The Spin-Foam Approach to Quantum Gravity

In the paper

“The reduced phase space is four dimensional and therefore the resulting number of physical degrees of freedom is two, as expected.”

In the paper

“For compact G it is easy to prove using the invariance (and normalization) of the integration measure (84) that is the projector onto . In this way the spin-foam amplitudes of SO(4) BF theory reduce to 88”

In the paper

“The first is to take γ = ±1, in which case the constraint (186) can be imposed strongly. More precisely, for the plus sign it reduces to the condition (or equivalently for the other sign ), which amounts to setting all the left quantum numbers in the BF state-sum for the group Spin(4) to zero.”

In the paper

“SU(2) coherent states have the usual semiclassical properties. Indeed, if one considers the operators Ji generators of su(2) one has 112 where ni is the corresponding three-dimensional unit vector for n ∈ S2. The fluctuations of are also minimal with ΔJ2 = ℏ2j, where we have restored ℏ for clarity.”

• Also stated in: § The coherent-states representation · plain

  “The fluctuations go to zero in the limit ℏ → 0 and j → ∞ while ℏj is kept constant.”

In the paper

“Unlike the Riemannian model, all values of spin j are allowed, and no restrictions on the Immirzi parameter arise.”

In the paper

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

In the paper

“Surprisingly, the conditions p = γj and k = j minimizing the Master constraint, solve its Lorentz invariant version strongly! This resolves the cylindrical consistency issue pointed out above.”

In the paper

“Direct evaluation shows that the previous diagram is proportional to , which vanishes in the semiclassical limit ℏ → 0, j → ∞ with ℏj = constant. We use the notation already adopted in (140) and call such a quantity . This concludes the proof that the triangle Plebanski constraints are satisfied in the semiclassical sense.”

In the paper

“Case γ > 1: In this case, γ ∈ ℕ would allow any j ∈ ℕ/2. Rational choices of γ have the same effect as in the previous case.”

In the paper

“Case γ < 1: If one wants the spin j ∈ ℕ/2 to be arbitrary, then the only possibilities are γ = 0 or γ = 1. Rational γ values would restrict the spins j to a subclass in ℕ/2.”

In the paper

“Case γ > 1: In this case, the minimum eigenvalue is obtained for 134 for which mj = ℏ2(γ2 − 1)j, and thus vanishes in the semiclassical limit ℏ → 0 and k → ∞ with ℏk = constant.”

In the paper

“Notice however, that the Master constraint is only invariant under the subgroup SU(2) ⊂ Spin(4) chosen at each tetrahedron. Its lack of Lorentz invariance is clear from the fact that its validity is justified from the canonical analysis in the time gauge.”

In the paper

“The solution is obtained by setting jf = 0 in all faces at the level of the amplitude (148). An immediate consequence of this is that and all tetrahedra are assigned a unique intertwiner: the Barrett-Crane intertwiner.”

In the paper

“Case γ < 1: In this case the minimum eigenvalue is obtained for 133 i.e., one has where mj = ℏ2(1 − γ2)j, where we have restored the explicit dependence on ℏ so that it is apparent that the selected eigenvalue vanishes in the semiclassical limit ℏ → 0 and k → ∞ with ℏk = constant.”

In the paper

“However, the representations j+ = j− + j are of maximum weight. Therefore, we can insert a coherent states resolution of the identity on the right representations and get: 177”

In the paper

“The vertex amplitude is divergent due to the presence of a redundant integration over SL(2, ℂ), it becomes finite by dropping an arbitrary integration, i.e., removing any of the 5 boxes in the vertex expression [167].”

What the sources say

• Not checked yetcites [167] Engle, J. and Pereira, R., “Regularization and finiteness of the Lorentzian loop quantum gravity vertices”, Phys. Rev. D, 79, 084034, (2009). [DOI], [arXiv:0805.4696]. (Cited on pages 68 and 73.) source

In the paper

“Moreover, by construction the coherent states participating in the previous amplitude satisfy the linear constraints (127) in expectation values, namely Thus the coherent states participating in the above representation of the EPRL amplitudes solve the linear simplicity constraints in the usual semiclassical sense.”

In the paper

“The number of quantum numbers necessary to label the basis element is 6g − 6, corresponding to the dimension of the moduli space of SU(2) flat connections on a Riemann surface of genus g. This is the number of degrees of freedom of the classical theory.”

