# Proof that gamma matrices form a complete basis

1. Jul 20, 2010

### joe_blogs

Hi all,

I'm interested in proving/demonstrating/understanding why the Dirac gamma matrices, plus the associated tensor and identity, form a complete basis for $4\times4$ matrices.

In my basic QFT course, the Dirac matrices were introduced via the Dirac equation, and we proved various properties. After doing this, we were presented with this table:
$$\begin{tabular}{|c|c|c|} \hline Form of element &Transforms as &\# of components\\ \hline \mathds{I} &scalar & 1 \\ \gamma^\mu & vector & 4 \\ \sigma^{\mu\nu} & tensor & 6 \\ \gamma^5\gamma^\mu& pseudo-vector & 4 \\ \gamma^5 & pseudo-scalar & 1 \\ \hline \end{tabular}$$
and told that these elements formed a complete basis for $4\times4$ matrices. I've used this fact, and am now employing it in studying the effective weak Hamiltonian as part of an introduction to particle phenomenology. I'm now interested in understanding why it is true.

I've looked through these forums and my searching hasn't turned up a complete answer, or enough of a hint to figure it out. One suggestion I found involved showing the gammas form a Clifford algebra, which can be represented by the matrices over the quaternions. The suggestion was then that the move to matrices over the complex numbers involved the addition of the $\gamma^5$, but I don't know how to work out the detail here.

I'm happy to be directed to textbooks/online sources.

Thanks.

P.S. I'm a grad student, so while I don't have homework I guess I'll mention in the interests of full disclosure that this isn't any sort of assignment for credit, but rather something I want to grasp to further my understanding of a subject I am new to.

2. Jul 20, 2010

### humanino

Did you notice that 1+4+6+4+1 = 16 = 4 x 4 ?
Once you notice that, does it make sense that if those matrices are independent, they must form a complete basis ?

3. Jul 20, 2010

### joe_blogs

Ah, that's very simple. I guess I was stuck on the fact that the matrices have multiple dependencies between them. Because of that I was fixed on showing their span was the whole space. But obviously they're linearly independent so I don't need that...

Ok, thanks. ^_^