Self-consistent frequencies of the electron-photon system

The Heisenberg equations describing the dynamics of coupled Fermion photon operators are solved self-consistently. Photon modes, for which ω≊kc, and particlelike Bohr modes with frequencies ${\mathrm{\omega }}_{\mathit{n}\mathit{I}}$≊(${\mathit{E}}_{\mathit{n}}$-${\mathit{E}}_{\mathit{I}}$)/ħ are both approximate solutions to the system of equations that results if the current density is the source in the operator Maxwell equations. Current fluctuations associated with the Bohr modes and required by a fluctuation-dissipation theorem are attributed to the point nature of the particle. The interaction energy is given by the Casimir-force-like expression ΔE=1/2ħtsum(Δ${\mathrm{\omega }}_{\mathit{n}\mathit{I}}$+Δ${\mathrm{\omega }}_{\mathit{k}\mathit{c}}$) or by the expectation value of 1/2(qcphi-qp^⋅A^/mc+${\mathit{q}}^2$${\mathit{A}}^2$/${\mathit{mc}}^2$). It is verified that the equal-time momentum-density and vector-potential operators commute if the contributions of both the Bohr modes and vacuum fluctuations are included. Both electromagnetic and Bohr or radiation-reaction modes are found to contribute equally to spontaneous emission and to the Lamb shift.