Philosophical perspectives on the emergence of molecular structure

 In philosophical discussions of emergence and reductionism in chemistry, molecular structure has received significant attention and debate. Sometimes molecular structure is used to justify strong emergence, i.e., that molecular cannot be predicted, even in principle, solely from quantum theory.

Primas was one of the first to claim that molecular structure could not be reduced to quantum physics. Consider the following statements.

“From a physical point of view the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one…

[the structure of benzene] does not exist in a full quantum-theoretical description since electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations. The concept of molecular structure does not follow from first principles - all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence. 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.”

I have italicised claims that are contestable and that I will discuss further below.

Cartwright has given philosophical arguments as to why chemistry cannot be reduced to physics. In this context, she claims (pp. 106-7)

“The typical method of quantum-mechanical treatment of molecules begins with the Born–Oppenheimer approximation…

This approximation treats the atomic nucleus as a classical particle. But this fundamentally violates quantum mechanics which, following the Heisenberg uncertainty principle, maintains that we cannot have a simultaneous assignment of fixed positions and fixed momenta. The approximations that provide the reduction violate the very theory that the chemistry is being reduced to… the success of quantum chemistry relies fundamentally on assumptions that belong to classical chemistry.” 

This claim that the BOA violates the Heisenberg uncertainty principle was rebutted in an earlier post and discussed in more detail by Huggett et al. Nevertheless, Lombardi et al. are not satisfied by the rebuttal.

Hendry claimed molecular structure is evidence of strong emergence and/or downward causation. In a similar spirit, Ellis and Drossel argued that crystal structures in solid state physics are evidence of strong emergence. The arguments of Hendry have been criticised by Seifert. The arguments centre around the fact that the molecular structure is a classical entity and a concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically. It is claimed that the existence of molecular structures is assumed and not derived in quantum chemistry calculations as they assume the Born-Oppenheimer approximation (BOA). 

Scerri responded to these arguments claiming that chemistry (particularly the concept of molecular structure) is irreducible to quantum physics. He claimed these arguments are not valid because they misunderstand the role of the BOA. It does not violate the uncertainty principle and in practice chemists can and do perform non-BOA approximations. 

Fortin et al. rejected the view that decoherence can explain isomerism, as decoherence does not resolve issues associated with the quantum measurement problem. 

Franklin and Seifert claim “the problem of molecular structure just is the quantum measurement problem.” This is debatable. Most molecular structures can be understood in terms of the nuclear probability density having a unique maximum and decoherence then is not relevant. Decoherence and the collapse of the nuclear wavefunction are only relevant in systems such as ammonia and stereoisomers in which there are molecular structures with equal energy and separated by high energy barriers.

I now respond to some of the contestable claims of Primas.

“electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations”

It is possible to quantify and calculate the amount of entanglement between electrons and nuclei in a specific quantum state. In an EPR state the entanglement is maximal. In the BOA wavefunction entanglement is present, but is absent in the crude BOA. The entanglement has been estimated in benzene and is very small. The only molecules where the entanglement may be significant are those, such as isomers, where there are two degenerate molecular structures and the overlap of the associated nuclear wavefunctions is small (i.e., the tunnel splitting is small). But then, in most chemical situations decoherence will wash out this entanglement.

“all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence.”

This is the problem of isomers. It is resolved because isomers are present in the solution to the Schrödinger equation, as I argued earlier.

“the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one… the main issue is not an approximation, but the breaking of a holistic symmetry.”

This seems subjective to me. I see the BOA as just a well-justified approximation. The electrons and nuclei are treated differently because they are. They have vastly different masses and this affects how they can be treated in any solution to the full Hamiltonian. Nevertheless, the BOA is a quantum theory and the nuclei are described by a wavefunction.

As discussed earlier, for small molecules in practise it is no longer necessary to use the BOA and the electrons and nuclei can be treated on an equal footing. Molecular structure is present in solutions to the full Schrödinger equation.

