Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

Abstract

In this note we present a geometric formulation of Maxwell's equations in Carnot groups (connected simply connected nilpotent Lie groups with stratified Lie algebra) in the setting of the intrinsic complex of differential forms defined by M. Rumin. Restricting ourselves to the first Heisenberg group H 1 , we show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations "in coordinates". Moreover, we analyze the notion of "vector potential", and we show that it satisfies a new class of 4th order evolution differential equations. Mathematics Subject Classification (2000). 43A80, 58A10, 83C22 .

Quasiregular maps and the conductivity equation in the Heisenberg group

“…This terminology is analogous to the plane case [1, Chapter 16.1.6] where it arises in connection with the Hodge * method. In classical electrodynamics E and JB give rise to Faraday's form as introduced in [9]. We will return to this facinating connection elsewhere.…”

“…We now recall the notion of horizontal curl of a horizontal vector field, introduced by Franchi, Tchou and Tesi in [8] and further studied in [9].…”

“…For the purposes of this paper, we briefly sketch the setup for the horizontal differential complex (Rumin complex) (E * 0 , d c ) in the first Heisenberg group. Here we follow the explicit representation in [9] based on M. Rumin's theory of intrinsic forms [15], [16].…”

“…For a general horizontal vector field V one can define (see [9]) an intrinsic divergence operator Div H by setting One can further introduce quantities…”

“…These systems are obtained from the usual Beltrami equation via the identification of the complex dilatation with a measurable conformal structure. In the Heisenberg setting the conductivity equation is formulated using the notion of the horizontal curl ∇ H × V of a horizontal vector field V , which has been recently studied by Franchi et al [8], [9].…”