One
of the problems in particle physics which has remained unsolved in the
past half century is the quark confinement problem. It is known that the
strong nuclear force is of short range. This implies that their force
carriers, namely the gluons, should be massive. Yet the gluon fields are
massless at the classical level. It is believed that in the quantum
theory they would gain mass by an unknown mechanism. The main part of
the confinement problem is to find an analytical approach to the
phenomena of mass generation for gluons in four dimensional (3 space + 1
time dimension) Yang-Mills theory. However the solution to this problem
is far from trivial in four dimensions.

Here
we will simplify this problem by studying a deformation of Yang-Mills
theory in four dimensions, but with one spacial dimension compactified
on a small circle, which allows for an analytical study of confinement.
Next we analyze the properties of the coefficients of the asymptotic
linear confining potential, known as the k-string tensions, for
different representations of the gauge group SU(N) with N-ality k.

In
the second part of this talk we discuss an improved perturbative
expansion for quantum mechanics and quantum field theory. It is known
that the perturbative expansions in quantum mechanics and quantum field
theory are asymptotic and are mainly applicable to the weak coupling
regime of the theory, however using this new method we obtain a
perturbative expansion that is convergent and can also be applied to the
strong coupling regime of the theory.