Consider a $U(1)$ symmetry of a complex scalar field realized as $$\phi\to\phi^\prime=e^{iQ\theta}\phi.$$ where $Q$ is the generator of the symmetry. The conserved Noether's charge (in $D$-spatial dimensions) corresponding to this symmetry is given by $$Q_N=\int j^0(\textbf{x},t) d^D\textbf{x}=iQ\int d^D\textbf{x}[(\partial_0\phi)\phi^*-(\partial_0\phi^*)\phi].$$ So it turns out that $Q_N\propto Q$ but not equal to $Q$ itself.
But in A. Zee's book QFT in a Nutshell, page 198, the Noether charge corresponding to a symmetry $$U(\theta)=e^{iQ\theta}$$ is written as $$Q_N=\int j^0(\textbf{x},t)d^D\textbf{x}=Q$$ But as I've shown $Q_N\propto Q$. So is it a mistake there? Or am I missing something?
Best Answer
Your confusion resides in the notation. To make it clear I'll use a hat over the operators. The Noether symmetry is $$ \hat\phi'\equiv\mathrm e^{i\theta\hat Q}\hat\phi\mathrm e^{-i\theta\hat Q}\equiv \mathrm e^{i\theta Q}\hat\phi $$ where $Q\in\mathbb R$ is a scalar. The generator of this symmetry is $\hat U(\theta)=\mathrm e^{i\theta\hat Q}$. The infinitesimal change in $\phi$ is $$ \theta\delta\hat\phi=[\hat Q,\hat\phi]=iQ\phi $$
The conserved charge is $$ \hat Q=\int \hat j^0(\textbf{x},t) d^D\textbf{x}=iQ\int d^D\textbf{x}[(\partial_0\hat\phi)\hat\phi^*-(\partial_0\hat\phi^*)\hat\phi]. $$
Sometimes some authors drop the factor of $Q$ in the definition of $\hat Q$, so that the infinitesimal change in $\phi$ reads $$ \theta\delta\hat \phi=i\phi $$