Galilean Electromagnetism Overview

• Galilean Electromagnetism is a non-relativistic formulation of Maxwell’s equations that emphasizes electric and magnetic limits under the Galilean symmetry group.
• It derives invariant field variables and conserved quantities through systematic non-relativistic contractions, highlighting applications in condensed matter physics and continuum mechanics.
• The framework connects infinite-dimensional symmetry extensions with practical models like Quantum Hall edge theory and pseudo-potential methods, advancing both theoretical and experimental insights.

Galilean Electromagnetism is the paper of electromagnetic theory in the non-relativistic regime where the relevant symmetry group is the Galilean group, as opposed to the Lorentz group underlying Special Relativity. The non-relativistic limits of Maxwell theory and their Galilean-covariant formulations not only illuminate foundational questions about the nature of electromagnetic phenomena at low velocities but also underpin effective descriptions relevant to condensed matter physics, continuum mechanics, and non-relativistic quantum field theory.

1. Structure of Galilean Limits of Electromagnetism

Galilean electromagnetism arises from specific non-relativistic limits of Maxwell’s equations, first systematically classified by Le Bellac and Lévy-Leblond. These limits are characterized by dominant field and current scales:

• Electric (Electroquasistatic) Limit: The electric field dominates (|E| ≫ |B|/c), charges dominate currents, and $\nabla \times \mathbf{E} = 0$ is imposed rather than the usual Faraday law.
• Magnetic (Quasimagnetostatic) Limit: The magnetic field dominates (|E| ≪ |B|c), currents dominate charges, and the displacement current in the Ampère–Maxwell law is neglected.
• Instantaneous Limit: Achieved by formally letting $c \to \infty$, making all field propagation instantaneous—mathematically similar to the electric limit, but physically distinct since there is no non-relativistic small-velocity constraint.

The Galilean limits are systematically extracted when Maxwell’s equations are written in unit-independent form (1012.1068).

$$\begin{align*} \text{(Electric Limit)} \quad & \nabla \cdot \mathbf{E} = \alpha \rho, \quad \nabla \cdot \mathbf{B} = 0, \ & \nabla \times \mathbf{E} = 0, \quad \nabla \times \mathbf{B} - \frac{\beta}{\alpha}\frac{\partial \mathbf{E}}{\partial t} = \beta \mathbf{J} \ \text{(Magnetic Limit)} \quad & \nabla \cdot \mathbf{E} = \alpha \rho, \quad \nabla \cdot \mathbf{B} = 0, \ & \nabla \times \mathbf{E} = -\frac{\alpha}{\beta c^2}\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \beta \mathbf{J} \end{align*}$$

Here, $\alpha$, $\beta$ refer to unit-system dependent constants.

A key insight is the “double role” of $c$ in electromagnetism (1012.1068): it acts both as a defining constant (e.g., relating units in Coulomb’s and Biot–Savart laws) and as the actual finite speed of field propagation. The “$c$ equivalence principle” ($c_\text{(units)} = c_\text{(propagation)}$) is required for the physical equivalence of electromagnetism and optics.

2. Galilean Invariance, Observables, and Noether Charges

In Galilean-invariant theories, conserved quantities follow directly from the underlying spacetime symmetry. Noether’s theorem applies as usual, with the boost symmetry yielding extra conservation laws beyond momentum and energy. For instance (1011.3057):

• For ordinary Galilean invariance under $x^i \rightarrow x^i + v^i t$, one obtains

$E(t) = E_0 + P t$

with $E(t)$ a collective coordinate (e.g., center-of-mass) and $P$ the conserved momentum. The boost Noether charge,

$Q_\mathrm{boost} = Pt - E$

is itself conserved.

Analogy with Galilean electromagnetism is direct: in non-relativistic EM, boost invariance “protects” the linear time evolution of charge moments (such as the charge dipole), so that the time evolution of the center-of-charge is linear, and the time derivative is a conserved quantity. This symmetry is explicitly broken in the relativistic (Lorentz-invariant) theory (1011.3057).

In effective theories with “internal” Galilean symmetries, such as galileon models, the shift $T(x) \to T(x) + b_\mu x^\mu$ forces the existence of a Noether current that can be written as a total divergence, $j^\mu = \partial_\nu \xi^{\mu\nu}$. This results in a “protected” dipole whose time evolution is linear (see equations (3)–(8) in (1011.3057)).

3. Scale Separation, Galilean Invariant Variables, and Semi-Relativistic Corrections

A systematic derivation of observer-invariant (objective) electromagnetic field variables within the semi-relativistic ($|v|/c \ll 1$ but nonzero) regime is given in (Song, 2013). Starting from Lorentz covariant tensors, a semi-relativistic Lorentz boost is used to define Galilean-invariant fields:

$$\begin{align*} \mathbf{E}^* &= \mathbf{E} + \mathbf{v} \times \mathbf{B} \ \mathbf{B}^* &= \mathbf{B} - \frac{\mathbf{v} \times \mathbf{E}}{c^2} \end{align*}$$

All electromagnetic objects (potentials, current density, stress tensor, etc.) can thus be redefined in terms of these invariants. Maxwell’s equations, Poynting’s theorem, and the electromagnetic momentum identity acquire Galilean invariant forms when rewritten with these variables. In “strong” electric or magnetic limits (where further scale separation exists), additional simplifications ensue—crucial for constitutive modeling of continua where EM and mechanical fields interact (Song, 2013).

