# Notes on Spinors and Polyforms II: Quaternions and Octonions

Niren Bhoja and Kirill Krasnov

School of Mathematical Sciences, University of Nottingham, NG7 2RD, UK

###### Abstract

Pauli matrices are $2\times 2$ tracefree matrices with a real diagonal and
complex (complex-conjugate) off-diagonal entries. They generate the Clifford
algebra ${\rm Cl}(3)$. They can be generalised by replacing the off-diagonal
complex number by one taking values in either quaternions or octonions (or
their split versions). These quaternionic and octonionic generalisations
generate well-known models of ${\rm Cl}(5)$ and ${\rm Cl}(9)$ respectively.
The main aim of the paper is to explicitly relate these models to the models
arising via the creation/annihilation operator construction. We describe in
details the models related to quaternions and octonions, as well as to the
split quaternions and octonions. In particular, we record the description of
the possible types of Weyl spinors of ${\rm Spin}(4,4)$, which does not seem
to have appeared in the literature.

##  1 Introduction

In an accompanying paper [1] we presented the construction of a set of
creation/annihilation operator models for the Clifford algebra ${\rm
Cl}(r,s),r+s=2m$. For some special values of $(r,s)$ the Clifford algebras
admit a description in terms of quaternions and octonions (or their split
versions). When such a description is available, it gives a formalism that is
preferred in view of its compactness and computational power. However, some
geometric aspects of spinors (for example the geometry related to pure
spinors) are more manifest in the creation/annihilation operator formalism.
For this reason, it is desirable to provide an explicit dictionary between the
two formalisms, when they both exist. This will translate the geometric
aspects that are manifest in the creation/annihilation operator picture to the
quaternionic/octonionic picture. This is the main aim of the present paper.
Thus, we provide an explicit dictionary between the description of ${\rm
Cl}(4),{\rm Cl}(2,2),{\rm Cl}(8)$ and ${\rm Cl}(4,4)$ in terms of
creation/annihilation operators and the description of the same Clifford
algebras in terms of quaternions, split quaterions, octonions and split
octonions respectively. The other aim is to record the possible types of
orbits in the space of Weyl spinors and discuss the corresponding geometry.

The creation/annihilation operator models are explained in the accompanying
paper, and will be reviewed below (for the cases of relevance). The starting
point of the quaternion/octonion description of Clifford algebras is the Pauli
matrices description of ${\rm Cl}(3)$. Thus, let

$\displaystyle\sigma^{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\qquad\sigma^{2}=\left(\begin{array}[]{cc}0&-{\rm i}\\\
{\rm i}&0\end{array}\right),\qquad\sigma^{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ (1.7)

be the usual Pauli matrices. These matrices anti-commute and square to plus
the identity. Therefore, they generate the Clifford algebra ${\rm
Cl}(3)\equiv{\rm Cl}(3,0)$, where our convention on ${\rm Cl}(r,s)$ is that
this is the algebra generated by
$\Gamma_{I}\Gamma_{J}+\Gamma_{J}\Gamma_{I}=2g_{IJ}$, where $g_{IJ}$ is a
metric of signature $r$ plusses and $s$ minuses.

We then note that a general linear combination of the Pauli matrices can be
written as

$\displaystyle X(r,q)=\left(\begin{array}[]{cc}r&\bar{q}\\\
q&-r\end{array}\right),$ (1.10)

with $q\in{\mathbb{C}},r\in{\mathbb{R}}$ and $X(r,q)={\rm
Re}(q)\sigma^{1}+{\rm Im}(q)\sigma^{2}+r\sigma^{3}$. We now note that
$q\in{\mathbb{C}}$ here can be replaced with $q\in{\mathbb{H}}$ or
$q\in{\mathbb{O}}$, where ${\mathbb{H}}$ and ${\mathbb{O}}$ are the spaces of
quaternions and octonions respectively. The corresponding matrices then
generate ${\rm Cl}(5)$ and ${\rm Cl}(9)$ respectively. A subtlety arises in
the case of octonions for these are non-associative. To deal with this, one
replaces the operation of multiplication with $q$ with the operator $L_{q}$ of
e.g. left multiplication. This is equivalent to a convention for the order of
multiplication and deals with the issue of non-associativity. The arising
description makes it clear that the spinors of ${\rm Cl}(3),{\rm Cl}(5),{\rm
Cl}(9)$ are two-component columns with values in ${\mathbb{C}},{\mathbb{H}}$
and ${\mathbb{O}}$ respectively. The described model of the Clifford algebras
${\rm Cl}(3),{\rm Cl}(5),{\rm Cl}(9)$ is not new and can be found in e.g. [2].

This construction can be further extended in several different ways. First,
one can consider $r=0$ off-diagonal $\Gamma$-matrices. These generate the
Clifford algebras ${\rm Cl}(2),{\rm Cl}(4),{\rm Cl}(8)$ for
$q\in{\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$ respectively. One can also choose
a preferred vector in ${\mathbb{R}}^{2},{\mathbb{R}}^{4},{\mathbb{R}}^{8}$ and
thus obtain a ${\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$-based model for ${\rm
Cl}(1),{\rm Cl}(3),{\rm Cl}(7)$ respectively. One can also apply the tensor
product construction and generate $\Gamma$-matrices for ${\rm Cl}(3,1),{\rm
Cl}(5,1),{\rm Cl}(9,1)$. These become $4\times 4$ matrices with entries in
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$ respectively. We will describe the
corresponding constructions in the main text. Finally, one can replace
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$ with their split versions. The
$X$-matrices (1.10) then generate ${\rm Cl}(2,1),{\rm Cl}(3,2)$ and ${\rm
Cl}(5,4)$ respectively and one can easily go either one dimension down or up
staying in the same formalism. Overall, we get a useful for computations and
economic description of a host of low (and not so low) dimensional Clifford
algebras.

It is desirable to relate the creation/annihilation operator construction and
the described quaternion/octonion one, when they are both applicable. This is
the aim of the present paper. We will see that each of the two constructions
emphasises a different geometrical aspect of spinors in the corresponding
dimension. And to fully understand what spinors are one needs to know both
constructions, when they are both available.

We also record the description of the various possible orbits of ${\rm
Spin}(4,4)$ on the space of its Weyl spinors, which appears to be new. The
space of ${\rm Spin}(4,4)$ Weyl spinors satisfying the reality condition (i.e.
Majorana-Weyl spinors) can be identified with split octonions. Thus, Weyl
spinors can be described as complexified split octonions. Null octonions (or
null complexified octonions) correspond to pure spinors, and their
classification is available from the general results on the classification of
pure spinors, see in particular [3]. Majorana-Weyl spinors are also easy to
classify. But we are not aware of any discussion of the possible types of
orbits in the general case, i.e. complexified split octonions that are not
null. In the analogous case of ${\rm Spin}(8)$ it is known that a general Weyl
spinor can be used to construct a certain pure spinor. It thus follows that a
general Weyl spinor continues to define a complex structure on
${\mathbb{R}}^{8}$ and its stabiliser is the same as that of a pure spinor,
which is ${\rm SU}(4)$. We observe that this phenomenon continues to the ${\rm
Spin}(4,4)$ setting, and that a general Weyl spinor that is of the form
$\alpha+{\rm i\,}\beta,\alpha,\beta\in{\mathbb{O}}^{\prime}$ with neither
$\alpha$ nor $\beta$ null continues to define a certain pair of pure spinors.
The possible stabilisers are then ${\rm SU}(2,2)$ when the corresponding pure
spinor is of the type that defines a complex structure, and ${\rm
SL}(4,{\mathbb{R}})$ when the pair of the arising pure spinors defines a
paracomplex structure. The new situation that has no ${\rm Spin}(8)$ analog is
when either $\alpha$ or $\beta$ is null. This describes an orbit with
interesting corresponding geometry, which we describe.

Tee organisation of the paper is as follows. We proceed by describing in turn
models related to quaternions in Section 2, split quaternions in 3 and then
octonions in and split octonions in Sections 4 and 5. We conclude with a
discussion.

##  2 Quaternionic models

The aim of this Section is to see how the quaternionic description of ${\rm
Spin}(4)$ arises within the creation/annihilation operator model. The Weyl
spinors of ${\rm Spin}(4)$ are seen to be quaternions. We then describe how
with similar methods one can describe ${\rm Cl}(6),{\rm Cl}(5,1),{\rm
Cl}(4,2)$. Note that the corresponding Spin groups all contain ${\rm
Spin}(4)$, and this is why the quaternionic description becomes possible. Only
in the case ${\rm Cl}(5,1)$ the arising quaternionic description is
sufficiently powerful to be useful. In the other two cases, while the
quaternionic language is possible, they are most usefully described using the
complex language.

###  2.1 Creation/annihilation operators construction of ${\rm Cl}(4)$

We choose a complex structure on ${\mathbb{R}}^{4}$ thus identifying
${\mathbb{R}}^{4}={\mathbb{C}}^{2}$. We will call the arising null complex
coordinates $z_{1,2}$, and the corresponding one-forms $dz_{1,2}$. We
introduce two pairs of creation/annihilation operators
$a_{1,2},a_{1,2}^{\dagger}$. The $\Gamma$ operators take the following form

$\begin{split}\Gamma_{4}&=a_{1}+a_{1}^{\dagger},\\\
\Gamma_{2}&=a_{2}+a_{2}^{\dagger},\\\
\end{split}\qquad\begin{split}\Gamma_{3}&=-i(a_{1}-a_{1}^{\dagger}),\\\
\Gamma_{1}&=-i(a_{2}-a_{2}^{\dagger}).\\\ \end{split}$ (2.1)

We remind that in our conventions $a$ is the creation and $a^{\dagger}$
annihilation operator. The $a_{1,2}$ act as operators wedging the polyform
they act on with $dz_{1,2}$ (from the left). The annihilation operators
$a^{\dagger}_{1,2}$ look for a copy of $dz_{1,2}$ in the polyform and erase
it. Some signs arise in this process, which can always be taken care of by
rewriting the differential form one acts on so that the one-forms $dz_{1,2}$
one wants to kill are in the leftmost position. For more details, see the
accompanying paper [1]. Note, however, that the notations in this paper are
somewhat changed as compared to [1]. We have adopted the numbering and the
signs in the imaginary $\Gamma$-matrices that become convenient below. A
generic Dirac spinor (general polyform) is given by

$\Psi=(u_{1}+u_{2}dz_{12})+(v_{1}dz_{1}+v_{2}dz_{2}),$ (2.2)

where $dz_{12}:=dz_{1}\wedge dz_{2}$ and $u_{1,2},v_{1,2}\in{\mathbb{C}}$. In
matrix notations, the Dirac spinor is a 4-component column. It is convenient
to adopt the $2\times 2$ block notations, in which Weyl spinors are
2-component. Thus, we write

$\displaystyle\Psi=\left(\begin{array}[]{c}\psi_{+}\\\
\psi_{-}\end{array}\right),\qquad\psi_{+}=\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right),\qquad\psi_{-}=\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right).$ (2.9)

With these conventions, the $\Gamma$ operators can be written in the matrix
notations

$\displaystyle\Gamma_{4}=\left(\begin{array}[]{cc}0&{\mathbb{I}}\\\
{\mathbb{I}}&0\end{array}\right),\quad\Gamma_{i}=\left(\begin{array}[]{cc}0&{\rm
i\,}\sigma^{i}\\\ -{\rm i\,}\sigma^{i}&0\end{array}\right),\quad i=1,2,3.$
(2.14)

Here $\sigma^{i}$ are the usual Pauli matrices. It is this simple form of the
resulting $\Gamma$-matrices that motivated the choices made in (2.1), (2.2).

The invariant inner product is determined by the following computation

$\displaystyle\langle\tilde{\Psi},\Psi\rangle=(\tilde{u}_{1}-\tilde{u}_{2}dz_{12}+\tilde{v}_{1}dz_{1}+\tilde{v}_{2}dz_{2})\wedge(u_{1}+u_{2}dz_{12}+v_{1}dz_{1}+v_{2}dz_{2})\Big{|}_{top}=$
(2.15)
$\displaystyle(\tilde{u}_{1}u_{2}-\tilde{u}_{2}u_{1})+(\tilde{v}_{1}v_{2}-\tilde{v}_{2}v_{1}).$

The sign changes in the polyform $\tilde{\Psi}$ are understood as coming from
rewriting this polyform with all decomposable summands written in the opposite
order, see the accompanying paper. The inner product is thus an anti-symmetric
pairing $\langle S_{+},S_{+}\rangle,\langle S_{-},S_{-}\rangle$. It can be
written in matrix terms as

$\displaystyle\langle\Psi_{1},\Psi_{2}\rangle=\Psi_{1}^{T}\left(\begin{array}[]{cc}\epsilon&0\\\
0&\epsilon\end{array}\right)\Psi_{2},\qquad\epsilon:={\rm
i\,}\sigma^{2}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right).$ (2.20)

For the possible reality conditions, both $R=\Gamma_{2}\Gamma_{4}\ast$ and
$R^{\prime}=\Gamma_{1}\Gamma_{3}\ast$ square to minus the identity, and so
there are no Majorana spinors in this case. These two operators only differ by
a sign in their action on $S_{+}$. Both of them can be used to define the hat
operator. We have

$\displaystyle R=\left(\begin{array}[]{cc}\epsilon&0\\\
0&-\epsilon\end{array}\right)\ast,\qquad
R^{\prime}=\left(\begin{array}[]{cc}\epsilon&0\\\
0&\epsilon\end{array}\right)\ast.$ (2.25)

Thus, they differ by a sign of their action on $S_{-}$. We then define the hat
operator

$\displaystyle\hat{\psi}_{+}=\epsilon\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)^{\ast}=\left(\begin{array}[]{c}u_{2}^{*}\\\
-u_{1}^{*}\end{array}\right),$ (2.30)

which squares to minus the identity. Using the hat operator we have the
following invariant norm on $S_{+}$

$\displaystyle\langle R(\psi_{+}),\psi_{+}\rangle=|u_{1}|^{2}+|u_{2}|^{2}.$
(2.31)

###  2.2 Quaternions and their complex description

Quaternions ${\mathbb{H}}$ is a normed division algebra. The three imaginary
unit quaternions satisfy

$\displaystyle{\bf i}^{2}={\bf j}^{2}={\bf k}^{2}=-{\mathbb{I}},\qquad{\bf
i}{\bf j}={\bf k}.$ (2.32)

A general quaternion is the linear combination

$\displaystyle q=q^{4}{\mathbb{I}}+q^{1}{\bf i}+q^{2}{\bf j}+q^{3}{\bf
k},\qquad q^{1,\ldots,4}\in{\mathbb{R}}.$ (2.33)

The squared norm is given by

$\displaystyle|q|^{2}=q\bar{q}=(q^{4})^{2}+\sum_{i=1}^{3}(q^{i})^{2}.$ (2.34)

Quaternions can be described in complex terms. To this end, one chooses a
complex structure on ${\mathbb{H}}$, which identifies it with a copy of
${\mathbb{C}}^{2}$. There are $S^{2}$ worth of complex structures on
${\mathbb{R}}^{4}$, and a general complex structure can be described as the
multiplication (left or right) by a unit imaginary quaternion. Let us choose
the right multiplication $R_{\bf k}$ by ${\bf k}$ as the complex structure.
The eigenspace of $R_{\bf k}$ of eigenvalue $-{\rm i\,}$ is spanned by
${\mathbb{I}}+{\rm i\,}{\bf k},{\bf i}-{\rm i\,}{\bf j}$. This means that we
can write a quaternion (2.33) as

$\displaystyle q(u_{1},u_{2})=u_{1}({\mathbb{I}}+{\rm i\,}{\bf k})+u_{2}({\bf
i}-{\rm i\,}{\bf j})+u_{1}^{*}({\mathbb{I}}-{\rm i\,}{\bf k})+u_{2}^{*}({\bf
i}+{\rm i\,}{\bf j}),$ (2.35)

where

$\displaystyle u_{1}=\frac{q_{4}-{\rm i\,}q_{3}}{2},\qquad
u_{2}=\frac{q_{1}+{\rm i\,}q_{2}}{2}.$ (2.36)

Note that the last two terms in (2.35) are the complex conjugates of the first
two terms, where the complex conjugation reverses the sign in front of ${\rm
i\,}$. The imaginary unit ${\rm i\,}$ should not be confused with the unit
imaginary quaternion $\bf i$. Thus, with the complex structure $R_{\bf k}$ at
hand, we get the isomorphism ${\mathbb{C}}^{2}\sim{\mathbb{H}}$ given by
$q(u_{1},u_{2})$.

For later purposes we take two quaternions with complex coordinates $u_{1,2}$
and $\tilde{u}_{1,2}$ and compute the pairing

$\displaystyle\tilde{u}_{1}u_{2}-\tilde{u}_{2}u_{1}=\frac{1}{4}(\tilde{q}_{4}q_{1}-\tilde{q}_{1}q_{4}+\tilde{q}_{3}q_{2}-\tilde{q}_{2}q_{3})+\frac{{\rm
i\,}}{4}(\tilde{q}_{4}q_{2}-\tilde{q}_{2}q_{4}+\tilde{q}_{1}q_{3}-\tilde{q}_{3}q_{1})=\frac{1}{4}(\tilde{q},q(-{\bf
i}))+\frac{{\rm i\,}}{4}(\tilde{q},q(-{\bf j})).$ (2.37)

It is also interesting to see what various operations on ${\mathbb{C}}^{2}$
corresponds to in ${\mathbb{H}}$. We have

$\displaystyle q(u_{2},-u_{1})=q_{1}{\mathbb{I}}-q_{4}{\bf i}+q_{3}{\bf
j}-q_{2}{\bf k}=(-{\bf i})q(u_{1},u_{2}),$ (2.38) $\displaystyle
q(u_{1}^{*},u_{2}^{*})=q_{4}{\mathbb{I}}+q_{1}{\bf i}-q_{2}{\bf j}-q_{3}{\bf
k}={\bf i}\,q(u_{1},u_{2})(-{\bf i}),$

and thus

$\displaystyle q(u_{2}^{*},-u_{1}^{*})=q_{1}{\mathbb{I}}-q_{4}{\bf
i}-q_{3}{\bf j}+q_{2}{\bf k}=q(u_{1},u_{2})(-{\bf i}).$ (2.39)

Thus, the anti-linear hat operation on ${\mathbb{C}}^{2}$ is just the right
multiplication by $-{\bf i}$ when ${\mathbb{C}}^{2}$ is viewed as
${\mathbb{H}}$. We then have two operations on ${\mathbb{C}}^{2}$: the complex
structure given by the multiplication by $-{\rm i\,}$, and the hat operator
(2.30). On ${\mathbb{H}}$ the first is the right multiplication $R_{\bf k}$
and the second is the right multiplication $-R_{\bf i}$. In their
${\mathbb{H}}$ versions these two operators clearly anti-commute. Having
$R_{\bf i}$ and $R_{\bf k}$ we have the quaternionic structure on
${\mathbb{H}}$. This discussion shows how this is encoded in the
${\mathbb{C}}^{2}$ picture in the hat operator together with the operator of
multiplication by $-{\rm i\,}$.

