Letters in Mathematical Physics

, Volume 18, Issue 2, pp 97–106 | Cite as

A topological theory of the electromagnetic field

  • Antonio F. Rañada


It is shown that Maxwell equations in vacuum derive from an underlying topological structure given by a scalar field ϕ which represents a map S3×RS2 and determines the electromagnetic field through a certain transformation, which also linearizes the highly nonlinear field equations to the Maxwell equations. As a consequence, Maxwell equations in vacuum have topological solutions, characterized by a Hopf index equal to the linking number of any pair of magnetic lines. This allows the classification of the electromagnetic fields into homotopy classes, labeled by the value of the helicity. Although the model makes use of only c-number fields, the helicity always verifies ∫ A·Bd3r=nα, n being an integer and α an action constant, which necessarily appears in the theory, because of reasons of dimensionality.

AMS subject classifications (1980)

78A25 81B 


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Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • Antonio F. Rañada
    • 1
  1. 1.Departamento de Fisica TeóricaUniversidad ComplutenseMadridSpain

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