In the paper

“The limit ϵ → 0 exists and one can give a closed expression for the physical inner product.”

In the paper

“In two dimensions the action of two neighboring plaquette-sums on the vacuum is equivalent to the action of a single larger plaquette action obtained from the fusion of the original ones. This implies the trivial scaling of the physical inner product under refinement of the regulator and the existence of a well-defined limit ϵ → 0.”

In the paper

“More generally, for arbitrary spin networks, the vertex amplitude corresponds to 3nj-symbols, and 〈s, s′〉p takes the same general form.”

In the paper

“Notice that the previous equation corresponds to the discrete version of Eq. (240).”

In the paper

“Notice that the assignment of intertwiners to edges is a consequence of the integration over the connection. This is not a particularity of this example, but rather a general property of local spin foams.”

In the paper

“One can also show that it is finite,28 and satisfies all the properties of an inner product [107].”

What the sources say

• Not checked yetcites [107] Buffenoir, E. and Noui, K., “Unfashionable observations about three dimensional gravity”, arXiv, e-print, (2003). [arXiv:gr-qc/0305079]. (Cited on page 87.)

In the paper

“The spin-foam amplitudes are purely combinatorial and can be explicitly computed from the simple action of the loop operator in .”

In the paper

“implies the asymptotic formula for the vertex amplitude to be given by the unbalanced square of the above formula [62], namely One can write the previous expression as 217”

• Also stated in: § Vertex amplitudes asymptotics · boosted

  “Remarkably, the above asymptotic formula is also valid for the case of γ > 1 [61].”

What the sources say

• Not checked yetcites [62] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “A Summary of the Asymptotic Analysis for the EPRL Amplitude”, in Kowalski-Glikman, J., Durka, R. and Szczachor, M., eds., The Planck Scale: Proceedings of the XXV Max Born Symposium, Wroclaw, Poland, 29 June–3 July 2009, AIP Conference Proceedings, 1196, pp. 36–43, (American Institute of Physics, Melville, NY, 2009). [DOI], [arXiv:0909.1882 [gr-qc]]. (Cited on pages 80 and 82.) source
• 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

“When Δ is a simplicial complex, all the edges in are 3-valent and vertices are 4-valent (one such vertex is emphasized in Figure 5, the intersection with the surrounding S2 is shown in dotted lines). Consequently, the vertex amplitude is given by the contraction of the corresponding four 3-valent intertwiners, i.e., a 6j-symbol. In that case, the partition function takes the familiar Ponzano-Regge”

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

“The sum over spins in (247) is typically divergent, as indicated by the previous equation. Divergences occur due to infinite volume factors corresponding to the topological gauge freedom (225) (see [186]).30 The factor τ in (248) represents such a volume factor. It can also be interpreted as a δ(0) coming from the existence of a redundant delta function in (242).”

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

“One can partially gauge fix this freedom at the level of the discretization. This has the effect of eliminating bubbles from the 2-complex.”

In the paper

“These conditions generally exist in spin-foam models and are used to regularize the vertex amplitudes of the Lorentzian models (we have illustrated this at the end of Section 8.3; for a general treatment 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 Lorentzian sector (internal gauge group SL(2, ℂ)), the internal gauge orbits have infinite volume and the lattice path integral would diverge without an appropriate gauge fixing condition.”

In the paper

“The transition amplitudes imply the skein relations that define the physical Hilbert space! The spin-foam quantization is equivalent to the canonical one.”

In the paper

“If this gauge group is compact, then anomaly free measures are defined using appropriate variables and invariant measures. In this case, gauge fixing is not necessary for the amplitudes to be well defined.”

In the paper

“However, a close look at the physical inner product defined by P permits one to conclude that the physical Hilbert space is one dimensional — the classical theory has zero degrees of freedom and so there is no non-trivial Dirac observable in the quantum theory.”

In the paper

“We can construct the physical Hilbert space by considering equivalence classes under states with zero transition amplitude with all the elements of , i.e., null states.”

In the paper

“Transition amplitudes can be defined along similar lines using a manifold with boundaries. Given Δ, then defines graphs on the boundaries. Consequently, spin foams induce spin networks on the boundaries.”

In the paper

“In the case of simply connected Σ, the gauge fixing is complete. One can eliminate bubbles and compute finite transition amplitudes.”

