Figure 3 : defined by its module v, β the zenithal angle and λ the azimuthal angle. Spin is defined by its zenithal angle θ and azimuthal angle ϕ.
v
S
CM
The initial (and constant, as we are in an inertial case without external forces) Center of Mass velocity is then :
V
0x
V
0y
V
0z
(
41
)If and are the time and position of the CC of the particle for the Center of Mass observer of this particle, and t and r are the time and position of the CC for any arbitrary inertial observer, they are related by the Poincaré transformation (1):
*
t
*
r
*
O
μ
x
μ
Λ
ν
ν
*
x
μ
a
and Λ = L () R (ψ, θ, ϕ) is a general Lorentz transformation, as a composition of a rotation R, followed by a boost of velocity , or pure Lorentz transformation L () .
We are going to describe now the rotation R (ψ, θ, ϕ), the boost or pure Lorentz transformation L (v) with arbitrary velocity v, the space translation T (d) of displacement d, and finally the time translation T (b), of value b .
Let us first analyze the rotation . Let us assume we have the Dirac particle in the center of mass reference frame, as depicted in the figure 5, with the spin along OZ axis .
v
v
v
We are going to describe now the rotation R (ψ, θ, ϕ), the boost or pure Lorentz transformation L (v) with arbitrary velocity v, the space translation T (d) of displacement d, and finally the time translation T (b), of value b .
Let us first analyze the rotation . Let us assume we have the Dirac particle in the center of mass reference frame, as depicted in the figure 5, with the spin along OZ axis .