Local and its alternatives

[joigus]

7 minutes ago, Markus Hanke said:

Something else just occurred to me earlier today - your mathematical definition of locality makes a tacit assumption: that the underlying manifold on which the field 'lives' has a trivial topology. But what happens if that is not the case? For example, what happens if the manifold is multiply connected, or has closed loops, or whatever else may be the case? Properly defining 'locality' becomes more difficult, then.

You're right. What else can I say? +1

Perhaps this:

I've been looking at Calabi-Yau manifolds for years and to me they look like things that, no matter where you're at in them, seem to try to twist spherically in one direction and hyperbollically in another. They're like (and I'm struggling for a rough picture here) trying to open holes in every point but never quite completely opening them, so what they "open" is just more and more saddle points. I don't know if it's clear what I'm trying to point at here. Irrespective of how sensible this picture is, you may be totally right that the topology may be totally crazy, and it's only the coarse-grained picture of it that gives the illusion of locality.

But somehow I don't think any of that touches much on Bell's theorem territory, so to speak. I could be wrong, of course.

ER=EPR is another issue I'm struggling to understand. It seems that experts themselves are trying to figure out what feature of BHs is really fundamental in the sense that other features like scrambling, or entropy, etc., are more naturally derived from the defining issue. I must look at ER=EPR in more detail. I've got some references but I never seem to get around to it.

[Markus Hanke]

32 minutes ago, joigus said:

But somehow I don't think any of that touches much on Bell's theorem territory, so to speak.

Actually, what I am hoping to find is precisely a connection between ER=EPR and the issue of locality/separability/realism. What if the violation of Bell's inequalities precisely implies that spacetime is in fact multiply connected? Shouldn't it be possible, at least in principle, to retain both locality and realism, while still violating Bell's inequalities, if the underlying spacetime is multiply connected in just the right ways?

[joigus]

On 8/15/2020 at 5:44 PM, Markus Hanke said:

Actually, what I am hoping to find is precisely a connection between ER=EPR and the issue of locality/separability/realism. What if the violation of Bell's inequalities precisely implies that spacetime is in fact multiply connected? Shouldn't it be possible, at least in principle, to retain both locality and realism, while still violating Bell's inequalities, if the underlying spacetime is multiply connected in just the right ways?

I understand what you mean. That's a possible avenue.

It's been said that in physics there are conservative revolutionaries and revolutionary conservatives (something like that).

I personally prefer to go back to things we think we know very well and think as deeply about them as I can, in search for hidden assumptions or unnoticed connections. So I suppose I would be a revolutionary conservative. Clearing up the landscape, if you know what I mean, rather than looking for new landscapes. But several strategies are possible as long as you firmly attach yourself to what's been gained already.

I would like to think more about this. I'm making this comment among other things because I would like to keep this conversation on the front burner.

[Markus Hanke]

5 hours ago, joigus said:

I personally prefer to go back to things we think we know very well and think as deeply about them as I can, in search for hidden assumptions or unnoticed connections.

Generally I'm the same, but I do enjoy some far-out speculation at times

[studiot]

You guys' wish to bring manifolds and fields into it got me thinking.

What does this mean for local and non local ?

What in fact are these terms applied to ?

The space or subspace itself, or some property or characteristic of the space?

A good working definition of global might be to take the expression used to define the property at some point in the space and see if it can be used at other points.

If it can be used at every point it could be considered global, otherwise local.

So let us apply that to a field which occupies the space.

First  a scalar field. Say temperature.

If we have a table of T at some points can this be used to determine T at the rest of the points ?

Interestingly for the other thread, isn't  the answer "not without time"  ?

In other words, not without additional information.

So temperature in a space that does not include time is local.

So how about a vector field ?

Well a vector is an object with a magnitude and a direction (isn't it  ??)

I am going to start with field that has no magnitude anywhere.

A direction field.

And my definition for the direction is derived from a purely random function.

So the direction at every point is purely random.

So the direction of not even neighbouring points can be determined from the direction at any point, although all directions at every point ae derived from the same function.

So is this field local or global ?

 

 

 

[joigus]

2 hours ago, studiot said:

A direction field.

And my definition for the direction is derived from a purely random function.

So the direction at every point is purely random.

So the direction of not even neighbouring points can be determined from the direction at any point, although all directions at every point ae derived from the same function.

So is this field local or global ?

In the framework of concepts that I've grown up with, it would be neither of them.

For me, for anything, (differential manifold, analytic manifold, space of solutions to a differential eq., etc.) for this space to be local, non local, or have global properties, it should have at least correlations from one point to another. If there are no correlations at all, well. I can see no meaningful way in which I (or anybody) can define any reasonable concept of locality. But I would be interested to know if anybody should put forward a sensible one.

Edit: On second thought, maybe you could come up with a sensible definition based on correlation functions...

I don't know. The concept is a bit alien to me, but maybe it's possible.

Edit 2: The more I think about it, the more I think it's an interesting idea. +1. Whenever I hear of an idea that I haven't heard before and I can see no reason why it doesn't make sense, it becomes automatically interesting to me.

So you would have a random angle at every point x:

\[\theta_{x}\]

And then you would have a density for these thetas:

\[\rho\left(\theta\right)\]

And yet you could have the thetas at distant points having well-defined, well-behaved, smooth correlations functions, even satisfying PDEs, if you want them to. Why not?:

\[\left\langle \theta_{x},\theta_{x'}\right\rangle =f\left(x,x'\right)\]

If anybody can think of any reason why this is not possible, please tell me.

It's a bit late. I'm going to bed.

Edited August 18, 2020 by joigus