We can now understand what the quaternion norm squared corresponds to in the
complex description. Taking $\tilde{q}=q(u_{1},u_{2})(-{\bf i})$ in (2.37) we
see that the imaginary term on the right-hand side $(q({-\bf i}),q(-{\bf
j}))=|q|^{2}({\bf i},{\bf j})=0$, and so drops out. What remains is

$\displaystyle(|u_{1}|^{2}+|u_{2}|^{2})=\frac{1}{4}|q|^{2}.$ (2.40)

It is also interesting to see what the octonionic conjugation does as a map on
${\mathbb{C}}^{2}$. We see from (2.36) that

$\displaystyle(u_{1},u_{2})(\bar{q})=(u_{1}^{*},-u_{2})(q).$ (2.41)

So, the quaternion conjguation does not commute with the complex structure
given by $R_{\bf k}$.

The left multiplication by an octonion $q$ commutes with $R_{\bf k}$. This
means that it can be described as a $2\times 2$ complex matrix acting on the
$-{\rm i\,}$ eigenspace of $R_{\bf k}$. A simple computation shows

$\displaystyle q(u_{1}({\mathbb{I}}+{\rm i\,}{\bf k})+u_{2}({\bf i}-{\rm
i\,}{\bf j}))=((q^{4}-{\rm i\,}q^{3})u_{1}-(q^{1}-{\rm
i\,}q^{2})u_{2})({\mathbb{I}}+{\rm i\,}{\bf k})+((q^{1}+{\rm
i\,}q^{2})u_{1}+(q^{4}+{\rm i\,}q^{3})u_{2})({\bf i}-{\rm i\,}{\bf j}).$
(2.42)

This means we can describe the operator $L_{q}$ of left multiplication by $q$
as a $2\times 2$ matrix acting on the column $(u_{1},u_{2})$

$\displaystyle L_{q}\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)=\left(\begin{array}[]{cc}q^{4}-{\rm
i\,}q^{3}&-(q^{1}-{\rm i\,}q^{2})\\\ q^{1}+{\rm i\,}q^{2}&q^{4}+{\rm
i\,}q^{3}\end{array}\right)\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right).$ (2.49)

Thus, if we use the ${\mathbb{C}}^{2}$ description, quaternions can be
identified with matrices of the type

$\displaystyle\left(\begin{array}[]{cc}a&-b^{*}\\\
b&a^{*}\end{array}\right),\qquad a,b\in{\mathbb{C}},$ (2.52)

with the left multiplication of quaternions described as multiplication of
such matrices from the left. Quaternions can also be identified with
2-component complex-valued columns. This is e.g. the first column of the
matrix (2.52). Note that the second column in (2.52) is the hat conjugate of
the first, we see that the left multiplication by a quaternion has the same
action on a 2-component column as on its hat conjugate. In other words, the
left multiplication by a quaternion commutes with the hat conjugation. This is
of course not surprising, because we already know that the hat conjugation
corresponds to the right multiplication by $-{\bf i}$, which clearly commutes
with $L_{q}$.

###  2.3 Spin(4) and quaternions

We now go back to the $4\times 4$ $\Gamma$-matrices derived from the
creation/annihilation operator construction. If we define

$\displaystyle E^{i}:=-{\rm i\,}\sigma^{i}$ (2.53)

we have

$\displaystyle E^{i}E^{j}=-\delta^{ij}{\mathbb{I}}+\epsilon^{ijk}E^{k},$
(2.54)

and so the objects $E^{i}$ can be identified with the unit quaternions ${\bf
i,j,k}$. This means that we can rewrite the $\Gamma$-matrices (2.14) as

$\Gamma_{4}=\begin{pmatrix}0&{\mathbb{I}}\\\ {\mathbb{I}}&0\end{pmatrix}\ ,\
\Gamma_{1}=\begin{pmatrix}0&-{\bf i}\\\ {\bf i}&0\end{pmatrix}\ ,\
\Gamma_{2}=\begin{pmatrix}0&-{\bf j}\\\ {\bf j}&0\end{pmatrix}\ ,\
\Gamma_{3}=\begin{pmatrix}0&-{\bf k}\\\ {\bf k}&0\end{pmatrix}$ (2.55)

To put it differently, the general element of the Clifford algebra ${\rm
Cl}(4)$ is

$\displaystyle
X(q):=q^{4}\Gamma_{4}+q^{i}\Gamma_{i}=\begin{pmatrix}0&\bar{q}\\\
q&0\end{pmatrix},$ (2.56)

where $q$ is a quaternion ${\mathbb{H}}\ni q=a^{4}{\mathbb{I}}+q^{i}E^{i}$.
Thus, the matrix $X(q)$ is of the type (1.10) with $r=0$. Such matrices act on
two-component columns with entries in ${\mathbb{H}}$, and so semi-spinors of
${\rm Spin}(4)$ are quaternions $S_{\pm}={\mathbb{H}}$. This shows how the
quaternionic model of ${\rm Cl}(4)$ explained in the Introduction arises from
the creation/annihilation operator model.

Let us compute the translation of the anti-linear map $R$ into the quaternion
notation. From (2.25) we see that it is given by the hat operator in the
${\mathbb{C}}^{2}$ description. From (2.39) we see that it translates into the
right multiplication by $-{\bf i}$ description. Thus, we can write

$\displaystyle R^{\prime}=\left(\begin{array}[]{cc}-R_{\bf i}&0\\\ 0&-R_{\bf
i}\end{array}\right),$ (2.59)

which acts on ${\mathbb{H}}^{2}$.

It is also important to describe how the inner product (2.20) gets encoded in
the quaternionic description. The inner product of two positive polyforms

$\displaystyle\psi_{+}=u_{1}+u_{2}dz_{12},\qquad\tilde{\psi}_{+}=\tilde{u}_{1}+\tilde{u}_{2}dz_{12}.$
(2.60)

is given by

$\displaystyle\langle\tilde{\psi}_{+},\psi_{+}\rangle=\left(\begin{array}[]{cc}\tilde{u}_{1}&\tilde{u}_{2}\end{array}\right)\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right).$ (2.66)

Using (2.37) this translates into a complex quantity.

$\displaystyle 4\langle\tilde{\psi}_{+},\psi_{+}\rangle=(\tilde{q},q(-{\bf
i}))+{\rm i\,}(\tilde{q},q(-{\bf j})).$ (2.67)

When $\tilde{\psi}_{+}=R^{\prime}\psi_{+}$ we get

$\displaystyle 4\langle R^{\prime}\psi_{+},\psi_{+}\rangle=|q|^{2}.$ (2.68)

Thus, the inner product of a Weyl spinor with its hat conjugate translates
into the norm squared of the corresponding quaternion.

###  2.4 Quaternionic description of ${\rm Cl}(6)$

The fact that even/odd polyforms in ${\mathbb{C}}^{2}$ can be identified with
quaternions allows us to give a quaternionic model of any Clifford algebra
that acts on polyforms with a ${\mathbb{C}}^{2}$ factor. Let us see how this
is possible for ${\rm Cl}(6)$. We start with the polyform description.

We choose a complex structure on ${\mathbb{R}}^{6}$, and introduce 3 complex
null coordinates $z_{1,2,3}$, as well as the corresponding one-forms
$dz_{1,2,3}$. We introduce 3 pairs of creation/annihilation operators
$a_{1,2,3},a_{1,2,3}^{\dagger}$. The Dirac spinor is a polyform which we
choose to write as $\Psi=\psi_{+}+\psi_{-}$ where

$\displaystyle\psi_{+}=u_{1}+u_{2}dz_{12}+(v_{1}dz_{1}+v_{2}dz_{2})\wedge
dz_{3},\quad\psi_{-}=(\tilde{u}_{1}+\tilde{u}_{2}dz_{12})\wedge
dz_{3}+\tilde{v}_{1}dz_{1}+\tilde{v}_{2}dz_{2}.$ (2.69)

The choices made here are motivated by the desire to have the Weyl spinors
(2.2) of ${\rm Spin}(4)$ recognisable here. Thus, we see that $\psi_{-}$ is
composed of a positive spinor $\tilde{u}_{1}+\tilde{u}_{2}dz_{12}$ and a
negative spinor $\tilde{v}_{1}dz_{1}+\tilde{v}_{2}dz_{2}$, and similarly for
$\psi_{+}$. The numbering of components in $\psi_{+}$ is such that the
invariant product is a pairing $\langle S_{+},S_{-}\rangle$ given by

$\displaystyle\langle\psi_{-},\psi_{+}\rangle=\tilde{u}_{1}u_{2}-\tilde{u}_{2}u_{1}+\tilde{v}_{1}v_{2}-\tilde{v}_{2}v_{1}=\left(\begin{array}[]{cc}\tilde{u}_{1}&\tilde{u}_{2}\end{array}\right)\epsilon\left(\begin{array}[]{c}u_{1}\\\
u_{2}\end{array}\right)+\left(\begin{array}[]{cc}\tilde{v}_{1}&\tilde{v}_{2}\end{array}\right)\epsilon\left(\begin{array}[]{c}v_{1}\\\
v_{2}\end{array}\right).$ (2.76)

We recognise this as being composed of the ${\rm Spin}(4)$ invariant product
of ${\rm Spin}(4)$ Weyl spinors. The $\Gamma$ operators are given by

$\begin{split}\Gamma_{4}&=a_{1}+a_{1}^{\dagger},\\\
\Gamma_{3}&=-i(a_{1}-a_{1}^{\dagger})\end{split}\quad,\qquad\begin{split}\Gamma_{2}&=a_{2}+a_{2}^{\dagger},\\\
\Gamma_{1}&=-i(a_{2}-a_{2}^{\dagger})\end{split}\quad,\qquad\begin{split}\Gamma_{5}&=a_{3}+a_{3}^{\dagger},\\\
\Gamma_{6}&=-i(a_{3}-a_{3}^{\dagger}),\end{split}$ (2.77)

where the numbering is to match that in (2.1). The anti-linear operator giving
the reality condition is the product of 3 imaginary $\Gamma$-operators
followed by the complex conjugation
$R^{\prime}=\Gamma_{1}\Gamma_{3}\Gamma_{6}\ast,(R^{\prime})^{2}={\mathbb{I}}$.

We now identify the pairs of complex numbers $u_{1,2},v_{1,2}$ and
$\tilde{u}_{1,2},\tilde{v}_{1,2}$ with quaternions that we call
$u,v,\tilde{u},\tilde{v}$. We can thus write

$\displaystyle\psi_{+}=u+v\wedge dz_{3},\qquad\psi_{-}=\tilde{u}\wedge
dz_{3}+\tilde{v},$ (2.78)

where it is understood that $u,\tilde{u}$ represent even polyforms and
$v,\tilde{v}$ represent odd in $\Lambda({\mathbb{C}}^{2})$.

We place these quaternions into a 4-component column

$\displaystyle\Psi=\left(\begin{array}[]{c}\psi_{+}\\\
\psi_{-}\end{array}\right),\qquad\psi_{+}=\left(\begin{array}[]{c}u\\\
v\end{array}\right),\qquad\psi_{-}=\left(\begin{array}[]{c}\tilde{u}\\\
\tilde{v}\end{array}\right).$ (2.85)

This allows to describe the $\Gamma$-matrices $\Gamma_{1,2},\Gamma_{4,5}$ in
quaternionic terms as $\Gamma_{p},p\in{\mathbb{H}}$. As we know from our
discussion of the ${\rm Spin}(4)$ case, the operator $\Gamma_{p}$ maps an even
quaternion $u$ to the odd quaternion $L_{p}u$, and an odd $v$ to the even
$L_{\bar{p}}v$. Thus, in matrix notations

$\displaystyle\Gamma_{p}=\left(\begin{array}[]{cccc}0&0&0&L_{\bar{p}}\\\
0&0&L_{p}&0\\\ 0&L_{\bar{p}}&0&0\\\ L_{p}&0&0&0\end{array}\right),\qquad
p\in{\mathbb{H}}.$ (2.90)

The other two $\Gamma$-matrices are worked out very easily using the (2.78)
representation. We have

$\displaystyle\Gamma_{3}\psi_{+}=u\wedge
dz_{3}-v,\qquad\Gamma_{3}\psi_{-}=-\tilde{v}\wedge dz_{3}+\tilde{u},$ (2.91)
$\displaystyle\Gamma_{6}\psi_{+}=-{\rm i\,}u\wedge dz_{3}-{\rm
i\,}v,\qquad\Gamma_{6}\psi_{-}={\rm i\,}\tilde{v}\wedge dz_{3}+{\rm
i\,}\tilde{u}.$

This corresponds to the following $\Gamma$-matrices

$\displaystyle\Gamma_{5}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\
0&-1&0&0\end{array}\right),\qquad\Gamma_{6}=\left(\begin{array}[]{cccc}0&0&{\rm
i}&0\\\ 0&0&0&{\rm i}\\\ -{\rm i}&0&0&0\\\ 0&-{\rm i}&0&0\end{array}\right).$
(2.100)

We can rewrite these $\Gamma$-matrices in the $2\times 2$ form with

$\displaystyle\Gamma_{I}=\left(\begin{array}[]{cc}0&\gamma_{I}\\\
\gamma_{I}&0\end{array}\right),\qquad I=1,\ldots,5$ (2.103)

where $\gamma_{I}$ are the generators of ${\rm Cl}(5)$ whose general linear
combination is the matrix of the form (1.10). To translate the last
$\Gamma$-matrix into quaternionic notation, we need to use the fact that the
multiplication by $-{\rm i\,}$ on ${\mathbb{C}}^{2}$ corresponds to the
complex structure $R_{\bf k}$ on ${\mathbb{H}}$. This means that

$\displaystyle\Gamma_{6}=\left(\begin{array}[]{cc}0&-R_{\bf k}{\mathbb{I}}\\\
R_{\bf k}{\mathbb{I}}&0\end{array}\right).$ (2.106)

We note that using the quaternionic formalism leads to an economy of
description. Thus, without quaternions the $\Gamma$-matrices are $8\times 8$
acting on ${\mathbb{C}}^{8}$. Using ${\mathbb{H}}={\mathbb{C}}^{2}$ we have
compacted them to $4\times 4$ matrices with quaternionic entries acting on
$S={\mathbb{H}}^{4}$.

It is important to compute the $R^{\prime}$ operator in the quaternionic
notation. We start by computing it in the polyform notation, and then
translate. Using (2.25) we have the result for $\Gamma_{1}\Gamma_{3}\ast$, and
so

$\displaystyle
R^{\prime}=\Gamma_{1}\Gamma_{3}\Gamma_{6}\ast=(\Gamma_{1}\Gamma_{3}\ast)(-\Gamma_{6})=\left(\begin{array}[]{cccc}\epsilon\ast&0&0&0\\\
0&\epsilon\ast&0&0\\\ 0&0&\epsilon\ast&0\\\
0&0&0&\epsilon\ast\end{array}\right)\left(\begin{array}[]{cccc}0&0&-{\rm
i}&0\\\ 0&0&0&-{\rm i}\\\ {\rm i}&0&0&0\\\ 0&{\rm i}&0&0\end{array}\right).$
(2.115)

We then know that $\epsilon\ast$ is the hat operator whose quaternion
translation is given by (2.39), which is the right multiplication by $-{\bf
i}$. The multiplication by $-{\rm i\,}$ is the action of the complex structure
on ${\mathbb{C}}^{2}$, and this translates as $R_{\bf k}$. Thus, the
quaternionic translation is

$\displaystyle R^{\prime}=\left(\begin{array}[]{cc}0&-R_{\bf i}R_{\bf
k}{\mathbb{I}}\\\ R_{\bf i}R_{\bf
k}{\mathbb{I}}&0\end{array}\right)=\left(\begin{array}[]{cc}0&-R_{\bf
j}{\mathbb{I}}\\\ R_{\bf j}{\mathbb{I}}&0\end{array}\right).$ (2.120)

where the off-diagonal blocks are operators on ${\mathbb{H}}^{2}$.

