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{\bf Current operators.}  We agree that $\rho_e = \rho_R + \rho_L = e \sum_I \left( \psi^\dagger_{R,I} \psi_{R,I} + \psi^\dagger_{L,I} \psi_{L,I} \right) = \frac{1}{2\pi} \sum_{I=1}^N \left( \partial_x \phi_I + \partial_x \phi_{N+I}  \right)$. Then the continuity equation implies  $$ j_e(x) = 2 e \sum_{I=1}^N \sum_{J=1}^{2N} \left[ V_{I,J} - V_{N+I,J} \right] \partial_x \phi_J(x) $$