G(F_{\mu\sigma}F_{\nu}^{\sigma}-g_{\mu\nu}\frac{1}{4}F_{\rho\sigma}F^{\rho\sigma}),$
(3.99)

where
$F_{\mu\sigma}F_{\nu}^{\sigma}-g_{\mu\nu}\frac{1}{4}F_{\rho\sigma}F^{\rho\sigma}$
is the electromagnetic stress-energy tensor. The spinor form of the Einstein-
Maxwell equation [15] is

$\displaystyle\Phi_{ABA^{\prime}B^{\prime}}=8\pi
G\varphi_{AB}\bar{\varphi}_{A^{\prime}B^{\prime}},$ (3.100)

where $\varphi_{AB},\bar{\varphi}_{A^{\prime}B^{\prime}}$ are decomposed
spinors of electromagnetic tensor $F_{\mu\nu}$, as following (3.18) and
(3.19). This can be deformed to

$\displaystyle\Phi_{A\quad B^{\prime}}^{\;\;\;BA^{\prime}}=8\pi
G\varphi_{A}^{\;\;B}\bar{\varphi}^{A^{\prime}}_{\;\;B^{\prime}}.$ (3.101)

From Eq. (3.86),

$\displaystyle-\frac{1}{4}\Theta_{kl}\sigma^{k\;\;\;B}_{\;A}\sigma^{l\,A^{\prime}}_{\;\;\;\;B^{\prime}}$
$\displaystyle=8\pi
G[\frac{1}{2}(F_{k0}-\frac{1}{2}i\,\epsilon_{\;\;k}^{ij}F_{ij})\sigma^{k\;\;\;B}_{\;A}]\times[\frac{1}{2}(F_{l0}+\frac{1}{2}i\,\epsilon_{\;\;\;l}^{pq}F_{pq})\sigma^{l\,A^{\prime}}_{\;\;\;\;B^{\prime}}]$
$\displaystyle=2\pi
G[(F_{k0}F_{l0}+\frac{1}{4}\epsilon_{\;\;k}^{ij}\epsilon_{\;\;\;l}^{pq}F_{ij}F_{pq})+\frac{i}{2}(F_{k0}\epsilon_{\;\;\;l}^{pq}F_{pq}-F_{l0}\epsilon_{\;\;k}^{ij}F_{ij})]\sigma^{k\;\;\;B}_{\;A}\sigma^{l\,A^{\prime}}_{\;\;\;\;B^{\prime}}~{}.$
(3.102)

Comparing the first line with the third line in (3.102), we get

$\displaystyle S_{kl}$ $\displaystyle=-8\pi
G(F_{k0}F_{l0}+\frac{1}{4}\epsilon_{\;\;k}^{ij}\epsilon_{\;\;\;l}^{pq}F_{ij}F_{pq}),$
(3.103) $\displaystyle P_{kl}$ $\displaystyle=-4\pi
G(F_{k0}\epsilon_{\;\;\;l}^{pq}F_{pq}-F_{l0}\epsilon_{\;\;k}^{ij}F_{ij}),$
(3.104)

and, from Eq. (3.104) we get

$\displaystyle\epsilon^{mkl}P_{kl}=-16\pi GF_{k0}F^{mk}.$ (3.105)

For $F^{\mu\nu}$ such that

$\displaystyle F^{\mu\nu}=\begin{pmatrix}0&-E_{1}&-E_{2}&-E_{3}\\\
E_{1}&0&-B_{3}&B_{2}\\\ E_{2}&B_{3}&0&-B_{1}\\\
E_{3}&-B_{2}&B_{1}&0&\end{pmatrix},$ (3.106)

we have $F_{k0}F^{mk}=(\vec{E}\times\vec{B})^{m}$ and
$(F_{k0}F_{l0}+\frac{1}{4}\epsilon_{\;\;k}^{ij}\epsilon_{\;\;\;l}^{pq}F_{ij}F_{pq})=E_{k}E_{l}+B_{k}B_{l}$.
From Eqs. (3.51) and (3.104) we get

