Yablon wrote:Joy,
As you know I have been busy the last two weeks with my Kaluza Klein paper, but I did want to take a moment to offer public congratulations! I know all too well the perils and pitfalls along the road you are traveling.
Jay
lkcl wrote:hooray, well done. so what's next on the list?
Yablon wrote:Joy, do you know if the listing they provide is for unique IP addresses? Or might there be a duplication if an individual goes back several times? Best, Jay
The triviality of the tangent bundle TS^3 means that the 3-sphere is parallelizable. A k-dimensional manifold is said to be parallelizable if it admits k vector fields that are linearly-independent everywhere. Thus on a 3-sphere we can always find three linearly-independent vector fields that are nowhere vanishing. These can then be used to define a basis of a tangent space at each of its points. As a result, a single coordinate chart can be defined on a 3-sphere that fixes each of its points uniquely. Informally, a manifold is said to be parallelizable if it is possible to set all of its points in a smooth flowing motion at the same time, in any direction. Rather astoundingly, this turns out to be possible only for the 0-, 1-, 3-, and 7-spheres. Thus parallelizability of these spheres happens to be an exceptionally special topological property. One way to appreciate it is by considering a manifold that is not parallelizable. For example, it is not possible to set every point of a 2-sphere in a smooth flowing motion, even in one direction. However you may try, there will always remain at least one fixed point --- a pole --- that will refuse to move. This makes it impossible, for example, to cover the Earth with a single coordinate chart. For similar reasons, parallelizability of the 3-sphere, or equivalently the triviality of its tangent bundle, turns out to be indispensable for respecting the completeness criterion of EPR. And since this criterion is the starting point of Bell’s theorem, understanding the parallelizability of 3-sphere turns out to be indispensable for understanding the topological error involved in all Bell-type arguments.
Joy Christian wrote:
This still does not answer your question: What is the significance of S^7. Well, S^7 I am working with is the algebraic representation space of the quaternionic 3-sphere described above. It is essential to understand the local origins and strengths of ALL quantum correlations, not just EPR-Bohm type singlet correlations. See theorem 3.1 of my RSOS paper.
Heinera wrote:Does anyone know of some good discussions about this grounbreaking paper on the internet? Links would be most welcome.
Mikko wrote:Heinera wrote:Does anyone know of some good discussions about this grounbreaking paper on the internet? Links would be most welcome.
There is some discussion on https://pubpeer.com/publications/170464 ... 5E2632D018
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