Figure 2. Initial position of the CM when the velocity of the CM and the Spin = direction form an angle α. It is perpendicular to the vectors and ×, of distance to the center O −CM = (v/2) sin α, in natural units, and is independent of the initial position and velocity of the CC.
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In general, the separation between both centers CC and CM, is in the interval | −| ∈ [1/2 −(v/2) sin α, 1/2 + (v/2) sin α] in natural units, so that this separation is not a constant of the motion.In general, the initial position of the CM for the laboratory observer , given in (18), is contained in the zitterbewegung plane (with d = 0 = b), is independent of the initial position of the CC, , and is depicted in the figure 2. The distance to the center is (v/2) sin α. If is perpendicular to the zitterbewegung plane, ·= ·= 0, and the CM is at the center of the circle and becomes equidistant of the trajectory of the CC. If , has a different orientation, the separation between both centers CC and CM, |−|, is not a constant of the motion. It is in the range |−| ∈ [−(v/2) sin α, +(v/2) sin α] in natural units.We must remark that the CC is the Poincaré transformed of the point , but is NOT the Poincaré transformed of the CM in . It is the CM of the particle in the laboratory frame O, at the same time than , while and are considered simultaneous at the frame . Simultaneous events in one frame are not simultaneous in another. The point represents the location of the CM of the particle simultaneous with , for the laboratory observer.
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8. Poincaré invariant interaction boundary conditions
8. Poincaré invariant interaction boundary conditions
9. Poincaré interaction examples
9. Poincaré interaction examples
We will now represent different interactions with different initial boundary conditions: