From how I've learned it in school the magnetic vector potential is used as a mathematical tool to simplify problems with current-carrying wires in classical electromagnetism, but is never treated as bearing any physical meaning. After doing some research on it myself, I found Feynman's lecture on the subject, which outlines how this vector potential actually takes center-stage in QED, not only featuring in its primary equations for concepts like phase change but also as capable of producing physical changes such the Aharonov-Bohm effect that can't be explained by the magnetic field alone. Are there any comparable physical effects the vector potential has in classical electromagnetism?
Electromagnetism – How to Understand the Reality of the Magnetic Vector Potential in Classical Electromagnetism?
aharonov-bohmelectromagnetismgauge-theorymagnetic fieldspotential
Related Solutions
We are allowed to use gauge invariance in quantum mechanics – even quantum mechanical theories with the electromagnetic 4-potential are gauge-invariant theories. However, it's not quite true that all gauge invariant quantities are functions or functionals of $F_{\mu\nu}$.
Instead, we may consider the phase $$ \exp\left(i\oint d\vec x\cdot \vec A\right) $$ where the integral goes along a circle surrounding the solenoid. The exponential above may be seen to be gauge-invariant (add the appropriate natural factor of $e$ or $e/c$ to the exponent to fit your normalizations) because $\vec A$ changes by $\nabla \lambda$ and the integral changes by the step of $\lambda$ between the beginning and end of the circular contour.
But this multiplicative factor is guaranteed to be a multiple of $2\pi$ because the charged fields of unit charge transform by getting multiplied by $\exp(i\lambda)$ and they have to stay single-valued in all directions of the solenoid. So the information given by the exponential – a complex number whose absolute value is one or, equivalently, the integral of $\vec A$ modulo $2\pi$ – remains the same under any gauge transformation. It has observable consequences in quantum mechanics. In particular, it affects the location of the interference patterns behind the solenoid.
Equivalently, you may rewrite the contour integral as $$\oint d\vec x\cdot \vec A = \int dS\cdot \vec B $$ which only depends on the gauge-invariant field strength $\vec B$. It's the magnetic flux through the solenoid. However, we must know the value of $\vec B$ even in – and especially in – regions that the electron never reaches, where it has a zero probability to be, namely inside the solenoid. Quantum mechanics is sensitive on the magnetic flux because it manifests itself as the relative phase of the wave function of the electron going around the left or right side of the solenoid, respectively.
The first explanation of the AB effect only uses quantities measured along the paths of the electron but one needs to use other gauge-invariant objects than the field strength; the second explanation agrees with the proposition that all gauge-invariant entities are functionals of the field strength but one must "nonlocally" consider the field strength's value in forbidden regions in the solenoid, too. They matter in quantum physics. In the classical limit, the interference patterns go away and the whole sensitivity on the magnetic flux is eliminated, too.
On point (1) I can see no reason why it should be impossible, but nobody has done it to the best of my knowledge.
On point (2), there is a paper claiming that the AB experiment is entirely a result of local interactions between fields, and that it does not occur if the field interactions are totally shielded. The author claims there was a flaw in the Tonomura experiment:
Experimentally, no experiments so far have been performed under the condition of perfect shielding of the field interactions. The most ideal one was the experiment performed by Tonomura et al. [10], where the magnetic flux is shielded by a superconductor from the moving electron’s path. Their setup is basically equivalent to Configuration I where the flux is confined in a superconducting shield. Contrary to the analysis for Configuration I, a clear AB phase shift was observed despite the presence of the superconducting shield. In this experiment, however, incident electrons with a speed of about $2.4 × 10^8$m/s were used. In fact, no superconducting material can shield the magnetic field produced by such fast electrons [2], and the ideal shielding analysis in Section IV-A cannot be applied to the experiment in Ref. 10. In other words, the shielding in the experiment of Ref. 10 was only one-sided where the incident electron is moving in a field-free region, whereas shielding of both sides is necessary to eliminate the Aharonov-Bohm effect. The experimental result of Ref. 10 can be fully understood in the framework of the local field interaction between the localized flux and the magnetic field produced by an incident electron.
Best Answer
The vector potential is gauge-dependent and unobservable in both classical and quantum mechanics. Only gauge-invariant quantities — including the electric and magnetic fields — are observable. Even in the Aharonov-Bohm effect, the vector potential is not directly observable; what we measure is the gauge-invariant quantity \begin{align} \Phi = \oint \vec{A}\cdot \mathrm{d}\vec{\ell} \end{align}