Role of the Electromagnetic Vacuum in the Transition from Classical to Quantum Mechanics

Abstract

We revisit the nonrelativistic problem of a bound, charged particle subject to the random zero-point radiation field (zpf), with the purpose of revealing the mechanism that takes it from the initially classical description to the final quantum-mechanical one. The combined effect of the zpf and the radiation reaction force results, after a characteristic time lapse, in the loss of the initial conditions and the concomitant irreversible transition of the dynamics to a stationary regime controlled by the field. In this regime, the canonical variables x, p become expressed in terms of the dipolar response functions to a set of field modes. A proper ordering of the response coefficients leads to the matrix representation of quantum mechanics, as was proposed in the early days of the theory, and to the basic commutator $\left[ {\hat{x}},{\hat{p}}\right] =i\hbar$. Further, the connection with the corresponding Fokker–Planck equation valid in the Markov approximation, allows one to obtain the (nonrelativistic) radiative corrections of qed. These results reaffirm the essentially electrodynamic and stochastic nature of the quantum phenomenon, as proposed by stochastic electrodynamics.