Trace of Dirac matrices

Question

I was calculating the trace of two Dirac matricies and I used their anti-commutation relations:

$$ Tr(\gamma^{\mu} \gamma^{\nu}) = -Tr(\gamma^{\nu} \gamma^{\mu}) - Tr(2\eta^{\mu\nu}) $$

$$ = - Tr(\gamma^{\mu} \gamma^{\nu}) - 8 $$

$$Tr(\gamma^{\mu} \gamma^{\nu}) = -4 $$

where I used that $2Tr(\eta^{\mu\nu})=8$, but I dont understand why it gives: $Tr(\gamma^{\mu} \gamma^{\nu}) = -4\eta^{\mu\nu}$, where does the metric come from?

By the same token, when calculating $ Tr(\gamma^{\mu} \gamma^{\nu}\gamma^{\alpha} \gamma^{\beta})$:

$$ Tr(\gamma^{\mu} \gamma^{\nu}\gamma^{\alpha} \gamma^{\beta}) = Tr( - \gamma^{\beta} \gamma^{\mu}\gamma^{\nu} \gamma^{\alpha} - 2\eta^{\mu\beta} \gamma^{\nu} \gamma^{\alpha} + 2\gamma^{\mu} \gamma^{\alpha} \eta^{\nu\beta} - 2\gamma^{\mu} \gamma^{\nu} \eta^{\alpha\beta}) $$

$$ 2Tr(\gamma^{\mu} \gamma^{\nu}\gamma^{\alpha} \gamma^{\beta}) = - 2\eta^{\mu\beta} Tr( \gamma^{\nu} \gamma^{\alpha}) + 2 \eta^{\nu\beta}Tr(\gamma^{\mu} \gamma^{\alpha}) - 2 \eta^{\alpha\beta}Tr(\gamma^{\mu} \gamma^{\nu} )$$

I dont understand why the Minkowski matrix comes out in this case in order to give the desired relation. Can someone tell me how to do it correctly?

Answer 1

Concerning the first calculation:

The problem here is that $\eta^{\mu\nu}$ does not denote the matrix, but a component of the Minkowski metric, see also the comments and the answer to this physics SE question.

Note that the anti-commutation relations of the Dirac matrices read

$$\{\gamma^\mu,\gamma^\nu\} = 2\, \eta^{\mu\nu} \,\mathbb{I} \quad ,$$

where $\mathbb{I}$ denotes the $4\times4$ identity matrix. Hence, we find that

$$ \mathrm{Tr}\, \gamma^\mu \gamma^\nu = \frac{1}{2} \mathrm{Tr}\left( \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu\right) =\frac{1}{2}\, \mathrm{Tr}\,\{\gamma^\mu,\gamma^\nu\} = \eta^{\mu\nu}\,\mathrm{Tr}\,\mathbb{I} = 4\, \eta^{\mu\nu} \quad.$$

In particular, since $\eta^{\mu\nu}$ is a number, you can pull it out of the trace operation. In the last step, we used that the trace of the $n\times n$ identity matrix is $n$.

Also note that Wikipedia contains some proofs of (trace) identities involving the gamma matrices.