Can Renormalization Group Flow End in a Big Mess?

TL;DR

This paper explores the possibility that renormalization group flows in quantum field theories can be chaotic and fractal-like, challenging traditional views of fixed point flows and suggesting complex behaviors in physical systems.

Contribution

It proposes that RG flows may lead to fractal attractors rather than fixed points, and discusses the implications and challenges of such chaotic flows in field theories.

Findings

Chaotic RG flows are compatible with effective action properties.

Such flows do not contradict the c-theorem or Lyapunov functions.

Constructing effective actions with chaotic flows is challenging.

Abstract

The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan-Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.