In the paper

“However, it is not yet clear if there is a sense in which these models correspond to a physical theory.”

In the paper

“Not much is known about gravity models in four dimensions at this stage.”

In the paper

“In fact, according to our tentative definition, an anomaly-free measure can be multiplied by any gauge invariant function and yield a new anomaly free measure. This kind of ambiguity is not wanted; however, it is, in fact, present in most of the spin-foam models defined so far”

In the paper

“In the context of LQG, a systematic first-principles analysis of such properties of the low-energy regime remains open.”

In the paper

“Since the theory has no local excitations (no gravitons), the result is independent of the chosen cellular decomposition. A single discretization suffices to capture the degrees of freedom of the topological theory.”

In the paper

“This cutoff is reflected by the fact that only a restrictive set of spin foams (spin network histories) is allowed in the path integral: those that can be obtained by all possible coloring of the underlying 2-complex. In addition, it restricts the number of possible 3-geometry states (spin network states) on the boundary by fixing a finite underlying boundary graph. This represents a truncation in”

In the paper

“However, transition amplitudes are discretization dependent now. A consistent definition of the path integral using spin foams should include a prescription to eliminate this discretization dependence.”

In the paper

“Loop quantum gravity (LQG) is a framework to address this task. The degrees of freedom of gravity are quantized in accordance to the principles of general covariance. At the present stage there are some indications that both the singularity problems of classical general relativity as well as the UV problem of standard QFTs might vanish in the framework.”

In the paper

“quantum field theory is plagued by singularities too (UV divergences) coming from the contribution of arbitrary high-energy quantum processes. This limitation of standard QFTs is expected to disappear once the quantum fluctuations of the gravitational field, involving the dynamic treatment of spacetime geometry, are appropriately taken into account.”

In the paper

“standard QFTs are not generally covariant, as they can only be defined if a non-dynamical spacetime geometry is provided”

• Also stated in: § Quantum Gravity: A Unifying Framework for Fundamental Physics · plain

  “the standard QFTs used to describe other fundamental forces are not appropriate to tackle the problem of quantum gravity.”

In the paper

“In 3 dimensions the radial integration can be done and the corresponding amplitudes coincide with the results of Section 14.5.”

In the paper

“The simple dimensional argument given for Yang-Mills theory is no longer valid in the case of gravity, and a full dynamical investigation of their effect in physical predictions must be investigated.”

In the paper

“Concrete quantizations producing well-defined operators are available: no UV divergences are encountered by these proposals; the fundamental discreteness of quantum geometry in LQG plays the role of physical regulator at the Planck scale [354].”

• Also stated in: § Loop quantum gravity · plain

  “make the issue of whether any of the proposed quantizations of the constraints are anomaly-free a difficult open question.”

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 basic idea behind the definition was that of the quantum tetrahedron introduced by Barbieri in [58] and generalized to 4D in [40].”

What the sources say

• Not checked yetcites [58] Barbieri, A., “Quantum tetrahedra and simplicial spin networks”, Nucl. Phys. B, 518, 714–728, (1998). [DOI], [arXiv:gr-qc/9707010]. (Cited on page 19.) source
• Not checked yetcites [40] Baez, J.C. and Barrett, J.W., “The Quantum Tetrahedron in 3 and 4 Dimensions”, Adv. Theor. Math. Phys., 3, 815–850, (1999). (Cited on page 19.) source

In the paper

“Unfortunately, the integration over the radial variables r involves products of Bessel functions and its behavior is not analyzed in detail.”

In the paper

“The non-renormalizability of perturbative gravity is often explained through an analogy with the (non-renormalizable) Fermi’s four-fermion effective description of the weak interaction.”

In the paper

“The theory is explicitly formulated in a background-independent, and therefore, non-perturbative fashion. The theory is based on the Hamiltonian (or canonical) quantization of general relativity in terms of variables that are different from the standard metric variables.”

In the paper

“Is this a complete answer to the non-renormalizability problem of gravity mentioned above? Unfortunately, the answer to this question is in the negative at this stage. The reason is that the true answer to the question of renormalizabity or not renormalizability is not in the presence of divergences but in the degree of intrinsic ambiguity of the quantization recipe applied to a field theory.”

In the paper

“the previous separation of degrees of freedom has no intrinsic meaning in general relativity”

• Also stated in: § Why non-perturbative quantum gravity? · plain

  “the splitting in Eq. (1) is inconsistent when considering general states with their arbitrary quantum excitations at all scales.”

