Relativity and conservation laws

[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.