Figure 3. Motion of the CC’s of the two Dirac particles with parallel spins, a phase shift of the CC’s, ψ1 −ψ2 = 180◦, and a separation of their CM’s of 3/4, in natural units. In part (a) the CC’s are closer, the repulsive force on particle 1 CC (e1) becomes an attractive force Fa between the CM’s. After half a turn of the charges, in part (b) the repulsive force between charges is also a repulsive force between their CM’s Fr but the intensity of this force is smaller Fr ≪Fa. In the Coulomb case Fa = 49Fr, while in the Poincaré interaction is greater of value Fa = 343Fr, in this example. The two Dirac particles form a stable bound state of spin 1.
The Poincaré force (29) has 4 terms. We analyze this force in the situation of figure 3.
In part (a) the repulsive force between the charges is an attractive force between the CMs.
The attractive force between the CM’s has a first contribution along-, of intensity
In part (a) the repulsive force between the charges is an attractive force
F
a
The attractive force between the CM’s has a first contribution along
r
1
r
2
F
a
α
2
2
(1/4)
α
2
because the term in the expression of the force = since the CC velocities are opposite to each other. The second term of the force ( = 2, so that it represents an attractive term between the CC’s but a repulsive force between the CM’s, of value
1-
u
1
u
2
2
29
) on particle 1 depends on the acceleration of the particle 2, with a minus sign, a
2
F
r
α
2(1/4)
2
a
2
α
2
The other two terms in the direction of the velocity of the other particle (3) = 0 and (4) = 0, vanish in this situation because in the third term, (-) · (-) = 0 as these two vectors are orthogonal to each other and in the fourth term = − = 0, and the same for = − = 0, because the velocity and acceleration of the CC’s are always orthogonal.The net force in situation 5-(a) is an attractive force between the CM’s of intensity
F
a
F
a
r
1
r
2
u
1
u
2
a
1
u
2
a
1
u
1
a
2
u
1
a
2
u
2
F
a
F
a
F
r
α
2
In the situation of figure 5-(b) the repulsive force between the CM’s is
F
r
α
2
2
(7/4)
16α
2
49
The term which depends on the acceleration is attractive between the CM’s of value
F
r
α
2(7/4)
2
a
2
2α
2
7
The remaining two terms also vanish, so that the total repulsive force is
F
r
F
r
F
r
16α
2
49
2α
2
7
2α
2
49
The ratio between the attractive force in 5-(a) to the repulsive force in 5-(b) is thus
F
a
F
r
14α
2
2α
2
/49This justifies that the Poincaré formation of bound pairs can be obtained even at great relative velocities.
In the Bookmark 4&5 we have different examples of the pairing, with phase shifts of the CC’s, − =18, but paired configurations can be found with phase differences quite far from the opposing 18, even with 9.
Regarding separation of the CMs, stable configurations are easily found up to a distance of approximately 0,41 between the CMs, close to the “radius” distance of 0,5.
The initial velocity of the CM’s can be raised up to approximately v = 0.06. Velocities above those values produce the separation of the particles.
This configuration is very sensitive to numerical errors on the integration because as soon as an infinitesimal error value is created in the coordinate Z, the charges repel each other in the Z axis and the pairing is quickly destroyed.
In the Bookmark 4&5 we have different examples of the pairing, with phase shifts of the CC’s,
ψ
1
ψ
2
◦
0
◦
0
◦
0
Regarding separation of the CMs, stable configurations are easily found up to a distance of approximately 0,41 between the CMs, close to the “radius” distance of 0,5.
The initial velocity of the CM’s can be raised up to approximately v = 0.06. Velocities above those values produce the separation of the particles.
This configuration is very sensitive to numerical errors on the integration because as soon as an infinitesimal error value is created in the coordinate Z, the charges repel each other in the Z axis and the pairing is quickly destroyed.