Field theories assume a space-time continuum in which the theory is expressed:
In classical field theory this continuum can host energy and momentum even in the absence of matter.
In quantum field theory the continuum is always and everywhere host to energy and momentum, regardless of the presence or absence of matter.
Direct Particle Interaction is constrained to predict the behavior of particles only in terms of their pair-wise interactions, with no physical role for a 4-dimensional continuum. More specifically, the laws governing their behavior depend on some function of the difference between qualities possessed by each member of the pair.
Since pair-wise interactions are denumerable DPI grants no physical role to a 4-dimensional space-time continuum.
The challenge for DPI is to be at least as good as field theory in explaining observations to date. If DPI succeeds in doing so it must be preferred over field theory because it has parsimony and Ockham's razor overwhelmingly on its side. This is because in DPI there are a finite number of degrees of freedom - a number proportional to the total number of particles - whereas the degrees of freedom in field theory are infinite and non-denumerable.
DPI and field theory cannot always agree however. So if DPI passes the test above, it is likely there is new physics to be discovered.