- #1
PLuz
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Homework Statement
So my question is related somehow to the Fierz Identities.
I'm taking a course on QFT. My teacher explained in class that instead of using the traces method one could use another, more intuitive, method. He said that we could use the fact that if we garante that we have the same number os indexes at each side of the expression and only use the base matrices (scalar, vector, pseudoscalar, tensor and axial) one would get the same results as using the traces method.
He then gave an example and advised for us to try with some other example.
I then tried to write [itex]\gamma_5\gamma^{\alpha}\gamma^{\mu}[/itex] using that method.
Homework Equations
[itex]\sigma^{\alpha\mu}=\frac{i}{2}[\gamma^{\alpha},\gamma^{\mu}][/itex]
[itex]\eta^{\alpha \mu}[/itex] is the minkowski metric and [itex]I_{4}[/itex] is the identity matrix in 4-spacetime.
The Attempt at a Solution
The attempt of a solution goes as:
[itex]\gamma_{5}\gamma^{\alpha}\gamma^{\mu}=
a*\eta^{\alpha\mu}I_{4}+b*\eta^{\alpha\mu}\gamma_{5}
+c*\sigma^{\alpha\mu}[/itex]
Is this correct?
If I contract [itex]\gamma_5\gamma^\alpha\gamma^\mu[/itex] with [itex]\eta_{\alpha \mu}[/itex] I get a=0 and b=1. But if the above expression is correct, how can I get [itex]c[/itex]?
Please, somebody help me.