# Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations

## Abstract

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. The proposed solution is applied to a simple, finite dimensional model of interacting spin systems, which serves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is explained how the content of this article and its companion affect the possible extraction of quantum field theory on curved spacetime from loop quantum gravity (including matter fields).

- Authors:

- Institut für Quantengravitation, Lehrstuhl für Theoretische Physik III, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7/B2, D-91058 Erlangen (Germany)

- Publication Date:

- OSTI Identifier:
- 22596687

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANNIHILATION OPERATORS; BORN-OPPENHEIMER APPROXIMATION; EIGENSTATES; LOOP QUANTUM GRAVITY; MATHEMATICAL SOLUTIONS; PERTURBATION THEORY; QUANTUM SYSTEMS; SPACE-TIME; TIME DEPENDENCE

### Citation Formats

```
Stottmeister, Alexander, E-mail: alexander.stottmeister@gravity.fau.de, and Thiemann, Thomas, E-mail: thomas.thiemann@gravity.fau.de.
```*Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations*. United States: N. p., 2016.
Web. doi:10.1063/1.4954228.

```
Stottmeister, Alexander, E-mail: alexander.stottmeister@gravity.fau.de, & Thiemann, Thomas, E-mail: thomas.thiemann@gravity.fau.de.
```*Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations*. United States. doi:10.1063/1.4954228.

```
Stottmeister, Alexander, E-mail: alexander.stottmeister@gravity.fau.de, and Thiemann, Thomas, E-mail: thomas.thiemann@gravity.fau.de. Wed .
"Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations". United States.
doi:10.1063/1.4954228.
```

```
@article{osti_22596687,
```

title = {Coherent states, quantum gravity, and the Born-Oppenheimer approximation. I. General considerations},

author = {Stottmeister, Alexander, E-mail: alexander.stottmeister@gravity.fau.de and Thiemann, Thomas, E-mail: thomas.thiemann@gravity.fau.de},

abstractNote = {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. The proposed solution is applied to a simple, finite dimensional model of interacting spin systems, which serves as a non-trivial, minimal model of the aforesaid problem. Furthermore, it is explained how the content of this article and its companion affect the possible extraction of quantum field theory on curved spacetime from loop quantum gravity (including matter fields).},

doi = {10.1063/1.4954228},

journal = {Journal of Mathematical Physics},

number = 6,

volume = 57,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}