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I'm having trouble unambiguously interpreting many answers here due to the fact that the terms locality and causality are sometimes used interchangeably, while other times seem to mean very different things to the author.

My current understanding is that 'a-causality' is "obviously forbidden" in a physical theory, because it violates Lorentz invariance and leads to logical paradoxes. (Although there are supposedly viable models such as De Broglie–Bohm that are a-causal and yet somehow OK because the a-causality is not accessible to experimental apparatus -- this only brings further confusion to the table.)

'Non-locality' on the other hand, seems to refer to, for example, correlations between events at space-like separation. In this sense 'non-locality' does not imply 'a-causality.'

On the other hand Bell (1964) uses the term 'non-local' to refer to, as he says in his conclusion, "not Lorentz invariant." So I think he is using the word 'locality' interchangeably with 'causality.'

This is the sense in which many, such as Luboš Motl seem to use the term, for example here, while others, such as Ron Maimon, seem to use the term differently, for example here, where he says:

The nonlocality of gravity doesn't mean that Lorentz invariance is broken

This is very confusing. Can someone give an authoritative set of definitions here that we can refer to?

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hoped for there to me more answers...:(..maybe this post from Lubos Motls blog would help, which highlights meaning and interconnection of locality, casuality, gauge and lorentz symmetries.. motls.blogspot.in/2010/02/… –  ramanujan_dirac Nov 23 '12 at 17:15

2 Answers 2

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 making the experiment. In a quantum theory, this implies that observables must commute at a space-like distance. A violation of this property in a Special Relativistic theory may give raise to time travels paradoxes.

Micro-causality (only used in field theory): A theory whose fundamental variables or the dynamical degrees of freedom commute at space-like distance is sometimes said to be local. Note that there are theories which violate this property without violating causality.

Lagrangian density (only used in field theory): A theory whose action functional is expressed as a space-time integral of a local Lagrangian density is sometimes said to be local. Note that there are theories which violate this property without violating causality.

Cluster decomposition principle: Results of experiments carried out at the same time (in a given reference-frame) but in different spatial regions are not correlated. This assumes that there are not previous correlations before making the experiment. This property together with Poincare invariance implies causality. Relativistic QFT and non-relativistic condensed matter QFT verify this notion of locality. This property imposes certain smooth dependence of the Hamiltonian density on the creation/annihilation operators.

Non-local wave-function's collapse (quantum notion): People sometimes say that quantum mechanics is non-local because the collapse is a non-local process. In my opinion, the wave-function collapse is not a physical process, but something that affects our mathematical description of the physical system. A short of algorithm to incorporate new information to the theory. So, in my opinion, this non-local collapse is not a signal of physical non-locality.

Non-local states and observables (quantum notion): In quantum mechanics there exist non-localized states and observables and I have heard people to call this non-locality. A particle whose linear momentum is very well defined may be an example of a non-localized state and the scattering operator may be an example of a non-local observable given that relate states at far past and future. Entangled states could be listed here.

Entanglement (or non-local correlations, quantum notion): In quantum mechanics there are quantum correlations which are non-local such as the spin correlations of the singlet state. One needs to measure the z component of both particles which can be very separated. A singlet state may also be called non-local. Since one cannot use these correlations to send information (a parallel classical channel is required, and this cannot be superluminal), therefore this property does not imply violations of causality.

Incompatibility of QM with local (causal) realism: QM and experiments violate Bell's inequalities. This leads to the incompatibility of QM and nature with either local (causal) realism (local hidden variables) or free will. People are currently discussing this on this site.

Only the first notion of locality (causality) must be required in a Poincare invariant theory. Semantic issue: Some people call a theory "special-relativistic" if the theory is Poincare invariant and causal, while other people by "special-relativistic" just mean Poincare invariant.

Examples:

  1. Non-relativistic quantum mechanics and non-relativistic QFT verify the cluster decomposition principle. However, there are non-local observables, entanglement, etc.
  2. Relativistic QFT verify causality ans the cluster decomposition principle.
  3. Quantum Electrodynamics in Coulomb gauge is a relativistic QFT, but it does not have a Lagrangian or Hamiltonian density.
  4. Para-statistics theories verify the cluster decomposition principle, but they do not verify micro-causality.
  5. Classical General Relativity has a local Lagrangian density and it is locally Lorentz invariant, but it allows time machines in some topologies (and in these cases is not a causal theory) and horizons have non-local properties.

I would be thankful if you correct this answer or make it more precise.

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How does your last type ('entanglement') relate to the first ('causality')? I ask because doesn't entanglement, for example in the measurement of spin correlations of a singlet space at space-like separation, imply that the observables (in this case, for example |z>) do not commute as space-like separation? Similarly for the 'Non-local states and observables' type you list, which might be grouped in with the last as well. –  user1247 Aug 21 '12 at 19:06
    
1) If I remember correctly, the spin operators corresponding to different particles commute. Entanglement is an essential property of quantum theory but it respects causality. 2) Entangled states are an example of non-localized states, but a simply superposition of position eigenstates is another example. A non-local observable is the scattering operator and also the total spin of a singlet. –  drake Aug 21 '12 at 19:26
    
Maybe it's just a question of semantics, since you list the total spin of a singlet as a non-local observable, but isn't measurement of entangled pairs using the same operator a clear example where the operators do not commute for a space-like interval? I mean, you cannot measure one to be spin up, and at the same time measure the other to be spin up. A measurement of one affects the result of the measurement of the other... –  user1247 Aug 21 '12 at 20:23
    
Instead of "A measurement of one affects the result of the measurement of the other", I prefer "the spin of one particle is correlated to the spin of the other". There is nothing traveling between the particles, it is not an interaction. –  drake Aug 21 '12 at 20:32
    
Other issues with your answer: you use 'local' in under your 'Incompatibility of QM with local realism' without it being clear which definition of 'local' you are referring to! You don't have an unambiguous 'locality' listed in your list. Also, under the 'Cluster decomposition principle' you require that space-like events be uncorrelated, and yet you contradict this in your 'example 1' saying that QM and QFT obey cluster decomposition but have non-local correlations. It would be great if we could craft an answer that was clear and unambiguous about these definitions. –  user1247 Aug 22 '12 at 8:13

non interchangeable...

Locality and Causality in the same dish...

Norsen:

“It isn’t necessarily that something in region 2 is causally influencing something in region 1, or vice versa. It is always possible that there is some other event, neither in region 1 nor region 2, which was not determined by [λ], and which itself causally influences both [beables in region 1] and [in region 2]. The point is, though, that this causal influence would have to be non-local"

Norsen, T. Found. Phys. 39, 273. 2009.


Zeilinger:

"What we cannot exclude, as with any experiment, is the possibility that an earlier common cause in the overlap of the backward light cones of the two events"

Zeilinger, A. New Journal of Physics 14 053030. 2012


and i add:

contextuality subsumes non-locality

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