Much of the material in this section is adapted from [arXiv:math-ph/9905024] .

The unit sphere $\SS^2\subset\RR^3$ is related to the Riemann sphere (the complex plane with a point at infinity added) via stereographic projection from the north pole, which takes the point $(x,y,z)$, with $x^2+y^2+z^2=1$, to the point \begin{equation} w = {x+iy \over 1-z} = {1+z \over x-iy} \label{SterDef} \end{equation} Under this transformation, the north pole is mapped to the point at infinity.

As discussed in detail by Penrose and Rindler [Penrose and Rindler] , we can regard $\SS^2$ as the set of future (or past) null directions, and specifically as the intersection of the future light cone of the origin in 4-dimensional Minkowski space with the hypersurface $t=1$. Other points on a given null ray are obtained by scaling with $t$, and we can extend stereographic projection to a map on the entire light cone via \begin{equation} w = {x+iy \over t-z} = {t+z \over x-iy} \end{equation} with the condition $x^2+y^2+z^2=t^2$.

As discussed in § 12.2, we can further identify $\SS^2$ with the complex projective space $\CP^1$, the space of complex lines in $\CC^2$, which is given by \begin{equation} \CP^1 = \{ [(b,c)] \in \CC^2: (b,c) \sim (\xi b,\xi c) \quad \forall\, 0\ne\xi\in\CC \} \label{CPDef} \end{equation} where the square brackets denote equivalence classes under the equivalence relation $\sim$. Then each $[(b,c)]\in\CP^1$ can be identified with the point $w$ in the complex plane given by \begin{equation} w = {b \over c} \label{ProjDef} \end{equation} which is further identified with a point in $\SS^2$ via (\ref{SterDef}); $[(b,0)]$ is to be identified with the north pole, corresponding to $w=\infty$. Stereographic projection (\ref{SterDef}) can be thought of as a special case of (\ref{ProjDef}) with $b$ or $c$ real.

The Möbius transformations in the complex plane are the complex mappings of the form 1) \begin{equation} w \mapsto {\alpha w+\beta \over \gamma w+\delta} \label{MobDef} \end{equation} where $\alpha \delta - \beta \gamma \ne 0$. It is usually assumed without loss of generality that the complex numbers $\alpha$, $\beta$, $\gamma$, $\delta$ satisfy \begin{equation} \alpha \delta - \beta \gamma = 1 \label{Normalize} \end{equation} Möbius transformations are the most general analytic transformations of the Riemann sphere to itself. Using (\ref{ProjDef}), we can rewrite (\ref{MobDef}) as \begin{equation} {b \over c} \mapsto {\alpha b + \beta c \over \gamma b + \delta c} \label{SpDef} \end{equation} The Möbius transformation (\ref{SpDef}) does not depend on the particular choice of $b$ and $c$ in the equivalence class $[(b,c)]$, which allows us to view the transformation as acting on $\CP^1$.

In § 12.2, we gave an alternate, matrix description of $\CP^1$ as \begin{equation} \CP^1 = \{\XX\in\CC^{2\times2}: \XX^\dagger=\XX, \XX^2 = \XX\} \label{cproj} \end{equation} where $X=vv^\dagger$ and $v=\begin{pmatrix}b\cr c\cr\end{pmatrix}$. From this point of view, $v$ is a spinor, whose square $\XX$ is a null vector. Using these various identifications, we can rewrite a Möbius transformation (\ref{MobDef}) as a map on spinors \begin{equation} v \mapsto M v \end{equation} where \begin{equation} M = \begin{pmatrix} \alpha& \beta\\ \noalign{\smallskip} \gamma& \delta\\ \end{pmatrix} \label{MatDef} \end{equation} and, imposing the condition (\ref{Normalize}), we see that $\det M = 1$. We thus see that Möbius transformations are exactly the same as Lorentz transformations. Note the key role played by associativity, which allows one to multiply numerator and denominator of a Möbius transformation by $c$, thus permitting a reinterpretation as a matrix equation.

But we know how to implement Lorentz transformations over the octonions. In § 9.3, we presented generators for the Lorentz group, originally given by Manogue and Schray [arXiv:hep-th/9302044] , whose components lie in a single complex subalgebra of $\OO$ (which may differ for different generators), and whose determinant is real. As discussed in § 12.5, Such generators also satisfy a compatibility condition, namely \begin{equation} (M v) (M v)^\dagger = M (v v^\dagger) M^\dagger \label{Compat} \end{equation} between the spinor ($v$) and vector ($\XX=vv^\dagger$) representations.

Putting this all together, we will invert the usual derivation that Lorentz transformations are the same as Möbius transformations. Rather, we will define octonionic Möbius transformations in terms of Lorentz transformations, and then show that these transformations can be rewritten in the form (\ref{SpDef}).

