ModMax Electrodynamics Overview

• ModMax electrodynamics is the unique continuously deformable nonlinear extension of Maxwell’s source-free theory that maintains full Poincaré, conformal, and electromagnetic duality invariance.
• The theory uses a single positive deformation parameter to regularize singular null configurations, doubling the class of knotted solutions compared to classical Maxwell knots.
• Employing Hamiltonian and Riemann–Silberstein formalisms, ModMax enables systematic construction of topologically nontrivial solutions with applications in quantum optics, condensed matter, and high-energy physics.

ModMax electrodynamics is the unique continuously deformable, nonlinear extension of Maxwell’s source-free electrodynamics in four-dimensional Minkowski spacetime that preserves the full suite of Poincaré and conformal symmetries as well as the continuous electromagnetic duality invariance. The deformation is realized by a single real, positive parameter, smoothly connecting the linear Maxwell theory with its maximally symmetric nonlinear generalization. Notably, while Maxwell’s theory admits null (E² = B², E·B = 0) field configurations—including knotted solutions (hopfion–Rañada knots)—these are singular in the Lagrangian density of any Poincaré and conformal invariant nonlinear electrodynamics; ModMax resolves this by introducing deformed configurations of the same topology, thereby regularizing otherwise singular solutions. For each classical electromagnetic knot, ModMax yields a doubled class of deformed knotted solutions, which continuously limit to the original null configurations as the deformation vanishes.

1. ModMax Electrodynamics: Symmetry Principles and Deformation Structure

ModMax theories are defined as the only continuous one-parameter deformation (parameterized by $y > 0$) of source-free Maxwell electrodynamics that preserve Poincaré invariance, conformal invariance, and electromagnetic duality. ModMax retains the structure of the Maxwell equations for $y \to 0$. The Lagrangian density is given by: $$L_y(S, P) = \cosh y \cdot S + \sinh y \cdot \sqrt{S^2 + P^2}$$ with Lorentz invariants $S = E^2 - B^2$ and $P = 2(E \cdot B)$. This form uniquely implements a hyperbolic mixing of invariants and is fixed by the requirement of symmetry. The continuous parameter $y$ controls the departure from linearity: setting $y=0$ strictly recovers Maxwell electrodynamics, while $y>0$ progressively activates the nonlinear character.

A critical property is that null configurations ($E^2 = B^2$, $E \cdot B = 0$) are singular points for any conformally invariant nonlinear Lagrangian; more precisely, the Lagrangian’s non-analyticity is unavoidable at $(E, B) = (0,0)$ for such field configurations. The ModMax deformation allows for topological objects analogous to Maxwell’s null knots, but these are non-null and hence regular for $y > 0$.

2. Hamiltonian Formalism and Riemann–Silberstein Representation

The Hamiltonian density in ModMax electrodynamics admits a natural expression in terms of the generalized Riemann–Silberstein (RS) vector, $\mathbf{R} = \mathbf{D} + i\mathbf{B}$. The Hamiltonian is

$$H_y(\mathbf{D}, \mathbf{B}) = \frac{1}{\cosh y} \left[ \cosh y (\mathbf{D}^2 + \mathbf{B}^2) - \sinh y (\mathbf{D}^2 - \mathbf{B}^2) + 4(\mathbf{D} \cdot \mathbf{B}) \right]$$

The deformation modifies the relation between field variables (e.g., the RS vector constructed from $\mathbf{H}$ and $\mathbf{E}$ versus $\mathbf{R}$): $$\mathbf{S}(\mathbf{R}, \mathbf{R}^*) = -i \cosh y \mathbf{R} + i \sinh y \left( \frac{\mathbf{R}}{|\mathbf{R}|} \right)^2 \mathbf{R}^*$$ The continuous deformation alters the underlying symplectic structure but preserves duality invariance at the Hamiltonian level, and the explicit RS vector construction facilitates classification and generation of topologically nontrivial solutions via methods such as the Bateman formalism.

