Towards the QFT on Curved Spacetime Limit of QGR. II: A Concrete Implementation

TL;DR

This paper develops a concrete method to derive effective quantum field theories on curved spacetimes from background independent quantum gravity, incorporating quantum gravitational fluctuations and background independence effects.

Contribution

It employs complexifier coherent states to explicitly compute expectation values, leading to effective matter Hamiltonians with quantum gravity corrections.

Findings

Derived explicit effective Hamiltonians for scalar and electromagnetic fields.

Identified quantum gravitational fluctuations as sources of corrections to standard QFT.

Predicted non-standard dispersion relations influenced by quantum gravity effects.

Abstract

The present paper is the companion of [1] in which we proposed a scheme that tries to derive the Quantum Field Theory (QFT) on Curved Spacetimes (CST) limit from background independent Quantum General Relativity (QGR). The constructions of [1] make heavy use of the notion of semiclassical states for QGR. In the present paper, we employ the complexifier coherent states for QGR recently proposed by Thiemann and Winkler as semiclassical states, and thus fill the general formulas obtained in [1] with life. We demonstrate how one can, under some simplifying assumptions, explicitely compute expectation values of the operators relevant for the gravity-matter Hamiltonians of [1] in the complexifier coherent states. These expectation values give rise to effective matter Hamiltonians on the background on which the gravitational coherent state is peaked and thus induce approximate notions of n-particle states and matter propagation on fluctuating spacetimes. We display the details for the scalar and the electromagnetic field. The effective theories exhibit two types of corrections as compared to the the ordinary QFT on CST. The first is due to the quantum fluctuations of the gravitational field, the second arises from the fact that background independence forces both geometry and matter to propagate on a spacetime that is the product of the real line and a (random) graph. Finally we obtain explicit numerical predictions for non-standard dispersion relations for the scalar and the electromagnetic field. They should, however, not be taken too seriously, due to the many ambiguities in our scheme, the analysis of the physical significance of which has only begun. We show however, that one can classify these ambiguities at least in broad terms.