  The theory of quantum optomechanics is reconstructed from first principles by
finding a Lagrangian from light's equation of motion and then proceeding to the
Hamiltonian. The nonlinear terms, including the quadratic and higher-order
interactions, do not vanish under any possible choice of canonical parameters,
and lead to coupling of momentum and field. The existence of quadratic
mechanical parametric interaction is then demonstrated rigorously, which has
been so far assumed phenomenologically in previous studies. Corrections to the
quadratic terms are particularly significant when the mechanical frequency is
of the same order or larger than the electromagnetic frequency. Further
discussions on the squeezing as well as relativistic corrections are presented.
