Integrability for relativistic spin networks

Baez, John C.; Barrett, John W.

doi:10.1088/0264-9381/18/21/316

Cited by 44 publications

(115 citation statements)

References 23 publications

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“…As in the Lorentzian case [19], one can show that if this integral converges, the result does not depend on our choice of the special vertex v 1 or the point x ∈ R 3 . Assuming the integral does converge, we can use the Kirillov trace formula (23) to reexpress it as:…”

Section: Degenerate Spin Networkmentioning

confidence: 71%

“…Comparing these asymptotics to those of the degenerate contribution, we can formulate the following: Here we say the spins labelling the edges incident to some vertex are 'admissible' if they sum to an integer and each is less than or equal to the sum of the rest. We do not yet have general criteria for when the integrals associated to Euclidean spin networks converge, and as we shall see, the relevant theorems are bound to be a bit different than in the Lorentzian case [19].…”

Section: Degenerate Spin Networkmentioning

confidence: 99%

“…Now let us turn to Lorentzian spin networks [12,19]. Each positive real number determines a unitary irreducible representation of the connnected Lorentz group SO 0 (3, 1) corresponding to an eigenspace of the Laplacian on H 3 ; however, we prefer to describe the evaluation using an integral formula.…”

Section: Lorentzian Spin Networkmentioning

confidence: 99%

“…This makes it more challenging to find criteria for convergence of degenerate spin networks, since we cannot simply mimic the theory that applies in the Lorentzian case [19]. Barrett and Crane also worked out the Lorentzian 4j symbols, obtaining:…”

Section: Lorentzian Spin Networkmentioning

confidence: 99%

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Asymptotics of 10 j symbols

Baez

Christensen

Egan

2002

Class. Quantum Grav.

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118

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Abstract. The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a 'degenerate spin network', where the rotation group SO (4) is replaced by the Euclidean group of isometries of R 3 . We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.

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Section: Degenerate Spin Networkmentioning

confidence: 71%

Section: Degenerate Spin Networkmentioning

confidence: 99%

Section: Lorentzian Spin Networkmentioning

confidence: 99%

Section: Lorentzian Spin Networkmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotics of 10 j symbols

Baez

Christensen

Egan

2002

Class. Quantum Grav.

Self Cite

118

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“…Note that we are following the conventions of [5,14] here. It should be mentioned that the integrals over H 3 + in (2.5) converge only after division by an infinite volume factor [14]. We keep this fact in mind, but leave our formulas unchanged in order to preserve their full symmetry.…”

Section: The Original Modelsmentioning

confidence: 99%

Causal Barrett-Crane model: Measure, coupling constant, Wick rotation, symmetries, and observables

Pfeiffer

2003

Phys. Rev. D

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We discuss various features and details of two versions of the Barrett-Crane spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian model and second of the SL(2, )-symmetric Lorentzian version in which all tetrahedra are space-like. Recently, Livine and Oriti proposed to introduce a causal structure into the Lorentzian Barrett-Crane model from which one can construct a path integral that corresponds to the causal (Feynman) propagator. We show how to obtain convergent integrals for the 10j-symbols and how a dimensionless constant can be introduced into the model. We propose a 'Wick rotation' which turns the rapidly oscillating complex amplitudes of the Feynman path integral into positive real and bounded weights. This construction does not yet have the status of a theorem, but it can be used as an alternative definition of the propagator and makes the causal model accessible by standard numerical simulation algorithms. In addition, we identify the local symmetries of the models and show how their four-simplex amplitudes can be re-expressed in terms of the ordinary relativistic 10j-symbols. Finally, motivated by possible numerical simulations, we express the matrix elements that are defined by the model, in terms of the continuous connection variables and determine the most general observable in the connection picture. Everything is done on a fixed two-complex.

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Lectures on Loop Quantum Gravity

2003

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Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures.Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry.Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures such as extra dimensions, extra symmetries or extra particle content beyond the standard model. While this is a very conservative approach and thus maybe not very attractive to many researchers, it has the advantage that pushing the theory to its logical frontiers will undoubtedly either result in a successful theory or derive exactly which extra structures are required, if necessary. Or put even more radically, it may show which basic principles of physics have to be given up and must be replaced by more fundamental ones.QGR therefore is, by definition, not a unified theory of all interactions in the standard sense since such a theory would require a new symmetry principle. However, it unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches.

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Integrability for relativistic spin networks

Cited by 44 publications

References 23 publications

Asymptotics of 10 j symbols

Asymptotics of 10 j symbols

Causal Barrett-Crane model: Measure, coupling constant, Wick rotation, symmetries, and observables

Lectures on Loop Quantum Gravity

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