We show that the electromagnetic radiation field, conventionally introduced as a perturbation in quantum mechanics, is actually at the basis of the operator formalism. We first analyze the linear resonant response of the (continuous) variables x(t), p(t) of a harmonic oscillator to the full radiation field, i.e. the zero-point field plus an applied field playing the role of the driving force, and then extend the analysis to the response of a charged particle bound by a non-linear force, typically an atomic (...) electron. This leads to the establishment of a one-to-one correspondence between the response functions and the respective quantum operators, and to the identification of the quantum commutator with the Poisson bracket of the response functions with respect to the normalized variables of the driving field. To complete the quantum description, a similar procedure is used to obtain the field operators as the response functions to the same normalized variables. The results allow us to draw important conclusions about the physical content of the quantum formalism, in particular about the meaning of the quantum expectation values and the coarse-grained nature of the quantum-mechanical description. (shrink)