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selfAdjoint
#73
Dec3-05, 08:48 AM
Emeritus
PF Gold
P: 8,147
Quote Quote by Careful
I think you understood me (and probably mike too) wrong. I asked about the possibility for a dynamics to be constructed which *generates* change of differentiable structure. This question was connected to your claim that you generate singular three surfaces which need the interpretation (by HAND) of a spacelike three surface associated to a matter distribution (say elementary particle). At that moment, I suggested : ``but what about the singular world 4 tube ? ┤┤, ``how to find a determinstic mechanism which allows for change of differentiable structure ?┤┤ since classically you would expect this already to be necessary (clutting of matter and so on). These objections have been left unanswered so far
There are obviously questions raised by this research, such as: according to theorems there are countably many differential structures, but we observe only a finite, and very specific, particle spectrum. What gives? This has been the bete noir of the string theorists. The mechanism for the origin of these curvatures can be pushed back to the big bang, or whatever, but we need a particle dynamics. All this will be a research program for the future.

Concerning the reference to the 96 torston paper (I note that I still did not recieve a definition of a singular connection here), let me ask some silly questions. For example at page 3, a map h: M -> N is constructed where M is an exotic 7 sphere and N the ordinary S^7, which is singular at one point, say x_0. You endow M and N with smooth Riemannian metrics, choose smooth frames e (in M) and f (in N) and claim that you can select the Riemannian metrics in such a way that dh(e)(x) = a(x) .f(h(x)) where a(x) is a S0(7) transformation in the specific orthogonal bundle over N related to coordinates in M. This seems wrong to me since dh(e)(x_0) = 0 and hence a(x_0) = 0 (could you clarify this??) which would lead to a zero curvature contribution (if I were to believe formula 9). Perhaps, I missed something but anyway...
Concerning the ``complex┤┤ curvature the authors get on page 10 in their example. It seems to me that they forgot to take the complex conjugate expression (a tangent basis in D^2 consists of d/dz and d/dz* which leads to matrix ( pz^{p-1} 0 )
( 0 pz*^{p-1})
and a^{-1} da = ( (p-1)dz/z 0 )
( 0 (p-1)dz*/z*)
(notice that there is NO division through p - these factors cancel out.) Since the authors are only interested in the trace, this gives:
(p-1)dz/z + (p-1)dz*/z*
which (in polar coordinates) gives : 2(p-1) dr/r which gives rise to zero curvature (at least when I would naively take the line integral of this around a circle).
I am sure these questions can be addressed. As I said before, careful critique is not at all objectionable.

I hope I made it more clear now why I insist upon a rigorous definition and example of a distributional connection related to a change of differentiable structure! I think this hardly classifies as ``frantic resistance┤┤.

It wasn't the requests for clarification but the insults ("sloppily written") that grated. The paper is up to the usual standards of preprint writing. Yes it will benefit by reconstructing some parts based on your remarks, but some of your comments seemed to demand they conduct a decade-long research program before publishing. Some physicists may work that way (Veltzmann comes to mind), but it is certainly not the community norm.