A Lorentz transformation is a linear transformation that maintains the length of paravectors. Lorentz transformations include rotations and boosts as proper Lorentz transformations and reflections and non-orthochronous transformations among the improper Lorentz transformations.
A proper Lorentz transformation can be written in spinorial form as
p
→
p
′
=
L
p
L
†
,
{\displaystyle p\rightarrow p^{\prime }=LpL^{\dagger },}
where the spinor
L
{\displaystyle L}
L
L
¯
=
1
{\displaystyle L{\bar {L}}=1}
In
C
l
3
{\displaystyle Cl_{3}}
L
{\displaystyle L}
W
{\displaystyle W}
L
=
e
W
{\displaystyle L_{{}_{}}=e^{W}}
If the biparavector
W
{\displaystyle W}
C
l
3
{\displaystyle Cl_{3}}
R
=
e
−
i
1
2
θ
{\displaystyle R=e^{-i{\frac {1}{2}}{\boldsymbol {\theta }}}}
for example, the following expression represents a rotor that applies a rotation angle
θ
{\displaystyle \theta }
e
3
{\displaystyle \mathbf {e} _{3}}
R
=
e
−
θ
2
e
12
=
e
−
i
θ
2
e
3
,
{\displaystyle R=e^{-{\frac {\theta }{2}}\mathbf {e} _{12}}=e^{-i{\frac {\theta }{2}}\mathbf {e} _{3}},}
applying this rotor to the unit vector along
e
1
{\displaystyle \mathbf {e} _{1}}
e
1
→
e
−
i
θ
2
e
3
e
1
e
i
θ
2
e
3
=
e
1
e
i
θ
2
e
3
e
i
θ
2
e
3
=
e
1
e
i
θ
e
3
=
e
1
(
cos
(
θ
)
+
i
e
3
sin
(
θ
)
)
=
e
1
cos
(
θ
)
+
e
2
sin
(
θ
)
{\displaystyle \mathbf {e} _{1}\rightarrow e^{-i{\frac {\theta }{2}}\mathbf {e} _{3}}\mathbf {e} _{1}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}=\mathbf {e} _{1}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}e^{i{\frac {\theta }{2}}\mathbf {e} _{3}}=\mathbf {e} _{1}e^{i\theta \mathbf {e} _{3}}=\mathbf {e} _{1}(\cos(\theta )+i\mathbf {e} _{3}\sin(\theta ))=\mathbf {e} _{1}\cos(\theta )+\mathbf {e} _{2}\sin(\theta )}
The rotor
R
{\displaystyle R}
Unimodular:
R
R
¯
=
1
{\displaystyle R{\bar {R}}=1}
Unitary:
R
R
†
=
1
{\displaystyle RR^{\dagger }=1}
In the case of rotors, the bar conjugation and the reversion have the same effect
R
¯
=
R
†
.
{\displaystyle {\bar {R}}=R^{\dagger }.}
If the biparavector
W
{\displaystyle W}
R
=
e
1
2
η
{\displaystyle R=e^{{\frac {1}{2}}{\boldsymbol {\eta }}}}
for example, the following expression represents a boost along the
e
3
{\displaystyle \mathbf {e} _{3}}
B
=
e
1
2
η
e
3
,
{\displaystyle B=e^{{\frac {1}{2}}\eta \,\mathbf {e} _{3}},}
where the real scalar parameter
η
{\displaystyle \eta }
The boost
B
{\displaystyle B}
Unimodular:
B
B
¯
=
1
{\displaystyle B{\bar {B}}=1}
Real:
B
†
=
B
{\displaystyle B^{\dagger }=B}
The Lorentz transformation as a composition of a rotation and a boost [ edit | edit source ] In general, the spinor of the proper Lorentz transformation can be written
as the product of a boost and a rotor
L
=
B
R
{\displaystyle L_{{}_{}}=BR}
The boost factor can be extracted as
B
=
L
L
†
{\displaystyle B={\sqrt {LL^{\dagger }}}}
and the rotor is obtained from the even grades of
L
{\displaystyle L}
R
=
L
+
L
¯
†
2
⟨
B
⟩
S
{\displaystyle R={\frac {L+{\bar {L}}^{\dagger }}{2\langle B\rangle _{S}}}}
Boost in terms of the required proper velocity [ edit | edit source ] The proper velocity of a particle at rest is equal to one
u
r
=
1
{\displaystyle u_{r_{}}=1}
Any proper velocity, at least instantaneously, can be obtained from an active Lorentz transformation from the particle at rest, such that
u
=
L
u
r
L
†
,
{\displaystyle u=Lu_{r_{}}L^{\dagger },}
that can be written as
u
=
L
L
†
=
B
R
(
B
R
)
†
=
B
R
R
†
B
†
=
B
B
=
B
2
,
{\displaystyle u=LL^{\dagger }=BR(BR)^{\dagger }=BRR^{\dagger }B^{\dagger }=BB=B^{2},}
so that
B
=
u
=
1
+
u
2
(
1
+
⟨
u
⟩
S
)
,
{\displaystyle B={\sqrt {u}}={\frac {1+u}{\sqrt {2(1+\langle u\rangle _{S})}}},}
where the explicit formula of the square root for a unit length paravector was used.
