A field-quantized electromagnetic propagator description emphasizing the spatial localization of light emitted from a single-electron atom is presented. With the aim of obtaining a better insight into the quantum electrodynamics in the near-field zone of the atom, two approaches, formally different but equivalent in respect to their physical predictions, are studied. From the Maxwell equations, among the transverse operators, a transverse propagator picture in which the source region for the field is identified with the spatial domain occupied by the transverse current density of the atom emerges. In this picture the field propagation is isotropic and always retarded. By identifying the source region with that of the total atomic current, the transverse propagator becomes anisotropic, and attains a nonretarded self-field part and a near-field part different from zero only for spacelike events. By changing the Coulomb Lagrangian, a nonrelativistic Hamiltonian formalism adequate for reconciling the photon concept with the anisotropic propagator picture is introduced. In this formalism the transverse self-field energy is transferred to the particle Hamiltonian, and the interaction Hamiltonian now includes an interaction between the retarded field and the transverse self-field dynamics. The link between the standard (nonpropagator) theory, often used to argue that the total electromagnetic field is always retarded, and the transverse propagator theory is established. The equivalence between the two approaches is proved by demonstrating that the standard theory in fact includes a nonretarded response in the spatial domain occupied by the longitudinal part of the induced atomic current density.
A field-quantized electromagnetic propagator description emphasizing the spatial localization of light emitted from a single-electron atom is presented. With the aim of obtaining a better insight into the quantum electrodynamics in the near-field zone of the atom, two approaches, formally different but equivalent in respect to their physical predictions, are studied. From the Maxwell equations, among the transverse operators, a transverse propagator picture in which the source region for the field is identified with the spatial domain occupied by the transverse current density of the atom emerges. In this picture the field propagation is isotropic and always retarded. By identifying the source region with that of the total atomic current, the transverse propagator becomes anisotropic, and attains a nonretarded self-field part and a near-field part different from zero only for spacelike events. By changing the Coulomb Lagrangian, a nonrelativistic Hamiltonian formalism adequate for reconciling the photon concept with the anisotropic propagator picture is introduced. In this formalism the transverse self-field energy is transferred to the particle Hamiltonian, and the interaction Hamiltonian now includes an interaction between the retarded field and the transverse self-field dynamics. The link between the standard (nonpropagator) theory, often used to argue that the total electromagnetic field is always retarded, and the transverse propagator theory is established. The equivalence between the two approaches is proved by demonstrating that the standard theory in fact includes a nonretarded response in the spatial domain occupied by the longitudinal part of the induced atomic current density.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.
A theoretical study of elements of the quantum electrodynamic interaction between two single-electron atoms in near-field contact is presented. The framework of the study is an electromagnetic propagator formalism that allows one to describe the near-field space-time interaction in such a manner that the Einstein causality and lack of photon localizability are manifest. First we set up the atom-field Hamiltonian in the so-called G-gauge, starting from the Coulomb Hamiltonian, and thereafter we show that this leads to a correct propagator description for the retarded part of the transverse electromagnetic field (operator). In the G-gauge approach, renormalization of the two-particle energy level structure stemming from the transverse self-field occurs. The intraparticle renormalization is calculated for a three-level atom (it is trivial for a two-level atom), and the interparticle renormalization, which depends on the atomic separation, is determined for two two-level atoms. The magnitude of the energy renormalization is always finite in our theory because we do not consider the atoms to be point-like objects from an electromagnetic point of view. Throughout, we relate our G-gauge formalism to the multipole theory often used in studies of interatomic electrodynamics.
Recent attempts to establish a space-time description of single-photon emission from single atoms (or molecules) are reviewed. In the wake of historical remarks on the so-called Lorenz zone of molecular influence, characteristics of the power-law confined birth domain of the photon are discussed within the framework of a field-unquantized formalism. In contrast to the common believe it is shown that the birth domain can be (and is) exponentially localized if the photon emerges from a pure spin transition. As a paradigm of exponential localization we discuss the emergence of a radio-frequency photon from the hydrogen hyperfine structure. Starting from the first-quantized description of the energy wave function of a polychromatic photon in vacuo, we introduce a propagator description of the photon embryo, and argue that the single-photon tunnelling process is a fingerprint of our inability to localize photons in space-time. With this understanding optical tunnelling appears as an indispensable part of all single photon-atom scattering processes. In the final part of the review we briefly describe how the formalism can be upgraded from single-photon wave mechanics to a field-quantized description based on wave packet (wave train) electrodynamics. In this new formalism, in which the polychromatic photon embryo is included as an integral part of the polychromatic field quanta, the electrons of the atom (or molecule) move not only in the self-consistent Coulomb potential but also in the transverse potential created by the particles themselves. A rigorous relativistic quantum electrodynamic theory of wave train photon emission from single atoms emerges in the horizon.
Elements of a new quantum electrodynamic theory which might enable one to obtain a better understanding of the linear and nonlinear interaction between atomic systems in near-field contact is presented. To follow the space-time dynamics in the atomic near-field zone an electromagnetic propagator formalism with polychromatic photons is constructed. A first-quantized description based on the photon energy wave function for free polychromatic photons is extended in such a manner that the birth process of the photon, which takes place in the near-field zone of the source, can be followed. In order to be able to describe harmonic generation processes in near-field optics polychromatic photons of the simple wave-train type are used. The wave mechanical (first-quantized) description based on wave-train photons is upgraded to a field-theoretic (second-quantized) formalism using a new so-called propagator gauge which is closely related to the Poincaré gauge underlying the multipole description of quantum electrodynamics.
The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2e2/3c3)(d〈v〉/dt)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.