arXiv subjects

Etera R. Livine

Publications and source records attributed to Etera R. Livine.

At least 73 records · Page 4Linked to original sources

arXiv2016

Surface state decoherence in loop quantum gravity, a first toy model

The quantum-to-classical transition through decoherence is a major facet of the semi-classical analysis of quantum models that are supposed to admit a classical regime, as quantum gravity should be. A particular problem of interest is the decoherence of black hole horizons and holographic screens induced by the bulk-boundary coupling with interior degrees of freedom. Here in this paper we present a first toy-model, in the context of loop quantum gravity, for the dynamics of a surface geometry as an open quantum system at fixed total area. We discuss the resulting decoherence and recoherence and compare the exact density matrix evolution to the commonly used master equation approximation {\it à la} Lindblad underlining its merits and limitations. The prospect of this study is to have a clearer understanding of the boundary decoherence of black hole horizons seen by outside observers.

gr-qc↗

arXiv2016

The Fock Space of Loopy Spin Networks for Quantum Gravity

In the context of the coarse-graining of loop quantum gravity, we introduce loopy and tagged spin networks, which generalize the standard spin network states to account explicitly for non-trivial curvature and torsion. Both structures relax the closure constraints imposed at the spin network vertices. While tagged spin networks merely carry an extra spin at every vertex encoding the overall closure defect, loopy spin networks allow for an arbitrary number of loops attached to each vertex. These little loops can be interpreted as local excitations of the quantum gravitational field and we discuss the statistics to endow them with. The resulting Fock space of loopy spin networks realizes new truncation of loop quantum gravity, allowing to formulate its graph-changing dynamics on a fixed background graph plus local degrees of freedom attached to the graph nodes. This provides a framework for re-introducing a non-trivial background quantum geometry around which we would study the effective dynamics of perturbations. We study how to implement the dynamics of topological BF theory in this framework. We realize the projection on flat connections through holonomy constraints and we pay special attention to their often overlooked non-trivial flat solutions defined by higher derivatives of the $δ$-distribution.

gr-qc↗

arXiv2015

Ising Spin Network States for Loop Quantum Gravity: a Toy Model for Phase Transitions

Non-perturbative approaches to quantum gravity call for a deep understanding of the emergence of geometry and locality from the quantum state of the gravitational field. Without background geometry, the notion of distance should entirely emerge from the correlations between the gravity fluctuations. In the context of loop quantum gravity, quantum states of geometry are defined as spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. Here, we develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: we introduce new Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. We construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize our states. We discuss their phase diagram and show how the distance can be reconstructed from the correlations in the various phases. Finally, we propose generalizations of these Ising states, which open the perspective to study the coarse graining and dynamics of spin network states using well-known condensed matter techniques and results.

gr-qc↗

arXiv2015

Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a Cut-off for Quantum Gravity

In the context of the geometrical interpretation of the spin network states of Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barbero connection on loops embedded in space-like hyperboloids. We use this simple setting to illustrate two points. First, the Ashtekar-Barbero connection is not a space-time connection, its holonomies depend on the spacetime embedding of the canonical hypersurface. This fact is usually interpreted as an inconvenience, but we use it to extract the extrinsic curvature from the holonomy and separate it from the 3d intrinsic curvature. Second, we show the limitations of this reconstruction procedure, due to a periodicity of the holonomy in the Immirzi parameter, which underlines the role of a real Immirzi parameter as a cut-off for general relativity at the quantum level in contrast with its role of a mere coupling constant at the classical level.

gr-qc↗

arXiv2015

Duality between Spin networks and the 2D Ising model

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories which couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.

math-ph↗

arXiv2015

Closure constraints for hyperbolic tetrahedra

We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2,C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

gr-qc↗

arXiv2014

Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries

In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the T*SU(2) phase space attached to each edge by SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant. Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries to a hyperbolic curvature.

