1,721,067 research outputs found
Noncommutative Fourier transform for the Lorentz group via the Duflo map
Full text linkWe defined a non-commutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the non-commutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map
Ryu-Takayanagi formula for symmetric random tensor networks
No full textWe consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the Rényi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research
Functional renormalization group analysis of tensorial group field theories on Rd
No full textRank-d tensorial group field theories are quantum field theories (QFTs) defined on a group manifold G×d, which represent a nonlocal generalization of standard QFT and a candidate formalism for quantum gravity, since, when endowed with appropriate data, they can be interpreted as defining a field theoretic description of the fundamental building blocks of quantum spacetime. Their renormalization analysis is crucial both for establishing their consistency as quantum field theories and for studying the emergence of continuum spacetime and geometry from them. In this paper, we study the renormalization group flow of two simple classes of tensorial group field theories (TGFTs), defined for the group G=R for arbitrary rank, both without and with gauge invariance conditions, by means of functional renormalization group techniques. The issue of IR divergences is tackled by the definition of a proper thermodynamic limit for TGFTs. We map the phase diagram of such models, in a simple truncation, and identify both UV and IR fixed points of the RG flow. Encouragingly, for all the models we study, we find evidence for the existence of a phase transition of condensation type
Functional Renormalization Group analysis of a Tensorial Group Field Theory on R
No full textWe study a model of Tensorial Group Field Theory (TGFT) on R3 from the point of view of the Functional Renormalization Group (FRG). This is the first attempt to apply a renormalization procedure to a TGFT model defined over a non-compact group manifold. IR divergences (with respect to the metric on R) coming from the non-compactness of the group are regularised via compactification, and a thermodynamic limit is then taken. We identify then IR and UV fixed points of the RG flow and find strong hints of a phase transition of the TGFT system from a symmetric to a broken or condensate phase in the IR
Statistical equilibrium of tetrahedra from maximum entropy principle
No full textDiscrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on nonlocal, combinatorial gluing constraints that are modeled as multiparticle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes's principle of constrained maximization of entropy, which has been shown recently to play an important role in characterizing equilibrium in background-independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory
Group field theory and tensor networks: Towards a Ryu-Takayanagi formula in full quantum gravity
No full textWe establish a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, we use this dictionary to compute the Rényi entropy of such states and recover the Ryu-Takayanagi formula, in two different cases corresponding to two different truncations/approximations, suggested by the established correspondence
Generalized quantum gravity condensates for homogeneous geometries and cosmology
Full text linkWe construct a generalized class of quantum gravity condensate states, that allows the description of continuum homogeneous quantum geometries within the full theory. They are based on similar ideas already applied to extract effective cosmological dynamics from the group field theory formalism, and thus also from loop quantum gravity. However, they represent an improvement over the simplest condensates used in the literature, in that they are defined by an infinite superposition of graph-based states encoding in a precise way the topology of the spatial manifold. The construction is based on the definition of refinement operators on spin network states, written in a second quantized language. The construction lends itself easily to be applied also to the case of spherically symmetric quantum geometries
Horizon entropy from quantum gravity condensates
Full text linkWe construct condensate states encoding the continuum spherically symmetric quantum geometry of a horizon in full quantum gravity, i.e., without any classical symmetry reduction, in the group field theory formalism. Tracing over the bulk degrees of freedom, we show how the resulting reduced density matrix manifestly exhibits a holographic behavior. We derive a complete orthonormal basis of eigenstates for the reduced density matrix of the horizon and use it to compute the horizon entanglement entropy. By imposing consistency with the horizon boundary conditions and semiclassical thermodynamical properties, we recover the Bekenstein-Hawking entropy formula for any value of the Immirzi parameter. Our analysis supports the equivalence between the von Neumann (entanglement) entropy interpretation and the Boltzmann (statistical) one. © 2016 American Physical Society
Group field theory and holographic tensor networks: Dynamical corrections to the Ryu-Takayanagi formula
No full textWe introduce a generalised class of (symmetric) random tensor network states in the framework of group field theory. In this setting, we compute the Rényi entropy for a generic bipartite state via a mapping to the partition function of a topological 3D BF theory, realised as a simple interacting group field theory. The expectation value of the entanglement entropy is calculated by an expansion into stranded Feynman graphs and is shown to be captured by a Ryu-Takayanagi formula. For the simple case of a 3D BF theory, we can prove the linear corrections, given by a polynomial perturbation of the Gaussian measure, to be negligible for a broad class of networks
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