The action of Maxwell theory has a field part and an interaction part:
In direct action the fields are replaced by an explicitly propagated source:
to give
Usually one chooses the Lorenz gauge and a Green's function that produces both advanced and retarded potentials, of equal magnitude, at every current:
Unlike Maxwell EM, this means that one is not free to specify an arbitrary 'in' field as a boundary condition at some initial time. Not only must any such fields be justified as originating at historical sources, they must also be consistent with the motion of the (local) current in question, and are not, therefore, independent of its motion.
A 1MB animation of a direct-action disturbance associated with a point source is here.
In un-renormalized EM there is no mechanical mass; all mass is electromagnetic. In conventional direct-action - having Maxwell ancestry - lack of mechanical mass means that the interaction above is the only action. The traditionally singular self-action and consequent infinite self-energy are still present. But with no mechanical action, there is no (mathametical) constraint against faster than light motion, and it becomes possible to 'balance' the singular self-action with singular action (creating an infinite force) from other particles moving faster than light. The required condition is that the interacting pair of charges in question must lie on each other's Cerenkov cone.
This state of affairs can be maintained continuously by a pair of particles in a common circular orbit, 180 degrees out of phase, provided the speed satisfies the eigenvalue condition:
For stability, the pair must be oppositely charged. Here is a figure showing the balance of forces. Note that each charge is visible in three places at any time due to its superluminal motion. At the present location each charge is singularly self-interacting and also experiences an infinite attractive force from the past and future locations of the other charge due to the Cerenkov cone interaction. These three images of a single charge are of course inseparable and add up - as seen from a distance - to a unit charge, suggesting a relationship between these superluminal images and individual quarks, and therefore, more generally, between the strong force and electromagnetic Cerenkov-cone interactions between superluminal charges.
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The forces at play between two charges in superluminal circular motion, 180 degrees apart. The black lines show the path of the singular electromagnetic interactions required to keep the charges on the circular path. The eigenvalue condition is the requirement that the projection of the circumferential velocity vector onto these lines, at the point of intersection, is equal to the speed of light. Here is a 72Kb animation. |
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The helical world lines of the two charges. The spheres are the charge 'events' that are visible as each world line passes through the light cone of the 'present' position of the positive charge. The two dark blue spheres are Cerenkov cone events. Interactions between these and the 'present' charge are shown by solid black lines. Notice the blue helix grazes the light cone at a tangent at the location of the Cernekov spheres. The pink and light blue spheres denote the presence of non-singular - non-Cerenkov - interactions which, to first order, play no role in the stability of the orbit. Here is a 6MB animation of the motion. |
The direct action kernel can be modified so that the self-action is zero and the eigenvalue condition for presumed circular motion becomes:
Light-speed circular motion is now a solution - here are 2 x 500kB animations of this 'classical zitterbewegung'. Actually, because the self-action vanishes identically at light speed, the mass is zero and the system is underconstrained in the absence of external fields, so that any motion at light speed is initially allowed. Mass appears and the system becomes properly constrained in the presence of external fields. Light speed motion is a special degenerate case of the eigensolutions for the velocity in that the three-way interaction between past present and future locations characteristic of the superluminal solutions collapses to a point at the present location, and cancels there exactly.
The modified theory also changes the way that superluminal charges interact, in particular changing the sign of the Cerenkov cone interaction force. A consequence is that stable self-binding orbits exist for a single charge in superluminal circular motion satisfying the new eigenvalue condition. Here are some animations of motion for a few modes satisfying the eigenvalue condition above. In these orbits the single charge appears as if in three places at once, corresponding to past present and future self-intersection of the lightcone and the Cerenkov cone. That is, a single charge generates the appearence of there being three charges, the sum of which adds up to a unit charge - suggesting a correspondence with quarks. In accord with that correspondence, here there is no possibility of separation of the three points of force, since (here) they are images of the same charge, suggesting in turn a connection between the strong force and electromagnetic interaction on the Cerenkov cone of a superluminal source.
It appears that the two classes of particles generated by modified direct action, i.e. associated with v = c, and discrete speeds v > c, have some characteristics corresponding to charged fermions and baryons, respectively.
