Fundamental particles may not be exactly (mathematical) point objects. They can have some extended structures, however small. In string theory a conserved quantity called R charge is often discussed. According to a paper by, Ori J. Ganor of UC Berkeley, fundamental particles may occupy a volume proportional to the R charge. If the particles are not point-like then their interactions are also not happening exactly at the same point, hence a small amount of non-locality is introduced. Therefore, I do not think non-locality always means action at a distance. It can also result from extended structures of elementary objects. 

Dear Biswajoy,

from the way in which you put this question I think that there is a misunderstanding in the concept of nonlocal interaction in the quantum world. In fact, whether you look to my quantum super Yang-Mills PDEs, say (YM), encoding quantum phenomena, you can see that there is not time there. Interactions are described by means of nonlinear quantum propagators. These are extended non-commutative geometric objects, hence quantum interactions are always of non-local nature. In this way one can state that quantum particles entering in interactions are always extended quantum entangled particles. This is a quantum phenomena that today has been verified at the experimental level too.

When one refers to an observer, namely when one considers an observed quantum super Yang-Mills PDEs, say (YM)[i], then you can consider the observer's proper time, but the non-local nature of interactions continues to work again, since reactions are encoded by means of observed nonlinear quantum propagators. These are also extended objects and quantum particles related by such geometric objects are extended quantum entangled also with respect to the observer. This is a natural algebraic topologic property, characteristic of observed nonlinear quantum propagators. This does not mean that there is an instantaneous propagation from the instant t_0, the initial cauchy data, and the time t_1>t_0, the final Cauchy data, since the nonlinear propagator does not describe any propagation, since it is what the observer see, when one projects on (YM)[i] the nonlinear propagator V contained in (YM). The lapse t_1-t_0 is the observed time of the reaction. This does not exist in the quantum world (YM). Furthermore, one can see that t_1-t_0 is neither the quantum time of the reaction, say \hat t, that is a noncommutative object, and that enters in the Heisenberg uncertainty relation. (To understand more on this subject please look to the following my paper: http://arxiv.org/abs/1205.2894 (II). )

Let me also add that the misunderstanding on the paradoxical nature of quantum entanglements, arises from the confusion that often one makes between mathematical model and observed mathematical model. In some sense, this confusion has been made also by Albert Einstein, when in the well known EPR paradox claimed the impossibility to consider quantum entangled phenomena. Really just Einstein's GR has taught us that space-time is a geometric object, say M, where the time does not exist ... In order to talk about time it is necessary to introduce an observer. The same holds for the Einstein PDEs (Ein). In order to talk of time it is necessary to consider the observed Einstein PDEs with respect to an observer, say (Ein)[i] ...

I second Agostino's mention of entanglement as an example of non-local interactions which as far as we know do not require more than 3 spatial dimensions.  But would seek to clarify a couple of other points assumed in the question.

• Communication faster than light (FTL) in the physical world, regardless of the dimensionality assumed, would allow detection of a preferred frame.  This would not change physics as much as some people think.  It would only mean we go back to Lorentzian Relativity.  I think General Relativity (GR) would remain as is.
• In my view (which certainly string theorists would disagree with) the existence of higher dimensions should be held to the same standard as the old Ether theories.  Even if they are consistent with observed physics, they should not be assumed except in the case of clear experimental evidence.  On this basis the Ether was discarded, not because it was ever found inconsistent.  This approach follows Carl Sagan's advice, extraordinary results require extraordinary proof.

That said, I realize my post does not address "the question" but is more of a commentary on it.

Dear Biswajoy and Robert,

let me spend some words more about extra-dimensions ... When one refers to extra-dimensions one generally thinks that a space-time solution, of an observed dynamic equation, has more than four dimension. This is not correct ! In fact any observed dynamic equation can be implemented on an observed fiber bundle \pi:W\to M, over an observed 4-dimensional space-time M. Here W can have the fiber of any dimension, depending on the particular physical system that one considers ... However, solutions remain of dimension 3+1. In other words so-called extra dimensions concern the fiber dimension only ...  For example in dynamics encoded by the observed Navier-Stokes PDEs, the fiber dimension is 3+2=5 (three components of the velocity flow + isobaric pressure + temperature). For the observed quantum Yang-Mills PDEs the fiber has a more complex structure ... let us skip on details here ... however solutions are always of dimension 3+1, namely three space-like dimensions and one time-like ... With this respect I agree with Robert's remark that to talk about extra-dimensions can be a naive idea considered as an unjustified use of Ether in the old electro-magnetic theories. Often one forgets that the real universe of a physical system is the geometric structures encoding its dynamic equation. This is a 'manifold' endowed of an infinite tower of all its prolongations ... There one can find more real dimensions than one can naively invent.

Thanks for clarification.  Unfortunately, I cannot contribute much to that interpretation of dimensions.

Fundamental particles may not be exactly (mathematical) point objects. They can have some extended structures, however small. In string theory a conserved quantity called R charge is often discussed. According to a paper by, Ori J. Ganor of UC Berkeley, fundamental particles may occupy a volume proportional to the R charge. If the particles are not point-like then their interactions are also not happening exactly at the same point, hence a small amount of non-locality is introduced. Therefore, I do not think non-locality always means action at a distance. It can also result from extended structures of elementary objects. 

This is correct, and it is related just to the algebraic topologic structure of nonlinear quantum propagators.

