Introductiong

Since the unification of the electromagnetic and weak interactions through the Glashow–Salam–Weinberg model [75], Yang–Mills theories [76] have been widely accepted as correctly describing elementary particle physics. This belief was reinforced when they proved to be renormalizable [77, 78]. Moreover, the discovery of color symmetry as the underlying gauge invariance associated with strong interactions raised the possibility that all interactions of nature could possibly be cast as Yang–Mills theories. This spawned interest in grand unified models and some partial successes were achieved in this direction.

A crucial ingredient in the description of elementary particle physics through gauge theories is the maintenance of the gauge invariance of physical results and the underlying theory and this is also crucial in order to be able to prove renormalizability.

The success of the electroweak model is yet to be achieved by the quark model of strong interactions. The reason is that perturbative techniques, which were adequate for the electroweak model, are only appropriate in the high energy regime of strong interactions. This motivated the interest in non-perturbative techniques, especially to prove the existence of a confining phase. A great effort took place in the late 1970s and suggestive arguments were put forward but a rigorous proof of quark confinement is still lacking.

In several of these attempts the use of loops played an important role. Loops were used in a variety of contexts and approaches including the one we are focusing on in this book, the loop representation.