Local and its alternatives

[studiot]

This thread was inspired by comments from Markus Hanke about how important 'local' is.

3 hours ago, Markus Hanke said:

I should mention here again (because this is really important) that the GR field equations are a purely local constraint.

 

So if some effect is either local or non local what does these mean ?

For instance how big a region does a local effect affect ?

Does it make any difference whether we are talking galaxy sized, or microbe sized ?

Is there any relationship between the size of the region and the effect?

 

Similarly if something is not local (non-local ?) what is then affected ?

I have in mind that a non local effect may be distributed without affecting the whole universe.

And also the difference between extrinsic and intrinsic properties.

[MigL]

I would think that while a global effect is easy to define, a local effect, not so much.
I have always considered a local effect to be where global effects become trivial.
Intrinsic, properties within the manifold while extrinsic, properties arising outside the manifold.

But its a good thing that we are now clarifying definitions so we are all on the same page, Studiot.

Edited August 10, 2020 by MigL

[joigus]

36 minutes ago, studiot said:

This thread was inspired by comments from Markus Hanke about how important 'local' is.

 

So if some effect is either local or non local what does these mean ?

For instance how big a region does a local effect affect ?

Does it make any difference whether we are talking galaxy sized, or microbe sized ?

Is there any relationship between the size of the region and the effect?

 

Similarly if something is not local (non-local ?) what is then affected ?

I have in mind that a non local effect may be distributed without affecting the whole universe.

And also the difference between extrinsic and intrinsic properties.

Very interesting question. +1. The concept of locality in field theory that I'm familiar with is better characterised with a precise mathematical definition. A typical non-local evolution equation would be,

\[f\left(x+a,t\right)=L\varphi\left(x,t\right)\]

where L is some differential operator, \varphi is your field and f is a source. The values of the field depend on distant values of the source. Any change in f would affect your field instantaneously. This can also be characterised by the dependence of the evolution on arbitrarily high orders in the spacial derivatives of the source (it could be the field itself). If you want to express the evolution in a local reading (values at x, and not at x+a), you would have,

\[f\left(x+a,t\right)=f\left(x,t\right)+af'\left(x,t\right)+\frac{1}{2!}a^{2}f''\left(x,t\right)+\cdots\]

You could have more complicated patterns of non-locality. For example, if your source term were of the form,

\[\int_{-a}^{a}dx'f\left(x'-x,t\right)=L\varphi\left(x,t\right)\]

There is usually a parameter like a here, which tells you how far away this range of non-local influence is.

People talk about non-locality in relation with Bell's theorem, but they are confusing this concept with that of non-separability, which is very different.

Edit: Another possibility for your source term:

\[f\left(x,t\right)=\int_{-\infty}^{\infty}daf\left(a\right)F\left(x-a,t\right)\]

Range a: the influence is exponentially suppressed by an a-dependent factor. f(a) falls off to a certain range.

36 minutes ago, studiot said:

I have in mind that a non local effect may be distributed without affecting the whole universe.

And also the difference between extrinsic and intrinsic properties.

Edit 2: I'm not so sure about what you mean here. I would have to think about it.

Edited August 10, 2020 by joigus
Addition/corrected math mistake

[Strange]

47 minutes ago, joigus said:

Any change in f would affect your field instantaneously.

Or, presumably, at any speed faster than light?

[joigus]

1 minute ago, Strange said:

Or, presumably, at any speed faster than light?

Exactly. In the Taylor expansion that I wrote before that infinite speed would be implicit in a² /dt², a³/dt³ etc. (powers of velocity) as compared to the values of the d⁽ⁿ⁾f's. You can make this propagation as fast as you want in principle.

If there were such a thing, it would reflect instantly in the values of the fields everywhere.

To the extent that I'm aware, nobody has taken this idea seriously, but every now and then there are claims that some quantity or other could have a non-local definition lying somewhere. I think the word non-local is one of the most abused terms during the last decades. Some rigour in the definitions is necessary so that everybody understands what they're talking about. But that's really my two cents about the matter...

[studiot]

Thanks for the replies and the support.

