# A Covariant gamma matrices

1. Dec 5, 2016

### spaghetti3451

Covariant gamma matrices are defined by

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

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

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

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

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

2. Dec 5, 2016

### 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

3. Dec 5, 2016

### spaghetti3451

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.

4. Dec 5, 2016

### 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.

5. Dec 5, 2016

### spaghetti3451

So, you mean

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

6. Dec 5, 2016

### 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.

Last edited: Dec 6, 2016
7. Dec 5, 2016

### spaghetti3451

But equation (36.46) in Srenicki has

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

8. Dec 6, 2016

### strangerep

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".