When is a phenomena truly emergent?
Is there some objective quantitative criteria that one might use to decide?
This is an issue because sometimes discussions of emergence are pretty fuzzy and even flaky.
I have now read the full article Asymptotics, Singularities, and the Reduction of Theories
I highly recommend the article as I think it has a very important insight: singular asymptotic expansions provide a concrete criteria for emergence.
Berry considers the specific problem:
I recommend reading this article before the one by Hans Primas (reviewed in the previous post) as the latter is more technical and philosophical than Berry's.
One thing I think this highlights is that the problem of emergence in quantum systems is neither more or less challenging or interesting than in classical systems, something I argued before.
I have one minor addition to Berry. In quantum many-body systems the singular parameter delta may not just be 1/N, where N = number of particles. It can also be the coupling constant, lambda. Emergent phenomena are associated with non-perturbative effects. Concrete examples are in the BCS theory of superconductivity and the Kondo effect. In both there is an emergent energy scale exp(-1/lambda). There is no convergent expansion in powers of lambda. Taylor series around lambda =0 is singular.