On the Connections between the Non-linearity of Equations of Motion in Quantum Theory,Asymptotic Behavior and Renormalization

It is shown that the ordinary perturbation expressions used in quantum mechanics lead to the wrong asymptotic behavior of the Heisenberg observables as function of time. This difficulty is traced to the non-linearity of the Heisenberg equations of motion and is studied in the context of a one-dimensional non-linear oscillator problem. It is found that the correct asymptotic behavior can be obtained by a process of renormalization analogous to renormalization theory in quantum field theory. It turns out that the renormalized parameters analogous to mass and wave-function renormalization are not c-numbers but are instead q-numbers. It is suggested that the renormalization parameters of quantum field theory are also q-numbers.