The Spin-Foam Approach to Quantum Gravity

In the paper

“For this reason it is said that the constraint algebra does not close.”

In the paper

“The previous action corresponds to the standard Hamiltonian formulation of general relativity in terms of SU(2) connection variables.”

In the paper

“As it follows from (42), encodes the full background-independent Riemannian geometry of Σ.”

In the paper

“We have seven (first class) constraints for the 18 phase-space variables ().”

In the paper

“The constraint (49) coincides with the standard Gauss law of Yang-Mills theory (e.g., in electromagnetism).”

In the paper

“The general solution of the simplicity constraints (26) requires, in the 3+1 setting, the introduction of four new fields: the lapse N and the shift vectors Na.”

In the paper

“first, quantum amplitudes are independent of the regulating triangulation used in their definition”

In the paper

“Spin-foam amplitudes vanish unless all the spins assigned to faces are equal, i.e., jf = j for all f ∈ Δ*.”

In the paper

“It is immediate to see that if we impose such restriction on the Spin(4) BF amplitude (106) it simply reduces to the amplitude (102) defining SU(2) BF theory.”

In the paper

“Finally, an important property of SU(2) coherent states stemming from the fact that is that 114”

In the paper

“From the commutation relations (120) of Section 7.1 we can easily compute the commutator of the previous tetrahedron constraints and conclude that in fact it does not close, namely 128”

In the paper

“The previous expression is exactly equal to the form (86) of the BF amplitude.”

In the paper

“second, their discretization-independent value depends only on global properties of the spacetime-defining topological invariants of the manifold .”

In the paper

“In other words, the BF amplitude associated to a two complex Δ* is simply given by summing over the numbers obtained by the natural contraction of the network of projectors according to the pattern provided defined by the two-complex Δ*. The sum is over all possible assignments of irreducible representations of G to faces.”

In the paper

“The states |j,n⟩ form (an over-complete) basis in ℋj.”

In the paper

“then it follows that the BF partition function becomes 95 where χ is the Euler character of the two-dimensional manifold .”

In the paper

“Case γ < 1: Following [142], in this case one restricts the Spin(4) representations to 130 which amounts to choosing the maximum weight component j = j+ + j− in the expansion (182). Simple algebra shows that condition (188) is met.”

What the sources say

• Not checked yetcites [142] Ding, Y. and Rovelli, C., “The volume operator in covariant quantum gravity”, Class. Quantum Grav., 27, 165003, (2010). [DOI], [arXiv:0911.0543 [gr-qc]]. (Cited on pages 52 and 78.) source

In the paper

“The study of the coherent state representation of the FK model for γ > 1, and comparison with Eq. (177) for the EPRL model, clearly shows the difference between the two models in this regime.”

In the paper

“The Barrett-Crane models [60, 59] can easily be obtained in the present framework as a suitable limiting case γ → ∞ of the previous constructions.”

What the sources say

• Not checked yetcites [60] Barrett, J.W. and Crane, L., “A Lorentzian Signature Model for Quantum General Relativity”, Class. Quantum Grav., 17, 3101–3118, (2000). [DOI]. (Cited on pages 70, 76, and 79.) source
• Not checked yetcites [59] Barrett, J.W. and Crane, L., “Relativistic spin networks and quantum gravity”, J. Math. Phys., 39, 3296–3302, (1998). [DOI]. (Cited on pages 19, 52, and 76.) source

In the paper

“However, is not a projector, namely 165 Technically, this follows from (146) and the fact that 166 i.e., the projection imposing the linear constraints (defined on the frame of a tetrahedron or edge) and the Spin(4) (or Lorentz) group averaging — rendering the result gauge invariant — do not commute.”

In the paper

“Interestingly, the states that satisfy the previous semiclassical criterion satisfy the Gupta-Bleuler criterion not exactly and only in the large spin sense.”

In the paper

“One can verify for the cases γ < 1 and γ > 1 the Lorentz invariant criterion is just equivalent to the Master-constraint analysis above.”

In the paper

“It follows from the discussion in the previous sections that states satisfy the constraints (186) in the following semiclassical sense: 140 where the symbol (order semiclassical) denotes a quantity that vanishes in limit ℏ → 0, j → ∞ with ℏj = constant.”

In the paper

“The previous expression is exactly equal to the form (86) of the BF amplitude.”

In the paper

“Again we see that the important condition 192 holds (recall the notion from Section 7.2.4).”

In the paper

“Case γ > 1: In this case [13] one restricts the Spin(4) representations to 131 which amounts to choosing the minimum weight component j = j+ − j− in the expansion (182). This choice of is the solution to condition (188).”

What the sources say

• Not checked yetcites [13] Alexandrov, S., “The new vertices and canonical quantization”, Phys. Rev. D, 82, 024024, (2010). [DOI], [arXiv:1004.2260 [gr-qc]]. (Cited on pages 52 and 68.) source

In the paper

“In particular, given j, one has k = j − r for all possible half-integers such that k > 0 [140].”

What the sources say

• Not checked yetcites [140] Ding, Y., Han, M. and Rovelli, C., “Generalized spinfoams”, Phys. Rev. D, 83, 124020, (2011). [DOI], [arXiv:1011.2149]. (Cited on pages 52, 71, and 78.) source

In the paper

“The admissible quantum states Ψ are elements of the subspace (i.e., minimum weight states) and satisfy the constraints (186) in the following semiclassical sense: 196 where, as in the Riemannian case, the symbol (order semiclassical) denotes a quantity that vanishes in the limit ℏ → 0, j → ∞ with ℏj = constant.”

In the paper

“However, notice that different weights are assigned to edges in the FK model. This is due to the fact that one is restricting the Spin(4) resolution of identity in the coherent basis in the previous expression, while in the EPRL model the coherent state resolution of the identity is used for SU(2) representations.”

In the paper

“The proof of the validity of the tetrahedra constraints (156).”

In the paper

“The linear simplicity constraint is satisfied as long as p = γj(j+1)/k.”

In the paper

“Another difficulty associated with is the failure of the amplitudes of the EPRL model, as defined here, to be consistent with the abstract notion of spin foams as defined in [38].”

What the sources say

• Not checked yetcites [38] Baez, J.C., “Spin Foam Models”, Class. Quantum Grav., 15, 1827–1858, (1998). [DOI]. (Cited on pages 19, 34, 63, 80, and 99.) source

In the paper

“Indeed, it has been shown that the state space of strong solutions of the quadratic quantum Plebanski constraints is one dimensional and it is given by the BC intertwiner |BC⟩ [313].”

What the sources say

• Not checked yetcites [313] Reisenberger, M.P., “On relativistic spin network vertices”, J. Math. Phys., 40, 2046–2054, (1999). [DOI]. (Cited on page 77.) source

In the paper

“This leads to the following Lorentz invariant condition in terms of the SL(2, ℂ) Casimirs 193”

In the paper

“As the continuum labels pf are restricted to pf = γjf, the Lorentzian EPRL model becomes a state-sum model, as its Riemannian relative.”

In the paper

“For the Riemannian model, the result is: Case γ < 1: 137Case γ > 1: 138 where j ∈ ℕ/2 is arbitrary — except from the requirement that j± ∈ ℕ/2, which depends on the value of the Immirzi parameter according to Section 7.2.2.”

In the paper

“For γ < 1, the vertex amplitude is identical to the EPRL model.”

• Also stated in: § The case of γ < 1 · plain

  “the vertex amplitudes of the FK and EPRL model coincide for γ < 1”

In the paper

“The reason is that step (172) directly leading to the discrete action in the previous case is no longer valid, as the representations of the subgroup SU(2) ∈ Spin(4) are now minimum instead of maximum weight.”

In the paper

“In this section, we prove that the quadratic constraints are satisfied in the sense that their path integral expectation value and fluctuation vanish in the appropriate semiclassical limit.”

In the paper

“For given j the minimum eigenvalue is obtained when k = j and p = γj in which case one gets mp,k,j ≥ mj for mj = ℏ2((1 + γ2)j + 1).”

In the paper

“Observe that the vertex amplitude depends only on the 10 corresponding , this is the Riemannian Barrett-Crane vertex.”

In the paper

“In the Lorentzian models one restricts the SL(2, ℂ) representations of BF theory to those satisfying 194 for j ∈ ℕ/2.”

In the paper

“The two point function for the EPRL model was calculated in [75] and was shown to produce a result in agreement with that of Regge calculus [80, 260] in the limit γ → 0.”

What the sources say

• Not checked yetcites [75] Bianchi, E., Magliaro, E. and Perini, C., “LQG propagator from the new spin foams”, Nucl. Phys. B, 822, 245–269, (2009). [DOI], [arXiv:0905.4082 [gr-qc]]. (Cited on page 83.) source
• Not checked yetcites [80] Bianchi, E. and Satz, A., “Semiclassical regime of Regge calculus and spin foams”, Nucl. Phys. B, 808, 546–568, (2009). [DOI], [arXiv:0808.1107 [gr-qc]]. (Cited on page 83.) source
• Not checked yetcites [260] Magliaro, E. and Perini, C., “Comparing LQG with the linearized theory”, Int. J. Mod. Phys. A, 23, 1200–1208, (2008). [DOI]. (Cited on page 83.) source

In the paper

“It was then applied to the Barrett-Crane model in [5, 4, 77], where it was discovered that certain components of the two-point function could not yield the expected result compatible with Regge gravity in the semiclassical limit.”

What the sources say

• Not checked yetcites [5] Alesci, E. and Rovelli, C., “Complete LQG propagator. II. Asymptotic behavior of the vertex”, Phys. Rev. D, 77, 044024, (2008). [DOI], [arXiv:0711.1284 [gr-qc]]. (Cited on page 83.) source
• Not checked yetcites [4] Alesci, E. and Rovelli, C., “Complete LQG propagator: Difficulties with the Barrett-Crane vertex”, Phys. Rev. D, 76, 104012, (2007). [DOI], [arXiv:0708.0883 [gr-qc]]. (Cited on page 83.) source
• Not checked yetcites [77] Bianchi, E., Modesto, L., Rovelli, C. and Speziale, S., “Graviton propagator in loop quantum gravity”, Class. Quantum Grav., 23, 6989–7028, (2006). [DOI], [arXiv:gr-qc/0604044 [gr-qc]]. (Cited on page 83.) source

In the paper

“As discussed in [321, 359], spin network states solve the Gauss constraint — — as they are manifestly SU(2) gauge invariant.”

What the sources say

• Not checked yetcites [321] Rovelli, C., Quantum Gravity, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 2004). [Google Books]. (Cited on pages 12, 15, 22, and 86.) source
• Not checked yetcites [359] Thiemann, T., Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 2007). [Google Books]. (Cited on pages 12, 22, and 86.) source

In the paper

“The fact that these special subclasses of projected spin networks appear naturally as boundary states of the new spin foams is shown in [158].”

What the sources say

• Not checked yetcites [158] Dupuis, M. and Livine, E.R., “Lifting SU(2) Spin Networks to Projected Spin Networks”, Phys. Rev. D, 82, 064044, (2010). [DOI], [arXiv:1008.4093]. (Cited on page 78.) source

In the paper

“The physical inner product takes the standard Ponzano-Regge form when the spin-network states s and s′ have only 3-valent nodes.”

In the paper

“The simple BF measure chosen in the presentation of the amplitudes in the previous sections satisfies these requirements.”

What the sources say

• Not checked yetcites [90] Bojowald, M. and Perez, A., “Spin foam quantization and anomalies”, Gen. Relativ. Gravit., 42, 877–907, (2010). [DOI], [arXiv:gr-qc/0303026]. (Cited on pages 63, 78, and 99.) source

In the paper

“Even though the ordering of the plaquette actions does not affect the amplitudes, the spin-foam representation of the terms in the sum (238) is highly dependent on that ordering.”

In the paper

“Moreover, geometric operators such as volume and area defined in the covariant context are shown to coincide with the corresponding operators defined in the canonical formulation [141, 142].”

What the sources say

• Not checked yetcites [141] Ding, Y. and Rovelli, C., “Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory”, Class. Quantum Grav., 27, 205003, (2010). [DOI], [arXiv: 1006.1294 [gr-qc]]. (Cited on page 78.) source
• Not checked yetcites [142] Ding, Y. and Rovelli, C., “The volume operator in covariant quantum gravity”, Class. Quantum Grav., 27, 165003, (2010). [DOI], [arXiv:0911.0543 [gr-qc]]. (Cited on pages 52 and 78.) source

In the paper

“For such boundary data, the action (215) has exactly two critical points leading to the asymptotic formula 216”

What the sources say

• Not checked yetcites [61] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “Asymptotic analysis of the EPRL four-simplex amplitude”, J. Math. Phys., 50, 112504, (2009). [DOI], [arXiv:0902.1170 [gr-qc]]. (Cited on pages 67, 80, 81, and 82.) source

In the paper

“If configurations are geometric (i.e., Regge-like) one has two kinds of contributions to the amplitude asymptotics: those coming from degenerate and non-degenerate configurations.”

What the sources say

• Not checked yetcites [122] Conrady, F. and Freidel, L., “Semiclassical limit of 4-dimensional spin foam models”, Phys. Rev. D, 78, 104023, (2008). [DOI], [arXiv:0809.2280 [gr-qc]]. (Cited on pages 80, 81, 82, and 83.) source

In the paper

“Even though the ordering of the plaquette actions does not affect the amplitudes, the spin-foam representation of the terms in the sum (238) is highly dependent on that ordering.”

In the paper

“The partition function corresponds to the ‘volume’ of the space of flat connections on .”

In the paper

“In fact, one can explicitly construct a basis of by choosing a linearly-independent set of representatives of the equivalence classes defined in (231).”

In the paper

“Higher-order correlation functions have been computed in [337]; the results are in agreement with Regge gravity in the γ → 0 limit.”

