We present a new formulation of self-dual nonlinear electrodynamics in which interactions are determined by an auxiliary-field potential, with causality ensuring a unique solution to the auxiliary-field equation. The long-standing problem of an explicit Lagrangian for the generic ‘analytic’ theory is simply solved by restriction to potentials that are even functions of the auxiliary field. In this case the Lagrangian can be linearised in quadratic field-strength scalars by the introduction of an additional pseudoscalar auxiliary field; this generalises, to all analytic self-dual theories, a well-known construction of the Born-Infeld theory.

Spain

Jorge Russo

United Kingdom

Paul Townsend

Simplified self-dual electrodynamics

The nonlinear electrodynamics proposed by Bandos, Lechner, Sorokin and Townsend is a remarkable theory that unifies Maxwell, Bialynicki-Birula and ModMax theories, which are known theories invariant under conformal transformations and electromagnetic duality transformations. In the Bandos-Lechner-Sorokin-Townsend nonlinear electrodynamics, we calculate the energy flux density, dispersion relations, refractive indices, phase and group velocities of plane waves as well as the changes of the photon wavelength in the Compton scattering process in the presence of a constant uniform electromagnetic background. Our results are useful for testing and constraining this new theory of nonlinear electrodynamics.

China

Yang Shi

Towe Wang

Compton scattering in Bandos-Lechner-Sorokin-Townsend nonlinear electrodynamics

Generic nonlinear theories of chiral 2-form electrodynamics allow superluminal propagation in some stationary homogeneous backgrounds and are therefore acausal. We find a simple parameterisation of the Hamiltonian for causal theories, and we use it to show that the stress-energy tensor satisfies both the Dominant and Strong energy conditions. We also revisit the Perry-Schwarz formulation, clarifying aspects of it and of its relation to the Hamiltonian formulation.

Causal chiral 2-form electrodynamics

For any causal nonlinear electrodynamics theory that is “self-dual” (electromagnetic U(1)-duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities   <math> <mfenced> <mi>L</mi> <mi>H</mi> </mfenced> </math>  $$ \left\{\mathcal{L},\mathcal{H}\right\} $$  are constructed from functions   <math> <mfenced> <mi>ℓ</mi> <mi>h</mi> </mfenced> </math>  $$ \left\{\ell, \mathfrak{h}\right\} $$  on ℝ + related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to   <math> <mi>h</mi> </math>  $$ \mathfrak{h} $$  implies that   <math> <mi>L</mi> </math>  $$ \mathcal{L} $$  and   <math> <mi>H</mi> </math>  $$ \mathcal{H} $$  are related by a simple map between appropriate pairs of variables. We also discuss Born’s “Legendre self-duality” and implications of a new “Φ-parity” duality. Our results are illustrated with many examples.

Dualities of self-dual nonlinear electrodynamics

For the general theory of nonlinear electrodynamics (NLED) we prove that causality implies both the Dominant Energy Condition (DEC) and, surprisingly, the Strong Energy Condition (SEC). This has implications for gravitational applications, such as regular black holes supported by NLED matter. For self-dual NLED theories, weak-field causality alone implies both the DEC and SEC, as we illustrate with Born-Infeld and ModMax electrodynamics.

Causality and energy conditions in nonlinear electrodynamics

Using Hamiltonian methods, we find six relativistic theories of nonlinear electrodynamics for which plane wave perturbations about a constant uniform background are not birefringent. All have the same conformal strong-field limit to Bialynicki-Birula (BB) electrodynamics, but only four avoid superluminal propagation: Born-Infeld (BI), its non-conformal “extreme” limits (electric and magnetic) and the conformal BB limit. The quadratic dispersion relation of BI is shown to degenerate in the extreme limits to a pair of linear relations, which become identical in the BB limit.

United States

Luca Mezincescu

Hamiltonian birefringence and Born-Infeld limits

The Hamiltonian formulation for nonlinear chiral 2-form electrodynamics in six-dimensional Minkowski spacetime is used to show that small-amplitude plane-wave perturbations of a generic uniform constant ‘magnetic’ background exhibit trirefringence: all three independent wave-polarisations have distinct dispersion relations. While two coincide for Lorentz invariant theories, all three coincide uniquely for the chiral 2-form theory on the worldvolume of the M5-brane of M-theory. We argue that this is because, in this M-theory context, the waves propagate in a planar M5-M2-M2 bound-state preserving 16 supersymmetries. We also show how our results imply analogous results for nonlinear electrodynamics in a Minkowski spacetime of five and four dimensions.

Igor Bandos

Italy

Kurt Lechner

Dmitri Sorokin

Trirefringence and the M5-brane

All solutions of the no-birefringence conditions for nonlinear electrodynamics are found. In addition to the known Born-Infeld and Plebanski cases, we find a “reverse Born-Infeld” case, which has a limit to Plebanski, and an “extreme-Born-Infeld” case, which arises as a Lagrangian constraint. Only Born-Infeld has a weak-field limit, and only Born-Infeld and extreme-Born-Infeld avoid superluminal propagation in constant electromagnetic backgrounds, but all cases have a conformal strong-field limit that coincides with the strong-field limit of Born-Infeld found by Bialynicki-Birula.

Nonlinear electrodynamics without birefringence

In the Hamiltonian formulation of chiral 2k-form electrodynamics, the 2k-form potential on the (4k + 1)-space is defined up to the addition of either (i) a closed 2k-form or (ii) an exact 2k-form, depending on the choice of chirality constraint. Case (i) is realized by the Floreanini-Jackiw 2D chiral boson (for k = 0) and its Henneaux-Teitelboim generalisation to k &gt; 0. For all k, but focusing on the 6D case, we present a simple Lorentz-invariant Hamiltonian model that realizes case (ii), and we derive it from Siegel’s manifestly Lorentz invariant Lagrangian formulation.

On chiral bosons in 2D and 6D

We give a prescription for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetrization of any (four-dimensional) nonlinear electrodynamics theory with a Lagrangian density satisfying a convexity condition that we relate to semi-classical unitarity. We apply it to the one-parameter ModMax extension of Maxwell electrodynamics that preserves both electromagnetic duality and conformal invariance, and its Born-Infeld-like generalization, proving that duality invariance is preserved. We also establish superconformal invariance of the superModMax theory by showing that its coupling to supergravity is super-Weyl invariant. The higher-derivative photino-field interactions that appear in any supersymmetric nonlinear electrodynamics theory are removed by an invertible nonlinear superfield redefinition.