gr-qc↗

arXiv2014

Deformed phase space for 3d loop gravity and hyperbolic discrete geometries

We revisit the loop gravity space phase for 3D Riemannian gravity by algebraically constructing the phase space $T^*\mathrm{SU}(2)\sim\mathrm{ISO}(3)$ as the Heisenberg double of the Lie group $\mathrm{SO}(3)$ provided with the trivial cocyle. Tackling the issue of accounting for a non-vanishing cosmological constraint $Λ\ne 0$ in the canonical framework of 3D loop quantum gravity, $\mathrm{SL}(2,\mathbb{C})$ viewed as the Heisenberg double of $\mathrm{SU}(2)$ provided with a non-trivial cocyle is introduced as a phase space. It is a deformation of the flat phase space $\mathrm{ISO}(3)$ and reproduces the latter in a suitable limit. The $\mathrm{SL}(2,\mathbb{C})$ phase space is then used to build a new, deformed LQG phase space associated to graphs. It can be equipped with a set of Gauss constraints and flatness constraints, which form a first class system and Poisson-generate local 3D rotations and deformed translations. We provide a geometrical interpretation for this lattice phase space with constraints in terms of consistently glued hyperbolic triangles, i.e. hyperbolic discrete geometries, thus validating our construction as accounting for a constant curvature $Λ<0$. Finally, using ribbon diagrams, we show that our new model is topological.

gr-qc↗

arXiv2013

Deformation Operators of Spin Networks and Coarse-Graining

In the context of loop quantum gravity, quantum states of geometry are mathematically defined as spin networks living on graphs embedded in the canonical space-like hypersurface. In the effort to study the renormalisation flow of loop gravity, a necessary step is to understand the coarse-graining of these states in order to describe their relevant structure at various scales. Using the spinor network formalism to describe the phase space of loop gravity on a given graph, we focus on a bounded (connected) region of the graph and coarse-grain it to a single vertex using a gauge-fixing procedure. We discuss the ambiguities in the gauge-fixing procedure and their consequences for coarse-graining spin(or) networks. This allows to define the boundary deformations of that region in a gauge-invariant fashion and to identify the area preserving deformations as U(N) transformations similarly to the already well-studied case of a single intertwiner. The novelty is that the closure constraint is now relaxed and the closure defect interpreted as a local measure of the curvature inside the coarse-grained region. It is nevertheless possible to cancel the closure defect by a Lorentz boost. We further identify a Lorentz-invariant observable related to the area and closure defect, which we name "rest area". Its physical meaning remains an open issue.

gr-qc↗

arXiv2013

Deformations of Polyhedra and Polygons by the Unitary Group

We introduce the set of framed convex polyhedra with N faces as the symplectic quotient C^2N//SU(2). A framed polyhedron is then parametrized by N spinors living in C^2 satisfying suitable closure constraints and defines a usual convex polyhedron plus a phase for each face. We show that there is an action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any polyhedron onto any other with the same total area. This realizes the isomorphism of the space of framed polyhedra with the Grassmannian space U(N)/SU(2)*U(N-2). We show how to write averages and correlations of geometrical observables over the ensemble of polyhedra as polynomial integrals over U(N) and we use the Itzykson-Zuber formula from matrix models as the generating function for them. In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners. The individual face areas are quantized as half-integers (spins) and the Hilbert spaces for fixed total area form irreducible representations of U(N). We define coherent intertwiner states peaked on classical framed polyhedra and transforming consistently under U(N) transformations. The U(N) character formula can be considered as an extension of the Itzykson-Zuber to the quantum level and generates the traces of all polynomial observables over the Hilbert space of intertwiners. We finally show that classical (convex) polygons can be described in a similar fashion trading U(N) for O(N). We conclude with a discussion of the possible (deformation) dynamics that one can define on the space of polygons or polyhedra. This work is a priori useful in the context of discrete geometry but it should also be relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum geometry is defined in terms of gluing of (quantized) polygons and polyhedra.