Here's a couple of thoughts to add

If a function vanishes at some points i.e. is zero, does that make it local or non-local at these points?

 

[joigus]

Well, with the characterization of non-locality that I was talking about, you could still have non-local influences. Maybe your function (I'm assuming it's a source, or perhaps a coupling) is zero at x (the field-point you're reading your fields in terms of). But f', f'', f''' don't have to be zero.

Take for example the inversion of the Earth's magnetic field. Some crazy theory could come up tomorrow, saying that the Earth's magnetic field is extremely sensitive to the seventh-order derivative of the local density of matter at some point in the Andromeda galaxy. Almost anything that you can implement by an explicit dependence on time, you could equally well do with some crazy assumption of the form f(x+a,t) with a translating you field point x to the Andromeda.

It would be very difficult to disprove quickly. That's what I think anyway.

But I remember you once pointed out how local conservation laws (the continuity equation) are ubiquitous in physics (something like that, I don't remember the precise point now). The reasons for rejecting the idea would rather be Ockam-based, I think. Also, anything that happens here, if it involves energy, must have come from the surroundings...

The possibility of non-locality opens a really frightful can of worms, IMO...

Edit: Hopefully interesting related note...

Some decades ago people became heavily involved in models that explained the wave packet reduction in terms of a non-local modification of Schrödinger equation. Something along the lines of,

\[-\frac{\hbar^{2}}{2m}\nabla^{2}\varphi\left(x,t\right)+f\left(x+a,t\right)\varphi\left(x,t\right)=i\hbar\frac{\partial}{\partial t}\varphi\left(x,t\right)\]

with a special non-local potential that acts at some point (where the measurement is performed) and kills the wave function at distant points. The Coleman-Hepp model I think is the most famous one. Bell proved* that this is not possible unless you're willing to sacrifice unitarity (infinite evolution times, singularities in the Hamiltonian, horrible things like those).

Something that, IMHO, should have been obvious from the start, as the Copenhaguen rule for normalising the state violates linearity, and a simple projection without normalisation (giving up the normalisation factor), violates the isometric character (probability conservation). But, as nobody presses this point anymore, I never use this argument anymore.

Sorry for the off-topic excursion.

Edit2: People said very crazy things about non-locality several decades ago, and they kind of got away with it.

*Edit 3: Bell proved that for the Coleman-Hepp model.

Edited August 10, 2020 by joigus

[MigL]

OK, I guess I'm the first to fall for the different definitions of 'local'.
Local or non-local can refer to subluminal or superluminal separations.
Or local and global can refer to small scale approximations ( such as a 'local', flat, Minkowsky approximation of a globally curved space-time )

I assume the two can be related, but it would still help to know which we are discussing.
( keep in mind I'm not a mathematician )

Edited August 10, 2020 by MigL

[joigus]

1 hour ago, MigL said:

OK, I guess I'm the first to fall for the different definitions of 'local'.
Local or non-local can refer to subluminal or superluminal separations.
Or local and global can refer to small scale approximations ( such as a 'local', flat, Minkowsky approximation of a globally curved space-time )

I assume the two can be related, but it would still help to know which we are discussing.
( keep in mind I'm not a mathematician )

Very good point. +1 (Sorry, can't give more rep-points today, I owe you one).

"Local" and "non-local" are used with at least somewhat different meanings in different contexts. One of them, as you point out, is "local" as opposed to "global".

This "local" as opposed to "global" has to do with properties at a point or at the vicinity, as opposed to properties of the whole tapestry, so to speak.

In field theories the latter always (AFAIK) are integrals of the field variables. For example, in GR a very famous one is the genus of the manifold (the number of holes). It's to do with the integral of the Ricci scalar to the whole manifold. The value of R itself at a point would be a local property.

But they're related. A local PDE would be one in which all the variables involved are expressed in terms of their values at one point. It's a point by point statement. If you force to be involved arbitrarily high order of the spacial derivatives, that's another way of invoking very far away phenomena at point x.

