## Motivation.

The notation I prefer for relativistic geometric algebra uses Hestenes’ space time algebra (STA) [2], where the basis is a four dimensional space $$\setlr{ \gamma_\mu }$$, subject to Dirac matrix like relations $$\gamma_\mu \cdot \gamma_\nu = \eta_{\mu \nu}$$.

In this formalism a four vector is just the sum of the products of coordinates and basis vectors, for example, using summation convention

\label{eqn:boostToParavector:160}
x = x^\mu \gamma_\mu.

The invariant for a four-vector in STA is just the square of that vector

\label{eqn:boostToParavector:180}
\begin{aligned}
x^2
&= (x^\mu \gamma_\mu) \cdot (x^\nu \gamma_\nu) \\
&= \sum_\mu (x^\mu)^2 (\gamma_\mu)^2 \\
&= (x^0)^2 – \sum_{k = 1}^3 (x^k)^2 \\
&= (ct)^2 – \Bx^2.
\end{aligned}

Recall that a four-vector is time-like if this squared-length is positive, spacelike if negative, and light-like when zero.

Time-like projections are possible by dotting with the “lab-frame” time like basis vector $$\gamma_0$$

\label{eqn:boostToParavector:200}
ct = x \cdot \gamma_0 = x^0,

and space-like projections are wedges with the same

\label{eqn:boostToParavector:220}
\Bx = x \cdot \gamma_0 = x^k \sigma_k,

where sums over Latin indexes $$k \in \setlr{1,2,3}$$ are implied, and where the elements $$\sigma_k$$

\label{eqn:boostToParavector:80}
\sigma_k = \gamma_k \gamma_0.

which are bivectors in STA, can be viewed as an Euclidean vector basis $$\setlr{ \sigma_k }$$.

Rotations in STA involve exponentials of space like bivectors $$\theta = a_{ij} \gamma_i \wedge \gamma_j$$

\label{eqn:boostToParavector:240}
x’ = e^{ \theta/2 } x e^{ -\theta/2 }.

Boosts, on the other hand, have exactly the same form, but the exponentials are with respect to space-time bivectors arguments, such as $$\theta = a \wedge \gamma_0$$, where $$a$$ is any four-vector.

Observe that both boosts and rotations necessarily conserve the space-time length of a four vector (or any multivector with a scalar square).

\label{eqn:boostToParavector:260}
\begin{aligned}
\lr{x’}^2
&=
\lr{ e^{ \theta/2 } x e^{ -\theta/2 } } \lr{ e^{ \theta/2 } x e^{ -\theta/2 } } \\
&=
e^{ \theta/2 } x \lr{ e^{ -\theta/2 } e^{ \theta/2 } } x e^{ -\theta/2 } \\
&=
e^{ \theta/2 } x^2 e^{ -\theta/2 } \\
&=
x^2 e^{ \theta/2 } e^{ -\theta/2 } \\
&=
x^2.
\end{aligned}

## Paravectors.

Paravectors, as used by Baylis [1], represent four-vectors using a Euclidean multivector basis $$\setlr{ \Be_\mu }$$, where $$\Be_0 = 1$$. The conversion between STA and paravector notation requires only multiplication with the timelike basis vector for the lab frame $$\gamma_0$$

\label{eqn:boostToParavector:40}
\begin{aligned}
X
&= x \gamma_0 \\
&= \lr{ x^0 \gamma_0 + x^k \gamma_k } \gamma_0 \\
&= x^0 + x^k \gamma_k \gamma_0 \\
&= x^0 + \Bx \\
&= c t + \Bx,
\end{aligned}

We need a different structure for the invariant length in paravector form. That invariant length is
\label{eqn:boostToParavector:280}
\begin{aligned}
x^2
&=
\lr{ \lr{ ct + \Bx } \gamma_0 }
\lr{ \lr{ ct + \Bx } \gamma_0 } \\
&=
\lr{ \lr{ ct + \Bx } \gamma_0 }
\lr{ \gamma_0 \lr{ ct – \Bx } } \\
&=
\lr{ ct + \Bx }
\lr{ ct – \Bx }.
\end{aligned}

Baylis introduces an involution operator $$\overline{{M}}$$ which toggles the sign of any vector or bivector grades of a multivector. For example, if $$M = a + \Ba + I \Bb + I c$$, where $$a,c \in \mathbb{R}$$ and $$\Ba, \Bb \in \mathbb{R}^3$$ is a multivector with all grades $$0,1,2,3$$, then the involution of $$M$$ is

\label{eqn:boostToParavector:300}
\overline{{M}} = a – \Ba – I \Bb + I c.

Utilizing this operator, the invariant length for a paravector $$X$$ is $$X \overline{{X}}$$.

