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arXiv · 1112.1825

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

Karim Noui·Alejandro Perez·Daniele Pranzetti
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Abstract

We investigate the canonical quantization of 2+1 gravity with Λ > 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 \surdΛe, 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_λ = A + λe 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.

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Karim Noui, Alejandro Perez, Daniele Pranzetti. 2011-12-08. Non-commutative holonomies in 2+1 LQG and Kauffman's brackets. https://doi.org/10.1088/1742-6596%2F360%2F1%2F012040

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