## 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].

### MSC:

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