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Relativistic Spin Networks and Quantum Gravity
 J. Math Phys
, 1998
"... Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geome ..."
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Cited by 180 (17 self)
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Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3simplex. This leads us to suggest that there may be a 4dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3dimensional quantum gravity from ordinary spin networks.
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 87 (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.
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 58 (12 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Structural Issues in Quantum Gravity
, 1995
"... A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of bo ..."
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Cited by 30 (2 self)
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A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twentyfive years of research in quantum gravity, and concludes with speculations on what a future theory might look like.
A holographic formulation of quantum general relativity”, Phys
 Rev. D
"... We show that there is a sector of quantum general relativity, in the Lorentzian signature case, which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)L ⊕ ..."
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Cited by 25 (12 self)
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We show that there is a sector of quantum general relativity, in the Lorentzian signature case, which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)L ⊕ SU(2)R, WZW field theory on the npunctured sphere, where n is related to the area of the boundary. The Bekenstein bound is explicitly satisfied. These results are based on a new lagrangian and hamiltonian formulation of general relativity based on a constrained Sp(4) topological field theory. The hamiltonian formalism is polynomial, and also leftright symmetric. The quantization uses balanced SU(2)L ⊕SU(2)R spin networks and so justifies the state sum model of Barrett and Crane. By extending the formalism to Osp(4/N) a holographic formulation of extended supergravity is obtained, as will be described in detail in a subsequent paper.
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1 ..."
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Cited by 22 (0 self)
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This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1
The Bekenstein bound, topological quantum field theory and pluralistic quantum cosmology
, 2008
"... ..."
Algebraic description of spacetime foam
 Classical and Quantum Gravity
, 2001
"... A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its eve ..."
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Cited by 17 (10 self)
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A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions. Physical Motivation In this paper we present an algebraic model of spacetime foam. The notion of spacetime foam has manifold and somewhat ambiguous meaning in the literature if only because the models vary. There is no unanimous agreement about what foam ‘really ’ pertains to mainly due to the fact that each of the mathematical models highlights different aspects of that concept. Here we use the term ‘foam ’ along the concrete but general lines originally introduced by Wheeler [42] who intended to refer to a spacetime with a dynamically variable, because quantally fluctuating, topology. The basic intuition is that at quantum scales even the topology of spacetime is subject to dynamics and interference. This conception of foam is in glaring contrast with general relativity, the classical theory of gravity, where spacetime is fixed to a topological manifold once and forever so that the sole dynamical variable is a higher level structure, namely, the spacetime geometry. It seems theoretically lame and rather ad hoc to regard the geometry of spacetime as being a dynamical variable that can in principle be measured (ie, an observable), while at the same time to think of its topology as a structure a priori fixed by the theoretician, an inert etherlike absolute background that is not liable to experimental investigation thus effectively an unobservable theoretical entity [9]. Especially in the quantum realm where everything seems
A candidate for a background independent formulation of M theory
"... It is proposed that the background independent membrane field theory constructed with Markopoulou is, when the algebra of observables is based on Osp(1016), a background independent form of M theory. The space of states is the space of conformal blocks on all compact twosurfaces of finite genus. H ..."
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Cited by 11 (9 self)
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It is proposed that the background independent membrane field theory constructed with Markopoulou is, when the algebra of observables is based on Osp(1016), a background independent form of M theory. The space of states is the space of conformal blocks on all compact twosurfaces of finite genus. Histories are sequences of certain local moves acting on the states, and have dynamical causal structure. All observables arise by decomposing the states along arbitrary surfaces. A map is constructed which projects the bulk states onto finite dimensional state spaces defined on these surfaces, which we then call holographic surfaces. String worldsheets arise from small perturbations of the histories. D0 branes are identified as points on the holographic surfaces where the strings end. It is found that their kinematics is identical, in the limit of vanishing cosmological constant, to that of the dWHNBFSS matrix model. The requirement that the dynamics of the matrix model be reproduced leads to a proposal for the fundamental evolution law.