Understanding the Electromagnetic
Vector Potential as a Discrete Velocity Potential The wave function is probablistic and as is well known is the basis of the Copenhagen Interpretation. There is another means to interpret quantum behavior from a strictly classical viewpoint which is antithetical to the positivist Copenhagen School but which nevertheless yields insight into the structure of matter and the mechanics of processes. In Goldstein's Jan 10, 1996 paper Quantum Philosophy: "The
Flight from Reason in Science." available from
http://xxx.lanl.gov
he mentions a comment by John Bell that This interpretation rests upon two assumptions; 2) there are a finite number of particles in the universe.
In a universe of n particles each particle has n-1
momentum states as opposed to an infinite number of momentum states
expressed by the probablistic psi (wave) function. This
is a direct and classical interpretation using the simplest of
axioms (1 above). The objection which might be raised is
that the momenta between two particles is a function of their
masses and their absolute relative velocity. As such
it may be argued that the relative velocity may be distributed
abitrarily into parts to each particle in such a manner so as
to provide a particle with an infinite number of momentum states.
The argument against this is that experiment does not demonstrate
the chaotic distribution of momenta between particles which experience
elastic collisions. In fact, the momenta distribution always
follows Newton's third law so that it is distributed equally and
oppositely between the particles. The distribution of momenta
to particles always produces a center of momentum frame which
can only occur if the absolute velocity between the particles
is distributed inversely proportional to their masses. If
this didn't happen then the existence of some preferential or
absolute rest frame(s) would be implied. The manifestation of
Newton's third law is the prima facie evidence that such absolute
frames do not exist. The fact that the velocity of light
is found to be the same in any frame is a reification of this
same principle. Therefore, we can dispose of the argument
that a single particle with respect to some one other particle
can have an infinite spectrum of momenta at any given instant.
In fact, a single particle with respect to some one other particle
can, at any given instant, have only a single momentum which must
be equal and opposite to that of the particle to which it is compared.
So, we are able to return to the assertion that at a given instant
a single particle in the universe must have n-1 momenta
in a universe of n particles, no more and no less.
For the purpose of clarity I accept the meaning of local
as being entirely relativistic. Since a particle has
a total of n-1 momentum states of which only one is 'local'
to some one other particle it is then evident that it has n-2
momentum states which are not local to that same one other
particle. Thus, it is clear that since with respect to any
single particle in the universe there are n-1 other particles
each of which have n-2 momentum states which are all nonlocal
to that single considered particle that there are (n-1)*(n-2)
or nmomentum
states which are nonlocal to any single given particle.
That certainly doesn't mean they do not exist. To say that
those states don't exist is to intellectually choose to ignore
more than 99.9999....% of the dynamics of the universe for relatively
small values of ^{2}-3n+2 n.The act of measurement (observation) brings to the local framework
a single state (which we call an observable) but does not dictate
the nonexistence of states which cannot be observed. Again,
each particle has Each of those nonlocal states can be considered to be a potential state (because it isn't manifested in a local frame in a straightforward manner that is easily or quickly understood). Each state is also a momentum state which means that it is related to a velocity so that we can understand them as 'velocity potentials'. Interpreting the electromagnetic
vector potentialAccording to Y. Aharonov and David Bohm (see Aharonov and Bohm's
classic paper "Significance of Electromagnetic Potentials
in the Quantum Theory" published in The Physical Review,
Second Series, Vol. 115, No. 3, August 1, 1959, Pages 485-491)
the potentials play a role in Schrödinger's equation which
is analogous to the index of refraction in optics. Eleven
years earlier W. Ehrenberg and R. W. Siday, Proc. Phys. Soc. London,
B62, 8 (1948) formulated electron optics with refractive index
represented by the potentials. This means that the
Liu has written several papers {quant-ph/9510004 and quant-ph/9506038) available at http://xxx.lanl.gov concerning the significance of the vector potentials referring not only to Aharonov and Bohm's work but also to the earlier work of Ehrenberg and Siday which both characterize the potentials as the media of propagation of charged particles.
From here we can make the logical leap (which is really just
a small step) that the unit charge is
Now the exciting part here is that not only have we arrived at the fundamental nature of charge but we have a global metric, (n-1), for counting purposes, for the unit charge.
From this point on we can see that we can also, in principle,
manufacture particles which are composites of the two types of
unit charge. For each velocity potential which is mapped onto
its conjugate we get a null line or a null velocity potential
line. Whatever geometrical arrangement of the potentials
that the fundamental charged particle might have we can also give
to our composite particle of n-1 null potential lines so
that we obtain a structure (electromagnetotoroid)
which has layers of closed equipotential surfaces of increasing
curvature (just like we assume charged particles have).
The electromagnetotoroid or Archetype structure is composed of
the complete metric set (n-1) of such potentials and their conjugates
organized into a toroidal geometry. Such a geometry where the
potentials and their conjugates are mapped to a ring produces
a structure which has layers of closed equipotential surfaces
of increasing curvature (just as we assume charged particles have).
Since each null line is a null motion line we end up with null
motion equipotential surfaces. This structure then has a null
motion gradient which is the same thing as a 'time-gradient'
field. A 'time-gradient' field is a gravitational
field. Thus we end up with a particle which has the unit gravitational
'charge'. One might argue that even if we could map
a sink type particle onto a source type particle that we would
end up with a charge of zero but this supposes that the 'field'
of a charged particle is a continous subtance. In fact we
see that the charge of a particle is composed of a finite number
of velocity potentials and therefore is not continuously differentiable
in the sense that a surface or solid volume is continuously differentiable.
We see that mapping of two oppositely charged particles onto each
other requires the one to one mapping of the potentials with their
conjugates so that n-1 null lines are produced. Clearly
there is a difference between no charge which is the absence of
a charge and a null charge which is the presence of a structure
which has a null velocity gradient. A null velocity
gradient structure does not exist in the absence of a charge.
In other words two charges, a charge and its conjugate, must be
present for there to exist a null velocity gradient structure.
So it is that we can provide a mathematical description of
physical reality in terms of continuous functions which includes
structures that can be counted but that are not separated from
one another by absolute boundaries. Daniel M. Pisello,
in his book, |