Further discussions and detailed calculations on the problem of the spontaneous, acceleration of free electromagnetically inter-acting particles by the zero-point field in the light of a quantum version of the Einstein-Hopf model are presented. It, is shown that acceleration occurs if the zero-point field is represented in a time-symmetric fashion within the viewpoint of the Wheeler-Feynman radiant-absorber theory. However, if the zero-point field is represented in the time-asymmetric form, the quantum Einstein-Hopf model yields no translational kineticenergy growth in disagreement with the previous prediction and with the result of the classical version of the zero-point field in stochastic electrodynamics. The calculations are clear and compelling. Despite that the last no-acceleration result is germane to phenomenological thermodynamics expectations and to a more consistent perspective of quantum theory, the second quantization, that leads to the time-symmetric zeropoint field yields a conceptually more satisfactory view of this background field which is no longer a free virtual field but becomes, a real field which is originated in and is associated with particles. The discussion is based on the different boundary conditions for the electromagnetic-field tensor that the zero-point field (asymmetricvs. symmetric) requires in quantum and in classical theory: time symmetry presupposes a universe that is opaque. If this condition, does not hold, we are forced to ordinary time asymmetry and, if a correspondence with quantum electrodynamics is desired, some modification of the hypothesis of stochastic electrodynamics would be required to prevent acceleration. The possible form of that modification is suggested.