A set of Feynman amplitudes (connected Feynman amplitudes) with no irrelevant dependences due to vacuum effects is introduced systematically in conjuction with the definition of connected parts of Green's functions. Coupled differential equations of these amplitudes are derived, and as an illustration of the method, equations for the anharmonic oscillator are studied. Detailed investigations on the asymptotic behaviour of Feynman amplitudes show that only the connected F. A. provides S-matrices for arbitrarily complicated processes in unambiguous way.