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$$ P = - \sum_I k_{F,I} ( \psi^\dagger_{R,I} \psi_{R,I} - \psi^\dagger_{L,I} \psi_{L,I} ) + \sum_I (\psi^\dagger_{R,I} (i\partial_x) \psi_{R,I} + \psi^\dagger_{L,I} (i\partial_x) \psi_{L,I} )$$  Bosons (we have a factor of 2 difference in the $k_F$ term Mike, eq 1.22)  $$ P = \frac{1}{4\pi} \sum_I k_{F,I} \left(\partial_x \phi_I - \partial_x \phi_{N+I} \right) + \frac{1}{4\pi} \sum_I \left( (\partial_x \phi_I)^2 - (\partial_x \phi_{N+I})^2 \right) $$ Fermion density. Giamarchi claims, for a non-chiral boson, $ \rho_R(x) + \rho_L(x) = \frac{-1}{\pi}\partial_x \theta$. In addition $\phi_R = K\theta - \phi, \phi_L = K \theta + \phi$. His final action for the chiral bosons looks like  $$ -S_G = \frac{1}{4\pi} \int d\tau dx \left[