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) BornOppenheimer 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).

Some physical implications of regularization ambiguities in SU(2) gauge-invariant loop quantum cosmology

“…for cµ 0 ∈ (−π, π]. Obviously the conditions (27) for c are necessary, irrespective of whether one uses the former constraint (26) or the one using gauge-covariant fluxes, i.e. (25).…”

“…The precise rescaling (33) will be derived below. In contrast to this, the remaining solutions for c in (27) can be matched to classical solutions in which a new form of matter appears in the effective Friedmann equations. It is hence necessary to view these points as corresponding to the asymptotic regime of the pre-bounce universe.…”

Surface state decoherence in loop quantum gravity, a first toy model

“…A special focus is devoted to understanding the renormalization of the theory by defining properly the coarse-graining of spin network states (e.g. [11,[14][15][16][17]) and also by defining the proper notion of coherent states of the quantum geometry [18][19][20][21]. The emergence of a classical reality from a purely quantum one is an issue that dates back to the origin of quantum theory and is nowadays mostly understood thanks to decoherence, a phenomenon recently observed in (cavity) quantum electrodynamic and condensed matter experiments (for reviews and recent ideas, [22][23][24][25]).…”

Coherent state operators in loop quantum gravity

“…This problem is of course due to the non-separability of the kinematical Hilbert space, and might be solved at the physical level. However, a definition of coherent states even at the diffeomorphism invariant level is still missing, though proposals exist [20] and preliminary studies on collective variables appeared [21], and a new program to deal with coherent states in the context of a Born-Oppenheimer approximation has been settled [22][23][24]. For most present purposes, however, the fixedgraph coherent states seem to work fine.…”