Generalised Non-Linear Electrodynamics: classical picture and effective mass generation

Starting from a generic Lagrangian, we discuss the number of propagating degrees of freedom in the framework of generalised non-linear electrodynamics when a photon-background split is applied. We start by stating results obtained in a previous paper, before modifying the action to an equivalent form. Within this new formulation, we highlight the presence of an effective mass and consider the mechanical reduction of the model to ensure the positivity of said mass. We then study the constraint algebra of the model and show that we shift from a model with two first-class to two second-class constraints, which implies the propagation of an additional degree of freedom. We also show that the Hamiltonian is bound from below and thus does not suffer from Ostrogradski-type instabilities. We conclude by deriving the propagator for the model, and discussing the potential link between the nature of this additional polarisation and the mechanism behind the effective mass generation in this class of models.

💡 Research Summary

The paper investigates a broad class of theories called Generalised Non‑Linear Electrodynamics (GNLED), focusing on how an effective photon mass can arise when the electromagnetic field is split into a strong, non‑dynamical background and a weak photon perturbation. Starting from a generic Lagrangian L(F,G) that depends on the two Lorentz invariants F = –¼ F_{\mu\nu}F^{\mu\nu} and G = –¼ F_{\mu\nu}\tilde F^{\mu\nu}, the authors perform a Taylor expansion around the background field. The expansion introduces background‑dependent coefficients C₁, C₂, D₁, D₂, D₃ that multiply the photon field strength f_{\mu\nu} and its couplings to the background.

A crucial step is a field redefinition of the photon potential a_{\mu}: a’{\mu}=α a{\mu} with α² = C₁. Because α is generally a function of the background, this transformation breaks the usual U(1) gauge invariance. The derivative of α defines a purely spatial vector V_{\mu}=∂_{\mu}\ln α, and the Lagrangian in the new variables becomes

L₀ = –¼ f’{\mu\nu}f’^{\mu\nu} – a’{\mu}V_{\nu}f’^{\mu\nu} + ½ M a’_{\mu}a’^{\mu},

where M = (5/4) V² > 0. The first term is the standard Maxwell kinetic term, the second term mixes the photon with the background through V_{\mu}, and the third term is an explicit mass term for the photon.

The authors analyse the equations of motion both in full covariant form and in a simplified “mechanical reduction” where spatial momentum vanishes. In that limit the dispersion relation reduces to ω = √(9/5) M, confirming that the photon acquires a positive mass proportional to the background‑induced parameter M.

A Hamiltonian analysis follows. By breaking explicit covariance, the Lagrangian is written in 3+1 form, and canonical momenta π^{μ} are defined. The primary constraint π⁰ = 0 and the secondary (Gauss‑type) constraint ∂_iπ^i + V_iπ^i = 0 are identified. Because of the V‑dependent mass term, the Poisson brackets of these constraints no longer close as first‑class; instead they become second‑class. Consequently the count of physical degrees of freedom changes from the usual two transverse photon polarizations to three propagating modes: the two transverse modes plus an additional longitudinal (or scalar‑like) mode associated with the mass term.

The Hamiltonian is shown to be bounded from below:

H = ∫ d³x