How to detect an anti-spacetime

Is it possible, in principle, to measure the sign of the Lapse? We show that fermion dynamics distinguishes spacetimes having the same metric but different tetrads, for instance a Lapse with opposite sign. This sign might be a physical quantity not captured by the metric. We discuss its possible role in quantum gravity.Comment: Article awarded with an "Honorable Mention" from the 2012 Gravity Foundation Award. 6 pages, 8 (pretty) figure

From Dimensional Reduction of 4d Spin Foam Model to Adding Non-Gravitational Fields to 3d Spin Foam Model

A Kaluza-Klein like approach for a 4d spin foam model is considered. By applying this approach to a model based on group field theory in 4d (TOCY model), and using the Peter-Weyl expansion of the gravitational field, reconstruction of new non gravitational fields and interactions in the action are found. The perturbative expansion of the partition function produces graphs colored with su(2) algebraic data, from which one can reconstruct a 3d simplicial complex representing space-time and its geometry; (like in the Ponzano-Regge formulation of pure 3d quantum gravity), as well as the Feynman graph for typical matter fields. Thus a mechanism for generation of matter and construction of new dimensions are found from pure gravity.Comment: 11 pages, no figure, to be published in International Journal of Geometric Methods in Modern Physic

Semiclassical Analysis of the Wigner 12j12j Symbol with One Small Angular Momentum

We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner $9j$ symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave-functions to derive asymptotic formulas for the $9j$ symbol with small and large angular momenta. When applying the same technique to the $12j$ symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram for a $9j$ symbol. This illustrates a general geometric connection between asymptotic limits of the various $3nj$ symbols. This work contributes the first known asymptotic formula for the $12j$ symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner $15j$ symbol with two small angular momenta.Comment: 15 pages, 14 figure

3-dimensional Gravity from the Turaev-Viro Invariant

We study the $q$-deformed su(2) spin network as a 3-dimensional quantum gravity model. We show that in the semiclassical continuum limit the Turaev-Viro invariant obtained recently defines naturally regularized path-integral $\grave{\rm a}$ la Ponzano-Regge, In which a contribution from the cosmological term is effectively included. The regularization dependent cosmological constant is found to be ${4\pi^2\over k^2} +O(k^{-4})$, where $q^{2k}=1$. We also discuss the relation to the Euclidean Chern-Simons-Witten gravity in 3-dimension.Comment: 11page

N=2 supersymmetric spin foams in three dimensions

We construct the spin foam model for N=2 supergravity in three dimensions. Classically, it is a BF theory with gauge algebra osp(2|2). This algebra has representations which are not completely reducible. This complicates the procedure when building a state sum. Fortunately, one can and should excise these representations. We show that the restricted subset of representations form a subcategory closed under tensor product. The resulting state-sum is once again a topological invariant. Furthermore, within this framework one can identify positively and negatively charged fermions propagating on the spin foam. These results on osp(2|2) representations and intertwiners apply more generally to spin network states for N=2 loop quantum supergravity (in 3+1 dimensions) where it allows to define a notion of BPS states.Comment: 12 page

Holography for the Lorentz Group Racah Coefficients

A known realization of the Lorentz group Racah coefficients is given by an integral of a product of 6 ``propagators'' over 4 copies of the hyperbolic space. These are ``bulk-to-bulk'' propagators in that they are functions of two points in the hyperbolic space. It is known that the bulk-to-bulk propagator can be constructed out of two bulk-to-boundary ones. We point out that there is another way to obtain the same object. Namely, one can use two bulk-to-boundary and one boundary-to-boundary propagator. Starting from this construction and carrying out the bulk integrals we obtain a realization of the Racah coefficients that is ``holographic'' in the sense that it only involves boundary objects. This holographic realization admits a geometric interpretation in terms of an ``extended'' tetrahedron.Comment: 12 pages, 2 figures; v2: minor changes; v3: "extended" tetrahedron interpretation adde

Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity

We show that the $\star$-product for $U(su_2)$, group Fourier transform and effective action arising in [1] in an effective theory for the integer spin Ponzano-Regge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scalar field theory previously proposed for 2+1 Euclidean quantum gravity using quantum group methods in [2]. The two are related by a classicalisation map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the half-integer spin information. We argue that the anomalous extra `time' dimension seen in the noncommutative geometry should be viewed as the renormalisation group flow visible in the coarse-graining in going from $SU_2$ to $SO_3$. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum delta-functions and Gaussians, the Duflo map and elements of `noncommutative sampling theory'. This allows us to understand the bandwidth limitation in 2+1 quantum gravity arising from the bounded $SU_2$ momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalised twist operator for the $\star$-product.Comment: 53 pages latex, no figures; extended the intro for this final versio

Discrete Space-Time Volume for 3-Dimensional BF Theory and Quantum Gravity

The Turaev-Viro state sum invariant is known to give the transition amplitude for the three dimensional BF theory with cosmological term, and its deformation parameter hbar is related with the cosmological constant via hbar=sqrt{Lambda}. This suggests a way to find the expectation value of the spacetime volume by differentiating the Turaev-Viro amplitude with respect to the cosmological constant. Using this idea, we find an explicit expression for the spacetime volume in BF theory. According to our results, each labelled triangulation carries a volume that depends on the labelling spins. This volume is explicitly discrete. We also show how the Turaev-Viro model can be used to obtain the spacetime volume for (2+1) dimensional quantum gravity.Comment: 13 pages, Revtex, figure

The loop-quantum-gravity vertex-amplitude

Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy $n$-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.Comment: 6page

Asymptotics of LQG fusion coefficients

The fusion coefficients from SO(3) to SO(4) play a key role in the definition of spin foam models for the dynamics in Loop Quantum Gravity. In this paper we give a simple analytic formula of the EPRL fusion coefficients. We study the large spin asymptotics and show that they map SO(3) semiclassical intertwiners into $SU(2)_L\times SU(2)_R$ semiclassical intertwiners. This non-trivial property opens the possibility for an analysis of the semiclassical behavior of the model.Comment: 14 pages, minor change