Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Abstract

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A ± = A± √ Λe. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ R) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity (with q = exp(i λ/2)). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

Symmetries of quantum spacetime in three dimensions

“…3 and 4 that this is not the case in 3D, namely diffeomorphisms can be expressed in terms of non-commutative holonomies and these are exactly the generators that lead to a deformed constraint algebra symmetry. Whether a κ-Poincaré symmetry can indeed be derived also in the 4D case or not using techniques analogous to those introduced in [7,8] is hard to say, since the extension to four dimensions is highly non-trivial from a technical point of view (see [40] for a recent alternative attempt to circumvent the obstruction found in [39]). …”

“…In three dimensions, the Riemannian theory with vanishing cosmological constant can be quantized using LQG techniques both in the covariant and canonical formalisms and the two quantizations have been shown to be equivalent [13] (see also [14] for a review of this topic). In the case of Λ ≠ 0 the canonical quantization has been implemented in [7,8], and it was shown to reproduce the physical transition amplitudes of the Turaev-Viro state sum [15], which provides a covariant quantization of the theory (see also [16,17] for alternative approaches to the LQG quantization). Here we want to study the off-shell algebra of the constraints, which is the new result of this section.…”

“…The quantization of the holonomy of the non-commutative connection as an operator acting on the kinematical Hilbert space of LQG has been performed in [7]. In that analysis, one considers two holonomies, h η , h γ , of the non-commutative connection on intersecting paths η, γ acting on the vacuum.…”

“…To this end, we replace the classical generators of the su(2) gauge algebra with the corresponding quantum operators, using the Loop Quantum Gravity (LQG) techniques [6]. This construction relies on earlier results [7], [8]. Having defined the quantum constraints we turn to calculating their commutators.…”

Observables in loop quantum gravity with a cosmological constant

“…Just as in 3d, it is not clear at all why a quantum group structure should appear in the LQG framework. There exist few arguments to justify this postulate [22]. We include now a table summarizing the different quantum group models appearing in 4d quantum gravity.…”