[ExI] Aharonov-Bohm Effect

scerir scerir at libero.it
Sun Jul 29 19:11:25 UTC 2007


> 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'. 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) 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. 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).

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

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