Plebański-Type Nonlinear Electrodynamics

• Plebański-type NLED models are Lorentz and gauge invariant theories formulated with Lagrangians depending solely on electromagnetic invariants S and P, enabling systematic Hamiltonian analysis.
• The ModMax model exemplifies these theories by preserving conformal and SO(2) duality invariance while introducing nonlinear self-interactions that lead to unique wave propagation and birefringence effects.
• These frameworks yield bounded Hamiltonians with reduced degrees of freedom and support applications such as modified black hole solutions and spontaneous Lorentz symmetry breaking.

A nonlinear electrodynamics (NLED) model of Plebański type defines the class of gauge-invariant field theories whose dynamics follow from a Lorentz-invariant Lagrangian density depending only on the two scalar invariants constructed from the electromagnetic field strength: $S=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $P=-\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$, where $$\widetilde{F}^{\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$. The Plebański formalism enables a systematic Legendre-transform approach, where the theory may be recast in first-order Hamiltonian form via an auxiliary antisymmetric tensor $P^{\mu\nu}$, facilitating Hamiltonian analysis, constraint classification, and the paper of field discontinuities and causal propagation. Central paradigms include Born–Infeld and, more recently, the ModMax model, which is unique in being both conformal and SO(2) duality-invariant, and illustrates the power of the Plebański construction (Escobar et al., 2021).

1. General Structure and Formalism

A Plebański-type NLED theory is determined by a Lagrangian density

$\mathcal{L}(S,P)$

or more generally $\mathcal{L}(F,G)$, with the invariants $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $P = -\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}$. In the first-order “Legendre-transform” Plebański formalism, an auxiliary antisymmetric tensor $P^{\mu\nu}$ is introduced, along with a potential $V(P^{\mu\nu})$, yielding the dual (Hamiltonian) Lagrangian

$$\widehat{\mathcal{L}}(P, A) = -\frac12 P^{\mu\nu} F_{\mu\nu} - V(P),$$

where $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$. The constitutive relations and equations of motion then systematically split into

• The nonlinear constitutive law $$F_{\mu\nu} = -V_x P_{\mu\nu} - V_y \tilde{P}_{\mu\nu}$$, with $x = -\frac14 P_{\mu\nu}P^{\mu\nu}$ and $y = -\frac14 P_{\mu\nu}\tilde{P}^{\mu\nu}$;
• The “Maxwell-type” equations $\partial_\mu P^{\mu\nu}=0$;
• The Bianchi identities $\partial_{[\mu} F_{\nu\rho]}=0$ (Escobar et al., 2021).

The formalism is ideally suited to Hamiltonian analysis and to the classification of constraints via the Dirac–Bergmann algorithm (Escobar et al., 2021), as well as to the systematic paper of duality symmetries and stability.

2. ModMax: Conformal and Duality-Invariant NLED

The ModMax model exemplifies the Plebański paradigm as the unique analytic family of NLEDs simultaneously preserving both conformal invariance and continuous SO(2) electric–magnetic duality rotations (Escobar et al., 2021, Kosyakov, 2020, Avetisyan et al., 2021). Its Lagrangian reads: $$\mathcal{L}_{\rm ModMax}(S,P) = -S\cosh\gamma + \sqrt{S^2+P^2}\sinh\gamma$$ with dimensionless rapidity parameter $\gamma \geq 0$. At $\gamma=0$ this reduces to Maxwell’s theory.

Dual invariance follows from the Gaillard–Zumino criterion, and conformal invariance from the requirement that the stress tensor is traceless (Bessel–Hagen criterion). The model is “exactly conformal” ($T^\mu{}_\mu = 0$) for all $\gamma$, preserves SO(2) duality, and is genuinely nonlinear for $\gamma > 0$; at $\gamma = 0$ it reverts to linear electrodynamics (Kosyakov, 2020).

Its Plebański–Hamiltonian potential is

$V(x,y) = -x\cosh\gamma - (x^2 + y^2)\sinh\gamma$

and its constitutive law is explicitly

$$F_{\mu\nu} = \cosh\gamma\,P_{\mu\nu} + \frac{P^2}{\sqrt{P^2+\tilde{P}^2}}\,\sinh\gamma\,\tilde{P}_{\mu\nu}.$$

The effective Hamiltonian is strictly bounded from below, possessing a unique trivial vacuum (Escobar et al., 2021).

3. Hamiltonian Structure: Constraints and Degrees of Freedom

Applying the Dirac constraint analysis in the Plebański formalism yields:

• Two first-class constraints (generating U(1) gauge symmetry);
• Four second-class constraints (from the structure of the canonical momenta);
• A total phase-space reduction to 2 configuration-space degrees of freedom, matching the two polarization modes of the photon—one propagating on the light cone ($k^2 = 0$), and the other birefringent (Escobar et al., 2021).

The physical Hamiltonian density, after imposing all constraints, is a function of the electric displacement $D^i$ and the magnetic field $H^i$, and is given in closed analytic form for ModMax. Boundedness from below is rigorously established, and the stationary-point analysis shows that the physical vacuum has vanishing field ($D^i = H^i = 0$), with a positive-definite Hessian at this point (Escobar et al., 2021).