math-ph↗

arXiv2013

Classical Setting and Effective Dynamics for Spinfoam Cosmology

We explore how to extract effective dynamics from loop quantum gravity and spinfoams truncated to a finite fixed graph, with the hope of modeling symmetry-reduced gravitational systems. We particularize our study to the 2-vertex graph with N links. We describe the canonical data using the recent formulation of the phase space in terms of spinors, and implement a symmetry-reduction to the homogeneous and isotropic sector. From the canonical point of view, we construct a consistent Hamiltonian for the model and discuss its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the dynamics from the spinfoam approach. We compute exactly the transition amplitude between initial and final coherent spin networks states with support on the 2-vertex graph, for the choice of the simplest two-complex (with a single space-time vertex). The transition amplitude verifies an exact differential equation that agrees with the Hamiltonian constructed previously. Thus, in our simple setting we clarify the link between the canonical and the covariant formalisms.

gr-qc↗

arXiv2013

Some classes of renormalizable tensor models

We identify new families of renormalizable of tensor models from anterior renormalizable tensor models via a mapping capable of reducing or increasing the rank of the theory without having an effect on the renormalizability property. Mainly, a version of the rank 3 tensor model as defined in [arXiv:1201.0176 [hep-th]], the Grosse-Wulkenhaar model in 4D and 2D generate three different classes of renormalizable models. The proof of the renormalizability is fully performed for the first reduced model. The same procedure can be applied for the remaining cases. Interestingly, we find that, due to the peculiar behavior of anisotropic wave function renormalizations, the rank 3 tensor model reduced to a matrix model generates a simple super-renormalizable vector model.

hep-th↗

arXiv2013

Holonomy Operator and Quantization Ambiguities on Spinor Space

We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant `generalized ladder operators' recently introduced in the U(N) approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.

gr-qc↗

arXiv2012

On terminal forms for topological polynomials for ribbon graphs: The $N$-petal flower

The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph $\cG$, the related polynomial should be computable from the knowledge of the terminal forms of $\cG$ namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with $N\ge 1$ petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced.

math.CO↗

arXiv2012

Entropy in the Classical and Quantum Polymer Black Hole Models

We investigate the entropy counting for black hole horizons in loop quantum gravity (LQG). We argue that the space of 3d closed polyhedra is the classical counterpart of the space of SU(2) intertwiners at the quantum level. Then computing the entropy for the boundary horizon amounts to calculating the density of polyhedra or the number of intertwiners at fixed total area. Following the previous work arXiv:1011.5628, we dub these the classical and quantum polymer models for isolated horizons in LQG. We provide exact micro-canonical calculations for both models and we show that the classical counting of polyhedra accounts for most of the features of the intertwiner counting (leading order entropy and log-correction), thus providing us with a simpler model to further investigate correlations and dynamics. To illustrate this, we also produce an exact formula for the dimension of the intertwiner space as a density of "almost-closed polyhedra".

gr-qc↗

arXiv2012

Generating Functions for Coherent Intertwiners

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.

gr-qc↗

arXiv2012

Group theoretical Quantization of Isotropic Loop Cosmology

We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemeterizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the su(1,1) Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical non-vanishing minimum volume. Since the Hamiltonian is an element in the su(1,1) Lie algebra, the dynamics is now implemented as SU(1,1) transformations. At the quantum level, the system is quantized as a time-like irreducible representation of the group SU(1,1). These representations are labeled by a half-integer spin, which gives the minimal volume. They provide superselection sectors without quantization anomalies and no factor ordering ambiguity arises when representing the Hamiltonian. We then explicitly construct SU(1,1) coherent states to study the quantum evolution. They not only provide semiclassical states but truly dynamical coherent states. Their use further clarifies the nature of the bounce that resolves the big bang singularity.

gr-qc↗

arXiv2012

Holomorphic Lorentzian Simplicity Constraints

We develop an Hamiltonian representation of the sl(2,C) algebra on a phase space consisting of N copies of twistors, or bi-spinors. We identify a complete set of global invariants, and show that they generate a closed algebra including gl(N,C) as a subalgebra. Then, we define the linear and quadratic simplicity constraints which reduce the spinor variables to (framed) 3d spacelike polyhedra embedded in Minkowski spacetime. Finally, we introduce a new version of the simplicity constraints which (i) are holomorphic and (ii) Poisson-commute with each other, and show their equivalence to the linear and quadratic constraints.

gr-qc↗