  We construct a 4-dimensional quantum field theory on a Hilbert space,
dependent on a simple Lie Algebra of a compact Lie group, that satisfies
Wightman's axioms. This Hilbert space can be written as a countable sum of
non-separable Hilbert spaces, each indexed by a non-trivial, inequivalent
irreducible representation of the Lie Algebra.
  In each component Hilbert space, a state is given by a triple, a space-like
rectangular surface $S$ in $\mathbb{R}^4$, a measurable section of the Lie
Algebra bundle over this surface $S$, represented irreducibly as a matrix, and
a Minkowski frame. The inner product is associated with the area of the surface
$S$.
  In our previous work, we constructed a Yang-Mills measure for a compact
semi-simple gauge group. We will use a Yang-Mills path integral to quantize the
momentum and energy in this theory. During the quantization process,
renormalization techniques and asymptotic freedom will be used. Each component
Hilbert space is the eigenspace for the momentum operator and Hamiltonian, and
the corresponding Hamiltonian eigenvalue is given by the quadratic Casimir
operator. The eigenvalue of the corresponding momentum operator will be shown
to be strictly less than the eigenvalue of the Hamiltonian, hence showing the
existence of a positive mass gap in each component Hilbert space. We will
further show that the infimum of the set containing positive mass gaps, each
indexed by an irreducible representation, is strictly positive.
  In the last section, we will show how the positive mass gap will imply the
Clustering Theorem.
