Showing papers on "Quantum geometry published in 2016"

Liouville Quantum Gravity on the Riemann Sphere

Abstract: In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance un-der P SL 2 (C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quantum Gravity.

Diffeomorphism-invariant observables and their nonlocal algebra

Abstract: Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics. These can be thought of as operators that create a particle, together with its inseparable gravitational field, and reduce to usual field operators of quantum field theory in the weak-gravity limit; they include both Wilson-line operators and those creating a Coulombic field configuration. We also describe operators creating the field of a particle in motion; as in the electromagnetic case, these are expected to help address infrared problems. An important characteristic of the quantum theory of gravity is the algebra of its observables. We show that the commutators of the simple observables of this paper are nonlocal, with nonlocality becoming significant in strong field regions, as predicted previously on general grounds.

Quantum Cosmology from Group Field Theory Condensates: a Review

Abstract: We give, in some detail, a critical overview over recent work towards deriving a cosmological phenomenology from the fundamental quantum dynamics of group field the- ory (GFT), based on the picture of a macroscopic universe as a \condensate" of a large num- ber of quanta of geometry which are given by excitations of the GFT field over a o-space" vacuum. We emphasise conceptual foundations, relations to other research programmes in GFT and the wider context of loop quantum gravity (LQG), and connections to the quantum physics of real Bose{Einstein condensates. We show how to extract an effective dynamics for GFT condensates from the microscopic GFT physics, and how to compare it with predictions of more conventional quantum cosmology models, in particular loop quan- tum cosmology (LQC). No detailed familiarity with the GFT formalism is assumed.

Planck star tunneling time: An astrophysically relevant observable from background-free quantum gravity

Abstract: A gravitationally collapsed object can bounce out from its horizon via a tunnelling process that violates the classical equations in a finite region. Since tunnelling is a nonperturbative phenomenon, it cannot be described in terms of quantum fluctuations around a classical solution, and a background-free formulation of quantum gravity is needed to analyze it. Here, we use loop quantum gravity to compute the amplitude for this process, in a first approximation. The amplitude determines the tunnelling time as a function of the mass. This is the key information to evaluate the relevance of this process for the interpretation of fast radio bursts or high-energy cosmic rays. The calculation offers a template and a concrete example of how a background-free quantum theory of gravity can be used to compute a realistic observable quantity.

Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates

Abstract: We study the effective cosmological dynamics, emerging as the hydrodynamics of simple condensate states, of a group field theory model for quantum gravity coupled to a massless scalar field and reduced to its isotropic sector. The quantum equations of motion for these group field theory condensate states are given in relational terms with respect to the scalar field, from which effective dynamics for spatially flat, homogeneous and isotropic space-times can be extracted. The result is a generalization of the Friedmann equations, including quantum gravity modifications, in a specific regime of the theory. The classical Friedmann equations of general relativity are recovered in a suitable semi-classical limit for some range of parameters of the microscopic dynamics. An important result is that the quantum geometries associated with these GFT condensate states are non-singular: a bounce generically occurs in the Planck regime. For some choices of condensate states, these modified Friedmann equations are very similar to those of loop quantum cosmology.

Canonical energy is quantum Fisher information

Abstract: In quantum information theory, Fisher Information is a natural metric on the space of perturbations to a density matrix, defined by calculating the relative entropy with the unperturbed state at quadratic order in perturbations. In gravitational physics, Canonical Energy defines a natural metric on the space of perturbations to spacetimes with a Killing horizon. In this paper, we show that the Fisher information metric for perturbations to the vacuum density matrix of a ball-shaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge R B of Anti-de-Sitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive definite. Thus, for physical perturbations to anti-de-Sitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This second-order constraint on the metric extends the first order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein’s equations.

Network geometry with flavor: From complexity to quantum geometry.

Abstract: Network geometry is attracting increasing attention because it has a wide range of applications, ranging from data mining to routing protocols in the Internet. At the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity. In network geometry, simplicial complexes describing the interaction between two or more nodes play a special role. In fact these structures can be used to discretize a geometrical d-dimensional space, and for this reason they have already been widely used in quantum gravity. Here we introduce the network geometry with flavor s=-1,0,1 (NGF) describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics. The NGF can generate discrete geometries of different natures, ranging from chains and higher-dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution, and nontrivial community structure. The NGF admits as limiting cases both the Bianconi-Barabasi models for complex networks, the stochastic Apollonian network, and the recently introduced model for complex quantum network manifolds. The thermodynamic properties of NGF reveal that NGF obeys a generalized area law opening a new scenario for formulating its coarse-grained limit. The structure of NGF is strongly dependent on the dimensionality d. In d=1 NGFs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale-free degree distribution. Instead, for NGF with dimension d>1 it is not necessary to have an explicit preferential attachment rule to generate scale-free topologies. We also show that NGF admits a quantum mechanical description in terms of associated quantum network states. Quantum network states evolve by a Markovian dynamics and a quantum network state at time t encodes all possible NGF evolutions up to time t. Interestingly the NGF remains fully classical but its statistical properties reveal the relation to its quantum mechanical description. In fact the δ-dimensional faces of the NGF have generalized degrees that follow either the Fermi-Dirac, Boltzmann, or Bose-Einstein statistics depending on the flavor s and the dimensions d and δ.

