Graphical method in loop quantum gravity: II. The Hamiltonian constraint and inverse volume operators

Abstract

This is the second paper in the series to introduce a graphical method to loop quantum gravity. We employ the graphical method as a powerful tool to calculate the actions of the Euclidean Hamiltonian constraint operator and the so-called inverse volume operator on spin network states with trivalent vertices. Both of the operators involve the co-triad operator which contains holonomies by construction. The non-ambiguous, concise and visual characters of our graphical method ensure the rigour for our calculations. Our results indicate some corrections to the existing results in literatures.

New volume and inverse volume operators for loop quantum gravity

“…In this appendix, we calculate two matrix elements of the volume operator in Eq. ( 17) for I = 1, J = 2, K = 3 using the graphical method introduced in [12,13].…”

“…Denote the spins of the three edges e 1 , e 2 , e 3 incident at v by j 1 , j 2 , j 3 . Then the part of the spin-network state T v,s γ, j, i (A) (the notation s denotes the segments s I ) corresponding to v can be expressed graphically as (see [12,13] for an introduction to the graphical method in LQG)…”

“…In this paper, the cotriad operator is used to construct new alternative volume and inverse volume operators. The graphical method introduced in [12,13] brings much convenience to study the properties of these two operators in detail (see also [14,15] for the graphical method based on Tempereley-Lieb algebra). The new alternative volume operator shares the same properties with the standard volume operator in LQG [16].…”

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

“…Several closely related but slightly different versions of the graphical formalism can be found in the literature of quantum angular momentum [20,24,25], and in articles in which graphical methods are used for calculations in loop quantum gravity [29][30][31]33,34,64]. In these circumstances it seems preferable (and it certainly requires less effort) to develop a consistent set of conventions for the graphical formalism on one's own, using the available references as a guide, rather than trying to ensure that one is in full agreement with the conventions chosen in some previous treatment of the subject.…”

“…For a student of loop quantum gravity wishing to become familiar with the graphical techniques of SU (2) recoupling theory (such as the author in the beginning of his PhD studies), it can be difficult to find suitable references from which the subject could be efficiently learned. Information about the topic tends to be scattered around in the literature of quantum angular momentum [20,24,25], and in research articles where graphical methods are applied to calculations in loop quantum gravity [29][30][31]33,34,64], instead of being available in a single, comprehensive reference -especially one written from the perspective of loop quantum gravity.…”