The Wilsonian Renomalization Group in Gauge/ Gravity Duality

Abstract

Just as Quantum Mechanics has a deep impact on any serious student of physics, its application to the relativistic domain, that is, Quantum Field Theory (QFT), strikes those who face its study for the first time. Most notably, the appearance of infinities here and there has no parallel in classical physics or in Quantum Mechanics of discrete systems (except in particular cases with singular sources and boundaries). The \shell game" of renormalization, necessary to make sense of this situation, may look like the last resort of desperate scientists. This feeling was actually shared not so long ago by the early formulators of quantum electrodynamics. However, a better understanding of renormalization in QFT emerged later. The crucial insights were provided by the work of Kenneth G. Wilson and others on the Renormalization Group (RG), which studies the different appearance of a QFT at different energy scales. In the Wilsonian formulation, this is done by regularizing the theory with some cutoff and studying how the changes in this cutoff are exactly compensated by changes in the action. The RG allows to give a clear and deep meaning to the short-distance divergences of QFT, which essentially re ect sensitivity to higher scales, and to the process of renormalization, in which the unknown ultraviolet details are parametrized by a set of local operators with arbitrary parameters. The Wilsonian RG provides a rigorous non-perturbative deffnition of continuous renormalizable QFT as relevant deformations of conformal field theories. Ideas such as asymptotic safety are based on this picture. Also effective field theories, a modern paradigm of physics, are best understood from the Wilsonian point of view. And not only are the renormalization process and the RG well-defined; they introduce new concepts and tools with deep physical implications.