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 he mentions a comment by John Bell that "vagueness, subjectivity and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice."

This interpretation rests upon two assumptions; 1) that all motion is relative (more specifically that: quantum particles can only have motion with respect to other quantum particles and not with respect to any arbitrarily contrived coordinate system), and 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 n2-3n+2 momentum 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 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 n-1 states each of which is relative to one other particular particle out of n-1 other particles.  We can call those relativistic states or local states, that is, the relative motion between two particles is local to them but not local to some third particle which has motion with respect to both of them.    So, we see that any single particle has a local framework of n-1 states which are all relative.  But it has only one state which is local between it and some other single particle.    We can easily see that probabilistic interpretations are a useful tool when applied with reason.  The Copenhagen Interpretation, however, doesn't approach the use of reason but rather is the application of a deliberate theoretical choice to ignore, as a matter of policy, the nonlocal attributes of the universe.  The task of the theoretical physicist is quite daunting, then, if in trying to understand the universe he is required to intellectually dismiss the bulk  of its dynamics simply because he has found no way in which such dynamics might be manifested in his own frame.

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 potential

According 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 potentials are the media of propagation.  They also proposed:

"that, in quantum mechanics, the fundamental physical entities are the potentials, while the fields are derived from them by differentiations."

Liu has written several papers {quant-ph/9510004 and quant-ph/9506038) available at 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.

It turns out, then, that we are fully justified in generalizing the electromagnetic vector potential as a discrete velocity potential.

From here we can make the logical leap (which is really just a small step) that the unit charge is n-1 velocity potentials.  We have two types of charge, sinks, and sources.  The fact that the magnitude of the unit sink charge is exactly equal to the magnitude of the unit source charge discloses that for every sink velocity potential there is a source velocity potential.  But we already knew that because for every momentum state there is an equal and opposite momentum state.  The universe is constructed so that they exist in conjugate pairs.  If they didn't then the universe would not display relational features.

So we arrive in a rather straightforward manner with a new kind of metric for the unit charge (which is n-1 velocity potentials).

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, "Gravitation, Electromagnetism and Quantized Charge" subtitled "The Einstein Insight" identified this as the task of the theoretical physicist.  In generalizing the vector potential as a discrete velocity potential we see Pisello's vision fulfilled.