Covariant Bilinears: Fierz Expansion of Dirac gamma matrices products

[PLuz]

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.

 

[fzero]

From the definition of [itex]\sigma^{\alpha\mu}[/itex] and the anticommutation relations, you can see that there must be a term [itex]\gamma_5\sigma^{\alpha\mu}[/itex] in the expansion of [itex]\gamma_5\gamma^\alpha\gamma^\mu[/itex]. I guess we'd call this a pseudotensor term.

 

[PLuz]

Thanks for the prompt reply,

I thought so, but I have read that such term would not be independent of the other 16 matrices.

[itex]\sigma^{\alpha \mu}\gamma_{5}=\frac{i}{2}\epsilon^{\alpha \mu \nu \beta}\sigma_{\nu \beta}[/itex]

,where [itex]\epsilon^{\alpha \mu \nu \beta}[/itex] is the Levi-Civita symbol

So the above expression for [itex]\gamma_{5}\gamma^{\alpha} \gamma^{\mu}[/itex] is incomplete and I should add a extra term
[itex]d* \epsilon^{\alpha \mu \nu \beta}\sigma_{\nu \beta}[/itex] ?

 

[PLuz]

I got it now.

One as to add such term and then get the correct answer. For other readers with a similar doubt I got:

[itex]\gamma_{5}\gamma^{\alpha}\gamma^{\mu}=\eta^{\alpha\mu}I_{4}+\frac{1}{2}\epsilon^{\alpha\mu\nu\beta}σ_{\nu\beta}[/itex]

Which I believe, it's the correct answer.

 

[paranormal]

I would say that your approach is correct. The Fierz expansion is a useful tool in simplifying expressions involving Dirac gamma matrices, and it is often used in quantum field theory calculations. In this case, you are trying to express \gamma_5\gamma^\alpha\gamma^\mu in terms of base matrices, and your attempt at a solution is a valid way to do so.

To determine the value of c, you can use the Fierz identities to simplify the expression further. For example, you can use the identity \gamma^\mu\gamma^\nu = \eta^{\mu\nu}I_4 + i\sigma^{\mu\nu} to rewrite the \sigma^{\alpha\mu} term in terms of \gamma^\mu\gamma^\nu. This will allow you to express the entire expression in terms of base matrices, and you can then use the Fierz identities again to simplify it further.

In summary, your approach is correct but you can use the Fierz identities to simplify the expression further and determine the value of c. Keep in mind that there may be different ways to express the same expression using base matrices, so your final answer may differ from others. As long as it satisfies the Fierz identities, it is a valid solution.