  Axiomatic quantum field theory (QFT) provides a rigorous mathematical
foundation for QFT, and it is the basis for proving some important general
results, such as the well-known spin-statistics theorem. Free-field QFT meets
the axioms of axiomatic QFT, showing they are consistent. Nevertheless, even
after more than 50 years, there is still no known non-trivial theory of quantum
fields with interactions in four-dimensional Minkowski spacetime that meets the
same axioms.
  This paper provides a similar axiomatic basis for parameterized QFT, in which
an invariant, fifth path parameter is added to the usual four spacetime
position arguments of quantum fields. Dynamic evolution is in terms of the path
parameter rather than the frame-dependent time coordinate. Further, the states
of the theory are allowed to be off shell. Particles are therefore
fundamentally "virtual" during interaction but, in the appropriate
non-interacting, large-time limit, they dynamically tend towards "physical",
on-shell states.
  Unlike traditional QFT, it is possible to define a mathematically consistent
interaction picture in parameterized QFT. This may be used to construct
interacting fields that meet the same axioms as the corresponding free fields.
One can then re-derive the Dyson series for scattering amplitudes, but without
the mathematical inconsistency of traditional, perturbative QFT. The present
work is limited to the case of scalar fields, and it does not address remaining
issues of gauge symmetry and renormalization. Nevertheless, it still
demonstrates that the parameterized formalism can provide a consistent
foundation for the interpretation of QFT as used in practice and, perhaps, for
better dealing with its further mathematical issues.
