Hot answers tagged locality
53
Entanglement is being presented as an "active link" only because most people - including authors of popular (and sometimes even unpopular, using the very words of Sidney Coleman) books and articles - don't understand quantum mechanics. And they don't understand quantum mechanics because they don't want to believe that it is fundamentally correct: they always ...
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I wish to complete @Luboš Motl's answer, to which I agree. My point is on why people continue to make this mistake of an active link. This mistake is connected with one of the most interesting properties of quantum mechanics, Bell's theorem. One can argue that any physical theory is an hidden variable theory, the hidden variable being the description of the ...
19
Since general relativity is a local theory just like any good classical field theory, the Earth will respond to the local curvature which can change only once the information about the disappearance of the Sun has been communicated to the Earth's position (through the propagation of gravitational waves).
So yes, the Earth would continue to orbit what ...
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The higher the number of derivatives the more initial data you have to provide. If you have some Lagrangian that contains an infinite number of derivatives (or derivatives appearing non-polynomially, such as one over derivative) then you have to provide an infinite amount of initial data which amounts to non-local info, in the sense explained below.
If you ...
13
Gravitational influences do propagate at the speed of light, not instantaneously.
The question of what would happen if the Sun instantly disappeared is actually a funny one in general relativity. The equations of general relativity imply as a mathematical consequence that energy must be locally conserved. Therefore, there is no valid solution to the ...
10
Just a nice analogue Prof. Jürgen Audretsch told me once:
Imagine at home you put one glove in your coat without looking (and noticing it's only one of the two). After exiting the train you notice it's cold and you pull out that single glove. At this very instant you know it's either the left or the right glove, and you therefore know which one is left ...
10
All observations are consistent with standard GR so far, but I don't think the speed of gravity, in particular, has ever been measured.
Experimental measurements of the speed of gravity was quite a controversy a few years ago when a paper came out claiming that the speed of gravity was very close to $c$ as measured by the Shapiro delay. To see papers on the ...
9
In classical mechanics, the lagrangians of two particles may be added only if the particules do not interract.
I wouldn't say that. You can always write a Lagrangian $L$ for a system of two particles. In general, it takes the form
$$L = L_1 + L_2 + L_i$$
where $L_i$ is an interaction term that depends on the coordinates and/or velocities of both ...
7
You are confusing the concepts of "interactions" and "nonlocality". In realistic field theories, including all theories we ever used to study phenomena in the world around us, the interactions exist but they keep the physics local.
As David mentioned, the Lagrangian density takes the form
$${\mathcal L} = \sum_i \left[ (\partial_\mu \phi_i)^2 + m^2 \phi_i^2 ...
7
Disillusionment with systems described by higher order Lagrangians harks back to a 1950 paper by Pais and Uhlenbeck, in which they showed that such systems were prone to pathologies, including states with negative energy and states with negative norm. There's a more recent discussion of this in arXiv:hep-th/0408104.
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Clearly, an interaction involving $\phi(x+h)$ deserved to be called nonlocal. But since $\phi(x+h)=\sum_{k=0}^\infty \phi^{(k)}(x) h^k/k!$, any nonlocal interaction can be expressed as a power series involving arbitrarily many derivatives. Therefore an action (or Lagrangian) is called nonlocal if it involves infinitely many derivatives.
If there are only ...
5
This is a truly excellent question in my opinion. It is still being worked on.
Here are some professional references that will somewhat clarify the issue, or perhaps even confuse you further:
http://arxiv.org/abs/1102.4467
http://arxiv.org/abs/1007.5518
http://arxiv.org/abs/1006.3680
Michael J.W. Hall
http://arxiv.org/abs/0808.2178
Travis Norsen
...
5
There are a few papers in which topological field theories are constructed in terms of nets of algebras. The idea generally is that a net of algebras gives you a model for the higher category associated to a point by an extended TQFT. (Physicists would say that a 2d conformal net describes a 2d CFT which is related to a 3d TQFT.)
The first one that comes ...
4
The issue is locality of what? Which quantities are assumed to be local?
If you say the reults of all experiments, hypothetical as well as actual, and say that these must be assignable definite values, then this is in conflict with quantum mechanics. But this is the assumption that is called "hidden variables", the reason is that this is the assumption ...
