The Maxwell equations. (Les équations de Maxwell.) (French) Zbl 0896.53049

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 115-125 (1997).
This paper presents the Maxwell equations in the formalism of differential forms. In the stationary case (electrostatics and magnetostatics), these differential forms are considered in Euclidean space \(\mathbb{R}^3\) or, more generally, in a three-dimensional oriented Riemannian manifold. The electric field is given by a closed 1-form which satisfies \(dE=0\), and the magnetic field by a 2-form which fulfills \(dB=0\). The Hodge operator \(\star\) is defined for an oriented manifold of dimension \(n\) which allows, among other things, to write the Maxwell equations in a non-orientable spacetime. Next, \(B\) and \(E\) are constructed by means of a 2-form in \({\mathbb{R}}^4: F=B+E\wedge dt\), which together with the 3-form \(j=\rho dx\wedge dy\wedge dz - J\wedge dt\) leads to the following form of the Maxwell equations: \(dF=0, d^{\star}F=\pm 4\pi{\mu_0\over \varepsilon_0}\star j\). The second equation can also be derived as the Euler-Lagrange equations of a variational principle.
Introducing a fibre bundle \(E\) over a manifold, the covariant derivative or connection \(\nabla\) is introduced as the derivative of a section \(S\) evaluated at the point \(x_0\in M\), in the direction \(V\in T_{x_0}M\). In this way, if \(E=M\times{\mathbb{R}}^{n}\), \(\nabla_iS=\frac{\partial S} {\partial x_i}+A_i(x)\), where \(A_i(x)\) are endomorphisms of \({\mathbb{R}}^n\) depending on \(x\). For such a connection, considering a loop \(\gamma\), a holonomy \(e^{i\int_{\gamma}A}\) can be introduced. As an example, the author considers a three-dimensional manifold \(M\) and a connection which has curvature \(F=eB/\hbar\), where \(e,\hbar\) are physical constants and \(B\) is a stationary magnetic field. This will impose a quantization of the magnetic field \(e\int_{S}B\in hZ\) if we consider a 2-cycle \(S\) on \(M\). This condition is connected to the occurence of magnetic monopoles.
Another given example is the parallel transport performed by \(e^{iq\int_{\gamma}A/\hbar}\) which is related to the Aharonov-Bohm effect in quantum mechanics.
For the entire collection see [Zbl 0882.00016].


53Z05 Applications of differential geometry to physics
78A25 Electromagnetic theory (general)
58A14 Hodge theory in global analysis
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