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.