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The geometric algebra form of Maxwell’s equations in free space (or source free isotopic media with group velocity \( c \)) is the multivector equation

\begin{equation}\label{eqn:planewavesMultivector:20}

\lr{ \spacegrad + \inv{c}\PD{t}{} } F(\Bx, t) = 0.

\end{equation}

Here \( F = \BE + I c \BB \) is a multivector with grades 1 and 2 (vector and bivector components). The velocity \( c \) is called the group velocity since \( F \), or its components \( \BE, \BH \) satisfy the wave equation, which can be seen by pre-multiplying with \( \spacegrad – (1/c)\PDi{t}{} \) to find

\begin{equation}\label{eqn:planewavesMultivector:n}

\lr{ \spacegrad^2 – \inv{c^2}\PDSq{t}{} } F(\Bx, t) = 0.

\end{equation}

Let’s look at the frequency domain solution of this equation with a presumed phasor representation

\begin{equation}\label{eqn:planewavesMultivector:40}

F(\Bx, t) = \textrm{Re} \lr{ F(\Bk) e^{-j \Bk \cdot \Bx + j \omega t} },

\end{equation}

where \( j \) is a scalar imaginary, not necessarily with any geometric interpretation.

Maxwell’s equation reduces to just

\begin{equation}\label{eqn:planewavesMultivector:60}

0

=

-j \lr{ \Bk – \frac{\omega}{c} } F(\Bk).

\end{equation}

If \( F(\Bk) \) has a left multivector factor

\begin{equation}\label{eqn:planewavesMultivector:80}

F(\Bk) =

\lr{ \Bk + \frac{\omega}{c} } \tilde{F},

\end{equation}

where \( \tilde{F} \) is a multivector to be determined, then

\begin{equation}\label{eqn:planewavesMultivector:100}

\begin{aligned}

\lr{ \Bk – \frac{\omega}{c} }

F(\Bk)

&=

\lr{ \Bk – \frac{\omega}{c} }

\lr{ \Bk + \frac{\omega}{c} } \tilde{F} \\

&=

\lr{ \Bk^2 – \lr{\frac{\omega}{c}}^2 } \tilde{F},

\end{aligned}

\end{equation}

which is zero if \( \Norm{\Bk} = \ifrac{\omega}{c} \).

Let \( \kcap = \ifrac{\Bk}{\Norm{\Bk}} \), and \( \Norm{\Bk} \tilde{F} = F_0 + F_1 + F_2 + F_3 \), where \( F_0, F_1, F_2, \) and \( F_3 \) are respectively have grades 0,1,2,3. Then

\begin{equation}\label{eqn:planewavesMultivector:120}

\begin{aligned}

F(\Bk)

&= \lr{ 1 + \kcap } \lr{ F_0 + F_1 + F_2 + F_3 } \\

&=

F_0 + F_1 + F_2 + F_3

+

\kcap F_0 + \kcap F_1 + \kcap F_2 + \kcap F_3 \\

&=

F_0 + F_1 + F_2 + F_3

+

\kcap F_0 + \kcap \cdot F_1 + \kcap \cdot F_2 + \kcap \cdot F_3

+

\kcap \wedge F_1 + \kcap \wedge F_2 \\

&=

\lr{

F_0 + \kcap \cdot F_1

}

+

\lr{

F_1 + \kcap F_0 + \kcap \cdot F_2

}

+

\lr{

F_2 + \kcap \cdot F_3 + \kcap \wedge F_1

}

+

\lr{

F_3 + \kcap \wedge F_2

}.

\end{aligned}

\end{equation}

Since the field \( F \) has only vector and bivector grades, the grades zero and three components of the expansion above must be zero, or

\begin{equation}\label{eqn:planewavesMultivector:140}

\begin{aligned}

F_0 &= – \kcap \cdot F_1 \\

F_3 &= – \kcap \wedge F_2,

\end{aligned}

\end{equation}

so

\begin{equation}\label{eqn:planewavesMultivector:160}

\begin{aligned}

F(\Bk)

&=

\lr{ 1 + \kcap } \lr{

F_1 – \kcap \cdot F_1 +

F_2 – \kcap \wedge F_2

} \\

&=

\lr{ 1 + \kcap } \lr{

F_1 – \kcap F_1 + \kcap \wedge F_1 +

F_2 – \kcap F_2 + \kcap \cdot F_2

}.

\end{aligned}

\end{equation}

The multivector \( 1 + \kcap \) has the projective property of gobbling any leading factors of \( \kcap \)

\begin{equation}\label{eqn:planewavesMultivector:180}

\begin{aligned}

(1 + \kcap)\kcap

&= \kcap + 1 \\

&= 1 + \kcap,

\end{aligned}

\end{equation}

so for \( F_i \in F_1, F_2 \)

\begin{equation}\label{eqn:planewavesMultivector:200}

(1 + \kcap) ( F_i – \kcap F_i )

=

(1 + \kcap) ( F_i – F_i )

= 0,

\end{equation}

leaving

\begin{equation}\label{eqn:planewavesMultivector:220}

F(\Bk)

=

\lr{ 1 + \kcap } \lr{

\kcap \cdot F_2 +

\kcap \wedge F_1

}.

\end{equation}

For \( \kcap \cdot F_2 \) to be non-zero \( F_2 \) must be a bivector that lies in a plane containing \( \kcap \), and \( \kcap \cdot F_2 \) is a vector in that plane that is perpendicular to \( \kcap \). On the other hand \( \kcap \wedge F_1 \) is non-zero only if \( F_1 \) has a non-zero component that does not lie in along the \( \kcap \) direction, but \( \kcap \wedge F_1 \), like \( F_2 \) describes a plane that containing \( \kcap \). This means that having both bivector and vector free variables \( F_2 \) and \( F_1 \) provide more degrees of freedom than required. For example, if \( \BE \) is any vector, and \( F_2 = \kcap \wedge \BE \), then

\begin{equation}\label{eqn:planewavesMultivector:240}

\begin{aligned}

\lr{ 1 + \kcap }

\kcap \cdot F_2

&=

\lr{ 1 + \kcap }

\kcap \cdot \lr{ \kcap \wedge \BE } \\

&=

\lr{ 1 + \kcap }

\lr{

\BE

–

\kcap \lr{ \kcap \cdot \BE }

} \\

&=

\lr{ 1 + \kcap }

\kcap \lr{ \kcap \wedge \BE } \\

&=

\lr{ 1 + \kcap }

\kcap \wedge \BE,

\end{aligned}

\end{equation}

which has the form \( \lr{ 1 + \kcap } \lr{ \kcap \wedge F_1 } \), so the solution of the free space Maxwell’s equation can be written

\begin{equation}\label{eqn:planewavesMultivector:260}

\boxed{

F(\Bx, t)

=

\textrm{Re} \lr{

\lr{ 1 + \kcap }

\BE\,

e^{-j \Bk \cdot \Bx + j \omega t}

}

,

}

\end{equation}

where \( \BE \) is any vector for which \( \BE \cdot \Bk = 0 \).