I saw the definition of electric charge in David Tong's QFT lecture in the Dirac theory to be given as, $$Q=\int d^{3} x \bar{\psi} \gamma^{0} \psi=\int d^{3} x \psi^{\dagger} \psi.$$ However, the combination $\psi^{\dagger} \psi$ is not Lorentz invariant. Then how the charge is a Lorentz invariant quantity as it is supposed to be? Am I missing something? Can someone clarify these to me. Also in QFT what is the most rigorous definition of electric charge if the electrons are modeled as excitations of the Dirac field?
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