In the paper

“The properties of the kernel of have not been studied in detail.”

In the paper

“The definition of the physical inner product is still an open issue.”

In the paper

“In four dimensions, the spin-foam approach is a program in progress, where various issues still remain open.”

In the paper

“Gambini and Pullin [193] introduced a very simple model obtained by modification of the BF theory skein relations.”

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

“Similar ambiguities are present in LQG; they appear in the regularization of the constraints in the canonical formulation [303]. These ambiguities are also present in the spin-foam formulation, which is the central subject of this review.”

What the sources say

• Not checked yetcites [303] Perez, A., “On the regularization ambiguities in loop quantum gravity”, Phys. Rev. D, 73, 044007, (2006). [DOI], [arXiv:gr-qc/0509118]. (Cited on page 12.) source

In the paper

“The definition of this spin-foam model is not explicit since the theory is directly defined by the physical skein relations.”

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

“The main prediction of LQG is the discreteness [344] of the spectrum of geometrical operators such as area and volume.”

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.)

In the paper

“Iwasaki defines a spin-foam model of self-dual Riemannian gravity9 by a direct lattice discretization of the continuous Ashtekar formulation of general relativity.”

In the paper

“In generally-covariant systems, conventional time evolution is pure gauge”

• Also stated in: § The path integral for generally covariant systems · plain

  “There is no meaningful notion of transition amplitude between states at different times t1 > t0 or equivalently a notion of “unitary time evolution” represented by an operator U(t1 − t0).”

In the paper

“Gambini and Pullin define a model by modifying these amplitudes so that some of the skein relations are now forbidden. This simple modification frees local excitations of a field theory.”

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

“In a non-renormalizable theory (e.g., perturbative quantum gravity) the situation is similar except for the fact that there are infinitely many parameters to be fixed by renormalization conditions. As the latter must be specified by observations, a non-renormalizable theory has very limited predictive power.”

In the paper

“Nevertheless, a path integral representation of generally covariant systems exists and arises as a tool for implementing the constraints in the quantum theory as we argue below.”

In the paper

“Iwasaki writes down an expression for the spin-foam amplitudes in which the integration over the connection and the e’s can be computed explicitly.”

In the paper

“Some indirect evidence of a possible non-trivial continuum limit has been obtained in some versions of the model in 1+1 dimensions.”

In the paper

“These rules generate surface-like excitations of the same kind we encounter in the more standard spin-foam model, but endow the foam with a notion of causality.”

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

“At the Planck scale (ℓp) the quantum fluctuations of the gravitational field become so important that there is no way (not even in principle6) to make observations without affecting the gravitational field. The notion of test fields and test observer is lost and with it the usual spacetime representation.”

In the paper

“However, there is a difficulty with this procedure: the algebra of operators do not close so that imposing the constraints sharply becomes a too strong condition on the BF configurations.”

In the paper

“In loop quantum gravity there is strong indications that the mathematical framework of the theory provides a regularization of divergences. It remains to settle the crucial issue of how to fix the associated ambiguities.”

In the paper

“Their formulation can be used to describe various theories of interest such as BF theories with cosmological terms, Yang-Mills theories (in 2 dimensions) and Riemannian self-dual gravity.”

In the paper

“To conclude this section, let us point out that the real connection formulation of general relativity presented here is a peculiar property of four dimensions due to the special property (45).”

In the paper

“Indeed, these models are arguably the same as the EPRL model for a suitable range of the Immirzi parameter (the vertex amplitudes coincide for γ < 1 as we will discuss in the sequel).”

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 coherent intertwiner basis relevant for the definition of spin foams and the 4D quantum gravity amplitude was also derived by Livine and Speziale in [254].”

What the sources say

• 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

“However, the fundamental difference is the absence of any non-dynamical background field in the theory.”

In the paper

“LQG is defined using Dirac quantization.”

In the paper

“This is exactly the linear simplicity constraints that we will rediscover in the simplicial setting of Sections 7 and 8, and which will be imposed in a suitable sense at all tetrahedra of the cellular decomposition in order to satisfy the Plebanski constraints and produce a state sum model for gravity out of that of BF theory.”

In the paper

“LQG is a proposal for the implementation of the quantization program established in the 1960s by Dirac, Wheeler, and DeWitt, among others (for recent reviews see [35, 356, 323]).”