I wonder if the objection to use of the BOA is any different to the use of approximations in other theories? For example, consider theoretical treatments of the motion of planets in the solar system. The effects of all the planets are not treated on an equal footing. The effect of other planets on a planet of interest are treated perturbatively.

In conclusion, the arguments that molecular structure is evidence of strong emergence are weak. Some confusion may partly arise from misinterpreting the significance of the following valid observations.

i. Molecular structures were first conjectured before quantum theory was proposed.

ii. The BOA was proposed to explain molecular structure from quantum theory.

iii. Today, almost all calculations of molecular structure assume BOA.

iv. Chemists talk about molecular structures largely in classical not quantum terms.

However, the scientific reality is that for small molecules their structure, can be understood, described, and calculated in purely quantum terms. Yet, that understanding provides a strong justification for the validity of the BOA and for the convenience of using classical language to describe molecular structure.

I stress that the weakness of the arguments for the strong emergence of molecular structure, does not mean that an emergent perspective is not relevant to chemistry. Molecular structure is emergent, when defined in terms of novelty. This leads to effective theories defined in terms of potential energy surfaces. Furthermore, as the next section argues chemistry exhibits novel properties, concepts, and theories that are not present in physics.

This post is extracted from Section 15, of my review article "Emergence: from physics to biology, sociology, and computer science."

Are gravity and spacetime really emergent in AdS-CFT?

There is an interesting Scientific American article by Adam Becker

What Is Spacetime Really Made Of?

Spacetime may emerge from a more fundamental reality. Figuring out how could unlock the most urgent goal in physics—a quantum theory of gravity

It considers two different approaches to quantum gravity (loop quantum gravity and AdS-CFT beloved by string theorists). Compared to some Scientific American articles it is moderately balanced and low on hype. The article has a nice engagement with some philosophers of physics. It is clear to me how loop quantum gravity has a natural interpretation that gravity and space-time are emergent. However, that is not clear for AdS-CFT.

 The following paragraph is pertinent.

But there are other ways to interpret the latest findings. The AdS/CFT correspondence is often seen as an example of how spacetime might emerge from a quantum system, but that might not actually be what it shows, according to Alyssa Ney, a philosopher of physics at the University of California, Davis. 

“AdS/CFT gives you this ability to provide a translation manual between facts about the spacetime and facts of the quantum theory,” Ney says. “That’s compatible with the claim that spacetime is emergent, and some quantum theory is fundamental.” 

But the reverse is also true, she says. The correspondence could mean that quantum theory is emergent and spacetime is fundamental—or that neither is fundamental and that there is some even deeper fundamental theory out there. Emergence is a strong claim to make, Ney says, and she is open to the possibility that it is true. “But at least just looking at AdS/CFT, I’m still not seeing a clear argument for emergence.”

Spontaneous symmetry breaking and the thermodynamic limit

 Spontaneous symmetry breaking is a fundamental concept in condensed matter and quantum field theory. Amongst philosophers of science the concept is receiving increasing attention, particularly in the context of discussions about emergence.

How do we understand the following two observations about a system at zero temperature?

At zero temperature for a finite-sized system there is no symmetry breaking. The ground state transforms as the trivial representation of the symmetry group of the Hamiltonian. It is non-degenerate.

In the thermodynamic limit, there is a family of degenerate ground states. They are related to one another by a transformation of the symmetry group. This concept is captured in picture below of the Mexican hat potential.

Motion around the trough is associated with the Goldstone mode. Motion perpendicular to the trough is associated with the "Higgs boson".

How does this picture connect with a finite system?

An intuitive picture is that the ball in the trough has a finite mass and so motion in the trough is like the quantum mechanics of a rotor with finite moment of inertia. Then there is a non-generate ground state with equal probability to be located at any angle. What might the moment of inertia be? For reasons described below it turns out to be related to the superfluid stiffness.

For the case of a Heisenberg antiferromagnet, the physics was worked out by Phil Anderson in 1952 where he introduced the concept of a "tower of states" that become degenerate in the thermodynamic limit. 

