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Let $a^\mu$ be a spacetime vector in 2+1D: $(t,x,y)$.

What would be the spinor-matrix representation ${A^\alpha}_\beta$ of the spacetime vector $a^\mu$?

I have not been able to even find examples or clear definition of a spinor matrix. Any help is greatly appreciated.

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The real Clifford algebra of 2+1 dimensions with $\mathbf t^2=+1$ is isomorphic to $M_2(\mathbb C)$ and one possible representation is $$\begin{pmatrix}0&i\\-i&0\end{pmatrix} t \; + \; \begin{pmatrix}i&0\\0&-i\end{pmatrix} x \; + \; \begin{pmatrix}0&i\\i&0\end{pmatrix} y$$

In this representation the pseudoscalar is $i$ and the even subalgebra is the subalgebra of real matrices.

The real algebra with $\mathbf t^2=-1$ is isomorphic to $M_2(\mathbb R)\oplus M_2(\mathbb R)$ and one possible representation is $$\left(\begin{smallmatrix}0&-1\\1&0\\&&0&1\\&&-1&0\end{smallmatrix}\right) t \; + \; \left(\begin{smallmatrix}1&0\\0&-1\\&&-1&0\\&&0&1\end{smallmatrix}\right) x \; + \; \left(\begin{smallmatrix}0&1\\1&0\\&&0&-1\\&&-1&0\end{smallmatrix}\right) y$$

In this representation the chiral subalgebras are those in which the top or bottom $2{\times}2$ block is zero and the even subalgebra is that in which the blocks are equal.

The complex algebra is isomorphic to $M_2(\mathbb C)\oplus M_2(\mathbb C)$ and you can use the second basis.

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