What the sources say

• Not checked yetcites [35] Ashtekar, A. and Lewandowski, J., “Background independent quantum gravity: A status report”, Class. Quantum Grav., 21, R53–R152, (2004). [DOI], [arXiv:gr-qc/0404018]. (Cited on pages 12, 14, 22, 28, and 29.) source
• Not checked yetcites [356] Thiemann, T., “Introduction to Modern Canonical Quantum General Relativity”, arXiv, e-print, (2001). [arXiv:gr-qc/0110034]. (Cited on pages 23 and 29.)
• Not checked yetcites [323] Rovelli, C., “Loop Quantum Gravity”, Living Rev. Relativity, 11, lrr-2008-5 (2008). URL (accessed 12 September 2011): http://www.livingreviews.org/lrr-2008-5. (Cited on page 29.)

In the paper

“They are the most developed candidates in four dimensions and we will review their properties in detail in the following part.”

In the paper

“A modification of the Barrett-Crane model was recently introduced in [168, 169] and extended for arbitrary Immirzi parameter in [164].”

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
• Not checked yetcites [164] Engle, J., Livine, E.R., Pereira, R. and Rovelli, C., “LQG vertex with finite Immirzi parameter”, Nucl. Phys. B, 799, 136–149, (2008). [DOI], [arXiv:0711.0146 [gr-qc]]. (Cited on pages 20, 36, 49, 58, and 68.) source

In the paper

“As in the three-dimensional case, BF theory in any dimensions can be coupled to topological defects [41].”

What the sources say

• Not checked yetcites [41] Baez, J.C. and Perez, A., “Quantization of strings and branes coupled to BF theory”, Adv. Theor. Math. Phys., 11, 3, (2007). [arXiv:gr-qc/0605087]. (Cited on page 46.)

In the paper

“It is also possible to introduce one-dimensional particles in four-dimensional BF theory and gravity as shown in [180].”

What the sources say

• Not checked yetcites [180] Freidel, L., Kowalski-Glikman, J. and Starodubtsev, A., “Particles as Wilson lines of gravitational field”, Phys. Rev. D, 74, 084002, (2006). [DOI], [arXiv:gr-qc/0607014]. (Cited on page 46.) source

In the paper

“The spin-foam approach is an attempt to define the path integral quantization of gravity using what we have learned from LQG.”

In the paper

“on the other hand, it is related to usual 2D gravity via constraints in a way similar to the one exploited in four dimensions (see Section 7).”

What the sources say

• Not checked yetcites [295] Oriti, D., Rovelli, C. and Speziale, S., “Spinfoam 2d quantum gravity and discrete bundles”, Class. Quantum Grav., 22, 85–108, (2005). [DOI], [arXiv:gr-qc/0406063]. (Cited on page 42.) source

In the paper

“The quantum deformed version of the above amplitudes lead to the Turaev-Viro model [364], which is expected to correspond to the quantization of three-dimensional Riemannian gravity in the presence of a non-vanishing positive cosmological constant.”

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

“The notion of a quantum state of 3-geometry is rigorously defined in terms of spin-network states.”

In the paper

“using this graphical notation one can easily prove the discretization independence of the BF amplitudes.”

• Also stated in: § Special cases · plain

  “if an appropriate regularization of bubble divergences is provided, (86) is topologically invariant and the spin-foam path integral is discretization independent.”

What the sources say

• Not checked yetcites [286] Oeckl, R., Discrete gauge theory: From lattices to TQFT, (Imperial College Press, London, UK, 2005). [Google Books]. (Cited on page 40.) source
• Not checked yetcites [291] Oeckl, R. and Pfeiffer, H., “The dual of pure non-Abelian lattice gauge theory as a spin foam model”, Nucl. Phys. B, 598, 400–426, (2001). [DOI], [arXiv:hep-th/0008095]. (Cited on pages 40 and 103.) 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

“Due to the fact that the constraints do not form a closed (first class) algebra in the generic case, one needs to devise a weaker sense in which they are to be imposed.”

In the paper

“In the four-dimensional case, defects are string-like [175] and can carry extra degrees of freedom such as topological Yang-Mills fields [275].”