Thus, given an octonion $w$, define (generators of) Möbius transformations via (\ref{MobDef}), which we rewrite as \begin{equation} f_M(w) = (\alpha w+\beta) (\gamma w+\delta)^{-1} \label{WTrans} \end{equation} and where the matrix of coefficients $M$ defined by (\ref{MatDef}) is now not only octonionic, but is further required to be one of Manogue & Schray's compatible generators of the Lorentz group.

We would like to be able to construct more general Möbius transformations by nesting. However, it is not at all obvious that iterating (\ref{WTrans}) leads to a (suitably nested) transformation of the same type. We would really like to be able to use (an octonionic version of) (\ref{SpDef}) to define Möbius transformations, as this would make it apparent that iterating Möbius transformations corresponds directly to nesting Lorentz transformations. As previously noted, this construction requires (\ref{SpDef}) to be independent of the particular choice of $b$ and $c$. Remarkably, the octonionic generalization of (\ref{SpDef}) does have this property, as we now show.

Suppose that \begin{equation} w = b c^{-1} \label{ProjDefO} \end{equation} where now $b,c\in\OO$. Letting \begin{equation} v_0 = \begin{pmatrix}w\\ 1\\\end{pmatrix} \end{equation} we have \begin{equation} v = v_0 c \end{equation} and \begin{equation} v v^\dagger = |c|^2 v_0 v_0^\dagger \end{equation} since only two octonionic directions are involved.

We now write \begin{equation} V = Mv = \begin{pmatrix}B\\ C\\\end{pmatrix} = \begin{pmatrix}BC^{-1}\\ 1\\\end{pmatrix} C \end{equation} leading to \begin{equation} V V^\dagger = \begin{pmatrix} |B|^2& B\bar{C}\\ \noalign{\smallskip} C\bar{B}& |C|^2\\ \end{pmatrix} = |C|^2 \begin{pmatrix} {|B|^2\over|C|^2}& BC^{-1}\\ \noalign{\smallskip} \bar{BC^{-1}}& 1\\ \end{pmatrix} \end{equation} and similar relations for $V_0=Mv_0$. Compatibility now leads to \begin{align} V V^\dagger &= (Mv) (Mv)^\dagger \nonumber\\ &= M (v v^\dagger) M^\dagger = |c|^2 M (v_0 v_0^\dagger) M^\dagger \nonumber\\ &= |c|^2 (Mv_0) (Mv_0)^\dagger = |c|^2 V_0 V_0^\dagger \label{MVeq} \end{align} Comparing the offdiagonal entries of (\ref{MVeq}), we obtain \begin{equation} |C|^2 B C^{-1} = |c|^2 |C_0|^2 B_0 C_0^{-1} \end{equation} But direct computation shows that \begin{equation} |C|^2 = |\gamma b + \delta c|^2 = |\gamma w + \delta|^2 |c|^2 = |C_0|^2 |c|^2 \end{equation} provided \begin{equation} \Big\langle [b,c,\gamma] , \delta \Big\rangle = 0 \end{equation} which holds for compatible $M$ since $\gamma$ and $\delta$ lie in the same complex subspace of $\OO$. Finally, by construction we have \begin{equation} f_M(w) = B_0 C_0^{-1} \end{equation} and putting this all together results in \begin{equation} B C^{-1} = f_M(w) \end{equation} or equivalently \begin{equation} f_M(w) = (\alpha w+\beta) (\gamma w+\delta)^{-1} = (\alpha b+\beta c) (\gamma b+\delta c)^{-1} \end{equation} This is the desired result, since $b$ and $c$ were arbitrary (satisfying (\ref{ProjDefO})).

We have shown that the finite octonionic Lorentz transformations in 10 dimensions as given by Manogue & Schray [arXiv:hep-th/9302044] can be used to define octonionic Möbius transformations, thus recovering (and correcting) the earlier results of Dündarer, Gürsey, & Tze [Gürsey and Tze] . However, our approach differs significantly from theirs, as theirs corresponds to using (\ref{MobDef}), while ours uses (\ref{SpDef}). We have thus shown that octonionic Möbius transformations extend to the octonionic projective space $\OP^1$, defined by 2) \begin{equation} \OP^1 = \{ [(b,c)] \in \OO^2: (b,c) \sim \left((bc^{-1})\xi,\xi\right) \quad \forall\, 0\ne\xi\in\OO \} \end{equation} This description could be a key ingredient when attempting to generalize 4-dimensional twistor theory to 10 dimensions. Much recent research in superstrings, supergravity, and M-theory has emphasized the importance of lightlike objects in 10 dimensions. An appropriate octonionic generalization of twistor theory to 10 dimensions might allow powerful twistor techniques to be applied to these other theories.

A key role in our argument is the use of two fundamental properties of the octonionic Lorentz transformations in [arXiv:hep-th/9302044] , namely nesting and compatibility. Our results here support our view that these are essential features of any computation involving octonions. Otherwise, repeated transformations of the form (\ref{MobDef}) are not equivalent to those of the form (\ref{SpDef}), due to the lack of associativity.