Relativity and conservation laws

[atyy]

Yes, this is true, but these effects are not "nonlocal" in the sense of breaking local conservation laws. That's part of the point of the paper we were discussing earlier; yes, some effects can "appear" nonlocal, but no local conservation laws are ever violated, and no information is ever transmitted faster than light.

The CJS paper? I guess I don't quite understand what "local" means if there are not even gauge-invariant local observables. Also, you describe "local" as no information is ever transmitted faster than light, which is indeed a requirement of relativity. But as wave function collapse shows, one can have nonlocality without violating the restriction on faster than light transmission of information.

Wave function collapse is a very "fuzzy" concept--for one thing, not all interpretations of QM even have it (the MWI being the most obvious example of one that doesn't). Part of the reason it's a "fuzzy" concept is precisely the apparent incompatibility with relativistic invariance; in QFT (as opposed to non-relativistic QM), as I understand it, collapse doesn't really appear (Weinberg's classic text, for example, IIRC never brings it up or uses it), because it just doesn't work once you require your theory to be relativistically invariant. IMO collapse is best viewed as a heuristic, a way of extracting practical predictions from the theory even though we don't really understand how things work underneath.

Weinberg's classic text does bring up wave function collapse, and it is also mentioned (but not in a rigourous way) by the more rigourous text of Dimock. Even if one does not prefer Copenhagen, the fact that Copenhagen does have nonlocal wave function collapse and yet obeys the restriction on faster than light transmission of information shows that "nonlocal" and "no faster than light transmission of information" are two different concepts, and it is only the latter which is forbidden by relativity - as you say, the incompatibility with relativity is only "apparent".

 

[vanhees71]

Yes, but you did bring up the whole QFT thing in the post I was responding to.
Once again, I agree that the details of how local conservation laws work in specific cases are not trivial. But you don't need all that detail to evaluate the simple claim that the principle of relativity requires that all conservation laws be local conservation laws.

Ok, so what's the specific argument that all conservation laws must be local? Is Feynman's claim available in a scientific paper? To make it clear, I don't think that there's any need to give up the local (Q)FTs and extend them to something non-local, but I'd be highly interested in a proof that any non-local theory is flawed.

 

[vanhees71]

The CJS paper? I guess I don't quite understand what "local" means if there are not even gauge-invariant local observables. Also, you describe "local" as no information is ever transmitted faster than light, which is indeed a requirement of relativity. But as wave function collapse shows, one can have nonlocality without violating the restriction on faster than light transmission of information.

I don't want to discuss quantum collapse assumptions here. If at all, one should discuss this in the quantum mechanics subforum, where it belongs (if at all, because I consider collapse a flawed und fortunately unnecessary concept, as you know from our earlier discussions).

The apparently "nonlocal observable" is, as you stated, the interference pattern in the Aharonov-Bohm effect. But this is just the manifestation of a non-integrable phase factor, which is gauge invariant and expressible in a local form via the electromagnetic four-potential. So there is no violation of locality in the usual sense in the AB effect.

 

[martinbn]

Ok, so what's the specific argument that all conservation laws must be local? Is Feynman's claim available in a scientific paper?

There was a summer school "Ettore Majorana" (there is a book about it but I don't have a copy, you probably can see it on books.google), where Feynman talks about non-local conservation. I believe that the original question is about that, and his argument is what I wrote above.

 

[atyy]

The apparently "nonlocal observable" is, as you stated, the interference pattern in the Aharonov-Bohm effect. But this is just the manifestation of a non-integrable phase factor, which is gauge invariant and expressible in a local form via the electromagnetic four-potential. So there is no violation of locality in the usual sense in the AB effect.

Yes, in the "usual sense". But is the usual sense "physical" since the four-potential is not a gauge invariant quantity?

I don't know whether an analogy would be like in gravitation, where gravity does not have (gauge-invariant) local stress-energy, but it does have nonlocal energy like the ADM energy?

 

[atyy]

Is this relevant?

https://www.math.ubc.ca/~bluman/JMP1997.pdf
http://scitation.aip.org/content/aip/journal/jmp/38/7/10.1063/1.531866
"The conservation laws yield nine functionally independent constants of motion which cannot be expressed in terms of the constants of motion arising from local conservation laws for space–time symmetries."

 

[vanhees71]

It is physical, because your local laws, i.e., the Lagrangian of QED in this case, is also written in terms of the potentials. Any physically measurable quantity must of course be gauge invariant, and that's the case for the non-integrable phase factor
$$\exp(\mathrm{i} \oint \mathrm{d} \vec{x} \cdot \vec{A})=\exp(\mathrm{i} \int \mathrm{d}^2 \vec{F} \cdot \vec{B}),$$
relevant for the interference fringes observed in the AB experiment. In the manifestly gauge-invariant version the phase factor appears to be "non local", but it's "non local" in the same sense as a probability distribution of electrons is "non local", i.e., this is the usual non-local aspect of quantum theory, which however is always about correlations, never about causally connected events.

 

[atyy]

It is physical, because your local laws, i.e., the Lagrangian of QED in this case, is also written in terms of the potentials. Any physically measurable quantity must of course be gauge invariant, and that's the case for the non-integrable phase factor
$$\exp(\mathrm{i} \oint \mathrm{d} \vec{x} \cdot \vec{A})=\exp(\mathrm{i} \in \dd \vec{F} \cdot \vec{B},$$
relevant for the interference fringes observed in the AB experiment. In the manifestly gauge-invariant version the phase factor appears to be "non local", but it's "non local" in the same sense as a probability distribution of electrons is "non local", i.e., this is the usual non-local aspect of quantum theory, which however is always about correlations, never about causally connected events.

It's just one of those things where terminology varies. For example, footnote 4 of http://arxiv.org/abs/quant-ph/9508009 says "It is true that the electron interacts locally with a vector potential. However, the vector potential is not a physical quantity; all physical quantities are gauge invariant."

Did you see the paper in post #36?

 

[vanhees71]

The relevant quantity is the phase factor, and this is a gauge-invariant expression that involves only local expressions. That's of course very formal. Physically one should note that the AB experiment as any interference experiment needs the charged particles to be prepared in a state that's described by a wave packet that is pretty narrow in momentum space and thus pretty spread in position space. You wouldn't call a wide spread in the particle's position probability distribution a "non-locality". I guess, the whole discussion about these issues is usually due to an unsharp definition of the words used. As I said before, there is a difference between correlations of far-distant measurements which can occur in quantum theory due to entanglement and a non-local interaction. The former is well understood and described by local QFT ("local" in the sense of local interactions, meaning that the Hamiltonian is built from a polynomial of field operators with arguments at the same space-time point). The latter is at least problematic. I've not yet seen any non-local theory that really works, but I don't know, whether there is a theorem that excludes any Einstein-causal non-local theory. Only because it's difficult to formulate such a theory doesn't mean that on has established that it doesn't exist! I'll have a look at this paper about "non-local conservation laws". However, it seems not to deal with a non-local theory but with standard electrodynamics, which is the historically first and paradigmatic example for a relativistic local field theory.