What the sources say

• Not checked yetcites [175] Fairbairn, W.J. and Perez, A., “Extended matter coupled to BF theory”, Phys. Rev. D, 78, 024013, (2008). [DOI], [arXiv:0709.4235 [gr-qc]]. (Cited on page 46.) source
• Not checked yetcites [275] Montesinos, M. and Perez, A., “Two-dimensional topological field theories coupled to four-dimensional BF theory”, Phys. Rev. D, 77, 104020, (2008). [DOI], [arXiv:0711.2875 [gr-qc]]. (Cited on page 46.) source

In the paper

“Regularization choices have to be made and the result is not unique.”

What the sources say

• Not checked yetcites [352] Thiemann, T., “Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity”, Phys. Lett. B, 380, 257–264, (1996). [DOI]. (Cited on page 32.) source
• Not checked yetcites [201] Gaul, M. and Rovelli, C., “A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements”, Class. Quantum Grav., 18, 1593–1624, (2001). [DOI]. (Cited on page 32.) source
• Not checked yetcites [135] Di Bartolo, C., Gambini, R., Griego, J. and Pullin, J., “Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra”, Class. Quantum Grav., 17, 3239–3264, (2000). (Cited on pages 32 and 33.) source
• Not checked yetcites [136] Di Bartolo, C., Gambini, R., Griego, J. and Pullin, J., “Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants”, Phys. Rev. Lett., 84, 2314–2317, (2000). [DOI]. (Cited on pages 32 and 33.) source

In the paper

“Unitary irreducible representations of SL(2, ℂ) are infinite dimensional and are labelled by a positive real number p ∈ ℝ+ and a half-integer k ∈ ℕ/2.”

What the sources say

• Not checked yetcites [338] Ruhl, W., The Lorentz Group and Harmonic Analysis, (W.A. Benjamin, New York, 1970). (Cited on pages 69 and 72.)
• Not checked yetcites [203] Gel’fand, I.M., Graev, M.I. and Vilenkin, N.Y., Generalized Functions, Vol. 5: Integral geometry and representation theory, (Academic Press, New York; London, 1966). (Cited on pages 69 and 72.)
• Not checked yetcites [204] Gel’fand, I.M., Minlos, R.A. and Shapiro, Z.Y., Representations of the rotation and Lorentz groups and their applications, (Pergamon Press, London; MacMillan, New York, New York, 1963). Translated from the Russian edition (Moscow, 1958). (Cited on page 69.)

In the paper

“The amplitude (167) fails this requirement due to .”

In the paper

“Strong imposition of the constraints would amount to looking for the kernel of the Master constraint Mf. However, generically the positive operator associated with the Master constraint does not contain the zero eigenvalue in the spectrum due to the open nature of the constraint algebra (187).”

In the paper

“They also emphasize how the model [168] corresponds to the sector γ = ∞, which has been shown to be topological [246].”

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 [246] Liu, L., Montesinos, M. and Perez, A., “A topological limit of gravity admitting an SU(2) connection formulation”, Phys. Rev. D, 81, 064033, (2010). [DOI], [arXiv:0906.4524 [gr-qc]]. (Cited on page 74.) source

In the paper

“As we will see in what follows the best choice from the point of view of the Master-constraint criterion is k = j, from which it follows that 189”

In the paper

“However, the advantage of the Master-constraint criterion is clear from the point of view of cylindrical consistency as the choice p = γ(j + 1) obtained in the previous subsection would imply that j = 0 spin-network links would be have a non-trivial amplitude in the path integral defined in Section 7.3. Such thing would be in sharp conflict with diffeomorphism invariance in the model.”

In the paper

“The first one is to take γ = ±i, in which case the constraint (186) could, in principle, be imposed strongly. However, the self-dual sector is not present in .The second possibility is to work in the sector where . Where, as in the Riemannian case, it corresponds to the Barret-Crane model [60] with the various limitations discussed in Section 10.”

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

In the paper

“However, these restrictions seem so far of little physical importance as they are a special feature of the Riemannian model that is absent in its Lorentzian relative.”

In the paper

“As pointed out before, this question has not been completely settled yet in the spin-foam community.”

In the paper

“A rigorous quantization is achieved if the regulator can be removed without the appearance of infinities, and if the number of ambiguities appearing in this process is under control. We shall see that all this can be done in the simple toy example in this section.”

In the paper

“The GFT formulation resolves by definition the two fundamental conceptual problems of the spin-foam approach: diffeomorphism gauge symmetry and discretization dependence.”