We can now see how to translate the invariant pairing (2.76) into quaternionic
terms. We use (2.37) and have

$\displaystyle 4\langle\psi_{-},\psi_{+}\rangle=(\tilde{u},u(-{\bf
i}))+(\tilde{v},v(-{\bf i}))+{\rm i\,}(\tilde{u},u(-{\bf j}))+{\rm
i\,}(\tilde{v},v(-{\bf j})).$ (2.121)

It is in general complex-valued. We can also see how an invariant norm of a
Weyl spinor arises. We take $\psi_{+}$. We have $R^{\prime}\psi_{+}\in S_{-}$,
and the corresponding negative Weyl spinor is $(\tilde{u},\tilde{v})=(R_{\bf
j}u,R_{\bf j}v)$, and so

$\displaystyle 4{\rm i\,}\langle
R^{\prime}(\psi_{+}),\psi_{+}\rangle=|u|^{2}+|v|^{2}.$ (2.122)

It is instructive to compute the matrices representing the Lie algebra. The
general Lie algebra element is

$\displaystyle\omega^{6I}\Gamma_{6}\Gamma_{I}+\omega^{IJ}\Gamma_{I}\Gamma_{J}.$
(2.123)

The products arising here are block-diagonal. The matrix
$\omega^{IJ}\Gamma_{I}\Gamma_{J}$ has the same $S_{\pm}$ blocks given by

$\displaystyle\left(\begin{array}[]{cc}L_{\bf x}&L_{\overline{q}}\\\
-L_{q}&L_{\bf y}\end{array}\right),\qquad{\bf x,y}\in{\rm
Im}\,{\mathbb{H}},\quad q\in{\mathbb{H}}.$ (2.126)

The $S_{+}$ restriction of the matrix $\omega^{6I}\Gamma_{6}\Gamma_{I}$ is

$\displaystyle(-R_{\bf k})\left(\begin{array}[]{cc}r&L_{\overline{p}}\\\
L_{p}&-r\end{array}\right),\qquad r\in{\mathbb{R}},\quad p\in{\mathbb{H}}.$
(2.129)

The $S_{\pm}$ restrictions of the Lie algebra element can then be written as

$\displaystyle A_{\mathfrak{spin}(6)}=\left(\begin{array}[]{cc}A-rR_{\bf
k}&-R_{\bf k}L_{\bar{p}}+L_{\overline{q}}\\\ -R_{\bf
k}L_{p}-L_{q}&A^{\prime}+rR_{\bf k}\end{array}\right),\quad
A^{\prime}_{\mathfrak{spin}(6)}=\left(\begin{array}[]{cc}A+rR_{\bf k}&R_{\bf
k}L_{\bar{p}}+L_{\overline{q}}\\\ R_{\bf k}L_{p}-L_{q}&A^{\prime}-rR_{\bf
k}\end{array}\right).$ (2.134)

Here $A,A^{\prime}$ are the two chiral $\mathfrak{su}(2)$-valued parts of the
Lie algebra $\mathfrak{spin}(4)$. The presence of $R_{\bf k}$ in these
expressions makes this quaternionic description of $\mathfrak{spin}(6)$
somewhat cumbersome. The purpose of the discussion was that it is possible. We
note that we could similarly describe $\mathfrak{spin}(4,2)$ in quaternionic
terms, but this description will also contain the operator $R_{\bf k}$ in the
Lie algebra matrices, and so unlikely to be useful. The quaternionic formalism
becomes most useful for the case ${\rm Cl}(5,1)$ which we now turn to.

###  2.5 Quaternionic description of ${\rm Cl}(5,1)$

The described above quaternionic model of ${\rm Cl}(6)$ is plagued by the
presence of the operator $R_{\bf k}$ encoding the multiplication by $-{\rm
i\,}$ in the quaternionic language. This operator disappears from the
$\Gamma$-matrices of ${\rm Cl}(5,1)$, which gives a much more powerful
quaternionic description. In fact, we have ${\rm Spin}(5,1)={\rm
SL}(2,{\mathbb{H}})$.

We first review the polyform model, and then provide its quaternionic
translation. The only difference as compared to the ${\rm Cl}(6)$ case is that
the third null coordinate is now real $dz_{3}\to du$. So, we write the general
polyform similar to (2.69), but replacing $dz_{3}\to du$.

$\displaystyle\psi_{+}=u_{1}+u_{2}dz_{12}+(v_{1}dz_{1}+v_{2}dz_{2})\wedge
du,\quad\psi_{-}=(\tilde{u}_{1}+\tilde{u}_{2}dz_{12})\wedge
du+\tilde{v}_{1}dz_{1}+\tilde{v}_{2}dz_{2}.$ (2.135)

All $\Gamma$-operators are unchanged as compared to what they are in the ${\rm
Cl}(6)$ case, apart from the last $\Gamma$-operator is now
$\Gamma_{6}=-(b-b^{\dagger})$ instead. The Weyl spinors are still objects in
${\mathbb{C}}^{4}$, and the invariant inner product is still given by (2.76).

We can again identify ${\mathbb{C}}^{2}={\mathbb{H}}$, and describe Weyl
spinors as 2-component columns with entries in ${\mathbb{H}}$. The
quaternionic version of the inner product is (2.121). In matrix notations, the
first five $\Gamma$-matrices are still given by (2.103). The last
$\Gamma$-matrix is now given by

$\displaystyle\Gamma_{6}=\left(\begin{array}[]{cc}0&{\mathbb{I}}\\\
-{\mathbb{I}}&0\end{array}\right).$ (2.138)

It is instructive to compute the matrices forming the Lie algebra. Taking
products of distinct $\Gamma$-matrices we get a block-diagonal matrix with
$2\times 2$ blocks on the diagonal. The upper-left block is the representation
of $\mathfrak{spin}(5,1)$ on $S^{+}$, and the lower-right block gives the
representation on $S^{-}$. We get

$\displaystyle
A_{{\mathfrak{s}pin}(5,1)}=\omega^{6I}\gamma_{I}+\omega^{IJ}\gamma_{I}\gamma_{J},\qquad
A^{\prime}_{{\mathfrak{s}pin}(5,1)}=-\omega^{6I}\gamma_{I}+\omega^{IJ}\gamma_{I}\gamma_{J},$
(2.139)

A simple computation shows that these are matrices of the form

$\displaystyle
A_{{\mathfrak{s}pin}(5,1)}=\left(\begin{array}[]{cc}r{\mathbb{I}}+{\bf
x}&\overline{p+q}\\\ p-q&-r{\mathbb{I}}+{\bf y}\end{array}\right),\qquad
A^{\prime}_{{\mathfrak{s}pin}(5,1)}=\left(\begin{array}[]{cc}-r{\mathbb{I}}+{\bf
x}&\overline{-p+q}\\\ -p-q&r{\mathbb{I}}+{\bf y}\end{array}\right),\qquad
r\in{\mathbb{R}},p,q\in{\mathbb{H}},{\bf x},{\bf y}\in{\rm Im}\,{\mathbb{H}}.$
(2.144)

In other words, each is an arbitrary $2\times 2$ matrix with quaternionic
entries satisfying the condition that the real part of its trace is zero. This
is one way to argue that ${\rm Spin}(5,1)={\rm SL}(2,{\mathbb{H}})$. We note
that ${\rm SL}(2,{\mathbb{H}})$ acts transitively on the space
${\mathbb{H}}^{2}$ of Weyl spinors.

The possible anti-linear operators are
$R=\Gamma_{2}\Gamma_{4}\Gamma_{5}\Gamma_{6}\ast$ and
$R^{\prime}=\Gamma_{1}\Gamma_{3}\ast$. Both of them now square to minus the
identity, so there are no Majorana spinors in this signature. The $R^{\prime}$
acts on all copies of ${\mathbb{C}}^{2}$ as $\epsilon\ast$. The operator
$\epsilon\ast$ is the hat operator, which translated to quaternions becomes
$-R_{\bf i}$. Thus, we can write

$\displaystyle R^{\prime}\Big{|}_{S_{+}}=\left(\begin{array}[]{cc}-R_{\bf
i}&0\\\ 0&-R_{\bf i}\end{array}\right).$ (2.147)

Both $R,R^{\prime}$ operators preserve the helicity, which means that there is
no quadratic invariant that can be constructed for a Weyl spinor in this
signature, which agrees with the fact that the action of ${\rm Spin}(5,1)$ on
$S_{\pm}$ is transitive.

A positive Weyl spinor $\psi_{+}\in{\mathbb{H}}^{2}$ is pure. The geometric
information it describes is that of a real null vector in
${\mathbb{R}}^{5,1}$, as well as a pair of two complex null vectors. The real
null direction can be recovered by computing $\langle
R^{\prime}(\psi_{+}),\Gamma\psi_{+}\rangle$. Acting with $\Gamma_{p}$ on
$\psi_{+}=(u,v)$ we have the spinor $(L_{\bar{p}}v,L_{p}u)\in S_{-}$. Using
(2.121) we get

$\displaystyle
4\langle\Gamma_{p}\psi_{+},R^{\prime}(\psi_{+})\rangle=-(L_{\bar{p}}v,u)-(L_{p}u,v)+{\rm
i\,}(L_{\bar{p}}v,uR_{\bf k})+{\rm i\,}(L_{p}u,vR_{\bf k})=-2(v,L_{p}u).$
(2.148)

Similarly, $\Gamma_{5}\psi_{+}=(u,-v)\in S_{-}$ and

$\displaystyle
4\langle\Gamma_{5}\psi_{+},R^{\prime}(\psi_{+})\rangle=-|u|^{2}+|v|^{2}+{\rm
i\,}(u,uR_{\bf k})-{\rm i\,}(v,vR_{\bf k})=-|u|^{2}+|v|^{2}.$ (2.149)

Finally, we have $\Gamma_{6}\psi_{+}=(u,v)\in S_{-}$ and

$\displaystyle
4\langle\Gamma_{6}\psi_{+},R^{\prime}(\psi_{+})\rangle=-|u|^{2}-|v|^{2}+{\rm
i\,}(u,uR_{\bf k})+{\rm i\,}(v,vR_{\bf k})=-|u|^{2}-|v|^{2}.$ (2.150)

Thus, overall we get

$\displaystyle-4\langle\Gamma\psi_{+},R^{\prime}(\psi_{+})\rangle=(2(v,L_{p}u),|u|^{2}-|v|^{2},|u|^{2}+|v|^{2})\in{\mathbb{R}}^{1,5}$
(2.151)

The right-hand side is a null vector. Taking a unit spinor
$|u|^{2}+|v|^{2}=1$, the ${\mathbb{R}}^{5}$ part of this vector lies on
$S^{4}$, which exhibits the quaternionic Hopf fibration ${\mathbb{H}}^{2}\ni
S^{7}\to S^{4}={\mathbb{H}}$. To summarise, a quaternionic model is possible
for any Spin group that contains ${\rm Spin}(4)$. It is however most useful in
the case ${\rm Spin}(5,1)$ where we have ${\rm Spin}(5,1)={\rm
SL}(2,{\mathbb{H}})$.

##  3 Models related to split quaternions

The Clifford algebra ${\rm Cl}(2,2)$ is related to split quaternions. One can
also obtain the split quaternion models of all Clifford algebras that contain
${\rm Cl}(2,2)$. Of these, the most interesting model that arises is that of
${\rm Cl}(3,3)$, because in this case one can identify ${\rm Spin}(3,3)={\rm
SL}(2,{\mathbb{H}}^{\prime})$.

###  3.1 Creation/annihilation operator model of ${\rm Spin}(2,2)$

There are two possible models in this case. The model that exhibits the
maximum analogy to the case of ${\rm Spin}(4)$, and also identifies semi-
spinors with split quaternions is the one that endows ${\mathbb{R}}^{2,2}$
with a complex structure. Let $z_{1,2}$ be the corresponding complex
coordinates. A general Dirac spinor is still given by the polyform (2.2). The
$\Gamma$-operators are given by

$\begin{split}\Gamma_{4}&=a_{1}+a_{1}^{\dagger}\\\ \Gamma_{2}&=-{\rm
i\,}(a_{2}+a_{2}^{\dagger})\end{split}\quad,\qquad\begin{split}\Gamma_{3}&=-{\rm
i\,}(a_{1}-a_{1}^{\dagger})\\\ \Gamma_{1}&=a_{2}-a_{2}^{\dagger}.\end{split}$
(3.1)

Here $\Gamma_{4,3}$ are the same as in (2.1). The other choices are motivated
by a computation below. The $\Gamma$-matrices are then easily recoverable from
(2.14) and are given by

$\displaystyle\Gamma_{4}=\left(\begin{array}[]{cc}0&{\mathbb{I}}\\\
{\mathbb{I}}&0\end{array}\right),\quad\Gamma_{3}=\left(\begin{array}[]{cc}0&{\rm
i\,}\sigma^{3}\\\ -{\rm
i\,}\sigma^{3}&0\end{array}\right),\quad\Gamma_{1}=\left(\begin{array}[]{cc}0&-\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),\quad\Gamma_{2}=\left(\begin{array}[]{cc}0&\sigma^{2}\\\
-\sigma^{2}&0\end{array}\right).$ (3.10)

The invariant inner product is still given by (2.20).

The two non-trivial anti-linear operators that can be constructed are
$R=\Gamma_{4}\Gamma_{1}\ast$ and $R^{\prime}=\Gamma_{2}\Gamma_{3}\ast$. We use
$R^{\prime}$ as the reality condition operator. In matrix form we have

$\displaystyle R^{\prime}=\left(\begin{array}[]{cc}\sigma^{1}&0\\\
0&\sigma^{1}\end{array}\right)\ast.$ (3.13)

The action of $R^{\prime}$ preserves $S_{+}$ (and $S_{-}$), and allows us to
define Majorana-Weyl spinors. The action of $R$ on $S_{+}$ is the same as that
of $R^{\prime}$. The following invariant norm can be constructed

$\displaystyle\langle
R^{\prime}(\psi_{+}),\psi_{+}\rangle=|u_{2}|^{2}-|u_{1}|^{2}.$ (3.14)

###  3.2 Split quaternions and their complex description

Quaternions ${\mathbb{H}}^{\prime}$ is a normed composition algebra. The three
imaginary unit quaternions $\bf{\tilde{i}},\bf{\tilde{j}},\bf{\tilde{k}}$
satisfy

$\displaystyle\bf{\tilde{i}}^{2}=\bf{\tilde{j}}^{2}={\mathbb{I}},\qquad\bf{\tilde{k}}^{2}=-{\mathbb{I}},\qquad\bf{\tilde{i}}\bf{\tilde{j}}=\bf{\tilde{k}}.$
(3.15)

A general quaternion is the linear combination

$\displaystyle\tilde{q}=\tilde{q}^{4}{\mathbb{I}}+\tilde{q}^{1}{\bf\tilde{i}}+\tilde{q}^{2}{\bf\tilde{j}}+\tilde{q}^{3}{\bf\tilde{k}},\qquad\tilde{q}^{1,\ldots,4}\in{\mathbb{R}}.$
(3.16)

The squared norm is given by

$\displaystyle|\tilde{q}|^{2}=\tilde{q}\overline{\tilde{q}}=(\tilde{q}^{4})^{2}+(\tilde{q}^{3})^{2}-(\tilde{q}^{2})^{2}-(\tilde{q}^{1})^{2}.$
(3.17)

Similarly to ${\mathbb{H}}$, the split quaternions ${\mathbb{H}}^{\prime}$ can
be described in complex terms. We want to select a complex structure on
${\mathbb{H}}^{\prime}$ that would commute with the left multiplication. The
only available choice now is $J=R_{\bf\tilde{k}}$. The $-{\rm i\,}$ eigenspace
of $R_{\bf k}$ is now spanned by ${\mathbb{I}}+{\rm
i\,}{\bf\tilde{k}},{\bf\tilde{i}}+{\rm i\,}{\bf\tilde{j}}$. This means that we
can write a general split quaternion as

$\displaystyle q=u_{1}({\mathbb{I}}+{\rm
i\,}{\bf\tilde{k}})+u_{2}({\bf\tilde{i}}+{\rm
i\,}{\bf\tilde{j}})+u_{1}^{*}({\mathbb{I}}-{\rm
i\,}{\bf\tilde{k}})+u_{2}^{*}({\bf\tilde{i}}-{\rm i\,}{\bf\tilde{j}}),$ (3.18)

where

$\displaystyle u_{1}=\frac{q_{4}-{\rm i\,}q_{3}}{2},\qquad
u_{2}=\frac{q_{1}-{\rm i\,}q_{2}}{2}.$ (3.19)

The norm squared is then $|q|^{2}=4|u_{1}|^{2}-4|u_{2}|^{2}$.

Let us describe how the action of the complex conjugation and $\sigma^{1}\ast$
get translated to ${\mathbb{H}}^{\prime}$. We have

$\displaystyle
q(u_{1}^{*},u_{2}^{*})=q^{4}{\mathbb{I}}+q^{1}{\bf\tilde{i}}-q^{2}{\bf\tilde{j}}-q^{3}{\bf\tilde{k}}={\bf\tilde{i}}\,q(u_{1},u_{2}){\bf\tilde{i}},$
(3.20) $\displaystyle
q(u_{2},u_{1})=q^{1}{\mathbb{I}}+q^{4}{\bf\tilde{i}}+q^{3}{\bf\tilde{j}}+q^{2}{\bf\tilde{k}}={\bf\tilde{i}}\,q(u_{1},u_{2}).$

This means that the combination of these operations, which is the operation
$\sigma^{1}\ast$, is given by the right multiplication by ${\bf\tilde{i}}$

$\displaystyle R^{\prime}=R_{\bf\tilde{i}}.$ (3.21)

Let us also work out what the inner product (2.20) becomes in the split
quaternion notation. We have

$\displaystyle\tilde{u}_{1}u_{2}-\tilde{u}_{2}u_{1}=\frac{1}{4}(\tilde{q}_{4}q_{1}-\tilde{q}_{3}q_{2}-\tilde{q}_{1}q_{4}+\tilde{q}_{2}q_{3})+\frac{1}{4{\rm
i\,}}(\tilde{q}_{4}q_{2}+\tilde{q}_{3}q_{1}-\tilde{q}_{1}q_{3}-\tilde{q}_{2}q_{4})=\frac{1}{4}(\tilde{q},q{\bf\tilde{i}})+\frac{1}{4{\rm
i\,}}(\tilde{q},q{\bf\tilde{j}}).$ (3.22)

This shows how the inner product of two ${\mathbb{C}}^{2}$ vectors gets
translated into the split quaternion notation.

