As the time coordinate does not rotate, we can make the R4 rotation transformation using this R(
α
) matrix :
R
4
(α)=
1
0
0
0
0
·
·
·
0
·
R(α)
·
0
·
·
·
And we will apply the rotation transformation as:
μ'
x
=
R
4
(α)
μ
x
=
1
0
0
0
0
·
·
·
0
·
R(α)
·
0
·
·
·
ct
r
x
​​
r
y
​​
r
z
=
ct
[R(α)

r
]
x
[R(α)

r
]
y
[R(α)

r
]
z
Lorentz Boost: A pure Lorentz transformation of velocity
v
(velocity boost), is given by the 4 × 4 matrix
L(
v
)=
γ
γ
v
x
c
γ
v
y
c
γ
v
z
c
γ
v
x
c
1+
2
v
x
2
c
2
γ
1+γ
v
x
v
y
2
c
2
γ
1+γ
v
x
v
z
2
c
2
γ
1+γ
γ
v
y
c
v
y
v
x
2
c
2
γ
1+γ
1+
2
v
y
2
c
2
γ
1+γ
v
y
v
z
2
c
2
γ
1+γ
γ
v
z
c
v
z
v
x
2
c
2
γ
1+γ
v
z
v
y
2
c
2
γ
1+γ
1+
2
v
z
2
c
2
γ
1+γ
(
2
)
The constant γ=γ(
v
) where:
γ(
v
)=
1
1-
2
v
c
=
1
1-
2
v
x
2
c
-
2
v
y
2
c
-
2
v
z
2
c
So that applying the Lorentz boost (1) to the space-time event observed by O to transform it due to its speed
v
relative to O’ we have:
c

t

r
x
​​

r
y
​​

r
z
=
γ
γ
v
x
c
γ
v
y
c
γ
v
z
c
γ
v
x
c
1+
2
v
x
2
c
2
γ
1+γ
v
x
v
y
2
c
2
γ
1+γ
v
x
v
z
2
c
2
γ
1+γ
γ
v
y
c
v
y
v
x
2
c
2
γ
1+γ
1+
2
v
y
2
c
2
γ
1+γ
v
y
v
z
2
c
2
γ
1+γ
γ
v
z
c
v
z
v
x
2
c
2
γ
1+γ
v
z
v
y
2
c
2
γ
1+γ
1+
2
v
z
2
c
2
γ
1+γ
ct
r
x
​​
r
y
​​
r
z
And the complete Poincaré transformation is:
Λ=L(
v
)
R
4
(α)+a
c

t

r
x
​​

r
y
​​

r
z
=L(
v
)
R
4
(α)
ct
r
x
​​
r
y
​​
r
z
+a=
γ
γ
v
x
c
γ
v
y
c
γ
v
z
c
γ
v
x
c
1+
2
v
x
2
c
2
γ
1+γ
v
x
v
y
2
c
2
γ
1+γ
v
x
v
z
2
c
2
γ
1+γ
γ
v
y
c
v
y
v
x
2
c
2
γ
1+γ
1+
2
v
y
2
c
2
γ
1+γ
v
y
v
z
2
c
2
γ
1+γ
γ
v
z
c
v
z
v
x
2
c
2
γ
1+γ
v
z
v
y
2
c
2
γ
1+γ
1+
2
v
z
2
c
2
γ
1+γ
1
0
0
0
0
·
·
·
0
·
R(α)
·
0
·
·
·
ct
r
x
​​
r
y
​​
r
z
+
cb
d
x
​​
d
y
​​
d
z
and then:

ct
=γct+γ
v
x
c
[R(α)
r
]
x
+γ
v
y
c
[R(α)
r
]
y
+γ
v
z
c
[R(α)
r
]
z
+cb

t
=γt+
γ
2
c
v
R(α)
r
+b
(
3
)
and

r
x
=γ
v
x
c
ct+
1+
2
v
x
2
c
2
γ
1+γ
[R(α)
r
]
x
+
v
x
v
y
2
c
2
γ
1+γ
[R(α)
r
]
y
+
v
x
v
z
2
c
2
γ
1+γ
[R(α)
r
]
z
+
d
x

r
x
=
γv
x
t+
[R(α)
r
]
x
+
2
γ
(1+γ)
2
c

v
x
v
x
[R(α)
r
]
x
+
v
x
v
y
[R(α)
r
]
y
+
v
x
v
z
[R(α)
r
]
z
+
d
x

r
x
=[R(α)
r
]
x
+
γv
x
t+
2
γ
(1+γ)
2
c

v
x
[
v
R(α)
r
]+
d
x
and in vector notation we have :

r
=R(α)
r
+γ
v
t+
2
γ
(1+γ)
2
c
(
v
R(α)
r
)
v
+
d
(
4
)
The event (0,0) in O is described by (b,d) in

