= Solution
The <group velocity> is $v_g=\hbar^{-1}\partial_k\varepsilon_k$. Since it is positive everywhere on the branch, every wave packet travels in the same direction: this is a <Chiral Majorana edge mode>.
Near $k=0$, particle-hole symmetry gives $\varepsilon_k=\hbar vk+O(k^3)$. Let $c_k$ annihilate the positive-$k$ part of this branch. Particle-hole symmetry identifies $c_k^\dagger=c_{-k}$. Hence
$$
c(x)=\int\frac{dk}{2\pi}e^{ikx}c_k
$$
satisfies $c(x)^\dagger=c(x)$ and is a Majorana field. Fourier transforming the linear dispersion gives
$$
H^{(\rm lw)}=-i\hbar v\int dx\,c(x)\partial_xc(x).
$$
The continuum analogue of the real antisymmetric matrix $A$ is the real anti-self-adjoint differential kernel
$$
A(x,y)=-\hbar v\,\partial_x\delta(x-y),
$$
up to the normalization convention for the Majorana anticommutator.
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