# Spinor matrix representation of a spacetime vector in 2+1D

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.

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.