arXiv subjects

Florian Girelli

Publications and source records attributed to Florian Girelli.

At least 37 records · Page 2Linked to original sources

arXiv2021

Discretization of 4d Poincaré BF theory: from groups to 2-groups

We study the discretization of a Poincaré/Euclidean BF theory. Upon the addition of a boundary term, this theory is equivalent to the BFCG theory defined in terms of the Poincaré/Euclidean 2-group. At an intermediate step in the discretization, we note that there are multiple options for how to proceed. One option brings us back to recovering the discrete variables and phase space of BF theory. Another option allows us to rediscover the phase space related to the G-networks given in [2]. Indeed, our main result is that we are now able to relate the continuum fields with the discrete variables in [2]. This relation is important to determine how to implement the simplicity constraints to recover gravity using the BFCG action. In fact, we show that such relation is not as simple as in the BF discretization: the discretized variables on the triangles actually depend on several of the continuum fields instead of solely the continuum B-field. We also compare and contrast the discretized BF and BFCG models as pairs of dual 2-groups. This work highlights (again) how the choice of boundary term influences the resulting symmetry structure of the \textit{discretized theory} -- and hence ultimately the choice of quantum states.

hep-th↗

arXiv2020

2+1D Loop Quantum Gravity on the Edge

We develop a new perspective on the discretization of the phase space structure of gravity in 2+1 dimensions as a piecewise-flat geometry in 2 spatial dimensions. Starting from a subdivision of the continuum geometric and phase space structure into elementary cells, we obtain the loop gravity phase space coupled to a collection of effective particles carrying mass and spin, which measure the curvature and torsion of the geometry. We show that the new degrees of freedom associated to the particle-like elements can be understood as edge modes, which appear in the decomposition of the continuum theory into subsystems and do not cancel out in the gluing of cells along codimension 2 defects. These new particle-like edge modes are gravitationally dressed in an explicit way. This provides a detailed explanation of the relations and differences between the loop gravity phase space and the one deduced from the continuum theory.

gr-qc↗

arXiv2020

First-order formulation of teleparallel gravity and dual loop gravity

There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{loop gravity} kinematical phase space where the zero torsion condition is enforced {first} and the other is the \emph{dual loop gravity} kinematical phase space where curvature is imposed to vanish {first}. Our argument is based on the observation that the GR first-order Einstein--Cartan action can also be seen as a first-order action for teleparallel gravity up to a boundary term. The results of our paper suggest that the \emph{dual loop gravity} framework is a natural discretization of teleparallel gravity, whereas \emph{loop gravity} is naturally related to the standard GR metric description.

gr-qc↗

arXiv2019

Quantum geometry from higher gauge theory

Higher gauge theories play a prominent role in the construction of 4d topological invariants and have been long ago proposed as a tool for 4d quantum gravity. The Yetter lattice model and its continuum counterpart, the BFCG theory, generalize BF theory to 2-gauge groups and -- when specialized to 4d and the Poincaré 2-group -- they provide an exactly solvable topologically-flat version of 4d general relativity. The 2-Poincaré Yetter model was conjectured to be equivalent to a state sum model of quantum flat spacetime developed by Baratin and Freidel after work by Korepanov (KBF model). This conjecture was motivated by the origin of the KBF model in the theory of 2-representations of the Poincaré 2-group. Its proof, however, has remained elusive due to the lack of a generalized Peter-Weyl theorem for 2-groups. In this work we prove this conjecture. Our proof avoids the Peter-Weyl theorem and rather leverages the geometrical content of the Yetter model. Key for the proof is the introduction of a kinematical boundary Hilbert space on which 1- and 2-Lorentz invariance is imposed. Geometrically this allows the identification of (quantum) tetrad variables and of the associated (quantum) Levi-Civita connection. States in this Hilbert space are labelled by quantum numbers that match the 2-group representation labels. Our results open exciting opportunities for the construction of new representations of quantum geometries. Compared to loop quantum gravity, the higher gauge theory framework provides a quantum representation of the ADM-Regge initial data, including an identification of the intrinsic and extrinsic curvature. Furthermore, it leads to a version of the diffeomorphism and Hamiltonian constraints that acts on the vertices of the discretization, thus providing a prospect for a quantum realization of the hypersurface deformation algebra in 4d.

