# Luttinger Liquid Parameter Physical Meaning - Attractive or Repulsive

For 1+1D systems at long wavelength, it is known that the (Tomonaga)-Luttinger Liquid can be rewritten in terms of bosonic parameters, where the Hamiltonian densities can be written \begin{align*} H =& \left[\psi_L i\partial_x \psi_L - \psi_R i \partial_x \psi_R + 2\pi g_2 \rho_R \rho_L + \pi g_4 (\rho_R^2 + \rho_L^2) \right]\\ \rightarrow H_\text{boson} =& \frac{v}{2}\left[\frac{1}{g}\Pi^2 + g(\partial_x \phi)^2\right] \end{align*} where $$\Pi = \partial_x \theta$$ is the momentum conjugate to $$\phi$$.

I have been confused about the relationship of $$g$$ to the parameters $$g_2, g_4$$, where either \begin{align*} g = \sqrt{\frac{1 + g_4 \pm g_2}{1+ g_4 \mp g_2}} \end{align*} In particular, for repulsive interactions (i.e. $$g_2,g_4 > 0$$), the choice of sign determines whether or not $$g>1$$ or $$g<1$$. For instance, in the seminal work by Kane and Fisher, they write that $$g>1$$ for attractive interactions, while in the lecture notes by Senechal and lecture notes by Fradkin, they write the opposite.

Which one is correct, or am I just missing something silly here? Perhaps a more physical question is if there can be an unambiguous physical meaning assigned to g?

In terms of the physical interpretation, $$g$$ can be directly related to the compressibility of the system. Obviously I cannot say the exact nature of this relationship without going through exactly how $$g$$ and the bosonic fields have been defined, but it should be either directly proportional or inversely proportional (notice that your 2 sign conventions for $$g$$ amount to swapping $$g$$ with $$\frac{1}{g}$$)