Let’s consider how boosts and rotations can be expressed in the paravector form. The half angle operator for a boost along the spacelike $$\Bv = v \vcap$$ direction has the form

\label{eqn:boostToParavector:120}
L = e^{ -\vcap \phi/2 },

\label{eqn:boostToParavector:140}
\begin{aligned}
X’
&=
c t’ + \Bx’ \\
&=
x’ \gamma_0 \\
&=
L x L^\dagger \\
&=
e^{ -\vcap \phi/2 } x^\mu \gamma_\mu
e^{ \vcap \phi/2 } \gamma_0 \\
&=
e^{ -\vcap \phi/2 } x^\mu \gamma_\mu \gamma_0
e^{ -\vcap \phi/2 } \\
&=
e^{ -\vcap \phi/2 } \lr{ x^0 + \Bx } e^{ -\vcap \phi/2 } \\
&=
L X L.
\end{aligned}

Because the involution operator toggles the sign of vector grades, it is easy to see that the required invariance is maintained

\label{eqn:boostToParavector:320}
\begin{aligned}
X’ \overline{{X’}}
&=
L X L
\overline{{ L X L }} \\
&=
L X L
\overline{{ L }} \overline{{ X }} \overline{{ L }} \\
&=
L X \overline{{ X }} \overline{{ L }} \\
&=
X \overline{{ X }} L \overline{{ L }} \\
&=
X \overline{{ X }}.
\end{aligned}

Let’s explicitly expand the transformation of \ref{eqn:boostToParavector:140}, so we can relate the rapidity angle $$\phi$$ to the magnitude of the velocity. This is most easily done by splitting the spacelike component $$\Bx$$ of the four vector into its projective and rejective components

\label{eqn:boostToParavector:340}
\begin{aligned}
\Bx
&= \vcap \vcap \Bx \\
&= \vcap \lr{ \vcap \cdot \Bx + \vcap \wedge \Bx } \\
&= \vcap \lr{ \vcap \cdot \Bx } + \vcap \lr{ \vcap \wedge \Bx } \\
&= \Bx_\parallel + \Bx_\perp.
\end{aligned}

The exponential

\label{eqn:boostToParavector:360}
e^{-\vcap \phi/2}
=
\cosh\lr{ \phi/2 }
– \vcap \sinh\lr{ \phi/2 },

commutes with any scalar grades and with $$\Bx_\parallel$$, but anticommutes with $$\Bx_\perp$$, so

\label{eqn:boostToParavector:380}
\begin{aligned}
X’
&=
\lr{ c t + \Bx_\parallel } e^{ -\vcap \phi/2 } e^{ -\vcap \phi/2 }
+
\Bx_\perp e^{ \vcap \phi/2 } e^{ -\vcap \phi/2 } \\
&=
\lr{ c t + \Bx_\parallel } e^{ -\vcap \phi }
+
\Bx_\perp \\
&=
\lr{ c t + \vcap \lr{ \vcap \cdot \Bx } } \lr{ \cosh \phi – \vcap \sinh \phi }
+
\Bx_\perp \\
&=
\Bx_\perp
+
\lr{ c t \cosh\phi – \lr{ \vcap \cdot \Bx} \sinh \phi }
+
\vcap \lr{ \lr{ \vcap \cdot \Bx } \cosh\phi – c t \sinh \phi } \\
&=
\Bx_\perp
+
\cosh\phi \lr{ c t – \lr{ \vcap \cdot \Bx} \tanh \phi }
+
\vcap \cosh\phi \lr{ \vcap \cdot \Bx – c t \tanh \phi }.
\end{aligned}

Employing the argument from [3],
we want $$\phi$$ defined so that this has structure of a Galilean transformation in the limit where $$\phi \rightarrow 0$$. This means we equate

\label{eqn:boostToParavector:400}
\tanh \phi = \frac{v}{c},

so that for small $$\phi$$

\label{eqn:boostToParavector:420}
\Bx’ = \Bx – \Bv t.

We can solving for $$\sinh^2 \phi$$ and $$\cosh^2 \phi$$ in terms of $$v/c$$ using

\label{eqn:boostToParavector:440}
\tanh^2 \phi
= \frac{v^2}{c^2}
=
\frac{ \sinh^2 \phi }{1 + \sinh^2 \phi}
=
\frac{ \cosh^2 \phi – 1 }{\cosh^2 \phi}.

which after picking the positive root required for Galilean equivalence gives
\label{eqn:boostToParavector:460}
\begin{aligned}
\cosh \phi &= \frac{1}{\sqrt{1 – (\Bv/c)^2}} \equiv \gamma \\
\sinh \phi &= \frac{v/c}{\sqrt{1 – (\Bv/c)^2}} = \gamma v/c.
\end{aligned}

The Lorentz boost, written out in full is

\label{eqn:boostToParavector:480}
ct’ + \Bx’
=
\Bx_\perp
+
\gamma \lr{ c t – \frac{\Bv}{c} \cdot \Bx }
+
\gamma \lr{ \vcap \lr{ \vcap \cdot \Bx } – \Bv t }
.

Authors like Chappelle, et al., that also use paravectors [4], specify the form of the Lorentz transformation for the electromagnetic field, but for that transformation reversion is used instead of involution.
I plan to explore that in a later post, starting from the STA formalism that I already understand, and see if I can make sense
of the underlying rationale.

# References

[1] William Baylis. Electrodynamics: a modern geometric approach, volume 17. Springer Science \& Business Media, 2004.

[2] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

[3] L. Landau and E. Lifshitz. The Classical theory of fields. Addison-Wesley, 1951.

[4] James M Chappell, Samuel P Drake, Cameron L Seidel, Lachlan J Gunn, and Derek Abbott. Geometric algebra for electrical and electronic engineers. Proceedings of the IEEE, 102 0(9), 2014.