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You and I are having a conversation:

"Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles."

"Something like that"

"...And then we can add these as local symmetries to space by restricting a connection on some priciple bundle with a nice little lagrangian..."

"Again, something like that."

"But this all seems pretty 'top-down'- I mean, why aren't we trying to see what matter looks like?" I reel off a somewhat overblown soliloquy on that old John Wheeler quote about empty space (free wine from the afternoon's colloquium, evident in its effect), but you have stopped listening.

"String theory?"

"Don't get me started on string theory! A bunch of guys who never learned to apply Occam's razor is what that is- 'ooh it's not working, must be because we need more assumptions'- or something to keep the differential geometers busy 'til they unfreeze Einstein!" You look offended. "What?!! I'm joking!!"

"Sure."

"Still, though, they're wrong- I mean; Donaldson's Theorem is a dead give away isn't it?! Here we are hurtling through topological $\mathbb{R}^4$, having this conversation, surrounded by artefacts of differential structure: someone's proved that this is the only space of this sort in which these artefacts can occur in some sensibly invariant way- and with a proper continuum of possibilities, too- and we're talking about string theory! Why isn't everyone in the mathematical physics world trying to crack the puzzle of the different structures in topological $\mathbb{R}^4$? I'm not saying it's going to be electron= Casson handle, but it must be worth at least looking. For crying out loud, why aren't we all looking at exotic $\mathbb{R}^4$?!

You resist the temptation to give me a withering look, clear your throat, and say:

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NB: didn't make it wiki on the basis of the quite technical nature of the question (cf. mathoverflow.net/questions/30480/… may be a borderline case though- feel free to hit it with the wiki hammer if you disagree strongly. –  Tom Boardman Jul 5 '10 at 15:46
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+1 not least for style. –  Steve Huntsman Jul 5 '10 at 16:06
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By the way, is it known whether all exotic R^4s admit lorentzian metrics? –  José Figueroa-O'Farrill Jul 5 '10 at 21:04
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A smooth manifold has a Lorentzian (3,1)-type metric if and only if it has a 1-dimensional subbundle of its tangent bundle. Since $\mathbb R^4$ is contractible, its tangent bundle is trivial regardless of the chosen smooth structure. –  Ryan Budney Jul 6 '10 at 2:35

8 Answers 8

To play devil's advocate, you could easily turn around this line of thought. Since we live in a 3+1-dimensional space-time, we develop concepts that are sensitive to our experiences. So calculus and manifold theory single out dimensions 3 and 4 because that's the only way we know how to design geometric things. Of course the definitions of smooth structure does not mention 3+1-dimensional space-time, but somehow the ingredient concepts are intrinsically 3+1-dimensional, at least that's the assertion of this comment. i.e. the whole concept of linear approximation and smooth function would perhaps appear quaint, uninformative or just plain missing-the-point to beings that live in a 5+3-dimensional space-time, whatever that might mean.

The larger upshot of this is, IMO, good physics is really inspired by experiments, not fancy mathematical tools. Without fancy mathematical tools you've maybe got not so much to work with but coming at physics from the perspective of a mathematician wanting to use tool X might not be productive. There's this saying "if all you have is a hammer, everything starts to look like a nail" that applies. We should design our tools to purpose, not shape the purpose to the tools.

If you're going to use exotic smooth $\mathbb R^4$'s to construct a physical theory one major problem is that exotic smooth $\mathbb R^4$'s are locally just like $\mathbb R^4$. So if you want your theory to have non-trivial dependence on the smooth structure it has to be a global (non-local in a very significant way) theory -- taking account of behavior off at infinity.