gr-qc↗

arXiv2018

Dual loop quantizations of 3d gravity

The loop quantization of 3d gravity consists in defining the Hilbert space of states satisfying the Gauß constraint and the flatness constraint. The Gauß constraint is enforced at the kinematical level by introducing spin networks which form a basis for the Hilbert space of gauge invariant functionals. The flatness constraint is implemented at the dynamical level via the Ponzano-Regge state-sum model. We propose in this work a dual loop quantization scheme where the role of the constraints is exchanged. The flatness constraint is imposed first via the introduction of a new basis labeled by group variables, while the Gauß constraint is implemented dynamically using a projector which is related to the Dijkgraaf-Witten model. We discuss how this alternative quantization program is related to 3d teleparallel gravity.

gr-qc↗

arXiv2017

SO*(2N) coherent states for loop quantum gravity

A SU(2) intertwiner with N legs can be interpreted as the quantum state of a convex polyhedron with N faces (when working in 3d). We show that the intertwiner Hilbert space carries a representation of the non-compact group SO*(2N). This group can be viewed as the subgroup of the symplectic group Sp(4N,R) which preserves the SU(2) invariance. We construct the associated Perelomov coherent states and discuss the notion of semi-classical limit, which is more subtle that we could expect. Our work completes the work by Freidel and Livine which focused on the U(N) subgroup of SO*(2N).

math-ph↗

arXiv2017

Discretization of 3d gravity in different polarizations

We study the discretization of 3d gravity with $Λ=0$ following the loop quantum gravity framework. In the process, we realize that different choices of polarization are possible. This allows to introduce a new discretization based on the triad as opposed to the connection as in the standard loop quantum gravity framework. We also identify the classical non-trivial symmetries of discrete gravity, namely the Drinfeld double, given in terms of momentum maps. Another choice of polarization is given by the Chern-Simons formulation of gravity. Our framework also provides a new discretization scheme of Chern-Simons, which keeps track of the link between the continuum variables and the discrete ones. We show how the Poisson bracket we recover between the Chern-Simons holonomies allows to recover the Goldman bracket. There is also a transparent link between the discrete Chern-Simons formulation and the discretization of gravity based on the connection (loop gravity) or triad variables (dual loop gravity).

gr-qc↗

arXiv2015

3d Lorentzian loop quantum gravity and the spinor approach

We consider the generalization of the "spinor approach" to the Lorentzian case, in the context of 3d loop quantum gravity with cosmological constant $Λ=0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary finite-dimensional ones, obtained in the process of generalizing the Wigner-Eckart theorem to SU(1,1). We use SU(1,1) tensor operators to build observables and a solvable quantum Hamiltonian constraint, analogue of the one introduced by V. Bonzom and his collaborators in the Euclidean case (with both $Λ=0$ and $Λ\neq0$). We show that the Lorentzian Ponzano-Regge amplitude is solution of the quantum Hamiltonian constraint by recovering the Biedenharn-Elliott relation (generalized to the case where unitary and non-unitary SU(1,1) representations are coupled to each other). Our formalism is sufficiently general that both the Lorentzian and the Euclidean case can be recovered (with $Λ=0$).

gr-qc↗

arXiv2014

Changing quantum reference frames

We consider the process of changing reference frames in the case where the reference frames are quantum systems. We find that, as part of this process, decoherence is necessarily induced on any quantum system described relative to these frames. We explore this process with examples involving reference frames for phase and orientation. Quantifying the effect of changing quantum reference frames serves as a first step in developing a relativity principle for theories in which all objects including reference frames are necessarily quantum.

quant-ph↗

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

Towards the Turaev-Viro amplitudes from a Hamiltonian constraint

3D Loop Quantum Gravity with a vanishing cosmological constant can be related to the quantization of the $\textrm{SU}(2)$ BF theory discretized on a lattice. At the classical level, this discrete model characterizes discrete flat geometries and its phase space is built from $T^\ast \textrm{SU}(2)$. In a recent paper \cite{HyperbolicPhaseSpace}, this discrete model was deformed using the Poisson-Lie group formalism and was shown to characterize discrete hyperbolic geometries while being still topological. Hence, it is a good candidate to describe the discretization of $\textrm{SU}(2)$ BF theory with a (negative) cosmological constant. We proceed here to the quantization of this model. At the kinematical level, the Hilbert space is spanned by spin networks built on $\mathcal{U}_{q}(\mathfrak{su}(2))$ (with $q$ real). In particular, the quantization of the discretized Gauss constraint leads naturally to $\mathcal{U}_{q}(\mathfrak{su}(2))$ intertwiners. We also quantize the Hamiltonian constraint on a face of degree 3 and show that physical states are proportional to the quantum 6j-symbol. This suggests that the Turaev-Viro amplitude with $q$ real is a solution of the quantum Hamiltonian. This model is therefore a natural candidate to describe 3D loop quantum gravity with a (negative) cosmological constant.

