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Towards a Unified Field Theory for Classical Electrodynamics

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Abstract.

In this paper we introduce a model which describes the relation of matter and the electromagnetic field from a unitarian standpoint in the spirit of ideas of Born and Infeld. In this model, based on a semilinear perturbation of Maxwell equations, the particles are finite-energy solitary waves due to the presence of the nonlinearity. In this respect the matter and the electromagnetic field have the same nature. Finite energy means that particles have finite mass and this makes electrodynamics consistent with the special relativity. We analyze the invariants of the motion of the semilinear Maxwell equations (SME) and their static solutions. In the magnetostatic case (i.e., when the electric field E = 0 and the magnetic field H does not depend on time) SME are reduced to the semilinear equation where ∇× denotes the curloperator, f′ is the gradient of a strictly convex smooth function f:R3→R and A:R3→R3 is the gauge potential related to the magnetic field H (H = ∇× A). Due to the presence of the curl operator, (1) is a strongly degenerate elliptic equation. Moreover, physical considerations impel f to be flat at zero (f′′(0)=0) and this fact leads us to study the problem in a functional setting related to the Orlicz space Lp+Lq. The existence of a nontrivial finite- energy solution of (1) is proved under suitable growth conditions on f. The proof is carried out by using a suitable variational framework related to the Hodge splitting of the vector field A.

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Authors and Affiliations

  1. Dipartimento di Matematica, Applicata “U. Dini” Università degli Studi di Pisa, via Bonanno 25/b, 56126, Pisa, Italy

    Vieri Benci

  2. Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125, Bari, Italy

    Donato Fortunato

Authors
  1. Vieri Benci
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  2. Donato Fortunato
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Correspondence to Donato Fortunato.

Additional information

Communicated by C. A. Stuart

We thank Marino Badiale and Charles Stuart for their useful suggestions.

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Benci, V., Fortunato, D. Towards a Unified Field Theory for Classical Electrodynamics. Arch. Rational Mech. Anal. 173, 379–414 (2004). https://doi.org/10.1007/s00205-004-0324-7

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  • DOI: https://doi.org/10.1007/s00205-004-0324-7

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