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What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?

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It means that

  1. the charge operator $Q$ is a Lie algebra generator for some Lie group $G$.

  2. the field $\phi\in V$ takes values/transform in a representation $V$ of the Lie group. (Note that any Lie group representation $V$ is also Lie algebra representation of the corresponding Lie algebra.)

  3. the charge operator $\rho(Q)$ in the representation $\rho: G \to GL(V)$ is proportional to the identity. The proportionality factor/eigenvalue is the actual charge of the field.

To see an example of this, say the (strong) $u(1)$ hypercharge $Y$, see this answer.

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