The proper velocity is the square of the boost
u
=
B
2
,
{\displaystyle u=B^{2^{}},}
so that
γ
(
1
+
v
c
)
=
e
η
,
{\displaystyle \gamma (1+{\frac {\mathbf {v} }{c}})=e^{\boldsymbol {\eta }},}
rewriting the rapidity in terms of the product of its magnitude and respective
unit vector
η
=
η
η
^
{\displaystyle {\boldsymbol {\eta }}=\eta {\hat {\boldsymbol {\eta }}}}
the exponential can be expanded as
γ
+
γ
v
c
=
cosh
(
η
)
+
η
^
sinh
(
η
)
,
{\displaystyle \gamma +\gamma {\frac {\mathbf {v} }{c}}=\cosh(\eta )+{\hat {\boldsymbol {\eta }}}\sinh(\eta ),}
so that
γ
=
cosh
η
{\displaystyle \gamma _{{}_{}}=\cosh {\eta }}
and
γ
v
c
=
η
^
sinh
(
η
)
,
{\displaystyle \gamma {\frac {\mathbf {v} }{c}}={\hat {\boldsymbol {\eta }}}\sinh(\eta ),}
where we see that in the non-relativistic limit the rapidity becomes the velocity divided by the speed of light
v
c
=
η
^
η
{\displaystyle {\frac {\mathbf {v} }{c}}={\hat {\boldsymbol {\eta }}}\eta }
Lorentz transformation applied to biparavectors [ edit | edit source ] The Lorentz transformation applied to biparavector has a different form from the Lorentz transformation applied to paravectors. Considering a general biparavector written in terms of paravectors
⟨
u
v
¯
⟩
V
→
⟨
u
′
v
¯
′
⟩
V
{\displaystyle \langle u{\bar {v}}\rangle _{V}\rightarrow \langle u^{\prime }{\bar {v}}^{\prime }\rangle _{V}}
applying the Lorentz transformation to the component paravectors
⟨
u
′
v
¯
′
⟩
V
=
⟨
L
u
L
†
L
v
L
†
¯
⟩
V
=
⟨
L
u
L
†
L
¯
†
v
¯
L
¯
⟩
V
=
⟨
L
u
v
¯
L
¯
⟩
V
=
L
⟨
u
v
¯
⟩
V
L
¯
,
{\displaystyle \langle u^{\prime }{\bar {v}}^{\prime }\rangle _{V}=\langle LuL^{\dagger }\,\,{\overline {LvL^{\dagger }}}\rangle _{V}=\langle LuL^{\dagger }\,{\bar {L}}^{\dagger }{\bar {v}}{\bar {L}}\rangle _{V}=\langle Lu{\bar {v}}{\bar {L}}\rangle _{V}=L\langle u{\bar {v}}\rangle _{V}{\bar {L}},}
so that if
F
{\displaystyle F}
F
→
F
′
=
L
F
L
¯
{\displaystyle F\rightarrow F^{\prime _{}}=LF{\bar {L}}}