A useful way of understanding it, I think, is this "grading of the concept of locality". Imagine a world so local that's even more local than ours: nothing can propagate:

\[\frac{\partial}{\partial t}\varphi\left(x,t\right)=f\left(x,t\right)\]

Your evolution eq. does not involve any spacial derivatives at all. In that case, the configuration at point x and at point x' are not connected. Physical quantities evolve at every point independently.

Next step is propagating: the time derivative is involved with the spacial derivatives. You can assume first order, second, etc. in spacial derivatives. Everybody calls this local, but it's "less local" only in the sense that field variables get affected in far-away points if you wait long enough.

You could always call a theory in finite order of spacial derivatives "local". You would only have to extend the set of initial data to higher and higher order spacial derivatives. Your field variables would be now phi, phi', phi'', etc.

The problem is when the order of spacial derivatives is unbounded. Then there is no way that you can re-define your state as local in any reasonable sense. Your field variables are sensitive to arbitrarily-high-order inhomogeneities in the spacial variables. You would have to provide all the derivatives, which amounts to providing the function in all space.

This graded explanation of locality is not standard, but I think it clarifies (or could clarify) how the relation between spacial inhomogeneity and time evolution is related to the intuitive concept of what local evolution must be.

 

3 hours ago, Strange said:

Or, presumably, at any speed faster than light?

3 hours ago, joigus said:

In the Taylor expansion that I wrote before that infinite speed would be implicit in a² /dt², a³/dt³ etc. (powers of velocity) as compared to the values of the d⁽ⁿ⁾f's.

I made a mistake here. There are no dtⁿ terms in the expansion. The arbitrarily high speed is implied somewhere else. But I'm sure you're right.

I'll think about it later. Maybe somebody comes up with the right idea.

Edited August 10, 2020 by joigus
minor addition

[studiot]

As regards local v global I have

 

The plot of y = - x² has a global maximum, but no minimum.

The plot of y = x³ - x² has no global maximum or minimum, but has a local maximum and local minimum.

[joigus]

And there you are! Another different meaning for the words "local" and "global". 😫

[MigL]

I'm sure we'll get yet another when Markus logs in tonight.

[MigL]

5 hours ago, studiot said:

The plot of y = x³ - x² has no global maximum or minimum, but has a local maximum and local minimum.

Looking at it from a Physics point-of-view, if x and y are describing a physical system, it is only for very limited values of x that y is at a local max and min. For any values less than 0, it diverges to neg infinity, and for all values greater than approx. 0.7 it diverges to infinity.
If this was a physical system, we would probably say that the local min/max were trivial, unless we were specifically considering those particular x values.

[studiot]

3 hours ago, MigL said:

Looking at it from a Physics point-of-view, if x and y are describing a physical system, it is only for very limited values of x that y is at a local max and min. For any values less than 0, it diverges to neg infinity, and for all values greater than approx. 0.7 it diverges to infinity.
If this was a physical system, we would probably say that the local min/max were trivial, unless we were specifically considering those particular x values.

These were just examples, there are many shapes of wiggly lines.

But I beg to differ that these are unimportant.

If the lines represent energy what about the local maximum of activation energy ?

And if we move to multivariable situations what about saddle points/ Are there any maximum / minimum points there?
This represents a whole vital part of Physics.

[joigus]

There's also "local hero", "local customs", which involve time and memory.

[studiot]

14 minutes ago, joigus said:

There's also "local hero", "local customs", which involve time and memory.

In Physics  ?

 

Interestingly in German the terms

 

im kleinem :   often translated from german textbooks as in the small

and im grossen :  often translated from german textbooks as in the large

 

are used for locally and globally

 

In Mathematics the term local  <some property>  applies in some neighbourhood of a point but not for all points in the entire set.

That is reserved for global, where the property may be applied to any and every point in the set without restriction.

A slight point here arises because strictly the term neighbourhood may apply to the whole set as well.

So really local on its own is incomplete.  It requires a statement of restriction for that.

Edited August 11, 2020 by studiot

[joigus]

1 minute ago, studiot said:

In Physics  ?

T'was a joke, Studiot.

[Markus Hanke]

On 8/10/2020 at 2:01 PM, studiot said:

So if some effect is either local or non local what does these mean ?

