Dear Ben,
thanks a lot for your interest in my essay and exotic 4-smoothness. Your questions are excellent. I mean they indicate the essence of the approach.
Let me try to answer them as far as I can.
A.1. In fact the 'classical' non-standard models of reals used by me are those found by A. Robinson in 1960s (nonstandard analysis). They contain infinitesimals and infinite large numbers. However, the nonstandard models are related to the first-order language and properties, only. Topology on R, as a family of open subsets, requires 2-nd order language. There exists however different perspective, yielded by category theory especially toposes. E.g. in the smooth topos found by Moerdijk and Reyes, one has notion of differentiability, continuous topology, and reals in such topos are nonstandatrd reals by Robinson. Naturals as well. From that perspective one can approach more directly the fake differentiability on exotic R^4 (cf. J. Król, Exotic smoothness and non-commutative spaces. The model-theoretic approach, Found. Phys. 34, 843, 2004). However, logic and set theory are not any longer classical - they are intuitionistic.
It is fair to say that the relation of models constructions and exotic 4-smoothness is still rather conjectural from purly mathematical point of view, though work on completing this programm is under development and relation with models for spacetime in physics is promising.
A.2. There exists, more-or-less natural notion of order and topology on general families of small and large exotic R^4s. However in our, with Torsten, approach we propose to relate to the radial family of small exotic R^4s. In this case exotic R^4s are parametrized by the real radius and all are embedded as open subsets in standard R^4. The radius is, in the same time, the value of the GV invariant of codimension-1 foliations of certain 3-manifold, say S^3. This is, in fact our main technical construction allowing for the relation with QM. Geometry of foliations is non-commutative geometry of Connes.
A.3. Exotic R^4s can not be scaled smoothly, i.e. thgeir smoothness has to extend to infinity. Otherwise we would have immediately exotic S^4, but we do not (open 4-d smooth Poincare conj.).
A.4. Given the radial family, as above, of exotic R^4s we (with Torsten) showed that magnetic monopoles in 4-spacetime have similar action as open 4-regions with exotic smoothness in this spacetime. As the consequence the quantization of electric charge can be explained by the exotic 4-region in spacetime and not necessarily by the monopoles. There is also the connection of exotic R^4 with quantum (condensed) matter, heavy fermions etc. Interestingly, this kind of relation is obtained (the work in progress) from string theory and Seiberg-Witten YM theory via quasi-modular expressions. I think it is also fascinating thread (though, certainly, not in the main-stream). The exotic R^4s which were recognized as having the connection with Kondo state are those with integer GV invariant (the square-root of the radius in the radial family).
A.5. a) Even though the relation of exotic R^4s with model theory is still conjectural mathematically, there exists a considerable chance for its completion. All exotic R^4s have the leyer of 1-st order model theoretic properties, so the model theoretic-self-duality is the case for all of them. But, we do not have a proof or the construction at present. So, in principle 'yes' but, in fact, we do not know. On the other hand, maybe, there is something else in the model-theory constructions appearing here, which would be ineteresting and were not simply reduced to exotic R^4s.
b) In some natural understanding of the relation (Robinson's models) 'Yes'. But when turning to categories it is not so sure (I would have to give more precise statements and definitions).
c) I am not quite sure what you mean; if it is the commutativity of the diagram like:
N ---> R
| |
*N -->*R
then, it is indeed obviously commutative.
B.1. Exotic R^4s are ordinary, smooth, Euclidean 4-manifolds, though, necessary curved. The absolute Poincare invariance is not the case as usually on curved manifolds (the notion of particle is not absolute). However, exotic R^4s are connected with the non-perturbative regime of some QFTs (monopoles and others) in a curious way. This means that exotic 4-geometry is determined by effective matter (or couples to the effective matter) not just by energy-momentum tensor as in GR but in a different way. In fact this is our (with Torsten) current field of interest and we try to understand this better. We do not know precisely what is the fate of Poincare invariance with respect to the effective matter coupled, in the above sense, to exotic 4-geometry.
B.2. Exotic R^4 is just classical Riemannian 4-geometry (see 1 above), but it is related to foliatioons so to non-commutative geometry. When smoothness is standard, there is no foliation (and no wild embedding as appearing in Torsten's essay) and it remains only classical geometry. Exotic smoothness indeed crosses the classical-quantum border.
B.3. Certainly there should be relation to path integral - exotic smooth spaces should be present in QG path integral. Moreover, we made an attempt in calculation of path integral on exotic R^4 directly (arXive: ). Another impacts derives from string theory and quasi-modularity. By this one can find gravitational instantons associated with exotic R^4s such that the semi-classical calculation of path integral can be expanded on these instantons.
I hope that this helps and I was able to answer partly your very essential questions. I appreciate your interest and the questions.
Now, I turn to reading your essay. Please, give me a couple of days.
Regards,
Jerzy