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

[itex]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}' }}}[/itex]

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:

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

???

Thank' s in advance for your replies