I try to use the information from physics as very sophisticated computations that physicists do that are tested by experiments and that somehow reveal if one looks at them carefully enough from a matehmatical point of view reveal amazinsingly beautiful structure
Alain Connes (online.kitp.ucsb.edu/online/strings05/connes/rm/jwvideo.html)
-Do you have a preference for mathematics over physics?
-“My heart lies with both.”
The Interpretation of Quantum Mechanics. By Roland Omnès. Princeton University Press, Princeton, New Jersey, 1994,
This monograph is the first book-length treatment of the consistent histories approach to the interpretation of quantum mechanics which I initiated in 1984, and to which Omn`es (starting in 1987) and Gell-Mann and Hartle (starting in 1990) have made major contributions. While consistent historians do not agree on every detail, there is a common core of ideas which can be summarized as follows. A closed quantum system (the universe, if one is ambitious) is represented by a Hilbert space, and anything that can sensibly be said about it at a particular time is represented by some subspace of this Hilbert space; in other words, there are no hidden variables. A history consists of a sequence of subspaces E1, E2, . . . associated with times t1, t2, . . ., understood as events occurring, or properties which are true, at these times. Provided the history is a member of a consistent family of histories, it can be assigned a probability, and within a given consistent family these probabilities function in the same way as those of a classical stochastic theory (imagine a sequence of coin tosses): one and only one of them occurs, and the theory assigns a probability to each possibility. Inconsistent histories, those which do not belong to a consistent family, are meaningless. The unitary time evolution generated by the Schrödinger equation, without any stochastic or nonlinear modifications, is used both for determining the consistency conditions which define consistent families, and for calculating the probabilities of histories belonging to a particular family.
Measurements play no fundamental role in the consistent history interpretation; they simply correspond to sequences of events inside a closed system in which the measurement apparatus, along with everything else, is treated quantum mechanically. Thus a possible history for a closed system in which there is an apparatus for measuring the z component of the spin of a particle might include an initial state, a value of S z at a time shortly before the particle reaches the apparatus, and the position of a pointer on the apparatus at some later time. Using conditional probabilities one can show, under suitable conditions, that the particle earlier had the property indicated later by the pointer position. In this and various other ways the consistent history approach replaces the smoky dragons which inhabit textbook and other treatments of “measurement” with precise mathematical and logical rules yielding results which are often much closer to the intuition of experimental physicists than to what one finds in the literature on quantum foundations...
While it does have defects, I nevertheless admire this book as a bold attempt to give substance to a vision common to consistent historians, namely that our current scientific understanding of the physical world, macroscopic phenomena as well as microscopic, can be linked to a firm foundation of quantum mechanical principles by appropriate and precise rules of sound reasoning. There is no need to add hidden variables to the Hilbert space, or tinker with the Schrödinger equation, or restrict ourselves to talk about “measurements” in order to construct a coherent interpretation of quantum mechanics which overcomes the well-known conceptual problems which have given such trouble to those who have been attempting to understand the subject for the past seventy years. We are indebted to Omnès for showing how much of this program can actually be carried out within the consistent histories framework. Some imperfections are inevitable in a pioneering effort, and those portions of the book which I find problematical are nonetheless useful as indications of what needs to be better understood, or more clearly explained, or perhaps both.
Robert B. Griffiths (Carnegie-Mellon)(Submitted on 16 May 1995)
Que sont donc les mathématiques ? Après Platon et Russell ou Gödel, cette grande question suscite toujours plus de débats et de controverses et elle attend encore sa réponse. Il s’agirait d’un pur jeu formel pour les uns, de l’image d’une réalité immatérielle pour les autres... Et voici que les acquis récents de la physique des particules invitent à explorer une troisième voie. Car les lois qui gouvernent les quarks et autres constituants de la matière, la manière dont elles furent découvertes et les modes de pensée par lesquels il fallut passer sont autant de révélateurs d’une parenté intimement profonde entre physique et mathématique. Pourtant, si l’on pénètre davantage les rapports complexes qui s’exercent entre la physis – la réalité empirique – et le logos – la réalité des formes –, force est de constater aussi qu’un irréductible hiatus les sépare. Alors, l’un devient deux...
What is mathematics? After Plato and Russell or Gödel, this fundamental question is still the subject of more debate and controversy and is still waiting for an answer. It would be a pure formal game for some, the image of an immaterial reality for others... And now recent achievements in particle physics invite us to explore a third way. Because the laws that govern the quarks and other constituents of matter, the way they were discovered and the ways of thinking through which they had to pass are all revealing of a profound relationship between physics and mathematics. However, if we penetrate more deeply into the complex relationships between the physis - the empirical reality - and the logos - the reality of forms - we must also note that they are separated by an irreducible hiatus. So, one becomes two...
(Translated with the help of www.DeepL.com/Translator)
Dans ma thèse j'ai choisi la citation suivante :
Proposition 32
La connaissance de la physis ne peut provenir que d'une forme d'expérience. La disjonction de nature entre physis et logos se laisse voir dans la philosophie traditionnelle comme l'impossibilité de construire la science par pure induction
Knowledge of physis can only come from a form of experiment. The natural disjunction between physis and logos can be seen in traditional philosophy as the impossibility of building science by pure induction.
