## Monday, November 28, 2022

### Aharonov–Bohm effect does not Prove Nonlocality

I heard the suggestion that the Aharonov–Bohm effect proves a form of quantum nonlocality.
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.[1] The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.

The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally.[2]

So the effect depends on the potential, and not just the fields.

The potential and fields are all locally defined, so what is the problem?

The problem is that only the fields are directly observable, and there is considerable discretion in defining the potential. Sometimes the potential is defined to satisfy a distant condition. This is allowed, because gauge symmetry means it has the same physical effect.

From the viewpoint of differential geometry, the potential is a connection on a complex line bundle, and is a purely local object. It is more fundamental than the fields.

The paradox is that an electron can interfere with itself after going around a non-null-homotopic loop with a flat complex line bundle. Arguably there is something nonlocal about that. I don't think so. It is not like action-at-a-distance at all.

#### 1 comment:

1. This is just one more scenario which shows that the concepts from EM theory do not *directly* stand for anything *concretely* real.

Earlier, people would tend to say that the (scalar and vector) potentials don't exist and are mere tools of calculations, whereas the fields ($\vec{E}$ and $\vec{B}$) did physically exist (by which they meant that they were as concretely real as the mostly invisible air).

Well, it was always wrong to suppose that, because there is no unique $\vec{B}$ field for a charge once it is *seen* as moving --- the supposedly physical'' fields depend on the mere *mathematical* device of the reference frame! Even then, people thought (and taught) that the electric and magnetic fields were real!

In contrast, these days, they seem to show a tendency to turn around and say that it's really the *potentials* that may be real but not the fields.

... Well, that's just shifting around the burden, if you ask me... The fact is: you cannot see'' (measure) either fields or potentials, only their effects on massive charges.

It's high time people recognized that EM simply isn't satisfactory enough a theory. It fails on so many counts ...: the absence of mechanisms whereby the massless fields can impart forces to massive charged bodies; other inconsistencies thrown up in building any satisfactory ontology for the charges, fields, and potentials; the utter failure to explain the stability of matter.... Given this all, I don't understand why people still entertain the idea that some or the other constructs of this theory has to correspond, in one way or the other, to something concretely real. ... There's a lot of intrinsicism going around here.

... And then, we all know that it's the quantum mechanical world, and EM must be not only a higher-level description but also a necessarily abstract one.

... I've begun thinking that we would be better off regarding EM as *more* a theory of mathematical methods than of direct physical content! None thinks of it this way, but it's an idea worth giving a thought.

BTW, also see:

https://physics.stackexchange.com/questions/94170/is-the-concept-of-a-field-necessary-to-electrodynamics

There is even a text-book: Electrodynamics: The Field-Free Approach'' by Kjell Prytz.

If you eliminate fields (and BTW, even potentials are fields in their own right), then action at a distance (albeit with a delay) might go completely unexplained. But if you ask me, starting with a clean slate like that would be much better than merely imagining things. It's worth recalling the restraint that Newton showed in regard to the sought mechanism for gravity.

Of course, if you accept the fields-free formulation for EM, then one issue arises immediately: how come the EM-defined entities (fields *or* potentials) can at all enter the Hamiltonian (or the Lagrangian) as used in QM (or QFT). ... On this count, I think I've some ideas, but I won't bring them up because I'm still learning RQM/QED/QFT.

Best,
--Ajit