In the paper

“In this limit the quantum geometry approximates the classical one when tested with suitable states (e.g., coherent states). However, the geometry remains discrete during this limiting process, as the limit is taken on a fixed regulating cellular structure.”

In the paper

“Due to their similarity to γ < 1, the same relationship between boundary data and elements of the kinematical Hilbert space hold for the FK model. However, such a simple relationship does not hold for the model in the case of γ > 1.”

In the paper

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

In the paper

“Riemannian gravity in 3 dimensions is a theory with no local degrees of freedom, i.e., a topological theory.”

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

“Recursion relations for the 6j vertex amplitudes have been investigated in [98, 157]. They provide a tool for studying dynamics in spin foams of 3D gravity and might be useful in higher dimensions [99].”

What the sources say

• Not checked yetcites [98] Bonzom, V. and Livine, E.R., “A New Recursion Relation for the 6j-Symbol”, Ann. Henri Poincare, 13, 1083–1099, (2012). [DOI], [arXiv:1103.3415]. (Cited on pages 44 and 95.) source
• Not checked yetcites [157] Dupuis, M. and Livine, E.R., “The 6j-symbol: Recursion, C. and Asymptotics”, Class. Quantum Grav., 27, 135003, (2010). [DOI], [arXiv:0910.2425]. (Cited on pages 44 and 95.) source

In the paper

“Spin foams represent a gauge history of a kinematical state. A sum over gauge histories is what defines P as a means for extracting the true degrees of freedom from those that are encoded in the kinematical boundary states.”

In the paper

“It is well known that, in addition to the invariance of the measure, one must provide appropriate gauge-fixing conditions for the amplitudes to be well defined.”

In the paper

“Heuristically, the presence of the cosmological constant introduces a physical infrared cutoff.”

In the paper

“Eq. (248) has been proven in this case for the case of simplicial decompositions in [364]; see also [363, 238]. The generalization for arbitrary cellular decomposition was obtained in [206].”

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
• 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

“This means that, in general, (even when local excitations are present, as in 4D) the spacetime geometric interpretation of the spin-foam configurations is subtle. This is an important point that is often overlooked in the literature: one cannot interpret the spin-foam sum of Eq. (238) as a sum over geometries in any obvious way.”

In the paper

“This means that the sum over spin foams in Eq. (238) is nothing but a sum over pure gauge degrees of freedom and hence no physical interpretation can be associated to it.”

In the paper

“This factorization is well defined, since the number of equivalent spin foams can be characterized in a fully combinatorial manner, and is finite for any spin foam defined on a finite discretization.”

In the paper

“The discretization can be interpreted, in this way, as a gauge fixing of 3-diffeomorphisms on the boundary. This gauge fixing is partial in the sense that, generically, there will remain a discrete symmetry remnant given by the discrete symmetries of the spin network. This remaining symmetry has to be factored out when computing transition amplitudes”

In the paper

“The underlying 2-complex on which spin foams are defined represents a partial gauge fixing for the configurations (spin foams) entering in the path integral. The remaining symmetry, to be factored out in the computation of transition amplitudes, corresponds simply to the finite group of discrete symmetries of the corresponding spin foams.”

In the paper

“Moreover, no Euclidean formulation seems meaningful in a background-independent scenario.”

In the paper

“Fundamental excitations are intrinsically discrete.”

In the paper

“Finally, discretization independence is not only a problem of spin foams but also of Regge gravity.”

In the paper

“The difficulties are shifted to the question of the physical role of Δ and the convergence of the corresponding perturbative series.”

In the paper

“From this perspective (and leaving aside the issue of convergence) the sum of different order amplitudes corresponding to equivalent spin foams should be interpreted as the definition of the physical amplitude of that particular spin foam.”

In the paper

“This is not necessarily a definite obstruction as divergent series can often be given an asymptotic meaning and provide physical information.”

In the paper

“This makes them, in principle, negligible at low energies so that the potential quantum-gravity effects would be observed as corrections to the standard QFT well-tested physics.”

In the paper

“This is mainly due to the presently-unsolved difficulties associated with a quantitative dynamical description of physical states.”

In the paper

“New means to eliminate the truncation introduced by the lattice need to be developed.”

In the paper

“The lattice (triangulation) contains only topological information and there is no geometrical meaning associated to its components.”

