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Title: 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:
;  [1]
  1. 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. 2016. "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 = 2016,
month = 6
}
  • 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.
  • 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. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall “coherentmore » state” transform for compact Lie groups G, which we prove for G = U(1){sup n} and support by numerical evidence for G = SU(2). The reason for conjoining this conjecture with the main topic of this article originates in the observation that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups G. But, as Weyl and Berezin quantisation for ℝ{sup 2d} are intimately related by heat kernel evolution, it is natural to ask whether a similar connection exists for compact Lie groups as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former.« less
  • The Born-Oppenheimer approximation is used as an exploratory tool to study bound states, quasibound states, and scattering resonances in muon ([mu])--hydrogen ([ital x])--hydrogen ([ital y]) molecular ions. Our purpose is to comment on the existence and nature of the narrow states reported in three-body calculations, for [ital L]=0 and 1, at approximately 55 eV above threshold and the family of states in the same partial waves reported about 1.9 keV above threshold. We first discuss the motivation for study of excited states beyond the well-known and well-studied bound states. Then we reproduce the energies and other properties of these well-knownmore » states to show that, despite the relatively large muon mass, the Born-Oppenheimer approximation gives a good, semiquantitative description containing all the essential physics. Born-Oppenheimer calculations of the [ital s]- and [ital p]-wave scattering of [ital d]-([ital d][mu]), [ital d]-([ital t][mu]), and [ital t]-([ital t][mu]) are compared with the accurate three-body results, again with general success. The places of disagreement are understood in terms of the differences in location of slightly bound (or unbound) states in the Born-Oppenheimer approximation compared to the accurate three-body calculations.« less
  • Using the quantum trajectory approach, we extend the Born-Oppenheimer (BO) approximation from closed to open quantum systems, where the open quantum system is described by a master equation in Lindblad form. The BO approximation is defined and the validity condition is derived. We find that the dissipation in fast variables improves the BO approximation, unlike the dissipation in slow variables. A detailed comparison is presented between this extension and our previous approximation based on the effective Hamiltonian approach [X. L. Huang and X. X. Yi, Phys. Rev. A 80, 032108 (2009)]. Several additional features and advantages are analyzed, which showmore » that the two approximations are complementary to each other. Two examples are described to illustrate our method.« less