# Space time algebra isomorphic to matrix algebra

i have the following problem:

I already know that there exists representation of the Clifford Algebra of the Minkowski space $\mathcal{C}l(M,\eta)$. Here $M$ denotes the Minkowski space and $\eta$ is the metric tensor with the signatur $(+1,-1,-1,-1)$.

I know that a representation is given by by the map $\gamma:M\to M_4(\mathbb{C})$ by $\gamma(e_\mu)=\gamma_\mu$, where $\gamma_\mu$ denotes the gamma matrix.

Now I want to show, that the spaces are isomorphic. I've already shown the defining relation $\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\eta_{\mu\nu}$.

My problem is to show that the defined mapping $\gamma$ is bijective. An advice was given in the form that it is enough to show that the mapping is injective or surjective because of dimension of the spaces.

I'm happy about all your helping comments. If you need more information it is no problem to add these ones.

Greetings.

## 1 Answer

The gamma matrices are four $\Bbb C$-linearly independent matrices of $M_2(\Bbb C)$, a space with $\Bbb C$-dimension $4$, so the gamma matrices form a basis. Since the image of $\gamma$ contains this basis, it is the entire space.

Thus $\gamma$ is surjective and hence injective.