A Summary of the Asymptotic Analysis for the EPRL Amplitude

Barrett, John W.; Dowdall, Richard J.; Fairbairn, Winston J.; Gomes, Henrique; Hellmann, Frank; Kowalski-Glikman, Jerzy; Durka, Remigiusz; Szczachor, M.

doi:10.1063/1.3284398

Cited by 86 publications

(306 citation statements)

References 23 publications

(45 reference statements)

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“…In addition, the 6j-symbol has been taken as a test case for asymptotic studies of amplitudes that occur in quantum gravity (Barrett andSteele 2003, Freidel andLouapre 2003), in which the authors developed integral representations for the 6j-symbol as integrals over products of the group manifold. There have also been quite a few other studies of asymptotics of particular spin networks, including Barrett and Williams (1999), Baez et al (2002), Rovelli and Speziale (2006), Hackett and Speziale (2007), Conrady and Freidel (2008), Alesci et al (2008), Barrett et al (2009), among others. We also mention the works of Gurau (2008), which applies standard asymptotic techniques (Stirling's approximation, etc) directly to Racah's sum for the 6j-symbol; of Ragni et al (2010) on the computation of 6j-symbols and on the asymptotics of the 6j-symbol when some quantum numbers are large and others small; and of Littlejohn and Yu (2009) on uniform approximations for the 6j-symbol.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical mechanics of the Wigner 6j-symbol

Aquilanti¹,

Haggard²,

Hedeman³

et al. 2012

J. Phys. A: Math. Theor.

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Abstract. The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the PonzanoRegge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.

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Section: Introductionmentioning

confidence: 99%

Semiclassical mechanics of the Wigner 6j-symbol

Aquilanti¹,

Haggard²,

Hedeman³

et al. 2012

J. Phys. A: Math. Theor.

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“…In order to do this, various types of approximations are introduced which will be explained in detail. The main tool used is an appropriate application of the large spin analysis, first introduced in the context of the EPRL-FK model in [30,31], for a single vertex, and in [32,33], for a general two simplex. Thanks to the geometric content of the equations involved in the analysis, it will be possible to interpret the most diverging contribution to the self-energy as stemming from the constructive contribution of two parity-related "chunks" of spacetime (i.e.…”

Section: Introductionmentioning

confidence: 99%

Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity

Riello

2013

Phys. Rev. D

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We calculate the most divergent contribution to non-degenerate sectorof the self-energy (or "melonic") graph in the context of the Lorentzian EPRL-FK Spin Foam model of Quantum Gravity. We find that such a contribution is logarithmically divergent in the cut-off over the SU (2)-representation spins when one chooses the face amplitude guaranteeing the facesplitting invariance of the foam. We also find that the dependence on the boundary data is different from that of the bare propagator. This fact has its origin in the non-commutativity of the EPRL-FK Y γ -map with the projector onto SL(2, )-invariant states. In the course of the paper, we discuss in detail the approximations used during the calculations, its geometrical interpretation as well as the physical consequences of our result.

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“…We next describe spin foams for BF theory. These again arise as histories of labels of corresponding canonical for each face f for each edge e ∈ ∂f Euclidean j [22,23,24]. This choice will facilitate imposing the simplicity constraint, as well as be important for taking the semiclassical limit of the theory.…”

Section: Spin Foams Of Bf Theorymentioning

confidence: 99%

“…We just have discussed the classical implications of the linear simplicity constraint (23); however, it is the quantum implications for the BF spin foams that will yield us our quantum theory of gravity. From equations (19), (20), and (21) one can deduce the consequences of linear simplicity (23) for the quantum numbers labeling the BF spin foams. In the Euclidean case, these are precisely…”

Section: Simplicity and The Lqg Spin Foam Modelmentioning

confidence: 99%

Spin Foams

Engle¹

2014

Springer Handbook of Spacetime

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Ever since special relativity, space and time have become seamlessly merged into a single entity, and space-time symmetries, such as Lorentz invariance, have played a key role in our fundamental understanding of nature. Quantum mechanics, however, did not originally conform to this new way of thinking. The original formulation of quantum mechanics, called 'canonical', involves wavefunctions, operators, Hamiltonians, and time evolution in a way that treats time very differently from space. This situation was improved by Feynman, who formulated quantum mechanics in terms of probabilities calculated by summing over amplitudes associated to classical histories -the path integral formulation of quantum mechanics. As histories are naturally space-time objects in which space and time can be viewed 'on equal footing', the path integral formulation allowed, for the first time, space-time symmetries to be manifest in a general quantum theory.The key insight of Einstein's theory of gravity, general relativity, is that gravity is spacetime geometry. Space-time geometry, the one 'background structure' -i.e., non-dynamical space-time structure -remaining after special relativity, was discovered to be dynamical and to describe the gravitational field, revealing nature to be 'background independent.' Background independence can equivalently be expressed in terms of a profound enlargement of the basic spacetime symmetry group of physics: invariance under Lorentz transformations and translations is replaced by invariance under the much larger group of space-time diffeomorphisms.We have already seen in the chapter by Sahlmann on gravity, geometry and the quantum, a canonical quantization of Einstein's gravity, and hence of geometry, in which geometric operators are derived with discrete eigenvalues [1, 2, 3]. Instead of space being a smooth continuum, we see that it comes in discrete quanta -minimal 'chunks of space.' Furthermore, as discussed in the chapter by Agullo and Corichi, when applied to cosmology, this quantum theory of gravity leads to a new understanding of the Big Bang in which usually problematic infinities are resolved, and one can actually ask what happened before the Big Bang. In spite of these successes, because it is a canonical theory, it has as a drawback that space-time symmetries, in particular space-time diffeomorphism symmetry, are not manifest. Equivalently, the preferred separation between space and time prevents full background independence from being manifest.One can ask: Is there a way to construct a path-integral formulation of quantum gravity, in which the most radical discovery of general relativity, background independence, or equivalently, space-time diffeomorphism invariance, can be fully manifest, which nevertheless retains the successes of the canonical theory? This is the question leading to the spin foam program. In answering it one must understand more carefully the relationship between the canonical and path integral formulations of quantum mechanics, and in particular how these apply to g...

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A Summary of the Asymptotic Analysis for the EPRL Amplitude

Abstract: We review the basic steps in building the asymptotic analysis of the Euclidean sector of new spin foam models using coherent states, for Immirzi parameter less than one. We focus on conceptual issues and by so doing omit peripheral proofs and the original discussion on spin structures.

Cited by 86 publications

References 23 publications

Semiclassical mechanics of the Wigner 6j-symbol

Semiclassical mechanics of the Wigner 6j-symbol

Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity

Spin Foams

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