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

TL;DR

This paper develops a framework for applying the Born-Oppenheimer approximation within loop quantum gravity, using space adiabatic perturbation theory, and demonstrates its application to spin systems and quantum field theory on curved spacetime.

Contribution

It introduces a general method to implement the Born-Oppenheimer approximation in loop quantum gravity using space adiabatic perturbation theory, addressing complex quantum systems.

Findings

Successfully applies the method to a finite-dimensional spin system model

Provides a pathway for deriving quantum field theory on curved spacetime from loop quantum gravity

Offers a solution to applying the Born-Oppenheimer ansatz to complex quantum systems

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).