I'm studying Debye model and I realized that I don't know what the left side means, specially why there's a $3$.
For the 2 dimensional model we have $$\sum_{modes} 3 = 3N$$
$N$ could be the atoms and $3$ is the number of oscillators for each atom $(x,y,z)$. However, why on the left hand side we have to multiply each mode by $3$ ? what is the meaning of this physically?
https://en.wikipedia.org/wiki/Debye_model Section 2d crystal is an example.
As the wikipedia article explains, the $3$ of is due to the $3$ possible polarizations.
Mathematically, you’d represent your phonons as quantum excitations of a $3D$ vector field defined on a $2D$ plane. Explicitly, writing $\vec U=(U_1,U_2,U,_3)$ the displacement pf the atoms, then $\vec U$ is a field taking arguments $(x,y)$ denoting the position in the $2D$ crystal.
The fact that $\vec U$ is a $3D$ vector determines the fact that you have $3$ polarization states. The dimensionality of the underlying lattice is irrelevant.
Imagine for example that you wanted to describe phonons of a single sheet of graphene. The each atoms can move in the whole $3D$ space (even outside the plane of graphene). Thus, such a model would be appropriate.
Hope this helps.
