2-Group Representations for Spin Foams

Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, "2-categories of 2-group representations" may provide interesting state sum models for 4d quantum topology, if not quantum gravity. Here we focus on the "Euclidean 2-group", built from the rotation group SO(4) and its action on the group of translations of 4d Euclidean space. We explain its infinite-dimensional unitary representations, and construct a model based on the resulting representation 2-category. This model, with clear geometric content and explicit "metric data" on triangulation edges, shows up naturally in an attempt to write the amplitudes of ordinary quantum field theory in a background independent way.Comment: 8 pages; to appear in proceedings of the XXV Max Born Symposium: "The Planck Scale", Wroclaw, Polan

THE "ART OF WAR FRIEZE" IN URBINO: A BLEND OF VIRTUAL RECONSTRUCTION AND SCIENTIFIC ACCURACY

The Art of War Frieze was commissioned by Federico da Montefeltro, Duke of Urbino, to decorate the back of the «wing façade» of the Ducal palace. The Frieze decorated the façade from the time it was realised towards the end of the XVth century until 1756. The Frieze consists of a very particular series of seventy-two limestone bas-reliefs, whose iconographic repertoire represents numerous war and building machines as well as military and political symbols. After it broke away from the outdoor façade it was stored in different rooms in the Palace but despite the many documentary records available, the question of the original sequence of the basreliefs has never been resolved. The primary scope of this paper is to create a "virtual" reconstruction of the original sequence of the bas-reliefs, starting from historical and iconographic records, an analysis of the back and the individual panels using a laser scanner and fully automatic open source photo modelling technologies like the Arc3d, and photogrammetric systems like Image Master together with analyses of the state of conservation, type of degradation correlated to atmospheric parameters (sunlight, temperature, rain). Tests will then be carried out with different systems in order to confirm the accuracy of the model if it is decided to reproduce the individual panels using the rapid prototyping technique associated to a study of the execution techniques

3-d visualization and animation of architectonic elements for prehistoric megalithic temples of the island of Gozo: the temple of Ggantija

Laser scanning can now be defined without doubt as the newest frontier in the field of survey technique, and recent technological developments of instruments and processing software have encouraged the introduction of this technique in the world of applications connected to archaeological site and other related disciplines. The temple of Ggantija on the island of Gozo was considered to be representative of the entire series of temple complexes due to their particular architectural characteristics, their stage of evolution and form of deterioration, both material and structural. The survey was conducted by the use of the local geodetic network in the different phases: • Topographic survey • 3D laser scanner survey • Photographic Survey: both traditional and digital pictures will be taken in order to fully documentation internal and external surfaces of the site. The treatment and analysis of data collections was divided into the following sub-stages: elaboration and compensation of close polygonal, thickening polygonal and direct measurements; elaboration and compensation of altimetric network; linking of the above data with the existing Maltese national networks; elaboration of laser scanner positions and absolute orientations; elaboration of points coordinates for georeferencing and linking the point clouds coming from laser; final data verification end quality control; analysis of laser measured point clouds, for filtering and subsequent elaboration; scanning orientations and subdivision into “islands” (internal rooms and external sides); analysis of laser measured point clouds over the grid determined by the topographic survey.; modelling of the Archaeological site, elimination of noises and metric "pollution" by statistics and verification; accentuation and reduction of triangles on areas interested by complex geometries; triangles transformation into complex surfaces (mesh); model checking by topographic points; mapping of digital photocolors covering all the surfaces of the site. The digital model will be cut by vertical and horizontal section plans at heights requested by customer 2D graphic editing of the plans, sections and elevations. Finishing of vertical sections (sections and views) using the mapped model created by rendering calculated, generating contours lines from the 3D model; of a light model (low density model) of the laser scanner data using the filtering tools of the software package; of an virtual animation of the high density model; of a mapped VRML (Virtual Reality Modelling Language) model for a web interactive and hypertestual navigation, using the low density model. This part of the study was aimed at defining the architectural characteristics and mode of construction of this monument

Non-commutative flux representation for loop quantum gravity

The Hilbert space of loop quantum gravity is usually described in terms of cylindrical functionals of the gauge connection, the electric fluxes acting as non-commuting derivation operators. It has long been believed that this non-commutativity prevents a dual flux (or triad) representation of loop quantum gravity to exist. We show here, instead, that such a representation can be explicitly defined, by means of a non-commutative Fourier transform defined on the loop gravity state space. In this dual representation, flux operators act by *-multiplication and holonomy operators act by translation. We describe the gauge invariant dual states and discuss their geometrical meaning. Finally, we apply the construction to the simpler case of a U(1) gauge group and compare the resulting flux representation with the triad representation used in loop quantum cosmology.Comment: 12 pages, matches published versio

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.Comment: 28 pages (RevTeX4), 7 figures, references adde

Bubbles and jackets: new scaling bounds in topological group field theories

We use a reformulation of topological group field theories in 3 and 4 dimensions in terms of variables associated to vertices, in 3d, and edges, in 4d, to obtain new scaling bounds for their Feynman amplitudes. In both 3 and 4 dimensions, we obtain a bubble bound proving the suppression of singular topologies with respect to the first terms in the perturbative expansion (in the cut-off). We also prove a new, stronger jacket bound than the one currently available in the literature. We expect these results to be relevant for other tensorial field theories of this type, as well as for group field theory models for 4d quantum gravity.Comment: v2: Minor modifications to match published versio

Commuting Simplicity and Closure Constraints for 4D Spin Foam Models

Spin Foam Models are supposed to be discretised path integrals for quantum gravity constructed from the Plebanski-Holst action. The reason for there being several models currently under consideration is that no consensus has been reached for how to implement the simplicity constraints. Indeed, none of these models strictly follows from the original path integral with commuting B fields, rather, by some non standard manipulations one always ends up with non commuting B fields and the simplicity constraints become in fact anomalous which is the source for there being several inequivalent strategies to circumvent the associated problems. In this article, we construct a new Euclidian Spin Foam Model which is constructed by standard methods from the Plebanski-Holst path integral with commuting B fields discretised on a 4D simplicial complex. The resulting model differs from the current ones in several aspects, one of them being that the closure constraint needs special care. Only when dropping the closure constraint by hand and only in the large spin limit can the vertex amplitudes of this model be related to those of the FK Model but even then the face and edge amplitude differ. Curiously, an ad hoc non-commutative deformation of the $B^{IJ}$ variables leads from our new model to the Barrett-Crane Model in the case of Barbero-Immirzi parameter goes to infinity.Comment: 41 pages, 4 figure

Hidden Quantum Gravity in 3d Feynman diagrams

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.Comment: 35 pages, 4 figures, some comments adde

The 1/N expansion of colored tensor models in arbitrary dimension

In this paper we extend the 1/N expansion introduced in [1] to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres S^D contribute to the leading order in the large N limit.Comment: 4 pages, 3 figure

Group field theory formulation of 3d quantum gravity coupled to matter fields

We present a new group field theory describing 3d Riemannian quantum gravity coupled to matter fields for any choice of spin and mass. The perturbative expansion of the partition function produces fat graphs colored with SU(2) algebraic data, from which one can reconstruct at once a 3-dimensional simplicial complex representing spacetime and its geometry, like in the Ponzano-Regge formulation of pure 3d quantum gravity, and the Feynman graphs for the matter fields. The model then assigns quantum amplitudes to these fat graphs given by spin foam models for gravity coupled to interacting massive spinning point particles, whose properties we discuss.Comment: RevTeX; 28 pages, 21 figure