Intrinsic angular momentum of Dirac field

[jys34]

(I'm sorry about my pool English..)


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

$$S_3 = \frac{1}{2}\int d^3 x :\Psi^\dagger \Sigma_3 \Psi :$$

$$\Psi = \int \frac{d^3 k}{\left ( 2\pi \right )^3} \frac{m}{k_0} \left ( b_\alpha \left ( k \right ) u^\alpha\left ( k \right ) e^{-ikx} + d_\alpha ^\dagger \left ( k \right ) v^\alpha\left ( k \right )e^{ikx} \right )$$


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


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

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?

 

[AndyW]

First of all, there is no need to apologize for your English. Science is a universal language and as long as we can understand each other, it doesn't matter if it's not perfect.

Regarding your question about the exercise for intrinsic angular momentum of quantized Dirac field, it is important to note that the second and third terms in the last line do not necessarily have to vanish. This is because the operators b and d, which represent the creation and annihilation of particles and antiparticles, respectively, do not commute with each other. Therefore, the order in which they appear in the expression matters and can affect the final result.

I suggest carefully reviewing your calculations to ensure that you are taking into account the correct ordering of these operators. Additionally, it may be helpful to consult with a colleague or professor for further guidance and clarification.

I hope this helps and wish you all the best in your research. Keep up the good work!