What I meant by my original comment was that tensors in general are local objects - in the sense that, in a tensor field such as the ones we are dealing with in GR, there's a tensor 'attached to' every event in spacetime. So essentially, the GR field equations are a constraint that link the distribution of energy-momentum in a small local neighbourhood to the geometry (as measured by Einstein curvature) within that same small local neighbourhood. I think in a mathematical sense, 'small' here would mean infinitesimally small, but in many practical applications that can be a more extended region too, depending on circumstances. I should think that what constitutes 'small' depends on how the quantity in question changes with space and time.

In a more physical sense, locality simply means that what happens in one place at a given instant of time does not somehow simultaneously depend on some distant process; it depends only on local influences from its immediate surrounds. So causal influences must propagate along time-like or null intervals, never space-like ones. Non-locality then is the opposite - what happens in one place depends on what happens someplace else, which can be very distant in space and/or time. joigus has given the formal definition; I'd just like to add that non-locality isn't just a spatial concept, it can also apply to time.

On 8/10/2020 at 2:16 PM, joigus said:

People talk about non-locality in relation with Bell's theorem, but they are confusing this concept with that of non-separability, which is very different.

I am not so certain that they are very different. Does non-separability not fulfil that very mathematical definition you gave? I would say non-separability implies non-locality, whereas the reverse is of course not necessarily true. Note also that neither non-separability nor non-locality necessarily imply time-like signalling, so there's no issue in regards to Lorentz invariance.
Given the (by now well verified) violation of Bell's theorem, we are really only left with three choices - let go of locality, let go of realism, or let go of both. I personally think the first option is the least troublesome, over all.

On 8/10/2020 at 2:16 PM, joigus said:

The concept of locality in field theory that I'm familiar with is better characterised with a precise mathematical definition.

Well presented  +1

 

 

[joigus]

13 hours ago, Markus Hanke said:

I am not so certain that they are very different. Does non-separability not fulfil that very mathematical definition you gave? I would say non-separability implies non-locality, whereas the reverse is of course not necessarily true. Note also that neither non-separability nor non-locality necessarily imply time-like signalling, so there's no issue in regards to Lorentz invariance.
Given the (by now well verified) violation of Bell's theorem, we are really only left with three choices - let go of locality, let go of realism, or let go of both. I personally think the first option is the least troublesome, over all.

That's what many people say and I understand how they can say that, because it's a formidable conceptual mirage. The violation of Bell's theorem cannot be put into question. But this only means that it is totally impossible to even think consistently about S_x, S_z and S_45º at the same time.

That's the extremely subtle point that I have no hope of getting across ever, because everybody seems to be more willing to believe in magic than to seriously sit and think about the assumptions. I know full well I will never hear the end of it. It's been 20 years for me personally, and counting. The truth is even John Bell was very cautious about making big statements concerning the implications of his theorem. Feynman was too.

However weird the correlations are (and they certainly are), they were there when the singlet state was prepared, they were there a Plank's time worth of time after, and they will still be there when the particles are received seconds later. And they will keep there for as long as the overall U(1) symmetry of the state is not broken by interactions. They are initial correlations. Strange, puzzling correlations, yes; but initial, and generated locally when the system cooled down to a singlet.

I know I have a strong opinion about this. But it's a reasoned opinion and at no point contradicts what's experimentally known. It's a matter of what it is that you read into the data.

 

 

[joigus]

On 8/10/2020 at 3:01 PM, studiot said:

I have in mind that a non local effect may be distributed without affecting the whole universe.

Yes, you can cut off your non-local couplings at a certain distance.

An interesting idea could be assuming non-locality but cutting off at microscopic range. Somewhere in the junkyard of my mind I remember that idea. I must have read it somewhere. You would have unblemished Einstein locality and causality, but small violations for very small distances. You can play with the idea with some freedom.

Take the non-local term that I gave you,

\[f\left(x,t\right)=\int_{-\infty}^{\infty}daf\left(a\right)F\left(x-a,t\right)\]

and make the range of f(a) as small as you want.