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+1 I would +2 if I could. Thumbs up for the observation that the mathematical models of space we define are sensitive to our experiences and innate mental psyche. –  Colin Tan Jul 7 '10 at 15:19
    
But non-locality is one of the main theoretical aspects of quantum mechanics! –  Leo Alonso Mar 22 '11 at 10:56
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I've pondered this idea for quite some time, and it still doesn't completely make sense. The pattern seems to be dim 1 = trivial, dim 2 = easy, dim 3 = hard, dim 4 = impossibly hard, dim 5+ = medium. If our brains - and therefore our questions - were adapted to a four dimensional universe, then why are the dimensions which are most relevant to our physical experience the most difficult? –  Paul Siegel Mar 22 '11 at 11:16
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Perhaps because we don't know how to ask proper high-dimensional questions? Our high-dimensional questions are immediate generalizations of low-dimensional concerns, so they're simple. –  Ryan Budney Mar 22 '11 at 16:02
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@Kelly Davis: You seem to keep missing my point. This isn't the place for this kind of discussion. –  Ryan Budney Aug 21 '13 at 15:30

A quick answer...

Likely the most well-known and accepted paper in the physics community on this is, no surprise, by Witten Global gravitational anomalies.

There he argues, in dimensions higher than 4, that an exotic differential structure should be interpreted as a gravitational instanton. He punts on the four dimensional case as the generalized Poincaré conjecture is unsolved in dimension 4.

So, if the GPC finds that exotic differential structures exist on $S^4$, Witten's argument implies they should be interpreted as gravitational instantons.

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Thanks! Looks intriguing... –  Tom Boardman Jul 5 '10 at 18:54
    
You're welcome. Note, for this topic you need only read section III of the paper. –  Kelly Davis Jul 5 '10 at 20:13

I can answer your literal question. Not everyone studies exotic $\mathbb{R}^4$, because the universe of mathematical and theoretical physics is a big one with many interesting ideas, and there's no reason for everyone to drop all of their potentially fruitful projects to study one speculative idea that may be a dead end. Even though we have a continuum of exotic differentiable structures, we don't seem to have a way to pass from a particular differentiable structure to a description of reality, especially one that lets us describe observable consequences of a difference in differentiable structure. Since there doesn't seem to be much existing evidence that this is something likely to work, people don't feel like investing their time in it.

A more specific response concerning extra dimensions is that Kaluza-Klein theory was an impressively parsimonius way to produce electromagnetism from pure gravity. One could reasonably argue that the existence of compact extra dimensions in this style is an idea that is both concise and carries more descriptive power than an ad-hoc correspondence between exotic structures on topological 4-space and say, states of the universe.

Finally, we spend time studying the standard model with its principal bundles because it has been fantastically successful for predicting and interpreting the outcomes of experiments. In order for a competing theory to gain much professional attention, it has to either efficiently envelop large parts of the standard model, co-opting its experimental success, or offer a conceptually satisfactory description of a phenomenon that the standard model fails to treat well. These are big boots to fill, and so far, exotic $\mathbb{R}^4$ as a theory of everything (inasmuch as it has been defined) doesn't come close.

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From a physicist's perspective I think that the latter part Ryan's answer really goes to the heart of the matter. The point is that the VAST majority of physical phenomena are purely local. Consider for example General Relativity. An observer existing for a finite time will probe a finite patch of spacetime. To describe what he sees he solves Einstein's equation:

$ R_{\mu \nu}-\frac{1}{2}Rg_{\mu \nu}\sim T_{\mu \nu}$

where in the above $g$ is the metric, $R_{\mu \nu}$ its Ricci curvature and $T$ is a tensor field describing the distribution of energy and matter in spacetime. This is a local differential equation, and since the observer sees a small patch, for the most part he could care less whether the global structure of spacetime is $\mathbb{R}^{4}$ or any other smooth four-manifold.

A crucial point is that unlike derived physical equations, like say the heat equation, equations of fundamental physics (General Relativity, Electrodynamics, Quantum Field Theory, String Theory) are invariant under the Lorentz group of symmetries. This means that, for reasonable physical matter and energy distributions, there is a $finite$ signal propagation speed (the speed of light) and thus far away properties of the differentiable structure of spacetime take a very long time to have local consequences for any fixed observer.