What the sources say

• Not checked yetcites [337] Rovelli, C. and Zhang, M., “Euclidean three-point function in loop and perturbative gravity”, Class. Quantum Grav., 28, 175010, (2011). [DOI], [arXiv:1105.0566]. (Cited on page 83.) source

In the paper

“In a further modification of the previous model (known as colored tensor models [208]) new techniques based on a suitable 1/N expansion imply that amplitudes are dominated by spherical topology [207]”

What the sources say

• Not checked yetcites [207] Gurau, R., “The 1/N expansion of colored tensor models”, Ann. Henri Poincare, 12, 829–847, (2011). [DOI], [arXiv:1011.2726 [gr-qc]]. (Cited on pages 80 and 104.) source

In the paper

“To each solution one can associate a second solution corresponding to a parity related 4-simplex and, consequently, the asymptotic formula has two terms.”

What the sources say

• Not checked yetcites [64] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Hellmann, F. and Pereira, R., “Lorentzian spin foam amplitudes: graphical calculus and asymptotics”, Class. Quantum Grav., 27, 165009, (2010). [DOI], [arXiv:0907.2440 [gr-qc]]. (Cited on pages 80 and 82.) source

In the paper

“Non-geometric configurations are exponentially suppressed.”

What the sources say

• Not checked yetcites [61] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “Asymptotic analysis of the EPRL four-simplex amplitude”, J. Math. Phys., 50, 112504, (2009). [DOI], [arXiv:0902.1170 [gr-qc]]. (Cited on pages 67, 80, 81, and 82.) source

In the paper

“They show that those solutions of the equations of motion of the effective discrete action that are non-geometric (i.e., the opposite of Regge-like) are not critical and hence exponentially suppressed in the scaling jf → λjf with λ ≫ 1.”

What the sources say

• Not checked yetcites [122] Conrady, F. and Freidel, L., “Semiclassical limit of 4-dimensional spin foam models”, Phys. Rev. D, 78, 104023, (2008). [DOI], [arXiv:0809.2280 [gr-qc]]. (Cited on pages 80, 81, 82, and 83.) source

In the paper

“This amplitude is given, in general, by an SU(2) 3Nj-symbol corresponding to the flat evaluation of the spin network defined by the intersection of the corresponding vertex with a 2-sphere.”

In the paper

“as in BF theory, the EPRL amplitudes can be generalized to arbitrary complexes with boundaries given by graphs with nodes of arbitrary valence. The extension of the model to arbitrary complexes was first studied in [237, 236] and has been revisited in [140].”

What the sources say

• Not checked yetcites [237] Kaminski, W., Lewandowski, J. and Pawlowski, T., “Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC”, Class. Quantum Grav., 26, 245016, (2009). [DOI], [arXiv:0907.4322]. (Cited on page 78.) source
• Not checked yetcites [236] Kaminski, W., Kisielowski, M. and Lewandowski, J., “Spin-Foams for All Loop Quantum Gravity”, Class. Quantum Grav., 27, 095006, (2010). [DOI], [arXiv:0909.0939 [gr-qc]]. (Cited on page 78.) source
• Not checked yetcites [140] Ding, Y., Han, M. and Rovelli, C., “Generalized spinfoams”, Phys. Rev. D, 83, 124020, (2011). [DOI], [arXiv:1011.2149]. (Cited on pages 52, 71, and 78.) source

In the paper

“Transition amplitudes define the physical scalar product by reproducing the skein relations of the canonical analysis.”

In the paper

“We have seen in Section 14.3 that transition amplitudes are indeed finite in the canonical framework, where M = Σ × R.”

In the paper

“Divergences occur due to infinite volume factors corresponding to the topological gauge freedom (225) (see [186]).”

What the sources say

• Not checked yetcites [186] Freidel, L. and Louapre, D., “Diffeomorphisms and spin foam models”, Nucl. Phys. B, 662, 279–298, (2003). [DOI], [arXiv:gr-qc/0212001]. (Cited on pages 43, 92, 95, 99, and 100.) source

In the paper

“In the case of gravity with cosmological constant the state-sum generalizes to the Turaev-Viro model [364] defined in terms of SUq(2) with qn = 1, where the representations are finitely many.”

What the sources say

• Not checked yetcites [364] Turaev, V.G. and Viro, O.Y., “State sum invariants of 3-manifolds and quantum 6j-symbols”, Topology, 31, 865–902, (1992). [DOI]. (Cited on pages 43, 93, and 95.) source

In the paper

“A crucial property of the partition function (and transition amplitudes in general) is that it does not depend on the discretization Δ.”

In the paper

“The crucial property of this spin-foam model is that the amplitudes are independent of the chosen cellular decomposition [363, 206].”

What the sources say

• Not checked yetcites [363] Turaev, V.G., Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics, 18, (Walter de Gruyter, Berlin; New York, 1994). [Google Books]. (Cited on page 93.) source
• Not checked yetcites [206] Girelli, F., Oeckl, R. and Perez, A., “Spin foam diagrammatics and topological invariance”, Class. Quantum Grav., 19, 1093–1108, (2002). [DOI], [arXiv:gr-qc/0111022]. (Cited on pages 40 and 93.) source

In the paper

“The result is equivalent to the physical scalar product defined in the canonical picture in terms of the delta measure.”

In the paper

“We have seen that this requirement (advocated in [90]) can be met by suitable definitions of the transition amplitudes [49, 235]”

What the sources say

• Not checked yetcites [90] Bojowald, M. and Perez, A., “Spin foam quantization and anomalies”, Gen. Relativ. Gravit., 42, 877–907, (2010). [DOI], [arXiv:gr-qc/0303026]. (Cited on pages 63, 78, and 99.) source
• Not checked yetcites [49] Bahr, B., Hellmann, F., Kaminski, W., Kisielowski, M. and Lewandowski, J., “Operator Spin Foam Models”, Class. Quantum Grav., 28, 105003, (2011). [DOI], [arXiv:1010.4787 [gr-qc]]. (Cited on pages 63, 64, and 99.) source
• Not checked yetcites [235] Kaminski, W., Kisielowski, M. and Lewandowski, J., “The EPRL intertwiners and corrected partition function”, Class. Quantum Grav., 27, 165020, (2010). [DOI], [arXiv:0912.0540 [gr-qc]]. (Cited on pages 63, 64, and 99.) source

In the paper

“For any contractible Wilson loop in the j representation, the state 252 for any spin network state s, has vanishing transition amplitude with any element of .”

In the paper

“However, there are indications that the discretization procedures used generally do break the general covariance of gravity.”

In the paper

“The discretization reduces the infinite dimensional functional integral to a multiple integration over a finite number of variables.”

In the paper

“There is no evidence that the amplitudes of any of the present models for 4D converge in such a limit.”

In the paper

“In the simple case of 3D gravity with cosmological constant, this can be shown explicitly at the classical level [45] as well as at the quantum level [304].”

What the sources say

• Not checked yetcites [45] Bahr, B. and Dittrich, B., “Improved and Perfect Actions in Discrete Gravity”, Phys. Rev. D, 80, 124030, (2009). [DOI], [arXiv:0907.4323 [gr-qc]]. (Cited on page 100.) source
• Not checked yetcites [304] Perez, A. and Pranzetti, D., “On the regularization of the constraints algebra of Quantum Gravity in 2+1 dimensions with non-vanishing cosmological constant”, Class. Quantum Grav., 27, 145009, (2010). [DOI], [arXiv:1001.3292 [gr-qc]]. (Cited on page 100.) source

In the paper

“Diffeomorphism-equivalent configurations (in the discrete sense described above) appear at all orders in the perturbation series.39”

In the paper

“The perturbative series are generically divergent.”