Edited August 12, 2020 by joigus

[Markus Hanke]

On 8/12/2020 at 6:39 AM, joigus said:

They are initial correlations. Strange, puzzling correlations, yes; but initial, and generated locally when the system cooled down to a singlet.

Ok, that's a good point, and I think I get what you mean. I looked at the situation only spatially, but not along the time line. So in that sense, as you explained it, it is indeed local - later measurement outcomes do not depend on distant parts of the system, since the statistical correlation has already been there from the beginning, and thus remains local at each branch of the experiment. So the situation does not in fact fulfil the non-locality definition you gave. That's a pretty self-consistent view on this, as it avoids any clashes with SR. This would seem to imply then that we have to let go of realism; it also implies that the spatiotemporal embedding of the underlying wavefunction that describes the system is non-trivial - it cannot be located anywhere in spacetime in any self-consistent way.

This is consistent with my own view which I keep exploring - that the underlying structure of reality is not in any way spatiotemporal in nature.

[joigus]

1 minute ago, Markus Hanke said:

Ok, that's a good point, and I think I get what you mean. I looked at the situation only spatially, but not along the time line. So in that sense, as you explained it, it is indeed local - later measurement outcomes do not depend on distant parts of the system, since the statistical correlation has already been there from the beginning, and thus remains local at each branch of the experiment. So the situation does not in fact fulfil the non-locality definition you gave. That's a pretty self-consistent view on this, as it avoids any clashes with SR. This would seem to imply then that we have to let go of realism; it also implies that the spatiotemporal embedding of the underlying wavefunction that describes the system is non-trivial - it cannot be located anywhere in spacetime in any self-consistent way.

This is consistent with my own view which I keep exploring - that the underlying structure of reality is not in any way spatiotemporal in nature.

Oh, my, you're sharp, Hanke! I may be going nowhere, but you understand exactly what I mean. +1 You're worth 10 points here.

In fact, there is a kind of non-locality in my view, but it has nothing to do either with space nor with time. It's abstract, internal-space.

The functions you're trying to measure are not point-to-point (eigenvalue-to-eigenvalue) functions of one another. What some analysts call "non-local operators". Maybe the expression filtered out from there.

Same way x is non-local operator in p-eigenstates (it depends on all the spectrum) and vice-versa.

[Markus Hanke]

22 minutes ago, joigus said:

but you understand exactly what I mean

This may be so, but I'm somewhat troubled by the implications. What does it even mean for the fundamental elements of the world to be inconsistent with realism? My immediate impulse would be to say that it means they really aren't fundamental at all - but I will have to ponder this for a bit. 

[joigus]

18 hours ago, Markus Hanke said:

This may be so, but I'm somewhat troubled by the implications. What does it even mean for the fundamental elements of the world to be inconsistent with realism? My immediate impulse would be to say that it means they really aren't fundamental at all - but I will have to ponder this for a bit. 

But couldn't we say that for anything to be inconsistent with reality, we must have a picture of that whole reality, which we... don't?

I tend to see "reality" as a place holder for whatever we intuit the next level of description may be. Perhaps the next level of reality description implies dropping our naïveté about what it must look like. Complex numbers, non-trivial topologies... The works!!

I've read a comment by @swansont far back in the past forums about "physical reality" that I don't remember exactly but I found very interesting, and it's similar to what I've just said. Maybe he can help me recover it. Or rephrase it.

[Markus Hanke]

2 hours ago, joigus said:

I tend to see "reality" as a place holder for whatever we intuit the next level of description may be. Perhaps the next level of reality description implies dropping our naïveté about what it must look like. Complex numbers, non-trivial topologies... The works!!

That is essentially what I was hinting at, actually...I think our current concept of what is fundamental (quantum fields and their interactions and excitations) just doesn't really cut it, from a philosophical point of view (not saying those things don't work!!). I think there is a whole lot more going on than we currently realise, and it will need a major paradigm shift to reveal it. 

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.

P.S. I'm aware of course of ER=EPR, but haven't really arrived at a conclusion about what this really implies.

Edited August 15, 2020 by Markus Hanke