Bookmark 6: Spin with respect to CC,
S
, spin with respect to
CM
,
S
CM
, velocity
u
, and angular velocity
w
S
precesses around the CM spin
S
CM
that is conserved in free motion and is an average of the CC spin
S
during one turn of the motion of the CC.The Spinning Particle Model satisfies Dirac’s equation when quantized.CC circles at speed c around CM in the CM frame because |
r
| is constant and its acceleration
a
is orthogonal to its velocity
u
and thus colinear with
r
. When
v
=0,
r
describes a circle of radius
R
0
at speed c as in bookmark 6.The angular velocity
w
of the point
r
has two components:
w
p
responsible for the curvature and
w
u
responsible for the torsion of the CC trajectory, respectively.The total energy H contains two terms: the translational energy ∝
P
and the rotational energy ∝
S
w
​Energy and linear momentum take the same expressions as in the Point Particle (particle modelled without spin) in terms of CM velocity.Dirac was right: the internal frequency of the electron is
2
2
mc
h
, twice the De Broglie frequency
2
mc
h
​When the particle moves with speed
v
, the time it takes to complete a turn is longer and thus its internal clock slows down.​In the following sections we will elaborate on the rational to get to all the above findings.

3. Fundamental Principles basis of the Spinning Particle Model


4. The 10 Variables of the Poincaré Group

Let’s find out the variables that are needed to describe the state of an elementary particle.
If t and
r
are the time and position of a space-time event for the observer O, and

t
and


r
are the time and position of the same event for any arbitrary inertial observer

O
, they are related by the transformation of the Poincaré group:

μ
x
=
μ
Λ
ν
ν
x
+
μ
a
(
1
)
μ'
x
≡c

t
,


r
≡
c

t

r
x
​​

r
y
​​

r
z
ν
x
≡ct,

r
≡
ct
r
x
​​
r
y
​​
r
z
μ
a
≡cb,

d
≡
cb
d
x
​​
d
y
​​
d
z
where:​

μ
x
represents the transformed four-vector components (time and space coordinates) in the new reference frame.​
ν
x
represents the four-vector components in the initial reference frame.,​
μ
Λ
ν
is the Lorentz transformation matrix (which includes boosts and rotations),​
μ
a
is the space-time translation (which shifts the space-time coordinates).and a general Lorentz transformation, a composition of a rotation R(α), followed by a boost or pure Lorentz transformation L(
v
) is
μ
Λ
ν
=L(
v
)·
R
4
(α)
The complete Poincaré transformation from O reference frame to

O
reference frame, including translation, rotation and velocity boost is:​Λ = L(
v
)
R
4
(
α
) +
a
​with: Space-time Translation =
a
​ Rotation = R(
α
) Lorentz Velocity Boost = L(
v
)