Dirac Trace Algebra: Which Gamma Matrices Matter?

[Maurice7510]

Homework Statement

This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first question is: does it matter *which* gamma matrices? For example, one of my traces has
$$ \gamma_{\nu}\gamma^5\gamma^{\rho}\gamma^{\alpha}\gamma^{\sigma}\gamma^0\gamma^5\gamma^0\gamma^{\lambda}\gamma_{\mu}\gamma^{\beta} $$ Since each $$\gamma^5$$ is a product of 4 gamma matrices, altogether this would be a product of 17, which is odd. But some are $$\gamma^{\mu}$$, some $$\gamma^0$$ and some $$\gamma^5$$ and I'm just not sure what the rules are for this type of thing.

Homework Equations

$$Tr (ABC) = Tr (CAB) = Tr (BCA)$$
$$Tr (A +B) = Tr(A)+Tr(B)$$
$$Tr(aA) = aTr(A)$$

The Attempt at a Solution

I would think that this would, in fact, still be zero because there's an odd number of them. Otherwise I would probably use commutation/anticommutation relations to bring the $$gamma^0$$ and $$\gamma^5$$ together (respectively) to get unity, then just deal with the rest.

 

[Orodruin]

zero because there's an odd number of them.

And there is your answer.

 

[Maurice7510]

So does that mean that all the standard trace identies for gamma matrices (e.g. in Griffiths or Peskin Schroeder) hold for *any* gamma matrices, regardless of their index?

 

[Orodruin]

*any* gamma matrices, regardless of their index?

As long as that index is 0, 1, 2, or 3. You will need to treat $\gamma^5$ differently.

 

[Maurice7510]

Isn't the last one just a product of the first 4?

 

[Orodruin]

Isn't the last one just a product of the first 4?

Yes (up to a factor), so you need to treat it differently.