Let us understand what the left multiplication by a split quaternion
corresponds to in the ${\mathbb{C}}^{2}$ notation. We have

$\displaystyle\tilde{q}(\alpha({\mathbb{I}}+{\rm
i\,}{\bf\tilde{k}})+\beta({\bf\tilde{i}}+{\rm
i\,}{\bf\tilde{j}}))=((\tilde{q}^{4}-{\rm
i\,}\tilde{q}^{3})\alpha+(\tilde{q}^{1}+{\rm
i\,}\tilde{q}^{2})\beta)({\mathbb{I}}+{\rm
i\,}{\bf\tilde{k}})+((\tilde{q}^{1}-{\rm
i\,}\tilde{q}^{2})\alpha+(\tilde{q}^{4}+{\rm
i\,}\tilde{q}^{3})\beta)({\bf\tilde{i}}+{\rm i\,}{\bf\tilde{j}}).$ (3.23)

We can again describe the operator $L_{\tilde{q}}$ of left multiplication by
$\tilde{q}$ as a $2\times 2$ matrix acting on the column $(\alpha,\beta)$

$\displaystyle L_{\tilde{q}}\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=\left(\begin{array}[]{cc}\tilde{q}^{4}-{\rm
i\,}\tilde{q}^{3}&\tilde{q}^{1}+{\rm i\,}\tilde{q}^{2}\\\ \tilde{q}^{1}-{\rm
i\,}\tilde{q}^{2}&\tilde{q}^{4}+{\rm
i\,}\tilde{q}^{3}\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=(\tilde{q}^{4}{\mathbb{I}}+\tilde{q}^{i}\tilde{E}^{i})\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right),$ (3.32)

where

$\displaystyle\tilde{E}^{1}=\sigma^{1},\quad\tilde{E}^{2}=-\sigma^{2},\quad\tilde{E}^{3}=-{\rm
i\,}\sigma^{3}.$ (3.33)

Thus, split quaternions can be identified with matrices of the type

$\displaystyle\left(\begin{array}[]{cc}a&b^{*}\\\
b&a^{*}\end{array}\right),\qquad a,b\in{\mathbb{C}}.$ (3.36)

Split quaternions can also be identified with 2-component complex columns,
e.g. the first column of the above matrix. The determinant of the matrix
(3.36) is the norm (3.17).

###  3.3 ${\rm Spin}(2,2)$ and split quaternions

Allowing the off-diagonal entries in (1.10) to be split quaternions we get an
${\mathbb{H}}^{\prime}$ model for ${\rm Cl}(3,2)$, and when $r=0$ for ${\rm
Cl}(2,2)$. To see how the creation/annihilation operator model maps into this,
we just need to put the $\Gamma$-matrices (3.10) into the form (1.10). We note
that the matrices (3.33) anti-commute and satisfy
$\tilde{E}^{1}\tilde{E}^{2}=\tilde{E}^{3}$, as well as
$(\tilde{E}^{1})^{2}=(\tilde{E}^{2})^{2}={\mathbb{I}}$ and
$(\tilde{E}^{3})^{2}=-{\mathbb{I}}$. Thus, they satisfy all the properties of
$\bf{\tilde{i}},\bf{\tilde{j}},\bf{\tilde{k}}$ and can be identified with
them. Then the object

$\displaystyle q=q^{4}{\mathbb{I}}+q^{i}\tilde{E}^{i}$ (3.37)

can be identified as a split quaternion. Note that as a matrix it is a matrix
of the type (3.36). The general linear combination of the $\Gamma$-matrices
(3.10) is then a matrix of the type (1.10) with $r=0$. The $\Gamma$-matrices
(3.10) act on 4-component columns with complex entries. But we have seen that
the 2-component columns can be identified with split quaternions. Thus, after
$\Gamma$-matrices are identified with matrices of the form (1.10), they act on
2-component columns with entries in ${\mathbb{H}}^{\prime}$. In particular,
semi-spinors are $S_{\pm}={\mathbb{H}}^{\prime}$.

Let us also translate the inner product (3.14) to the quaternion notation.
Using the fact that $R^{\prime}$ acts as $R_{\bf\tilde{i}}$, and using (3.23)
we get

$\displaystyle\langle
R^{\prime}(\psi_{+}),\psi_{+}\rangle=\frac{1}{4}(q{\bf\tilde{i}},q{\bf\tilde{i}})+\frac{1}{4{\rm
i\,}}(q{\bf\tilde{i}},q{\bf\tilde{j}})=-\frac{1}{4}|q|^{2},$ (3.38)

which matches (3.14).

###  3.4 Split quaternion model of ${\rm Cl}(3,3)$

We can construct the split quaternion model of any Clifford algebra that
contains ${\rm Cl}(2,2)$. In six dimensions the two cases that are covered are
${\rm Cl}(4,2)$ and ${\rm Cl}(3,3)$. The last case is particularly
interesting, as it leads to the isomorphism ${\rm Spin}(3,3)={\rm
SL}(2,{\mathbb{H}}^{\prime})$. We only consider this case.

To split quaternion model is obtained by considering polyforms in
$\Lambda({\mathbb{C}}^{2}\oplus{\mathbb{R}})$. We write the general polyform
in exactly the same way (2.135) as we did in the case of ${\rm Cl}(5,1)$. The
first 4 $\Gamma$-operators are then the same as in the ${\rm Cl}(2,2)$ case
and are given by (3.1). The remaining $\Gamma$-operators are

$\displaystyle\Gamma_{5}=b+b^{\dagger},\qquad\Gamma_{6}=-(b-b^{\dagger}).$
(3.39)

Writing spinors as 8-component columns with entries in ${\mathbb{C}}$ we get
the following $\Gamma$-matrices

$\displaystyle\Gamma_{I}=\left(\begin{array}[]{cc}0&\gamma_{I}\\\
\gamma_{I}&0\end{array}\right),\quad
I=1,\ldots,5,\qquad\Gamma_{6}=\left(\begin{array}[]{cc}0&{\mathbb{I}}\\\
-{\mathbb{I}}&0\end{array}\right),$ (3.44)

where

$\displaystyle\gamma_{5}=\left(\begin{array}[]{cc}{\mathbb{I}}&0\\\
0&-{\mathbb{I}}\end{array}\right),\quad\gamma_{4}=\left(\begin{array}[]{cc}0&{\mathbb{I}}\\\
{\mathbb{I}}&0\end{array}\right),\quad\gamma_{3}=\left(\begin{array}[]{cc}0&{\rm
i\,}\sigma^{3}\\\ -{\rm
i\,}\sigma^{3}&0\end{array}\right),\quad\gamma_{1}=\left(\begin{array}[]{cc}0&-\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),\quad\gamma_{2}=\left(\begin{array}[]{cc}0&\sigma^{2}\\\
-\sigma^{2}&0\end{array}\right).$ (3.55)

We can now identify ${\mathbb{C}}^{2}={\mathbb{H}}^{\prime}$. Under this
identification the matrices $\gamma_{I}$ become the canonical matrices of the
type (1.10), so that

$\displaystyle
r\gamma_{5}+\sum_{a=1}^{4}q^{a}\gamma_{a}=\left(\begin{array}[]{cc}r{\mathbb{I}}&L_{\bar{q}}\\\
L_{q}&-r{\mathbb{I}}\end{array}\right),\qquad q\in{\mathbb{H}}^{\prime}.$
(3.58)

Overall, we get the model that is precisely analogous to the quaternionic
model of ${\rm Cl}(5,1)$, with the only difference being that quaternions are
replaced by the split quaternions. The computation of the Lie algebra is also
analogous, and shows that
$\mathfrak{spin}(3,3)=\mathfrak{sl}(2,{\mathbb{H}}^{\prime})$.

The reality condition operator is the product of all the imaginary
$\Gamma$-operators followed by the complex conjugation, and continues to be
given by what it is in the ${\rm Cl}(2,2)$ case, which is $\sigma^{1}\ast$. In
the split quaternion description this becomes $R_{\bf\tilde{i}}$.

It is also interesting to discuss the arising spinor geometry. The general
Weyl spinor is pure, and defines two complex and one real null directions. The
real null direction can be again recovered as $\langle
R^{\prime}(\psi_{+}),\psi_{+}\rangle$. Taking the spinor to be unit, we get
the split version of the quaternionic Hopf fibration.

We can also take the Weyl spinor to be null. In particular, Majorana-Weyl
spinors are null. A Majorana-Weyl spinor describes three real null directions,
which can be recovered by computing
$\langle\psi_{+},\Gamma\Gamma\Gamma\psi_{+}\rangle$. This 3-form in
${\mathbb{R}}^{3,3}$ is decomposable and is given by the product of the 3 null
real directions when $\psi_{+}$ is Majorana-Weyl.

##  4 The case of ${\rm Spin}(8)$

In the already described cases of ${\rm Cl}(4)$ and ${\rm Cl}(2,2)$ semi-
spinors are 2-component columns with complex entries, and these can be
identified with ${\mathbb{H}},{\mathbb{H}}^{\prime}$ respectively. The
difference with the cases of ${\rm Cl}(8)$, ${\rm Cl}(4,4)$ is that it is the
Majorana-Weyl spinors that can be identified with
${\mathbb{O}},{\mathbb{O}}^{\prime}$. Thus, the identification is only
possible after a reality condition is imposed. To see how this works we first
need to describe the Majorana-Weyl spinors explicitly.

###  4.1 Creation/annihilation operator construction

To construct ${\rm Cl}(8)$ we choose a complex structure, which identifies
${\mathbb{R}}^{8}\sim{\mathbb{C}}^{4}$. We denote the complex coordinates by
$z_{1,\ldots,4}$, and the coordinate one-forms by $dz_{1,\ldots,4}$. We
introduce four pairs of creation/annihilation operators
$a_{I},a_{I}^{\dagger},I=1,\ldots,4$. We introduce the following
$\Gamma$-operators

$\displaystyle\Gamma_{4+I}:=a_{I}+a_{I}^{\dagger},\qquad\Gamma_{I}:=-{\rm
i\,}(a_{I}-a_{I}^{\dagger}),\quad i=1,2,3,4.$ (4.1)

Spinors are polyforms, i.e. elements of $\Lambda({\mathbb{C}}^{4})$, in
general with complex coefficients. Weyl spinors are even or odd degree
polyforms. Note that we have chosen the same signs as in (2.1).

The interesting anti-linear operator that arises in this case is given by the
product of all four imaginary $\Gamma$-operators with the complex conjugation

$\displaystyle R:=\Gamma^{1}\Gamma^{2}\Gamma^{3}\Gamma^{4}*,$ (4.2)

It is easy to check that $R$ commutes with all the gamma-matrices. It is also
easy to check that $R^{2}={\mathbb{I}}$, so $R$ is a real structure. Since $R$
is composed of an even number of gamma-matrices, it preserves the spaces of
Weyl spinors. Thus, it allows us to define the notion of Majorana-Weyl spinors
for ${\rm Spin}(8)$. A simple computation shows that the action of $R$ is
complex conjugation followed by the Hodge star in ${\mathbb{C}}^{4}$, modulo
some signs.

###  4.2 Majorana-Weyl spinors explicitly

It is now convenient to choose a basis $e^{I}=dz_{I}$ of basic one-forms. Then
a general odd/even polyform that is also real can be written as follows

$\displaystyle\psi^{-}\equiv\psi^{-}(u)=u_{1}e^{1}+\bar{u}_{1}e^{234}+u_{2}e^{2}+\bar{u}_{2}e^{314}+u_{3}e^{3}+\bar{u}_{3}e^{124}+{\rm
i\,}\bar{u}_{4}e^{4}+{\rm i\,}u_{4}e^{123},$ (4.3)
$\displaystyle\psi^{+}\equiv\psi^{+}(u)=u_{1}e^{41}+\bar{u}_{1}e^{23}+u_{2}e^{42}+\bar{u}_{2}e^{31}+u_{3}e^{43}+\bar{u}_{3}e^{12}+{\rm
i\,}\bar{u}_{4}+{\rm i\,}u_{4}e^{4123},$

where the quantities $u_{I}$, with $I=1,\ldots,4$ are complex numbers. This
particular choice of the complex coordinates $u_{I}$, and in particular the
choices made for the coordinate $e^{4}$, will become justified below by the
desired form of the action of the $\Gamma$-matrices.

The inner product is a pairing $\langle S_{+},S_{+}\rangle,\langle
S_{-},S_{-}\rangle$. If we take two positive spinors
$\psi^{+}(\tilde{u}),\psi^{+}(u)$, the product
$\langle\psi^{+}(\tilde{u}),\psi^{+}(u)\rangle$ is computed by taking the
polyform $\psi^{+}(\tilde{u})$ in the reverse order, wedging with
$\psi^{+}(u)$ and projecting on the top component. A simple computation gives

$\displaystyle\langle\psi^{+}(\tilde{u}),\psi^{+}(u)\rangle=2{\rm
Re}\sum_{I=1}^{4}\tilde{u}_{I}\bar{u}_{I}.$ (4.4)

Thus, Majorana-Weyl spinors are identified $S_{\pm}\sim{\mathbb{C}}^{4}$, with
the invariant pairings on $S_{\pm}$ being given by the standard definite
Hermitian metric on ${\mathbb{C}}^{4}$.

The form of the inner product makes it clear that the basis
$e^{42},e^{42},e^{43},{\mathbb{I}}$ of $S^{+}$ in (4.3) is totally null. To
make contact with octonions that are usually described in a non-null basis, it
is necessary to switch to a different parametrisation of polyforms. We
parametrise the polyforms by the real and imaginary parts of $u_{I}$ and write

$\displaystyle u_{1}=\alpha_{1}+{\rm i\,}\alpha_{5},\quad
u_{2}=\alpha_{2}+{\rm i\,}\alpha_{6},\quad u_{3}=\alpha_{3}+{\rm
i\,}\alpha_{7},\quad u_{4}=\alpha_{0}+{\rm i\,}\alpha_{4}.$ (4.5)

We will denote the components of the positive polyform by $\alpha$ and the
negative polyform components by $\beta$. We then have

$\displaystyle\psi^{+}=\alpha_{1}(e^{41}+e^{23})+\alpha_{2}(e^{42}+e^{31})+\alpha_{3}(e^{43}+e^{12})+\alpha_{4}(1-e^{4123})$
(4.6) $\displaystyle+{\rm i\,}\alpha_{5}(e^{41}-e^{23})+{\rm
i\,}\alpha_{6}(e^{42}-e^{31})+{\rm i\,}\alpha_{7}(e^{43}-e^{12})+{\rm
i\,}\alpha_{0}(1+e^{4123}),$
$\displaystyle\psi^{-}=\beta_{1}(e^{1}+e^{423})+\beta_{2}(e^{2}+e^{431})+\beta_{3}(e^{3}+e^{412})+\beta_{4}(e^{4}-e^{123})$
(4.7) $\displaystyle+{\rm i\,}\beta_{5}(e^{1}-e^{423})+{\rm
i\,}\beta_{6}(e^{2}-e^{431})+{\rm i\,}\beta_{7}(e^{3}-e^{412})+{\rm
i\,}\beta_{0}(e^{4}+e^{123}).$

###  4.3 The action of $\Gamma$-matrices

We now introduce a 16-component column

$\displaystyle\Psi=\left(\begin{array}[]{c}\psi^{+}\\\
\psi^{-}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{0}\\\ \alpha_{1}\\\
\vdots\\\ \alpha_{7}\\\ \beta_{0}\\\ \beta_{1}\\\ \vdots\\\
\beta_{7}\end{array}\right).$ (4.18)

A computation shows that the $\Gamma$-operators become the following $16\times
16$ matrices

$\Gamma_{0}=\begin{pmatrix}0&{\mathbb{I}}\\\
{\mathbb{I}}&0\end{pmatrix},\quad\Gamma_{a}=\begin{pmatrix}0&-E_{a}\\\
E_{a}&0\end{pmatrix}\ \text{for}\ a\in\\{1,\ldots,7\\},$ (4.19)

where

$\begin{split}E_{1}&=-E_{01}+E_{27}-E_{36}+E_{45},\\\
E_{2}&=-E_{02}-E_{17}+E_{35}+E_{46},\\\
E_{3}&=-E_{03}+E_{16}-E_{25}+E_{47},\\\
E_{4}&=-E_{04}-E_{15}-E_{26}-E_{37},\\\
E_{5}&=-E_{05}+E_{14}+E_{23}-E_{67},\\\
E_{6}&=-E_{06}-E_{13}+E_{24}+E_{57},\\\
E_{7}&=-E_{07}+E_{12}+E_{34}-E_{56}.\end{split}$ (4.20)

###  4.4 Octonions

The space of octonions ${\mathbb{O}}$ is a normed algebra with the property
$|xy|=|x||y|$ (i.e. a composition algebra). The usual octonions (unlike split
octonions) also have the property that the norm of every non-zero element is
not zero, which makes them into a division algebra. It is non-commutative and
non-associative, but alternative, which can be stated as the property that the
subalgebra generated by any two elements is associative.