4. Symmetry and Infinite-Dimensional Extensions

A remarkable feature is the infinite-dimensional enhancement of symmetries in Galilean Electromagnetism as compared to relativistic cases. The process begins by performing a non-relativistic contraction (“Inönü–Wigner contraction”) of the relativistic conformal algebra. The resulting field theories—including the free Maxwell and Yang-Mills theories in D=4—are not merely Galilean invariant, but invariant under the infinite-dimensional Galilean Conformal Algebra (GCA) (Bagchi et al., 2014, Bagchi et al., 2015, Bagchi et al., 2017).

The essential construction is as follows:

• Perform distinct scaling of time vs. spatial coordinates, and of the gauge potentials: $A_0 \to A_0$, $A_i \to \epsilon A_i$ in the electric limit; $A_0 \to \epsilon A_0$, $A_i \to A_i$ in the magnetic limit.
• After contraction, the equations of motion retain invariance under both the finite GCA (translations, rotations, boosts, dilatation, special conformal) and a tower of infinite extension generators:

$$L^{(n)} = -[(n+1) t^n x^i \partial_i + t^{n+1} \partial_t], \qquad M_i^{(n)} = t^{n+1}\partial_i \qquad (n \in \mathbb{Z})$$

Both sectors (electric and magnetic) exhibit this infinite symmetry in D=4, constraining correlators and operator dynamics analogously to how 2D conformal symmetry powers CFT analyses (Bagchi et al., 2014).

Remarkably, when coupled to matter, such as Galilean fermions or scalars, the enhanced symmetry persists provided the parent relativistic theory is conformal (Bagchi et al., 2017).

5. Quantum Field Theory, Renormalization, and Fixed Points

The quantum properties of Galilean Electrodynamics take two distinct forms, depending on dimension and field content.

• In (2+1)D, for abelian GED minimally coupled to a Schrödinger scalar (derived via null reduction from relativistic Maxwell theory), the theory possesses an extra dimensionless, gauge-invariant scalar in the gauge multiplet, leading to an infinite tower of marginal couplings (Chapman et al., 2020).
• The background field method reveals that while these couplings ($\mathcal{J}[M], \mathcal{V}[M], \mathcal{E}[M]$) run logarithmically, the gauge coupling $e$ does not renormalize to any order (a consequence of particle number conservation and the structure of non-relativistic loops).
• The beta functions for these functions, e.g.

  $$\beta_{\mathcal{J}}[M] = \frac{e^2}{2\pi} \left( \mathcal{J}'[M] + \frac{5}{8}M^{-4} \right)$$

  admit power-law fixed-point solutions, e.g. $\mathcal{J}^*[M] = j^* M^{-3}$, and yield a continuous family (“conformal manifold”) of fixed points preserving the non-relativistic conformal (Schrödinger) symmetry.

• For (3+1)D, when GED is coupled to Galilean fermions, one-loop renormalizability holds, but the beta function for the coupling grows linearly ($\beta(g) = 2g$), precluding asymptotic freedom, in contrast to the situation in (2+1)D scalar-coupled GED (Banerjee et al., 2022).

This rich RG structure, with exactly marginal gauge couplings and flowing infinite couplings in the scalar sector, points to novel types of universality classes in nonrelativistic quantum critical systems (Chapman et al., 2020, Banerjee et al., 2022).

6. Physical Models: Quantum Hall Edge, Birefringence, and Pseudopotential Formalisms

Galilean invariant formulations are crucial for consistent physical predictions in condensed matter and effective field theories.

• Quantum Hall edge theory: The chiral Luttinger liquid edge action must be modified for Galilean invariance, notably in the identification of the statistical gauge field and kinetic terms. The resultant electromagnetic response (momentum- and frequency-dependent edge conductivity) (Moroz et al., 2015):

  $$\sigma(\omega, p_x) = \frac{\nu}{2\pi} \left[1 + \frac{\mathcal{S}}{4} \frac{p_x^2}{B}\right] \frac{(-i)c}{\omega - c p_x + i0^+} - i m \epsilon''(B) \frac{p_x}{B}$$

  allows experimental extraction of both the shift $\mathcal{S}$ and bulk Hall viscosity.