In the paper

“This is a major issue where concrete results have not been obtained so far beyond the topological case.”

In the paper

“Their results indicate that diff-invariance is indeed broken by the discretization in the sense that there is no infinitesimal generator of diffeomorphism.”

What the sources say

• Not checked yetcites [190] Gambini, R., Ponce, M. and Pullin, J., “Consistent discretizations: the Gowdy spacetimes”, Phys. Rev. D, 72, 024031, (2005). [DOI], [arXiv:gr-qc/0505043]. (Cited on page 100.) source
• Not checked yetcites [199] Gambini, R. and Pullin, J., “Consistent discretizations as a road to Quantum Gravity”, in Oriti, D., ed., Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, pp. 378–392, (Cambridge University Press, Cambridge; New York, 2009). [arXiv:gr-qc/0512065 [gr-qc]]. (Cited on page 100.) source
• Not checked yetcites [198] Gambini, R. and Pullin, J., “Consistent discretization and canonical, classical and quantum Regge calculus”, Int. J. Mod. Phys. D, 15, 1699–1706, (2006). [DOI], [arXiv:gr-qc/0511096]. (Cited on page 100.) source

In the paper

“The LIV effects obtained in LQG so far promote by assumption the kinematical discreteness of quantum geometry to discreteness of the physical background in a literal fashion”

In the paper

“A general covariant approach to the quantization of gravity is needed.”

In the paper

“Einstein’s theory is, in this sense, incomplete as a fundamental description of nature. A clear indication of such incompleteness is the generic prediction of spacetime singularities in the context of gravitational collapse.”

In the paper

“these indications are not conclusive, mainly because the systematic description of dynamical processes remains an open problem at the present stage.”

In the paper

“In general relativity, the gravitational field is encoded in the dynamical geometry of spacetime, implying a strong form of universality that precludes the existence of any non-dynamical reference system — or non-dynamical background — on top of which things occur.”

In the paper

“General relativity must be replaced by a more fundamental description that appropriately includes the quantum degrees of freedom of gravity.”

In the paper

“Using the kinematical setting of LQG with the assumption of the existence of a micro-local (in the sense of Planck scale) causal structure, Markopoulou and Smolin define a general class of (causal) spin-foam models for gravity [267, 264] (see also [268]).”

What the sources say

• Not checked yetcites [267] Markopoulou, F. and Smolin, L., “Causal evolution of spin networks”, Nucl. Phys. B, 508, 409–430, (1997). [DOI]. (Cited on page 19.) source
• Not checked yetcites [264] Markopoulou, F., “Dual formulation of spin network evolution”, arXiv, e-print, (1997). [arXiv:gr-qc/9704013]. (Cited on page 19.)
• Not checked yetcites [268] Markopoulou, F. and Smolin, L., “Quantum geometry with intrinsic local causality”, Phys. Rev. D, 58, 084032, (1998). [DOI]. (Cited on page 19.) source

In the paper

“Freidel and Krasnov [181] define a general framework to construct spin-foam models corresponding to theories, whose action has the general form 14 where the first term is the BF action, while Φ(B) is a certain polynomial function of the B field.”

What the sources say

• Not checked yetcites [181] Freidel, L. and Krasnov, K., “Spin Foam Models and the Classical Action Principle”, Adv. Theor. Math. Phys., 2, 1183–1247, (1999). (Cited on page 18.) source

In the paper

“In order to avoid this, Reisenberger defines a one-parameter family of models by inserting the operator 13 instead of the delta function above. In the limit z → ∞ the constraints are sharply imposed.”

In the paper

“the assumption that such structure exists all the way down to the Planck scale is inconsistent with what we know about gravity and quantum mechanics.”

In the paper

“As we argued in Section 14, skein relations defining the physical Hilbert space of BF theory are implied by the spin-foam transition amplitudes.”

In the paper

“The model obtained in this way is very similar to Reisenberger’s one. However, there are some technical differences. One of the more obvious is that the non-commutative invariant vector fields representing Bi are replaced here by the commutative functional derivatives δ/δJi.”

In the paper

“At the present stage of the approach, physical observables are explicitly known only in some special cases.”

In the paper

“The models are not related to any continuum action.”

What the sources say

• Not checked yetcites [267] Markopoulou, F. and Smolin, L., “Causal evolution of spin networks”, Nucl. Phys. B, 508, 409–430, (1997). [DOI]. (Cited on page 19.) source
• Not checked yetcites [264] Markopoulou, F., “Dual formulation of spin network evolution”, arXiv, e-print, (1997). [arXiv:gr-qc/9704013]. (Cited on page 19.)

In the paper

“Different choices of actions lead to different discrete theories. However, in the low-energy effective action, the differences appear only in local operators of dimension five or higher.”

• Also stated in: § Non-renormalizability of general relativity · plain

  “the ambiguities at the ‘microscopic level’ do not have any effect at low energies where one recovers quantum Yang-Mills theory.”

In the paper

“the fundamental discreteness of quantum geometry in LQG plays the role of physical regulator at the Planck scale [354].”

What the sources say

• Not checked yetcites [354] Thiemann, T., “Quantum spin dynamics (QSD): V: Quantum gravity as the natural regulator of matter quantum field theories”, Class. Quantum Grav., 15, 1281–1314, (1998). [DOI], [arXiv:gr-qc/9705019]. (Cited on pages 14 and 32.) source

In the paper

“The spin-foam approach was constructed as a means to tackle the difficult question of dynamics and the definition of observable quantities in LQG. It is an attempt to address the difficulties associated with the last three steps of the list given above by rethinking the problem of dynamics from a path-integral covariant perspective.”

In the paper

“The fundamental observation of [230] is that one can write the discrete action in a very compact form if we encode part of the degrees of freedom of the tetrad in an SU(2) group element.”

What the sources say

• Not checked yetcites [230] Iwasaki, J., “A lattice quantum gravity model with surface-like excitations in 4-dimensional space-time”, arXiv, e-print, (2000). [arXiv:gr-qc/0006088]. (Cited on page 18.)

In the paper

“A remarkable feature is that the corresponding physical states are (in a certain sense) solutions to various regularizations of the scalar constraint for (Riemannian) LQG.”

What the sources say

• Not checked yetcites [193] Gambini, R. and Pullin, J., “A finite spin-foam-based theory of three- and four-dimensional quantum gravity”, Phys. Rev. D, 66, 024020, (2002). [DOI], [gr-qc/0111089]. (Cited on page 20.) source

In the paper

“General considerations indicate that standard notions of space and time are expected to fail near the Planck scale ℓp.2”

In the paper

“The interest of the latter formulation is that it provides an action of gravity that is given by a BF term plus a genuine potential term instead of constraints.”

What the sources say

• Not checked yetcites [259] MacDowell, S.W. and Mansouri, F., “Unified Geometric Theory of Gravity and Supergravity”, Phys. Rev. Lett., 38, 739–742, (1977). [DOI]. Erratum: ibid. 38 (1977) 1376. (Cited on page 18.) source

In the paper

“However, there is a simple model that can be obtained as a systematic quantization of the simplicial SO(4) Plebanski’s action.”