A general octonion is an object

$\displaystyle q=q_{0}{\mathbb{I}}+\sum_{a=1}^{7}q_{a}{\bf e}^{a},$ (4.21)

where ${\bf e}^{a}$ are unit imaginary octonions. The unit octonions anti-
commute and square to minus the identity. The octonion conjugate changes the
sign of all the imaginary octonions. The octonionic pairing is

$\displaystyle(q,q)=|q|^{2}=q\overline{q}=(q_{0})^{2}+\sum_{a=1}^{7}(q_{a})^{2}.$
(4.22)

We encode the octonionic product by the cross-product in the space of
imaginary octonions. Thus, we write

$\displaystyle{\mathbb{O}}={\mathbb{R}}+{\rm Im}\,{\mathbb{O}}.$ (4.23)

Let ${\bf e}^{1,\ldots,7}$ be a basis in the space of imaginary octonions. The
cross-product in ${\rm Im}\,{\mathbb{O}}$ can be encoded by the following
3-form

$\displaystyle C={\bf e}^{567}+{\bf e}^{5}({\bf e}^{41}-{\bf e}^{23})+{\bf
e}^{6}({\bf e}^{42}-{\bf e}^{31})+{\bf e}^{7}({\bf e}^{43}-{\bf e}^{12}).$
(4.24)

This encodes the cross-product in the sense that $C({\bf e}^{a},{\bf
e}^{b},{\bf e}^{c})=({\bf e}^{a}\times{\bf e}^{b},{\bf e}^{c})$, where the
standard metric on ${\mathbb{R}}^{7}$ is used. So, for instance ${\bf
e}^{5}\times{\bf e}^{6}={\bf e}^{7}$.

###  4.5 Octonionic model for ${\rm Cl}(8)$

The octonionic product can also be encoded into $8\times 8$ matrices. To this
end we represent a general octonion as an 8-component column. Then the
operators of left multiplication by unit imaginary octonions can be checked to
be given by $L_{{\bf e}^{a}}=E_{a}$, where $E_{a}$ are precisely the same
matrices already encountered in (4.20). Coming back to our model for the
Clifford algebra ${\rm Cl}(8)$ we see that the general linear combination of
the $\Gamma$-matrices is

$\displaystyle
q_{0}\Gamma_{0}+\sum_{a=1}^{7}q_{a}\Gamma_{a}=\left(\begin{array}[]{cc}0&L_{\bar{q}}\\\
L_{q}&0\end{array}\right).$ (4.27)

These are matrices of the type (1.10) with $r=0$, which reproduces the model
explained in the Introduction for $q\in{\mathbb{O}}$. We thus see that ${\rm
Cl}(8)$ is generated by matrices (4.27) that act on 2-component columns with
entries in ${\mathbb{O}}$. Majorana-Weyl spinors are then identified with
copies of ${\mathbb{O}}$.

###  4.6 Pure spinors

In the creation/annihilation operator model, the pure spinors are decomposable
polyforms. In particular, the model comes with two preferred pure spinors, the
identity polyform, and the top polyform. For ${\rm Cl}(8)$, both are in
$S_{+}$, and both are null spinors. It is also clear from (4.3) that only the
linear combinations ${\rm i\,}(1+e^{4123})$ and $1-e^{4123}$ are Majorana-Weyl
spinors. Majorana-Weyl spinors can be identified with ${\mathbb{O}}$, and are
never null. Thus, we cannot see pure spinors if we restrict our attention to
Majorana-Weyl spinors.

In order to describe pure spinors we need arbitrary, not necessarily
satisfying the reality condition polyforms. These can be described as
complexified real polyforms, which are then identified with complexified
octonions. Thus, in the octonionic description of ${\rm Cl}(8)$ the general
Weyl spinor is a complexified octonion. In particular, pure spinors are
necessarily complexified octonions, because they are null.

Let us see how this works for the two canonical pure spinors. The Majorana-
Weyl identity and ${\bf e}^{e}$ octonion spinors are

$\displaystyle{\mathbb{O}}\ni{\mathbb{I}}={\rm
i\,}(1+e^{4123}),\qquad{\mathbb{O}}\ni{\bf u}:={\bf e}^{4}=1-e^{4123}.$ (4.28)

Note that we have introduced a convenient notation and denoted ${\bf u}$ the
imaginary unit octonion ${\bf e}^{4}$. This means that the pure spinors
$1,e^{4123}\in S_{+}$ are given by

$\displaystyle 1=\frac{1}{2{\rm i\,}}({\mathbb{I}}+{\rm i\,}{\bf u}),\qquad
e^{4123}=\frac{1}{2{\rm i\,}}({\mathbb{I}}-{\rm i\,}{\bf u}).$ (4.29)

We have the operation of complex conjugation that reverses the sign in front
of ${\rm i\,}$. This operation is not to be confused with the octonion
conjugation. It is clear that the two polyforms $1,e^{4123}$ are related by
the complex conjugation. This complex conjugation is the translation of the
$R$ anti-linear operator that acts on polyform into the octonion picture.

###  4.7 The complex structure defined by a pure spinor

By Cartan’s general result pure spinors of ${\rm Spin}(2n)$ are Weyl spinors
that have the property that all $B_{k}(\psi_{+},\psi_{+})$ are vanishing apart
from $k=n$. For ${\rm Spin}(8)$ the only possible number of insertions of
$\Gamma$-matrices between two Weyl spinors of the same helicity is zero, two
and four. The product of two $\Gamma$-matrices restricted to $S_{+}$ is
expressible as either $E_{a}$ or $E_{a}E_{b}$ with $a,b=1,\ldots,7$. Both are
anti-symmetric. Thus, $B_{2}(\psi_{+},\psi_{+})=0$. This shows that null
spinors with $B_{0}(\psi_{+},\psi_{+})=0$ are pure. Because the inner product
of a Majorana-Weyl spinor with itself is the norm squared of the corresponding
octonion, the Majorana-Weyl spinors are not pure.

In (4.29) we have an example of two canonical pure spinors. In the octonion
description they become complex linear combinations of two unit octonions.
From general considerations we know that either of these two pure spinors
defines a complex structure in ${\mathbb{R}}^{8}$ and that
$B_{4}(\psi_{+},\psi_{+})$ is the product of four null directions spanning one
of the eigenspaces of this complex structure. It is interesting to compute
both the complex structure and $B_{4}(\psi_{+},\psi_{+})$ explicitly.

The complex structure defined by the pure spinor $\psi_{+}=1$ is easiest
computed by computing $B_{2}(R(\psi_{+}),\psi_{+})$. We have $R(1)\sim
e^{4123}$ and so we need to compute

$\displaystyle B_{2}(R(\psi_{+}),\psi_{+})\sim\langle({\mathbb{I}}+{\rm
i\,}{\bf u}),\Gamma\Gamma({\mathbb{I}}-{\rm i\,}{\bf u})\rangle.$ (4.30)

The components of this 2-form in the ${\mathbb{I}},x\in{\rm Im}({\mathbb{O}})$
directions are

$\displaystyle\langle({\mathbb{I}}+{\rm i\,}{\bf
u}),\Gamma_{\mathbb{I}}\Gamma_{x}({\mathbb{I}}-{\rm i\,}{\bf
u})\rangle=(({\mathbb{I}}+{\rm i\,}{\bf u}),L_{x}({\mathbb{I}}-{\rm i\,}{\bf
u})),$ (4.31) $\displaystyle\langle({\mathbb{I}}+{\rm i\,}{\bf
u}),\Gamma_{x}\Gamma_{y}({\mathbb{I}}-{\rm i\,}{\bf
u})\rangle=(({\mathbb{I}}+{\rm i\,}{\bf u}),L_{\bar{x}}L_{y}({\mathbb{I}}-{\rm
i\,}{\bf u})).$

We also have

$\displaystyle(({\mathbb{I}}+{\rm i\,}{\bf u}),L_{x}({\mathbb{I}}-{\rm
i\,}{\bf u}))=2{\rm i\,}{\bf u},$ (4.32) $\displaystyle(({\mathbb{I}}+{\rm
i\,}{\bf u}),L_{x}L_{y}({\mathbb{I}}-{\rm i\,}{\bf u}))=2{\rm
i\,}(e^{15}+e^{26}+e^{37}):=2{\rm i\,}\omega.$

Thus, overall, we have the following 2-form in ${\mathbb{R}}^{8}$

$\displaystyle\frac{1}{2{\rm i\,}}(({\mathbb{I}}+{\rm i\,}{\bf
u}),\Gamma\Gamma({\mathbb{I}}-{\rm i\,}{\bf u}))={\mathbb{I}}\wedge{\bf
u}-\omega.$ (4.33)

This is the $(1,1)$ form for the arising complex structure, which allows us to
recover the complex structure (e.g. by raising one of the indices on the two-
form and thus interpreting it as an endomorphism of ${\mathbb{R}}^{8}$).

To see what the complex structure is explicitly, it helps to start by
considering the complex structure on ${\mathbb{O}}$ given by $L_{\bf u}$, i.e.
the left multiplication by the unit imaginary octonion $\bf u$. The
eigenvectors of eigenvalue $-{\rm i\,}$ are given by

$\displaystyle{\mathbb{I}}+{\rm i\,}{\bf u},\qquad{\bf e}^{1}+{\rm i\,}{\bf
e}^{5},\qquad{\bf e}^{2}+{\rm i\,}{\bf e}^{6},\qquad{\bf e}^{3}+{\rm i\,}{\bf
e}^{7}.$ (4.34)

The Kähler form arising as $({\rm i\,}/2)dz\wedge d\bar{z}$ is
${\mathbb{I}}\wedge{\bf u}+\omega$, which is (4.33) up to the sign in front of
the last term. This tells us that we need to consider a slightly different
complex structure on ${\mathbb{O}}$ to get (4.33). Let us instead consider the
operator $R_{\bf u}$ of right multiplication by ${\bf u}$. Its $-{\rm i\,}$
eigenvectors are now

$\displaystyle{\mathbb{I}}+{\rm i\,}{\bf u},\qquad{\bf e}^{1}-{\rm i\,}{\bf
e}^{5},\qquad{\bf e}^{2}-{\rm i\,}{\bf e}^{6},\qquad{\bf e}^{3}-{\rm i\,}{\bf
e}^{7},$ (4.35)

and the corresponding Kähler form is precisely (4.33).

We can alternatively recover the complex structure with its $(0,1)$ and
$(1,0)$ directions by computing $B_{4}(\psi_{+},\psi_{+})$ or
$B_{4}(R(\psi_{+}),R(\psi_{+}))$. We know that both are decomposable and are
given by the product of four $(0,1)$ null directions, or four $(1,0)$
directions.

Thus, we want to compute

$\displaystyle\langle({\mathbb{I}}-{\rm i\,}{\bf
u}),\Gamma\Gamma\Gamma\Gamma({\mathbb{I}}-{\rm i\,}{\bf
u})\rangle\in\Lambda^{4}({\mathbb{R}}^{8}).$ (4.36)

The various components of this 4-form that we need are given by

$\displaystyle\langle\psi,\Gamma_{\mathbb{I}}\Gamma_{x}\Gamma_{y}\Gamma_{z}\psi\rangle=(\psi,L_{x}L_{\bar{y}}L_{z}\psi),$
(4.37)
$\displaystyle\langle\psi,\Gamma_{x}\Gamma_{y}\Gamma_{z}\Gamma_{w}\psi\rangle=(\psi,L_{x}L_{\bar{y}}L_{z}L_{\bar{w}}\psi),$

where on the right-hand side the spinor $\psi$ is interpreted as a
(complexified) octonion. A computation gives

$\displaystyle(({\mathbb{I}}-{\rm i\,}{\bf
u}),L_{x}L_{y}L_{z}({\mathbb{I}}-{\rm i\,}{\bf u}))=2{\rm i\,}\Omega,$ (4.38)
$\displaystyle(({\mathbb{I}}-{\rm i\,}{\bf
u}),L_{x}L_{y}L_{z}L_{w}({\mathbb{I}}-{\rm i\,}{\bf u}))=2{\bf
u}\wedge\Omega,$

where

$\displaystyle\Omega:=(e^{1}-{\rm i\,}e^{5})\wedge(e^{2}-{\rm
i\,}e^{6})\wedge(e^{3}-{\rm i\,}e^{7}).$ (4.39)

This means that

$\displaystyle\frac{{\rm i\,}}{2}\langle({\mathbb{I}}-{\rm i\,}{\bf
u}),\Gamma\Gamma\Gamma\Gamma({\mathbb{I}}-{\rm i\,}{\bf
u})\rangle=({\mathbb{I}}+{\rm i\,}{\bf u})\wedge\Omega.$ (4.40)

This is precisely the wedge product of the directions (4.35), as expected. The
4-form obtained as $\langle({\mathbb{I}}+{\rm i\,}{\bf
u}),\Gamma\Gamma\Gamma\Gamma({\mathbb{I}}+{\rm i\,}{\bf u})\rangle$ is given
by the product of the complex conjugate directions.

To summarise, we learn that the complex structure on
${\mathbb{R}}^{8}\sim{\mathbb{O}}$ that corresponds to the complex conjugate
pair of pure spinors ${\mathbb{I}}+{\rm i\,}{\bf u},{\mathbb{I}}-{\rm i\,}{\bf
u}$ is given by $R_{\bf u}$, the right multiplication by ${\bf u}$.

###  4.8 Majorana spinors

Now that we understand the octonionic description of pure spinors, we can come
back to Majorana-Weyl spinors. We already know that such spinors cannot be
pure. In fact, we see from (4.29) that they are given by a linear combination
of two pure spinors.

It is important to discuss the geometry of a Majorana-Weyl spinor $\psi_{M}\in
S_{+}$. First, such a spinor has a non-vanishing norm
$\langle\psi_{M},\psi_{M}\rangle$, which coincides with the norm squared of
the corresponding octonion. Second, the group ${\rm Spin}(8)$ acts on the
space of Majorana-Weyl spinors of fixed norm and of one helicity transitively,
with the stabiliser ${\rm Spin}(7)$. So, we have

$\displaystyle S^{7}={\rm Spin}(8)/{\rm Spin}(7).$ (4.41)

This can be seen from the fact that the Majorana-Weyl representation is
isomorphic to the vector representation $S_{+}\sim{\mathbb{R}}^{8}$.

A Majorana-Weyl spinor of ${\rm Spin}(8)$ thus has ${\rm Spin}(7)$ as the
stabiliser, and endows ${\mathbb{R}}^{8}$ with a ${\rm Spin}(7)$ structure.
This is the 4-form $B_{4}(\psi_{M},\psi_{M})$, whose stabiliser in ${\rm
GL}(8,{\mathbb{R}})$ is ${\rm Spin}(7)$. It is instructive to compute this
4-form explicitly for $\psi_{M}={\mathbb{I}}$. Any unit Majorana-Weyl spinor
is in the ${\rm Spin}(8)$ orbit of this spinor. The components of this 4-form
are given by (4.37), and we have

$\displaystyle({\mathbb{I}},L_{x}L_{y}L_{z}{\mathbb{I}})=-C,$ (4.42)
$\displaystyle({\mathbb{I}},L_{x}L_{y}L_{z}L_{w}{\mathbb{I}})=-{}^{*}C,$

where

$\displaystyle
C:=e^{567}+e^{5}(e^{41}-e^{23})+e^{6}(e^{42}-e^{31})+e^{7}(e^{43}-e^{12}),$
(4.43)
$\displaystyle{}^{*}C:=e^{1234}+e^{67}(e^{41}-e^{23})+e^{75}(e^{42}-e^{31})+e^{56}(e^{43}-e^{12}).$

Note that $C$ is just the 3-form (4.24) that encodes the octonion product, and
${}^{*}C$ is its Hodge dual in ${\mathbb{R}}^{7}$. Thus, overall, we have

$\displaystyle\Lambda^{4}({\mathbb{R}}^{8})\ni\langle{\mathbb{I}},\Gamma\Gamma\Gamma\Gamma{\mathbb{I}}\rangle={\mathbb{I}}\wedge
C-{}^{*}C.$ (4.44)

We note, for future use, that

$\displaystyle C={\bf u}\wedge\omega+{\rm Im}(\Omega),\qquad{}^{*}C={\rm
Re}(\Omega)\wedge{\bf u}+\frac{1}{2}\omega\omega.$ (4.45)

###  4.9 A general spinor

A general Weyl spinor of ${\rm Spin}(8)$ is a complexified octonion. When the
spinor is not null, we can always rescale it by a complex number to make it
unit. As is explained in [4], see page 33, a complex impure spinor defines a
certain pure spinor. Indeed, assuming $\psi$ is unit and denoting
$\lambda=\langle R(\psi),\psi\rangle$, consider

$\displaystyle\tilde{\psi}=\frac{\lambda\psi-R(\psi)}{\sqrt{\lambda^{2}-1}}.$
(4.46)

Then $\langle\tilde{\psi},\tilde{\psi}\rangle=-1$ and

$\displaystyle\psi+\tilde{\psi}$ (4.47)

is null and therefore pure. Because the stabiliser of $R(\psi)$ is the same as
the stabiliser of $\psi$ (because the stabiliser is real), this stabiliser
also coincides with that of $\tilde{\psi}$ and thus $\psi+\tilde{\psi}$. This
means that the stabiliser of $\psi$ is that of a pure spinor, which is ${\rm
SU}(4)$.

