Emergence and singular asymptotic expansions

Seth Olsen kindly lent me his copy of Chemistry, Quantum Mechanics, and Reductionism by Hans Primas, published in 1981. I has a Foreword by Paul Feyerabend! 

[Primas died last October and there will be a symposium in his honour later this year]
This is a book I had wanted to read for a while since I had seen it referenced in various philosophical contexts. Besides some deep philosophy he has lots of polemical statements about theoretical chemistry.

Wanting to find an electronic version I could copy choice quotes from led me to a more dense, broader, and more recent (1998) article Emergence in exact natural science.

Here I mention a few highlights.

emergence and theory reduction are related.

Theory reduction is the process where a more general theory, such as quantum mechanics or special relativity, "reduces" in a particular mathematical limit to a less general theory such as classical mechanics. This is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who sometimes overstate the problem]. The subtleties arise because the two different theories usually involve concepts that are "incommensurate" with one another.

the distinction inside/outside is not covered by the most fundamental context-independent natural laws (first principles of physics). 

Here Primas is stresses the sometimes "arbitrary" value judgements that are made in distinguishing a "system" and its "environment". This involves distinguishing "patterns" and invoking "symmetry breaking".  He introduces notions of topology to try and make such distinctions more rigorous. I found this too technical to appreciate.

 Many inter-theoretical relations can be mathematically described by asymptotic expansions. Singular asymptotic expansions are never uniformly convergent in the intrinsic topology of the basic theory. This nonuniformity is not a disaster but an indication that the limiting case represents a caricature, suppressing irrelevant details and enhancing contextually relevant features. The discontinuous change in the limit leads to a discontinuous change in the semantics and therewith to a description in a new language in terms of emergent properties. In the same sense as a photograph can never replace a brilliant caricature, an asymptotic description can – for the intended purpose – be more adequate than the exact description.

Michael Berry also has a 1994 article that takes a similar point of view.

The assertions  

“something consists of elementary systems”,   

“something can be decomposed into “elementary systems”, 

“something can be described in terms of “elementary systems”, 

are not equivalent. 

When a light wave passes an object, a typical discontinuity – called the shadow – can be observed. However, in Maxwell’s electrodynamics – the fundamental theory for the propagation of light – shadows do not exist. Maxwell’s electrodynamics is governed by partial differential equations which have only continuous solutions. The discontinuities associated with shadows appear only in geometric optics, the limiting case of vanishing wavelength l , l → 0 .

He discusses at length how the notion of "molecular structure" in chemistry is an emergent concept. This relates to the issue of quantum entanglement between electrons and nuclei.

In a quantum theoretical description the molecular shape emerges by abstracting from the actually existing Einstein–Podolsky–Rosen correlations between the electrons and the nuclei. Historically, the structure concept has been introduced into quantum chemistry by the so-called Born-Oppenheimer approximation. But this terminology is misleading since the main issue is not an approximation, but the breaking of a holistic symmetry. A more proper appreciation of the Born–Oppenheimer-description stresses its singular nature: it is an expansion about the singular point of infinite nuclear masses. An asymptotic expansion can be formulated in terms of the ratio e = (m/M)^1/4, where m is the mass of an electron and M is a mean nuclear mass of the molecular system. In the limiting case e = 0 the holistic correlations between nuclei and electrons are suppressed so the description of a molecule reduces to the description of the motion of electrons in the electric field of a classical nuclear framework. In this description the molecular structure is a property described by an emergent classical observable. The singular limiting case     e = 0 leads to a discontinuous change in the description and is the starting point for an asymptotic expansion in terms of the emergent property at higher levels of description. 

He then gives a another example that was new to me.

The transition from the more fundamental Lorentz-relativistic quantum mechanics to Galilei-relativistic quantum mechanics is governed by the contraction of the Lorentz group to the Galilei group – a highly singular limit. While the Lorentz group is semisimple, the Galilei group is not but has a more complicated mathematical structure. The emergent quantity associated with this contraction is the mass in the sense of a classical observable (which commutes with all other observables and can therefore be treated as a real parameter).

He also discusses how the concept of temperature is emergent, emphasising the centrality of the zeroth law of thermodynamics.