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Born–Infeld model

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In theoretical physics, the Born–Infeld model or the Dirac–Born–Infeld action is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 1930s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric field at the origin. It was introduced by Max Born and Leopold Infeld in 1934,[1] with further work by Paul Dirac in 1962.[2][3][4][5][6]

Overview

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Born–Infeld electrodynamics is named after physicists Max Born and Leopold Infeld, who first proposed it. The model possesses a whole series of physically interesting properties.

In analogy to a relativistic limit 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.[1]

Born–Infeld electrodynamics displays good physical properties concerning wave propagation, such as the absence of shock waves and birefringence. A field theory showing this property is usually called completely exceptional, and Born–Infeld theory is the only[7] completely exceptional regular nonlinear electrodynamics.

This theory can be seen as a covariant generalization of Mie's theory and very close to Albert Einstein's idea of introducing a nonsymmetric metric tensor with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.

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.[8]

Since 1985 there was a revival of interest on Born–Infeld theory and its nonabelian extensions, as they were found in some limits of string theory. It was discovered by E.S. Fradkin and A.A. Tseytlin[9] 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.

Equations

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We will use the relativistic notation here, as this theory is fully relativistic.

The Lagrangian density is

where η is the Minkowski metric, F is the Faraday tensor (both are treated as square matrices, so that we can take the determinant of their sum), and b is a scale parameter. The maximal possible value of the electric field in this theory is b, and the self-energy of point charges is finite. For electric and magnetic fields much smaller than b, the theory reduces to Maxwell electrodynamics.

In 4-dimensional spacetime the Lagrangian can be written as

where E is the electric field, and B is the magnetic field.

In string theory, gauge fields on a D-brane (that arise from attached open strings) are described by the same type of Lagrangian:

where T is the tension of the D-brane and is the invert of the string tension.[10][11]

References

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  1. ^ a b Born, 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. Bibcode:1934RSPSA.144..425B. doi:10.1098/rspa.1934.0059.
  2. ^ Dirac, Paul (1962-06-19). "An extensible model of the electron". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 268 (1332): 57–67. Bibcode:1962RSPSA.268...57D. doi:10.1098/rspa.1962.0124. ISSN 0080-4630. S2CID 122728729.
  3. ^ Han, Xiaosen (2016-04-01). "The Born–Infeld vortices induced from a generalized Higgs mechanism". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 472 (2188): 20160012. doi:10.1098/rspa.2016.0012. ISSN 1364-5021. PMC 4892282. PMID 27274694.
  4. ^ Liu, Chien-Hao; Yau, Shing-Tung (2016-06-28). "Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term". arXiv:1606.08529 [hep-th].
  5. ^ "Dirac-Born-Infeld action in nLab". ncatlab.org. Retrieved 2023-11-01.
  6. ^ Dymnikova, Irina (2021). "Image of the Electron Suggested by Nonlinear Electrodynamics Coupled to Gravity". Particles. 4 (2): 129–145. Bibcode:2021Parti...4..129D. doi:10.3390/particles4020013. ISSN 2571-712X.
  7. ^ Bialynicki-Birula, I (1983). "3. Nonlinear Electrodynamics: Variations on a Theme by Born and Infield". In Jancewicz, B.; Lukierski, J. (eds.). Quantum Theory of Particles and Fields: Festschrift of J. Lopuszanski. World Scientific. pp. 31–42. ISBN 9971-950-77-4. OCLC 610059703.
  8. ^ Soff, Gerhard; Rafelski, Johann; Greiner, Walter (1973). "Lower Bound to Limiting Fields in Nonlinear Electrodynamics". Physical Review A. 7 (3): 903–907. Bibcode:1973PhRvA...7..903S. doi:10.1103/PhysRevA.7.903. ISSN 0556-2791.
  9. ^ Fradkin, E.S.; Tseytlin, A.A. (1985). "Non-linear electrodynamics from quantized strings". Physics Letters B. 163 (1–4): 123–130. Bibcode:1985PhLB..163..123F. doi:10.1016/0370-2693(85)90205-9.
  10. ^ Leigh, R.G. (1989). "DIRAC-BORN-INFELD ACTION FROM DIRICHLET σ-MODEL". Modern Physics Letters A. 04 (28): 2767–2772. doi:10.1142/S0217732389003099.
  11. ^ Tseytlin, A. A. (2000). "Born-Infeld Action, Supersymmetry and String Theory". The Many Faces of the Superworld. pp. 417–452. arXiv:hep-th/9908105. doi:10.1142/9789812793850_0025. ISBN 978-981-02-4206-0. S2CID 9569497.