Three dimensional Loop Quantum Gravity: towards a self-gravitating Quantum Field Theory

TL;DR

This paper develops a self-gravitating quantum field theory in three-dimensional Riemannian space using the Drinfeld double DSU(2), constructing the Fock space, defining the quantum field, and analyzing quantum gravity corrections.

Contribution

It introduces a novel self-gravitating quantum field theory model based on DSU(2), including the construction of Fock space, interaction potential, and explicit calculations of quantum gravity effects.

Findings

Constructed the Fock space for the self-gravitating field.

Defined a DSU(2)-covariant quantum field with interactions.

Calculated lowest order quantum gravity corrections to propagator and three-point functions.

Abstract

In a companion paper, we have emphasized the role of the Drinfeld double DSU(2) in the context of three dimensional Riemannian Loop Quantum Gravity coupled to massive spinless point particles. We make use of this result to propose a model for a self-gravitating quantum field theory (massive spinless non-causal scalar field) in three dimensional Riemannian space. We start by constructing the Fock space of the free self-gravitating field: the vacuum is the unique DSU(2) invariant state, one-particle states correspond to DSU(2) unitary irreducible simple representations and any multi-particles states is obtained as the symmetrized tensor product between simple representations. The associated quantum field is defined by the usual requirement of covariance under DSU(2). Then, we introduce a DSU(2)-invariant self-interacting potential (the obtained model is a Group Field Theory) and compute explicitely the lowest order terms (in the self-interaction coupling constant $\lambda$) of the propagator and of the three-points function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the three-points function.