# Electromagnetic action and Lagrangian formulation
**Electromagnetic action and Lagrangian formulation** refers to the variational formulation of classical electrodynamics, in which the theory is packaged into a single Lorentz-scalar action, such as \(S = \int d^{4}x[-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} - j_{\mu}A^{\mu}]\) in natural (Heaviside–Lorentz) units.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf) Varying this action with respect to the four-potential \(A_{\nu}\) yields the inhomogeneous Maxwell equations, \(\partial_{\mu}F^{\mu\nu} = j^{\nu}\), while the homogeneous pair is satisfied identically by the construction of \(F_{\mu\nu}\) from the potential and requires no Lagrangian term.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf) Applying Noether's theorem to the translation-invariant action delivers the conservation laws of energy and momentum.[2](https://doi.org/10.1093/oso/9780192867421.003.0008) The consistency condition \(\partial_{\mu}J^{\mu} = 0\), expressing local charge conservation, follows from the field equation itself.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
The variational route packages all of classical electrodynamics into a single Lorentz-scalar expression, and [Noether's theorem](https://www.edgepedia.org/noethers-theorem) applied to it delivers the conservation laws of energy and momentum.[2](https://doi.org/10.1093/oso/9780192867421.003.0008)
| Key fact | Detail |
|---|---|
| Free-field Lagrangian density | \(\mathcal{L}_{\mathrm{field}} = -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}\) (SI); the only quadratic Lorentz invariant that gives nontrivial equations, up to a total derivative[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) |
| Homogeneous equations | ∇·B = 0 and Faraday's law are satisfied identically by F_μν built from A_μ; no Lagrangian term is needed[4](http://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node142.html) |
| 3+1 reduction | \(-\tfrac{1}{4}F^2 - j \cdot A = \tfrac{1}{2}(E^2 - B^2) - j_{\mu}A^{\mu}\) in natural units[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf) |
| Consistency condition | Applying \(\partial_{\mu}\) to the field equation forces \(\partial_{\mu}J^{\mu} = 0\), charge conservation[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) |
| Normalization | The equations fix only the relative coefficient of the two terms; the overall scale is fixed by matching the standard field energy[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) |
## Ingredients and why F² is essentially unique
The field tensor transforms as \(F_{\mu\nu} \rightarrow \Lambda_{\mu}^{\rho}\Lambda_{\nu}^{\sigma}F_{\rho\sigma}\) under Lorentz transformations, so any scalar contracted from its components is itself Lorentz invariant.[5](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf)
[Gauge invariance](https://www.edgepedia.org/gauge-invariance) restricts the free-field term heavily: it must depend on \(A_{\mu}\) only through \(F_{\mu\nu}\), so that shifting \(A_{\mu}\) by a gradient leaves \(\mathcal{L}\) unchanged. The only Lorentz-invariant quadratic forms are then \(F_{\mu\nu}F^{\mu\nu}\) and \(\varepsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\). The second, the dual contraction, is a total derivative: it integrates to a boundary term and contributes nothing to the local equations of motion.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) This is why the θ-term familiar from non-Abelian gauge theories is invisible classically in electrodynamics: it changes the action, but not the Euler–Lagrange equations.
The prefactor depends on the unit system. In [Gaussian units](https://www.edgepedia.org/gaussian-units) the Lagrangian density is postulated as \(\mathcal{L} = -(c^2/16\pi)F_{\mu\nu}F^{\mu\nu} + J_{\mu}A^{\mu}\), with coefficient and sign chosen so the [Euler–Lagrange equation](https://www.edgepedia.org/euler-lagrange-equation) is \(\partial_{\nu}F^{\mu\nu} = (4\pi/c^2)J^{\mu}\).[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) The Gaussian Maxwell equations themselves carry factors of 4π, \(\nabla \cdot \mathbf{e} = 4\pi\rho\) and \(\nabla \times \mathbf{b} = (1/c)\partial\mathbf{e}/\partial t + (4\pi/c)\mathbf{j}\), and this unit dependence is a recognized inconvenience for the relativistic formulation.[6](http://scholarpedia.org/article/Special_relativity:_electromagnetism) Many textbooks accordingly rewrite [Maxwell's equations](https://www.edgepedia.org/maxwells-equations) in Gaussian units before demonstrating Lorentz invariance.[7](https://physics.uwo.ca/~mhoude2/courses/phy502b/Covariant_formulation.pdf) Note also that the sign of the interaction term differs across sources: the KU Leuven notes write \(+J_{\mu}A^{\mu}\) in Gaussian units, while the SI convention writes \(-J^{\mu}A_{\mu}\); the two agree once the index placement and unit conventions are translated.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
## Euler–Lagrange derivation of Maxwell's equations
Varying the action with respect to \(A_{\nu}\) gives two derivatives of the Lagrangian. In natural units, \(\partial \mathcal{L}/\partial A_{\nu} = -j^{\nu}\) and \(\partial \mathcal{L}/\partial(\partial_{\mu}A_{\nu}) = -F^{\mu\nu}\), so the Euler–Lagrange equation \(\partial \mathcal{L}/\partial A_{\nu} - \partial_{\mu}[\partial \mathcal{L}/\partial(\partial_{\mu}A_{\nu})] = 0\) becomes \(\partial_{\mu}F^{\mu\nu} = j^{\nu}\).[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
The homogeneous pair (∇·**B** = 0 and Faraday's law) never enters the variational calculation. It is satisfied automatically and identically by the construction of F_μν from the four-potential, independent of any dynamics; in the covariant form of the homogeneous equations, the pairing involves a symmetric tensor contracted against the antisymmetric F_μν, which vanishes identically.[4](http://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node142.html) The homogeneous equations are therefore not independent as far as the action principle is concerned, which is why there is no Lagrangian term for them.[4](http://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node142.html)