I once considered with Robert Marnelius the interaction of fundamental or  elementary particles as extended objects by developing a bi-local quantum filed theory: In those days we were considering an alternative to string theory. see:

A Quantum Field Theory for Scattering of Hadrons with Infinite Mass spectrum 

   Neil Craigie      R. Marnelius                  

Il Nuovo Cimento A (Nuovo Cimento) Publisher: Società italiana di fisica, Società Italiana di Fisica  ISSN: 0369-3546    

ABSTRACT We examined the possibility of constructing a quantum field theory of the scattering of a system of hadrons, which lie on Regge trajectories, by associating the spectrum with a mass operator involving internal coordinates. We show that, providing the construction ensures no tachyonic states, the latter can be used to derive a causal bare theory, which leads to a generalised Feynman graph expansions of the S matrix. In an explicit example involving an oscillator mass spectrum and generalise scalar field cubed interaction, the tree graph are shown to be multi Reggie behaved. However, the latter is linked to unphysical infrared singularities coming from the structure of the mass operator. We try to resolve the latter problem in the renormalisation program, where we indicate that the theory is renormalizable and that the loop insertions in the generalise propagator leave its structure basically unchanged. As far as the physical interpretability of the theory is concerned, we propose a solution, which merges the bootstrap idea with the Quark model type mass spectrum

A Quantum Field Theory for Scattering of Hadrons with Infinite Mass Spectrum. Available from: https://www.researchgate.net/publication/243192008_A_Quantum_Field_Theory_for_Scattering_of_Hadrons_with_Infinite_Mass_Spectrum [accessed Jun 8, 2015].

Article A quantum field theory for the scattering of hadrons with an...

Thanks a lot for all the responses. Please keep it up.

In string theory, the concept of world-sheet is used instead of the concept of world line, which is used in field theory. This is a generalization which was introduced by Susskind in 1967. String theory can bypass unwanted infrared divergences, which are a generic and problematic feature of quantum field theories. Divergences are to be renormalized for each order of perturbation theory. These divergences appear due to the underlying assumption that elementary objects are point-like. 

A question for Dr. Craigie: From the abstract of your paper it seems to me that bi-local field theories also produce divergences which are to be renormalized. You have mentioned that singularities do emerge from the structure of mass operator. Are there field theories with non-local interactions, which can bypass the problem of infrared divergences ?

The theory was renormailzable  and less divergent than the local Version, but in todays perspective,  the infra red singularities are probaly linked to zero modes of the mass Operator. In those days they had no place in hadron physics. In string theories, the zero modes are the elementary particles. However the paper I mentioned was written years ago, so I would have to look at it again to be sure of wjhat I am saying.

Dear Biswajoy and Neil ,

from your last posts I see that you are starting to understand why the nonlinear quantum propagator, considered in my quantum supergravity is the right geometric tool that solves any problem in the unification of Einstein's GR with QM. Let me add that string theory has solved only one aspect, namely to consider particles as extended objects. On the other hand just the phenomenology of Regge's trajectories can be now reinterpreted as non-local effects in interactions ... (See some my recent papers posted on arxiv and quoted below...)

By the way what is lacking in the string theory is that this is formulated as a classical theory that must be quantized. Really according to the old point of view, string theory aims to encode quantum world by means of a classical model, that after necessitates to be quantized. This is point of view cannot work in order to unify GR with QM. In fact quantize a classical model it means to believe that quantum world is a fluctuation of classical one. In quantum world there are nonlinear phenomena that cannot be approximated as quantum fluctuations around classical solutions of classical PDEs. Thus a new noncommutative geometric point of view must be adopted (quantum (super)manifolds) and a new geometric theory of PDEs in the category of quantum super manifolds it necessitates.

I have developed this program during many decades and a 'self-contained' theory of quantum-(super)gravity that extends the old quantum field theory, string theory and M-theory is contained in the paper  DOI: 10.1016/j.nonrwa.2012.02.014.

Further developments are also contained in the following trilogy: arXiv: 1205.2894 (I-II); arXiv: 1206.4856 (III).

With compliments.

Agostino Prástaro

An extended object cannot have a single coordinate. In classical mechanics sometimes we use coordinates of the center of gravity of a rigid body, but here in the extreme microscopic domain, there is no such thing as the center of gravity which can specify the location of an extended object such as an one dimensional string. An one dimensional string can be thought of as a collection of infinite number of mathematical points. Therefore if I say that X0, X1, X2,.......,XN-1 are the coordinates of a string (where N is the dimensionality), then for each component of X, there are infinite number of values which run along the string. In this way we can think of Xi  where i=0,N-1 as functions of some parameters. Here Xi are dependent variables. Then what are independent ones? In string theory these parameters are called  σ and  τ. Here sigma and tau vary and consequently Xi coordinates also vary along the length of the string. String action is then constructed in such a way that it remains unchanged under reparametrization.

These are the coordinates which only locate the string. But the real dynamics is encoded in the variation of string coordinates with respect to time.

Dear Biswajoy,

observed dynamics of observed quantum extended objects are solutions of observed quantum super Yang-Mills PDEs ... Please look to my previous posts.

By the way to seriously understand this subject it is necessary to read my quoted papers.

When writing ``extra-dimensional theories'', one question that must be addressed is, whether these are invariant under Lorentz transformations in four dimensions, or in 4+n dimensions (where n is the number of extra dimensions). In the latter case, at the classical level, the theory is local and there isn't any issue of action at a distance. In the former case, there can be, unless the extra dimensions decouple. How this decoupling might be achieved isn't obvious. If the extra dimensions are compact, then one doesn't have full Lorentz invariance and it isn't that obvious that one can even  maintain four-dimensional Lorentz invariance, precisely because signals can travel along the extra dimensions-if they're compact, by winding around them. Of course these statements assume point like excitations. ``Action at a distance'' are words-that must be translated to mathematical statements. 

The most (only?) consistent non-local theory with extra dimensions is known under the name String Theory