Why Might Dirac Bilinears Not Zero Out for Non-Diagonal Components?

[grelade]

Hi,

I have free solutions of Dirac equation in the form (without exp(), i also use the contracted 2 dimensional convection ):

for r=1,2 $$E=E_p$$

$$\omega^{r=1,2}(\vec{p}) = N \[ \left( \begin{array}{c} \phi_r \\ \phi_r c \frac{\vec{\sigma}\cdot\vec{p}}{E_p + mc^2} \end{array} \right)\]$$

r=3,4 $$E=-E_p$$

$$\omega^{r=3,4}(\vec{p}) = N \[ \left( \begin{array}{c} \phi_r c \frac{\vec{\sigma}\cdot\vec{p}}{E_p + mc^2} \\ \phi_r \end{array} \right)\]$$

I want to calculate one of Dirac bilinears (antisymmetric tensor) for these solutions, namely:

$$b^{\mu\nu}_r = \bar{\omega}^r(\vec{p}) \sigma^{\mu\nu}\omega^r(\vec{p})$$

where
$$\sigma^{\mu\nu} = i/2 \left[\gamma^\mu, \gamma^\nu \right]$$

On lectures it was said that it would be zero. However, i couldn't zero it out for $$\mu \neq \nu$$. My calculations goes like this (for now i will post only case $$b^{j0}, j=1,2,3$$):
$$b^{j0}_r = \bar{\omega}^r(\vec{p}) \sigma^{j0}\omega^r(\vec{p}) = i\omega^{r+}\gamma^0\gamma^j\gamma^0\omega^{r} = -i\omega^{r+}\gamma^j\omega^{r}$$
Now, for r=1,2 i put explicit matrices:
$$= i N^2 \[ \left( \begin{array}{cc} \phi_r^+ & \phi_r^+ c \frac{\vec{\sigma}\cdot\vec{p}}{E_p + mc^2} \end{array} \right)\] \[ \left( \begin{array}{cc} 0 & -\sigma_j \\ -\sigma_j & 0 \end{array} \right)\] \[ \left( \begin{array}{c} \phi_r \\ \phi_r c \frac{\vec{\sigma}\cdot\vec{p}}{E_p + mc^2} \end{array} \right)\]$$
This gives:
$$= -i N^2 (\phi_r^+ \frac{\sigma_j \sigma_i p^i}{E_p+mc^2}\phi_r - \phi_r^+ \frac{\sigma_i \sigma_j p^i}{E_p + mc^2}\phi_r) = -i N^2 (\phi_r^+ \frac{\left [\sigma_j, \sigma_i \right] p^i}{E_p+mc^2}\phi_r)$$
Commutator of Pauli matrices:
$$\left [\sigma_j,\sigma_i \right] = 2i \epsilon_{jik}\sigma_k$$
So after inserting this we have something like this:
$$= \frac{2 N^2\epsilon_{jik}\sigma_k p^i}{E_p + mc^2}$$
Unfortunately i cannot find any reason why this should be zero. Maybe someone could help me by telling what is wrong or tell that it shouldn't be zero in the first place?

 

[eljose79]

Hello,

Thank you for sharing your solutions and calculations with us. Based on my understanding of the Dirac equation and bilinears, I believe that your calculation is correct and that the result should not be zero. This is because the Dirac equation and its solutions describe the behavior of fermions, which are particles with half-integer spin. These particles have intrinsic angular momentum and thus have a non-zero value for their spin tensor. Therefore, the bilinear you are calculating should also have a non-zero value for \mu\neq\nu, as long as the solutions you are using accurately describe the spin state of the particle.

In fact, the non-zero value of this bilinear is important for understanding the behavior of fermions in electromagnetic fields, as it contributes to the magnetic moment of the particle. I would recommend double-checking your calculations and seeking feedback from your peers or a professor to confirm your results. Keep up the good work!