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Through our physics education, we first learn that "the gauge potential $A_\mu$ is just a mathematical tool and does not have any physical consequences."

When we study quantum theory we begin to learn that it does have physical effects and I am confused.

The following is what I understand:

"The gauge potential is just a mathematical tool and does not have any physical effect in classical physics. But it has physical effects in the quantum theory."

Is there anything wrong or missing in my above statement?

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I believe this is a common misconception. Potentials are not any more or less real quantum mechanically than they are in classical physics. There are two aspects of this that I would like to point out.

Aharonov-Bohm effect:

It is usually argued that the Aharonov-Bohm effect provides an example of why gauge potentials are more real in quantum physics than in classical physics. This is an artefact of the semiclassical approximation used in the textbook treatment of the Aharonov-Bohm Effect. In fact, it is true that the test particle in the Aharonov-Bohm effect experiences a zero field, but the sources creating the potential in the usual treatment do not. In fact, the sources creating the field experience a non-zero field and that field depends on which path the test particle takes. It is the non-zero field of the source charges, and not the vector potential, which is responsible for the phase shift. So actually the story of the Aharonov-Bohm effect is not that the potential became real because of quantum physics. The real story is more similar to entanglement: the quantum wave function describes the sources and the testcharge together.

For more details, see this paper: https://arxiv.org/abs/1110.6169 or/and this blog post: http://blog.jessriedel.com/2014/06/24/bad-ways-to-think-about-quantum-mechanics-part-1-potentials-and-the-aharonov-bohm-effect/

Quantum gauge theories, like QED and QCD:

An other annoying thing for a long time was that unlike classical electrodynamics, which you could perfectly well describe just using the fields $\vec{E}$ and $\vec{B}$, quantum electrodynamics had no formulation like that, and it was only possible to use the $A_\mu$ to do calculations in QED. Similar things can be said about QCD. This was somewhat annoying, since a massless spin-1 particle like the photon or gluon has helicity, so only physical 2 degrees of freedom, and not 4, like the mathematical object we use to describe it. This has also changed somewhat in recent years by the introduction of spinor helicity formalism. A textbook treatment of this is near the end of A.Zee - Quantum Field Theory in a Nutshell. The chapter is called gluon scattering, I think.

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  • $\begingroup$ There's a fair bit of truth to that argument, but it's also premature to say that it covers all aspects of the effect. In particular, as per physics.stackexchange.com/q/56926/8563, AB experiments have also included a layer of superconductor around the solenoid, which then experiences no magnetic interaction with the electron. That's not to say that there isn't an equivalent description but to my knowledge it has yet to be worked out. $\endgroup$ – Emilio Pisanty Nov 26 '16 at 15:03

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