The extent of their validity is their ability to correctly predict and agree with experimental results.
When particles and fields propagate at speeds close to light, Lorentz symmetry is one of the most common settings because the equation and its solutions are then consistent with special relativity.
Another symmetry arises from gauge freedom, which is intrinsic to the field equations.
Gauge fields may be classified as in group theory, as abelian or nonabelian.
Alternatively, given suitable Lagrangian or Hamiltonian densities and using the principle of stationary action, the wave equations can be obtained also.
They are extra equations to provide additional constraints for a given physical system.
They are also creation and annihilation operators which satisfy commutation relations and are subject to the spin–statistics theorem.