• Galilean Carroll–Field–Jackiw and Podolsky electrodynamics: Non-relativistic models with Chern–Simons-like or higher-derivative Galilean symmetry–breaking terms yield birefringence (double-mode splitting in plane waves) and topological mass generation in the planar regime (Belich et al., 1 Mar 2025). The structure of dispersion relations and emergent topological phenomena is governed by the details of the Galilean-covariant Lagrangian.
• Moving pseudopotentials in electronic structure theory: The correct adaptation of nonlocal pseudopotentials to nuclear motion is essential to maintain Galilean covariance in the Schrödinger equation (Stengel et al., 24 Mar 2025). This ensures a precise equivalence between the mechanical response to rototranslations and electromagnetic perturbations—manifested, for instance, by an unequivocal connection between the Drude weight and the total electronic inertia,

  $$\sum_{IJ} M_{I\alpha, J\beta}^{(\mathrm{el})} = N \delta_{\alpha\beta} - \Omega D_{\alpha\beta} / \pi$$

  reconciling inertial (mechanical) and electrical definitions across metals.

7. Modern Mathematical Frameworks: Covariance and Geometric Algebra

Coordinate-free, covariant reformulations have unified Galilean and Lorentz-covariant electrodynamics.

• Reference frames as affine space maps (Saatkamp, 2023): Galilean group transformations are realized via frame-changing diffeomorphisms $F':M \to A^1 \times A^3$, with absolute time and velocity reciprocity $v(F'|F) = -v(F|F')$, and velocity addition rules. Maxwell’s equations are recast as differential forms, e.g. $F = \mathcal{R}^{-1}(-E^\flat/\alpha, *B^\flat)$, with observer independence manifest.
• Geometric algebra and Galilean spacetime algebra (GSTA) (Petronilo, 8 Jan 2024): Maxwell’s equations are encoded in a five-dimensional geometric algebra, with Faraday bivector $F = -E_e Y_4 + E_m Y_5 + iB - aK$, and Galilean spinors entering via minimal left ideals. The Levy-Leblond (non-relativistic Dirac) equation is written using Galilean gamma matrices and a properly constructed pseudoscalar.

8. Advances and Open Directions

• Geometric gauge theories (Marsh, 24 Sep 2024) recast both particle mechanics and classical electromagnetism in a bundle framework, with gauge-covariant derivatives ensuring observer and gauge independence. The Lagrangians reduce consistently from Einstein bundles to relativistic dust, then to relativistic and finally non-relativistic particle mechanics.
• Nonlinear extensions: Galilean versions of ModMax and Born-Infeld theories retain (remnants of) duality and conformal invariance (Banerjee et al., 2022, Banerjee et al., 2023), with duality transformations relating electric and magnetic Galilean sectors just as in the Maxwell limit, but supplemented by higher-order nonlinearities.
• Uniqueness and Degrees of Freedom: The action principle exists for the magnetic sector (requiring an auxiliary scalar), but in the electric sector does not admit an action, breaking the electromagnetic duality present in relativistic Maxwell theory. The resulting Galilean magnetic sector is topological, with no local degrees of freedom and only global/infinite-dimensional symmetry realizations (Banerjee et al., 2019).

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Summary Table: Galilean Electromagnetism Key Aspects

Concept/Aspect	Main Result or Principle	arXiv Paper(s)
Nonrelativistic Limits	Electric, magnetic, and instantaneous limits of Maxwell's equations, with distinct physical origins	(1012.1068, Song, 2013)
Symmetry Structure	Infinite-dimensional Galilean Conformal Algebra in D=4 for both free and interacting theories	(Bagchi et al., 2014, Bagchi et al., 2015, Bagchi et al., 2017)
Gauge Field Reduction	Electric/magnetic limits via scaling; unique NR relations among gauge field components	(Saha et al., 25 Jul 2025)
Noether Charges & Conserved Quantities	Linear evolution of center-of-mass/charge, protected dipole moments, link to internal symmetries	(1011.3057)
Quantum Properties	Vanishing or linear beta functions, conformal manifolds, structure of marginal operators	(Chapman et al., 2020, Banerjee et al., 2022)
Nonrelativistic QED	Schrödinger and Pauli–Schrödinger theories coupled to Galilean EM fields, NR Feynman rules	(Saha et al., 25 Jul 2025, Banerjee et al., 2022)
Topological/Edge Effects	Universal edge conductivity, link between shift and Hall viscosity in QH systems	(Moroz et al., 2015)
Mathematical Frameworks	Affine/frame-covariant, geometric algebra, geometric gauge theory	(Saatkamp, 2023, Petronilo, 8 Jan 2024, Marsh, 24 Sep 2024)
Sum Rules and Inertial/Electric Equivalence	Moving pseudopotentials restore Galilean covariance of Schrödinger equation and unify Drude weight definitions	(Stengel et al., 24 Mar 2025)

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Galilean Electromagnetism thus constitutes a coherent body of theoretical methods, symmetry principles, and effective models that underpin non-relativistic limits of electromagnetic theory. Its modern formulations—ranging from infinite-dimensional conformal symmetries to geometric gauge frameworks and coupled matter–field systems—offer both rigorous foundational insights and powerful tools for condensed matter, field theory, and mathematical physics.