I am exploring the behavior of multivectors in 2D geometric algebra, specifically examining the product $\mathbf{u}^\ddagger \mathbf{u}$, where $\mathbf{u}=a+xe_1+ye_2+be_{12}$ and its Clifford conjugate $\mathbf{u}^\ddagger=a-xe_1-ye_2-be_{12}$.
I defined an operator $g\mathbf{u}= I \mathbf{u} I^{-1}$, with $I=e_{12}$, leading to:
$$ I (a+xe_1+ye_2+be_{12})I^{-1}=a-xe_1-ye_2+be_{12} $$
I presumed $g$ is self-adjoint, since $g^\ddagger \mathbf{u} = (-I)\mathbf{u} (-I)^{-1} = I \mathbf{u}I^{-1}$.
However, I'm encountering different results based on the order of operation:
When applying $g$ to the right-most term of $\mathbf{u}^\ddagger (g\mathbf{u})$:
$$ \mathbf{u}^\ddagger (g\mathbf{u}) = (a-xe_1-ye_2-be_{12})(a-xe_1-ye_2+be_{12}) $$
When applying $g$ immediately to the left-most term $\mathbf{u}$:
$$ (g\mathbf{u})^\ddagger \mathbf{u} = (a+xe_1+ye_2-be_{12})(a+xe_1+ye_2+be_{12}) $$
These results aren't aligning as I expected. I am puzzled by this discrepancy. Since $(g\mathbf{u})^\ddagger= \mathbf{u}^\ddagger g^\ddagger$ and $g^\ddagger=g$, I anticipated that all expressions should yield the same result : $(g\mathbf{u})^\ddagger \mathbf{u} = \mathbf{u}^\ddagger g \mathbf{u}$ . Am I misunderstanding something in my approach, or is there a subtlety in the application of the self-adjoint operator in this context? Any insights or corrections to my method would be greatly appreciated.
The scalar product corresponding to the quadratic form $\|u\|^2 = u^{\ddagger}u$ is $(u, v) = \mathrm{re}(u^{\ddagger}v)$, where $\mathrm{re}(a + xe_1 + ye_2 + be_{12}) = a$. Self-adjointness of $g$ means $(u, gv) = (gu, v)$, that is, $\mathrm{re}(u^{\ddagger}(gv)) = \mathrm{re}((gu)^{\ddagger}v)$, not $u^{\ddagger}(gu) = (gu)^{\ddagger}u$