The above discussion suggests that a general complex spinor continues to
define a complex structure on ${\mathbb{R}}^{8}$. Let us see how this arises.
A complex spinor has a well-defined real and imaginary parts $\psi=\alpha+{\rm
i\,}\beta,\alpha,\beta\in{\mathbb{O}}$. There are two invariant scalars that
can be constructed

$\displaystyle\langle\psi,\psi\rangle=|\alpha|^{2}-|\beta|^{2}+2{\rm
i\,}(\alpha,\beta),$ (4.48) $\displaystyle\langle
R(\psi),\psi\rangle=|\alpha|^{2}+|\beta|^{2}.$

Rescaling the spinor to make it unit, we assume
$|\alpha|^{2}-|\beta|^{2}=1,(\alpha,\beta)=0$. Also, using the action of ${\rm
Spin}(8)$ we can make $\alpha$ a multiple of ${\mathbb{I}}$, and then use
${\rm Spin}(7)$ that stabilises ${\mathbb{I}}$ to make $\beta$ (which is
orthogonal to $\alpha$) to be a multiple of ${\bf u}$. This leads us to
consider the unit spinor

$\displaystyle\psi=\cosh\tau{\mathbb{I}}+{\rm i\,}\sinh\tau{\bf u}.$ (4.49)

Then $\lambda=\cosh^{2}\tau+\sinh^{2}\tau=\cosh(2\tau)$ and

$\displaystyle\tilde{\psi}=\sinh\tau{\mathbb{I}}+{\rm i\,}\cosh\tau{\bf u},$
(4.50)

so that

$\displaystyle\psi+\tilde{\psi}=(\cosh\tau+\sinh\tau)({\mathbb{I}}+{\rm
i\,}{\bf u}),$ (4.51)

which is the pure spinor we already considered above. And indeed, we have the
following results

$\displaystyle((\cosh\tau{\mathbb{I}}+{\rm i\,}\sinh\tau{\bf
u}),L_{x}(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf u}))={\rm
i\,}\sinh(2\tau){\bf u},$ (4.52) $\displaystyle((\cosh\tau{\mathbb{I}}+{\rm
i\,}\sinh\tau{\bf u}),L_{x}L_{y}(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u}))={\rm i\,}\sinh(2\tau)\omega,$

which shows that

$\displaystyle\langle(\cosh\tau{\mathbb{I}}+{\rm i\,}\sinh\tau{\bf
u}),\Gamma\Gamma(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf u})={\rm
i\,}\sinh(2\tau)({\mathbb{I}}\wedge{\bf u}-\omega).$ (4.53)

This is a multiple of the Kähler 2-form for the complex structure defined by
the pure spinor ${\mathbb{I}}+{\rm i\,}{\bf u}$.

To compute the 4-form $B_{4}(\psi,\psi)$ we need the following results

$\displaystyle((\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u}),L_{x}L_{y}L_{z}(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf u}))={\rm
i\,}\sinh(2\tau){\rm Re}(\Omega)-\cosh(2\tau){\rm Im}(\Omega)-{\bf
u}\wedge\omega,$ $\displaystyle((\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u}),L_{x}L_{y}L_{z}L_{w}(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u}))=-\cosh(2\tau){\rm Re}(\Omega)\wedge{\bf u}-{\rm i\,}\sinh(2\tau){\rm
Im}(\Omega)\wedge{\bf u}-\frac{1}{2}\omega\wedge\omega.$

This means that

$\displaystyle\langle(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u}),\Gamma\Gamma\Gamma\Gamma(\cosh\tau{\mathbb{I}}-{\rm i\,}\sinh\tau{\bf
u})\rangle=\cosh(2\tau)({\mathbb{I}}\wedge{\rm Im}(\Omega)+{\bf u}\wedge{\rm
Re}(\Omega))$ (4.54) $\displaystyle-{\rm
i\,}\sinh(2\tau)({\mathbb{I}}\wedge{\rm Re}(\Omega)-{\bf u}\wedge{\rm
Im}(\Omega))-\frac{1}{2}({\mathbb{I}}\wedge{\bf
u}-\omega)\wedge({\mathbb{I}}\wedge{\bf u}-\omega).$

We note that the blocks that appear here, namely ${\mathbb{I}}\wedge{\rm
Im}(\Omega)+{\bf u}\wedge{\rm Re}(\Omega)$ and ${\mathbb{I}}\wedge{\rm
Re}(\Omega)-{\bf u}\wedge{\rm Im}(\Omega)$ are the imaginary and real parts of
the holomorphic 4-form (4.40). Both are thus ${\rm SU}(4)$ invariant. The last
term is the product of the two copies of the Kähler 2-form, which is again
${\rm SU}(4)$ invariant. This shows how the 4-form $B_{4}(\psi,\psi)$ is ${\rm
SU}(4)$ invariant.

##  5 Spin(4,4) and Split Octonions

###  5.1 Real model

The link to split octonions arises if we consider Majorana-Weyl spinors. These
are easiest to describe in the real model that starts by selecting a
paracomplex structure on ${\mathbb{R}}^{4,4}$. However, Majorana-Weyl real
spinors do not capture all possible spinor types that arise, as we have
witnessed in the previous section. For this reason, it is better to develop
everything in a complex model from the beginning, and then impose the Majorana
condition if needed. Nevertheless, we start by describing the simpler real
model, and then switch to the complex description.

The real model arises by selecting a paracomplex structure on
${\mathbb{R}}^{4,4}$. Let ${\mathbb{R}}^{4}$ be one of the arising totally
null subspaces, and let $u_{I},I=1,\ldots,4$ be the null coordinates and
$du_{I}$ the basic one-forms. It is more convenient to use the notation
$du_{I}=e_{I}$. We introduce four pairs of creation/annihilation operators
$a_{I},a_{I}^{\dagger}$. We define the $\Gamma$-operators as follows

$\begin{split}\Gamma_{0}&=a_{4}+a_{4}^{\dagger},\\\
\Gamma_{1}&=a_{1}+a_{1}^{\dagger},\\\ \Gamma_{2}&=a_{2}+a_{2}^{\dagger},\\\
\Gamma_{3}&=a_{3}+a_{3}^{\dagger},\\\
\end{split}\qquad\begin{split}\Gamma_{4}&=a_{4}-a_{4}^{\dagger},\\\
\Gamma_{5}&=a_{1}-a_{1}^{\dagger},\\\ \Gamma_{6}&=a_{2}-a_{2}^{\dagger},\\\
\Gamma_{7}&=a_{3}-a_{3}^{\dagger}.\\\ \end{split}$ (5.1)

The Majorana-Weyl spinors are even and odd polyforms in
$\Lambda({\mathbb{R}}^{4})$ with real coefficients. The basic polyforms
$e_{IJ\ldots}:=e_{I}\wedge e_{J}\wedge\ldots$ are all null with respect to the
invariant inner product that will be written below. For this reason, to make
link with the split octonions in the usual non-null basis we introduce a non-
null basis in $\Lambda({\mathbb{R}}^{4})$. We write

$\begin{split}S^{+}\ni\psi^{+}=\alpha_{1}(e_{41}-e_{23})+\alpha_{2}(e_{42}-e_{31})+\alpha_{3}(e_{43}-e_{12})+\alpha_{4}(1-e_{4123})\\\
+\alpha_{5}(e_{41}+e_{23})+\alpha_{6}(e_{42}+e_{31})+\alpha_{7}(e_{43}+e_{12})+\alpha_{0}(1+e_{4123}),\\\
S^{-}\ni\psi^{-}=\beta_{1}(e_{1}-e_{423})+\beta_{2}(e_{2}-e_{431})+\beta_{3}(e_{3}-e_{412})+\beta_{4}(e_{4}-e_{123})\\\
+\beta_{5}(e_{1}+e_{423})+\beta_{6}(e_{2}+e_{431})+\beta_{7}(e_{3}+e_{412})+\beta_{0}(e_{4}+e_{123}).\end{split}$
(5.2)

The invariant inner product is a pairing $\langle S^{+},S^{+}\rangle,\langle
S^{-},S^{-}\rangle$. A simple computation gives

$\displaystyle\langle\psi^{+},\psi^{+}\rangle=2((\alpha_{0})^{2}+(\alpha_{1})^{2}+(\alpha_{2})^{2}+(\alpha_{3})^{2}-(\alpha_{4})^{2}-(\alpha_{5})^{2}-(\alpha_{6})^{2}-(\alpha_{7})^{2}),$
(5.3)
$\displaystyle\langle\psi^{-},\psi^{-}\rangle=2((\beta_{0})^{2}+(\beta_{1})^{2}+(\beta_{2})^{2}+(\beta_{3})^{2}-(\beta_{4})^{2}-(\beta_{5})^{2}-(\beta_{6})^{2}-(\beta_{7})^{2}).$

We now form a Dirac spinor, which is a 16-component column

$\displaystyle\Psi=\left(\begin{array}[]{c}\psi^{+}\\\
\psi^{-}\end{array}\right)=\left(\begin{array}[]{c}\alpha_{0}\\\ \alpha_{1}\\\
\vdots\\\ \alpha_{7}\\\ \beta_{0}\\\ \beta_{1}\\\ \vdots\\\
\beta_{7}\end{array}\right),$ (5.14)

where we put the 8 $\alpha$ components of $\psi^{+}$ on top and 8 $\beta$
components of $\psi^{-}$ at the bottom of the column. The order in which the
components appear is $0,1,\ldots,7$. The $\Gamma$-operators become the
following $\Gamma$-matrices in this basis

$\Gamma_{0}=\begin{pmatrix}0&{\mathbb{I}}\\\
{\mathbb{I}}&0\end{pmatrix},\quad\Gamma_{a}=\begin{pmatrix}0&-\tilde{E}_{a}\\\
\tilde{E}_{a}&0\end{pmatrix}\ \text{for}\ a\in\\{1,\ldots,7\\},$ (5.15)

where

$\begin{split}\tilde{E}_{1}&=-E_{01}-E_{23}-E_{45}+E_{67}\\\
\tilde{E}_{2}&=-E_{02}-E_{31}-E_{46}+E_{75}\\\
\tilde{E}_{3}&=-E_{03}-E_{12}-E_{47}+E_{56}\\\
\tilde{E}_{4}&=S_{04}-S_{15}-S_{26}-S_{37}\\\
\tilde{E}_{5}&=S_{05}+S_{14}-S_{27}+S_{36}\\\
\tilde{E}_{6}&=S_{06}+S_{17}+S_{24}-S_{35}\\\
\tilde{E}_{7}&=S_{07}-S_{16}+S_{25}+S_{34}.\end{split}$ (5.16)

###  5.2 Split octonions

Split octonions ${\mathbb{O}}^{\prime}$ form a non-associative normed
composition algebra. It is not a division algebra because there are null
elements. A split octonion is an object

$\displaystyle\tilde{q}=\tilde{q}_{0}{\mathbb{I}}+\sum_{a=1}^{7}\tilde{q}_{a}{\bf\tilde{e}}^{a}.$
(5.17)

The unit imaginary octonions ${\bf\tilde{e}}^{a}$ anti-commute and satisfy

$\displaystyle({\bf\tilde{e}}^{1})^{2}=({\bf\tilde{e}}^{2})^{2}=({\bf\tilde{e}}^{3})^{2}=-{\mathbb{I}},\quad({\bf\tilde{e}}^{4})^{2}=({\bf\tilde{e}}^{5})^{2}=({\bf\tilde{e}}^{6})^{2}=({\bf\tilde{e}}^{7})^{2}={\mathbb{I}}.$
(5.18)

Thus, the split octonions
${\mathbb{I}},{\bf\tilde{e}}^{1},{\bf\tilde{e}}^{2},{\bf\tilde{e}}^{3}$
generate a copy of ${\mathbb{H}}\subset{\mathbb{O}}^{\prime}$. The octonion
pairing is given by

$\displaystyle(\tilde{q},\tilde{q})=\tilde{q}\overline{\tilde{q}}=(\tilde{q}_{0})^{2}+(\tilde{q}_{1})^{2}+(\tilde{q}_{2})^{2}+(\tilde{q}_{3})^{2}-(\tilde{q}_{4})^{2}-(\tilde{q}_{5})^{2}-(\tilde{q}_{6})^{2}-(\tilde{q}_{7})^{2},$
(5.19)

where the conjugation denoted by overbar changes the signs of all the
imaginary generators.

The product rules are most efficiently encoded into the following 3-form on
${\rm Im}({\mathbb{O}}^{\prime})={\mathbb{R}}^{7}$

$\tilde{C}={\bf\tilde{e}}^{123}-{\bf\tilde{e}}^{1}({\bf\tilde{e}}^{45}-{\bf\tilde{e}}^{67})-{\bf\tilde{e}}^{2}({\bf\tilde{e}}^{46}-{\bf\tilde{e}}^{75})-{\bf\tilde{e}}^{3}({\bf\tilde{e}}^{47}-{\bf\tilde{e}}^{56}).$
(5.20)

This encodes the vector product via

$\displaystyle(u\times v,w)=w\lrcorner v\lrcorner u\lrcorner\tilde{C}.$ (5.21)

Here $u\lrcorner$ is the operator of insertion of a vector field $u$ into a
differential form. For example
${\bf\tilde{e}}^{1}\times{\bf\tilde{e}}^{2}={\bf\tilde{e}}^{3}$, but
${\bf\tilde{e}}^{1}\times{\bf\tilde{e}}^{6}=-{\bf\tilde{e}}^{7}$ because the
octonion pairing is negative-definite on directions $4,5,6,7$.

###  5.3 Octonionic model for ${\rm Cl}(4,4)$

One can encode the operators of left multiplication by a unit octonion into
$8\times 8$ matrices. Indeed, we encode an octonion into an 8-component column

$\displaystyle\tilde{q}\to\left(\begin{array}[]{c}\tilde{q}_{0}\\\
\tilde{q}_{1}\\\ \vdots\\\ \tilde{q}_{7}\end{array}\right).$ (5.26)

It is then a straightforward computation to see that the operators of left
multiplication by a unit octonion precisely match the matrices in (5.16)
$L_{\bf{\tilde{e}}^{a}}=\tilde{E}_{a}$. This shows that a general linear
combination of the $\Gamma$-matrices (5.15) is a matrix of the form (1.10)
with $r=0$

$\displaystyle\tilde{q}_{0}\Gamma_{0}+\sum_{a=1}^{7}\tilde{q}_{a}\Gamma_{a}=\left(\begin{array}[]{cc}0&L_{\overline{\tilde{q}}}\\\
L_{\tilde{q}}&0\end{array}\right),$ (5.29)

where $L_{\tilde{q}}$ is the operator of left multiplication by a split
octonion $\tilde{q}\in{\mathbb{O}}^{\prime}$. This shows how the model (1.10)
described in the Introduction, with $q\in{\mathbb{O}}^{\prime}$ and $r=0$,
arises from the creation/annihilation operator model, in the version of it
which uses real polyforms.

###  5.4 The complex model

We now develop the complex model of ${\rm Cl}(4,4)$. The starting point is a
complex structure on ${\mathbb{R}}^{4,4}$, so that ${\mathbb{R}}^{4,4}$ gets
identified with ${\mathbb{C}}^{4}$. Let $z_{I},I=1,\ldots,4$ be the
corresponding complex (null) coordinates, and $e^{I}=dz_{I}$ the basic one-
forms. The metric on ${\mathbb{R}}^{4,4}$ becomes the following indefinite
Hermitian metric

$\displaystyle|dz_{3}|^{2}+|dz_{4}|^{2}-|dz_{1}|^{2}-|dz_{2}|^{2}.$ (5.30)

A general Dirac spinor is a polyform in $\Lambda({\mathbb{C}}^{4})$, with
complex coefficients. The $\Gamma$-operators that square to plus the identity
are given by

$\displaystyle\Gamma_{0}=a_{4}+a_{4}^{\dagger},\qquad\Gamma_{3}={\rm
i\,}(a_{4}-a_{4}^{\dagger}),$ (5.31)
$\displaystyle\Gamma_{2}=a_{3}+a_{3}^{\dagger},\qquad\Gamma_{1}={\rm
i\,}(a_{3}-a_{3}^{\dagger}).$

Note that these generate a copy of ${\rm Cl}(4)$, and act only on the
$e_{3},e_{4}$ polyform directions. The $\Gamma$-operators that square to minus
the identity are

$\displaystyle\Gamma_{4}={\rm
i\,}(a_{2}+a_{2}^{\dagger}),\qquad\Gamma_{7}=a_{2}-a_{2}^{\dagger},$ (5.32)
$\displaystyle\Gamma_{6}={\rm
i\,}(a_{1}+a_{1}^{\dagger}),\qquad\Gamma_{5}=a_{1}-a_{1}^{\dagger}.$

The link to split octonions arises if we consider Majorana-Weyl spinors, so we
must understand the reality conditions first.

###  5.5 Reality conditions

There are two anti-linear operators, the product of all real
$\Gamma$-operators followed by the complex conjugation, and the product of the
imaginary ones followed by the complex conjugation. Both square to plus the
identity, and so give a possible reality condition. They only differ in their
action (by a sign) on odd polyforms, and agree on even polyforms. It turns out
to be better to use

$\displaystyle R^{\prime}=\Gamma_{3}\Gamma_{1}\Gamma_{4}\Gamma_{6}\ast$ (5.33)

as the reality condition. A simple calculation shows that the following
polyforms parametrised by ${\mathbb{C}}^{4}$ are real

$\displaystyle\psi_{-}=u_{1}e^{1}-u_{1}^{*}e^{423}+u_{2}e^{2}-u_{2}^{*}e^{431}+u_{3}e^{3}+u_{3}^{*}e^{412}+iu_{4}^{*}e^{4}+{\rm
i\,}u_{4}e^{123},$ (5.34)
$\displaystyle\psi_{+}=u_{1}e^{41}-u_{1}^{*}e^{23}+u_{2}e^{42}-u_{2}^{*}e^{31}+u_{3}e^{43}+u_{3}^{*}e^{12}+{\rm
i\,}u_{4}^{*}+{\rm i\,}u_{4}e^{4123}.$

This should be compared to (4.3). Only some signs are different as compared to
the ${\rm Cl}(8)$ situation. We parametrise the even polyforms by the real and
imaginary parts of $u_{I}$

$\displaystyle u_{1}=\alpha_{5}+{\rm i\,}\alpha_{6},\quad
u_{2}=\alpha_{7}+{\rm i\,}\alpha_{4},\quad u_{3}=\alpha_{2}+{\rm
i\,}\alpha_{1},\quad u_{4}=\alpha_{3}+{\rm i\,}\alpha_{0}.$ (5.35)

We get the following real parametrisation of even and odd polyforms

$\displaystyle\psi^{+}=\alpha_{5}(e^{41}-e^{23})+\alpha_{7}(e^{42}-e^{31})+\alpha_{2}(e^{43}+e^{12})+\alpha_{0}(1-e^{4123})$
(5.36) $\displaystyle+{\rm i\,}\alpha_{6}(e^{41}+e^{23})+{\rm
i\,}\alpha_{4}(e^{42}+e^{31})+{\rm i\,}\alpha_{1}(e^{43}-e^{12})+{\rm
i\,}\alpha_{3}(1+e^{4123}),$
$\displaystyle\psi^{-}=\beta_{5}(e^{1}-e^{423})+\beta_{7}(e^{2}-e^{431})+\beta_{2}(e^{3}+e^{412})+\beta_{0}(e^{4}-e^{123})$
(5.37) $\displaystyle+{\rm i\,}\beta_{6}(e^{1}+e^{423})+{\rm
i\,}\beta_{4}(e^{2}+e^{431})+{\rm i\,}\beta_{1}(e^{3}-e^{412})+{\rm
i\,}\beta_{3}(e^{4}+e^{123}).$

The spinor norms are then

$\displaystyle\frac{1}{2}\langle\psi_{+},\psi_{+}\rangle=(\alpha_{0})^{2}+(\alpha_{1})^{2}+(\alpha_{2})^{2}+(\alpha_{3})^{2}-(\alpha_{4})^{2}-(\alpha_{5})^{2}-(\alpha_{6})^{2}-(\alpha_{7})^{2},$
(5.38)
$\displaystyle\frac{1}{2}\langle\psi_{-},\psi_{-}\rangle=(\beta_{0})^{2}+(\beta_{1})^{2}+(\beta_{2})^{2}+(\beta_{3})^{2}-(\beta_{4})^{2}-(\beta_{5})^{2}-(\beta_{6})^{2}-(\beta_{7})^{2}.$

We now place the components $\alpha,\beta$ into a 16-component column (5.14),
and work out the matrix representation of the $\Gamma$-operators. We get
precisely the matrices of the form (5.15) with (5.16), which also justifies
the choices for the signs of the $\Gamma$-operators. So, we again reproduce
the model (1.10) explained in the Introduction, with
$q\in{\mathbb{O}}^{\prime}$.

