What is the true definition of the covariant gamma matrix ##\gamma_{5}##?

[spaghetti3451]

Covariant gamma matrices are defined by

$$\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\{\gamma^{0},-\gamma^{1},-\gamma^{2},-\gamma^{3}\}.$$

-----------------------------------------------------------------------------------------------------------------------------------------------------------

The gamma matrix $\gamma^{5}$ is defined by

$$\gamma^{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}.$$

-----------------------------------------------------------------------------------------------------------------------------------------------------------

Is the covariant matrix $\gamma_{5}$ then defined by

$$\gamma_{5} = i\gamma_{0}(-\gamma_{1})(-\gamma_{2})(-\gamma_{3})?$$

 

[dextercioby]

No, typically the chirality matrix has the index "downstairs" and is defined in terms of the "downstair" gammas. So the three minuses in your last equality should be omitted.k

 

[spaghetti3451]

No, typically the chirality matrix has the index "downstairs" and is defined in terms of the "downstair" gammas. So the three minuses in your last equality should be omitted.k

But, in Peskin and Schroeder, page 50, $\gamma^{5}$ is defined as

$$\gamma^{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$

and the downstairs index is not used on the chirality matrix.

 

[dextercioby]

Iirc, there's only one matrix being used (either with the index "down" or "up"), not both in a book. I don't have a statistics in my head, but the lower 5 is prevalent.

 

[spaghetti3451]

So, you mean

$$\gamma_{5} \equiv \gamma^{5} \equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}?$$

 

[dextercioby]

No, to have a consistent definition, you have gamma_5 = i gamma_0 * gamma_1 *...
And separately gamma^5 = i gamma^0 * gamma^1 *...
Because of the sign ambiguity (the metric has either 1 or 3 minuses), books will choose to use only one type of 5.

 

[spaghetti3451]

But equation (36.46) in Srenicki has

$$\gamma_{5} = i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$

 

[strangerep]

But equation (36.46) in Srenicki has$$\gamma_{5} = i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$$

That's because the '5' is just a dummy name, not a legitimate index. The 2nd part of Srednicki's (36.46) is actually $$\gamma_5 ~=~ -\,\frac{i}{24}\, \epsilon_{\mu\nu\rho\sigma} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma ~.$$Peskin & Schroeder do something similar on p49 where they write $$\gamma^{\mu\nu\rho\sigma} ~=~ \gamma^{[\mu} \gamma^\nu \gamma^\rho \gamma^{\sigma]} ~,$$but then introduce a $\gamma^5$ in eq(3.68). Whichever place you put the "5" index, the 5th gamma is a pseudo-scalar. It should probably be called something else not involving the index "5".