Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity

2019/10/15 by Ilkka Mäkinen, Mäkinen, Ilkka · 2 voices · 1 citation
Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #gr-qc

paper · pdf · doi:10.48550/arxiv.1910.06821

arxiv published 2019/10/15 · arxiv updated 2019/10/18

Abstract

We present a pedagogical introduction to SU(2) recoupling theory, focusing on those aspects of the topic which are useful for practical calculations in loop quantum gravity. In particular, we give a self-contained presentation of the powerful graphical formalism, which is an indispensable tool for performing computations in the spin network basis of loop quantum gravity. The use of the graphical techniques in loop quantum gravity is illustrated by several detailed example calculations. Plenty of exercises are included for the benefit of the ambitious student.

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Discussions

• Let $D^{(j)\; m}_n(g)$ be the Wigner matrix (unitary representation of $SU(2)$ on a $d_j=2j+1$ dimensional representation space $H_j$ and let $(C^{j_1\;j_2}_{j_3})$ be the Clebsch-Gordan coefficient c [stackexchange, 3 points]
• Here is a sketched proof of the permutation symmetry of the Wigner $3j$ symbols . Ref.1 defines the $3j$ symbols $^1$ $$(123)~=~\frac{(-1)^{j_1-j_2-j_3}}{\sqrt{2j_3+1}}\langle 12 |T|3\rangle \tag{2.13 [stackexchange, 2 points]

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• Introduction to Loop Quantum Gravity and Spin Foams