###  5.6 Pure spinors

In contrast to the ${\rm Spin}(8)$ case, we now have several different types
of pure spinors. We describe them all on the basis of the complex model.

The complex model was obtained by choosing a complex structure on
${\mathbb{R}}^{4,4}$, and so the canonical pure spinors that this model comes
with, namely $1,e^{4123}\in S_{+}$ give back this complex structure. To see
this, we translate the polyforms into split octonions. As in the case of ${\rm
Spin}(8)$ we need to complexify the octonions to see (at least certain types
of) the pure spinors. We have

$\displaystyle{\bf\tilde{e}}^{3}={\rm
i\,}(1+e^{4123}),\qquad{\mathbb{I}}=(1-e^{4123}),$ (5.39)

and so

$\displaystyle 1=\frac{1}{2}({\mathbb{I}}-{\rm i\,}{\bf\tilde{e}}^{3}),\qquad
e^{4123}=-\frac{1}{2}({\mathbb{I}}+{\rm i\,}{\bf\tilde{e}}^{3}).$ (5.40)

We now need the following results

$\displaystyle(({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}),L_{x}({\mathbb{I}}-{\rm i\,}{\bf\tilde{e}}^{3}))=2{\rm
i\,}{\bf\tilde{e}}^{3},$ (5.41) $\displaystyle(({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}),L_{x}L_{y}({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3}))=2{\rm
i\,}({\bf\tilde{e}}^{12}+{\bf\tilde{e}}^{74}+{\bf\tilde{e}}^{56}).$

This means that

$\displaystyle\langle({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}),\Gamma\Gamma({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3})\rangle=2{\rm
i\,}({\mathbb{I}}\wedge{\bf\tilde{e}}^{3}-({\bf\tilde{e}}^{12}+{\bf\tilde{e}}^{74}+{\bf\tilde{e}}^{56}).$
(5.42)

This is a $(1,1)$ form of the complex structure that this (complex conjugate)
pair of pure spinors defines. To see what this complex structure is, let us
consider the right multiplication by ${\bf\tilde{e}}^{3}$. The $-{\rm i\,}$
eigenvectors of $R_{{\bf\tilde{e}}^{3}}$ are

$\displaystyle z_{3}:={\mathbb{I}}+{\rm i\,}{\bf\tilde{e}}^{3},\qquad
z_{4}:={\bf\tilde{e}}^{1}-{\rm i\,}{\bf\tilde{e}}^{2},\qquad
z_{1}:={\bf\tilde{e}}^{7}+{\rm i\,}{\bf\tilde{e}}^{4},\qquad
z_{2}:={\bf\tilde{e}}^{5}+{\rm i\,}{\bf\tilde{e}}^{6}.$ (5.43)

We then have

$\displaystyle\langle({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}),\Gamma\Gamma({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3})\rangle=z_{3}^{*}\wedge z_{3}+z_{4}^{*}\wedge
z_{4}-z_{1}^{*}\wedge z_{1}-z_{2}^{*}\wedge z_{2},$ (5.44)

and so indeed $R_{{\bf\tilde{e}}^{3}}$ is the complex structure that
corresponds to the pure spinors $1,e^{4123}$.

Let us also state the result for the stabiliser of the spinor
${\mathbb{I}}-{\rm i\,}{\bf\tilde{e}}^{3}$. The general Lie algebra element on
$S_{+}$ is

$\displaystyle
A_{\mathfrak{spin}(4,4)}=w^{a}\tilde{E}_{a}-w^{ab}\tilde{E}_{a}\tilde{E}_{b}.$
(5.45)

The stabiliser of ${\mathbb{I}}-{\rm i\,}{\bf\tilde{e}}^{3}$ is the subalgebra
determined by the following equations

$\displaystyle w^{3}=w^{12}+w^{56}+w^{74},$ (5.46) $\displaystyle
w^{1}=w^{23},\quad w^{45}=w^{67},\quad w^{2}=w^{31},\quad w^{46}=w^{75},$
$\displaystyle w^{4}=w^{73},\quad w^{51}=w^{26},\quad w^{7}=w^{34},\quad
w^{16}=w^{25},$ $\displaystyle w^{5}=w^{36},\quad w^{14}=w^{27},\quad
w^{6}=w^{53},\quad w^{71}=w^{24}.$

The stabiliser is thus $28-13=15$ dimensional. Given that it preserves a
complex structure in ${\mathbb{R}}^{4,4}$, the stabiliser coincides with
$\mathfrak{su}(2,2)$.

A different type of pure spinors that is not difficult to describe corresponds
to a paracomplex structure on ${\mathbb{R}}^{4,4}$. Consider the null split
octonions

$\displaystyle\frac{1}{2}({\mathbb{I}}+{\bf\tilde{e}}^{4})=1-e^{4123}+{\rm
i\,}(e^{42}+e^{31}),\qquad\frac{1}{2}({\mathbb{I}}-{\bf\tilde{e}}^{4})=1-e^{4123}-{\rm
i\,}(e^{42}+e^{31}),$ (5.47)

where we also indicated the corresponding polyforms. We have

$\displaystyle(({\mathbb{I}}-{\bf\tilde{e}}^{4}),L_{x}({\mathbb{I}}+{\bf\tilde{e}}^{4}))=2{\bf\tilde{e}}^{4},$
(5.48)
$\displaystyle(({\mathbb{I}}-{\bf\tilde{e}}^{4}),L_{x}L_{y}({\mathbb{I}}+{\bf\tilde{e}}^{4}))=-2({\bf\tilde{e}}^{15}+{\bf\tilde{e}}^{26}+{\bf\tilde{e}}^{37}).$

Therefore

$\displaystyle\langle({\mathbb{I}}-{\bf\tilde{e}}^{4}),\Gamma\Gamma({\mathbb{I}}+{\bf\tilde{e}}^{4})\rangle=2({\mathbb{I}}\wedge{\bf\tilde{e}}^{4}+{\bf\tilde{e}}^{15}+{\bf\tilde{e}}^{26}+{\bf\tilde{e}}^{37}).$
(5.49)

We thus see that the pair of pure spinors
${\mathbb{I}}-{\bf\tilde{e}}^{4},{\mathbb{I}}+{\bf\tilde{e}}^{4}$ defines the
paracomplex structure whose real null eigenvectors are

$\displaystyle{\mathbb{I}}+{\bf\tilde{e}}^{4},\qquad{\bf\tilde{e}}^{1}+{\bf\tilde{e}}^{5},\qquad{\bf\tilde{e}}^{2}+{\bf\tilde{e}}^{6},\qquad{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{7}.$
(5.50)

As an operator on ${\mathbb{O}}^{\prime}$ this paracomplex structure is
described by $R_{{\bf\tilde{e}}^{4}}$.

Let us state the stabiliser in this case. The stabiliser of
${\mathbb{I}}+{\bf\tilde{e}}^{4}$ is given by the following equations

$\displaystyle w^{4}+w^{15}+w^{26}+w^{37}=0,$ (5.51) $\displaystyle
w^{1}-w^{14}+w^{5}+w^{45}=0,\quad w^{23}+w^{27}-w^{36}+w^{67}=0,$
$\displaystyle w^{2}-w^{24}+w^{6}+w^{46}=0,\quad
w^{13}+w^{17}-w^{35}+w^{57}=0,$ $\displaystyle
w^{3}-w^{34}+w^{7}+w^{47}=0,\quad w^{12}+w^{16}-w^{25}+w^{56}=0,$

and so is $28-7=21$ dimensional. It can be understood geometrically as ${\rm
SL}(4,{\mathbb{R}})$ mixing up the four real null directions of this pure
spinor, semi-direct product with a copy of nilpotent ${\rm SO}(4)$. If we in
addition impose the condition that ${\mathbb{I}}-{\bf\tilde{e}}^{4}$ is fixed,
we get six new equations

$\displaystyle w^{1}+w^{14}-w^{5}+w^{45}=0,\quad
w^{23}-w^{27}+w^{36}+w^{67}=0,$ (5.52) $\displaystyle
w^{2}+w^{24}-w^{6}+w^{46}=0,\quad w^{13}-w^{17}+w^{35}+w^{57}=0,$
$\displaystyle w^{3}+w^{34}-w^{7}+w^{47}=0,\quad
w^{12}-w^{16}+w^{25}+w^{56}=0,$

which together with the previous set gives a subalgebra of dimension
$28-7-6=15$. It is clear that the stabiliser is ${\rm SL}(4,{\mathbb{R}})$
that mixes the four real null directions of the null subspaces of both pure
spinors.

Let us now consider the spinors

$\displaystyle\frac{1}{2}({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3})-\frac{1}{2}({\bf\tilde{e}}^{4}+{\rm
i\,}{\bf\tilde{e}}^{7})=1+{\rm i\,}e^{42},\qquad\frac{1}{2}({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3})+\frac{1}{2}({\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7})=-e^{4123}+{\rm i\,}e^{31}.$ (5.53)

Both are null spinors, with a non-vanishing inner product between them. The
first of them is annihilated by

$\displaystyle\Gamma_{1}-{\rm i\,}\Gamma_{2},\quad\Gamma_{5}+{\rm
i\,}\Gamma_{6},\quad\Gamma_{0}+\Gamma_{4},\quad\Gamma_{3}+\Gamma_{7},$ (5.54)

and the second by the complement of these four vectors. So, they are a pair of
pure spinors with null subspace spanned by two real and two complex vectors.
Note that we can rewrite these pure spinors as

$\displaystyle\frac{1}{2}({\mathbb{I}}-{\bf\tilde{e}}^{4})-\frac{{\rm
i\,}}{2}({\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{7}),\qquad\frac{1}{2}({\mathbb{I}}+{\bf\tilde{e}}^{4})+\frac{{\rm
i\,}}{2}({\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{7}).$ (5.55)

Both are of the form $\alpha+{\rm i\,}\beta$ where $\alpha,\beta$ are real
pure spinors with $(\alpha,\beta)=0$.

We want to see how the structure of the mixed type gets produced by these
spinors. We have

$\displaystyle(({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7}),L_{x}({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7}))=4{\bf\tilde{e}}^{4},$ (5.56)
$\displaystyle(({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7}),L_{x}L_{y}({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}-{\rm i\,}{\bf\tilde{e}}^{7}))=-4{\rm
i\,}({\bf\tilde{e}}^{12}+{\bf\tilde{e}}^{56}-{\rm i\,}{\bf\tilde{e}}^{37}).$

This means that we have

$\displaystyle\langle({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7}),\Gamma\Gamma({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7})\rangle=4({\mathbb{I}}\wedge{\bf\tilde{e}}^{4}+{\bf\tilde{e}}^{37}+{\rm
i\,}{\bf\tilde{e}}^{12}+{\rm i\,}{\bf\tilde{e}}^{56}).$ (5.57)

This is a complex tensor of a mixed type. Raising one of the indices gives a
complex endomorphism of ${\mathbb{R}}^{4,4}$ that squares to minus the
identity, and is a sum of a paracomplex structure in the directions
${\mathbb{I}},4,3,7$ and the imaginary unit times the complex structure in the
directions $1,2,5,6$.

Let us determine the stabilisers in this case. The stabiliser of
$({\mathbb{I}}-{\rm i\,}{\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7})$ is given by the following set of equations

$\displaystyle w^{16}-w^{25}=0,\quad w^{12}+w^{56}=0,\quad
w^{15}+w^{26}=0,\quad w^{37}+w^{4}=0,$ (5.58) $\displaystyle
w^{1}+w^{14}=0,\quad w^{27}-w^{23}=0,\quad w^{2}+w^{24}=0,\quad
w^{35}+w^{57}=0,$ $\displaystyle w^{5}-w^{45}=0,\quad w^{36}+w^{67}=0,\quad
w^{6}-w^{46}=0,\quad w^{13}-w^{17}=0,$ $\displaystyle
w^{3}+w^{34}+w^{47}-w^{7}=0.$

The dimension of the stabiliser is thus $28-13=15$. Demand that also the
complementary spinor $({\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}-{\rm i\,}{\bf\tilde{e}}^{7})$ is
stabilised reproduces the equations in the first line, while gives for the
other lines

$\displaystyle w^{1}-w^{14}=0,\quad w^{27}+w^{23}=0,\quad w^{2}-w^{24}=0,\quad
w^{35}-w^{57}=0,$ (5.59) $\displaystyle w^{5}+w^{45}=0,\quad
w^{36}-w^{67}=0,\quad w^{6}+w^{46}=0,\quad w^{13}+w^{17}=0,$ $\displaystyle
w^{3}-w^{34}+w^{47}+w^{7}=0.$

Thus, together the equations imply

$\displaystyle w^{16}-w^{25}=0,\quad w^{12}+w^{56}=0,\quad
w^{15}+w^{26}=0,\quad w^{4}=w^{73},\quad w^{3}=w^{74},\quad w^{7}=w^{34},$
(5.60)

as well as

$\displaystyle w^{1,2,5,6}=0,\quad w^{41}=w^{42}=w^{45}=w^{46}=0,\quad
w^{27}=w^{23}=w^{35}=w^{57}=w^{36}=w^{67}=w^{13}=w^{17}=0.$ (5.61)

The stabiliser of both complementary pure spinors is thus 6-dimensional. The
geometric interpretation of the stabiliser is that it consists of ${\rm
SU}(1,1)$ mixing up the two complex null directions ${\bf\tilde{e}}^{1}-{\rm
i\,}{\bf\tilde{e}}^{2},{\bf\tilde{e}}^{5}+{\rm i\,}{\bf\tilde{e}}^{6}$, as
well as ${\rm SL}(2,{\mathbb{R}})$ mixing up the two real null directions
${\mathbb{I}}+{\bf\tilde{e}}^{4},{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{7}$.

We have thus described the three different types of pure spinors of ${\rm
Spin}(4,4)$. One gives a complex structure, one paracomplex, and the third
type gives a structure of the mixed type. Of these only the pure spinor giving
the paracomplex structure is real.

###  5.7 Majorana spinors

The Majorana spinors are simply the split octonions. There are three possible
types of such spinors. Non-null spacelike or timelike, and null. We have
determined the stabiliser of a null octonion in (5.51). As we have seen in the
previous subsection, a null octonion is a pure spinor. The stabilisers of
spacelike or timelike octonions are also 21 dimensional and in both cases are
given by ${\rm Spin}(4,3)$.

For future usage we compute explicitly the stabiliser subalgebra of
$\psi={\mathbb{I}}$. It is given by the following relations

$\displaystyle w^{1}=w^{23}-w^{45}+w^{67},\quad
w^{2}=-w^{13}-w^{46}-w^{57},\quad w^{3}=w^{12}-w^{47}+w^{56},$ (5.62)
$\displaystyle w^{4}=-w^{15}-w^{26}-w^{37},\quad
w^{5}=w^{14}-w^{27}+w^{36},\quad w^{6}=w^{17}+w^{25}-w^{35},$ $\displaystyle
w^{7}=-w^{16}+w^{25}+w^{34}.$

###  5.8 General spinors

We now enter into a less familiar territory, as there seems to be no known
classification of the orbits of the real ${\rm Spin}(4,4)$ on the complex Weyl
spinors, apart from already considered case of pure and Majorana spinors. This
is in contrast to the case of ${\rm Spin}(8)$, where there is only one
possible type of general spinors, with the stabiliser ${\rm SU}(4)$.

To classify general ${\rm Spin}(4,4)$ spinors we use the same idea that worked
in the ${\rm Spin}(8)$ case. We consider a general complex spinor, which is a
complexified split octonion $\psi=\alpha+{\rm i\,}\beta$. The relations (4.48)
are still valid. The only novelty now is that the norm squared does not need
to be positive. We assume that the spinor is not null (because if it is null
it is pure). We again rescale the spinor to make it unit, so that
$|\alpha|^{2}-|\beta|^{2}=1$ and $(\alpha,\beta)=0$. This gives $\langle
R(\psi),\psi\rangle=1+2|\beta|^{2}$. The novelty now is that this quantity
does not need to be greater than one.

