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This implies that total momentum of fermions is  $$ P = - \sum_I k_{F,I} ( \psi^\dagger_{R,I} \psi_{R,I} - \psi^\dagger_{L,I} \psi_{L,I} ) + \sum_I (\psi^\dagger_{R,I} (i\partial_x) \psi_{R,I} + \psi^\dagger_{L,I} (i\partial_x) \psi_{L,I} )$$  If I use the non-chiral action (Mike's eqn 1.23 except with an overall minus sign in the definition of $\Theta_I$) $\Theta_I$, this is correct given $K = K_R \oplus K_L$ and $K_R = 1$)  \begin{align}  \Theta_I &=& \phi_I - \phi_{N+I} \\  \Phi_I &=& \phi_I + \phi_{N+I} \\