In this book we have attempted to present in a structured fashion the various aspects of the use of loops in the quantization of gauge theories and gravitation. The discussion mixed historical and current developments and we rewrote many results in a more modern language. In this chapter we would like to concentrate on the outlook arising from the material presented and focus on current developments and on possible future avenues of work. We will divide the discussion into gauge theories and gravity, since the kinds of developments in these two fields follow naturally somewhat disjoint categories.

Gauge theories

Overall, the picture which emerges is satisfying in the sense that the bulk of the techniques developed can be applied systematically to the construction of loop representations for almost any theory based on a connection as the main canonical variable, either free or interacting with various forms of matter. In this respect we must emphasize the developments listed in chapters 1, 2 and 3 which are the main mathematical framework that we used to understand the physical applications. Many of these aspects, as we have mentioned, have been studied with mathematical rigor by various authors in spite of the fact that the presentation we have followed here is oriented towards physicists.

The main conclusion to be drawn from this book is that loop techniques are at present a practical tool for the analysis of the quantum mechanics of gauge theories.