On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables

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,C)-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.Comment: 22 pages, LaTeX 2e, 1 figur

Asymptotics of 10j symbols

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.Comment: 25 pages LaTeX with 8 encapsulated Postscript figures. v2 has various clarifications and better page breaks. v3 is the final version, to appear in Classical and Quantum Gravity, and has a few minor corrections and additional reference

Positivity of Spin Foam Amplitudes

The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e^{iS}) rather than imaginary-time (e^{-S}) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett-Crane model.Comment: 15 pages LaTeX. v3: Final version, with updated conclusions and other minor changes. To appear in Classical and Quantum Gravity. v4: corrects # of samples in Lorentzian tabl

Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity

The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for every triangulated closed 4-manifold is a non-negative real number. Roughly speaking, this means that if one sums over triangulations, there is no interference between the different triangulations. We prove non-negativity by transforming the model into a ``dual variables'' formulation in which the amplitude for a given triangulation is expressed as an integral over three copies of hyperbolic space for each tetrahedron. Then we prove that, expressed in this way, the integrand is non-negative. In addition to implying that the amplitude is non-negative, the non-negativity of the integrand is highly significant from the point of view of numerical computations, as it allows statistical methods such as the Metropolis algorithm to be used for efficient computation of expectation values of observables.Comment: 13 page

Asymptotics of Relativistic Spin Networks

The stationary phase technique is used to calculate asymptotic formulae for SO(4) Relativistic Spin Networks. For the tetrahedral spin network this gives the square of the Ponzano-Regge asymptotic formula for the SU(2) 6j symbol. For the 4-simplex (10j-symbol) the asymptotic formula is compared with numerical calculations of the Spin Network evaluation. Finally we discuss the asymptotics of the SO(3,1) 10j-symbol.Comment: 31 pages, latex. v3: minor clarification

Spin Foam Models of Yang-Mills Theory Coupled to Gravity

We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidian gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity.Comment: 10 page

A Lorentzian Signature Model for Quantum General Relativity

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation. Version 2 is a major revision with explicit formulae included for the evaluation of relativistic spin networks and the computation of examples which have finite value

Topological Aspects of Spin and Statistics in Nonlinear Sigma Models

We study the purely topological restrictions on allowed spin and statistics of topological solitons in nonlinear sigma models. Taking as space the connected $d$-manifold $X$, and considering nonlinear sigma models with the connected manifold $M$ as target space, topological solitons are given by elements of $pi_d(M)$. Any topological soliton $\alpha \in \pi_d(M)$ determines a quotient \Stat_n(X,\alpha) of the group of framed braids on $X$, such that choices of allowed statistics for solitons of type $\alpha$ are given by unitary representations of \Stat_n(X,\alpha) when $n$ solitons are present. In particular, when $M = S^2$, as in the $O(3)$ nonlinear sigma model with Hopf term, and $\alpha \in \pi_2(S^2)$ is a generator, we compute that \Stat_n(\R^2,\alpha) = \Z, while \Stat_n(S^2,\alpha) = \Z_{2n}. It follows that phase $\exp(i\theta)$ for interchanging two solitons of type $\alpha$ on $S^2$ must satisfy the constraint $\theta = k\pi/n$, $k \in \Z$, when $n$ such solitons are present.Comment: 14 page

Dual variables and a connection picture for the Euclidean Barrett-Crane model

The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett-Crane model can be understood as a sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the model are balanced representations of SO(4) which describe the quantized areas of the triangles. We present an exact duality transformation for the full quantum theory and reformulate the model in terms of new variables which can be understood as variables conjugate to the quantized areas. The new variables are pairs of S^3-values associated to the tetrahedra. These S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally embedded in R^4), and the fact that there is a pair of variables for each tetrahedron can be viewed as a consequence of an SO(4)-valued parallel transport along the edges dual to the tetrahedra. We reconstruct the parallel transport of which only the action of SO(4) on S^3 is physically relevant and rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the 2-complex dual to the triangulation subject to suitable constraints whose form we derive at the quantum level. Our reformulation of the Barrett-Crane model in terms of continuous variables is suitable for the application of various analytical and numerical techniques familiar from Statistical Mechanics.Comment: 33 pages, LaTeX, combined PiCTeX/postscript figures, v2: note added, TeX error correcte

An Invitation to Higher Gauge Theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry at the 2009 Corfu Summer Institut