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.