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

Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime

We offer a perspective on some recent results obtained in the context of the group field theory approach to quantum gravity, on top of reviewing them briefly. These concern a natural mechanism for the emergence of non-commutative field theories for matter directly from the GFT action, in both 3 and 4 dimensions and in both Riemannian and Lorentzian signatures. As such they represent an important step, we argue, in bridging the gap between a quantum, discrete picture of a pre-geometric spacetime and the effective continuum geometric physics of gravity and matter, using ideas and tools from field theory and condensed matter analog gravity models, applied directly at the GFT level.Comment: 13 pages, no figures; uses JPConf style; contribution to the proceedings of the D.I.C.E. 2008 worksho

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

Quasitriangular structure and twisting of the 3D bicrossproduct model

We show that the bicrossproduct model...SCOAP

Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagatorsComment: 46 pages, the wrong file was first submitte

Gravity induced from quantum spacetime

We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model quantum spacetime algebra [x,t]=\lambda x drastically reduce the moduli of possible metrics g up to normalisation to a single real parameter which we interpret as a time in the past from which all timelike geodesics emerge and a corresponding time in the future at which they all converge. Our analysis also implies a reduction of moduli in n-dimensions and we study the suggested spherically symmetric classical geometry in n=4 in detail, identifying two 1-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density with ratio w_Q=-{1\over 2}. The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra. The noncommutative Riemannian geometry is fully solved for $n=2$ and includes the quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita connection which blows up as \lambda\to 0. We also propose a `quantum Einstein tensor' which is identically zero for the main part of the moduli space of connections (as classically in 2D). However, when the quantum Ricci tensor and metric are viewed as deformations of their classical counterparts there would be an O(\lambda^2) correction to the classical Einstein tensor and an O(\lambda) correction to the classical metric.Comment: 42 pages LATEX, 4 figures; expanded on the physical significanc

Relative Locality in κ\kappa-Poincar\'e

We show that the $\kappa$-Poincar\'e Hopf algebra can be interpreted in the framework of curved momentum space leading to the relativity of locality \cite{AFKS}. We study the geometric properties of the momentum space described by $\kappa$-Poincar\'e, and derive the consequences for particles propagation and energy-momentum conservation laws in interaction vertices, obtaining for the first time a coherent and fully workable model of the deformed relativistic kinematics implied by $\kappa$-Poincar\'e. We describe the action of boost transformations on multi-particles systems, showing that in order to keep covariant the composed momenta it is necessary to introduce a dependence of the rapidity parameter on the particles momenta themselves. Finally, we show that this particular form of the boost transformations keeps the validity of the relativity principle, demonstrating the invariance of the equations of motion under boost transformations.Comment: 24 pages, 4 figures, 1 table. v2 matches accepted CQG versio

Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity

In a recent work, a dual formulation of group field theories as non-commutative quantum field theories has been proposed, providing an exact duality between spin foam models and non-commutative simplicial path integrals for constrained BF theories. In light of this new framework, we define a model for 4d gravity which includes the Immirzi parameter gamma. It reproduces the Barrett-Crane amplitudes when gamma goes to infinity, but differs from existing models otherwise; in particular it does not require any rationality condition for gamma. We formulate the amplitudes both as BF simplicial path integrals with explicit non-commutative B variables, and in spin foam form in terms of Wigner 15j-symbols. Finally, we briefly discuss the correlation between neighboring simplices, often argued to be a problematic feature, for example, in the Barrett-Crane model.Comment: 26 pages, 1 figur

3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory

An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in hep-th/0512113. In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recasted as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models.Comment: 17 pages, revtex4, 1 figur

Holonomy observables in Ponzano-Regge type state sum models

We study observables on group elements in the Ponzano-Regge model. We show that these observables have a natural interpretation in terms of Feynman diagrams on a sphere and contrast them to the well studied observables on the spin labels. We elucidate this interpretation by showing how they arise from the no-gravity limit of the Turaev-Viro model and Chern-Simons theory.Comment: 15 pages, 2 figure