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I have a question about some exercise for intrinsic angular momentum part of quantized Dirac field.

[tex]S_3 = \frac{1}{2}\int d^3 x :\Psi^\dagger \Sigma_3 \Psi : [/tex]

[tex]\Psi = \int \frac{d^3 k}{\left ( 2\pi \right )^3} \frac{m}{k_0}

\left ( b_\alpha \left ( k \right ) u^\alpha\left ( k \right ) e^{-ikx} + d_\alpha ^\dagger \left ( k \right ) v^\alpha\left ( k \right )e^{ikx} \right ) [/tex]

[tex]S_3 = \frac{1}{2}\int \frac{d^3 x d^3 k d^3 q}{\left ( 2\pi \right )^6} \frac{m^2}{k_0 q_0} : \left ( b^\dagger _{\alpha k} u^\dagger _{\alpha k} e^{ikx} + d_{\alpha k} v^\dagger_{\alpha k} e^{-ikx} \right ) \Sigma_3 \left ( b_{\beta q} u_{\beta q} e^{-iqx} + d^\dagger _{\beta q} v_{\beta q} e^{iqx} \right ) : [/tex]

[tex]S_3 = \frac{1}{2}\int \frac{d^3 k}{\left ( 2\pi \right )^3} \frac{m^2}{k_0 ^2 }\left ( b^\dagger _{\alpha k} b_{\beta k} u^\dagger _{\alpha k} \Sigma_3 u_{\beta k} + b^\dagger _{\alpha k} d^\dagger _{\beta\left ( -k \right )} u^\dagger _{\alpha k} \Sigma_3 v_{\beta \left ( -k \right )} + d_{\alpha k} b_{\beta \left ( -k \right )} v^\dagger_{\alpha k} \Sigma_3 u_{\beta \left ( -k \right )} - d^\dagger _{\beta k} d_{\alpha k} v^\dagger_{\alpha k} \Sigma_3 v_{\beta k}\right )[/tex]

In the last line, second term and third term seems to vanish, like U(1) Noether Charge calculation.

However, I had some calculations, they aren't zero, generally.

What is my mistake?

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# Intrinsic angular momentum of Dirac field

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