...once one took String Theory seriously, one soon found a lot of reasons that physicists would have to pay attention to previously unfamiliar topics in more or less modern mathematics. For example, at a basic level, a string moving in spacetime sweeps out a two-dimensional surface with the property of what mathematicians call a Riemann surface. Riemann surfaces are an important topic in the mathematics of the last century, and they became important for physicists primarily because of String Theory.From my vantage point, all this made the interaction of physics with more or less contemporary mathematics far more robust and significant. Opportunities to apply physics-based insights to “purely mathematical” problems stopped seeming like exceptions.In my own work in the years just after 1984, the development that is most worth mentioning here involves topological quantum field theory. This was partly motivated by hints and suggestions by the mathematician Michael Atiyah, who pointed to mathematical developments that he suggested should be better understood using physical insight. Other hints came from developments in physics.Each problem here involved applying physics ideas to a problem that traditionally would have been viewed as a math problem, not a physics problem. These were all problems that I would not have seriously considered working on until String Theory broadened our horizons concerning the relations between mathematics and physics. In each case, the aim of my work was to try to show how a problem that naively is “purely mathematical” could be approached by methods of physicists.I will just tell you about one of these problems. It involved knots in ordinary three-dimensional space. A tangled piece of string is a familiar thing in everyday life, but probably most of us are not aware that in the 1900's, mathematicians built a deep and subtle theory of knots. By the time I became involved, which was in 1987-8, there was a puzzle, which Atiyah helped me appreciate. The mathematician Vaughn Jones had discovered a marvelous new way of studying knots – for which he later received the Fields Medal. Vaughn Jones had proved that his formulas worked, but “why they worked was mysterious.
It may be hard for someone who does not work in mathematics or science to fully appreciate the difference between understanding “what” is true and understanding “why” it is true. But this difference is an important part of the fascination of physics and mathematics, and I guess all of science. I will say, however, that the difference between “what” and “why” depends on the level of understanding one has at a given level of time. One generation may be satisfied with the understanding of “why” something is true, and the next generation may take a closer look.
Anyway, getting back to knots, I was able to get a new explanation of Vaughn Jones's formulas by thinking of a knot as the trajectory followed by an elementary particle in a three-dimensional spacetime. There were a few tricks involved, but many of the ideas were standard ideas of physicists. Much of the novelty was just to apply the techniques of physicists to a problem that physicists were not accustomed to thinking about.
This work became one of my best-known contributions, among both mathematicians and physicists. But it is also an excellent illustration of something I said in my acceptance speech the other night. No matter how clever we are, what we can accomplish depends on the achievements of our predecessors and our contemporaries and the input we get from our colleagues. My ability to do this work depended very much on clues I got from work of other scientists. In several cases, I knew of these clues because colleagues pointed out the right papers to me or because the work was being done right around the corner from me by colleagues at the Institute for Advanced Study in Princeton. It also helped at a certain point to remember some of what I had learned from Sidney Coleman back when I had been at Harvard, involving yet another insight of Albert Schwarz.
Edward Witten
En 1979, j'étais à une conférence sur les théorie de jauge à Cargèse où l'un des vedettes était ... Ed Witten. Il se trouve qu'on était dans le même hôtel et que j'ai eu donc souvent l'occasion de l'écouter ; et il y a une chose qui m'a beaucoup frappé. Tandis que les autres physiciens ... parlaient toujours en termes de phénomènes, éventuellement en termes de modèles concrets testés sur ordinateurs, Witten jonglait tout le temps avec les théories ellees-mêmes, leur manière d'intéragir, de se compléter ou tout simplement d'exploser. Il me semble qu'il y a un lien direct entre cette manière de penser et la M-théorie d'aujourd'hui.
In 1979, I was at a lecture on gauge theory at Cargèse where one of the stars was ... Ed Witten. It turns out we were in the same hotel and so I often had the opportunity to listen to him ; and there is one thing that struck me a lot. While other physicists ... always spoke in terms of phenomena, possibly in terms of concrete models tested on computers, Witten juggled all the time with theories themselves, their way of interacting, of complementing each other or simply of exploding. It seems to me that there is a direct link between this way of thinking and today's M-theory. (blogger's translation)
Valentin Poénaru
As a mathematician and outsider to the world of physics, I feel that gives me a perspective which to some extent is above the fray which is taking place in theoretical physics. Experience shows that successful physical theories follow a fairly well defined sequence of major steps. The first step is the elucidation of the essential physical ideas in purely physical terms which one needs to describe the theory. The second step is to describe mathematically a system which corresponds with the physical ideas resulting from the first step. The third step is to use the mathematical system resulting in step two to make calculations which make predictions of physically interesting quantities. The fourth step is to experimentally test the predictions resulting from the third step. One might introduce a fifth step which is to modify physical ideas in step one and go through the sequence of steps again in order to make improvements in the theory. In short, the steps which can be considered as a cycle, are
(1) Physics(2) Mathematics(3) Calculation(4) Experiment
Some comments on these steps are now in order. It is important that the physical ideas of (1) not be overly influenced by mathematics. In step one, physics must be the main consideration. In (2) on the other hand, the level of rigour should not be too high as it can prevent progress and get in the way of progress to step three. Likewise, in (3), for calculations, similarly sometimes physical ideas can be used to aid in calculation where it would not be permitted in pure mathematics. For instance, if the Weierstrass level of rigour had been required of Newton, it could have prevented the development of Newtonian mechanics. The reason that the requirements of rigour can be relaxed here is due to the final step (4) which will be the final arbiter of success. Notice that this means that if (1), (3), and (4) are omitted, all that remains is sloppy mathematics...
Maurice J. Dupré, Department of Mathematics New Orleans, LA 7011818 September 2013
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.
(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...
[Submitted on 3 Dec 1992]
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]).
(Submitted on 24 Jan 2016)
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.
(Submitted on 8 Feb 2016)
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"
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 EinsteinSubmitted 4 November 1915, Published 11 November 1915
