Born-Oppenheimer applied to quantum geometry

[marcus]

Atyy spotted 3 potentially important LQG papers and added them to the biblio thread. The B-O approximation has proven essential in quantum chemistry and similar applications dealing with wave functions of complex systems with many components. Geometry is such a system and it seems reasonable that B-O, applied to QG, could play a key role in LQG.
I'll list the three papers, and then give some overview and introduction to the Born-Oppenheimer approximation, excerpting and referencing the Wikipedia article.
==excerpts==
http://arxiv.org/abs/1504.02169
Coherent states, quantum gravity and the Born-Oppenheimer approximation, I: General considerations
Alexander Stottmeister, Thomas Thiemann
(Submitted on 9 Apr 2015)
This article, as the first of three, aims at establishing the (time-dependent) Born-Oppenheimer approximation, in the sense of space adiabatic perturbation theory, for quantum systems constructed by techniques of the loop quantum gravity framework, especially the canonical formulation of the latter. The analysis presented here fits into a rather general framework, and offers a solution to the problem of applying the usual Born-Oppenheimer ansatz for molecular (or structurally analogous) systems to more general quantum systems (e.g. spin-orbit models) by means of space adiabatic perturbation theory. ...

http://arxiv.org/abs/1504.02170
Coherent states, quantum gravity and the Born-Oppenheimer approximation, II: Compact Lie Groups
Alexander Stottmeister, Thomas Thiemann
(Submitted on 9 Apr 2015)
In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. ...

http://arxiv.org/abs/1504.02171
Coherent states, quantum gravity and the Born-Oppenheimer approximation, III: Applications to loop quantum gravity
Alexander Stottmeister, Thomas Thiemann
(Submitted on 9 Apr 2015)
In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.
==endquote==

 

[marcus]

Now for some introductory overview of the Born-Oppenheimer approximation.
For more information check out the full Wikipedia article.
http://en.wikipedia.org/wiki/Born–Oppenheimer_approximation
I'll quote a portion to have handy for reference in this thread.
==quote==
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be separated. The approach is named after Max Born and J. Robert Oppenheimer. In mathematical terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components.

Computation of the energy and the wavefunction of an average-size molecule is simplified by the approximation. For example, the benzene molecule consists of 12 nuclei and 42 electrons. The time independent Schrödinger equation, which must be solved to obtain the energy and wavefunction of this molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei. The BO approximation makes it possible to compute the wavefunction in two less complicated consecutive steps. This approximation was proposed in 1927, in the early period of quantum mechanics, by Born and Oppenheimer and is still indispensable in quantum chemistry.

In the first step of the BO approximation the electronic Schrödinger equation is solved, yielding the wavefunction [PLAIN]http://upload.wikimedia.org/math/0/0/7/0076c10e3b86b41c636daae1252c01fc.pngdepending on electrons only. For benzene this wavefunction depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be in nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables.

The success of the BO approximation is due to the difference between nuclear and electronic masses. The approximation is an important tool of quantum chemistry; without it only the lightest molecule, H2, could be handled, and all computations of molecular wavefunctions for larger molecules make use of it. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations.

The electronic energies consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions. In accord with the Hellmann-Feynman theorem, the nuclear potential is taken to be an average over electron configurations of the sum of the electron–nuclear and internuclear electric potentials.

In molecular spectroscopy, because the ratios of the periods of the electronic, vibrational and rotational energies are each related to each other on scales in the order of a thousand, the Born–Oppenheimer name has also been attached to the approximation where the energy components are treated separately.

The nuclear spin energy is so small that it is normally omitted.
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==endquote==

BTW "Born-Oppenheimer approximation" was picked up as a catch phrase by the writers of the TV series "The Big Bang Theory" and was one of the topics touched on in an interview with the show's science advisor David Salzburg. He is "D" in this portion of the interview
http://www.symmetrymagazine.org/article/march-2012/the-brain-behind-tvs-the-big-bang-theory
==quote==
B: Do the writers ever ask you to explain the science and it goes completely over their heads?

D: We respond by email so I don't really know. But I don't think it goes over their heads because you can Wikipedia anything.

One thing was a little difficult for me: they asked for a spoof of the Born-Oppenheimer approximation, which is harder than it sounds. But for the most part it's just a matter of narrowing it down to a few choices. There are so many ways to go through it and I deliberately chose things that are current.

First of all, these guys live in our universe—they're talking about the things we physicists are talking about. And also, there isn't a whole lot of science journalism out there. It's been cut back a lot. In getting the words out there, whether it's "dark matter" or "topological insulators," hopefully some fraction of the audience will Google it.
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==endquote==

 

[DrDu]

Interesting, however most of the physics in these articles seem to be hidden behind mathematical fuzz.
The important step in the modern formulation of the BO approximation was the clean separation of adiabatic from semiclassical effects in the expansion in the mass ratio. This has first been carried trough in the very readable paper by
Weigert, Stefan, and Robert G. Littlejohn. "Diagonalization of multicomponent wave equations with a Born-Oppenheimer example." Physical Review A 47.5 (1993): 3506
although it has been proposed before.