For a gauge invariant system of quantum fields to be a self-consistent framework for describing various interactions, some mechanisms for short-range interactions must be found (sections 10.1 and 10.2) and its renormalizability proved (section 10.3). In addition, non-Abelian gauge theories have exhibited some novel features, which have suggested certain interpretations concerning the structure of the vacuum state and the conditions for the quantization of physical parameters such as charges. Thus a new question, which never appeared in the traditional foundational investigations of (Abelian-gaugeinvariant) QED or other non-gauge-invariant local field theories, has posed itself with a certain urgency, attracted intense attention, and become a favorite research topic among a sizable portion of mathematics-oriented physicists in recent years. This is the question of the global features of non-Abelian gauge field theories (section 10.4). This chapter will review the formation of these conceptual foundations of gauge theories, both as a theoretical framework and as a research programme, and will register some open questions that remain to be addressed by future investigators.

Mechanisms for short-range interactions (I): spontaneous symmetry breaking

The original Yang–Mills theory failed to be an improvement on the already existing theories of strong nuclear interactions, because it could not reproduce the observed short-range behavior of the nuclear force without explicitly violating gauge symmetry. A major obstacle to be overcome in the further development of gauge theories was then the need to have a consistent scheme with massive gauge quanta while retaining (in some sense) gauge invariance. One way of solving the problem is so-called spontaneous symmetry breaking (SSB).