They are described by the following effective Hamiltonian

c is the speed of magnons (Goldstone bosons). vec(S) is the total spin, V is the volume of the system, and rho_s is the spin stiffness associated with the broken-symmetry. As the thermodynamic limit is approached the energy of these states scale with L^-d where d is the dimension of the system. In contrast, the magnon states scale with L^-1. Thus, for exact diagonalisation of sufficiently large systems, the "tower of states" should be clearly be below the magnons states.

 In 1992 all of the above was confirmed for the triangular lattice in numerical work by Bernu, Hluillier, and Pierre.  

In the figure below, the top panel shows the low-lying eigenstates. The lowest energy states do scale with S^2. The middle panel shows how these states do separate from the magnon states. 

The figure below shows how the moment of inertia [proportional to the denominator in the tower of states equation above] does scale with the system size.

 

More recently there has been some interesting work that explores how the tower of states appears in the entanglement entropy.

Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry

Max A. Metlitski, Tarun Grover

But for now, discussing that is above my pay grade 😀

I thank Gerard Milburn for asking me questions that led to me finally getting a better physical picture of the issues discussed here.

Nobel Prize predictions for 2020

It is that time of year again. My physics predictions are the same as last year.

For physics this year I predict
Experiments for testing Bell inequalities and elucidating the role of entanglement in quantum physics
Alain Aspect, John Clauser, and Anton Zeilinger
They received the Wolf Prize in 2010, a common precursor to the Nobel. 

My personal preference for the next Nobel for CMP would be centred around Kondo physics since that is such a paradigm for many-body physics, maybe even comparable to BCS.

Kondo effect and heavy fermions
Jun Kondo, Frank Steglich, David Goldhaber-Gordon

Arguably the latter two might be replaced with others who worked on heavy fermions and/or Kondo in quantum dots.
Steglich discovered heavy fermion superconductivity.
Goldhaber-Gordon realised tuneable Kondo and Anderson models in quantum dots (single-electron transistors).

Unlike many, I still remain to be convinced that topological insulators are worthy of a Nobel.

How about other prizes?

The nomination deadline was January 31, before most people appreciated the significance of covid-19. I predict next year that their will be at least one prize (Chemistry, Medicine, Economics, or Peace) relating to public health and/or viruses. One possibility would be Bill and Melinda Gates for Peace.

Here are a few unusual suggestions.

Literature: Lin-Manuel Miranda for Hamilton

Peace (more likely next year): Colin Kaepernick and/or Black Lives Matter, Joshua Wong and/or other Hong Kong protestors, Jacinda Ardern.

On peace, here are some other ideas.

What do you think?

Emergent quasi-particles and gauge fields in quantum matter

Unfortunately, there is a paucity of good review articles that give gentle introductions to current research in condensed matter, both for beginning graduate students and for curious non-experts. Too many reviews are exhaustive, in both senses of the word! Contemporary Physics is a journal that aims to address this problem. I should look at it more often. In 2009, there was a nice 50th-anniversary issue, featuring some significant articles, with retrospective commentary. For example, there is a fascinating article about Snow Crystals by F.C. Franks.

My UQ colleague, Ben Powell recently submitted a nice review to the journal.

Emergent particles and gauge fields in quantum matter 

I give a pedagogical introduction to some of the many particles and gauge fields that can emerge in correlated matter. The standard model of materials is built on Landau's foundational principles: adiabatic continuity and spontaneous symmetry breaking. These ideas lead to quasiparticles that inherit their quantum numbers from fundamental particles, Nambu-Goldstone bosons, the Anderson-Higgs mechanism, and topological defects in order parameters. I then describe the modern discovery of physics beyond the standard model. Here, quantum correlations (entanglement) and topology play key roles in defining the properties of matter. This can lead to fractionalised quasiparticles that carry only a fraction of the quantum numbers that define fundamental particles. These particles can have exotic properties: for example Majorana fermions are their own antiparticles, anyons have exchange statistics that are neither bosonic nor fermionic, and magnetic monopoles do not occur in the vacuum. Gauge fields emerge naturally in the description of highly correlated matter and can lead to gauge bosons. Relationships to the standard model of particle physics are discussed.