3. Topological Knots: Hopfion–Rañada Solutions in the Linear Theory

Hopfion–Rañada knots are exact, null solutions of the Maxwell equations characterized by closed, linked, and knotted field lines. These are constructed using the Bateman method, in which a pair of self-dual complex scalar potentials parametrize all null configurations. The double Hopf fibration of $S^3$ provides a geometric underpinning for their topological structure. Such solutions are noteworthy for encoding nontrivial topology within linear electrodynamics, and their existence is tightly linked to the full conformal invariance of Maxwell’s equations.

The physical significance of these knots lies both in the explicit realization of nontrivial electromagnetic topology and in their connection to conserved quantities associated with duality symmetry. Their null character, however, makes them inaccessible or ill-defined in conformally invariant nonlinear theories at the level of the Lagrangian.

4. Deformation of Knots in the ModMax Nonlinear Framework

ModMax theory regularizes the pathological nature of null knots through a constructive symmetry protocol:

• Begin with a static, homogeneous (hence non-null) electromagnetic field configuration.
• Apply a conformal inversion, yielding a new solution (due to conformal covariance of the dynamics).
• Subsequently, perform a pure imaginary time translation, producing a configuration whose field lines remain topologically knotted.

The resulting solutions are “deformed hopfion–Rañada knots”: while they share the knotted topology of the original null solutions, their invariants $S$ and $P$ are no longer strictly null, and the resulting field lies outside the non-analytic regions of the Lagrangian. The deformation eliminates the Lagrangian’s singularity for such configurations. Each classical knot yields a doubled class of physical solutions, distinguished by a discrete parameter $d = \pm 1$, which arises from the underlying transformation properties of the RS vectors and Bateman potentials. In the limit $y \to 0$, these deformed solutions coalesce smoothly to the original null hopfion–Rañada knot.

5. Theoretical Consequences and Applications

The persistence of knotted, topologically nontrivial lattice of solutions in ModMax electrodynamics underscores the robustness of electromagnetic topology under continuous, maximally symmetric nonlinear deformations. This directly demonstrates that:

• Nonlinear generalizations preserving duality and conformal invariance retain the same complex knotted structures as in the linear theory, provided configurations are regularized appropriately.
• The explicit deformation parameter $y$ regularizes previously ill-defined (null) configurations, providing a controlled manner to explore how non-trivial topology and symmetry content interact under nonlinear corrections.

The Hamiltonian and RS vector formalisms enable the systematic construction of further exotic solutions: arbitrary holomorphic transformations of the Bateman potentials generate an infinite family of $(p, q)$ knots, augmenting the already rich structure of electromagnetic knots. Potential avenues for application include:

• Condensed matter systems and magnetohydrodynamics that support topologically nontrivial solitons.
• Quantum vacuum nonlinear optics, leveraging ModMax as a symmetry-respecting testbed for investigating higher-order quantum corrections and vacuum birefringence.
• Early universe and high-energy physics scenarios, where nonlinear electrodynamics may capture quantum gravity and string-theoretic corrections.

6. Future Directions and Mathematical Topology

The moduli space of deformed knots in ModMax theory raises fundamental questions regarding the classification and stability of electromagnetic knots under symmetric nonlinear deformations. The continuous interpolability between null and non-null knotted solutions suggests the possibility of a broader topological classification of electromagnetic fields, possibly leveraging modern knot invariants and techniques from differential geometry and algebraic topology.

Additionally, the existence of a doubled class of solutions for each knot, with their merger in the linear limit, points to previously underexplored discrete degrees of freedom in the spectrum of nonlinear electromagnetic knots. Investigating their quantum mechanical implications, scattering properties, and stability remains an open thematic area.

---

In summary, ModMax electrodynamics furnishes the only continuous, single-parameter, maximally symmetric nonlinear extension of Maxwell theory that supports regularized, topologically knotted electromagnetic solutions with an enriched, doubled spectrum. The ModMax deformation allows the explicit realization of classically forbidden configurations, preserves the intricate symmetry structure of the electromagnetic field even in the presence of nonlinearity, and paves the way for both theoretical and applied advances in understanding the intersection of field topology and nonlinear dynamics (Dassy et al., 2021).