Three-dimensional Loop Quantum Gravity Research Articles

Spinor representation of the Hamiltonian constraint in 3D loop quantum gravity with a nonzero cosmological constant

We develop in a companion paper the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation that is well known at zero cosmological constant. In this paper, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$ deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints, therefore, build a Turaev-Viro model at real $q$.

Spectra of geometric operators in three-dimensional loop quantum gravity: From discrete to continuous

We study and compare the spectra of geometric operators (length and area) in the quantum kinematics of two formulations of three-dimensional Lorentzian loop quantum gravity. In the $\mathrm{SU}(2)$ Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $\ensuremath{\gamma}$ exactly like in the four-dimensional case. However, we show that when working with the self-dual variables and imposing the reality conditions the spectra become continuous and $\ensuremath{\gamma}$ independent.

The Hilbert space of 3d gravity: quantum group symmetries and observables

We relate three-dimensional loop quantum gravity to the combinatorial quantisation formalism based on the Chern-Simons formulation for three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological constant. We compare the construction of the kinematical Hilbert space and the implementation of the constraints. This leads to an explicit and very interesting relation between the associated operators in the two approaches and sheds light on their physical interpretation. We demonstrate that the quantum group symmetries arising in the combinatorial formalism, the quantum double of the three-dimensional Lorentz and rotation group, are also present in the loop formalism. We derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs. This establishes a direct link between the two quantisation approaches and clarifies the role of quantum group symmetries in three-dimensional gravity.

Three-dimensional loop quantum gravity: towards a self-gravitating quantum field theory

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 are 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 explicitly compute the lowest order terms (in the self-interaction coupling constant λ) of the propagator and of the three-point function. Finally, we compute the lowest order quantum gravity corrections (in the Newton constant G) to the propagator and to the three-point function.

Three-dimensional loop quantum gravity: Particles and the quantum double

It is well known that the quantum double structure plays an important role in three-dimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of three-dimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surfaced Σ and the Hilbert structure is defined by the Ashtekar-Lewandowski measure. In the case where Σ is the sphere S2, we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations, and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface Σ.

Three-dimensional loop quantum gravity: coupling to point particles

We consider the coupling between three-dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin-foam amplitudes. In addition we provide an explicit expression of the classical distance operator between two particles which is defined as an observable.

Three-dimensional loop quantum gravity: physical scalar product and spin-foam models

In this paper, we address the problem of the dynamics in three-dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space—corresponding to the quantization of the infinite-dimensional kinematical configuration space of the theory—to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three-dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard ‘lattice’ methods), and second that no ill-defined sum over spins (‘bubble’ divergences) are present in the spin-foam representation.