- #1
Ghost Repeater
- 32
- 5
I'm reading a book on gauge symmetry, and in the discussion of the Aharanov-Bohm effect, the author says the following:
But a paragraph later, he goes on to say:
It seems to me like there is a contradiction here (indicated by phrases in bold). How can the a change in potential be accompanied by a change in relative phase that results in an observable change in the interference pattern, but at the same time 'no physical result is changed' and 'nothing observable is changed by the gauge transformation.' Isn't the shift in the interference pattern observable, and isn't it due to the gauge transformation?!
Also, what are these 'changes elsewhere' which 'conspire' to give the same diffraction pattern after the phase change?
My understanding of gauge invariance in EM is that, although a change in the EM potential (a gauge transformation) doesn't lead to a change in the EM field, and therefore doesn't change the equations of motion of a charged particle wrt space-time, the point of the AB effect was to show that such a gauge transformation DOES alter the 'internal motion' of the phase, and that this DOES show up experimentally, precisely in the form of a shifted interference pattern. But parts of what the author is saying here seem to contradict that (as well as other parts of what he's said).
Can anyone help clarify what the author is saying here?
The shielded solenoid magnet will not generate a magnetic field outside the magnetic shield [...], but the magnet will induce a vector potential that will produce opposite phase changes in the de Broglie waves passing from the two slits to the detector. The change in the relative phase of the de Broglie waves from the two slits in turn modifies the interference pattern.
But a paragraph later, he goes on to say:
Although the phase of the de Broglie wave is changed at every space-time point by an amount λ(x,y,z,t), which can have quite different values at each space-time point, the changes must be so correlated that no physical result is changed. The phases of the de Broglie waves in front of the two slits are changed by the transformation , but the sum of the changes elsewhere conspires to give exactly the same diffraction pattern. Nothing observable is changed by the gauge transformation. The matter field, represented by the de Broglie wave, exhibits a local symmetry under the gauge transformation that changes the phase of the wave everywhere and at every time by an angle λ(x,y,z,t). The effects of the changes in the electromagnetic potentials induced by the gauge transformation field compensate for the phase changes in such a manner as to hold physical observables invariant under the transformation.
It seems to me like there is a contradiction here (indicated by phrases in bold). How can the a change in potential be accompanied by a change in relative phase that results in an observable change in the interference pattern, but at the same time 'no physical result is changed' and 'nothing observable is changed by the gauge transformation.' Isn't the shift in the interference pattern observable, and isn't it due to the gauge transformation?!
Also, what are these 'changes elsewhere' which 'conspire' to give the same diffraction pattern after the phase change?
My understanding of gauge invariance in EM is that, although a change in the EM potential (a gauge transformation) doesn't lead to a change in the EM field, and therefore doesn't change the equations of motion of a charged particle wrt space-time, the point of the AB effect was to show that such a gauge transformation DOES alter the 'internal motion' of the phase, and that this DOES show up experimentally, precisely in the form of a shifted interference pattern. But parts of what the author is saying here seem to contradict that (as well as other parts of what he's said).
Can anyone help clarify what the author is saying here?