4 ms·

How does the Aharonov-Bohm effect avoid fixing the gauge? e.g. if I can find a potential at which the particles act like there’s no cylinder there at all, isn’t

How does the Aharonov-Bohm effect avoid fixing the gauge? e.g. if I can find a potential at which the particles act like there’s no cylinder there at all, isn’t that equivalent to fixing the gauge of the potential, by finding a privileged 0?

antognini 4y ago
You never measure the potential directly. The potential causes a phase shift between the two paths that the particles can take. Different gauges will produce the same phase shift. So if you turn the solenoid off, you could fix the gauge by setting the magnetic potential to zero, but you don't have to. You could still add any function whose curl is zero and have a valid magnetic potential. This would not change the observed phase difference. So there's still no preferred gauge. (Some choices of gauge may make the math easier in the same way that some choices of coordinate systems are more natural to the experiment, but any choice of gauge is valid, just as any choice of coordinate system is.)
- lalaithion 4y ago
  But doesn’t > If you turn the solenoid on and off you would think it would make no difference because no matter what you do, the magnetic field the particle encounters is zero. The key, though, is that the magnetic potential is not zero along the particle's path, and the magnetic potential affects the particle's phase in measurable way when the two states recombine. Imply that you can measure the potential somehow? That the particles phase is affected by the potential being non-zero, and therefore the gauge is at least somewhat fixed?
  - antognini 4y ago
    The issue is that the choice of gauge changes the equation between the value of the potential and all the observable quantities. So you can go ahead and measure the phase difference, the charge of the particle, and the value of the magnetic flux within the solenoid, but in order to turn that into a value of the magnetic potential, you need an equation relating them all. And different choices of gauges give you different equations. So there's no unique way to go from the quantities you've observed to a value of the magnetic potential.

antognini 4y ago

You never measure the potential directly. The potential causes a phase shift between the two paths that the particles can take. Different gauges will produce the same phase shift. So if you turn the solenoid off, you could fix the gauge by setting the magnetic potential to zero, but you don't have to. You could still add any function whose curl is zero and have a valid magnetic potential. This would not change the observed phase difference. So there's still no preferred gauge. (Some choices of gauge may make the math easier in the same way that some choices of coordinate systems are more natural to the experiment, but any choice of gauge is valid, just as any choice of coordinate system is.)

lalaithion 4y ago
But doesn’t > If you turn the solenoid on and off you would think it would make no difference because no matter what you do, the magnetic field the particle encounters is zero. The key, though, is that the magnetic potential is not zero along the particle's path, and the magnetic potential affects the particle's phase in measurable way when the two states recombine. Imply that you can measure the potential somehow? That the particles phase is affected by the potential being non-zero, and therefore the gauge is at least somewhat fixed?
- antognini 4y ago
  The issue is that the choice of gauge changes the equation between the value of the potential and all the observable quantities. So you can go ahead and measure the phase difference, the charge of the particle, and the value of the magnetic flux within the solenoid, but in order to turn that into a value of the magnetic potential, you need an equation relating them all. And different choices of gauges give you different equations. So there's no unique way to go from the quantities you've observed to a value of the magnetic potential.