Understanding the Composition Rule and Transformation Law in Weinberg's QFT

[cosmic dust]

Hello, a confusion has arose during my so far study of the above book.

According to the composition rule (2.3.11) it should be: U\left( \Lambda ,a \right)=U\left( \mathbf{1},a \right)U\left( \Lambda \right) and according to transformation law (2.5.3) and the eigenvalue equation which follows (2.5.1), it should be:

U\left( \Lambda ,a \right){{\Psi }_{p,\sigma }}={{e}^{-i\left( \Lambda p \right)\cdot a}}\sum\limits_{{{\sigma }'}}{{{C}_{{\sigma }'\sigma }}\left( \Lambda ,p \right){{\Psi }_{\Lambda p,{\sigma}' }}}​

Right? If yes, then my question is: how are the above compatible with eq. (3.1.1), since the phase factor that appears in the beginning of the RHS of this equation, contains the untrasformed four-momentums of the particles? Shouldn’t the phase of this factor be the following:

-i{{a}_{\mu }}\left[ {{\left( \Lambda {{p}_{1}} \right)}^{\mu }}+{{\left( \Lambda {{p}_{2}} \right)}^{\mu }}+... \right]
​

?

Thank' s in advance for your replies

 

[Bill_K]

Isn't that what it is? That's what it says in my book. Maybe there was a misprint that got corrected.

 

[cosmic dust]

Then obviously my print has some mistakes. Thank’ s a lot Bill_K , now I can continue my study…