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You are sweeping a VAST amount of subtlety under the rug. The heat equation is also a "local differential equation", but each point feels the influence of other points infinitely soon. One can appeal to some sort of anthropic principle to claim that all reasonable laws of physics must have a domain of dependence property, but it certainly does not follow just from Einstein's equation or Lagrangian field theory, or a PDE formulation of the laws. –  Willie Wong Jul 6 '10 at 11:22
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Lorentz invariance does not imply finite signal propagation speed. (If it did, the dominant energy condition would not be a physical assumption, rather a theorem.) For certain linear theories, because of the lack of a dispersion relation (at the level of the principal symbol, so something like Klein-Gordon doesn't count), the characteristic cone is frequency independent and must be preserved under the Lorentz transform, and heuristically you can argue those theories can only have finite speed of propagation: indeed the propgation speed must be the speed of gravity. –  Willie Wong Jul 6 '10 at 15:40
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But for quasilinear systems (non-linear electrodynamics, non-linear field theories from high energy physics and cosmology), this argument doesn't work. My point in my original comment was precisely this: finite speed of propagation is not immediately guaranteed just because you wrote down a Lorentz-invariant Lagrangian. Take something like the Skyrme Model with a negative coupling constant and consider the small-field limit. The equation of motion is Lorentz invariant, with a bit of work one can show small data global wellposedness, and it is a reasonable model of neucleons. –  Willie Wong Jul 6 '10 at 15:50
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But it admits acausal propagation compared to the gravity cone: the characteristic cone of the principal symbol (which determines the propagation of singularities and hence, the speed of propagation) lies outside of the gravity one. –  Willie Wong Jul 6 '10 at 15:54
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Now, just to confuse people even more: a domain of dependence property is virtually required for one to be able to prove well-posedness of the Cauchy problem for the mathematical model. But the domains of dependence guaranteed by the Lagrangian theory may or may not coincide with the background Lorentzian metric. So for the modified Skyrme model I mentioned, the initial value problem can be solved for space-like slices that are also space-like relative to the causal character of the model, which is a smaller class than general space-like hypersurfaces. –  Willie Wong Jul 6 '10 at 15:59

I am not an expert but I guess there is a number of "historical" reasons explaining the lack of exploration of physical consequences of exotic differential structures:

1) many physicists are inclined to keep things as simple as possible;

2) the existence of exotic differential structures on $\mathbb{R}^4$ is just not sufficiently well known in the physics community (how many textbooks on differential geometry mention this?).

That said, a bit of googling has brought up (in addition to the book pointed out by Steve Huntsman in the comments) a 1989 paper Topological defects and differential structures by R. Rohm exploring some of the possible physical manifestations of exotic differential structures of spacetime, the 1994 paper Exotic Smoothness and Physics by C. Brans, the 1996 paper Exotic smoothness, noncommutative geometry, and particle physics by J. Sladkowski (preprint version is on arXiv), and, more recently, Exotic Smoothness and Quantum Gravity by Asselmeyer-Maluga and other preprints by him and his coauthors, and the papers Exotic Smoothness and Noncommutative Spaces. The Model-Theoretical Approach and Exotic Smooth 4-Manifolds and Gerbes as Geometry for Quantum Gravity by Król.

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«many physicists are inclined to keep things as simple as possible» That's quite an odd claim, in view of what they manage to come up with! :D –  Mariano Suárez-Alvarez Jul 5 '10 at 17:00
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Oh, plenty of us know about the exotic structures on $R^4$. It's a not entirely uncommon joke that the uncountable number of them dominates the path integral, and that's the reason why we live in 4 dimensions. The problem with such jokes and other things along these lines is that's it's been pretty much impossible to turn them into an actual theory. –  Aaron Bergman Jul 5 '10 at 17:02
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@Tom B. Just in case, the preface and chapter 1 of the book are available here: worldscibooks.com/physics/4323.html –  mathphysicist Jul 5 '10 at 17:18
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@Mariano: I still stick to many (especially in the fields closer to experiments), but definitely not all :) –  mathphysicist Jul 5 '10 at 17:19
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@Aaron--your joke reminds me of this: arxiv.org/abs/gr-qc/9702052 (Tegmark, M. “On the dimensionality of spacetime”. Classical and Quantum Gravity 14, L69 (1997).) –  Steve Huntsman Jul 5 '10 at 19:29

Why isn't everyone in the mathematical physics world trying to crack the puzzle of the different structures in topological R4? I'm not saying it's going to be electron= Casson handle, but it must be worth at least looking

How can you tell that nobody is looking? The only way I know to ascertain something like that is to look there myself and to see something I can show the others. Then I can check whether they've seen it and only if they haven't I can safely assume they weren't looking. So, you asked your final question a bit prematurely. Your opponent can easily answer that he just doesn't see anything worth his or anyone else's attention there at the moment, but he would certainly be interested if you show him anything. You look straight into his eyes and say: (your turn).

