  We formulate an S-matrix theory in which localisation effects of the particle
interactions involved in a scattering process are consistently taken into
account. In the limit of an infinite spread of all interactions, the S-matrix
assumes its standard form. To better understand the significance of the
emerging quantum phenomena in this formalism, we consider a solvable
field-theoretic model with spatial Gaussian spreads at the interaction
vertices. This solvable model, which was previously introduced in the
literature, enables accurate descriptions of detection regions that are either
close to or far from the source. In close analogy with light diffraction in
classical optics, we call these two regions near-field and far-field zones, or
the Fresnel and Fraunhofer regions. We revisit the question whether mixed
mediators produce an oscillating pattern if their detection occurs in the
Fresnel region. Besides corroborating certain earlier findings of the S-matrix
amplitude in the forward Fresnel and Fraunhofer regimes, we observe several
novel features with respect to its angular dependence which have not been
accounted before in the literature. In particular, we obtain a ``quantum
obliquity factor'' that suppresses particle propagation in the backwards
direction, thereby providing an explicit quantum field-theoretic description
for its origin in diffractive optics. Present and future colliders, as well as
both short and long baseline neutrino experiments, would greatly benefit from
the many predictions that can be offered from such a holistic localised
S-matrix theory.