$\displaystyle-\frac{1}{2}\epsilon^{mkl}P_{kl}$
$\displaystyle=R_{0i}^{\;\;\;\;mi}=8\pi G(\vec{E}\times\vec{B})^{k}~{}.$
(3.107)

This is a momentum of electromagnetic tensor and it shows that $P_{ij}/(8\pi
G)$ is related to momentum. From (3.49) and (3.103),

$\displaystyle S_{kl}$
$\displaystyle=R_{0i\;0j}+\frac{1}{4}\epsilon_{pqi}\epsilon_{rsj}R_{pq\;rs}=8\pi
G(-E_{k}E_{l}-B_{k}B_{l}),$ (3.108) $\displaystyle S_{l\,l}$
$\displaystyle=8\pi G(|\vec{E}|^{2}+|\vec{B}|^{2}).$ (3.109)

Those are the shear stress and the energy of electromagnetic field. It shows
that $S_{ij}/(8\pi G)$ is related to stress-energy.

####  3.6.3 The quaternion form of differential Bianchi identity: Another
Example of Section V

The spinor form of Bianchi identity (3.27) is known [15] as

$\displaystyle\nabla^{A}_{B^{\prime}}X_{ABCD}=\nabla^{A^{\prime}}_{B}\Phi_{CDA^{\prime}B^{\prime}},$
(3.110)

which can be deformed to

$\displaystyle\nabla^{B^{\prime}A}X_{A\;\;C}^{\;\;B\;\;D}=\nabla^{BA^{\prime}}\Phi_{C\;\;A^{\prime}}^{\;\;D\;\;B^{\prime}}.$
(3.111)

In flat coordinate, $\nabla^{A^{\prime}A}$ equals to

$\displaystyle\partial^{A^{\prime}A}=g^{\mu\;A^{\prime}A}\partial_{\mu}=\frac{1}{\sqrt{2}}\sigma^{\mu
A^{\prime}A}\partial_{\mu}=\frac{1}{\sqrt{2}}\sigma^{\tilde{\mu}A^{\prime}A}\partial_{\tilde{\mu}}$
$\displaystyle=\frac{1}{\sqrt{2}}\bar{q}^{\mu\;A^{\prime}A}\tilde{\partial}_{\mu}=\frac{1}{\sqrt{2}}q^{\mu\;A^{\prime}A}\partial^{\prime}_{\mu}~{},$
(3.112)

where $\tilde{\mu}$ is tilde-spacetime indices which is defined as
$O^{\tilde{\mu}}=(O^{0},i\,O^{1},i\,O^{2},i\,O^{3})$,
$O_{\tilde{\mu}}=(O_{0},-i\,O_{1},-i\,O_{2},-i\,O_{3})$ for any
$O^{\mu}=(O^{0},O^{1},O^{2},O^{3})$, $O_{\mu}=(O_{0},O_{1},O_{2},O_{3})$ [28];
$\bar{q}^{\mu}$ is
$\bar{q}^{\mu}=\sigma^{\tilde{\mu}A^{\prime}A}=(\sigma^{0},i\sigma^{1},i\sigma^{2},i\sigma^{3})$
which is isomorphic to $(1,-\mathbf{i},-\mathbf{j},-\mathbf{k})$,
$\tilde{\partial}_{\mu}=\partial_{\tilde{\mu}}=(\partial_{0},-i\partial_{1},-i\partial_{2},-i\partial_{3})$,
and
$\partial^{\prime}_{\mu}=(\partial_{0},i\partial_{1},i\partial_{2},i\partial_{3})$.
We used the property $A_{\mu}B^{\mu}=A_{\tilde{\mu}}B^{\tilde{\mu}}$ [28].
$\partial^{A^{\prime}A}$ can be expanded to

$\displaystyle\partial^{A^{\prime}A}$
$\displaystyle=\partial_{0}\delta^{A^{\prime}A}+\partial^{\prime}_{k}q^{k\;A^{\prime}A}$
(3.113)

and, considering matrix representation, $\partial^{AA^{\prime}}$ can be
represented as

$\displaystyle\partial^{AA^{\prime}}$
$\displaystyle=\partial_{0}\delta^{AA^{\prime}}+\partial^{\prime}_{k}q^{\bar{k}\;AA^{\prime}}=\partial_{0}\delta^{AA^{\prime}}+\partial^{\prime}_{\bar{k}}q^{k\;AA^{\prime}}~{},$
(3.114)

where the bar index $A^{\bar{k}}$ means the opposite-handed quantity of
$A^{k}$, which is $A^{\bar{1}}=A^{1},A^{\bar{2}}=-A^{2},A^{\bar{3}}=A^{3}$ ;
when $k=2$, $\bar{k}$ index change sings of $A^{k}$. It has following
properties,

$\displaystyle A^{\bar{k}}B_{k}=A^{k}B_{\bar{k}},\quad
A^{\bar{k}}B_{\bar{k}}=A^{k}B_{k},\quad\varepsilon_{pqr}A^{\bar{q}}B^{\bar{r}}=-\varepsilon_{\bar{p}qr}A^{q}B^{r},\quad\varepsilon_{\bar{p}\bar{q}\bar{r}}=-\varepsilon_{pqr}.$
(3.115)

Then Eq. (3.111) can be written as

$\displaystyle(\partial_{0}\delta^{B^{\prime}A}+\partial^{\prime}_{k}q^{k\;B^{\prime}A})\,\Xi_{ir}q^{i\;\;B}_{\;A}q^{r\;\;D}_{\;C}=(\partial_{0}\delta^{BA^{\prime}}+\partial^{\prime}_{\bar{k}}q^{k\;BA^{\prime}})\,\Phi_{r\bar{s}}q^{r\;\;D}_{\;C}q^{s\;\;B^{\prime}}_{\;A^{\prime}},$
(3.116)

since
$q^{\bar{s}\;\;B^{\prime}}_{\;A^{\prime}}=\varepsilon^{B^{\prime}D^{\prime}}\varepsilon_{C^{\prime}A^{\prime}}q^{s\,C^{\prime}}_{\;\;\;\;D^{\prime}}$.
Using Eq. (3.89),

$\displaystyle\partial_{0}\Xi_{ir}q^{i\;B^{\prime}B}+\partial^{\prime}_{k}\Xi_{kr}\delta^{B^{\prime}B}+\varepsilon_{pki}\partial^{\prime}_{k}\Xi_{ir}q^{p\;B^{\prime}B}$
$\displaystyle=\partial_{0}\Phi_{rs}q^{\bar{s}\;BB^{\prime}}+\partial^{\prime}_{k}\Phi_{rk}\delta^{BB^{\prime}}+\varepsilon_{pks}\partial^{\prime}_{\bar{k}}\Phi_{r\bar{s}}q^{p\;B^{\prime}B}~{},$
(3.117)

which can be rearranged as

$\displaystyle\partial^{\prime}_{k}(\Xi_{kr}-\Phi_{rk})\delta^{B^{\prime}B}+[\partial_{0}(\Xi_{sr}-\Phi_{rs})+\varepsilon_{ski}\partial^{\prime}_{k}(\Xi_{ir}+\Phi_{ri})]q^{s\;B^{\prime}B}=0~{}.$
(3.118)

This is the quaternion form of Bianchi identities.

###  3.7 Conclusion

In chpater 3, we established a new method to express curvature spinors, which
allows us to grasp components of the spinors easily in a locally inertial
frame. During such a process, we technically utilized modified sigma matrices
as a basis, which are sigma matrices multiplied by $\varepsilon$, and
calculated the product of sigma matrices with mixed spinor indices. Using
those modified sigma matrices as a basis can be regarded as the rotation of
the basis of four sigma matrices
$(\sigma^{0},\sigma^{1},\sigma^{2},\sigma^{3})$ to $(s^{0},s^{1},s^{2},s^{3})$
defined in Eq. (3.14), similar to a rotation of quaternion basis as shown in
our previous work [28]. By comparing the Ricci spinor with the spinor form of
Einstein equation, we could appreciate the roles of each component of the
Riemann curvature tensor. Furthermore, from the representation of Weyl
conformal spinor, we find that the components of Weyl tensor can be replaced
by complex quantities $\Xi_{ij}$, which are defined in Eq. (3.63).

We represented the elements of sedenion basis as the direct product of
elements of quaternion basis themselves. And then we defined a new algebra
‘sedon’, which has the same basis representation except slightly modified
multiplication rule from the multiplication rule of sedenion. The relations
between sedon and the curvature spinors are derived for a general gravitaional
field, not just for a weak gravitational field. We calculated multiplications
of spinors with a quaternion form, and observed that the results of the
multiplications are also represented in a sedon form.

We observed that many gravitational quantities can be represented with
3-dimensional basis. It shows that time and space may be interpreted
differently from conventional interpretations in which time and space are
treated as the same. And the relations among quaternion, sedon and the
representations of curvature spinors imply that gravity may come from a
combination of right-handed and left-handed abstract rotational operations.

##  Chapter 4 Conclusion

Our research verified the relationship between quaternion and 2-spinor
formalism in electromagnetism, and applied the relationship to the gravity.
From the amazing fitting of quaternion with physics, we could seek more
fundamental interpretations for the origin of electromagnetic force,
gravitational force, and spacetime.

We showed the reason why quaternion is useful to represent electromagnetic
law. The reason comes from the equivalence between the quaternion
representation and the 2-spinor formalism representation about electromagnetic
fields. Then using tilde-spacetime index notation, it is shown that all
tensors with spacetime indices and the spinors which is equivalent to the
tensors are trivially related to quaternion. We explained that quaternion can
be considered as not only a Wick rotated spacetime basis, which rotated in the
direction of space, but also an operator changing spinor basis. We suggested
that, since the Cayley-Dickson algebras can be expressed as many levels of
imaginary numbers and quaternion is one of algebras which can be represented
with two different imaginary number, quaternion can be thought of 2-folded
rotation. All of these gives the rotational interpretation about
electromagnetic forces and spacetime.

The relationship between quaternion representation and 2-spinor formalism
representation about electromagnetic fields was applied to representing the
components of curvature spinors as matrices in locally flat coordinate. The
representation has been formed with ‘3dim$\otimes$3dim’ basis, which implies
that there may be other interpretations that do not treat time as the 0-th
component of 4 dimensional spacetime. From the matrix representation of
spinors, we also could see that all information of Riemann curvature tensors
and Weyl tensors were put in more compact factors in locally flat coordinate.
We saw that, since the representation can be considered as sedon whose basis
is consisted of ‘quaternion basis $\otimes$ quaternion basis’, curvature
spinors can be thought of a combination of abstract left-handed rotation and
right-handed rotation. As shown these things, using the relationship between
quaternion and 2-spinor formalism, we get many different aspects about
gravitational forces.

It is attractive that structure of quaternion is similar to a spacetime
structure that is interpreted as having three dimensions of space and one-
dimensional time. We have given the meaning of the rotating operator to
quaternion and interpreted it as the operator applied to the base of spinor
which is also related to rotation. In addition, Cayley-Dickson algebras like
complex number, quaternion, octonion and sedenion was given the meaning of
n-folded rotations and tried to associate these with physics. This gives the
impression that some spins are the source of nature’s fundamental things. It
also gives the impression that it can be related to Yinyang, the four
phenomena(Sìxiàng) or eight trigrams(Bagua) in Oriental philosophy.

