algebras and spin representation. Let $V$ be a vector space of dimension $n$ endowed with a non-degenerate symmetric bilinear form ${(\cdot,\cdot)}$. Its Clifford algebra ${\mathcal{C}(V)}$ is defined as the quotient of the tensor algebra $\oplus_{k=0}^n V^{\otimes k}$ by the relations $x\otimes y+y\otimes x=2{(x,y)}$, with $x,y\in V$. It is naturally an associative ${\mathbb{Z}}_2$-graded algebra. Up to isomorphisms, there exists a unique irreducible module $S$ of ${\mathcal{C}(V)}$ called spin representation [@MR1636473]. The vectors of $S$ are called spinors.

An operator $O$ on $S$ is called even (or of degree 0) if $O(S^i)\subset S^{i}$ and odd (or of degree 1) if $O(S^i)\subset
S^{i+1}$. Here $i\in{\mathbb{Z}}_2$. If $O_1$ and $O_2$ are operators of degree $d_1$ and $d_2$ respectively, then their commutator is the operator $${\lfloor O_1,O_2\rfloor}= O_1\circ O_2-(-1)^{d_1 d_2}O_2\circ O_1 .$$

\[translagr\] Let $W$ be a vector space of dimension $r$. We can endow $V=W\oplus
W^*$ with the non-degenerate pairing $$\label{5} {(u_1+\xi_1,u_2+\xi_2)}={\tfrac{1}{2}}\big(\xi_1(u_2)+
\xi_2(u_1)\big) ,$$ where $u_1,u_2\in W$ and $\xi_1,\xi_2\in W^*$. The representation of ${\mathcal{C}(V)}$ on $S=\oplus_{k=0}^r \wedge^k W$ defined by $u\cdot w=u\wedge w$ and $\xi\cdot w={\iota_{\xi}}w$, where $u\in W$, $\xi\in W^*$ and $w\in S$, is the spin representation. Note that $S$ is ${\mathbb{Z}}$- and thus also ${\mathbb{Z}}_2$-graded.

Recall that ${\boldsymbol{E}}={T_{{\mathbb{C}}}{\boldsymbol{X}}}\oplus{T^*_{{\mathbb{C}}}{\boldsymbol{X}}}$ admits the standard pseudo-metric 