Extended wave propagators as pulsed-beam communication channels

Abstract

Let P_0(x_r-x_e) be the causal propagator for the wave equation, representing the signal received at the spacetime point x_r due to an impulse emitted at the spacetime point x_e. 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 P_0 to an analytic function P(z_r - z_e), where z_e=x_e+iy_e represents a circular pulsed-beam emitting antenna centered at x_e and radiating in the spatial direction of y_e while z_r=x_r-iy_r represents a circular pulsed-beam receiving antenna centered at x_r and receiving from the spatial direction of y_r. The space components of y_e and y_r give the spatial orientations and radii of the antennas, while their time components s_e, s_r represent the time a signal takes to propagate along the antennas between the center and the boundary. The analytic propagator P(z_r - z_e) represents the transmission amplitude, forming a communication channel. Causality requires that the extension/orientation 4-vectors y_e and y_r belong to the future cone V, so that z_e and z_r belong to the future tube and the past tube in complex spacetime, respectively. The invariance of P(z_r - z_e) under imaginary spacetime translations has nontrivial consequences.