Here we linked quaternion with electromagnetism and sedenion with gravity, but
we did not discussed much about octonion. There are already some papers which
discuss about the relationship between octonion and strong interaction [25].
Since quaternion is linked to SU(2) and octonion is linked to SU(3), it makes
us to guess that sedenion is linked to SU(4). In view of these things, though
we have found the many meaningful connections between Cayley-Dickson algebra
and fundamental forces, there should be more researches which give more
consistent and general structure of connection between fundamental forces and
Cayley-Dickson algebra.

## Appendix

chapter[0pt]

###  A.1 Expansions of a few Quaternion Products in Equation (2.6)

####  A.1.1 Expansions of Products in (3), (4) and (5) of Equation (2.6)

Using the multiplication expression shown in Equation (1.4), the left sides of
the relations (3), (4) and (5) in Equation (2.6) are expanded as follows.

$\displaystyle\@text@daccent{d}\bar{\d{A}}$
$\displaystyle=(\frac{\partial}{\partial
t}-i\nabla)(V-i\vec{A})=(\frac{\partial V}{\partial
t}+\nabla\cdot\vec{A})+i\,(-\nabla V-\frac{\partial\vec{A}}{\partial
t})-(\nabla\times\vec{A})$ (4.1) $\displaystyle\@text@daccent{d}\bar{\d{F}}$
$\displaystyle=(\frac{\partial}{\partial t}-i\nabla)(-i\vec{E}-\vec{B})$ (4.2)
$\displaystyle=\nabla\cdot\vec{E}-i\nabla\cdot\vec{B}-(\nabla\times\vec{E}+\frac{\partial\vec{B}}{\partial
t})+i\,(-\frac{\partial\vec{E}}{\partial t}+\nabla\times\vec{B})$
$\displaystyle\@text@daccent{d}\bar{\d{J}}$
$\displaystyle=(\frac{\partial}{\partial
t}-i\nabla)(\rho-i\vec{J})=(\frac{\partial\rho}{\partial
t}+\nabla\cdot\vec{J})+i\,(-\nabla\rho+\frac{\partial\vec{J}}{\partial
t})-(\nabla\times\vec{J})$ (4.3)

####  A.1.2 The proof of (7) in Equation (2.6)

Here we show that $(\d{F}\d{d})\bar{\d{F}}$ is equal to
$\d{F}(\d{d}\bar{\d{F}})$ where $(\d{F}\d{d})$ is the quaternion differential
operator.

$\displaystyle(\d{F}\d{d})\bar{\d{F}}$
$\displaystyle=\left[(i\vec{E}-\vec{B})(\partial_{t}-i\nabla)\right](-\vec{E}i-\vec{B})$
$\displaystyle=\left[-\vec{E}\partial_{t}\cdot\vec{E}-(\vec{B}\times\nabla)\cdot\vec{E}-\vec{B}\partial_{t}\cdot\vec{B}+(\vec{E}\times\nabla)\cdot\vec{B}\right]$
$\displaystyle+\left[-i\vec{B}\partial_{t}\cdot
E+i(\vec{E}\times\nabla)\cdot\vec{E}+i\vec{E}\partial_{t}\cdot\vec{B}+i(\vec{B}\times\nabla)\cdot\vec{B}\right]$
$\displaystyle+\left[\vec{E}\partial_{t}\times\vec{E}-(\vec{B}\cdot\nabla)\vec{E}+(\vec{B}\times\nabla)\times\vec{E}+\vec{B}\partial_{t}\times\vec{B}+(\vec{E}\cdot\nabla)\vec{B}-(\vec{E}\times\nabla)\times\vec{B}\right]$
$\displaystyle+\left[i\vec{B}\partial_{t}\times\vec{E}+i(\vec{E}\cdot\nabla)\vec{E}-i(\vec{E}\times\nabla)\times\vec{E}-i\vec{E}\partial_{t}\times\vec{B}+i(\vec{B}\cdot\nabla)\vec{B}-i(\vec{B}\times\nabla)\times\vec{B}\right]$
(4.4)
$\displaystyle=\vec{J}\cdot\vec{E}+i\vec{B}\cdot\vec{J}+\rho\vec{B}+\vec{J}\times\vec{E}+i(\rho\vec{E}+J\times\vec{B})$
$\displaystyle=\d{F}\d{J},$

where we have used the following relations:

$\displaystyle(\vec{B}\times\nabla)\cdot\vec{E}$
$\displaystyle=\epsilon_{ijk}B_{i}\nabla_{j}\vec{E}_{k}=\vec{B}\cdot(\nabla\times\vec{E}),$
(4.5) $\displaystyle\left[(\vec{B}\times\nabla)\times\vec{E}\right]_{i}$
$\displaystyle=\epsilon_{ipq}(\epsilon_{pjk}B_{j}\nabla_{k})E_{q}=(\delta_{qj}\delta_{ik}-\delta_{qk}\delta_{ij})(B_{j}\nabla_{k}\vec{E}_{q})$
(4.6) $\displaystyle=B_{j}\nabla_{i}E_{j}-B_{i}(\nabla\cdot\vec{E}),$
$\displaystyle\left[\vec{B}\times(\nabla\times\vec{E})\right]_{i}$
$\displaystyle=\epsilon_{iqp}B_{q}(\epsilon_{pjk}\nabla_{j})E_{k}=(\delta_{ij}\delta_{qk}-\delta_{ik}\delta_{qj})B_{q}\nabla_{j}E_{k}$
(4.7) $\displaystyle=B_{j}\nabla_{i}E_{j}-(\vec{B}\cdot\nabla)E_{i},$
$\displaystyle(\vec{B}\times\nabla)\times\vec{E}$
$\displaystyle=\vec{B}\times(\nabla\times\vec{E})+(B\cdot\nabla)\vec{E}-\vec{B}(\nabla\cdot\vec{E}).$
(4.8)

###  B.2 The Proof of Equation (2.19)

$\nabla\times\mathfrak{p}=\nabla\times(\vec{E}\times\vec{B})=(\vec{E}\cdot\nabla)\vec{B}-(\vec{B}\cdot\nabla)\vec{E}+\vec{E}(\nabla\cdot\vec{B})-\vec{B}(\nabla\cdot\vec{E}).$
(4.9) $\displaystyle\nabla(\vec{E}\cdot\vec{B})$
$\displaystyle=\vec{E}\nabla\vec{B}+\vec{B}\nabla\vec{E}$ (4.10)
$\displaystyle=\vec{E}\times(\nabla\times\vec{B})+\vec{B}\times(\nabla\times\vec{E})+(\vec{E}\cdot\nabla)\vec{B}+(\vec{B}\cdot\nabla)E\;\;\;\text{(from
(\ref{df}))}$
$\displaystyle=\vec{E}\times(\partial_{t}\vec{E})+\vec{E}\times\vec{J}+\vec{B}\nabla\vec{E}+(\vec{E}\cdot\nabla)\vec{B}$
$\displaystyle=-\vec{B}\times(\partial_{t}\vec{B})+\vec{E}\nabla\vec{B}+(\vec{B}\cdot\nabla)\vec{E}$

where $(\vec{A}\nabla\vec{B})_{i}\equiv A_{j}(\nabla_{i}B_{j})$ for vector
fields $\vec{A}$ and $\vec{B}$. Substituting this into Equation (4.9), we get
Equation (2.19).