O
, so that these values represent the time and position of the origin of O, measured by

O
when the clock of O marks zero. The velocity
v
is the velocity of the origin of O as measured by

O
. Finally R(α) is the rotation

O
has to perform to its own Cartesian frame to obtain the Cartesian frame of O before the Lorentz boost L(
v
). The values of the variables (b,
d
,
v
,
α
) of a Poincaré group element is the way the observer

O
describes any other observer O. This is the maximal set of variables we can use to describe the variables of any elementary particle. We do not make any assumption about the size or shape of the particle.​Then the state of an elementary particle is characterized at most by the ten variables
x
i
≡ (t,
r
,
u
,
α
), that are interpreted as the time t, the position of a single point
r
, the velocity of this point
u
, and finally the orientation
α
, of a comoving Cartesian frame attached to the point
r
. And the same variables, with different values, for the final point
x
f
of the variational evolution. If we restrict ourselves to a system where the initial state is given by the variables
x
i
≡ (t,
r
), which span a homogeneous space of the Poincaré group, we are describing the spinless Point Particle. ​It is in terms of the extra variables (
u
,
α
), that we will be able to describe the spin structure. ​We shall call kinematical variables to the variables (t,
r
,
u
,
α
). Elementary particles are localized by a single point in space-time, move and rotate.What we see is that we have three different cases depending on whether the variable |
u
| < c, |
u
| = c or |
u
| > c. Among the models of spinning particles this formalism predicts, the only one that satisfies Dirac’s equation when quantized, is that model in which |
u
| = c, the point
r
is moving at the speed of light, as is demonstrated in the article reference [4].The Lagrangian depends on the variables used to describe the system (t,
r
,
u
,
α
) and also depends on their next order derivatives, therefore the Lagrangian must depend on the acceleration of the point
a
=

u
t
, and on the angular velocity
ω
=

α
t
. The point
r
, satisfies therefore a system of fourth-order differential equations and, as it is the only point where the external fields will be defined, we interpret the point
r
, as the location of the Center of Charge (CC) of the particle.

5. Homogeneity of the Lagrangian


6.
r
moves at the speed of light for all inertial observers


7. Calculating the acceleration transformation


8. Constants of the motion, The Four Momenta

The kinematical variables which define the variables of the variational description of an elementary particle are
t,
r
,
u
,
α

. The generalized coordinates are thus the same variables with the time variable excluded (
r
,
u
,
α
).
Generalized coordinates are the set of variables used to describe the configuration of a system in terms of its degrees of freedom, defining the system’s state without explicitly involving time, independently of its dynamics.
Hamiltonian mechanics describe a system in terms of the total energy of the system H(t,x,p) =
∑
i
p
i
x
i
- L(t,x,

x
)
In Hamiltonian mechanics, each generalized coordinate
x
i
has a corresponding canonical conjugate momentum
p
i
, defined for L(t,
r
,
u
) as:​
p
i
=
∂L
∂

x
i
​where​

x
i
=

x
i
t
​The conjugate momentum has a longer expression for L(t,
r
,
u
,
a
,
α
) ​The canonical conjugate momentum tells us how the system evolves in response to changes in the coordinate
x
i
.In Hamiltonian mechanics the system is fully described by generalized coordinates
x
i
and their conjugate momenta
p
i
and the Hamilton equations are:​​

x
i
t
=
∂H
∂
p
i
​​​

p
i
t
=-
∂H
∂
x
i
​​Therefore there will be a canonical conjugate momentum of each of the variables
r
,
u
and
α
. The conjugate momentum of
r
variables is the vector​​
p
r
=
∂L
∂
u
-

∂L
∂
a
t
=
R
-

U
t
​​the conjugate momentum of
u
variables is the vector​​
p
u
=
∂L
∂
a
=
U
​​and finally the conjugate momentum of
α
variables is the vector ​​
p
α
=
∂L
∂
ω
=
W​​
​​The Hamiltonian of this particle is by definition:​​
H=
p
r
u
+
p
u
a
+
p
α
ω
-L=-T-
u

U
t
​​If under some one-dimensional infinitesimal transformation of a group of infinitesimal parameter δg, each of the kinematical variables transform as follows:​​

t
=t+
M
0
∂g​​

r
=
r
+
M
r
∂g​​

u
=
u
+
M
u
∂g​​

α
=
α
+
M
α
∂g
​​If the dynamical equations remain invariant under the above infinitesimal transformation delta g, Noether’s theorem gives us a conserved quantity:​​
N=
HM
0
-
p
r
M
r
-
p
u
M
u
-
p
α
M
α
​​If the particle is free, the Lagrangian
L
o
, will be invariant under all transformations of the Poincaré group. As derived in the referred book [1], if we analyze the constants of the motion under infinitesimal time translation, space translation, Lorentz boost and rotation, respectively, we find the following constants of the motion to which we give the names indicated below: