Effective theories in classical and quantum mechanics

Working in quantum many-body theory, I slowly learned that many key concepts and techniques have predecessors and analogues in classical systems and one-body quantum systems. Examples include Green's functions, path integrals, cumulants, the linked cluster theorem, Hubbard-Stratonavich transformation (completing the square), mean-field theory, localisation due to disorder, and BBGKY hierarchy. Learning a full-blown quantum many-body version is easier if you first understand simpler analogues.

This post is about effective theories in classical systems and one-body quantum systems, following my earlier post about effective theories in quantum field theories of elementary particles.

Michèle Levi has a pedagogical article

Effective field theories of post-Newtonian gravity: a comprehensive review

This is motivated by the use of EFTs to describe gravitational waves produced by the inspiraling and merging of binary black holes and neutron stars. She discusses the different scales involved and how there are effective theories at each scale. She also puts these EFTs in the broader context of other fields.

Analogues in one-body quantum mechanics are also discussed  in

Effective Field Theories, Reductionism and Scientific Explanation, by Stephan Hartmann

"In his beautiful book Qualitative Methods in Quantum Theory, Migdal (1977) discusses an instructive example from quantum mechanics. Let S be a system which is composed of a fast subsystem Sf and a slow subsystem Ss, characterised by two frequencies of and os. It can be shown that the effects of Sf on Ss can be taken into account effectively by adding a potential energy term to the Hamiltonian operator of Ss. In this case, as well as in many other cases, one ends up with an effective Hamiltonian operator for the subsystem characterised by the smaller frequency (or energy)."

An important example of this is the Born-Oppenheimer approximation which is based on the separation of time and energy scales associated with electronic and nuclear motion. It is used to describe and understand the dynamics of nuclei and electronic transitions in solids and molecules. The potential energy surfaces for different electronic states define an effective theory for the nuclei. Without this concept, much of theoretical chemistry and condensed matter would be incredibly difficult.