Remnant Geometric Hall Response in a Quantum Quench

Abstract: Out-of-equilibrium systems can host phenomena that transcend the usual restrictions of equilibrium systems. Here, we unveil how out-of-equilibrium states, prepared via a quantum quench in a two-band system, can exhibit a nonzero Hall-type current—a remnant Hall response—even when the instantaneous Hamiltonian is time reversal symmetric (in contrast to equilibrium Hall currents). Interestingly, the remnant Hall response arises from the coherent dynamics of the wave function that retain a remnant of its quantum geometry postquench, and can be traced to processes beyond linear response. Quenches in two-band Dirac systems are natural venues for realizing remnant Hall currents, which exist when either mirror or time-reversal symmetry are broken (before or after the quench). Its long time persistence, sensitivity to symmetry breaking, and decoherence-type relaxation processes allow it to be used as a sensitive diagnostic of the complex out-of-equilibrium dynamics readily controlled and probed in cold-atomic optical lattice experiments.

Investigation of the spinfoam path integral with quantum cuboid intertwiners

Abstract: In this work, we investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the spinfoam model by Engle, Pereira, Rovelli, Livine, Freidel and Krasnov. To tackle the problem, we restrict to a set of quantum geometries that reflects the large amount of lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, i.e. coherent intertwiners which describe a cuboidal geometry in the large-$j$ limit. Using asymptotic expressions for the vertex amplitude, we find several interesting properties of the state sum. First of all, the value of coupling constants in the amplitude functions determines whether geometric or nongeometric configurations dominate the path integral. Secondly, there is a critical value of the coupling constant $\ensuremath{\alpha}$, which separates two phases. In both phases, the diffeomorphism symmetry appears to be broken. In one, the dominant contribution comes from highly irregular, in the other from highly regular configurations, both describing flat Euclidean space with small quantum fluctuations around them, viewed in different coordinate systems. On the critical point diffeomorphism symmetry is nearly restored, however. Thirdly, we use the state sum to compute the physical norm of kinematical states, i.e. their norm in the physical Hilbert space. We find that states which describe boundary geometry with high torsion have an exponentially suppressed physical norm. We argue that this allows one to exclude them from the state sum in calculations.

Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory

Abstract: We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin–Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.

Hofstadter’s butterfly in quantum geometry

Abstract: We point out that the recent conjectural solution to the spectral problem for the Hamiltonian $H=e^{x}+e^{-x}+e^{p}+e^{-p}$ in terms of the refined topological invariants of a local Calabi-Yau geometry has an intimate relation with two-dimensional non-interacting electrons moving in a periodic potential under a uniform magnetic field. In particular, we find that the quantum A-period, determining the relation between the energy eigenvalue and the Kahler modulus of the Calabi-Yau, can be found explicitly when the quantum parameter $q=e^{i\hbar}$ is a root of unity, that its branch cuts are given by Hofstadter's butterfly, and that its imaginary part counts the number of states of the Hofstadter Hamiltonian. The modular double operation, exchanging $\hbar$ and $4\pi^2/\hbar$, plays an important role.

Testing Quantum Gravity Induced Nonlocality via Optomechanical Quantum Oscillators

Abstract: Several quantum gravity scenarios lead to physics below the Planck scale characterized by nonlocal, Lorentz invariant equations of motion. We show that such nonlocal effective field theories lead to a modified Schrodinger evolution in the nonrelativistic limit. In particular, the nonlocal evolution of optomechanical quantum oscillators is characterized by a spontaneous periodic squeezing that cannot be generated by environmental effects. We discuss constraints on the nonlocality obtained by past experiments, and show how future experiments (already under construction) will either see such effects or otherwise cast severe bounds on the nonlocality scale (well beyond the current limits set by the Large Hadron Collider). This paves the way for table top, high precision experiments on massive quantum objects as a promising new avenue for testing some quantum gravity phenomenology.

Loop Quantum Cosmology: A brief review

Abstract: In the last decade, progress on quantization of homogeneous cosmological spacetimes using techniques of loop quantum gravity has led to insights on various fundamental questions and has opened new avenues to explore Planck scale physics. These include the problem of singularities and their possible generic resolution, constructing viable non-singular models of the very early universe, and bridging quantum gravity with cosmological observations. These results, which emerge from an interplay of sophisticated analytical and numerical techniques, has also led to valuable hints on loop quantization of black hole and inhomogeneous spacetimes. In this review, we provide a summary of this progress while focusing on concrete examples of the quantization procedure and phenomenology of cosmological perturbations.

