  We consider a tracer particle coupled to a Bose scalar field and study the
regime where the field's propagation speed approaches infinity. For initial
states devoid of field excitations, we introduce an effective approximation of
the time-evolved wave function and prove its validity in Hilbert space norm. In
this approximation, the field remains in the vacuum state while the tracer
particle propagates with a modified dispersion relation. Physically, the new
dispersion relation can be understood as the effect of radiative corrections
due to interactions with virtual bosons. Mathematically, it is defined as the
solution of a self-consistent equation, whose form depends on the relevant time
scale.
