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<title>Born–Infeld model</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Born–Infeld model</span></span>

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<p>In <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a>, the <b>Born–Infeld model</b> or the <b>Dirac–Born–Infeld action</b> is a particular example of what is usually known as a <a href="Nonlinear_electrodynamics" title="Nonlinear electrodynamics">nonlinear electrodynamics</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It was historically introduced in the 1930s to remove the divergence of the electron's <a href="Self-energy" title="Self-energy">self-energy</a> in <a href="Classical_electromagnetism" title="Classical electromagnetism">classical electrodynamics</a> by introducing an upper bound of the electric field at the origin. It was introduced by <a href="Max_Born" title="Max Born">Max Born</a> and <a href="Leopold_Infeld" title="Leopold Infeld">Leopold Infeld</a> in 1934,<sup id="cite_ref-M._Born,_L._Infeld_2-0" class="reference"><a href="#cite_note-M._Born,_L._Infeld-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> with further work by <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in 1962.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<p>Born–Infeld electrodynamics is named after physicists <a href="Max_Born" title="Max Born">Max Born</a> and <a href="Leopold_Infeld" title="Leopold Infeld">Leopold Infeld</a>, who first proposed it. The model possesses a whole series of physically interesting properties.

</p><p>In analogy to a <a href="Ultrarelativistic_limit" title="Ultrarelativistic limit">relativistic limit</a> on velocity, Born–Infeld theory proposes a limiting force via limited electric field strength. A maximum electric field strength produces a finite electric field self-energy, which when attributed entirely to electron mass-produces maximum field.<sup id="cite_ref-M._Born,_L._Infeld_2-1" class="reference"><a href="#cite_note-M._Born,_L._Infeld-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle E_{\rm {BI}}=1.187\times 10^{20}\,\mathrm {V} /\mathrm {m} .}</annotation>

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</math></span><img src="./0a892d74a0299863e5717cdcd08e4080af447900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.017ex; height:3.176ex;" alt="{\displaystyle E_{\rm {BI}}=1.187\times 10^{20}\,\mathrm {V} /\mathrm {m} .}" loading="lazy"></span></dd></dl>

<p>Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of <a href="Shock_waves" class="mw-redirect" title="Shock waves">shock waves</a> and <a href="Birefringence" title="Birefringence">birefringence</a>. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> completely exceptional <i>regular</i> <a href="Nonlinear_electrodynamics" title="Nonlinear electrodynamics">nonlinear electrodynamics</a>.

</p><p>This theory can be seen as a covariant generalization of Mie's theory and very close to <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s idea of introducing a nonsymmetric <a href="Metric_tensor" title="Metric tensor">metric tensor</a> with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.

</p><p>The compatibility of Born–Infeld theory with high-precision atomic experimental data requires a value of a limiting field some 200 times higher than that introduced in the original formulation of the theory.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>Since 1985 there was a revival of interest on Born–Infeld theory and its nonabelian extensions, as they were found in some limits of <a href="String_theory" title="String theory">string theory</a>. It was discovered by <a href="Efim_Fradkin" title="Efim Fradkin">E.S. Fradkin</a> and <a href="Arkady_Tseytlin" title="Arkady Tseytlin">A.A. Tseytlin</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> that the Born–Infeld action is the leading term in the low-energy effective action of the open string theory expanded in powers of derivatives of gauge field strength.

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<div class="mw-heading mw-heading2"><h2 id="Equations">Equations</h2></div>

<p>We will use the <a href="Theory_of_relativity" title="Theory of relativity">relativistic</a> notation here, as this theory is fully relativistic.

</p><p>The <a href="Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a> is

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {-\det \left(\eta +{\frac {F}{b}}\right)}}+b^{2},}</annotation>

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</math></span><img src="./ac88e582caec4bd6f47bcb48a7ccbc721fcdb01b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.245ex; height:7.509ex;" alt="{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {-\det \left(\eta +{\frac {F}{b}}\right)}}+b^{2},}" loading="lazy"></span></dd></dl>

<p>where <i>η</i> is the <a href="Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a>, <i>F</i> is the <a href="Faraday_tensor" class="mw-redirect" title="Faraday tensor">Faraday tensor</a> (both are treated as square matrices, so that we can take the <a href="Determinant" title="Determinant">determinant</a> of their sum), and <i>b</i> is a scale parameter. The maximal possible value of the electric field in this theory is <i>b</i>, and the <a href="Self-energy" title="Self-energy">self-energy</a> of point charges is finite. For electric and magnetic fields much smaller than <i>b</i>, the theory reduces to <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell electrodynamics</a>.