What the sources say

• Not checked yetcites [59] Barrett, J.W. and Crane, L., “Relativistic spin networks and quantum gravity”, J. Math. Phys., 39, 3296–3302, (1998). [DOI]. (Cited on pages 19, 52, and 76.) source
• Not checked yetcites [38] Baez, J.C., “Spin Foam Models”, Class. Quantum Grav., 15, 1827–1858, (1998). [DOI]. (Cited on pages 19, 34, 63, 80, and 99.) source

In the paper

“In [311], Reisenberger constructs a spin-foam model for this simplicial theory by imposing the constraints Ωij directly on the SU(2) BF amplitudes.”

What the sources say

• Not checked yetcites [311] Reisenberger, M.P., “A lattice worldsheet sum for 4-d Euclidean general relativity”, arXiv, e-print, (1997). [arXiv:gr-qc/9711052]. (Cited on page 17.)

In the paper

“According to Plebanski [307], the action of self-dual Riemannian gravity can be written as a constrained SU(2) BF theory”

What the sources say

• Not checked yetcites [307] Plebański, J.F., “On the Separation of Einsteinian Substructures”, J. Math. Phys., 18, 2511, (1977). [DOI]. (Cited on pages 17, 22, and 23.) source

In the paper

“If we ignore all these issues and try to set up a naive perturbative quantization of gravity, we find that the theory is non-renormalizable. This can be expected from dimensional analysis as the quantity playing the role of the coupling constant turns out to be the Planck length ℓp.”

In the paper

“The representation of the basic unconstrained phase-space variables as suitable operators in used in LQG — which has the additional key property of containing a special state that is diffeomorphism invariant — has been shown to be unique [245].”

What the sources say

• Not checked yetcites [245] Lewandowski, J., Okołów, A., Sahlmann, H. and Thiemann, T., “Uniqueness of Diffeomorphism Invariant States on Holonomy-Flux Algebras”, Commun. Math. Phys., 267, 703–733, (2006). [DOI], [arXiv:gr-qc/0504147]. (Cited on pages 13 and 30.) source

In the paper

“At the heart of the completion of the definition of the theory, the clear-cut definition of quantum dynamics remains open.”

In the paper

“In the Capovilla-Dell-Jacobson [110, 109] formulation of general relativity this ‘integration’ is partially performed at the continuum level.”

What the sources say

• Not checked yetcites [110] Capovilla, R., Jacobson, T.A. and Dell, J., “A Pure spin connection formulation of gravity”, Class. Quantum Grav., 8, 59–73, (1991). [DOI]. (Cited on page 20.) source
• Not checked yetcites [109] Capovilla, R., Jacobson, T.A. and Dell, J., “General relativity without the metric”, Phys. Rev. Lett., 63, 2325, (1989). [DOI]. (Cited on page 20.) source

In the paper

“The spin-foam representation arises naturally as the path-integral representation of the field theoretical analog of P in the context of LQG.”

In the paper

“In the case of gravity there is evidence indicating that the theory might be non-perturbatively renormalizable coming from the investigations of non trivial fix points of the renormalization group flow in truncated models (see [317] and references therein).”

What the sources say

• Not checked yetcites [317] Reuter, M. and Saueressig, F., “Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity”, arXiv, e-print, (2007). [arXiv:0708.1317 [hep-th]]. (Cited on page 11.)

In the paper

“It is well known that here too the regulating procedure leads to ambiguities; the relevance of the example resides in the fact that these ambiguities resemble in nature those appearing in LQG.”

In the paper

“The fact that physical states of BF theory solve the scalar constraint is well known [353], since roughly F(A) = 0 implies EEF(A) = 0.”

What the sources say

• Not checked yetcites [353] Thiemann, T., “Quantum spin dynamics (QSD): IV. 2+1 Euclidean quantum gravity as a model to test 3+1 Lorentzian quantum gravity”, Class. Quantum Grav., 15, 1249–1280, (1998). [DOI]. (Cited on page 20.) source

In the paper

“Our starting point is Plebanski’s Riemannian action for general relativity, which can be thought of as Spin(4) BF theory plus constraints on the B-field.”

What the sources say

• Not checked yetcites [307] Plebański, J.F., “On the Separation of Einsteinian Substructures”, J. Math. Phys., 18, 2511, (1977). [DOI]. (Cited on pages 17, 22, and 23.) source

In the paper

“Modern LQG and spin foams are based on a certain relative of these variables, that is often referred to as Ashtekar-Barbero variables [56, 227].”

What the sources say

• Not checked yetcites [56] Barbero G.J.F., “Real Ashtekar variables for Lorentzian signature space times”, Phys. Rev. D, 51, 5507–5510, (1995). [DOI]. (Cited on pages 23 and 29.) source
• Not checked yetcites [227] Immirzi, G., “Real and complex connections for canonical gravity”, Class. Quantum Grav., 14, L177–L181, (1997). [DOI]. (Cited on pages 23 and 32.) source

In the paper

“One serious problem of this formulation is that it corresponds to a sector of gravity where the Weyl tensor satisfies certain algebraic requirements.”

What the sources say

• Not checked yetcites [110] Capovilla, R., Jacobson, T.A. and Dell, J., “A Pure spin connection formulation of gravity”, Class. Quantum Grav., 8, 59–73, (1991). [DOI]. (Cited on page 20.) source
• Not checked yetcites [109] Capovilla, R., Jacobson, T.A. and Dell, J., “General relativity without the metric”, Phys. Rev. Lett., 63, 2325, (1989). [DOI]. (Cited on page 20.) source

In the paper

“In a generally covariant system quantum dynamics is fully governed by constraint equations.”

In the paper

“There is a natural way to relax the constraints and this leads to a simple model that has a more clear relationship with the canonical picture provided by LQG.”

What the sources say

• Not checked yetcites [168] Engle, J., Pereira, R. and Rovelli, C., “Loop-Quantum-Gravity Vertex Amplitude”, Phys. Rev. Lett., 99, 161301, (2007). [DOI], [arXiv:0705.2388 [gr-qc]]. (Cited on pages 20, 36, 49, and 74.) source
• Not checked yetcites [169] Engle, J., Pereira, R. and Rovelli, C., “Flipped spinfoam vertex and loop gravity”, Nucl. Phys. B, 798, 251–290, (2008). [DOI], [arXiv:0708.1236 [gr-qc]]. (Cited on pages 20 and 36.) source

In the paper

“A very similar set of models were independently introduced by Freidel and Krasnov in [182].”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“The diffeomorphism constraint does not exist in the quantum theory.”

In the paper

“The idea is to express the BF path integral in terms of the coherent state representation, in order to impose the Plebanski constraints semiclassically in terms of expectation values.”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“It has been shown that the associated (Poisson) holonomy-flux algebra admits a unique quantization in a Hilbert space with diffeomorphism-invariant states [176, 245].”

