  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.
