ELECTROMAGNETIC WAVELETS AS HERTZIAN PULSED BEAMS IN COMPLEX SPACETIME

Abstract

Electromagnetic wavelets are a family of 3 × 3 matrix fields Wz(x ′ ) parameterized by complex spacetime points z = x + iy with y timelike. They are translates of a basic wavelet W(z) holomorphic in the future-oriented union T of the forward and backward tubes. Applied to a polarization vector p = pm − ipe, W(z) gives an anti-selfdual solution W(z)p derived from a selfdual Hertz potential ˜ Z(z) = −iS(z)p, where S is the Synge function acting as a Whittaker-like scalar Hertz potential. Resolutions of unity exist giving representations of sourceless electromagnetic fields as superpositions of wavelets. With the choice of a branch cut, S splits into a difference S + (z) −S − (z) of retarded and advanced pulsed beams whose limits as y → 0 give the propagators of the wave equation. This yields a similar splitting of the wavelets and leads to their complete physical interpretation as pulsed beams absorbed and emitted by a disk source D(y) representing the branch cut. The choice of y determines the beam’s orientation, collimation and duration, giving beams as sharp with pulses as short as desired. The sources are computed as spacetime distributions of electric and magnetic dipoles supported on D(y). The wavelet representation of sourceless electromagnetic fields now splits into representations with advanced and retarded sources. These representations are the electromagnetic counterpart of relativistic coherent-state representations previously derived for massive Klein-Gordon and Dirac particles.