p 3:
II. METHOD AND MAIN RESULTS
In four dimensional flat spacetime ##M=\left(R^4, \eta_{a b}\right)##, the spinor equations describing free massless spin ##s \geq 1 / 2## fields are given by
$$
\begin{equation*}
\partial_{A^{\prime}}^{A_1} \phi_{A_1 \cdots A_{2 s}}(x)=0, \tag{2.1}
\end{equation*}
$$
where ##\phi_{A_1 \cdots A_{2 s}}## is a symmetric spinor representing the spin ##s## field strength [1].
Here ##\partial_{A A^{\prime}}= e^a{ }_{A A^{\prime}} \partial_a## is the spinorial derivative associated with the metric compatible derivative ##\partial_a## on ##M##, i.e., ##\partial_c \eta_{a b}=0 ; e^a{ }_{A A^{\prime}}## is the soldering form given by a complex-valued null tetrad basis for the metric ##\eta_{a b}## satisfying ##e^a{ }_{A A^{\prime}} e^b{ }_{B B^{\prime}} \eta_{a b}=\epsilon_{A B} \epsilon_{A^{\prime} B^{\prime}}##, where ##\epsilon_{A B}## is the spin metric.
Throughout this paper we use the index notation and conventions of Ref. [1];
the metric signature is ##(+,-,-,-)##, and the spin metric is used for raising/lowering indices.
Note that, in standard Minkowski coordinates ##x^\mu##, the components of the derivative operators ##\partial_a## and ##\partial_{A A^{\prime}}## are simply the coordinate partial derivatives ##\partial_\mu=\partial / \partial x^\mu## and ##e^\mu{ }_{A A^{\prime}} \partial_\mu=\partial / \partial x^{A A^{\prime}}##, respectively, with ##x^{A A^{\prime}}=e_\mu^{A A^{\prime}} x^\mu##.
...
p 18: A. Electric and magnetic spin ##s \geq 1/2## fields
As a preliminary, we consider the familiar case ##s=1##.
Using a timelike unit vector ##t^{A A^{\prime}}##, we define the electric and magnetic parts of the ##s=1## field strength ##\phi_{A B}## by ##\vec{E}_{A A^{\prime}}+\mathrm{i} \vec{B}_{A A^{\prime}}= t_A{ }^{B^{\prime}} \bar{\phi}_{A^{\prime} B^{\prime}}##, where ##\vec{E}_{A A^{\prime}}, \vec{B}_{A A^{\prime}}## represent real-valued vectors satisfying ##t^{A A^{\prime}} \vec{E}_{A A^{\prime}}=t^{A A^{\prime}} \vec{B}_{A A^{\prime}}=## 0 .
Thus ##\vec{E}_{A A^{\prime}}, \vec{B}_{A A^{\prime}}## have no time component, and we thereby obtain the decomposition ##\bar{\phi}_{A^{\prime} B^{\prime}}=2 t_{A^{\prime}}{ }^A\left(\vec{E}_{A B^{\prime}}+\mathrm{i} \vec{B}_{A B^{\prime}}\right)##.
Now, we split the ##s=1## field equations ##\bar{\phi}_{A^{\prime} B^{\prime}, A}{ }^{B^{\prime}}=0## into time and space components with respect to ##t^{A A^{\prime}}##.
Let ##D_t=t^{A A^{\prime}} D_{A A^{\prime}}## denote the total time derivative, and ##\vec{D}_{A A^{\prime}}=D_{A A^{\prime}}-t_{A A^{\prime}} D_t## denote the total spatial gradient.
The splitting then yields
$$
\begin{align*}
& D_t \vec{E}_{A A^{\prime}}=\vec{D} \times \vec{B}_{A A^{\prime}}, \quad \vec{D} \cdot \vec{E}=0 \tag{5.2}\\
& D_t \vec{B}_{A A^{\prime}}=-\vec{D} \times \vec{E}_{A A^{\prime}}, \quad \vec{D} \cdot \vec{B}=0 \tag{5.3}
\end{align*}
$$
where ##\vec{D}\cdot ## and ##\vec{D} \times## denote the standard spatial divergence and curl operators which act on spinorial vector functions ##v_{A A^{\prime}}## satisfying ##t^{A A^{\prime}} v_{A A^{\prime}}=0## on ##J^{\infty}(\phi)## by
$$
\begin{equation*}
\vec{D} \cdot v=\vec{D}^{A A^{\prime}} v_{A A^{\prime}}, \quad \vec{D} \times v_{A A^{\prime}}=\mathrm{i} t^{B B^{\prime}}\left(\vec{D}_{A B^{\prime}} v_{B A^{\prime}}-\vec{D}_{B A^{\prime}} v_{A B^{\prime}}\right) \tag{5.4}
\end{equation*}
$$
Thus, equations (5.2) and (5.3) describe an electric-magnetic formulation of the ##s=1## field equations, comprising a spinorial version of the Maxwell equations.
We now proceed analogously for integer spins ##s=1,2, \ldots##. Write
$$
\begin{equation*}
t_{A_1}{ }^{B_1^{\prime}} \cdots t_{A_s}{ }^{B_s^{\prime}} \bar{\phi}_{A_1^{\prime} \cdots A_s^{\prime} B_1^{\prime} \cdots B_s^{\prime}}=\vec{E}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}+\mathrm{i} \vec{B}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}, \tag{5.5}
\end{equation*}
$$
where ##\vec{E}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}, \vec{B}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}## are real symmetric spinors satisfying
$$
\begin{equation*}
t^{A_1 A_1^{\prime}} \vec{E}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}=t^{A_1 A_1^{\prime}} \vec{B}_{A_1 \cdots A_s A_1^{\prime} \cdots A_s^{\prime}}=0 . \tag{5.6}
\end{equation*}
$$ ...
[1] Penrose, R. and Rindler, W. , Spinors and Space-time, Volumes I, II, (Cambridge University Press 1986)