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Conference Paper

Non-commutative holonomies in 2+1 LQG and Kauffman's brackets


Pranzetti,  Daniele
Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Noui, K., Perez, A., & Pranzetti, D. (2012). Non-commutative holonomies in 2+1 LQG and Kauffman's brackets. Journal of Physics: Conference Series, 360: 012040.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000E-EE6E-6
We investigate the canonical quantization of 2+1 gravity with {\Lambda} > 0 in the canonical framework of LQG. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A\pm = A \PM \surd{\Lambda}e, where the SU(2) connection A and the triad field e are the conjugated variables of the theory. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A_{\lambda} = A + {\lambda}e acting on spin network links of the kinematical Hilbert space of LQG. We provide an explicit construction of the quantum holonomy operator, exhibiting a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity. The crucial difference is that the result is completely described in terms of standard SU(2) spin network states.