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$$ \rho_L(x) = \frac{-1}{2\pi} \partial_x \phi_L^{(G))} = \frac{-1}{2\pi} \partial_x \phi_L $$  From the commutation relation of Giamarchi (2.25) $[ \phi(x), \partial_y \theta (y) ] = i \pi \delta(x-y)$ we can backtrack the commutation relations of the chiral fields after we substitute the field redefinition into the above commutator:  $$ \phi = \frac{\phi_R^{(G)} \frac{\phi_L^{(G)}  - \phi_L^{(G)}}{2} \phi_R^{(G)}}{2} $$  $$ \theta = \frac{\phi_L^{(G)} + \phi_R^{(G)}}{2K} $$  $$ \frac{1}{4K} \left[ \phi_L^{(G)}, \partial_x \phi_L^{(G)} \right] + \frac{1}{4K} \left[ -\phi_R^{(G)}, \partial_x \phi_R^{(G)} \right] = i \pi \delta(x-y)  $$