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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The question to which my question is being said to be duplicate deals with classical definition of charge. Whereas my question is concerned with Dirac spinors. Dirac spinor combination $\psi^\dagger \psi$ has different transformation property than say classical charge density. I want to see how in quantum theory the same result can be obtained via explicit calculation. Therefore, the question is not duplicate and I request to review and reopen the question. – fogof mylife Mar 22 '20 at 09:18
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1Yes this quantity is a scalar. For any conserved quantity, $\partial_\mu M^\mu =0$, you can prove that $\int d^3 x M^0$ is a scalar. – my2cts Mar 22 '20 at 09:44
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1I agree that the application to qed means that the questions are not identical. Charge density is not a scalar. It is the time component of current density $j^a = \bar{\psi} \gamma^{a} \psi$, which is a vector operator. One may easily show $\partial_a j^a =0$ . – Charles Francis Mar 22 '20 at 09:49
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@my2cts Yes it is indeed the proof I want see. But the question is closed now. Can you please give me a reference where I can find the proof for $\int d^3x M^0$ being a scalar given $\partial_\mu M^\mu=0$. – fogof mylife Mar 22 '20 at 10:07