## Complex-pair representation of GA(2,0) multivectors

We found previously that a complex pair representation of a GA(2,0) multivector had a compact geometric product realization. Now that we know the answer, let’s work backwards from that representation to verify that everything matches our expectations.

We are representing a multivector of the form
\label{eqn:bicomplexCl20:20}
M = a + b \Be_1 \Be_2 + x \Be_1 + y \Be_2,

as the pair of complex numbers
\label{eqn:bicomplexCl20:40}
M \sim \lr{ a + i b, x + i y }.

Given a pair of multivectors with this complex representation
\label{eqn:bicomplexCl20:60}
\begin{aligned}
M &= \lr{ z_1, z_2 } \\
N &= \lr{ q_1, q_2 },
\end{aligned}

we found that our geometric product representation was
\label{eqn:bicomplexCl20:80}
M N \sim
\lr{ z_1 q_1 + z_2^\conj q_2, z_2 q_1 + z_1^\conj q_2 }.

Our task is now to verify that this is correct. Let’s set
\label{eqn:bicomplexCl20:100}
\begin{aligned}
z_1 &= a + i b \\
q_1 &= a’ + i b’ \\
z_2 &= x + i y \\
q_2 &= x’ + i y’,
\end{aligned}

and proceed with an expansion of the even grade components
\label{eqn:bicomplexCl20:120}
\begin{aligned}
z_1 q_1 + z_2^\conj q_2
&=
\lr{ a + i b } \lr{ a’ + i b’ }
+
\lr{ x – i y } \lr{ x’ + i y’ } \\
&=
a a’ – b b’ + x x’ + y y’
+ i \lr{ b a’ + a b’ + x y’ – y x’ } \\
&=
x x’ + y y’ + i \lr{ x y’ – y x’ } + \quad a a’ – b b’ + i \lr{ b a’ + a b’ }.
\end{aligned}

The first terms is clearly the geometric product of two vectors
\label{eqn:bicomplexCl20:140}
\lr{ x \Be_1 + y \Be_2 } \lr{ x’ \Be_1 + y’ \Be_2 }
=
x x’ + y y’ + i \lr{ x y’ – y x’ },

and we are able to verify that the second parts can be factored too
\label{eqn:bicomplexCl20:160}
\lr{ a + b i } \lr{ a’ + b’ i }
=
a a’ – b b’ + i \lr{ b a’ + a b’ }.

This leaves us with
\label{eqn:bicomplexCl20:180}

as expected. This part of our representation checks out.

Now, let’s look at the vector component of our representation. First note that to convert from our complex representation of our vector $$z = x + i y$$ to the standard basis representation of our vector, we need only multiply by $$\Be_1$$ on the left, for example:
\label{eqn:bicomplexCl20:220}
\Be_1 \lr{ x + i y } = \Be_1 x + \Be_1 \Be_1 \Be_2 y = \Be_1 x + \Be_2 y.

So, for the vector component of our assumed product representation, we have
\label{eqn:bicomplexCl20:200}
\begin{aligned}
\Be_1 \lr{ z_2 q_1 + z_1^\conj q_2 }
&=
\Be_1 \lr{ x + i y } \lr{ a’ + i b’ }
+
\Be_1 \lr{ a – i b } \lr{ x’ + i y’ } \\
&=
\Be_1 \lr{ x + i y } \lr{ a’ + i b’ }
+
\lr{ a + i b } \Be_1 \lr{ x’ + i y’ } \\
&=
\end{aligned}

as expected.

Our complex-pair realization of the geometric product checks out.

## Motivation.

Suppose that we want to represent GA(2,0) (Euclidean) multivectors as a pair of complex numbers, with a structure like
\label{eqn:bicomplexGA20:20}
M = (m_1, m_2),

where
\label{eqn:bicomplexGA20:40}
\begin{aligned}
\end{aligned}

Specifically
\label{eqn:bicomplexGA20:60}
\begin{aligned}
\gpgrade{M}{1} \cdot \Be_1 &= \textrm{Re}(m_2) \\
\gpgrade{M}{1} \cdot \Be_2 &= \textrm{Im}(m_2) \\
\end{aligned}

where $$i \sim \Be_1 \Be_2$$.