</p><p>In 4-dimensional spacetime the Lagrangian can be written as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {1-{\frac {E^{2}-B^{2}}{b^{2}}}-{\frac {(\mathbf {E} \cdot \mathbf {B} )^{2}}{b^{4}}}}}+b^{2},}</annotation>

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</math></span><img src="./c8f0a31d60e8ee94d2d8187114dd9f0ed3f4529a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.648ex; height:7.676ex;" alt="{\displaystyle {\mathcal {L}}=-b^{2}{\sqrt {1-{\frac {E^{2}-B^{2}}{b^{2}}}-{\frac {(\mathbf {E} \cdot \mathbf {B} )^{2}}{b^{4}}}}}+b^{2},}" loading="lazy"></span></dd></dl>

<p>where <b>E</b> is the electric field, and <b>B</b> is the magnetic field.

</p><p>In <a href="String_theory" title="String theory">string theory</a>, gauge fields on a <a href="D-brane" title="D-brane">D-brane</a> (that arise from attached open strings) are described by the same type of Lagrangian:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-T{\sqrt {-\det(\eta +2\pi \alpha 'F)}},}</annotation>

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</math></span><img src="./104455528378f9b1891e0f8dba48381fd56b9985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:28.769ex; height:4.676ex;" alt="{\displaystyle {\mathcal {L}}=-T{\sqrt {-\det(\eta +2\pi \alpha 'F)}},}" loading="lazy"></span></dd></dl>

<p>where <i>T</i> is the tension of the D-brane and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \alpha '}">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi \alpha '}</annotation>

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</math></span><img src="./e8f2b33bbc0627195980e3057342b49f60ed9fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.667ex; height:2.509ex;" alt="{\displaystyle 2\pi \alpha '}" loading="lazy"></span> is the invert of the <a href="String_theory" title="String theory">string tension</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFFradkinTseytlin1985" class="citation journal cs1">Fradkin, E.S.; Tseytlin, A.A. (1985). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/162803">"Non-linear electrodynamics from quantized strings"</a>. <i>Physics Letters B</i>. <b>163</b> (<span class="nowrap">1–</span>4): <span class="nowrap">123–</span>130. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985PhLB..163..123F">1985PhLB..163..123F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2885%2990205-9">10.1016/0370-2693(85)90205-9</a>.</cite></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeigh1989" class="citation journal cs1">Leigh, R.G. (1989). "DIRAC-BORN-INFELD ACTION FROM DIRICHLET σ-MODEL". <i>Modern Physics Letters A</i>. <b>04</b> (28): <span class="nowrap">2767–</span>2772. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217732389003099">10.1142/S0217732389003099</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFTseytlin2000" class="citation book cs1">Tseytlin, A. A. (2000). "Born-Infeld Action, Supersymmetry and String Theory". <i>The Many Faces of the Superworld</i>. pp. <span class="nowrap">417–</span>452. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9908105">hep-th/9908105</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789812793850_0025">10.1142/9789812793850_0025</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-02-4206-0</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9569497">9569497</a>.</cite></span>

</li>

</ol></div>

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</style><div id="Quantum_field_theories202" style="font-size:114%;margin:0 4em"><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theories</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li>

<li><a href="Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li>

<li><a href="Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li>

<li><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>

<li><a href="Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li>

<li><a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul>

<li><a href="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li>

<li><a href="Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li>

<li><a href="Proca_action" title="Proca action">Proca</a></li>

<li><a href="Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li>

<li><a href="Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li>

<li><a href="Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li>

<li><a href="Soler_model" title="Soler model">Soler</a></li>

<li><a href="Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li>

<li><a href="Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li>

<li><a href="Yukawa_interaction" class="mw-redirect" title="Yukawa interaction">Yukawa</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li>

<li><a href="Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li>

<li><a href="Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li>

<li><a href="Schwinger_model" title="Schwinger model">Schwinger</a></li>

<li><a href="Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li>

<li><a href="Thirring_model" title="Thirring model">Thirring</a></li>

<li><a href="Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li>

<li><a href="Liouville_field_theory" title="Liouville field theory">Liouville</a></li>

<li><a href="Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li>

<li><a href="Polyakov_action" title="Polyakov action">Polyakov</a></li>

<li><a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li>

<li><a href="Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li>

<li><a href="Super_QCD" title="Super QCD">Super QCD</a></li>

<li><a href="Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chiral_model" title="Chiral model">Chiral</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi</a></li>

<li><a href="Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li>

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Standard_Model" title="Standard Model">Standard Model</a></li>

<li><a href="Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Cosmic_string" title="Cosmic string">Cosmic string</a></li>

<li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li>

<li><a href="On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_foam" title="Quantum foam">Quantum foam</a></li>

<li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li>

<li><a href="Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"></span></span> Template:Quantum mechanics topics</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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