We develop a perturbative description of vacuum radiative processes in quantum electrodynamics with a prescribed external electromagnetic background capable of producing electron-positron pairs. Since the initial vacuum is then unstable and the in- and out-vacua are inequivalent, radiative observables require a real-time formulation beyond the ordinary in-out approach of vacuum-stable QED. Using the Keldysh-Schwinger-Fradkin nonequilibrium technique, we derive the mean number density of emitted photons through the second nonvanishing order in the fine-structure constant. The leading term, of order $α$ , reproduces the known vertex and tadpole mechanisms, while the complete order- $α^2$ correction contains interference, loop, and induced-current contributions. We also give an independent derivation based on the spectral decomposition of the identity operator in the in-Fock space, where the photon number density is represented as a sum of squared transition amplitudes and vacuum-disconnected terms are canceled by the optical theorem generalized to an unstable vacuum. In addition, we compute the mean electromagnetic field through order $e^3$ , including the electromagnetic dressing of the induced vacuum current, and verify it using the corresponding Schwinger-Dyson equations. The final formulas are expressed in terms of exact solutions and propagators of the Dirac equation in the external background and apply to general spacetime-dependent field configurations.We develop a perturbative description of vacuum radiative processes in quantum electrodynamics with a prescribed external electromagnetic background capable of producing electron-positron pairs. Since the initial vacuum is then unstable and the in- and out-vacua are inequivalent, radiative observables require a real-time formulation beyond the ordinary in-out approach of vacuum-stable QED. Using the Keldysh-Schwinger-Fradkin nonequilibrium technique, we derive the mean number density of emitted photons through the second nonvanishing order in the fine-structure constant. The leading term, of order $α$, reproduces the known vertex and tadpole mechanisms, while the complete order-$α^2$ correction contains interference, loop, and induced-current contributions. We also give an independent derivation based on the spectral decomposition of the identity operator in the in-Fock space, where the photon number density is represented as a sum of squared transition amplitudes and vacuum-disconnected terms are canceled by the optical theorem generalized to an unstable vacuum. In addition, we compute the mean electromagnetic field through order $e^3$, including the electromagnetic dressing of the induced vacuum current, and verify it using the corresponding Schwinger-Dyson equations. The final formulas are expressed in terms of exact solutions and propagators of the Dirac equation in the external background and apply to general spacetime-dependent field configurations.

hep-thCross-listedarXiv:2606.12547Jun 12, 2026

First- and second-order phase transitions in the Dicke model: Relation to optical bistability

C. M. Bowden, Chi C. Sung

The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.

Physical Review AJun 1, 1979125 citations

Integral equation for a particle in an electric field

Curt A. Moyer

Annals of PhysicsNov 1, 19703 citations

Foldy-Wouthyusen transformation and semiclassical limit for relativistic particles in strong external fields

Alexander J. Silenko

A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows one to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows one to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.

arXiv:0710.4218Physical Review AJan 31, 2008117 citations

α

, reproduces the known vertex and tadpole mechanisms, while the complete order-

α^2

correction contains interference, loop, and induced-current contributions. We also give an independent derivation based on the spectral decomposition of the identity operator in the in-Fock space, where the photon number density is represented as a sum of squared transition amplitudes and vacuum-disconnected terms are canceled by the optical theorem generalized to an unstable vacuum. In addition, we compute the mean electromagnetic field through order

e^3

, including the electromagnetic dressing of the induced vacuum current, and verify it using the corresponding Schwinger-Dyson equations. The final formulas are expressed in terms of exact solutions and propagators of the Dirac equation in the external background and apply to general spacetime-dependent field configurations.