Extended wave propagators as pulsed-beam communication channels

Let P(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 to an analytic function Pmacr(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 y_e,y_r of y_e,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 Pmacr 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 space-time, respectively. The imaginary "retarded time" T_e=s _e-|y_e|/c represents the duration of the emitted pulse, and T_r=s_r-|y_r|/c represents the integration time for the received pulse. The bandwidths of the antennas are 1/T_e and 1/T_r, respectively. The invariance of Pmacr under imaginary spacetime translations (z_erarrz_e+ieta, z_rrarrz_r +ieta) has nontrivial consequences