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

Observables in Loop Quantum Gravity with a cosmological constant

An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant $Λ$. In 3d, Chern-Simons theory provides some guiding lines: $Λ$ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of $Λ\neq0$. In this article, we introduce by hand the quantum group $\mathcal{U}_{q}(\mathfrak{su}(2))$ into the LQG framework, that is we deal with $\mathcal{U}_{q}(\mathfrak{su}(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for $\mathcal{U}_{q}(\mathfrak{su}(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U($n$) formalism in this deformed case, which is given by the quantum group $\mathcal{U}_{q}(\mathfrak{u}(n))$. We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory.

gr-qc↗

arXiv2013

Quantum hyperbolic geometry in loop quantum gravity with cosmological constant

Loop Quantum Gravity (LQG) is an attempt to describe the quantum gravity regime. Introducing a non-zero cosmological constant $Λ$ in this context has been a withstanding problem. Other approaches, such as Chern-Simons gravity, suggest that quantum groups can be used to introduce $Λ$ in the game. Not much is known when defining LQG with a quantum group. Tensor operators can be used to construct observables in any type of discrete quantum gauge theory with a classical/quantum gauge group. We illustrate this by constructing explicitly geometric observables for LQG defined with a quantum group and show for the first time that they encode a quantized hyperbolic geometry. This is a novel argument pointing out the usefulness of quantum groups as encoding a non-zero cosmological constant. We conclude by discussing how tensor operators provide the right formalism to unlock the LQG formulation with a non-zero cosmological constant.

gr-qc↗

arXiv2012

Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics

Research during the last decade demonstrates that effects originating on the Planck scale are currently being tested in multiple observational contexts. In this review we discuss quantum gravity phenomenological models and their possible links to loop quantum gravity. Particle frameworks, including kinematic models, broken and deformed Poincaré symmetry, non-commutative geometry, relative locality and generalized uncertainty principle, and field theory frameworks, including Lorentz violating operators in effective field theory and non-commutative field theory, are discussed. The arguments relating loop quantum gravity to models with modified dispersion relations are reviewed, as well as, arguments supporting the preservation of local Lorentz invariance. The phenomenology related to loop quantum cosmology is briefly reviewed, with a focus on possible effects that might be tested in the near future. As the discussion makes clear, there remains much interesting work to do in establishing the connection between the fundamental theory of loop quantum gravity and these specific phenomenological models, in determining observational consequences of the characteristic aspects of loop quantum gravity, and in further refining current observations. Open problems related to these developments are highlighted. characteristic aspects of loop quantum gravity, and in further refining current observations. Open problems related to these developments are highlighted.

gr-qc↗

arXiv2011

Spinors and Voros star-product for Group Field Theory: First Contact

In the context of non-commutative geometries, we develop a group Fourier transform for the Lie group SU(2). Our method is based on the Schwinger representation of the Lie algebra su(2) in terms of spinors. It allows us to prove that the non-commutative R^3 space dual to the SU(2) group is in fact of the Moyal-type and endowed with the Voros star-product when expressed in the spinor variables. Finally, from the perspective of quantum gravity, we discuss the application of these new tools to group field theories for spinfoam models and their interpretation as non-commutative field theories with quantum-deformed symmetries.

gr-qc↗

arXiv2011

Diffeomorphisms in group field theories

We study the issue of diffeomorphism symmetry in group field theories (GFT), using the recently introduced noncommutative metric representation. In the colored Boulatov model for 3d gravity, we identify a field (quantum) symmetry which ties together the vertex translation invariance of discrete gravity, the flatness constraint of canonical quantum gravity, and the topological (coarse-graining) identities for the 6j-symbols. We also show how, for the GFT graphs dual to manifolds, the invariance of the Feynman amplitudes encodes the discrete residual action of diffeomorphisms in simplicial gravity path integrals. We extend the results to GFT models for higher dimensional BF theories and discuss various insights that they provide on the GFT formalism itself.

hep-th↗

arXiv2010

Snyder Space-Time: K-Loop and Lie Triple System

Different deformations of the Poincare symmetries have been identified for various non-commutative spaces (e.g. $κ$-Minkowski, $sl(2,R)$, Moyal). We present here the deformation of the Poincare symmetries related to Snyder space-time. The notions of smooth "K-loop", a non-associative generalization of Abelian Lie groups, and its infinitesimal counterpart given by the Lie triple system are the key objects in the construction.

math-ph↗