Two ways of looking at weak current?

Question

A weak current of the form: $$j^\mu=\bar u_e \gamma^\mu \frac{1}{2}(1-\gamma^5)u_\nu$$ implies that it automatically selects the left-handed neutrino. This current can also be written as: $$j^\mu=\bar u_e \frac{1}{2}(1+\gamma^5)\gamma^\mu u_\nu$$ which seems to imply that only left-handed electrons take part in weak interactions. Which statement is right?

Answer 1

Both statements are correct. Only left-handed electrons and left-handed neutrinos participate in weak interactions. The projection operators $$ P_L = \frac{1}{2}(1-\gamma^5)\\ P_R = \frac{1}{2}(1+\gamma^5)\\ $$ satisfy the relations $$ P_L \gamma_\mu = \gamma_\mu P_R\\ P_LP_R=0\\ P_LP_L=P_L\\ P_L + P_R =1 $$ From this it follows that $$ j^\mu=\bar u_e \gamma^\mu P_L u_\nu\\ = \bar u_e \gamma^\mu P_L P_L u_\nu\\ = \bar u_e P_R \gamma^\mu P_L u_\nu\\ = \bar u_{Le} \gamma^\mu u_{L\nu} $$ where in the final line I use the fact that $$ \bar u_{Le} = \overline{(P_L u_e)} = u_e^\dagger P_L^\dagger \gamma_0 = \bar u_e P_R $$