Anatoly E. Shabad

Effective Lagrangian in nonlinear electrodynamics and its properties of causality and unitarity

In nonlinear electrodynamics, by implementing the causality principle as the requirement that the group velocity of elementary excitations over a background field should not exceed the speed of light in the vacuum c=1, and the unitarity principle as the requirement that the residue of the propagator should be nonnegative, we establish the positive convexity of the effective Lagrangian on the class of constant fields, also the positivity of all characteristic dielectric and magnetic permittivity constants that are derivatives of the effective Lagrangian with respect to the field invariants. Violation of the general principles by the one-loop approximation in QED at exponentially large magnetic field is analyzed, resulting in complex energy ghosts that signal the instability of the magnetized vacuum. Superluminal excitations (tachyons) appear, too, but for the magnetic field exceeding its instability threshold. Also other popular Lagrangians are tested to establish that the ones leading to spontaneous vacuum magnetization possess wrong convexity.

Finite field-energy of a point charge in QED

We consider a simple nonlinear (quartic in the fields) gauge-invariant modification of classical electrodynamics, to show that it possesses a regularizing ability sufficient to make the field energy of a point charge finite. The model is exactly solved in the class of static central-symmetric electric fields. Collation with quantum electrodynamics (QED) results in the total field energy of a point elementary charge about twice the electron mass. The proof of the finiteness of the field energy is extended to include any polynomial selfinteraction, thereby the one that stems from the truncated expansion of the Euler–Heisenberg local Lagrangian in QED in powers of the field strength.

A note on “Electron self-energy in logarithmic electrodynamics” by P. Gaete and J. Helayël-Neto

We propose an identification of the free parameter in the model of nonlinear electrodynamics proposed in [1] by equating the second term in the power expansion of its Lagrangian with that in the expansion of the Heiseberg-Euler Lagrangian. The resulting value of the field-energy of a point-like charge makes 0.988 of the electron mass, if the charge is that of the electron. Recently Gaete and Helayël-Neto [1] proposed a nonlinear Lagrangian for what they called the logarithmic electrodynamics, which, in the special case of presence of only electric field E, has the form L = −β ln [

QED with an external field: Hamiltonian treatment of Lorentz-non-invariant background as an anisotropic medium

Nonlinear electrodynamics, QED included, is considered against the Lorentz-noninvariant external field background, treated as an anisotropic medium. Hamiltonian formalism is applied to electromagnetic excitations over the background, and entities of electrodynamics of media, such as field inductions and intensities, are made sense of in terms of canonical variables. Both conserved and nonconserved generators of space-time translations and rotations are defined on the phase space, and their Hamiltonian equations of motion and Dirac bracket relations, different from the Poincaré algebra, are established. Nonsymmetric, but–in return–gauge-invariant, energy-momentum tensor suggests a canonical momentum density other than the Poynting vector. A photon magnetic moment is found to govern the evolution of the photon angular momentum. It is determined by the antisymmetric part of the energy-momentum tensor.

Nonlinear (magnetic) correction to the field of a static charge in an external field

We find first nonlinear correction to the field, produced by a static charge at rest in a background constant magnetic field. It is quadratic in the charge and purely magnetic. The third-rank polarization tensor the nonlinear response function is written within the local approximation of the effective action in an otherwise modeland approximation-independent way within any P-invariant nonlinear electrodynamics, QED included.

Real and virtual photons in an external constant electromagnetic field of most general form

The photon behavior in an arbitrary superposition of constant magnetic and electric fields is considered on most general grounds based on the first principles like Lorentz, gauge, charge, and parity invariance. We make model-independent and approximation-independent, but still rather informative, statements about the behavior that the requirement of causal propagation prescribes to massive and massless branches of dispersion curves, and describe the way the eigenmodes are polarized. We find, as a consequence of Hermiticity in the transparency domain, that adding a smaller electric field to a strong magnetic field in parallel to the latter causes enhancement of birefringence. We find the magnetic field produced by a point electric charge far from it--a manifestation of magnetoelectric phenomenon. We establish degeneracies of the polarization tensor that--under special kinematical conditions--occur due to space-time symmetries of the vacuum left after the external field is imposed.

Magnetic response to applied electrostatic field in external magnetic field

We show, within QED and other possible nonlinear theories, that a static charge localized in a finite domain of space becomes a magnetic dipole, if it is placed in an external (constant and homogeneous) magnetic field in the vacuum. The magnetic moment is quadratic in the charge, depends on its size and is parallel to the external field, provided the charge distribution is at least cylindrically symmetric. This magneto-electric effect is a nonlinear response of the magnetized vacuum to an applied electrostatic field. Referring to the simple example of a spherically symmetric applied field, the nonlinearly induced current and its magnetic field are found explicitly throughout the space; the pattern of the lines of force is depicted, both inside and outside the charge, which resembles that of a standard solenoid of classical magnetostatics.

Coulomb field in a constant electromagnetic background

Nonlinear Maxwell equations are written up to the third-power deviations from a constant-field background, valid within any local nonlinear electrodynamics including QED with a Euler-Heisenberg (EH) effective Lagrangian. The linear electric response to an imposed static finite-sized charge is found in the vacuum filled by an arbitrary combination of constant and homogeneous electric and magnetic fields. The modified Coulomb field and corrections to the total charge and to the charge density are given in terms of derivatives of the effective Lagrangian with respect to the field invariants. These are specialized for the EH Lagrangian.

Classical noncommutative electrodynamics with external source

In a U(1)⋆-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate – to the first order in the NC parameter – gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size a, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, 1/r included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size a. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.

Electric charge is a magnetic dipole when placed in a background magnetic field

It is demonstrated, owing to the nonlinearity of QED, that a static charge placed in a strong magnetic field\