Renormalization in physics vs. dynamical systems

Question

I am studying complex dynamics, so to me renormalization of a dynamical system means something like a rescaled first-return map on (a subset of) the underlying space. I understand that in quantum field theory there are also notions of renormalization and universality having to do with eliminating divergences in certain perturbative integrals, and in statistical physics, where they help explain phase transitions.

What I don’t see is the connection between the physicists‘ usage and what people in dynamical systems call renormalization. How are they related? Any references would be appreciated

Answer 1

The renormalization approach to dynamical systems pioneered by Chen, Goldenfeld and Oono [1] applies the Gell-Mann and Low renormalization group from quantum physics [2] to extract the global behavior of a function known locally from perturbation theory.

We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slow motion dynamics in nonequilibrium phenomena are RG equations. The renormalized perturbation approach may be simpler to use than other approaches, because it does not require the use of asymptotic matching and yields practically superior approximations.

For a more recent overview, see Renormalization Group as a Probe for Dynamical Systems.

[1] L.Y. Chen, N. Goldenfeld, and Y. Oono, Renormalization Group Theory for Global Asymptotic Analysis (1994).
[2] M. Gell-Mann and F.E. Low, Quantum Electrodynamics at Small Distances (1954).

Answer 2

There’s a bit of a terminological collision going on here. Physicists often use the term “renormalization” to refer the process of removing infinities from QFT calculations and to renormalization or “renormalization group” where you understand how theories behave at different scales, which is also how it is used in statistical physics. I prefer to call the removal of infinites “regularization” to distinguish the two.

The connection between the two uses of the term is that the (UV) divergences of the theory arise due to the short scale behavior of the theory. However, we don’t actually know the theory at short distances. We can assume that some unknown theory cuts off the divergences and ask about how the theory behaves at lower energy scales using renormalization. You can show that only a subset of terms in the Lagrangian govern the theory and that other terms are suppressed at low energies. The terms that survive are precisely the “renormalizable” terms for which the various infinity removal processes work.

In other words, understanding renormalization (a la Wilson) explains (some of) the divergences in QFT and what we’re really doing when we remove them.