Solving Gamma Matrices Identity Problems in Particle Physics

[Safinaz]

Hi all,
I make some exercises in particle physics but I'm stuck in two problems related to Gamma matrices identities,
First: the Fermion propagator $\frac {i } { /\!\!\!p - m} = i \frac { /\!\!\!p + m } { p^2 - m^2}$ So how $/ \!\!\!\!p ^2 = p^2$ ? Where $/\!\!\!p = \gamma_\mu p^\mu$.

I think $/ \!\!\!\!p ^2 = \gamma_\mu p^\mu \gamma_\nu p^\nu = \gamma_\mu p^\mu \gamma_\nu g^{\mu\nu} p_\mu = \gamma_\mu \gamma^\mu p^2 = 4 p^2$, so I got a factor 4 ! What's wrong here?

Second: It's related to the helicity operator, $h= S . \bf{p}$ ( where S is 2 by 2 matrix, with $\sigma^i$ on the diagonal ), that as mentioned in a reference as [arXiv:1006.1718], it commutes with the Dirac Hamiltonian $H = \gamma^0 ( \gamma^i p^i + m )$ equ. (3.3), due to Gamma matrices anticommutation relation, but this isn't clear for me at all..

Thanx

 

[phyzguy]

I'm not sure about the second one, but on the first one, you've just been sloppy with your summation indices, and you've used μ as a summation index twice. If you do what you were trying to do correctly, you get:
$$\not p \not p = \gamma_\mu p^\mu \gamma_\nu p^\nu = \gamma_\mu p^\mu \gamma_\nu g^{\nu \lambda}p_\lambda = \gamma_\mu p^\mu \gamma^\lambda p_\lambda= ?$$

An easier way to see it is as follows:

$$\not p \not p= \frac{1}{2}(\gamma^\mu p_\mu \gamma^\nu p_\nu + \gamma^\nu p_\nu \gamma^\mu p_\mu)= \frac{1}{2}p_\mu p_\nu(\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu) = \frac{1}{2}p_\mu p_\nu 2 g^{\mu \nu} = p_\mu p^\mu = p^2$$

 

[MisterX]

For the first question:
$/ \!\!\!\!p ^2 = / \!\!\!\!p ^2$ so by simple relabling
$\gamma_\mu p^\mu \gamma_\nu p^\nu = \gamma_\nu p^\nu \gamma_\mu p^\mu$
now $/ \!\!\!\!p ^2 = \frac{1}{2}( / \!\!\!\!p ^2 + / \!\!\!\!p ^2)$
$\gamma_\mu p^\mu \gamma_\nu p^\nu = \frac{1}{2}( \gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) = \frac{1}{2}\left\{ \gamma_\mu,\, \gamma_\nu\right\}p^\nu p^\mu$
since $[p^\nu, p^\mu ] = 0$
You should be able to finish the last one or two steps from here.

Edit: I didn't refresh my browser to see someone beat me to it before posting. Oh well.