What the sources say

• Not checked yetcites [176] Fleischhack, C., “Representations of the Weyl Algebra in Quantum Geometry”, Commun. Math. Phys., 285, 67–140, (2009). [DOI], [arXiv:math-ph/0407006]. (Cited on page 30.) source
• Not checked yetcites [245] Lewandowski, J., Okołów, A., Sahlmann, H. and Thiemann, T., “Uniqueness of Diffeomorphism Invariant States on Holonomy-Flux Algebras”, Commun. Math. Phys., 267, 703–733, (2006). [DOI], [arXiv:gr-qc/0504147]. (Cited on pages 13 and 30.) source

In the paper

“The anomalous commutation relations of the B10 field in the quantum theory imply that the commutation of the Plebanski constraints does not define a closed algebra.”

In the paper

“In particular, flat geometries are not contained in this sector.”

What the sources say

• Not checked yetcites [110] Capovilla, R., Jacobson, T.A. and Dell, J., “A Pure spin connection formulation of gravity”, Class. Quantum Grav., 8, 59–73, (1991). [DOI]. (Cited on page 20.) source
• Not checked yetcites [109] Capovilla, R., Jacobson, T.A. and Dell, J., “General relativity without the metric”, Phys. Rev. Lett., 63, 2325, (1989). [DOI]. (Cited on page 20.) source

In the paper

“The new spin-foam models are based on Plebanski’s realization that one can obtain classical general relativity by suitably constraining the variations of the B field in the simple kind of topological theory called BF theory [307].”

What the sources say

• Not checked yetcites [307] Plebański, J.F., “On the Separation of Einsteinian Substructures”, J. Math. Phys., 18, 2511, (1977). [DOI]. (Cited on pages 17, 22, and 23.) source

In the paper

“LQG and spin foams are based on the possibility of formulating Hamiltonian general relativity in terms of Yang-Mills-like connection variables.”

What the sources say

• Not checked yetcites [23] Ashtekar, A., “New Variables for Classical and Quantum Gravity”, Phys. Rev. Lett., 57, 2244–2247, (1986). [DOI]. (Cited on pages 23 and 29.) source
• Not checked yetcites [56] Barbero G.J.F., “Real Ashtekar variables for Lorentzian signature space times”, Phys. Rev. D, 51, 5507–5510, (1995). [DOI]. (Cited on pages 23 and 29.) source
• Not checked yetcites [227] Immirzi, G., “Real and complex connections for canonical gravity”, Class. Quantum Grav., 14, L177–L181, (1997). [DOI]. (Cited on pages 23 and 32.) source

In the paper

“Imposing the constraints strongly, as in the Barrett-Crane model, implies the imposition of additional conditions that are not present in the classical theory.”

What the sources say

• Not checked yetcites [168] Engle, J., Pereira, R. and Rovelli, C., “Loop-Quantum-Gravity Vertex Amplitude”, Phys. Rev. Lett., 99, 161301, (2007). [DOI], [arXiv:0705.2388 [gr-qc]]. (Cited on pages 20, 36, 49, and 74.) source
• Not checked yetcites [169] Engle, J., Pereira, R. and Rovelli, C., “Flipped spinfoam vertex and loop gravity”, Nucl. Phys. B, 798, 251–290, (2008). [DOI], [arXiv:0708.1236 [gr-qc]]. (Cited on pages 20 and 36.) source

In the paper

“However, the logic used in the derivation is different.”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“Freidel and Krasnov introduced11 the linear version of the Plebanski constraints that later used in order to provide a simpler derivation and generalization of the EPRL model to arbitrary Immirzi parameter.”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“In this sense we can regard general relativity in the new variables as a background independent relative of SU(2) Yang-Mills theory.”

In the paper

“As we show below, the Hamiltonian analysis of such a formulation of gravity is at the heart of the replacement of the Plebanski quadratic constraints by linear ones that has been so useful in the definition of the new vertex amplitudes [182].”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“The basic idea was to relax the imposition of the Plebanski constraints that reduce BF theory to general relativity in the quantum theory.”

What the sources say

• Not checked yetcites [168] Engle, J., Pereira, R. and Rovelli, C., “Loop-Quantum-Gravity Vertex Amplitude”, Phys. Rev. Lett., 99, 161301, (2007). [DOI], [arXiv:0705.2388 [gr-qc]]. (Cited on pages 20, 36, 49, and 74.) source
• Not checked yetcites [169] Engle, J., Pereira, R. and Rovelli, C., “Flipped spinfoam vertex and loop gravity”, Nucl. Phys. B, 798, 251–290, (2008). [DOI], [arXiv:0708.1236 [gr-qc]]. (Cited on pages 20 and 36.) source

In the paper

“The area operator gives a clear geometrical interpretation to spin-network states: the fundamental 1-dimensional excitations defining a spin-network state can be thought of as quantized ‘flux lines’ of area.”

What the sources say

• Not checked yetcites [344] Smolin, L., “Recent developments in nonperturbative quantum gravity”, in Pérez-Mercader, J., Sola, J. and Verdaguer, E., eds., Quantum Gravity and Cosmology, Proceedings of the XXII GIFT International Seminar on Theoretical Physics, Sant Feliu de Guíxols, Catalonia, Spain, 3–8 June 1991, (World Scientific, Singapore, 1992). [hep-th/9202022]. (Cited on pages 13 and 32.)
• Not checked yetcites [329] Rovelli, C. and Smolin, L., “Discretneess of the area and volume in quantum gravity”, Nucl. Phys. B, 442, 593–619, (1995). [DOI], [arXiv:gr-qc/9411005]. Erratum: ibid. 456 (1995) 753. (Cited on page 32.) source
• Not checked yetcites [33] Ashtekar, A. and Lewandowski, J., “Quantum Theory of Gravity: I. Area Operators”, Class. Quantum Grav., 14, A55–A81, (1997). [DOI]. (Cited on page 32.) source

In the paper

“The generalized states described above solve all of the constraints (64) but the scalar constraint.”

In the paper

“The spectrum of the volume operator is also discrete [344, 329, 258, 34].”

What the sources say

• Not checked yetcites [344] Smolin, L., “Recent developments in nonperturbative quantum gravity”, in Pérez-Mercader, J., Sola, J. and Verdaguer, E., eds., Quantum Gravity and Cosmology, Proceedings of the XXII GIFT International Seminar on Theoretical Physics, Sant Feliu de Guíxols, Catalonia, Spain, 3–8 June 1991, (World Scientific, Singapore, 1992). [hep-th/9202022]. (Cited on pages 13 and 32.)
• Not checked yetcites [329] Rovelli, C. and Smolin, L., “Discretneess of the area and volume in quantum gravity”, Nucl. Phys. B, 442, 593–619, (1995). [DOI], [arXiv:gr-qc/9411005]. Erratum: ibid. 456 (1995) 753. (Cited on page 32.) source
• Not checked yetcites [258] Loll, R., “Simplifying the spectral analysis of the volume operator”, Nucl. Phys. B, 500, 405–420, (1997). [DOI]. (Cited on page 32.) source
• Not checked yetcites [34] Ashtekar, A. and Lewandowski, J., “Quantum Theory of Gravity: II. Volume Operators”, Adv. Theor. Math. Phys., 1, 388–429, (1997). [arXiv:gr-qc/9711031]. (Cited on page 32.) source

In the paper

“on the one hand it is equivalent to the Jackiw-Teitelboim model [232, 351]”

What the sources say

• Not checked yetcites [232] Jackiw, R., “Liouville field theory: a two-dimensional model for gravity?”, in Christensen, S.M., ed., Quantum Theory of Gravity: Essays in Honor of the 60th Birthday of Bryce S. De Witt, pp. 403–420, (Adam Hilger, Bristol, 1984). (Cited on page 42.)
• Not checked yetcites [351] Teitelboim, C., “The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly”, in Christensen, S.M., ed., Quantum Theory of Gravity: Essays in Honor of the 60th Birthday of Bryce S. De Witt, pp. 327–344, (Adam Hilger, Bristol, 1984). (Cited on page 42.)

