  We have developed a microscopic theory on phonon energy dispersion in chiral
crystals within a harmonic approximation. One of the main issues is about the
splitting of sound velocity of acoustic phonons with opposite ``crystal''
angular momentum. We have shown that the splitting must be zero even in chiral
crystals and the difference starts from the order of at least $k^2$ or higher
in their energy dispersion. Splitting is evident for chiral optical phonons,
and we have derived a formula for their $k$-linear splitting. Another important
finding is about possible interactions of atomic displacements in microscopic
models. We have found that antisymmetric interactions of $\mathbf{D} _{ij}
\cdot (\mathbf{d} _i \times \mathbf{d} _j) $ type are not allowed in
microscopic Hamiltonians for chiral phonons in compatible with the stability
against the Nambu-Goldstone mode. We have identified that the splitting in both
acoustic and optical modes arises from the harmonic potentials with the
electric toroidal quadrupole of $G_{u}$-type symmetry. These constraint are
important for modeling real materials. Most of our microscopic calculations
have been performed for (quasi-)one-dimensional systems with a trigonal crystal
symmetry including Te, but these results generally hold also for other chiral
phonon systems.
