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Conference Paper

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

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##### Fulltext (public)

1112.1825

(Preprint), 128KB

CQG_360_1_012040.pdf

(Any fulltext), 379KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

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: http://hdl.handle.net/11858/00-001M-0000-000E-EE6E-6

##### Abstract

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.