Peskin's Introduction to QFT 3.82 - What Has He Done?

[sunkesheng]

peskin`s introduction to QFT [3.82],do not know what he had done to the underline

 

[ansgar]

write out the components of each entry in the matrices

 

[sunkesheng]

write out the components of each entry in the matrices

i see ,but why the \singma[\nu]\sigma[\lambta] in the first row can be write to that in the second row,

 

[ansgar]

what properties of the sigma can u think of?

 

[sunkesheng]

what properties of the sigma can u think of?

many thanks .it is my fault,i don`t express my question clearly,so
sigma is the 2*2 pauli matrix ,but subscript of the front two sigma matrixs in the first row is \beta ,in the second it changes to \delta,that is what i puzzled.

 

[ansgar]

just write out every dirac index and massage it

 

[sunkesheng]

just write out every dirac index and massage it

get it ,there is a change which is important and very simple ,and it is until now that i realized it :
\sigma^{\mu}_{\alpha\beta}=\sigma^{\mu}^{T}_{\beta\alpha}
thanks a lot!

 

[ansgar]

ah great, I was about to do this all for myself so that I could pin-point exactly what is going on ;)

It "always" work to write out the Dirac indices

One thing I don't like is that Peskin uses greek letter for Dirac index sometimes.. like here, very annoying and confusing

 

[oopsitbroke]

For posterity, here's a step-by-step derivation of line 1 to line 2 of eqn. 3.82 for Peskin & Schroeder QFT.