The Continuum Limit of Causal Fermion Systems

Abstract: This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and "quantum geometries." The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book provides all the necessary mathematical background and explains how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. The focus is on getting a mathematically sound connection between causal fermion systems and physical systems in Minkowski space. The book is intended for graduate students entering the field, and is furthermore a valuable reference work for researchers in quantum field theory and quantum gravity.

An elementary introduction to loop quantum gravity

Abstract: An introduction to loop quantum gravity is given, focussing on the fundamental aspects of the theory, different approaches to the dynamics, as well as possible future directions. It is structured in five lectures, including exercises, and requires only little prior knowledge of quantum mechanics, gauge theory, and general relativity. The main aim of these lectures is to provide non-experts with an elementary understanding of loop quantum gravity and to evaluate the state of the art of the field. Technical details are avoided wherever possible.

Impact of nonlinear effective interactions on group field theory quantum gravity condensates

Abstract: We present the numerical analysis of effectively interacting Group Field Theory (GFT) models in the context of the GFT quantum gravity condensate analogue of the Gross-Pitaevskii equation for real Bose-Einstein condensates including combinatorially local interaction terms. Thus we go beyond the usually considered construction for free models. More precisely, considering such interactions in a weak regime, we find solutions for which the expectation value of the number operator N is finite, as in the free case. When tuning the interaction to the strongly nonlinear regime, however, we obtain solutions for which N grows and eventually blows up, which is reminiscent of what one observes for real Bose-Einstein condensates, where a strong interaction regime can only be realized at high density. This behaviour suggests the breakdown of the Bogoliubov ansatz for quantum gravity condensates and the need for non-Fock representations to describe the system when the condensate constituents are strongly correlated. Furthermore, we study the expectation values of certain geometric operators imported from Loop Quantum Gravity in the free and interacting cases. In particular, computing solutions around the nontrivial minima of the interaction potentials, one finds, already in the weakly interacting case, a nonvanishing condensate population for which the spectra are dominated by the lowest nontrivial configuration of the quantum geometry. This result indicates that the condensate may indeed consist of many smallest building blocks giving rise to an effectively continuous geometry, thus suggesting the interpretation of the condensate phase to correspond to a geometric phase.

Observational exclusion of a consistent loop quantum cosmology scenario

Abstract: It is often argued that inflation erases all the information about what took place before it started. Quantum gravity, relevant in the Planck era, seems therefore mostly impossible to probe with cosmological observations. In general, only very \textit{ad hoc} scenarios or hyper fine-tuned initial conditions can lead to observationally testable theories. Here we consider a well-defined and well motivated candidate quantum cosmology model that predicts inflation. Using the most recent observational constraints on the cosmic microwave background B modes, we show that the model is excluded for all its parameter space, without any tuning. Some important consequences are drawn for the \textit{deformed algebra approach} to loop quantum cosmology. We emphasize that neither loop quantum cosmology in general nor loop quantum gravity are disfavored by this study but their falsifiability is established.

Quantum gravity kinematics from extended TQFTs

Abstract: We show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously-introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum, which are essential to allow for a representation of the holonomies and fluxes. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to (2+1)-dimensional gravity with a cosmological constant. The new representation presents a number of advantages over the representations which exist so far. It possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. The notion of basic excitations leads to a fusion basis which offers exciting possibilities for constructing states with interesting global properties. The work presented here showcases how the framework of extended TQFTs can help design new representations and understand the associated notion of basic excitations. This is essential for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.

Probing the geometry of the Laughlin state

Abstract: It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantumHall states—the filling v = 1/3 Laughlin state.We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a ‘pair amplitude’ operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. Furthermore, these various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.

The Fock space of loopy spin networks for quantum gravity

Abstract: 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 $$\delta $$ -distribution.

Symmetries of quantum spacetime in three dimensions

Abstract: By applying loop quantum gravity techniques to 3D gravity with a positive cosmological constant $\Lambda$, we show how the local gauge symmetry of the theory, encoded in the constraint algebra, acquires the quantum group structure of $so_q(4)$, with $ q = \exp{(i\hbar \sqrt{\Lambda}/2\kappa)}$. By means of an Inonu-Wigner contraction of the quantum group bi-algebra, keeping $\kappa$ finite, we obtain the kappa-Poincare algebra of the flat quantum space-time symmetries.

