Wavelet Electrodynamics II: Atomic Composition of Electromagnetic Waves

Abstract

The representation of solutions of Maxwell's equations as superpositions of scalar wavelets with vector coefficients developed earlier is generalized to wavelets with polarization, which are matrix-valued. The construction proceeds in four stages: (1) A Hilbert space H of solutions is considered, based on a conformally invariant inner product. (2) The analytic-signal transform extends solutions from real space-time to a complex space-time domain T (double tube). The evaluation map E_z, which sends any solution F=B+iE in H to the value F(z) at z\in T, is bounded. The electromagnetic wavelets are defined as the adjoints the Ψ_z=E_z^*. (3) The eight real parameters z=x+iy\in T are given a complete physical interpretation: x\in R^4 is interpreted as a space-time point about which Ψ_z is focussed, and the timelike vector y gives its scale and velocity. Thus wavelets parameterized by the set of {\sl Euclidean} points (real space, imaginary time) have stationary centers, and the others are Doppler-shifted versions of the former. All the wavelets can be obtained from a single "mother wavelet" by conformal transformations. (4) A resolution of unity is established in H, giving a representation of solutions as "atomic compositions" of wavelets parameterized by z\in E. This yields a constructive method for generating solutions with initial data specified locally in space and by scale. Other representations, employing wavelets with moving centers, are obtained by applying conformal transformations to the stationary representation. This could be useful in the analysis of electromagnetic waves reflected or emitted by moving objects, such as radar signals.