[ExI] Aharonov-Bohm Effect

Lee Corbin lcorbin at rawbw.com
Sun Jul 29 21:48:04 UTC 2007

Serafino writes

>> The more I understand about this, indeed the more
>> outrageous it is. But I have one more question, below.
> The A-B effect is only one of the many quantum mysteries.
> Someone begins to think that QM could be a sort of
> 'operating system'.

You're not a believer in MWI?

> That is to say, not a theory about
> physical 'behaviours' in space-time. But a 'syntax', a 
> compendium of abstract rules. Like any other rule, or
> like any other operating system, it cannot be 'explained'.
> One can only judge its efficiency, in terms of complexity,
> of informations, of probabilities, of evolutions, etc.
> (The speculation above might be relevant while studying
> quantum gravity).
>> So in the A-B effect, is the vector potential outside
>> the (shielded) solenoid different along the path that
>> the electron takes? That is, if X and Y are two points
>> of the path, is there or isn't there a difference in the
>> strength of the vector potential?   Surely the answer
>> must be that there is *no* change!  (Else we would
>> have to say that the EM field itself was there, right?)
> The magnetic flux within a long solenoid of radius R
> is given by the magnetic field strenght x pi x R^2.
> Outside the solenoid the magnetic field is (fapp)
> null. However the vector potential forms *cylindrical
> equipotential surfaces* outside (and also inside)

Ah, that's great. That can be nicely visualized.

> the solenoid, with a sense of circulation which is
> opposite to that of the electron current in the solenoid.
> According to Maxwell the vector potential was a
> measurable quantity related to momentum ('electromagnetic
> momentum at a point'). It seems that the importance
> of the vector potential, in the quamtum domain, 
> has been established by Dirac (in the '30s) and then 
> by Aharonov and Bohm (in the '50s). 


> Now, if you have a two-slit interferometer and many
> electrons entering the interferometer, you get (for
> each electron) two 'amplitudes', one for each slit. 
> You can compose the two 'amplitudes' at a point on 
> a screen (and you get an interference pattern). If, 
> between the two 'amplitudes' (or the two possible paths 
> of the electron), you insert a vector potential field,
> you'll find a different interference pattern. The effect
> might be thought as a force-free interaction with a vector
> potential field (which is 'local' ghost) or as a force-free 
> interaction with a 'non-local' (and unknown) magnetic field.
> Note however that the force-free interaction with a 
> 'local' ghost, like a vector potential field, or a
> 'non-local' one, like an unknown magnetic field,
> is *not* sufficient to produce the A-B effect. 
> You also need that the allowed paths of the electrons
> (in the two-slit example) *circumscribe* the region
> in which the solenoid, or the shielded magnetic field,
> is located.

Oh, very good. This path dependence is familiar from
curved space phenomena. That does explain it.

Now, may I visualize one of the two paths (of one 
particle) emanating from one slit and, as it nears the
solenoid, penetrating one after the other of the
cylindrical surfaces and then un-penetrating them
in turn as the electron goes further away?  Or is
that the wrong idea?

> The A-B effect is then related to the geometry
> (or, better, the topology) of the space accessible
> to the particles.    
> Given this topological factor, it could be interesting
> to study a gravitational extension of the A-B effect.
> But you need at least two paths which *circumscribe*
> the region in which the gravitational potential resides.
> (I think it has been done, in different contexts,
> maybe also within neutron interferometry).

In principle it sounds simple:  just lower a gravitating
field inside the two paths of an interstellar split beam
experiment. (Wheeler loved to describe such a large
"apparatus" wherein the two possibilities interfere
after traveling many light years.)  So I would think
that this latter experiment would not involve EM,
but rather gravitation, and might show just the same

> [The impression is that this post is rather chaotic,
> or worse. Bah.]

Nothing could be further from the truth!  Your 
explanation above is the clearest I can imagine.
Many thanks.


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