2011/05/31 by Karim Noui, K. Noui, Alejandro Perez +3 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Canonical quantization #Canonical quantum gravity #Hilbert space #Holonomy #Loop quantum cosmology #Loop quantum gravity #Noncommutative and Quantum Gravity Theories #Quantization (signal processing) #Quantum Mechanics and Applications #Quantum gravity #Spin foam #Spin network #gr-qc #hep-th
paper · pdf · doi:10.1007/jhep10(2011)036
published as JHEP 1110 (2011) 036 · 19 pages, references added. Published version
arxiv created 2011/09/12 · openalex publication_date 2011/10/01 · arxiv updated 2012/08/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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± √ Λ e . 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 ( iℏ 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.