Consistency check on the fundamental and alternative flux operators in loop quantum gravity

Abstract

There are different constructions of the flux of triad in loop quantum gravity, namely the fundamental and alternative flux operators. In parallel to the consistency check on the two versions of operator by the algebraic calculus in the literature, we check their consistency by the graphical calculus. Our calculation based on the original Brink graphical method is obviously simpler than the algebraic calculation. It turns out that our consistency check fixes the regulating factor ${\kappa }_{\mathrm{reg}}$ of the Ashtekar-Lewandowski volume operator as $\frac{1}{2}$, which corrects its previous value in the literature.

arXiv comment12 pages, 4 figures; minor updates

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• Polytopes in all dimensional loop quantum gravity
• Investigating the shadows of new regular black holes with a Minkowski core: Effects of spherical accretion and core type differences
• Scalar curvature operator for quantum-reduced loop gravity
• Relating spin-foam to canonical loop quantum gravity by graphical calculus
• Loop quantum cosmology from an alternative Hamiltonian. II. Including the Lorentzian term