Relativistic Dispersion In One Space Dimension I'm now reading Three Lectures On Topological Phases Of Matter by
Edward Witten and face some statements that are unclear to me.
According to lectures: 
As I understand, electronic excitations in boundary of fermi-surface  are described by continuous Hamiltonian (1.2) + (1.3). 
I am not a condensed matter physicist, so this statement is very unclear to me. Could somebody provide more detail explanation of this statement?
More concretely, I don't understand why we can describe discrete electrons, which rules out by many-body quantum mechanics in terms of field theory and why the Hamiltonian has such form.
Also, how to describe this system if $v=0$? As I think, one need consider quadratic terms, but what kind of Hamiltonian will describe this?
 A: Given your equation (1.1), we can start from a dispersion of the shape $\varepsilon(k) = k + O(k^2)$ (setting $v=1$). The context is a one-dimensional lattice model in the thermodynamic limit, which means that $k$ is defined on a compact domain: $k \in \left[ - \frac{\pi}{a} , \frac{\pi}{a} \right]$ where $a$ is the lattice spacing. This compactness in momentum space means that its dual variable is discrete. Taking the continuum limit $a\to 0$, we have the effective dispersion
$$ \varepsilon(k) = k + O(k^2) \qquad \textrm{with } k \in \mathbb R. $$
Having unbounded momentum means that its conjugate variable $x$ is now a continuous variable! Hence, we can now use the familiar substitution $k \mapsto - i \partial_x$. We thus get
$$ D_x := \varepsilon(-i\partial_x) = -i\partial_x + O(\partial_x^2) \qquad \textrm{with } x \in \mathbb R. $$
This is all that Witten used to go from Eq.(1.1) to Eq.(1.2). Note that he dropped the $O(\partial^2)$ since such higher-order derivatives are RG-irrelevant (i.e., they naturally disappear under RG flow).
Finally, to answer your question about the case $v=0$: following the same logic, the RG-dominant part would now give us $D_x = \partial_x^2$. This is also gapless and corresponds to a quadratic touching point. In other words, this is not relativistic but instead has a dynamical critical exponent $z_\textrm{dyn} = 2$.