There are five cases to consider. First, when $|\beta|^{2}>0$ both
$\alpha,\beta$ have positive norm and can be chosen to be multiples of
${\mathbb{I}},{\bf\tilde{e}}^{3}$. We get the unit spinor

$\displaystyle\psi=\cosh\tau{\mathbb{I}}+{\rm
i\,}\sinh\tau{\bf\tilde{e}}^{3}.$ (5.63)

It is clear that the analysis in the case of ${\rm Spin}(8)$ is unchanged, and
this spinor defines a pure spinor that is a multiple of ${\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}$, whose stabiliser is ${\rm SU}(2,2)$. Thus, the
general spinor of this type still defines a complex structure, and its
stabiliser is ${\rm SU}(2,2)$.

Another case is when $-1<|\beta|^{2}<0$. This means that $|\alpha|^{2}>0$, and
we can still choose $\alpha$ to be a multiple of ${\mathbb{I}}$. This leads us
to consider the unit spinor

$\displaystyle\psi=\cos\theta{\mathbb{I}}+{\rm
i\,}\sin\theta\,{\bf\tilde{e}}^{4}.$ (5.64)

We have $\langle
R(\psi),\psi\rangle=\cos^{2}\theta-\sin^{2}\theta=\cos(2\theta)$. Denoting
$\lambda=\cos(2\theta)$ we can form

$\displaystyle\tilde{\psi}=\frac{\lambda\psi-R(\psi)}{{\rm
i\,}\sqrt{1-\lambda^{2}}}={\rm
i\,}\sin\theta\,{\mathbb{I}}+\cos\theta\,{\bf\tilde{e}}^{4}.$ (5.65)

This is again a spinor of norm minus one, and so

$\displaystyle\psi\pm\tilde{\psi}=(\cos\theta+{\rm
i\,}\sin\theta)({\mathbb{I}}\pm{\bf\tilde{e}}^{4})$ (5.66)

are both null and thus pure spinors. They are (complex) multiples of the
spinors $({\mathbb{I}}\pm{\bf\tilde{e}}^{4})$ that we already encountered
before. This pair of pure spinor defines a paracomplex structure on
${\mathbb{R}}^{4,4}$. Thus, the general spinors of this type defines two pure
spinors of the type $({\mathbb{I}}\pm{\bf\tilde{e}}^{4})$, and thus defines a
paracomplex structure on ${\mathbb{R}}^{4,4}$. The stabiliser of a general
spinor of this type is thus ${\rm SL}(4,{\mathbb{R}})$.

Yet another case is $|\beta|^{2}<-1$. This means that both $\alpha,\beta$ have
negative norms. For example we can choose
$\alpha\sim{\bf\tilde{e}}^{4},\beta\sim{\bf\tilde{e}}^{7}$. This leads us to
consider the unit spinor

$\displaystyle\psi=\sinh\tau\,{\bf\tilde{e}}^{4}+{\rm
i\,}\cosh\tau\,{\bf\tilde{e}}^{7}.$ (5.67)

Then $\lambda=\langle R(\psi),\psi\rangle=-\cosh(2\tau)$ and

$\displaystyle\tilde{\psi}=\frac{\lambda\psi-R(\psi)}{\sqrt{\lambda^{2}-1}}=-\cosh\tau\,{\bf\tilde{e}}^{4}-{\rm
i\,}\sinh\tau\,{\bf\tilde{e}}^{7}.$ (5.68)

This is again a spinor of norm minus one, and the spinor

$\displaystyle\psi+\tilde{\psi}=(\sinh\tau-\cosh\tau)({\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7})$ (5.69)

is pure. This spinor defines a complex structure on ${\mathbb{R}}^{4,4}$ with
the Kähler form

$\displaystyle\langle({\bf\tilde{e}}^{4}-{\rm
i\,}{\bf\tilde{e}}^{7}),\Gamma\Gamma({\bf\tilde{e}}^{4}+{\rm
i\,}{\bf\tilde{e}}^{7})\rangle=2{\rm
i\,}({\mathbb{I}}\wedge{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{12}+{\bf\tilde{e}}^{47}+{\bf\tilde{e}}^{56}).$
(5.70)

Thus, the general spinor of this type still defines a complex structure, and
its stabiliser is ${\rm SU}(2,2)$.

There are also two cases when a null split octonion arises. When $\beta$ is
null, the octonion $\alpha$ is unit, and we are led to consider the spinor

$\displaystyle\psi={\mathbb{I}}+{\rm
i\,}({\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}).$ (5.71)

This is complex spinor whose real part is an identity octonion, and the
imaginary part is a null octonion. Then $\lambda=\langle
R(\psi),\psi\rangle=1$ and the construction of $\tilde{\psi}$ is no longer
applicable. It is clear that the geometry arising in this case knows both
about the geometry related to ${\mathbb{I}}$, and that of the real pure spinor
${\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}$. The identity octonion defines the
${\rm Spin}(7)$ invariant 4-form
$-{\mathbb{I}}\wedge\tilde{C}+{}^{*}\tilde{C}$ on ${\mathbb{R}}^{4,4}$. The
pure spinor ${\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}$ defines its null subspace
that can be seen to be spanned by

$\displaystyle{\mathbb{I}}-{\bf\tilde{e}}^{7},\quad{\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4},\quad{\bf\tilde{e}}^{2}-{\bf\tilde{e}}^{5},\quad{\bf\tilde{e}}^{1}+{\bf\tilde{e}}^{6}.$
(5.72)

To understand the geometry arising better we compute

$\displaystyle\langle({\mathbb{I}}+{\rm i\,}{\bf\tilde{e}}^{3}+{\rm
i\,}{\bf\tilde{e}}^{4}),\Gamma\Gamma({\mathbb{I}}-{\rm
i\,}{\bf\tilde{e}}^{3}-{\rm i\,}{\bf\tilde{e}}^{4})\rangle=2{\rm
i\,}(({\mathbb{I}}+{\bf\tilde{e}}^{7})\wedge({\bf\tilde{e}}^{3}-{\bf\tilde{e}}^{4})+({\bf\tilde{e}}^{2}+{\bf\tilde{e}}^{5})\wedge({\bf\tilde{e}}^{1}-{\bf\tilde{e}}^{6})).$
(5.73)

The 2-form that arises is thus a sum of two decomposable pieces, each built
entirely from null vectors that are complementary to those in (5.72).

It is also interesting to compute the stabiliser of ${\mathbb{I}}+{\rm
i\,}{\bf\tilde{e}}^{3}+{\rm i\,}{\bf\tilde{e}}^{4}$. It is clear that it is
the intersection of the stabilisers of ${\mathbb{I}}$ and
${\bf\tilde{e}}^{3}+{\bf\tilde{e}}^{4}$. Both are 21-dimensional. An explicit
calculation shows that this intersection is given by (5.62) supplemented by
the following relations

$\displaystyle w^{16}=w^{25},\quad w^{12}+w^{15}+w^{26}+w^{56}=0,\quad
w^{23}-w^{24}-w^{35}+w^{45}=0,\quad-w^{13}+w^{14}-w^{36}+w^{46}=0.$ (5.74)

It is thus $21-4=17$ dimensional.

##  6 Discussion

The Clifford algebras ${\rm Cl}(3),{\rm Cl}(5),{\rm Cl}(9)$ and ${\rm
Cl}(2,1),{\rm Cl}(3,2),{\rm Cl}(5,4)$ admit a uniform description as generated
by matrices of the form (1.10), where $q$ takes values in
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$ for the first list and the split
composition algebras
${\mathbb{C}}^{\prime},{\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$ for the
second list. It is also easy to generate related models for Clifford algebras
in one dimension up and one (or more) dimensions down. To go one dimension up
one uses the tensor product construction and generates off-diagonal $2\times
2$ matrices where the off-diagonal block entries are the matrices (1.10) as in
(2.103). The additional $\Gamma$-matrix is then the matrix (2.138). This again
gives a uniform description of ${\rm Cl}(3,1),{\rm Cl}(5,1),{\rm Cl}(9,1)$ and
${\rm Cl}(2,2),{\rm Cl}(3,3),{\rm Cl}(5,5)$. To go one dimension down one just
sets $r=0$ in (1.10), getting a description of ${\rm Cl}(2),{\rm Cl}(4),{\rm
Cl}(8)$ and ${\rm Cl}(1,1),{\rm Cl}(2,2),{\rm Cl}(4,4)$. It is possible to go
down in dimension even further. Thus, one can omit the $\Gamma$-matrix
corresponding to the identity element in
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}},{\mathbb{C}}^{\prime},{\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$.
This gives a model for ${\rm Cl}(1),{\rm Cl}(3),{\rm Cl}(7)$ and ${\rm
Cl}(0,1),{\rm Cl}(1,2),{\rm Cl}(3,4)$. One can go down in dimension even
further by selecting a unit imaginary element in
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}},{\mathbb{C}}^{\prime},{\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$
and then omitting the corresponding $\Gamma$-matrix. This gives a model for
${\rm Cl}(0),{\rm Cl}(2),{\rm Cl}(6)$ and ${\rm Cl}(0,0)$ for
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}},{\mathbb{C}}^{\prime}$. Of these only
the model for ${\rm Cl}(6)$ is of interest, and gives a useful and powerful
octonionic description of ${\rm Spin}(6)$. In the case of
${\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$ there is a choice of which
imaginary element to omit, and one can get models for either ${\rm Cl}(0,2)$
or ${\rm Cl}(1,1)$ in the case of ${\mathbb{H}}^{\prime}$, or ${\rm Cl}(2,4)$
and ${\rm Cl}(3,3)$ for ${\mathbb{O}}^{\prime}$. This gives a split octonionic
description of ${\rm Spin}(2,4)$ and ${\rm Spin}(3,3)$. Both are powerful and
useful descriptions of these groups. Especially the first of these should be
noted, as it is the conformal group of the Minkowski space.

The described constructions are relatively well-known, see e.g. [2], even
though their split versions are rarely discussed in the literature. The
novelty of this paper is that we have shown how the models based on
${\mathbb{C}},{\mathbb{H}},{\mathbb{O}},{\mathbb{C}}^{\prime},{\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$
arise from the creation/annihilation operator description of the relevant
Clifford algebras. Thus, we have seen that even and odd polyforms on
${\mathbb{C}}^{2}$ can be identified with either quaternions or split
quaternions. The difference between these two cases arises only in the form of
the anti-linear hat operator that acts on spinors and commutes with all
$\Gamma$-matrices. In the case of ${\rm Cl}(4)$ the hat operator squares to
minus the identity. It anti-commutes with the complex structure that acts on
the components of spinors as the operator of multiplication by $-{\rm i\,}$.
In the quaternionic description the complex structure that is the operator of
multiplication by $-{\rm i\,}$ gets translated into the operator $R_{\bf k}$
of right multiplication by a unit imaginary quaternion. The hat operator gets
translated into $-R_{\bf i}$. The two operators together generate the
quaternionic structure. In the case of ${\rm Cl}(2,2)$ the hat operator
squares to plus the identity. In the split quaternionic description we again
have the complex structure that translates as $R_{\bf\tilde{k}}$. The hat
operator translates into $R_{\bf\tilde{i}}$. Again this generates the
quaternionic structure, but this time it is the split quaternions that appear.

The difference in the cases of ${\rm Cl}(8)$ and ${\rm Cl}(4,4)$ is that even
and odd polyforms on ${\mathbb{C}}^{4}$ satisfying a reality condition are
identified with octonions and split octonions. The anti-linear operator
imposing the reality condition in both cases acts as the operator related to
the Hodge star operator on $\Lambda({\mathbb{C}}^{4})$. The difference between
${\rm Cl}(8)$ and ${\rm Cl}(4,4)$ is in some signs arising. After the reality
condition gets imposed, we can identify real even and odd polyforms with
either octonions or split octonions. Thus, Majorana-Weyl spinors of ${\rm
Cl}(8)$ and ${\rm Cl}(4,4)$ are octonions and split octonions respectively.
This gives a powerful description, because a general Weyl spinor can then be
described as a complexified octonion or a split octonion. The derivation of
the octonionic description from the one in terms of the creation/annihilation
operators makes the description of pure spinors of ${\rm Cl}(8)$ and ${\rm
Cl}(4,4)$ particularly clear. Pure spinors are null objects. In the case of
${\rm Cl}(8)$ they are null octonions and thus necessarily null complexified
octonions. In the case of ${\rm Cl}(4,4)$ there are three types of pure
spinors. There are now null split octonions, and they are pure spinors of real
index four. The other two types of pure spinors are described by complexified
octonions. There are two types that arise this way. In one type, the
complexified null octonion has real and imaginary parts that are not null, see
(5.40) for a representative of this orbit. This type of pure spinor defines a
complex structure on ${\mathbb{R}}^{4,4}$. In the other case the real and
imaginary parts are both null, see (5.55). This type of pure spinor defines
the structure of a mixed type with two real and two complex null directions.
We have also described the possible types of general spinors for both ${\rm
Cl}(8)$ and ${\rm Cl}(4,4)$. In the case of ${\rm Cl}(4,4)$ there is an orbit
that does not have a ${\rm Cl}(8)$ analog. It is the orbit when one of the two
octonions in $\alpha+{\rm i\,}\beta$ is null. We have characterised its
stabiliser and computed the arising in this case 2-form, see (5.73).

The two constructions we described, namely the general description of the
Clifford algebras using creation/annihilation operators, and the description
of some of the Clifford algebras using
${\mathbb{K}}={\mathbb{C}},{\mathbb{H}},{\mathbb{O}},{\mathbb{C}}^{\prime},{\mathbb{H}}^{\prime},{\mathbb{O}}^{\prime}$
allows to produce $\mathbb{K}$-based models for Clifford algebras other than
those mentioned above. The idea is very simple. As discussed, even and odd
polyforms on ${\mathbb{C}}^{2}$ and ${\mathbb{C}}^{4}$ can be identified with
quaternions or split quaternions and with (complexified) octonions or split
octonions. This means that polyforms on spaces containing the
${\mathbb{C}}^{2}$ and ${\mathbb{C}}^{4}$ factors can also be given a
description based on
${\mathbb{H}},{\mathbb{H}}^{\prime},{\mathbb{O}},{\mathbb{O}}^{\prime}$. We
have already seen how this works for ${\rm Cl}(6)$ in Section 2.4. One can
apply essentially the same construction to ${\rm Cl}(10)$ and generate its
octonionic description. The resulting octonionic formalism has already been
described in [5]. The semi-spinors of ${\rm Spin}(10)$ are then 2-component
columns with complexified octonions as entries. If desired, one can go in
dimension even higher and describe Clifford algebras such as ${\rm Cl}(12)$
and ${\rm Cl}(11,1)$ using octonions. Semi-spinors of both are 4-component
columns with complexified octonions as entries. We hope to return to the
octonionic description of the phenomenologically important cases of ${\rm
Spin}(10)$ and ${\rm Spin}(11,1)$ in another publication.

The starting point of a creation/annihilation operator model of ${\rm
Cl}(r,s)$ is a mixed type structure on ${\mathbb{R}}^{r,s}$, see the
accompanying paper [1]. This selects two complementary totally null subspaces
$E^{\pm}$ of the complexification of ${\mathbb{R}}^{r,s}$. The spinors in the
creation/annihilation operator description are then polyforms on one of the
two totally null subspaces. In particular, even model comes with two
”canonical” pure spinors whose annihilator subspaces are $E^{\pm}$.

The models based on composition algebras $\mathbb{K}$ are different. The
natural question is what is the geometric structure (if any) that a model of
this type introduces. For concreteness, let us discuss this question for the
case of ${\rm Spin}(8)$. As we have seen, the ${\mathbb{O}}$-based model
identifies Majorana-Weyl spinors of ${\rm Spin}(8)$ with octonions. A
Majorana-Weyl spinor of ${\rm Spin}(8)$ is never pure, and so there are no
preferred pure spinors that arise from the model. The spinors that can be said
to come with the model are the positive and negative helicity Majorana-Weyl
spinors that correspond to the identity octonion. The stabiliser of one of
them is ${\rm Spin}(7)$, the stabiliser of both of them is the group of
automorphisms of the octonions ${\rm G}_{2}$. So, it could be said that the
geometry on ${\mathbb{R}}^{8}$ that comes with the ${\mathbb{O}}$-based model
of ${\rm Spin}(8)$ is the one that reduces the group ${\rm Spin}(8)$ to ${\rm
G}_{2}$. Geometrically, a pair $\psi_{\pm}$ of Majorana-Weyl spinors of
opposite helicity defines a vector in ${\mathbb{R}}^{8}$, via
$\langle\psi_{+},\Gamma\psi_{-}\rangle$. Also, each of the two Majorana-Weyl
spinors defines the 4-form (4.44), with the relative sign between the two
terms in this form different for the two spinors. Inserting the vector
${\mathbb{I}}\in{\mathbb{O}}\sim{\mathbb{R}}^{8}$ into this 4-form one gets
the 3-form $C\in\Lambda^{3}({\mathbb{R}}^{7})$, whose stabiliser is ${\rm
G}_{2}$. This is the geometry defined by the two ”canonical” Majorana-Weyl
spinors ${\mathbb{I}}\in S_{\pm}$ that the ${\mathbb{O}}$-based model of ${\rm
Spin}(8)$ comes with.

## Acknowledgement

KK is grateful to F. Reese Harvey for correspondence.

## References

  * [1] N. Bhoja and K. Krasnov, ”Notes on spinors and polyforms I: General case.”
  * [2] F. Reese Harvey, ”Spinors and Calibrations”, Academic Press, 1990.
  * [3] W. Kopczynski and A. Trautman, ”Simple spinors and real structures”, J. Math. Phys. 33 (2) (1992) 550-559.
  * [4] P. Charlton, ”The Geometry of Pure Spinors, with Applications”, available as http://csusap.csu.edu.au/~pcharlto/charlton_thesis.pdf
  * [5] R. Bryant, ”Notes on spinors in low dimension,” arXiv:2011.05568 [math.GR].