  It is demonstrated that the nature of optical parametric amplification is a
quantum phenomenon. The system Lagrangian can be constructed by the path
integral of coherent state. The equations of motion for photon operators are
indeed the Euler-Lagrange equations of a Lagrangian. The quantum state
evolution equation can also be obtained without resorting to quantum
Hamiltonian or Lagrangian. Starting with classical Newton equation, quantum
transition amplitude of the system can be educed by surface integral.
