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malawi_glenn

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## Main Question or Discussion Point

**gamma matrix trace Paradox??**

Hello, i tried to evaluate this particular little guy:

[tex]\text{Tr} (\gamma ^0 p_\mu \gamma ^\mu \gamma ^0 q_\nu \gamma ^\nu )[/tex]

using these identities:

[tex]\gamma^0 \gamma^0 = I[/tex]

[tex]\text{Tr} (\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma) = 4 (g^{\rho \sigma} g^{\mu \nu} - g^{\nu \sigma} g^{\mu \rho} + g^{\mu \sigma}g^{\nu \rho} ) [/tex]

[tex] \text{Tr} (\gamma^\mu\gamma^\nu) = 4\eta^{\mu\nu} [/tex]

[tex] g^{00} = 1, \quad g^{ii} = -1 [/tex]

using that second relation, I get:

[tex] p_\mu q_\nu \text{Tr} (\gamma ^0 \gamma ^\mu \gamma ^0 \gamma ^\nu ) = p_\mu q_\nu 4 (g^{0\mu} g^{0 \nu} - g^{0 0} g^{\mu \nu } + g^{\mu 0}g^{\nu 0} ) = [/tex]

[tex] p_\mu q_\nu (8\delta ^{0\mu}\delta ^{0\nu} - 4g^{\mu \nu } ) = 4p^0q^0 + 4\vec{q}\cdot \vec{p}[/tex]

Using the first and third, and the fact the traces are invariant under cyclic permutations of matrices.

[tex]p_\mu q_\nu\text{Tr} (\gamma^0 \gamma ^0 \gamma ^\mu \gamma ^\nu ) = p_\mu q_\nu 4g^{\mu \nu } = 4p^0q^0 - 4\vec{q}\cdot \vec{p}[/tex]

What happened?