nLab electromagnetism

Idea

Electromagnetism or electrodynamics is the gauge theory whose field is the electromagnetic field, see there for more details.

The corresponding quantum field theory is quantum electrodynamics.

Related concepts

• Maxwell's equations

• dielectric medium, pre-metric electromagnetism

• electromagnetic radiation

  • light

• Hall effect, quantum Hall effect

  spin Hall effect, quantum spin Hall effect

• topological photonics

• magnetohydrodynamics

• Dirac charge quantization

• uncertainty of fluxes

• Einstein-Maxwell theory

• cosmic electromagnetism

• Born-Infeld theory

• vector meson dominance

• higher Maxwell-type gauge theory

• quantum field theory

  • gauge theory

    • classical electrodynamics

    • electromagnetism

    • Yang-Mills theory

    • electroweak field

  • higher U(1)-gauge theory

    • B-field

    • C-field

References

Classical electromagnetism

(Pre-)history:

• Edmund T. Whittaker:

  A History of the Theories of Aether and Electricity

  • Vol. 1: The Classical Theories

    From the age of Descartes to the close of the Nineteenth century

    (1910, 1951) [pdf]

  • Vol. 2: The Modern Theories 1900-1926

    (1953) [pdf]

  reprinted by Humanities Press (1973)

  [Wikipedia entry]

Textbook accounts of classical electrodynamics:

• V. C. A. Ferraro, Electromagnetic Theory, The Athlone Press (1954, 1956)

• Lev Landau, Evgeny Lifshitz, Electrodynamics of Continuous Media, volume 8 of: Course of Theoretical Physics, Pergamon Press (1960, 1984) [ISBN:9780750626347, archive, pdf]

• David Griffiths: Introduction to Electrodynamics, Prentice Hall (1981, 1989, 1999), Pearson (2013), Cambridge University Press (2023, 2025) [pdf, Wikipedia entry]

• Julian Schwinger et al., Classical Electrodynamics, based on Schwinger’s 1976 lecture, Routledge (1999) [ISBN:9780738200569, pdf]

• John D. Jackson, Classical Electrodynamics, Wiley (1962, 1975, 1998) [ISBN:978-0-471-30932-1, Wikipedia entry]

• Friedrich W. Hehl, Yuri N. Obukhov, Foundations of Classical Electrodynamics – Charge, Flux, and Metric, Progress in Mathematical Physics 33, Springer (2003) [doi:10.1007/978-1-4612-0051-2]

  (from the pre-metric perspective)

• Walter Greiner, Classical Electrodynamics, Springer (1998) [doi:10.1007/978-1-4612-0587-6]

• Mikio Nakahara, Section 10.5 of: Geometry, Topology and Physics, IOP (2003) [doi:10.1201/9781315275826, pdf]

• Robert Wald, Advanced Classical Electromagnetism, Princeton University Press (2022) [ISBN:9780691220390]

Maxwell’s equations via differential forms

On the expression of classical electromagnetism, and especially of Maxwell's equations, in terms of differential forms, the de Rham differential and Hodge star operators:

• Élie Cartan, §80 in: Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite), Annales scientifiques de l’É.N.S. 3e série, tome 41 (1924) 1-25 [numdam:ASENS_1924_3_41__1_]

  (already in pregeometric form)

• Charles Misner, Kip Thorne, John Wheeler, §3.4 and §4.3 in: Gravitation, W. H. Freeman, San Francisco (1973) [ISBN:9780716703440]

• Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly 81 4 (1974) [doi:10.1080/00029890.1974.11993557, jstor:2318995, doi:10.2307/2318995]

• Walter Thirring, vol 2 §1.3 in: A Course in Mathematical Physics – 1 Classical Dynamical Systems and 2 Classical Field Theory, Springer (1978, 1992) [doi:10.1007/978-1-4684-0517-0]

• Theodore Frankel, §§9-10 in: Gravitational Curvature, Freeman, San Francisco (1979) [ark:13960/t58d7nn19]

• Dominic G. B. Edelen, §9.2 in: Applied exterior calculus, Wiley (1985) [GoogleBooks]

• Theodore Frankel, §3.5 & §7.2b in: The Geometry of Physics - An Introduction, Cambridge University Press (1997, 2004, 2012) [doi:10.1017/CBO9781139061377]

• Gregory L. Naber, §2.2 in: Topology, Geometry and Gauge fields – Interactions, Applied Mathematical Sciences 141 (2011) [doi:10.1007/978-1-4419-7895-0]

• Masao Kitano, Reformulation of Electromagnetism with Differential Forms, Chapter 2 in: Trends in Electromagnetism – From fundamentals to applications, InTech (2012) 21-44 [ISBN:978-953-51-0267-0, pdf]

• Sébastien Fumeron, Bertrand Berche, Fernando Moraes, Improving student understanding of electrodynamics: the case for differential forms, American Journal of Physics 88 (2020) 1083 [doi:10.1119/10.0001754, arXiv:2009.10356]

Phase space and Poisson brackets

• Daniel N. Blaschke, François Gieres, On the canonical formulation of gauge field theories and Poincaré transformations, Nuclear Physics B 965 (2021) 115366 [doi:10.1016/j.nuclphysb.2021.115366, arXiv:2004.14406]

• Corichi 1998, §III

For more see the references at Yang-Mills theory – References – Phase space and canonical quantization.

Quantum electromagnetism

Discussion of Fock space quantization based on careful analysis of the covariant phase space:

• Alejandro Corichi, Introduction to the Fock Quantization of the Maxwell Field, Rev. Mex. Fis. 44 4 (1998) 402-412 [arXiv:physics/9804018]

Discussion of electromagnetism in the generality of quantum field theory on curved spacetimes originates with

• Jonathan Dimock, Quantized Electromagnetic Field on a Manifold, Reviews in Mathematical Physics, Volume 04, Issue 02, 1992 (doi:10.1142/S0129055X92000078)

and is further developed in (e.g. regarding Hadamard states) in

• K. Sanders, Claudio Dappiaggi, T.-P. Hack, Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss’ law, Commun. Math. Phys. 328(2), 625–667 (2014)

• Claudio Dappiaggi, D. Siemssen, Hadamard states for the vector potential on asymptotically flat spacetimes, Rev. Math. Phys. 25, 1 (2013)

• Claudio Dappiaggi, B. Lang, Quantization of Maxwell’s equations on curved backgrounds and general local covariance, Lett. Math. Phys. 101(3), 265–287 (2012)

Discussion of global topological effects due to flux quantization:

• Daniel Freed, Greg Moore, Graeme Segal, The Uncertainty of Fluxes, Commun. Math. Phys. 271 (2007) 247-274 [arXiv:hep-th/0605198, doi:10.1007/s00220-006-0181-3]

• Daniel Freed, Greg Moore, Graeme Segal, Heisenberg Groups and Noncommutative Fluxes, Annals Phys. 322 (2007) 236-285 [arXiv:hep-th/0605200, doi:10.1016/j.aop.2006.07.014]

• Alexei Kitaev, Gregory W. Moore, Kevin Walker, Noncommuting Flux Sectors in a Tabletop Experiment [arXiv:0706.3410]