Here Θ Θ ( x ) := { 1 x ≥ ≥ 0 0 x < 0 is the Heaviside step function, τ τ x y := ( x 0 − − y 0 ) 2 − − ( x → → − − y → → ) 2 is the proper time from x to y, and J 1 is a Bessel function of the first kind. The propagator is non-zero only if y ≺ ≺ x , i.e., y causally precedes x, which, for Minkowski spacetime, means