 

Quantum matters for the public

I have now finished my first draft of  Chapter 7 of Condensed Matter Physics: A Very Short Introduction. The main purpose of the chapter is to introduce quantum states of matter. It is arguably the most challenging of the chapters to write and to understand. But, it is potentially the most fascinating.

I welcome comments and suggestions. However, bear in mind that my target audience is not the typical reader of this blog, but rather your non-physicist friends and family.

I think it still needs a lot of work, particularly to be less technical. For example, I should probably drop Aharonov Bohm ...

The goal is for the chapter to be interesting, accessible, and bring out the excitement and importance of condensed matter physics.

2019 Nobel Predictions

It is that time of year again. I have not made predictions for a few years.

For physics this year I predict
Experiments for testing Bell inequalities and elucidating the role of entanglement in quantum physics
Alan Aspect, John Clauser, and Anton Zeilinger
They received the Wolf Prize in 2010, a common precursor to the Nobel.

My personal preference for the next Nobel for CMP would be centred around Kondo physics, since that is such a paradigm for many-body physics, maybe even comparable to BCS.

Kondo effect and heavy fermions
Jun Kondo, Frank Steglich, David Goldhaber-Gordon

Arguably the latter two might be replaced with others who worked on heavy fermions and/or Kondo in quantum dots.
Steglich discovered heavy fermion superconductivity.
Goldhaber-Gordon realised tuneable Kondo and Anderson models in quantum dots (single-electron transistors).

Unlike many, I still remain to be convinced that topological insulators is worthy of a Nobel.

For chemistry, my knowledge is more limited. However, I would go for yet another condensed matter physicist to win the chemistry prize: John Goodenough, inventor of the lithium battery.
He also made seminal contributions to magnetism, random access memories, and strongly correlated electron materials.

What do you think?

Postscripts (October 10).

I got confused about the day of the physics prize and I think when I posted my ``prediction'' the prize may have already been announced.

A few years ago I read Goodenough's fascinating autobiography. It was actually in that book that I learned about U. Chicago requiring PhD students to publish a single author paper. This observation featured in my much commented on recent post about PhD theses.

I also have a prediction for the Peace Prize. First, I hope it is not Greta Thunberg, as much as I admire her and agree with the importance of her cause. I worry whether it may ruin her life.
My wife suggested the Prime Minister of Ethiopia, Abiy Ahmed and the President of Eritrea, Isaias Afwerki. I find it truly amazing what Ahmed has achieved.
Another great choice would be some of the leaders of Armenia, which has seen significant increases in human rights, political freedoms, and freedom the press. It was selected as The Economist's country of the year in 2018.

Postscript (October 30).
I was really happy about the economics prize. Six years ago, I read Poor Economics, by Banerjee and Duflo, with my son (an economics student), and blogged about it. Below a respond to a commenter who was critical of this prize.

Laughing at your life in science

I watched Ph.D Movie 2: Still in Grad School, which is based on the legendary Ph.D comics, written by Jorge Cham.



It is worth watching as it is quite funny. On the other hand, some of the caricatures are getting a little too close to reality....

While on the funny side of science, my wife and I have been enjoying watching Young Sheldon. I am a Big fan of The Big Bang Theory, but was not sure whether this new show would be as good. This was partly influenced by a moderately negative review in the New York Times (albeit based on one episode). I disagree as I think that both shows do have an interesting cast of characters.
On the other hand, Young Sheldon does not have as much science as TBBT, at least for the first six episodes that I have seen. Here Schrodinger's cat gets discussed.

How are DMFT, DMET, and slave bosons related?

Several years ago I posted about Density Matrix Embedding Theory (DMET), proposed as a (computationally cheaper) alternative to Dynamical Mean-Field Theory (DMFT).

