80] or by using string theory [12]). 

Smooth quantum gravity: Exotic smoothness and Quantum gravity

Torsten Asselmeyer-Maluga

(Submitted on 24 Jan 2016)

... initiating a long march through Grothendieck theory of topoï

Currently it is a bit of a folklore to say that dimension 4 is exceptional both in physics and mathematics. On the one hand this is the dimension where Einstein theories of relativity were formulated, where the physics of particles and quantum fields found their marvelous realization on (curved) Minkowski spacetimes, and where the cosmological evolution of our world is to be described. On the other hand, many curious mathematical facts, like the existence of exotic R⁴, or in fact, of a continuum many of them, take place exactly in this dimension. It was a big effort of many mathematicians in 1980’s like Donaldson, Freedman, Gompf, Taubes and many others whose work on topology and geometry of manifolds in dimension 4 opened our eyes on the unique 4-dimensional topological and ‘smooth’ world and help in its understanding. However, taking seriously advanced and technical mathematical findings as applicable to physics, required much scientific imagination and courage in those days. It was Carl Brans who took the step in a series of papers [8, 9, 10, 11]. Soon after, there appeared the work of Torsten Asselmeyer-Maluga (e.g. [1]) and Jan S ladkowski (e.g. [42, 43]) who approached the role of exotic R⁴’s in physics from various perspectives. Carl’s Brans ideas and the papers above were an inspiration to me and I have been lucky as a researcher to work together with Torsten and Jan within the recent years. It is a big honor and pleasure to me to contribute to the volume celebrating the work of Carl Brans. 

Exotic smoothness structures on R⁴ are just Riemannian, curved smooth 4-manifolds (exotic R⁴) which topologically are (homeomorphic to) R⁴. In this chapter, I will show that the perspective of set theory and Grothendieck toposes, hence foundations of mathematics, is the right one when considering physical applications of exotic, open 4-smoothness. Even though this is neither obvious nor widely accepted approach, the use of model and set-theoretic methods in physics has a firm and vivid tradition arisen from the foundations of mathematics (e.g. [39, 12, 44, 29]). That was developed substantially further in recent years (e.g. [13, 14, 18, 25, 23, 31]).

...

Corollary 6 The renormalization problem of some perturbative QFT can be translated into the geometry of some (Euclidean) exotic R⁴ background which complements the Minkowski flat spacetime.  

One can restate the corollary as: Ultraviolet (UV) divergencies in some perturbative QFT determine exotic smoothness of the Euclidean R⁴ background. We expect that ultraviolet divergencies counterterms of some perturbative QFT’s on Minkowski spacetime are expressible in terms of the Riemannian (sectional) curvature of R⁴ 1,2 . This Euclidean curved 4-background complements the Minkowski’s one. Recall that exotic R⁴’s are just Riemannian smooth 4-manifolds which can not be flat. Thus the Corollary 6 indicates that a curvature in spacetime, hence nonzero density of gravitational energy emerges, when renormalization problem is solved geometrically. This connection with gravity is a rather universal, non-perturbative phenomenon of different perturbative QFT’s and it is an important feature of the approach.

Model and Set-Theoretic Aspects of Exotic Smoothness Structures on R4

Jerzy Król

(Submitted on 8 Feb 2016)

Qu'est ce qu'on fête aujourd'hui ?

La magie des mathématiques naturellement!

Today is November 25th 2015. I have decided to celebrate here the 100th anniversary of the publication of the last of the four papers Einstein wrote the same month about the general theory of relativity. In this paper he overcame the remaining central tension in the relation between mathematical formalism and physical interpretation. This fits nicely with the theme of this blog thus let's go a little further quoting the interesting book "The road to relativity" by Hanoch Gutfreund and Jürgen Renn:

The tension expressed itself either in a physically meaningless coordinate restriction (in the case of the theory of November 4th [a coordinate restriction followed from the {energy-momentum} conservation principle requirement]) or in a speculative hypothesis about the structure of matter (in the case of the theory presented ... on November 11th {a gravitational field equation based on the Ricci tensor was derived from the assumption that the only fields occurring as sources of gravitation are electromagnetic ones})... All that was required to achieve this final version was to change the way in which the sources of the gravitational field were inserted on the right-hand side of the gravitational field equation. If the trace of the energy-momentum tensor... is appropriately added to the source term on the right-hand side of the field equation then all the additional conditions become superfluous. In particular, the conservation principle is also satisfied as an automatic consequence of the modified field equation... 

In his latter writings Einstein frequently emphasized that the new solution of the problem of gravitation is a natural consequence of the mathematical theory centered on the Riemann tensor... So he himself described the breakthrough of late 1915 not as the result of a convergence of physical and mathematical strategies but an exclusive success of the latter. Even in his first November paper, Einstein was fascinated by the power of mathematical formalism to lead to the correct theory: "Nobody who really grasped  [the general theory of relativity] can escape from its charm, because it signifies a real triumph of the general differential calculus as founded by GAUSS, RIEMANN, CHRISTOFFEL, RICCI AND LEVI-CIVITA"

Here is the original quotation in German of the last sentence from the wonderful website The collected papers of Albert Einstein

Dem Zauber dieser Theorie wird sich kaum jemand entziehen können, der sie wirklich erfasst hat; sie bedeutet einen wahren Triumph der durch GAUSS, RIEMANN, CHRISTOFFEL, RICCI AND LEVI-CIVITA begründeten Methode des allgemeinen Differentialkalküls.

Zur allgemainen Relativitätstheorie, Albert Einstein

Submitted 4 November 1915, Published 11 November 1915

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) :