Geometry and dynamics of one-norm geometric quantum discord

Abstract: We investigate the geometry of one-norm geometric quantum discord and present a geometric interpretation of one-norm geometric quantum discord for a class of two-qubit states. It is found that one-norm geometric quantum discord has geometric behavior different from that described in Lang and Caves (Phys Rev Lett 105:150501, 2010), Li et al. (Phys Rev A 83:022321, 2011) and Yao et al. (Phys Lett A 376:358---364, 2012). We also compare the dynamics of the one-norm geometric quantum discord and other measures of quantum correlations under correlated noise. It is shown that different decoherent channels bring different influences to quantum correlations measured by concurrence, entropic quantum discord and geometric quantum discord, which depend on the memory parameter and decoherence parameter. We lay emphasis on the behaviors such as entanglement sudden death and sudden transition of quantum discord. Finally, we study the dynamical behavior of one-norm geometric quantum discord in one-dimensional anisotropic XXZ model by utilizing the quantum renormalization group method. It is shown that the one-norm geometric quantum discord demonstrates quantum phase transition through renormalization group approach.

Quantum self-gravitating collapsing matter in a quantum geometry

Abstract: The problem of how space–time responds to gravitating quantum matter in full quantum gravity has been one of the main questions that any program of quantization of gravity should address. Here we analyze this issue by considering the quantization of a collapsing null shell coupled to spherically symmetric loop quantum gravity. We show that the constraint algebra of canonical gravity is Abelian both classically and when quantized using loop quantum gravity techniques. The Hamiltonian constraint is well defined and suitable Dirac observables characterizing the problem were identified at the quantum level. We can write the metric as a parameterized Dirac observable at the quantum level and study the physics of the collapsing shell and black hole formation. We show how the singularity inside the black hole is eliminated by loop quantum gravity and how the shell can traverse it. The construction is compatible with a scenario in which the shell tunnels into a baby universe inside the black hole or one in which it could emerge through a white hole.

Inflation with the Starobinsky potential in Loop Quantum Cosmology

Abstract: A self-consistent pre-inflationary extension of the inflationary scenario with the Starobinsky potential, favored by Planck data, is studied using techniques from loop quantum cosmology (LQC). The results are compared with the quadratic potential previously studied. Planck scale completion of the inflationary paradigm and observable signatures of LQC are found to be robust under the change of the inflation potential. The entire evolution, from the quantum bounce all the way to the end of inflation, is compatible with observations. Occurrence of desired slow-roll phase is almost inevitable and natural initial conditions exist for both the background and perturbations for which the resulting power spectrum agrees with recent observations. There exist initial data for which the quantum gravitational corrections to the power spectrum are potentially observable.

Quantum Geometry A Statistical Field Theory Approach

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Interferometric Tests of Planckian Quantum Geometry Models

Abstract: The effect of Planck scale quantum geometrical effects on measurements with interferometers is estimated with standard physics, and with a variety of proposed extensions. It is shown that effects are negligible in standard field theory with canonically quantized gravity. Statistical noise levels are estimated in a variety of proposals for nonstandard metric fluctuations, and these alternatives are constrained using upper bounds on stochastic metric fluctuations from LIGO. Idealized models of several interferometer system architectures are used to predict signal noise spectra in a quantum geometry that cannot be described by a fluctuating metric, in which position noise arises from holographic bounds on directional information. Lastly, predictions in this case are shown to be close to current and projected experimental bounds.

Measuring finite quantum geometries via quasi-coherent states

Abstract: We develop a systematic approach to determine and measure numerically the geometry of generic quantum or 'fuzzy' geometries realized by a set of finite-dimensional Hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace- or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.

Quantum self-gravitating collapsing matter in a quantum geometry

Abstract: The problem of how space-time responds to gravitating quantum matter in full quantum gravity has been one of the main questions that any program of quantization of gravity should address. Here we analyze this issue by considering the quantization of a collapsing null shell coupled to spherically symmetric loop quantum gravity. We show that the constraint algebra of canonical gravity is Abelian both classically and when quantized using loop quantum gravity techniques. The Hamiltonian constraint is well defined and suitable Dirac observables characterizing the problem were identified at the quantum level. We can write the metric as a parameterized Dirac observable at the quantum level and study the physics of the collapsing shell and black hole formation. We show how the singularity inside the black hole is eliminated by loop quantum gravity and how the shell can traverse it. The construction is compatible with a scenario in which the shell tunnels into a baby universe inside the black hole or one in which it could emerge through a white hole.

Quantum Reduced Loop Gravity and the foundation of Loop Quantum Cosmology

Abstract: Quantum Reduced Loop Gravity is a promising framework for linking Loop Quantum Gravity and the effective semiclassical dynamics of Loop Quantum Cosmology. We review its basic achievements and its main perspectives, outlining how it provides a quantum description of the Universe in terms of a cuboidal graph which constitutes the proper framework for applying loop techniques in a cosmological setting.