4. Propagation, Dispersion, and Causality

A key property of Plebański-type NLED is the emergence of birefringence for generic backgrounds: the high-frequency (geometric optics) limit yields two effective metrics, and propagation is determined by a quartic Fresnel equation that factorizes into two distinct null cones (Escobar et al., 2021, Schellstede et al., 2016). In ModMax, the wavefront analysis shows:

• Two polarization modes: one with dispersion $k^2=0$ (Maxwellian), the other with $G^{\mu\nu}k_\mu k_\nu=0$, where the effective metric $G^{\mu\nu}$ depends on the background field and parameter $\gamma$;
• Nontrivial birefringence for $\gamma > 0$, vanishing in the Maxwell limit (Escobar et al., 2021).

The causal structure is controlled by algebraic inequalities on derivatives of the Lagrangian, ensuring the effective optical metrics’ null cones remain within (or coincide with) the spacetime light cone (Schellstede et al., 2016). For the entire Plebański class, necessary and sufficient algebraic conditions on $L_{FF}, L_{GG}, L_{FG}$ (and their physical domain of definition) guarantee the absence of superluminal propagation and causality violation across all backgrounds.

5. Weak-Field Limit and Post-Maxwellian Corrections

For small fields, Plebański-type NLED recovers Maxwell theory at leading order, with nonlinear self-interaction effects manifesting at higher orders. The systematic expansion (Schellstede, 2016): $$F_{mn} = f_{mn}^{(0)} + \epsilon\,f_{mn}^{(1)} + \epsilon^2 f_{mn}^{(2)} + \dots$$ yields a hierarchy of Maxwell-like equations in which higher-order field corrections act as sources for themselves. This structure allows explicit calculation of post-Maxwellian corrections, including measurable self-interaction effects such as vacuum polarization. For example, the leading correction in a charged sphere or wire modifies the field profile near the center but vanishes at large distance (Schellstede, 2016).

The scale of these effects is linked to the characteristic nonlinearity amplitude (e.g. Born–Infeld or Heisenberg–Euler constants), which are orders of magnitude larger than present day laboratory field strengths, rendering these effects experimentally tiny but conceptually crucial.

6. Coupling to Gravitation and Black Hole Solutions

When coupled to General Relativity, Plebański-type NLED sources rich families of charged and magnetized black hole solutions. The field equations retain the same general structure, with the NLED energy-momentum tensor modifying the geometry and regularizing singularities. Noteworthy properties include:

• Regular black hole solutions for both dyonic and purely magnetic configurations, with the NLED scale parameter controlling the near-center geometry and enabling finite electromagnetic mass (Kruglov, 2017, Fathi et al., 2 Dec 2025);
• Explicit analytic forms for metric functions, with corrections to the Reissner–Nordström geometry appearing at subleading orders (e.g., $O(r^{-4})$ or $O(r^{-5})$);
• Rich black hole thermodynamics, including second-order phase transitions, bounded total mass, and new stability regimes (Kruglov, 2017, Kruglov, 2020);
• In the self-dual case (equal electric and magnetic charge), the solutions revert exactly to the Reissner–Nordström metric, demonstrating the NLED's subtle effect on electromagnetic duality in gravitational backgrounds (Kruglov, 2020).

The Legendre-transformed structural (Hamiltonian) function $H(P,Q)$ provides a direct route for constructing and analyzing such solutions in more general $f(R)$ gravity frameworks (0807.2325).

7. Spontaneous Lorentz Symmetry Breaking and Related Phenomena

For generic Plebański potentials $V(P,Q)$, stationary points of the effective Hamiltonian correspond to vacua with nonzero expectation values of the field invariants, yielding spontaneous breaking of Lorentz symmetry (Escobar et al., 2018). These vacua can be fully classified: trivial (no breaking), magnetic-like (field aligned), and generic (mixed electric/magnetic backgrounds), with each phase exhibiting distinct symmetry-breaking patterns and numbers of Nambu–Goldstone (NG) modes (Escobar et al., 2018).

Physical implications extend to Standard Model Extensions (SME) and potential loop-induced Pauli couplings. Causality becomes subtle: in spontaneously broken vacua, the effective metric for wave propagation can admit shock or superluminal solutions, with the eikonal approximation's validity subject to breakdown (Escobar et al., 2018).

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References:

• "Hamiltonian analysis of ModMax nonlinear electrodynamics in the first order formalism" (Escobar et al., 2021).
• "Nonlinear electrodynamics with the maximum allowable symmetries" (Kosyakov, 2020).
• "On causality in nonlinear vacuum electrodynamics of the Plebański class" (Schellstede et al., 2016).
• "On the weak-field limit of Plebański class electrodynamics" (Schellstede, 2016).
• "Nonlinear vacuum electrodynamics and spontaneous breaking of Lorentz symmetry" (Escobar et al., 2018).
• "Nonlinear electrodynamics and magnetic black holes" (Kruglov, 2017).
• "Dyonic and magnetized black holes based on nonlinear electrodynamics" (Kruglov, 2020).
• "Quasinormal modes of a static black hole in nonlinear electrodynamics" (Fathi et al., 2 Dec 2025).
• "Exact solutions of f(R) gravity coupled to nonlinear electrodynamics" (0807.2325).
• "Democratic Lagrangians for Nonlinear Electrodynamics" (Avetisyan et al., 2021).