Infinitesimal transformation of a field

[marcom]

Hi,

Could you please explain me why, under the transformation of a complex valued field Φ→e^iαΦ, for an infinitesimal transformation we have the following relation?

δΦ=iαΦ

Thanks a lot

 

[samalkhaiat]

Hi,

Could you please explain me why, under the transformation of a complex valued field Φ→e^iαΦ, for an infinitesimal transformation we have the following relation?

δΦ=iαΦ

Thanks a lot

When $\alpha$ is infinitesimal, i.e. $\alpha^{2} = 0$, you can write $e^{i\alpha} = 1 + i\alpha$. So, $\Phi \to \Phi^{'} = \Phi + i\alpha \Phi$, leads to $\delta\Phi = \Phi^{'} - \Phi = i\alpha \Phi$.