An important question is what is the relationship (if any) between the two methods?
Given that the two methods are formulated in quite different ways it was not clear to me at all whether these question could be answer in any sort of definitive way.

There is a very nice paper which does answers this question in a precise way, with the bonus of also giving the relationship of both methods with rotationally invariant slave bosons (RISB).

Dynamical mean-field theory, density-matrix embedding theory, and rotationallyinvariant slave bosons: A unified perspective 
Thomas Ayral, Tsung-Han Lee, and Gabriel Kotliar

The main results are summarised in the Figure below. A result that is useful and insightful is that DMET corresponds to RISB with the quasi-particle weight set to unity (Z=1). There is then no band narrowing associated with the correlations. Given this is a key aspect of strongly correlated electron systems, I think this is a significant shortcoming of DMET. I am also a bit confused about how DMET can then capture a Mott transition.

Did Schrodinger's cat explore Tolkien's garden?

In 1935 Schrodinger wrote his famous paper (with the cat) introducing the term entanglement, in response to the Einstein-Podolsky-Rosen paper published earlier that year.

When Schrodinger wrote the paper he was living in a house on Northmoor Road, Oxford. This was the same house where Schrodinger learned he had been awarded the Nobel Prize.

I recently learned some fascinating historical trivia.
Schrodinger was a neighbour of J.R.R. Tolkien, who during that time was finishing up work on The Hobbit.

It would be nice to see this landmark honoured, such as the one on Tolkien's house. However, it seems Schrodinger's house does not meet the criteria of Oxfordshire Blue Plaques Board, because he lived there for three years, less than the required minimum of five years.

Another option would be a plaque of the Institute of Physics, such as this one.

Entering the strange world of Kurt Godel

The picture below is of Godel's rotating universe. It represents an exact solution to Einstein's gravitational field equations and has the strange property of closed timelike curves (i.e. one can travel into the past!). This mathematical solution was found by Kurt Godel while he was employed by the Institute for Advanced Study at Princeton.

I think I first encountered this picture in my final undergraduate year in the classic book, The Large Scale Structure of Space-Time by Hawking and Ellis, while working on a research project in general relativity.

Godel's universe is just one example of the fascinating science and stories recounted in the book
Who Got Einstein's Office? Eccentricity and Genius at the Institute for Advanced Study by Ed Regis, first published 30 years ago.

I only read the book this past week and loved it. It is a captivating blend of science, mathematics, personalities, history, philosophy, humorous anecdotes, gossip, eccentricities ...
I was so captivated that I read it during two situations I would not normally read something so "heavy": during a long flight [normally I watch reruns of The Big Bang Theory or Upper Middle Bogan [need to laugh!] or recently a Warren Buffett documentary... sorry better not mention that again...], and during "down time" in the evening after a busy day.

Regis nicely describes the continuum hypothesis, Einstein-Podolsky-Rosen (EPR) "paradox" in quantum theory, von Neumann machines, cellular automata, the Bourbaki seminar, parity violation, the solar neutrino problem, fractals, the stability of matter, ...

The personalities covered include Godel, Einstein, Herman Weyl, John von Neumann, J. Robert Oppenheimer, Freeman Dyson, T.D. Lee,  C.N. Yang, Andre Weil, John Bahcall, Stephen Wolfram, Ed Witten, .....

It is amazing how much Regis packs into less than 300 pages (in a paperback).

The tragic mental health problems of Godel are described in a sensitive manner.

One pathetic story concerns the endless quibbles of T.D. Lee and C.N. Yang.
(Aside: They actually did their Nobel Prize winning work on parity violation at the IAS. This is in contrast to the countless Nobel laureates who at one time have been affiliated with the IAS but did not do their prize work there.)
Lee and Yang (or is it Yang and Lee?) argued constantly about the order in which their names should be listed, not just as co-authors, and at the Nobel ceremony, but even in newspaper and magazine articles about them. Furthermore, it is crazy to read the wildly different and self-serving accounts of certain concrete events. Great scientists are all too human ......

