# Trace of a product of gamma matrices

1. May 28, 2013

### BVM

1. The problem statement, all variables and given/known data
A proof of equality between two traces of products of gamma matrices.

$$Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = 2Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu)$$

Where no special property of A is given, so we must assume it is just a random 4x4 matrix.
$1_4$ represents the 4x4 unity matrix.

2. Relevant equations
$$\gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3$$
The usual product and trace identities as found here.

3. The attempt at a solution
I have been able to prove that
$$Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu) + Tr(\gamma^5 \gamma^\mu A (1_4-\gamma^5) \gamma^\nu)$$
using the definition of the trace and the anticommutativity of the $\gamma^\mu$ and $\gamma^5$.

Last edited: May 28, 2013
2. May 28, 2013

### TSny

For your last term, try using the fact that the trace is invariant under any cyclic permutation of the matrices inside the trace. So, you can move the γ5 on the left to the far right. Then simplify (14 - γ5) γ$^\nu$ γ5.

3. May 28, 2013

### BVM

Got it! That hint really helped. Thanks.

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