4
Presuming that there aren't nonlocal constraints, a differential operator that is polynomial in differential operators is local, it doesn't have to be quadratic. My understanding is that irrational or transcendental functions of differential operators are generally nonlocal (though that's perhaps a Question for math.SE).
A given space of solutions implies a ...
4
Different regions of a general spacetime that are Minkowskian to $O(Delta x^2)$ can have light cones which have null rays that point in different directions. A particle in this spacetime moves from one such region to another by connection coefficients, sometimes called the Christoffel symbols, which patch together these different locally flat regions. This ...
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"How can a point-like particle “feel” gravity"
It can't. That's in fact true not only for perfect point like particles, but also for sufficiently small objects, like... Can you feel gravity? I don't think so. What you can feel is the ground pushing against your feet, but there is no way of knowing whether this is because of gravity or because the earth ...
4
Let me give a tentative answer. I agree that these terms —especially 'locality'— are used for different concepts and this is annoying. I will list several notions of locality.
Causality (or Einsteinian locality): Results of experiments carried out at a space-like distance are not correlated. This assumes that there are not previous correlations before ...
4
A quantity is local if it is a finite linear combination $\sum_k g_k P_k(x)~~$ of products $P_k(x)$ (or other pointwise functions, such as $\sin \Phi(x)~$ for sine-Gordon theory) of field operators or their derivatives at the same point $x$.
A quantum field theory is local if its classical Lagrangian density is local.
(By abuse of terminology, an action or ...
4
In Newtonian physics, there was no problem with action at a distance, and indeed Newton explicitly formulated his theory of gravitation in such terms. It may be that this was criticised from a philosophical standpoint (I don't know whether it was or not), but there were no fundamental mathematical difficulties with the idea.
However, in relativity the ...
4
Most of the AQFT toolkit is about algebras of local operators. There aren't any physical local operators in topological QFTs whose interesting observables are global - topological - so AQFT, TQFT have almost nothing to do with each other. TQFT are QFTs that may be made pretty rigorous which is why e.g. Witten could get a Fields medal for such things but AQFT ...
4
In your comment you ask:
That question only addresses gravitational force. Is it the same for all kind of forces?
Generally speaking a force transmitted by massless particles like the photon and graviton obeys an inverse square law while a force transmitted by massive particles falls off exponentially with distance. This means that only the forces ...
3
I think that the best picture to understand this correlation is given by many-worlds interpretation:
A singlet decomposes in a coupled pair of particles superposition |+⟩(A)|-⟩(B) + |-⟩(A)|+⟩(B)
So observer A sees a simple superposition of |+⟩ + |-⟩ (which is a partial trace of the global density matrix) and so do B
in many worlds interpretation; ...
3
Your question was first asked by Laplace. The following is from the Wikipedia article on "The speed of gravity"
"Laplace
The first attempt to combine a finite gravitational speed with Newton's theory was made by Laplace in 1805. Based on Newton's force law he considered a model in which the gravitational field is defined as a radiation field or fluid. ...
3
What I take to be elementary significant papers on this question pre-date arXiv, so they are unfortunately usually available only behind paywalls. I've always found the simplicity of Willem de Muynck's argument in Physics Letters A 114, 65 (1986), "THE BELL INEQUALITIES AND THEIR IRRELEVANCE TO THE PROBLEM OF LOCALITY IN QUANTUM MECHANICS", somewhat ...
3
The majority opinion is that Einstein was wrong. However, I see some problems with the standard quantum mechanics (SQM) approach that you outlined. SQM contains two major parts: unitary evolution (described, e.g., by the Dirac equation) and the measurement theory (e.g., collapse, or the projection postulate, which, loosely speaking, states that, after ...
3
I wouldn't say that "holographic theories are non-local by definition". On the contrary, in AdS/CFT the CFT is completely local and satisfies cluster decomposition.
The cluster decomposition property in AdS can be proved using the CFT bootstrap for all CFTs in $d > 2$ (see http://arxiv.org/abs/arXiv:1212.3616, the proof only requires CFT `axioms', ...
3
Part A
A. This "simple would-be relativistic quantum theory" indeed violates locality, but not for the reason you write. You wrote that the energy i.e. the Hamiltonian is defined as
$$ H =\sqrt{p^2+m^2} $$
so it cannot be "set as we want to", independently of $k$. In fact, the problem is that $H$ is too constrained, not that it is too free. By your rules, ...
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