Some people consider the book is a bit of a "hatchet" job and has a mocking tone that paints the IAS in a poor light and questions its value and existence. I would not agree. I think it does show that the IAS has produced a lot of important scholarship. Regis does raise some important questions I mention below. But, I did think that he did refer to the IAS as "the One True Platonic Heaven" too many times.

Regis is implicitly critical of the fact that there is very little interaction between different research groups and disciplines within IAS. However, there is one important story he missed: when Freeman Dyson and the number theorist Hugh Montgomery were introduced at tea at the IAS and they made a connection between random matrix theory (quantum physics) and zeros of the Riemann zeta function.

Some questions the book raises for me include:

Can you really "manage" genius?

How do you create an institutional environment that increases the likelihood of truly great discoveries and scholarship?

What is the best way to hire "great" people?

What is a good mix of young and old staff?

What is a good mix of permanent faculty, postdocs, and short term senior visitors?

When is the absence of students in a research institute good or bad?

When is the absence of experimentalists in an institution bad/good for theoretical physics?

How do you foster a healthy synergy between pure mathematics and theoretical physics?

How might you foster some constructive interaction between distinct disciplines: philosophy, mathematics, theoretical physics, economics, history, ....?

Here is Feynman's perspective (partly quoted in the book):

I don't believe I can really do without teaching. The reason is, I have to have something so that when I don't have any ideas and I'm not getting anywhere I can say to myself, "At least I'm living; at least I'm doing something; I am making some contribution" -- it's just psychological. 

When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don't get any ideas for a while: They have every opportunity to do something, and they are not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come. 

Nothing happens because there's not enough real activity and challenge: You're not in contact with the experimental guys. You don't have to think how to answer questions from the students. Nothing!

Governments have less and less interest in "research for its own sake" and "without constraints" [hallmarks of the IAS]. However, there is an increasing number of generous and wealthy philanthropic organisations who are very interested. These are important questions for them.

Although I lived in Princeton for four years around the time the book was being written I only recall going inside "the Brain Farm" [as a friend called it] once, and that was for a music concert. Nevertheless, I spent many pleasant hours walking, jogging, bird watching, and skiing in the beautiful woods located behind the IAS.

I thank Ben Powell for a conversation about the IAS, stimulating me to remember I had inherited a copy of the book from my parents.

I welcome thoughts on any of the questions and any good IAS stories...

A lucid lecture on the last 50 years of superconductivity

At the weekly condensed matter theory cake meeting today we watched a video of a KITP blackboard talk given by Piers Coleman in 2015.
Superconducting Surprises: five decades of discovery, in both temperature and time!

It is a very nice exposition of the history and some of the key physics.

A couple of minor comments.

Organic superconductors were discovered in 1980 not 1973.

Piers claims that the difference between the thermodynamic entropy of the superconducting and metallic states (determined from integrating the temperature dependent specific heat) is related to the quantum entanglement entropy of the superconducting ground state.
The relationship between entanglement entropy (defined on a pure quantum state (at zero temperature) which is divided in two) and thermal entropies (defined for a bulk system in a mixed state at finite temperature) is an incredibly subtle and complex issue that I don't think is resolved. See for example the discussion in this paper.

Quantum entanglement technology hype

Last month The Economist had a cover story and large section on commercial technologies based on quantum information.

To give the flavour here is a sample from one of the articles

Very few in the field think it will take less than a decade [to build a large quantum computer], and many say far longer. But the time for investment, all agree, is now—because even the smaller and less capable machines that will soon be engineered will have the potential to earn revenue. Already, startups and consulting firms are springing up to match prospective small quantum computers to problems faced in sectors including quantitative finance, drug discovery and oil and gas. .... Quantum simulators might help in the design of room-temperature superconductors allowing electricity to be transmitted without losses, or with investigating the nitrogenase reaction used to make most of the world’s fertiliser.

