Trace Theorems and Dirac Matrices

[dm4b]

I think I'm missing something real simple on trace theorems and Dirac matrices, but am just not seeing it.

In the Peskin and Schroeder QFT text on page 135 we have:

gamma^(mu)*gamma^(nu)*gamma_(mu) = -2*gamma^(nu)

But, why can't we anti-commute and obtain the following:

gamma^(mu)*gamma^(nu)*gamma_(mu)
= -gamma^(nu)*gamma^(mu)*gamma_(mu)
= -4*gamma^(nu)

which contradicts the equation above?

Any help would be much appreciated. Thanks!

P.S. ^=superscript, and _=subscript (LATEX wasn't working for me)

 

[fzero]

But, why can't we anti-commute and obtain the following:

gamma^(mu)*gamma^(nu)*gamma_(mu)
= -gamma^(nu)*gamma^(mu)*gamma_(mu)
= -4*gamma^(nu)

You missed part of the anticommutation relation:

$$\gamma^\mu \gamma^\nu \gamma_\mu = \left( -\gamma^\nu \gamma^\mu + 2\eta^{\mu\nu} \right) \gamma_\mu.$$

 

[dm4b]

doh! thanks ... maybe I better take a break from doing physics now ;-)