deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 005e765..a821753 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
(opposite velocities to PM paper)  $$ \mathcal{L}_f = \psi^\dagger_R i(\partial_t + v_F \partial_x) \psi_R + \psi^\dagger_L i(\partial_t - v_F \partial_x) \psi_L $$  I agree with your equation 1.7  I want $\psi^\dagger 1.7.   $$\psi^\dagger  = e^{-i k_F x} \psi^\dagger_R + e^{i k_F x} \psi^\dagger_L$ \psi^\dagger_L $$  Just like in the perfect metal paper:  \begin{align}