Erratum

In loop quantum gravity, basic geometric quantities have been quantized and found to have discrete spectra.

“Basic geometric quantities have been quantized [10][11][12][13][14] and found to have discrete spectra.”

The area and volume operators of Rovelli and Smolin differ from those of Ashtekar, Lewandowski and collaborators, but for spin networks with generic 4-valent vertices their formulae agree up to constant factors.

“The area and volume operators first introduced by Rovelli and Smolin [56] differ from those later studied by Ashtekar, Lewandowski and collaborators [6][7][8][9]33].However, these differences do not arise in situations involving spin networks with generic 4-valent vertices (i.e.those for which no triple of incident edges have linearly dependent tangent vectors).In such situations, their formulae”

Area and volume operators for triangles and tetrahedra in this picture are closely related to those introduced by Rovelli and Smolin and by Ashtekar, Lewandowski and collaborators.

“These operators are closely related to the area and volume operators first introduced by Rovelli and Smolin [56], as well as the slightly different ones studied by Ashtekar, Lewandowski and collaborators [6][7][8][9]33].”

The Dirac algebra is not faithfully represented, but the algebra is non-anomalous in the sense that a consistent quantum theory is obtained.

“(i) As we will see in the next section, the Dirac algebra is indeed not faithfully represented (one could, however, use the arguments of [27] to improve this), but still the algebra is non-anomalous in the sense that we obtain a consistent quantum theory”

The classical limit schemes lim_{ε→0} lim_{δ→0} and (lim_{δ→0})|_{ε=δ} are identical and both give back the classical magnetic Hamiltonian constraint.

“This is justified because the classical limit schemes lim →0 lim δ→0 and (lim δ→0 ) | = (δ), (0)=0 are identical, both give back the classical magnetic Hamiltonian constraint.”

The EPRL model uses only simplicial complexes (or cubulations) and the spin-networks defined on their boundaries.

“The EPRL model on the other hand, uses only simplicial complexes (or, recently cubulations [28]) and the spin-networks defined on their boundaries.”

The LQG result fixes the proportionality constant for area as A_j = 8πħG√j(j+1).

“The LQG result assures us that we can interpret it as a physical quantization and not an artifact of the discretization and fixes the proportionality constant: A j = 8 π ¯ hG √ j ( j +1).”

The quantization of the physical area, with eigenvalues proportional to √j(j+1), is a key result of LQG, reappearing here in the context of the spinfoam models.

“The quantization of the physical area, with eigenvalues proportional to √ j ( j +1) is a key result of LQG [11], reappearing here in the context of the spinfoam models.”