In the paper

“There exists an over-complete basis ∣j, g⟩ ∈ ℋj labeled by g ∈ SU(2) such that 108”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“The state sum as presented above matches the quantum amplitudes first proposed by Ponzano and Regge in the 1960s based on their discovery of the asymptotic expressions of the 6j symbols [309] and is often referred to as the Ponzano-Regge model.”

What the sources say

• Not checked yetcites [309] Ponzano, G. and Regge, T., “Semiclassical limit of Racah Coeficients”, in Bloch, F., Cohen, S.G., de Shalit, A., Sambursky, S. and Talmi, I., eds., Spectroscopy and Group Theoretical Methods in Physics: Racah Memorial Volume, pp. 1–58, (North-Holland, Amsterdam, 1968). (Cited on pages 43, 80, 92, and 95.)

In the paper

“There is no well-defined notion of kinematical state ∣[qab]〉 as the notion of kinematical Hilbert space in standard metric variables has never been defined.”

In the paper

“The non-compactness of the gauge group leads to certain infrared divergences of transition amplitudes (infinite volume factors).”

In the paper

“As first emphasized in [182], the quadratic simplicity constraints (25) — and more precisely in their dual version presented below (154) — are equivalent in the discrete setting to the linear constraint on each face of a given tetrahedron 126”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

In the paper

“This subgroup will be shown, a posteriori, to be the internal gauge group of the gravitational phase space in connection variables (appearing in the classical canonical study of Section 3.2 and leading to the quantization of Section 4).”

In the paper

“The theory has no local excitations: all solutions of the equations of motion are locally related by gauge transformations.”

• Also stated in: § Spin-foam Quantization of BF Theory · plain

  “The gauge symmetries are so vast that all the solutions to the equations of motion are locally pure gauge. The theory has only global or topological degrees of freedom.”

What the sources say

• Not checked yetcites [39] Baez, J.C., “An Introduction to Spin Foam Models of Quantum Gravity and BF Theory”, in Gausterer, H., Grosse, H. and Pittner, L., eds., Geometry and Quantum Physics, Proceedings of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999, Lecture Notes in Physics, 543, pp. 25–94, (Springer, Berlin; New York, 2000). [gr-qc/9905087]. (Cited on pages 17, 37, and 85.) source

In the paper

“The relationship with general relativity stems from the fact that constraining the field in the action (76) — where is interpreted as the tetrad co-frame — produces the action of general relativity in four dimensions.”

In the paper

“Three dimensional BF theory and the spin-foam quantization presented above is intimately related to classical and quantum gravity in three dimensions (for a classic reference see [112]).”

What the sources say

• Not checked yetcites [112] Carlip, S., Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 1998). [Google Books]. (Cited on pages 43, 85, and 95.) source

In the paper

“However, this implies that Ŝ acts only on spin network nodes.”

In the paper

“The second possibility is to work in the sector where . We will show later that this corresponds to the famous Barret-Crane model [59] with the various limitations discussed in Section 10.”

What the sources say

• Not checked yetcites [59] Barrett, J.W. and Crane, L., “Relativistic spin networks and quantum gravity”, J. Math. Phys., 39, 3296–3302, (1998). [DOI]. (Cited on pages 19, 52, and 76.) source

In the paper

“We will show in Section 7.4 that the models constructed on the requirement of a suitable imposition of the linear constraints (126) satisfy the usual quadratic Plebanski constraints in the path-integral formulation (up to quantum corrections, which are neglected in the usual semiclassical limit).”

In the paper

“Thus, this possibility does not lead to an acceptable model for quantum gravity in 4D.”

In the paper

“Such choice of the rotation subgroup is the precise analog of the time gauge in the canonical analysis of Section 3.”

In the paper

“Moreover, the linear simplicity constraint above is the simplicial version of the canonical primary constraint (37).”

In the paper

“One is the discovery that smooth loop states naturally solve the scalar constraint operator [234, 328].”

What the sources say

• Not checked yetcites [234] Jacobson, T.A. and Smolin, L., “Nonperturbative quantum geometries”, Nucl. Phys. B, 299, 295–345, (1988). [DOI]. (Cited on page 33.) source
• Not checked yetcites [328] Rovelli, C. and Smolin, L., “Loop space representation of quantum general relativity”, Nucl. Phys. B, 331, 80–152, (1990). [DOI]. (Cited on page 33.) source

In the paper

“The partition function corresponds to the ‘volume’ of the space of flat connections on .”

In the paper

“In contrast to the Gauss and vector constraints, the scalar constraint does not have a simple geometrical meaning.”

In the paper

“In the special case G = SU(2) and d = 3, BF theory is (Riemannian) general relativity, where the field is given by the co-tetrad frames of general relativity in the first-order formalism.”

In the paper

“The state |j,n⟩ is a semiclassical state describing a vector in ℝ3 of length j and of direction n.”

In the paper

“We will see in Part IV that natural regularizations are available and that the model is well defined [67, 281, 186].”

What the sources say

• Not checked yetcites [67] Barrett, J.W. and Naish-Guzman, I., “The Ponzano-Regge model”, Class. Quantum Grav., 26, 155014, (2009). [DOI], [arXiv:0803.3319 [gr-qc]]. (Cited on pages 43 and 95.) source
• Not checked yetcites [281] Noui, K. and Perez, A., “Three dimensional loop quantum gravity: Physical scalar product and spin foam models”, Class. Quantum Grav., 22, 1739–1762, (2005). [DOI], [arXiv:gr-qc/0402110]. (Cited on pages 43, 78, and 95.) source
• Not checked yetcites [186] Freidel, L. and Louapre, D., “Diffeomorphisms and spin foam models”, Nucl. Phys. B, 662, 279–298, (2003). [DOI], [arXiv:gr-qc/0212001]. (Cited on pages 43, 92, 95, 99, and 100.) source

In the paper

“The previous commutation relations imply that the constraint algebra is not closed and cannot therefore be imposed as operator equations on the states summed over in the BF partition function in general.”

In the paper

“In addition, and very importantly, these states are intrinsically discrete (colored graphs on Σ), suggesting a possible solution to the functional measure problem, i.e., the possibility of constructing a notion of a Feynman ‘path integral’ in a combinatorial manner involving sums over spin-network world-sheet amplitudes.”

In the paper

“The EPRL model is obtained by restricting the representations appearing in the expression of the BF partition function, so that at each tetrahedron the linear constraints (186) are satisfied the strongest possible way that is compatible with the uncertainties relations stemming from (187).”

