Dirac Matrix Property? Possible Book mistake?

[silence11]

Dirac Matrix Property? Possible Book mistake? Derive KG from Dirac

I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu} \gamma^{\sigma})

and i am trying to figure out why this is because the only reason I could see why it's true is if \gamma^{\mu} \gamma^{\nu} = \gamma^{\nu} \gamma^{\mu} which for the love of my brain I can't figure out why that would be true, I'm pretty sure it's not. Is this a book mistake. For reference what he is doing is deriving the KG equation starting from Dirac.

on another note, regardless of the answer what i am actually looking for is a derivation of the kg equation starting from dirac, or perhaps the other way around. if someone can point me to that, that is a fine answer as well.

 

[Muphrid]

Just guessing here, but isn't g_{\mu \sigma} symmetric, so any antisymmetric terms would cancel?

 

[silence11]

i don't see what you mean

 

[Muphrid]

Yeah, I think what I was thinking isn't relevant after all. If anything, I had thought the gammas anticommute.

 

[akhmeteli]

I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu} \gamma^{\sigma})

.

To prove the formula, you just need to sum over the dummy indices in the right-hand side: remember that the metric tensor g_{\nu \sigma} lowers indices.

 

[fzero]

I had a look at that book, he's being careless in writing that equation. The important thing is that the term he wants to simplify can be written as $\gamma^\mu \gamma^\nu \partial_\mu \partial_\nu$. The derivatives here are symmetric in $\mu\nu$, so we want to compute the symmetric part of

$$ \gamma^\mu \gamma^\nu = \frac{1}{2} \{ \gamma^\mu, \gamma^\nu\} + \frac{1}{2} [\gamma^\mu ,\gamma^\nu].$$

The first term is symmetric, while the second, commutator, part is antisymmetric. The antisymmetric part vanishes when we sum against $\partial_\mu \partial_\nu$.

The formula in your OP does not hold in general, only in a sum against a symmetric object.

As for deriving the KG equation from the Dirac eq, the method in this book is fine as long as you realize the sloppiness. Usually, we just note that, from

$$ (i\gamma^\mu \partial_\mu - m) \psi =0,$$

we can just compute

$$ 0 = (i\gamma^\mu \partial_\mu + m) (i\gamma^\mu \partial_\mu - m) \psi = - (\partial^\mu\partial_\mu + m^2 ) \psi,$$

which is the KG equation. This is entirely equivalent to the derivation given in the book.

 

[akhmeteli]

I am sorry, I was not careful enough: indeed, only the first equality is correct.

 

[silence11]

okie doke, thanks frank.