1

Famously, one of the objections against Newtons theory of Gravity was that it instantaneously acted at a distance. The question here is a physically philosophical one - one expects an influence to propagate. Now if the influence is instantaneous it seems to imply either an infinite speed of propagation or that distant points are somehow connected - that is non-local. Either of these seem problematical.

The solution was found in Einsteins GR, where it was seen that it was spacetime itself that propagated the influence.

The violation of Bells inequalities implies that if there is an ontologically real (in the sense of having definite values) quantum reality it must be non-local. One such proposal in Bohmian Mechanics.

Now one of the insights of the mathematical investigation of space, is that there are two structures - the space of points and the space of contiguities ie its topology. This means that distant points can be connected directly without there being a propagating influence. (Another insight is that points can be discarded).

Does this mean that one should consider a radically different topology on quantum reality (which in Bohmian Mechanics is real) such that every point is connected? But in conventional topology this means that the topology is trivial. Does this then mean one needs a 'quantum topology' that incorporates somehow the mechanics into quantum reality so that the topology is not trivial?

In short, should a physical reality which presents itself as spacetime should also have the topology we expect that spacetime itself has: the points we expect to be next to each other actually are?

3

"Famously, one of the objections against Newton's theory of Gravity was that it instantaneously acted at a distance."

Firstly, historically, I'd say it's not necessarily an objection against Newton's gravity but against special relativity. Secondly, it's worth pointing out that the problems in considering the two theories next to each other are not merely of philosophical nature. If two observers in special relativity move with non-vanishing velocity relative to each other, their conception of spatial now is different, geometrically tilted to one another. And if you have instantaneous action, you might be able to affect events in the others past. The concept of the Tachyonic antitelephone is similar.

The solution was found in Einstein's GR, where it was seen that it was spacetime itself that propagated the influence.

I have some minor issues with some formulation. E.g., the expression, "propagated the influence", seems to take one concept and make two out of it for the sake of rhetoric.

Also, and this is just an digression, I always like to point out that there were earlier attempts to do geometric gravity. And many later attempts, although they were never able to overtake Einstein's construction. The alternative proposals seem endless. This list contains some classical ones.

"Now one of the insights of the mathematical investigation of space, is that there are two structures - the space of points and the space of contiguities ie its topology. This means that distant points can be connected directly without there being a propagating influence."

Again, minor issue: To a contemporary mathematican, calling a mere set of points a structure sounds like calling a group of people (e.g. Madonna, Obama, Bob and you) a team. Dedekind said: "I imagine a set to be like a bag of things."* And then such a set is considered a space as soon as you equip it with a topology.

This is still a broad notion compared to what one would usually take to be space. When you speak about propagation of the gravitational effect, the sense in which "points can be connected" by topology is rather loose. Topology is able to capture a sense of neighborhood. This MathOverflow thread is of interest, but it is also a lot broader than that.

Connectedness is a feature. You say they can be connected directly, but without some additional structure, such as a notion of distance, this sort of connection has not much effect. Conversely, a distance function (metric) induces a topology, but the topology induced by a classical metric hardly leaves space open for playing around.

*Cantor said: "I imagine a set to be like an abyss".

Does this mean that one should consider a radically different topology on quantum reality (which in Bohmian Mechanics is real) such that every point is connected? But in conventional topology this means that the topology is trivial.

By definition, the whole set is part of the topology in any case.

Does this then mean one needs a 'quantum topology' that incorporates somehow the mechanics into quantum reality so that the topology is not trivial?

Again, the notion lets one restrict the class of functions (one can restrict oneself to investigate continuous functions, a concept which depends on the chosen topology) but a topology alone doesn't do much on its own.

Contemporary theories are build upon spaces which admit a map to R^n, and a spacetime metric like general relativity produces topologies which are very classical. Nobody really knows how to improve this successfully, but of course, there are people who are of the opinion that the problem of matching the quantum world with the classical space concept is by not starting with space in the first place. I can't really say anything about the question if there is a natural topology for, e.g., emergent gravity scenarios which differ much from the ones which are well investigated. I also don't know to what extent the notion is even necessary if, e.g., spacetime is just the spectrum of an abstract operator. I think noncommutative geometry goes along these lines.

Lastly, although this kind of work is arguably more far fetched than the already highly hypotetical attempts above, the authors of this paper argue for non-set foundations for physics and in the first two chapters, they give their arguments.

  • Interesting answer which covers a lot of ground - but a few pointers: special relativity doesn't involve gravity - so action at a distance is of no consequence here. The list of other geometric theories of gravity are not earlier than Einsteins - they're actually inspired by him. There is an earlier conception by William Clifford of a possible geometric theory of gravity but no actual theory - this was after Riemann had formulated the idea of a manifold in complex geometry. – Mozibur Ullah Jun 21 '13 at 23:55
  • It is true that the whole set is always part of the topology, but if its the only part - then it is called trivial. If it includes every possible part then it is also called trivial. One of the motivations for topology is realising that its possible to work with actual distances - what is generally called a metric. The point of physics in Topos theory is via the quantum logic of von Neumann and that of pointless topology. Its true that Topos Theory isn't set-based but it is inspired by it. A couple of extra axioms thrown in makes it equivalent to such a theory. – Mozibur Ullah Jun 22 '13 at 0:03
  • @MoziburUllah: SR doesn't "involve" it because of the described problems. But if you intend to use SR for your underlying mechanics, then it doesn't match up and the devide is artificial. There were earlier theories, see here. And in your post you conclude that every point is "connected" leads to a "trivial topology". In the context to your reply, I don't see what this should mean unless you equate the terms. – Nikolaj-K Jun 22 '13 at 10:23
1

There have recently been some fascinating theoretical developments in physics, centered around an attempt to make consistent sense of issues pertaining to information transfer around black holes. These developments have led to the most astounding proposal: That the superficially distinct concepts of quantum entanglement and spacetime wormhole are in fact identical! If this is correct, then yes, we will need to consider a radically different topology.

  • Would you have references to support these positions? This would give the reader a place to go for more information and strengthen the answer. – Frank Hubeny Aug 17 '18 at 16:02
0

"Does this mean that one should consider a radically different topology on quantum reality (which in Bohmian Mechanics is real) such that every point is connected?"

I would say yes, we do need to consider a different topology. But this would be a misleading way of presenting the issue. If every point is connected then every point is in the same place, and in this case they are not connected but identical. Thus for me nonlocality implies the truth of the Perennial philosophy. It is also suggested by the absurdity of the notion that space is made out of points.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.