The proper velocity can be expressed in terms of the associated Lorentz transformation spinor called eigenspinor .
The proper time derivative of the eigenspinor can be expressed in terms of the
spacetime rotation biparavector
Ω
{\displaystyle \Omega }
d
Λ
d
τ
=
1
2
Ω
Λ
{\displaystyle {\frac {d\Lambda }{d\tau }}={\frac {1}{2}}\Omega \Lambda }
The spacetime rotation biparavector contains the information of the interaction that the particle is subject to.
The dynamics of a classical particle can be expressed in terms of a differential equation involving the eigenspinor.
d
p
d
τ
=
m
c
d
d
τ
(
Λ
Λ
†
)
=
m
c
(
d
Λ
d
τ
Λ
†
+
Λ
d
Λ
†
d
τ
)
,
{\displaystyle {\frac {dp}{d\tau }}=mc{\frac {d}{d\tau }}(\Lambda \Lambda ^{\dagger })=mc({\frac {d\Lambda }{d\tau }}\Lambda ^{\dagger }+\Lambda {\frac {d\Lambda ^{\dagger }}{d\tau }}),}
using the definition of the spacetime rotation biparavector
m
c
1
2
(
Ω
Λ
Λ
†
+
Λ
Λ
†
Ω
†
)
=
m
c
⟨
Λ
Ω
⟩
ℜ
,
{\displaystyle mc{\frac {1}{2}}(\Omega \Lambda \Lambda ^{\dagger }+\Lambda \Lambda ^{\dagger }\Omega ^{\dagger })=mc\langle \Lambda \Omega \rangle _{\Re {}},}
that finally gives the dynamic equation
d
p
d
τ
=
m
c
⟨
Ω
Λ
⟩
ℜ
,
{\displaystyle {\frac {dp}{d\tau }}=mc\langle \Omega \Lambda \rangle _{\Re {}},}
where the role of the force is now played by the spacetime rotation biparavector.
Once the spacetime rotation is identified with the electromagnetic force,
the result is the spinorial form of the Lorentz force.
d
Λ
d
τ
=
e
2
m
c
F
Λ
{\displaystyle {\frac {d\Lambda }{d\tau }}={\frac {e}{2mc}}F\Lambda }
Constant Electric Field
edit
The eigenspinor form of the Lorentz force can be solved analytically in many interesting cases. One of the simplest is the case of the constant electric field. If the electric field is directed along the z direction, the equation to solve is
d
Λ
d
τ
=
e
2
m
c
E
e
3
Λ
{\displaystyle {\frac {d\Lambda }{d\tau }}={\frac {e}{2mc}}E\mathbf {e} _{3}\Lambda }
and the solution is
Λ
=
e
e
2
m
c
E
e
3
τ
Λ
0
.
{\displaystyle \Lambda =e^{{\frac {e}{2mc}}E\mathbf {e} _{3}\tau }\Lambda _{0}.}
The proper velocity is calculated as
u
=
e
e
2
m
c
E
e
3
τ
u
0
e
e
2
m
c
E
e
3
τ
.
{\displaystyle u=e^{{\frac {e}{2mc}}E\mathbf {e} _{3}\tau }u_{0}e^{{\frac {e}{2mc}}E\mathbf {e} _{3}\tau }.}
If the particle starts at rest, the proper velocity is simply
u
=
e
e
m
c
E
e
3
τ
.
{\displaystyle u=e^{{\frac {e}{mc}}E\mathbf {e} _{3}\tau }.}
An additional integration gives the spacetime position
x
=
x
0
+
m
c
2
e
E
e
3
(
e
e
m
c
E
e
3
τ
−
1
)
{\displaystyle x=x_{0}+{\frac {mc^{2}}{eE}}\mathbf {e} _{3}(e^{{\frac {e}{mc}}E\mathbf {e} _{3}\tau }-1)}