I know people are making advances [which are interesting from a fundamental science point of view] but it seems to me we are a very long way from doing anything cheaper [both financially and computationally] than a classical computer.

Doug Natelson noted that at the last APS March Meeting, John Martinis said that people should not believe the hype, even from him!

Normally The Economist gives a hard-headed analysis of political and economic issues. I might not agree with it [it is too neoliberal for me] but at least I trust it to give a rigorous and accurate analysis. I found this section to be quite disappointing. I hope uncritical readers don't start throwing their retirement funds into start-ups that are going to develop the "quantum internet" because they believe that this is going to be as important as the transistor (a claim the article ends with).

Maybe I am missing something.
I welcome comments on the article.

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. 

Diabatic states rock!

Physical Chemistry Chemical Physics has just published a series of four articles by Jeff Reimers, Laura McKemmish, Noel Hush, and myself.

A unified diabatic description for electron transfer reactions, isomerization reactions, proton transfer reactions, and aromaticity"

Non-adiabatic effects in thermochemistry, spectroscopy and kinetics: the general importance of all three Born-Oppenheimer breakdown corrections

Electron-vibration entanglement in the Born-Oppenheimer description of chemical reactions and spectroscopy

Bond angle variations in XH3 [X=N,P,As,Sb,Bi]: the critical role of Rydberg orbitals exposed using a diabatic state model

It took a number of years to finish these papers. I am certainly the junior co-author and I commend my co-authors for all their hard work and perseverance.

The four papers have two common related themes, that are hopefully not lost in all the technical detail.

1. Diabatic states provide a powerful scheme, both conceptually and quantitatively, to describe a wide range of chemical phenomena.
2. This can be nicely illustrated using a simple model Hamiltonian describing the coupling of two electronic states to a single vibrational mode.
In chemistry language this is a E x beta Jahn-Teller model. In condensed matter language, is a two-site spinless fermion Holstein model.

We welcome comments.

Update (24 September, 2015). The papers made it to the cover of the print edition.

What is so great about the von Neumann entropy?

I got a referee report for a paper submitted to PhysChemChemPhys that looks at the quantum entanglement of electronic and nuclear degrees of freedom in molecules. The paper goes beyond the calculations considered here, and explores subtle issues about how entanglement may or may not be related to the breakdown of the Born Oppenheimer approximation.

One referee asked a good but basic question, "Why is the von Neumann entropy the appropriate measure of entanglement to consider here?"

Here is my answer. I think experts could do better and so I welcome suggestions.

The von Neumann entropy is widely accepted as the best measure of quantum entanglement for pure quantum states defined on bipartite systems, such as that considered here. This is because the von Neumann entropy satisfies certain desired criteria, including vanishing for separable states, monotonicity (it does not increase under local operations or classical communication between the subsystems), additivity, convexity, and continuity.

It was a bit of work to come up with this answer, because this is all second nature to people who work in quantum information. It is hard to find a place where this is clearly stated and discussed in detail. The Quantiki wiki entry on entanglement measures  and the axiomatic approach, along with the review article by the Horodecki family helps.

Anyone suggest a place where this basic issue is discussed and worked out in detail?

The big challenge is defining entanglement measures for multi-partite systems and for mixed quantum states.

Is quantum entanglement really needed?

On tuesday at UQ Carl Caves gave a Quantum Science Seminar "Quantum metrology meets Quantum Information Science".

One side point he made was that just because quantum entanglement is glamorous and beloved by luxury journals does not mean that you actually always need it to optimise any and every task. A specific example is in this paper which states:

The Heisenberg limit is thus achieved without any entanglement between the arms of the interferometer. In fact, Jiang, Lang, and Caves [4] showed that the state ∣∣ψin⟩optis the only nonclassical product state, i.e., not a coherent state, that produces no modal entanglement after a beam splitter. These results indicate that, as in Ref. [18], modal entanglement is not a crucial resource for quantum-enhanced interferometry.