Abstract
Abstract Chapter 15 examines the conditions under which it is possible to define renormalized quantum field theories (QFTs) consistent on all scales (a property not necessary for effective QFTs and not shared by ‘trivial’ theories like quantum electrodynamics (the triviality issue)). It has been conjectured that a necessary and, perhaps, sufficient condition is the existence of ultraviolet fixed points, a property called asymptotic safety. Examples are “asymptotically free” theories like non–Abelian gauge theories (quantum chromodynamics, or QCD) in four dimensions or the non–linear sigma model in two dimensions. These theories are characterized by a crossover scale between universal infrared and large momentum behaviour behaviours. The non–linear sigma and Gross–Neveu models in dimensions greater than 2 are examined in detail. For example, the non–linear sigma model exhibits, below the critical temperature, a crossover between an infrared behaviour dominated by Goldstone modes and a universal, large momentum, critical behaviour.
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