In plain English: it makes sense to promote "crazy" (=unusual, shocking; no negative connotation implied) ideas only after you can show that something new (however insignificantly looking, but new) can be squeezed from them.

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I wish that last paragraph could be shouted from the rooftops, stencilled on the walls, etc etc –  Yemon Choi Aug 19 '13 at 0:10
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For those who aren't deaf and blind, it is. For the rest it won't help. –  fedja Aug 19 '13 at 0:41

The idea: exotic smoothness=matter is fascinating and cannot wiped off by the locality argument. A dynamical process is a submanifold of the spacetime. We found this page accidentally but had parallel ideas. In a recent paper geometrization of matter by exotic smoothness we realize the idea: matter = Casson handle, i.e. we showed that the action of the fermion and gauge fields follow from exotic smoothness by considering the structure of the Casson handle. In dimension 4 one has the special fact that a local change of the 4-manifold can change the smoothness, i.e. from the physical point of view we have a local theory.

Now some words about quantization:

In a paper Exotic smooth R^4, noncommutative algebras and quantization we found a close relation between codimension-1 foliations and exotic smoothness. Especially I'm interested in the question: given a 4-manifold with boundary, how can I detect exotic smoothness on the boundary? The answer is also strongly related to the exotic R^4 where one has also a kind of localization. Surprisingly we found an answer: on the boundary (i.e. a 3-manifold) one has a codimension-1 foliation and its cobordism class (detected by the Godbillon-Vey invariant) gives you the exotic smoothness class. But by Connes work, foliations are strongly connected to C* algebras and thus we have the link to QFT. In the paper above we construct an example for the exotic R^4 and show that this C* algebra appears by deformation quantization of a Poisson algebra, all details can be read there.

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This sounds thoroughly exciting! I shall tuck into the paper immediately. –  Tom Boardman Jul 14 '10 at 12:59
    
@Aaron -- exotic smoothness is a part of the path integral independent of the integration over the possible geometries see [[arxiv.org/abs/1003.5506]] for the details –  Torsten Asselmeyer-Maluga Jul 16 '10 at 8:32
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Hopefully, a constructive comment on why exotic smoothness != matter. Let $M$ be a smooth closed simply connected 4-manifold, and $M'$ be an exotic copy of $M$. There exists an Akbulut cork, a compact contractible 4-submanifold $W \subset M$ with complement $N$ and an involution $f : \partial W \rightarrow \partial W$ such that $M = N \cup_{id} W$ and $M' = N \cup_{f} W$. [arXiv:0807.4248v1]. A world line of a particle is an embedding $g: \mathbb{R} \rightarrow M$ that, observation implies, need not lie in a compact set, but the Akbulut cork lies in a compact set. –  Kelly Davis Jul 16 '10 at 19:35
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For example, a rock sitting on a table traces out world line that is not in a compact set. An Akbulut cork could not describe a rock, or anything else, sitting on a table. –  Kelly Davis Jul 16 '10 at 23:21

Coming somewhat late to the party, I have two answers.

I once asked a general relativist this and my understanding of what he said was that the way modern mathematical GR proceeds is by solving the Cauchy problem for the Einstein equations. So you start with a Cauchy hypersurface and let things evolve. You don't impose the topology of all spacetime a priori and singularities or topology might develop (e.g. the work of Christodoulou).

Perhaps a more concrete answer is provided by this paper:

http://arxiv.org/abs/1201.6070

which says that exotic spacetimes would violate global hyperbolicity (building on work of Bernal and Sanchez).

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