Taking \(\partial_{\mu}\) of the field equation produces a consistency requirement: the left side vanishes by the symmetry–antisymmetry argument on \(F_{\mu\nu}\), so the current must obey \(\partial_{\mu} J^{\mu} = 0\), the continuity equation expressing local charge conservation.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
## Gauge invariance, gauge fixing, and the cost of Lorenz gauge
The canonical momentum conjugate to \(A_0\) vanishes identically, a direct reflection of the gauge redundancy inherent in \(A_\mu\): gauge-related potentials describe the same physical fields.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf) Two remedies exist. One is to fix the gauge in advance, eliminating the unphysical degrees of freedom at the price of losing manifest [Lorentz covariance](https://www.edgepedia.org/lorentz-covariance). The other is to add a gauge-fixing term, for example a Lagrange-multiplier term imposing the Lorenz condition \(\partial_\mu A^\mu = 0\) (Feynman gauge corresponds to \(\lambda = 1\) in the family \(-(\lambda/2)(\partial_\mu A^\mu)^2\)), with spurious degrees of freedom removed at the end of the calculation.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
With the Lorenz condition imposed, the field equation reduces to a wave equation \(\Box A^{\mu}\) proportional to \(J^{\mu}\) (in Gaussian units, \(\Box A^{\mu} = -(4\pi/c^{2})J^{\mu}\)), making the propagation of the potential explicit.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
## Noether's theorem: stress–energy and conserved momentum
Because the action is translation invariant (when the current is not a fixed external function of position), Noether's theorem supplies a conserved quantity for each spacetime direction. The field part is packaged in the electromagnetic energy–momentum tensor, whose divergence expresses local conservation of field energy and momentum.[2](https://doi.org/10.1093/oso/9780192867421.003.0008)
The tensor obtained directly from the Lagrangian by Noether's procedure has defects: it depends explicitly on A_ρ, so it is not gauge invariant, and it is not symmetric.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) The Belinfante improvement repairs both, producing the gauge-invariant, symmetric tensor built from **E**, **B**, and the stress components.
With an external current \(j_{\mu}(x)\), translation invariance is broken and the canonical tensor is not conserved: \(\partial_{\mu} T^{\mu\nu} = (\partial_{\nu} j_{\alpha}) A^{\alpha} \neq 0\). The field alone does not conserve energy–momentum because the prescribed source can inject or absorb both.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
## Adding matter: sources, double counting, and self-force
In the simplest action the particles appear only as sources of the four-current \(J^{\mu}\). The [KU Leuven](https://www.edgepedia.org/ku-leuven) notes flag this as incomplete: a full classical treatment should contain three types of terms, a free-field Lagrangian, a free-particle Lagrangian, and an interaction Lagrangian, with variation over the fields giving Maxwell's equations and variation over the particles giving the [Lorentz force](https://www.edgepedia.org/lorentz-force).[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf) The IST Lisbon notes describe the same situation as a "compromise": the static source \(j_{\mu}(x)\) is a stand-in that must eventually be eliminated, because in a fully interacting theory the current should emerge from other dynamical fields.[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
Beyond the source question lie the self-energy and radiation-reaction problems (the Larmor/Abraham–Lorentz territory of runaway solutions and pre-acceleration). Point-particle self-energy and radiation-reaction problems are notoriously delicate within the classical action framework, requiring regularization or extended formalisms beyond the simple single-action setup described here.[3](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
## Comparison with the 3+1 and sibling formulations
The covariant density reduces directly to the familiar one. In natural units,[1](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
\(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - j_{\mu}A^{\mu} = \frac{1}{2}(E^2 - B^2) - j_{\mu}A^{\mu}\),
which is equivalent to the standard 3+1 Euler–Lagrange treatment of **E** and **B**. The analogy with classical mechanics is structural: where the mechanical Lagrangian is kinetic minus potential energy, here one integrates a Lagrangian density over spacetime.[5](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf)
## References
1. [QFT lecture notes, Chapter 5: Electromagnetic field (IST Lisbon)](http://cftp.ist.utl.pt/~gernot.eichmann/2015-qft/qft-5.pdf)
2. [Covariant formulation of electrodynamics (Oxford University Press chapter)](https://doi.org/10.1093/oso/9780192867421.003.0008)
3. [Mathematical methods in physics — electromagnetic Lagrangian (KU Leuven lecture notes)](https://fys.kuleuven.be/itf/staff/toine/files/electromagn.pdf)
4. [Building a Relativistic Field Theory (Duke University, PHY319)](http://webhome.phy.duke.edu/~rgb/Class/phy319/phy319/node142.html)
5. [Lorentz Invariant Formulation of Electromagnetism (University of Virginia)](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/18_7420/18_7420_17_Covariant_Electromagnetism.pdf)
6. [Special relativity: electromagnetism (Scholarpedia)](http://scholarpedia.org/article/Special_relativity:_electromagnetism)
7. [Covariant Formulation of Electrodynamics (Western University PHY502B)](https://physics.uwo.ca/~mhoude2/courses/phy502b/Covariant_formulation.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Covariant formulation of electromagnetism › Covariant action and Lagrangian formulation*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 11, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*
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