Spin foam perturbation theory

Baez, John C.

doi:10.1090/conm/318/05541

Cited by 3 publications

(12 citation statements)

References 32 publications

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“…Similarly to [Ba2], to eliminate the triangulation dependence of V (n) TV (M, ∆), we want to consider the limit…”

Section: Dilute Gas Limitmentioning

confidence: 99%

“…Here N = N ∆ denotes the number of tetrahedra of a triangulation ∆ of M . The case considered in [Ba2] is the limit when the maximal diameter of each tetrahedra of a triangulation ∆ of M tends to zero, called there the "Dilute Gas Limit. "…”

Section: Dilute Gas Limitmentioning

confidence: 99%

“…It corresponds to Conjecture B in page 8 of [Ba2]. In Subsection 5.2 we explain how we can go around it by restricting the class of triangulations with which we work, so that all calculations are valid.…”

Section: Preliminary Approachmentioning

confidence: 99%

“…of n non-intersecting tetrahedra of M ; compare with Conjecture A on page 8 of [Ba2]. Note that space orderings are not relevant in this case.…”

Section: Restriction 2 Choose a Positive Integer D = D(m )mentioning

confidence: 99%

“…Since General Relativity in 3 and 4 dimensions can be represented as a perturbed BF theory, see [FK,M2], then, in order to find the corresponding Quantum Gravity theory, one would need a spin foam perturbation theory. Baez has analysed the spin foam perturbation theory from a general point of view in [Ba2], and he was able to show that, under certain reasonable assumptions, the perturbed spin foam state sum Z can be calculated in the dilute gas limit. In this limit, the number N of tetrahedra of a manifold triangulation ∆ tends to infinity and λ, the perturbation theory parameter, tends to zero, in a way such that the effective coupling constant g = λN is finite.…”

Section: Introductionmentioning

confidence: 99%

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Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Martins

Mikovic

2009

Commun. Math. Phys.

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We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold, is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.

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“…Similarly to [Ba2], to eliminate the triangulation dependence of V (n) TV (M, ∆), we want to consider the limit…”

Section: Dilute Gas Limitmentioning

confidence: 99%

Section: Dilute Gas Limitmentioning

confidence: 99%

Section: Preliminary Approachmentioning

confidence: 99%

“…of n non-intersecting tetrahedra of M ; compare with Conjecture A on page 8 of [Ba2]. Note that space orderings are not relevant in this case.…”

Section: Restriction 2 Choose a Positive Integer D = D(m )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Martins

Mikovic

2009

Commun. Math. Phys.

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Spin foam models of Riemannian quantum gravity

Baez

Christensen

Halford

et al. 2002

Class. Quantum Grav.

131

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Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model.

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Four-Dimensional Spin Foam Perturbation Theory

Martins

Mikovic

2011

SIGMA

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Abstract. We define a four-dimensional spin-foam perturbation theory for the BF-theory with a B ∧ B potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M . This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group U q (g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ ⊗ Λ → A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M . We prove that the firstorder perturbative contribution vanishes for finite triangulations, so that we define a dilutegas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the F ∧ F theory.

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Spin foam perturbation theory

Cited by 3 publications

References 32 publications

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Spin foam models of Riemannian quantum gravity

Four-Dimensional Spin Foam Perturbation Theory

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