A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications

TL;DR

This paper demonstrates that the diffeomorphism invariant Hilbert space in loop quantum gravity can be structured as a Fock space, enabling new insights into quantum states and their geometric interpretations.

Contribution

It establishes a Fock space framework for the diffeomorphism invariant states in LQG and connects this to condensate states and quantum geometry.

Findings

Diffeomorphism invariant states can be organized into a Fock space structure.

Some condensate states of group field theory are represented as coherent states in this framework.

The quantum geometry of multi-particle states shows resemblance to multi-component geometries.

Abstract

Loop quantum gravity (LQG) is a quantization program for gravity based on the principles of QFT and general covariance of general relativity. Quantum states of LQG describe gravitational excitations based on graphs embedded in a spatial slice of spacetime. We show that, under certain assumptions on the class of diffeomorphisms, the space of diffeomorphism invariant states carries a Fock space structure. The role of one-particle excitations for this structure is played by the diffeomorphism invariant states based on graphs with a single (linked) component. This means, however, that a lot of the structure of the diffeomorphism invariant Hilbert space remains unresolved by this structure. We show how the Fock structure allows to write at least some condensate states of group field theory as diffeomorphism invariant coherent states of LQG in a precise sense. We also show how to construct other interesting states using this Fock structure. We finally explore the quantum geometry of single- and multi-particle states and tentatively observe some resemblance to geometries with a single or multiple components, respectively.