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I want $K = K_R \oplus - K_L$. And$ K_R And $K_R  = K_L = 1$. What does this imply about the signs of the velocities in the fermionic Langrangian? I think the correct thing is $\psi^\dagger_R i( \partial_t - v \partial_x) \psi_R$. This is what the perfect metal paper has.  Mike, this goes against your equation 1.8.