I was paging thru my text Introduction to Electrodynamics by Griffiths, third edition, page 420, and stumbled across this odd statement:
First of all I can’t see how the speed of the scalar potential can depend on the gauge. Or can it? Can a gauge transformation change the propagation speed of V? Just on the face of it that sounds ridiculous.
Secondly, in light of the Aharonov-Bohm effect how can this be true? If the scalar potential can alter the phase of the wavefunction then it must be real, but if it’s real it can’t propagate faster than c
Why not? By choice of gauge I can add any time derivative (psi-dot) to V, as long as I compensate A appropriately. For example, in one dimension I might take something like
psi(x,t) = ct-x , ct>x
psi(x,t) = 0 , ct<x
with time derivative a step function traveling rightward with speed c. If I want I can add a couple of these traveling pulses together to get a gauge in which V propagates at 1m/s. (Just don’t name it after me.)
The Aharonov-Bohm effect includes contributions from both V and A, in a gauge-independent way. Changing V by a constant amount will, if you like, give a global phase to the wavefunction; but the global phase of the wavefunction is not observable, so this doesn’t matter.
The point that Griffiths is trying to make is not that “the speed of the scalar potential depends on the gauge”, but that “the speed of the scalar potential is not a well-defined concept.” In fact, if you give me a solution to Maxwell’s equations expressed in some gauge, for which the scalar potential as a function of space and time is V[sub]0/sub, I can actually perform a gauge transformation and change the scalar potential to any function of space and time that I want. Formally, this transformation would be accomplished by picking the gauge function to be
V(x,t) here is the desired function of space and time. So you want a gauge in which V is zero everywhere? Done. (This is the so-called temporal gauge.) In which V equals 5 (in appropriate units) on Tuesdays in Schenectady and is 3 at all other places and times? Easy-peasy. In which V(x,t) is something that looks, for all the world, like a scalar wave propagating at ten thousand times the speed of light? No problem. The electric fields and magnetic fields corresponding to V(x,t) will be exactly the same as those corresponding to V[sub]0/sub — it’ll just look like different “parts” of the physical fields arise from the scalar vs. the vector potentials.
As far as the Aharanov-Bohm effect goes, I don’t see where there’s a paradox. The phase shift can be written entirely in terms of the magnetic field flux between the two slits, and this quantity is gauge-invariant. It’s true that the absolute phase of the paths depends on the actual values of the scalar potential — but you can’t measure absolute phases in QM, only relative phase differences.
Well I’m still confused. The canonical example of the Aharonov-Bohm effect is the double slit experiment with a solenoid between the slits. With this set-up the magnetic field is confined to the interior of the solenoid. So the only thing that could produce the phase and associated interference pattern shifts is the vector potential.
There is no field in the space accessible to the particle.
In order for this to occur it would seem A would have to be real. And real things don’t exceed c. (Or we have some sort of action at a distance)
The above example uses the vector potential but the effect can also be demonstrated with V and no E
Oh, and thanks for explanation of V propagating faster than c.
The classical force law, from which all classically-observable behavior can be derived, depends locally only on the fields E and B and not on the potentials V and A; but the Aharonov-Bohm effect is a quantum observable which cannot be determined using only the local field values, but can be determined in terms of the local potential values. So the Aharonov-Bohm effect is often described as demonstrating “the reality of the potentials”—otherwise we have to get rid of locality.
But this is a slight simplification. The potentials are sufficient for locally determining all quantum observables, but they are still not completely defined. In particular, their gauge freedom is still not a quantum observable. As MikeS said, you can still predict the A-B effect using, for example, the integral of B over the surface enclosed by the two paths; the choice of gauge does not affect the observable.
For example, the magnetic A-B equation (when V=0) predicts a relative phase proportional to the integral of A.dx around a closed path. The gauge freedom lets us add the gradient of a time-constant scalar field to A; but the integral of a gradient around a closed path is zero, so this gauge change does not affect the observable relative phase.
It is possible to think about this topologically. Suppose the B field is restricted to some small flux tubes (for example, a collection of toroidal solenoids) so that B=0 elsewhere. Now let’s make measurements restricted to the region where B=0. Because flux lines do not end, this region is not simply connected (unless there is no field anywhere). The loop integral of A.dx is equal to the flux enclosed by the loop; because this flux is zero everywhere on this region, this integral is topological in nature: that is, it depends only on which flux tubes are enclosed by the loop. So magnetic flux induces a sort of topological defect in the zero-field space that we’re interested in, and this can be measured by taking a particle around closed loops in the space.