In the paper

“Even when the regulator (or the discretization dependence) eventually has to be removed (see next Section 17), the theory is presumed to remain discrete at the fundamental level. The smooth-manifold diffeomorphism-invariant description is expected to emerge in the (low energy) continuum limit.”

In the paper

“For example, such a possibility has been considered in [200], where it is shown that the large LIV effects can be avoided in the context of Euclidean field theory (the protecting mechanism being here the trading of the non-compact Poincaré group by the Euclidean group).”

What the sources say

• Not checked yetcites [200] Gambini, R., Rastgoo, S. and Pullin, J., “Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects?”, Class. Quantum Grav., 28, 155005, (2011). [DOI], [arXiv:1106.1417 [gr-qc]]. (Cited on pages 101 and 105.) source

In the paper

“An example where this procedure is well defined is Zapata’s treatment of the Turaev-Viro model [372]. The key point in this case is that amplitudes are independent of the discretization (due to the topological character of the theory), so that the refinement limit is trivial.”

What the sources say

• Not checked yetcites [372] Zapata, J.A., “Spin foam model for 3D gravity in the continuum”, in García-Compeán, H., Mielnik, B., Montesinos, M. and Przanowski, M., eds., Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebański, Proceedings of 2002 International Conference, Cinvestav, Mexico City, 17–20 September 2002, pp. 495–502, (World Scientific, Singapore; Hackensack, NJ, 2006). [DOI]. (Cited on pages 93, 94, and 103.) source

In the paper

“However, the nature of this regulator is fundamentally different from the standard concept in the background-independent framework: since geometry is encoded in the coloring (that can take any spin values) the configurations involve fluctuations all the way to Plank scale.”

In the paper

“A prescription that avoids this last issue is the radical proposal of Freidel [179], where the tree-level GFT amplitudes are used to define the physical inner product of quantum gravity.”

What the sources say

• Not checked yetcites [179] Freidel, L., “Group field theory: An overview”, Int. J. Theor. Phys., 44, 1769–1783, (2005). [DOI], [arXiv:hep-th/0505016]. (Cited on page 103.) source

In the paper

“From this viewpoint, the precise meaning of the gauge symmetries of such a theory would have to be formulated directly at the discrete level. We have seen that this can be achieved in the case of 3-dimensional gravity (recall Section 14.5.1).”

In the paper

“The discussion of Section 16 does not apply in the GFT formulation, i.e., there is no need for gauge fixing.”

In the paper

“The GFT formulation could resolve by definition the two fundamental conceptual problems of the spin-foam approach: diffeomorphism gauge symmetry and discretization dependence.”

In the paper

“In the absence of detailed formulations, certain early pioneering model calculations [192] (see also general treatment [340, 341]) indicate the possibility for certain Lorentz-invariance-violation (LIV) effects associated to a quantum-gravity granularity of spacetime measured at rest by a preferred frame of observers.”

What the sources say

• Not checked yetcites [192] Gambini, R. and Pullin, J., “Nonstandard optics from quantum space-time”, Phys. Rev. D, 59, 124021, (1999). [DOI], [arXiv:gr-qc/9809038 [gr-qc]]. (Cited on page 105.) source
• Not checked yetcites [340] Sahlmann, H. and Thiemann, T., “Towards the QFT on curved space-time limit of QGR: I. A general scheme”, Class. Quantum Grav., 23, 867–908, (2006). [DOI], [arXiv:gr-qc/0207030 [gr-qc]]. (Cited on page 105.) source
• Not checked yetcites [341] Sahlmann, H. and Thiemann, T., “Towards the QFT on curved space-time limit of QGR: II. A concrete implementation”, Class. Quantum Grav., 23, 909–954, (2006). [DOI], [arXiv:gr-qc/0207031 [gr-qc]]. (Cited on page 105.) source

In the paper

“This would mean that the way the fundamental Planckian discreteness of LQG must enter physics is through reference-frame-independent phenomena (as mass enters a Klein-Gordon theory).”

In the paper

“on the other, at the level of gauge-invariant observables, some discrete aspects could disappear [154]”

What the sources say

• Not checked yetcites [154] Dittrich, B. and Thiemann, T., “Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?”, J. Math. Phys., 50, 012503, (2009). [DOI], [arXiv:0708.1721 [gr-qc]]. (Cited on page 106.) source