In the paper

“More recently, it has been shown that the constraint solution p = γj and k = j also follows naturally from a spinor formulation of the simplicity constraints [369, 156, 256].”

What the sources say

• Not checked yetcites [369] Wieland, W.M., “Twistorial phase space for complex Ashtekar variables”, Class. Quantum Grav., 29, 045007, (2012). [DOI], [arXiv:1107.5002 [gr-qc]]. (Cited on page 72.) source
• Not checked yetcites [156] Dupuis, M., Freidel, L., Livine, E.R. and Speziale, S., “Holomorphic Lorentzian Simplicity Constraints”, J. Math. Phys., 53, 032502, (2012). [DOI], [arXiv:1107.5274 [gr-qc]]. (Cited on pages 68 and 72.) source
• Not checked yetcites [256] Livine, E.R., Speziale, S. and Tambornino, J., “Twistor networks and covariant twisted geometries”, Phys. Rev. D, 85, 064002, (2012). [DOI], [arXiv:1108.0369 [gr-qc]]. (Cited on page 72.) source

In the paper

“As in the Riemannian case, it can be shown as in Section (7.4) that the quadratic simplicity constraints (154) are implied, in the discrete setting, by the linear constraint on each face 186”

In the paper

“Nevertheless, as shown in [333], one can also write down a criterion (equivalent to the Master-constraint one) in a manifestly Spin(4) invariant way.”

What the sources say

• Not checked yetcites [333] Rovelli, C. and Speziale, S., “Lorentz covariance of loop quantum gravity”, Phys. Rev. D, 83, 104029, (2011). [DOI], [arXiv:1012.1739 [gr-qc]]. (Cited on pages 53 and 71.) source

In the paper

“Freidel and Krasnov [182] introduced a set of new spin-foam models for four-dimensional gravity using the coherent state basis of the quantum tetrahedron of Livine and Speziale [254].”

What the sources say

• Not checked yetcites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source
• Not checked yetcites [254] Livine, E.R. and Speziale, S., “New spinfoam vertex for quantum gravity”, Phys. Rev. D, 76, 084028, (2007). [DOI], [arXiv:0705.0674 [gr-qc]]. (Cited on pages 21, 47, 65, and 74.) source

In the paper

“The Lorentzian version of the EPRL model can be obtained from the imposition of the linear simplicity constraints to this formal expression, as discussed in Section 8.2.”

In the paper

“In the FK approach, one simply accepts from the starting point that, due to their property of not defining the set that is closed under commutation relations, the Plebanski constraints are to be imposed semiclassically.”

In the paper

“One way to get it is to take the limit γ → ∞ of the linear simplicity constraint (126) at the classical level. In this way the constraint becomes 206 with which the constraint algebra closes. One can impose the constraints strongly in this case.”

In the paper

“As proposed in [49, 235] one can obtain a consistent modification of the EPRL model by replacing (167) by a genuine projector , graphically 168”

What the sources say

• Not checked yetcites [49] Bahr, B., Hellmann, F., Kaminski, W., Kisielowski, M. and Lewandowski, J., “Operator Spin Foam Models”, Class. Quantum Grav., 28, 105003, (2011). [DOI], [arXiv:1010.4787 [gr-qc]]. (Cited on pages 63, 64, and 99.) source
• Not checked yetcites [235] Kaminski, W., Kisielowski, M. and Lewandowski, J., “The EPRL intertwiners and corrected partition function”, Class. Quantum Grav., 27, 165020, (2010). [DOI], [arXiv:0912.0540 [gr-qc]]. (Cited on pages 63, 64, and 99.) source

In the paper

“Notice that this modification does not interfere with the results of the semiclassical limit (to leading order) as reviewed in Section 13.”

What the sources say

• Not checked yetcites [1] Alesci, E., Bianchi, E., Magliaro, E. and Perini, C., “Asymptotics of LQG fusion coefficients”, Class. Quantum Grav., 27, 095016, (2010). [DOI], [arXiv:0809.3718 [gr-qc]]. (Cited on page 65.) source

In the paper

“As we show below, this last set of constraints follow from the Spin(4) gauge invariance of the EPRL node (i.e., the validity of the Gauss law) plus the validity of the tetrahedra constraints (156).”

In the paper

“The previous equation will be of great importance in the graphical calculus that will allow us to show that the linear constraint imposed here at the level of states implies the vanishing of the quadratic Plebanski constraints (25) and their fluctuations, computed in the path integral sense, in the appropriate large spin semiclassical limit.”

In the paper

“This shows that the EPRL model can be considered as a lattice definition of quantum-gravity theory.”

In the paper

“The fact that, for the new model, the double scaling limit γ → 0 and j → ∞ with γj = constant defines the appropriate regime, where the fluctuation behaves as in Regge gravity (in the leading order) has been further clarified in [261].”

What the sources say

• Not checked yetcites [261] Magliaro, E. and Perini, C., “Regge gravity from spinfoams”, arXiv, e-print, (2011). [arXiv:1105.0216]. (Cited on page 83.)

In the paper

“In this way, the boundary Hilbert space associated with ∂Δ* is isomorphic (if one used the natural Ashtekar-Lewandowski measure) with the Hilbert space of LQG truncated to that fixed graph.”

In the paper

“One can see this freedom of representation as an analogy of the gauge freedom in the spacetime representation in the classical theory.”

In the paper

“In this way we arrive at a fully combinatorial definition of the standard by reducing the infinite degrees of freedom of the kinematical phase space to finitely many by the action of the generalized projection operator P.”

In the paper

“However, in the path integral formulation, the presence of constraints has the additional effect of modifying the weights with which those histories are to be summed: second-class constraints modify the path-integral measure (in the spin-foam context this issue was raised in [90]).”

What the sources say

• Not checked yetcites [90] Bojowald, M. and Perez, A., “Spin foam quantization and anomalies”, Gen. Relativ. Gravit., 42, 877–907, (2010). [DOI], [arXiv:gr-qc/0303026]. (Cited on pages 63, 78, and 99.) source

In the paper

“According to the construction of the model (Sections 7 and 8), boundary states are in one-to-one correspondence with SU(2) spin networks.”

In the paper

“we now present the results on the large-spin asymptotics of the spin-foam amplitudes suggesting that, on a fixed discretization, the semiclassical limit of the EPRL-FK models is given by Regge’s discrete formulation of general relativity [61, 64].”

What the sources say

• Not checked yetcites [61] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “Asymptotic analysis of the EPRL four-simplex amplitude”, J. Math. Phys., 50, 112504, (2009). [DOI], [arXiv:0902.1170 [gr-qc]]. (Cited on pages 67, 80, 81, and 82.) source
• Not checked yetcites [64] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Hellmann, F. and Pereira, R., “Lorentzian spin foam amplitudes: graphical calculus and asymptotics”, Class. Quantum Grav., 27, 165009, (2010). [DOI], [arXiv:0907.2440 [gr-qc]]. (Cited on pages 80 and 82.) source

In the paper

“Roughly speaking, the degrees of freedom of e are now encoded in the representation being summed over in (244).”