The purpose of this paper is to construct a quantum field theory suitable for describing quantum electrodynamics
and Yang-Mills theory in a form which satisfies the conditions of the Millennium prize offered by
the Clay Mathematics Institute as described by Jaffe and Witten [12], by showing that it satisfies ‘axioms at
least as strong as those cited by’ Wightman [18] and by Osterwalder and Schrader [14], and by observing
that this form of field theory has no mass gap. The definitions provide a model for relativistic quantum
mechanics which supports a form of relativistic quantum field theory, but which does not depend on the second
quantisation of a “matter wave”. Continuous laws of wave mechanics are found in model of discrete
particle interactions which does not involve waves, or the quantisation of interacting fields. Newton’s first
law and conservation of momentum are established from the principle of homogeneity. Maxwell’s equations
are derived from the simple interaction in which a Dirac particle emits or absorbs a photon, showing that
the renormalised mass and coupling constant are equal to their bare values. Feynman rules are calculated
for the discrete theory and give the predictions of the standard renormalised theory. Quark confining interactions are described for qed and for an adaptation of Yang-Mills theory.