## Multivector product representation.

Let’s figure out how we can represent the various GA products, starting with the geometric product. Let
\label{eqn:bicomplexGA20:80}
\begin{aligned}
M &= \gpgrade{M}{0,2} + \gpgrade{M}{1} = (m_1, m_2) = (m_{11} + m_{12} i, m_{21} + m_{22} i) \\
N &= \gpgrade{N}{0,2} + \gpgrade{N}{1} = (n_1, n_2) = (n_{11} + n_{12} i, n_{21} + n_{22} i),
\end{aligned}

so
\label{eqn:bicomplexGA20:200}
\begin{aligned}
M N
\end{aligned}

The first two terms have only even grades, and the second two terms are vectors. The complete representation of the even grade components of this multivector product is
\label{eqn:bicomplexGA20:240}
\gpgrade{M N}{0,2} \sim \lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), 0 },

or
\label{eqn:bicomplexGA20:260}
\begin{aligned}
\gpgrade{M N}{0} &= \textrm{Re}\lr{ m_1 n_1 + m_2 n_2^\conj } \\
\gpgrade{M N}{2} i^{-1} &= \textrm{Im}\lr{ m_1 n_1 – m_2 n_2^\conj }.
\end{aligned}

For the vector components we have
\label{eqn:bicomplexGA20:280}
\begin{aligned}
&=
+
\end{aligned}

For these,
\label{eqn:bicomplexGA20:300}
\begin{aligned}
&= \lr{ m_{21} \Be_1 + m_{22} \Be_2 } \Be_{12}
&= – m_{22} \Be_1 + m_{21} \Be_2,
\end{aligned}

and
\label{eqn:bicomplexGA20:320}
\begin{aligned}
&= \Be_{12} \lr{ n_{21} \Be_1 + n_{22} \Be_2 }
&=
n_{22} \Be_1 – n_{21} \Be_2.
\end{aligned}

Comparing to
\label{eqn:bicomplexGA20:340}
i (a + i b)
= -b + i a,

we see that
\label{eqn:bicomplexGA20:360}
\sim
\lr{ 0, n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

If we want the vector coordinates, those are
\label{eqn:bicomplexGA20:380}
\begin{aligned}
\gpgrade{M N}{1} \cdot \Be_1 &= \textrm{Re} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 } \\
\gpgrade{M N}{1} \cdot \Be_2 &= \textrm{Im} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.
\end{aligned}

## Summary.

\label{eqn:bicomplexGA20:n}
M N \sim
\lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

A sample Mathematica implementation is available, as well as an example notebook (that also doubles as a test case.)

## Clarification.

I skipped a step above, showing the correspondances to the dot and wedge product.

Let $$z = a + bi$$, and $$w = c + di$$. Then:
\label{eqn:bicomplexGA20:420}
\begin{aligned}
z w^\conj
&=
\lr{ a + b i } \lr{ c – d i } \\
&= a c + b d -i \lr{ a d – b c }.
\end{aligned}

Compare that to the geometric product of two vectors $$\Bx = a \Be_1 + b \Be_2$$, and $$\By = c \Be_1 + d \Be_2$$, where we have
\label{eqn:bicomplexGA20:440}
\begin{aligned}
\Bx \By &= \Bx \cdot \By + \Bx \wedge \By \\
&= \lr{ a \Be_1 + b \Be_2 } \lr{ c \Be_1 + d \Be_2 } \\
&= a c + b d + \Be_1 \Be_2 \lr{ a d – b c }.
\end{aligned}

So we have
\label{eqn:bicomplexGA20:460}
\begin{aligned}
a b + cd &= \Bx \cdot \By = \textrm{Re} \lr{ z w^\conj } \\
a d – b c &= \lr{ \Bx \wedge \By } \Be_{12}^{-1} = – \textrm{Im} \lr{ z w^\conj }.
\end{aligned}

We see that $$\lr{z w^\conj}^\conj = z^\conj w$$ can be used as a representation of the geometric product of two vectors (setting $$i = \Be_1 \Be_2$$ as usual.)

## Another simplification.

We have sums of the form
\label{eqn:bicomplexGA20:480}
\textrm{Re}(z) w \pm \textrm{Im}(z) i w

above. Let’s see if those can be simplified. For the positive case we have
\label{eqn:bicomplexGA20:500}
\begin{aligned}
\textrm{Re}(z) w + \textrm{Im}(z) i w
&=
\inv{2} \lr{ z + z^\conj } w + \inv{2} \lr{ z – z^\conj } w \\
&=
z w,
\end{aligned}

and for the negative case, we have
\label{eqn:bicomplexGA20:520}
\begin{aligned}
\textrm{Re}(z) w – \textrm{Im}(z) i w
&=
\inv{2} \lr{ z + z^\conj } w – \inv{2} \lr{ z – z^\conj } w \\
&=
z^\conj w.
\end{aligned}

This, with the vector-vector product simplification above, means that we can represent the full multivector product in this representation as just
\label{eqn:bicomplexGA20:540}
M N \sim
\lr{ m_1 n_1 + m_2^\conj n_2, m_2 n_1 + m_1^\conj n_2 }.