###  C.3 The Proof of Equation (2.23)

As we mentioned on Equation (2.3), $\d{q}=q_{0}+i\vec{q}$ is isomorphic to
$q_{\mu}\sigma^{\mu}$ where
$q_{\mu}=(q_{0},\vec{q})=(q_{0},q_{1},q_{2},q_{3})$,
$\sigma^{\mu}=\\{\sigma^{0},\sigma^{1},\sigma^{2},\sigma^{3}\\}$ and
$g_{\mu\nu}=(1,-1,-1,-1)$.

Let us introduce some quantities that are isomorphic to some quaternions

$\displaystyle\partial_{\mu}\bar{\sigma}^{\mu}\;\quad\sim\;\;\d{d}=\partial_{t}-i\nabla$
(4.11) $\displaystyle
M_{\mu}\bar{\sigma}^{\mu}\quad\sim\;\;\d{F}=i\vec{E}-\vec{B}$ (4.12)
$\displaystyle N_{\mu}\sigma^{\mu}\quad\sim\;\;\bar{\d{F}}=-i\vec{E}-\vec{B},$
(4.13)

where $\partial_{\mu}=(\partial_{t},\partial_{x},\partial_{y},\partial_{z})$,
$M_{\mu}=(0,-\vec{E}-i\vec{B})$, $N_{\mu}=(0,-\vec{E}+i\vec{B})$.

Then,

$\displaystyle\@text@daccent{d}(\bar{\d{F}}\d{F})$
$\displaystyle\sim\partial_{\rho}\bar{\sigma}^{\rho}(N_{\mu}\sigma^{\mu}M_{\nu}\bar{\sigma}^{\nu})=(\partial_{\rho}N_{\mu}M_{\nu})\bar{\sigma}^{\rho}\sigma^{\mu}\bar{\sigma}^{\nu}$
(4.14)
$\displaystyle=(\partial_{\rho}N_{\mu})M_{\nu}\bar{\sigma}^{\rho}\sigma^{\mu}\bar{\sigma}^{\nu}+N_{\mu}(\partial_{\rho}M_{\nu})\bar{\sigma}^{\rho}\sigma^{\mu}\bar{\sigma}^{\nu}$
$\displaystyle=(\partial_{\rho}N_{\mu})M_{\nu}\bar{\sigma}^{\rho}\sigma^{\mu}\bar{\sigma}^{\nu}+N_{\mu}(\partial_{\rho}M_{\nu})(2g^{\mu\rho}-\bar{\sigma}^{\mu}\sigma^{\rho})\bar{\sigma}^{\nu}$
$\displaystyle=(\partial_{\rho}N_{\mu})M_{\nu}\bar{\sigma}^{\rho}\sigma^{\mu}\bar{\sigma}^{\nu}+2(N^{\mu}\partial_{\mu})M_{\nu}\bar{\sigma}^{\nu}-N_{\mu}(\partial_{\rho}M_{\nu})\bar{\sigma}^{\mu}\sigma^{\rho}\bar{\sigma}^{\nu}$
$\displaystyle=(\partial_{\rho}\bar{\sigma}^{\rho}N_{\mu}\sigma^{\mu})M_{\nu}\bar{\sigma}^{\nu}-N_{\mu}\bar{\sigma}^{\mu}(\partial_{\rho}\sigma^{\rho}M_{\nu}\bar{\sigma}^{\nu})+2(N^{\mu}\partial_{\mu})M_{\nu}\bar{\sigma}^{\nu}$
$\displaystyle\sim(\d{d}\bar{\d{F}})\d{F}+\bar{\d{F}}(\bar{\d{d}}\d{F})+2i(\bar{\d{F}}\cdot\nabla)\d{F}.$

We have used the relations
$(\bar{\sigma}^{\mu}\sigma^{\nu}+\bar{\sigma}^{\nu}\sigma^{\mu})^{A^{\prime}}_{\;\;B^{\prime}}=2g^{\mu\nu}\delta^{A^{\prime}}_{\;\;B^{\prime}}$
[52] for spinor indices $A,B,A^{\prime}$ and $B^{\prime}$.

chapter[0pt]

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