Results 1  10
of
53
Spin networks in nonperturbative quantum gravity, in The Interface of Knots and
, 1996
"... A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3dimensional topological quantum field theory, functional integration on th ..."
Abstract

Cited by 50 (8 self)
 Add to MetaCart
A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L 2 (A/G) is spanned by states labelled by spin networks. Then we explain the ‘new variables ’ for general relativity in 4dimensional spacetime and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states. 1
Quantum gravity with a positive cosmological constant
, 2002
"... A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, dis ..."
Abstract

Cited by 48 (9 self)
 Add to MetaCart
A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the WheelerdeWitt equation corrections to the energymomentum relations for matter fields of the form E 2 = p 2 +m 2 +αlPlE 3 +... where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary ChernSimons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.
StateSum Invariants of 4Manifolds
 J. Knot Theory Ram
, 1997
"... Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery ..."
Abstract

Cited by 29 (6 self)
 Add to MetaCart
Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semisimple subquotient of Rep(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4manifolds equipped with 2dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1
Spin Network States in Gauge Theory
 Adv. Math
, 1996
"... Given a realanalytic manifold M, a compact connected Lie group G and a principal Gbundle P → M, there is a canonical ‘generalized measure ’ on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L 2 (A/G). Here we construct a set of vect ..."
Abstract

Cited by 29 (0 self)
 Add to MetaCart
Given a realanalytic manifold M, a compact connected Lie group G and a principal Gbundle P → M, there is a canonical ‘generalized measure ’ on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L 2 (A/G). Here we construct a set of vectors spanning L 2 (A/G). These vectors are described in terms of ‘spin networks’: graphs φ embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin networks associated to any fixed graph φ. We conclude with a discussion of spin networks in the loop representation of quantum gravity, and give a categorytheoretic interpretation of the spin network states. 1
Minimal surface representations of virtual knots and
"... Equivalence classes of virtual knot diagrams are in a one to one correspondence with decorated immersions of S 1 into orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to an immersion of S 1 with over/under markings in a unique ..."
Abstract

Cited by 27 (19 self)
 Add to MetaCart
Equivalence classes of virtual knot diagrams are in a one to one correspondence with decorated immersions of S 1 into orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to an immersion of S 1 with over/under markings in a unique minimal surface. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. We use minimal surfaces and a generalized version of the bracket polynomial for surfaces to determine when a virtual knot diagram is nontrivial and nonclassical. 1 1
The classical evaluation of relativistic spin networks
 Adv. Theor. Math. Phys
, 1998
"... The evaluation of a relativistic spin network for the classical case of the Lie group is given by an integral formula over copies of SU(2). For the graph determined by a 4simplex this gives the evaluation as an integral over a space of geometries for a 4simplex. 1 ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
The evaluation of a relativistic spin network for the classical case of the Lie group is given by an integral formula over copies of SU(2). For the graph determined by a 4simplex this gives the evaluation as an integral over a space of geometries for a 4simplex. 1
Spin Networks in Gauge Theory
, 1996
"... Given a realanalytic manifold M , a compact connected Lie group G and a principal Gbundle P ! M , there is a canonical `generalized measure' on the space A=G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L 2 (A=G). Here we construct a set ..."
Abstract

Cited by 25 (7 self)
 Add to MetaCart
Given a realanalytic manifold M , a compact connected Lie group G and a principal Gbundle P ! M , there is a canonical `generalized measure' on the space A=G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L 2 (A=G). Here we construct a set of vectors spanning L 2 (A=G). These vectors are described in terms of `spin networks': graphs OE embedded in M , with oriented edges labelled by irreducible unitary representations of G, and with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges. We also describe an orthonormal basis of spin network states associated to any fixed graph OE. We conclude with a discussion of spin networks in the loop representation of quantum gravity, and give a categorytheoretic interpretation of the spin network states. 1 Introduction Penrose [14] introduced the notion of a ...
Spin foam models of Riemannian quantum gravity
, 2002
"... Using numerical calculations, we compare three versions of the Barrett– Crane model of 4dimensional 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 tria ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
Using numerical calculations, we compare three versions of the Barrett– Crane model of 4dimensional 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 4manifolds. 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 spinzero 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.
On the TQFT representations of the mapping class groups
 Pacific J. Math
"... We prove that the image of the mapping class group by the representations arising in the SU(2)TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the rth ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
We prove that the image of the mapping class group by the representations arising in the SU(2)TQFT is infinite, provided that the genus g ≥ 2 and the level of the theory r ̸ = 2,3,4,6 (and r ̸ = 10 for g = 2). In particular it follows that the quotient groups Mg/N (tr) by the normalizer of the rth power of a Dehn twist t are infinite if g ≥ 3 and r ̸ = 2,3,4,6,8,12. 1. Introduction. Witten [50] constructed a TQFT in dimension 3 using path integrals and afterwards several rigorous constructions arose, like those using the quantum group approach ([39, 25]), the TemperleyLieb algebra ([30, 31]), the theory based on the Kauffman bracket ([4, 5]) or that obtained from the mapping
Supersymmetric spin networks and quantum supergravity
 Phys.Rev
, 2000
"... We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the nonperturbative quantization of supergravity. The supersymmetric extension of th ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the nonperturbative quantization of supergravity. The supersymmetric extension of the area operator is defined and partly diagonalized. The spectrum is discrete as in quantum general relativity, and the two cases could be distinguished by measurements of quantum geometry. 1