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Physics
Beyond the Standard Models
Non-locality: sheaf cohomology
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[QUOTE="Auto-Didact, post: 6113010, member: 590399"] [B][I]Seeing the thread hasn't gotten a lot of replies yet, I want to make some clarifying comments about what is meant by nonlocality and why cohomology (a topological concept) is relevant w.r.t. the topic of nonlocality.[/I][/B] There seems to be a notorious semantic issue - frequently leading to huge yet completely avoidable misconceptions and misunderstandings - when discussing Bell's theorem and the related nonlocality which is also characteristic of some aspects of QM. First, we should make a clear distinction between I) [I]mathematical locality[/I] - i.e. locality as nearness or proximity in a geometric or topological sense, [B]let's call it [I]topologic locality[/I][/B] - and II) [I]physical locality[/I] - i.e. locality referring to SR's speed of light ##c## as the maximum speed limit at which influences can travel, [B]let's call it [I]relativistic locality[/I][/B]. One might argue that it is somewhat peculiar to refer to there being a finite ##c## as 'locality' and they wouldn't be mistaken; it is a somewhat strange and unfortunate accident of history that this custom became standard terminology in SR and physics following Einstein, Lorentz et al. [B]It is however absolutely paramount to recognize and realize that definition I is far more general than definition II, i.e. topologic locality is independent of relativistic locality but not vice versa[/B]; this is because [I]relativistic locality necessarily refers to temporally occurring processes in space(time)[/I] while [I]topologic locality can be completely atemporal, i.e. a necessary consequence of the topology of the space.[/I] I think not carefully acknowledging this distinction directly leads to much of the confusion surrounding the issue of nonlocality in QM; I think it is quite clear that the [I]nonlocality in QM is a form of geometric or topologic nonlocality[/I], perhaps one even (fully) consistent with relativistic locality. The reason for this claim is that nonlocal influences in QM (e.g. entanglement) seem to be [B]occurring [I]instantaneously[/I] between coupled distant objects while [I]no information can be transmitted[/I] using such instantaneous effects; this implies that [I]this nonlocality is a consequence of properties i.e. the topology of spacetime itself. [/I][/B] This is where cohomology comes in in the present story. The problem is that topology - even without focussing on cohomology - is a difficult and abstruse topic, which many physicists simply may not be as familiar with as with other forms of mathematics, unfortunately halting further enquiry. Luckily, the authors in the second paper in the OP (Abramsky et al. 2015) not only carefully explained why cohomology is so important in this context, but they even referred to a much more accessible paper which must be the best exposition of cohomology I have ever seen: [URL='http://upcommons.upc.edu/revistes/bitstream/2099/1062/1/st17-05-a2-ocr.pdf']Penrose 1992, On the cohomology of impossible figures.[/URL] [/QUOTE]
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Beyond the Standard Models
Non-locality: sheaf cohomology
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