Part I.Bookmark 1: In this figure we represent the Center of Charge (CC), the Center of Mass (CM), the spin relative to the CC, the spin relative to the CM, the velocity and acceleration of the CC, and the angular velocity (= + )
S
S
CM
u
a
ω
ω
p
ω
u
As shown in Part I, the position of the CC, satisfies a system of fourth-order differential equations [5] (thus it has torsion, giving rise to the spin), that can be separated into a system of second-order differential equations for the CC and CM positions, where the CM position , is defined in terms of the motion of the CC, , by (Part I.27)
r
q
r
q
r
(-.)
2
c
v
u
2
a
a
(
1
)where :
u
r
t
(
2
)v
q
t
(
3
)a
2
2
t
r
u
t
(
4
)The relativistic differential equations in the presence of an external electromagnetic field (t, ) and (t, ), defined at the CC position , and in any arbitrary inertial reference frame are [5] as derived also in the previous post Classical Dirac particle I: the electron spinning particle model for dynamic simulations Part II. (9-12):
E
r
B
r
r
2
2
t
q
1
mγ(v)
F
em
1
2
c
v
F
em
v
F
em
E
u
B
(
5
)2
2
t
r
2
c
v
u
2
(-)
q
r
q
r
(
6
)always with the constraint || = c and || < c.
u
v
To integrate the system of differential equations for two particles, we have to establish the appropriate boundary conditions for the positions and velocities of both points, the CC and the CM of each particle . These are the 12 boundary values for the variables , , and . These parameters are those parameters that define the relationship between the Center of Mass observer of the particle and any arbitrary inertial observer or laboratory observer who sees the CM of the particle moving at the speed .
r
0
u
0
q
0
v
0
v
The initial or boundary conditions of the particle are calculated as a function of the parameters that describe the system (user defined parameters in the code of section): - The initial location of the Center of Mass of the particle is defined by the coordinates =(dx, dy, dz). - The velocity of the Center of Mass of the particle is defined by its module v, β the zenithal angle and λ the azimuthal angle. - The Center of Mass Spin is defined by its zenithal angle θ and azimuthal angle ϕ. - The Center of Charge initial location is defined through the initial phase ψ.These initial conditions are the same no matter if the particle is free or is in the presence of external fields, and they have been derived in Part I.Section17
d
v
The energy H, and the linear momentum , are finally written, as in the case of the point particle model, in terms of the center of mass velocity , as Part I.(17 & 18):
p
v
H=γ()=γ()m=γ(v)=
v
2
mc
P
v
v
H
2
c
v
1
1-
2
v
2
c
(
7
)The expression of the spin with respect to the CC, , can be written as: Part I.(20)
S
r
S
r
q
u
(
8
)while the spin with respect to the CM, , is defined by: Part I.(21)
q
S
CM
S
r
q
p
r
q
u
v
(
9
)The spins (8) and (9) are finally expressed in terms of the instantaneous separation between both centers and of the velocities of both points, while it is only the velocity of the CM , which appears in the definition of H and , (7).
v
p
H=+
P
u
1
2
c
u
t
u
S
(
10
)which contains two terms: the first term, proportional to the linear momentum, represents the translational energy, while the second, proportional to the spin and to the motion of the CC (zitterbewegung) represents the rotational energy, which never vanishes. If we introduce in (9) the expressions of and , given in (7), respectively, we get H = γ(v)m. It is this linear expression in terms of H and , which will give rise to Dirac’s Hamiltonian when quantizing the model [4]. In the quantum description H = iℏ∂/∂t and = −iℏ∇ and (10) is transformed into the Dirac equation.
p
S
2
c
p
p
As we will derive later, the dynamical equations, in natural units, for the two particle system taking into account their interaction and also the external electromagnetic fields are:
¨
q
a
a
a
1
γ
v
a
F
a
v
a
F
a
v
a
(
11
)¨
r
a
1-
u
a
v
a
2
-
q
a
r
a
q
a
r
a
(
12
)In the previous section we have the code to model the interaction and in later sections we will derive the expressions and observe some examples.
4.1 Poincaré invariant interaction
4.1 Poincaré invariant interaction
4.2 Poincaré Invariant interaction Lagrangian
4.2 Poincaré Invariant interaction Lagrangian
4.3 Interaction Force
4.3 Interaction Force
5. Conservation laws
5. Conservation laws
6. Natural units
6. Natural units
7. Introduction to Boundary Conditions: one particle
7. Introduction to Boundary Conditions: one particle
To integrate the system of differential equations (26) and (27), we have to establish the appropriate boundary conditions for the positions and velocities of both points, the CC and the CM of each particle. These 12 boundary values for the variables (0), (0), (0) and (0) of each particle, are going to be expressed in terms of 10 parameters.
This reduction is forced due to the 2 constraints: the absolute value of the CC velocity |(0)| = 1, and that in the dynamical equations the vector −, has the direction of the acceleration of the CC and thus is orthogonal to the CC velocity .
These 10 parameters are those parameters that define the relationship between the center of mass observer of the particle and any arbitrary inertial observer or laboratory observer who sees the CM of each particle moving at the speed.
r
u
q
v
This reduction is forced due to the 2 constraints: the absolute value of the CC velocity |
u
r
q
u
These 10 parameters are those parameters that define the relationship between the center of mass observer of the particle and any arbitrary inertial observer or laboratory observer who sees the CM of each particle moving at the speed
v
a