Extended wave propagators as pulsed-beam communication channels
Abstract
Let P0(xr-xe) be the causal propagator for the wave equation, representing the signal received at the spacetime point xr due to an impulse emitted at the spacetime point xe. Such processes are highly idealized since no signal can be emitted or received at a precise point in space and at a precise time. We propose a simple and compact model for extended emitters and receivers by continuing P0 to an analytic function P(zr - ze), where ze=xe+iye represents a circular pulsed-beam emitting antenna centered at xe and radiating in the spatial direction of ye while zr=xr-iyr represents a circular pulsed-beam receiving antenna centered at xr and receiving from the spatial direction of yr. The space components of ye and yr give the spatial orientations and radii of the antennas, while their time components se, sr represent the time a signal takes to propagate along the antennas between the center and the boundary. The analytic propagator P(zr - ze) represents the transmission amplitude, forming a communication channel. Causality requires that the extension/orientation 4-vectors ye and yr belong to the future cone V, so that ze and zr belong to the future tube and the past tube in complex spacetime, respectively. The invariance of P(zr - ze) under imaginary spacetime translations has nontrivial consequences.
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