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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 157 (14 self)
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For a copy with the handdrawn figures please email
Spin foam models
 Classical and Quantum Gravity
, 1998
"... While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a ‘spin foam ’ going from one spin network to another. Just as a spin network is a graph with e ..."
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Cited by 71 (2 self)
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While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a ‘spin foam ’ going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higherdimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a ‘spin foam model’, such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe ‘quantum 3geometries’, we describe how spin foams describe ‘quantum 4geometries’. We conclude by presenting a spin foam model of 4dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.
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 ..."
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Cited by 53 (8 self)
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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
Knot theory and quantum gravity in loop space: A premier, hepth/9301028, to appear
 in the proceedings of the Vth Mexican school on particles and
, 1993
"... These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar’s variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques o ..."
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Cited by 12 (0 self)
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These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar’s variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field. 1 Quantum Gravity: why and how? I wish to thank the organizers for inviting me to speak here. This may well be a sign of our times, that a person generally perceived as a “General Relativist ” would be invited to speak at a Particle Physics School. It just reflects the higher degree of interplay these two fields have enjoyed over the last years. In these lectures we will see more reasons for this enhanced interplay. We will see several notions from Gauge Theories, as Wilson Loops for instance, playing a central role in gravitation. An even greater interplay takes place with Topological Field Theories. We will see the important role that the ChernSimons form, the Jones Polynomial and other notions of knot theory seem to play in General Relativity. The quantization of General Relativity is a problem that has defied resolution for the last sixty years. In spite of the long time that has been invested in trying to solve it, we believe that several people do not necessarily fully appreciate the reasons of our failure and the magnitude of the problem. It is a general perception –especially among particle physicists – that “General Relativity is nonrenormalizable ” and that is the basic problem with the theory. This statement is misleading in three ways: a) The fact that a theory is nonrenormalizable does not necessarily mean that the theory has an intrinsic problem or is “bad ” in any way. It merely says that perturbation
4Dimensional BF Theory as a Topological Quantum Field Theory
 Lett. Math. Phys
, 1996
"... Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The c ..."
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Cited by 11 (6 self)
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Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal Gbundle. The case G = GL(4; R) is especially interesting because every 4manifold is then naturally equipped with a principal Gbundle, namely its frame bundle. In this case, the partition function of a compact oriented 4manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of 4manifolds. 1 Introduction In comparison to the situation in 3 dimensions, topological quantum field theories (TQFTs) in 4 dimensions are poorly understood. This is ironic, because the subject was initiated by an attempt to understand Donaldson theory in terms of a quantum field theory in 4 dimensions....
The geometry of quantum spin networks
 grqc/9512043. Anastasios Mallios and Ioannis Raptis
, 1996
"... The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation. The operators for area and volume defined in the qdeformation of the theory are partly diagonalized. The eigenstates are expressed in terms of qdeformed spin networks. ..."
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Cited by 7 (3 self)
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The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation. The operators for area and volume defined in the qdeformation of the theory are partly diagonalized. The eigenstates are expressed in terms of qdeformed spin networks. The qdeformation breaks some of the degeneracy of the volume operator so that trivalent spinnetworks have nonzero volume. These computations are facilitated by use of a technique based on the recoupling theory of SU(2)q, which simplifies the construction of these and other operators through diffeomorphism invariant regularization procedures.
Propagating spin modes in canonical quantum gravity”, grqc/9810024. 36 Penrose R
, 1998
"... One of the main results in canonical quantum gravity is the introduction of spin network states as a basis on the space of kinematical states. To arrive at the physical state space of the theory though we need to understand the dynamics of the quantum gravitational states. To this aim we study a mod ..."
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Cited by 6 (0 self)
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One of the main results in canonical quantum gravity is the introduction of spin network states as a basis on the space of kinematical states. To arrive at the physical state space of the theory though we need to understand the dynamics of the quantum gravitational states. To this aim we study a model in which we allow for the spins, labeling the edges of spin networks, to change according to simple rules. The gauge invariance of the theory, restricting the possible values for the spins, induces propagating modes of spin changes. We investigate these modes under various assumptions about the parameters of the model.
The extended Loop representation of quantum gravity
"... A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop d ..."
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A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum Gravity can be realized on the state space of extended loop dependent wavefunctions. The extended representation generalizes the loop representation and contains this representation as a particular case. The resulting diffeomorphism and hamiltonian constraints take a very simple form and allow to apply functional methods and simplify the loop calculus. In particular we show that the constraints are linear in the momenta. The nondegenerate solutions known in the loop representation are also solutions of the constraints in the new representation. The practical calculation advantages allows to find a new solution to the WheelerDeWitt equation. Moreover, the extended representation puts in a precise framework some of the regularization problems of the loop representation. We show that the solutions are generalized knot invariants, smooth in the extended variables, and any framing is unnecessary. 1 1
Spin Networks in Nonperturbative . . .
, 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 ..."
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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 L²(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.