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In `F33f`_`[high-energy physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=High-energy_physics]`_`f, `!nonlinear electrodynamics`! (`!NED`! or `!NLED`!) refers to a family of generalizations of `F33f`_`[Maxwell electrodynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Covariant_formulation_of_classical_electromagnetism]`_`f which describe `F33f`_`[electromagnetic fields`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Electromagnetic_field]`_`f that exhibit `F33f`_`[nonlinear dynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonlinear_system]`_`f.`:cite-ref-introductory-notes-1-0[`F5bf`_`[1`#cite-note-introductory-notes-1]`_`f] For a theory to describe the electromagnetic field (a `F33f`_`[U(1)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=U(1)]`_`f `F33f`_`[gauge field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauge_field]`_`f), its action must be `F33f`_`[gauge invariant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauge_invariant]`_`f; in the case of U ( 1 ) {\\displaystyle U(1)} , for the theory to not have `F33f`_`[Faddeev-Popov ghosts`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Faddeev-Popov_ghost]`_`f, this constraint dictates that the `F33f`_`[Lagrangian`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lagrangian_(field_theory)]`_`f of a nonlinear electrodynamics must be a function of only s ≡ ≡ − − 1 4 F α α β β F α α β β {\\displaystyle s\\equiv -{\\frac {1}{4}}F_{\\alpha \\beta }F^{\\alpha \\beta }} (the `F33f`_`[Maxwell Lagrangian`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Maxwell_Lagrangian]`_`f) and p ≡ ≡ − − 1 8 ϵ ϵ α α β β γ γ δ δ F α α β β F γ γ δ δ {\\displaystyle p\\equiv -{\\frac {1}{8}}\\epsilon ^{\\alpha \\beta \\gamma \\delta }F_{\\alpha \\beta }F_{\\gamma \\delta }} (where ϵ ϵ {\\displaystyle \\epsilon } is the `F33f`_`[Levi-Civita tensor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Levi-Civita_tensor]`_`f).`:cite-ref-introductory-notes-1-1[`F5bf`_`[1`#cite-note-introductory-notes-1]`_`f]`:cite-ref-holographic-dc-2-0[`F5bf`_`[2`#cite-note-holographic-dc-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] Notable NED models include the `F33f`_`[Born-Infeld model`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Born-Infeld_model]`_`f,`:cite-ref-m-born-l-infeld-4-0[`F5bf`_`[4`#cite-note-m-born-l-infeld-4]`_`f] the `F33f`_`[Euler-Heisenberg Lagrangian`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler-Heisenberg_Lagrangian]`_`f,`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f] and the CP-violating U ( 1 ) {\\displaystyle U(1)} `F33f`_`[Chern-Simons theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chern-Simons_theory]`_`f L = s + θ θ p {\\displaystyle {\\mathcal {L}}=s+\\theta p} .`:cite-ref-holographic-dc-2-1[`F5bf`_`[2`#cite-note-holographic-dc-2]`_`f]`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f]`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f]

Some recent formulations also consider nonlocal extensions involving fractional U(1) `F33f`_`[holonomies`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holonomies]`_`f on `F33f`_`[twistor space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Twistor_space]`_`f, though these remain speculative.

>>References

`:cite-note-introductory-notes-1`!1.`! `F0af`_`[↑`#cite-ref-introductory-notes-1-0]`_`f `:citerefsorokin2022`aSorokin, Dmitri P. (2022). "Introductory Notes on Non-linear Electrodynamics and its Applications". `*Fortschritte der Physik`*. `!70`! (7–8). `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:2112.12118. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1002/prop.202200092.

`:cite-note-holographic-dc-2`!2.`! `F0af`_`[↑`#cite-ref-holographic-dc-2-0]`_`f `:citerefbitao2021`aBi, Shihao; Tao, Jun (2021). "Holographic DC conductivity for backreacted NLED in massive gravity". `*Journal of High Energy Physics`* (6): 174. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:2101.00912. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2021JHEP...06..174B. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/JHEP06(2021)174.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefbruce2024`aBruce, Stanley A. (2024). "Nonlinear electrodynamics and its possible connection to relativistic superconductivity: An example". `*Zeitschrift für Naturforschung A`*. `!79`! (11): 1041–1046. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2024ZNatA..79.1041B. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1515/zna-2024-0136.

`:cite-note-m-born-l-infeld-4`!4.`! `F0af`_`[↑`#cite-ref-m-born-l-infeld-4-0]`_`f `:citerefborninfeld1934`aBorn, M.; Infeld, L. (1934). "Foundations of the New Field Theory". `*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences`*. `!144`! (852): 425–451. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1934RSPSA.144..425B. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1098/rspa.1934.0059.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefheisenbergeuler1936`aHeisenberg, W.; Euler, H. (1936). "Folgerungen aus der Diracschen Theorie des Positrons". `*Zeitschrift für Physik`* (in German). `!98`! (11–12): 714–732. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1936ZPhy...98..714H. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/bf01343663. `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 1434-6001.

`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f `:citereffuzhaoliu2021`aFu, Qi-Ming; Zhao, Li; Liu, Yu-Xiao (2021). "Weak deflection angle by electrically and magnetically charged black holes from nonlinear electrodynamics". `*Physical Review D`*. `!104`! (2): 024033. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:2101.08409. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2021PhRvD.104b4033F. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1103/PhysRevD.104.024033.

`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `:citerefdelphenich2003`aDelphenich, David (2003). "Nonlinear Electrodynamics and QED". `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:hep-th/0309108.

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