... 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.
(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)
... 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.
(Submitted on 8 Feb 2016)
