Showing papers on "Quantum geometry published in 2014"

Group Field Theory and Loop Quantum Gravity

Abstract: We introduce the group field theory formalism for quantum gravity, mainly from the point of view of loop quantum gravity, stressing its promising aspects. We outline the foundations of the formalism, survey recent results and offer a perspective on future developments.

Quantum black holes in loop quantum gravity

Abstract: We study the quantization of spherically symmetric vacuum spacetimes within loop quantum gravity. In particular, we give additional details about our previous work in which we showed that one could complete the quantization of the model and that the singularity inside black holes is resolved. Moreover, we consider an alternative quantization based on a slightly different kinematical Hilbert space. The ambiguity in kinematical spaces stems from how one treats the periodicity of one of the classical variables in these models. The corresponding physical Hilbert spaces solve the diffeomorphism and Hamiltonian constraint but their intrinsic structure is radically different depending on the kinematical Hilbert space one started from. In both cases there are quantum observables that do not have a classical counterpart. However, one can show that at the end of the day, by examining Dirac observables, both quantizations lead to the same physical predictions.

Why are the effective equations of loop quantum cosmology so accurate

Abstract: We point out that the relative Heisenberg uncertainty relations vanish for noncompact spaces in homogeneous loop quantum cosmology. As a consequence, for sharply peaked states quantum fluctuations in the scale factor never become important, even near the bounce point. This shows why quantum backreaction effects remain negligible and explains the surprising accuracy of the effective equations in describing the dynamics of sharply peaked wave packets. This also underlines the fact that minisuperspace models---where it is global variables that are quantized---do not capture the local quantum fluctuations of the geometry.

Statistics, holography, and black hole entropy in loop quantum gravity

Abstract: In loop quantum gravity the quantum states of a black hole horizon are produced by point-like discrete quantum geometry excitations (or {\em punctures}) labelled by spin $j$. The excitations possibly carry other internal degrees of freedom also, and the associated quantum states are eigenstates of the area $A$ operator. On the other hand, the appropriately scaled area operator $A/(8\pi\ell)$ is also the physical Hamiltonian associated with the quasilocal stationary observers located at a small distance $\ell$ from the horizon. Thus, the local energy is entirely accounted for by the geometric operator $A$. We assume that: In a suitable vacuum state with regular energy momentum tensor at and close to the horizon the local temperature measured by stationary observers is the Unruh temperature and the degeneracy of `matter' states is exponential with the area $\exp{(\lambda A/\ell_p^2)}$---this is supported by the well established results of QFT in curved spacetimes, which do not determine $\lambda$ but asserts an exponential behaviour. The geometric excitations of the horizon (punctures) are indistinguishable. In the semiclassical limit the area of the black hole horizon is large in Planck units. It follows that: Up to quantum corrections, matter degrees of freedom saturate the holographic bound, {\em viz.} $\lambda=\frac{1}{4}$. Up to quantum corrections, the statistical black hole entropy coincides with Bekenstein-Hawking entropy $S={A}/({4\ell_p^2})$. The number of horizon punctures goes like $N\propto \sqrt{A/\ell_p^2}$, i.e the number of punctures $N$ remains large in the semiclassical limit. Fluctuations of the horizon area are small while fluctuations of the area of an individual puncture are large. A precise notion of local conformal invariance of the thermal state is recovered in the $A\to\infty$ limit where the near horizon geometry becomes Rindler.

Quantization ambiguities and bounds on geometric scalars in anisotropic loop quantum cosmology

Abstract: We study quantization ambiguities in loop quantum cosmology that arise for space-times with non-zero spatial curvature and anisotropies. Motivated by lessons from different possible loop quantizations of the closed Friedmann–Lemaitre–Robertson–Walker cosmology, we find that using open holonomies of the extrinsic curvature, which due to gauge-fixing can be treated as a connection, leads to the same quantum geometry effects that are found in spatially flat cosmologies. More specifically, in contrast to the quantization based on open holonomies of the Ashtekar–Barbero connection, the expansion and shear scalars in the effective theories of the Bianchi type II and Bianchi type IX models have upper bounds, and these are in exact agreement with the bounds found in the effective theories of the Friedmann–Lemaitre–Robertson–Walker and Bianchi type I models in loop quantum cosmology. We also comment on some ambiguities present in the definition of inverse triad operators and their role.

Quantum Gravity via Causal Dynamical Triangulations

Abstract: Causal dynamical triangulations (CDT ) represent a lattice regularization of the sum over spacetime histories, providing us with a nonperturbative formulation of quantum gravity. The ultraviolet fixed points of the lattice theory can be used to define a continuum quantum field theory, potentially making contact with quantum gravity defined via asymptotic safety. We describe the formalism of CDT, its phase diagram, and the quantum geometries emerging from it. We also argue that the formalism should be able to describe a more general class of quantum-gravitational models of Hořava–Lifshitz type.

Non-Perturbative Quantum Geometry III

Abstract: The Nekrasov-Shatashvili limit of β-ensembles with polynomial potential and N = 2 supersymmetric gauge theories in the -background is intimately related to complex one-dimensional quantum mechanics. Multi-instanton corrections in quantum mechanics, inferable from exact quantization conditions, imply additional non-perturbative corrections to the Nekrasov-Shatashvili free energies. Besides filling some of the gaps in previous derivations, we present analytic expressions for such additional nonperturbative corrections in the case of SU(2) gauge theory expanded at strong coupling. In contrast, at weak coupling these additional non-perturbative corrections appear to be negligible.

Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers

Abstract: Interferometry with massive particles may have the potential to explore the limitations of standard quantum mechanics, in particular where it concerns its boundary with general relativity and the yet to be developed theory of quantum gravity. This development is hindered considerably by the lack of experimental evidence and testable predictions. Analyzing effects that appear to be common to many of such theories, such as a modification of the energy dispersion and of the canonical commutation relation within the standard framework of quantum mechanics, has been proposed as a possible way forward. Here we analyze in some detail the impact of a modified energy-momentum dispersion in a Ramsey-Bord\'e setup and provide achievable bounds of these correcting terms when operating such an interferometer with nanodiamonds. Thus, taking thermal and gravitational disturbances into account will show that without specific prerequisites, quantum gravity modifications may in general be suppressed requiring a revision of previously estimated bounds. As a possible solution we propose a stable setup which is rather insensitive to these effects. Finally, we address the problems of decoherence and pulse errors in such setups and discuss the scalings and advantages with increasing particle mass.

Quantum cosmology of (loop) quantum gravity condensates: an example

Abstract: Spatially homogeneous universes can be described in (loop) quantum gravity as condensates of elementary excitations of space. Their treatment is easiest in the second-quantized group field theory formalism, which allows the adaptation of techniques from the description of Bose–Einstein condensates in condensed matter physics. Dynamical equations for the states can be derived directly from the underlying quantum gravity dynamics. The analogue of the Gross–Pitaevskii equation defines an anisotropic quantum cosmology model, in which the condensate wavefunction becomes a quantum cosmology wavefunction on minisuperspace. To illustrate this general formalism, we give a mapping of the gauge-invariant geometric data for a tetrahedron to a minisuperspace of homogeneous anisotropic three-metrics. We then study an example for which we give the resulting quantum cosmology model in the general anisotropic case and derive the general analytical solution for isotropic universes. We discuss the interpretation of these solutions. We suggest that the WKB approximation used in previous studies, corresponding to semiclassical fundamental degrees of freedom of quantum geometry, should be replaced by a notion of semiclassicality that refers to large-scale observables instead.

From General Relativity to Quantum Gravity

Abstract: In general relativity (GR), spacetime geometry is no longer just a background arena but a physical and dynamical entity with its own degrees of freedom. We present an overview of approaches to quantum gravity in which this central feature of GR is at the forefront. However, the short distance dynamics in the quantum theory are quite different from those of GR and classical spacetimes and gravitons emerge only in a suitable limit. Our emphasis is on communicating the key strategies, the main results and open issues. In the spirit of this volume, we focus on a few avenues that have led to the most significant advances over the past 2-3 decades.

Exact results in N=8 Chern-Simons-matter theories and quantum geometry

Abstract: We show that, in ABJ(M) theories with N=8 supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group: they involve Jacobi theta functions on the spectral curve associated to the planar limit. Exact quantization conditions for the spectral problem of the Fermi gas are then obtained from the vanishing of the theta function. We also show that the partition function, as a function of N, can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.

Quantum Gravity, Dynamical Phase Space and String Theory

Abstract: In a natural extension of the relativity principle we argue that a quantum theory of gravity involves two fundamental scales associated with both dynamical space-time as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase space and in which space-time is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The space-time and momentum space dynamics, and thus dynamical phase space, is governed by a new version of the Renormalization Group.

The continuum limit of loop quantum gravity - a framework for solving the theory

Abstract: The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit. 1 Solving the dynamics of loop quantum gravity Loop quantum gravity led to a rigorous non–perturbative framework, in which to formulate the dynamics of quantum gravity. It allowed fascinating insights into quantum geometry and a possible structure of quantum space time. To get a complete picture of the theory – in the form of constructing the so–called physical Hilbert space – we need to construct the continuum limit. In the framework presented here physical states, i.e. solutions of the equations of motions of the theory, are constructed by taking the refinement limit via a coarse graining procedure. The conceptual underpinnings of this framework rely on the inductive limit Hilbert space construction used in loop quantum gravity to define the continuum (so far kinematical) Hilbert space. We point out the powerful concept of this inductive limit construction if one allows for a generalization of the refinement maps that define the inductive limit Hilbert spaces. It leads to a framework in which physical states are computed in a truncation scheme, where the type of truncation is determined by the dynamics itself. This procedure allows for an understanding of the dynamics of quantum gravity on all scales – which we here argue is to understand in terms of coarseness or fineness of configurations. The different scales of the theory are connected via the cylindrical consistency condition inherent in the inductive limit construction. This replaces the notion of renormalization flow in theories with a background scale. We start our considerations with a short explanation of the inductive limit construction in section 2 and discuss the difference between kinematical and dynamical understanding of the continuum limit. In section 3 we start with the task to construct the physical Hilbert space of the theory and explain that it necessitates the construction of the refinement limit for the dynamics of the theory. This results in an iterative coarse graining scheme, in which physical states – or amplitude maps – are constructed in a certain truncation, labelled by the coarseness or fineness of the discrete structures involved. The relation of this scheme with a renormalization flow is clarified in section 4. Concrete realizations of this scheme in the form of (decorated) tensor network methods are shortly explained in section 5. We then point out the powerful notion of diffeomorphism symmetry for discrete systems in section 6. The realization of this diffeomorphism symmetry is necessary for the definition of physical states, however also indicates that a continuum limit is reached. In this sense physical states can only be defined in the continuum limit. We end with a discussion and outlook of future developments in section 7. 2 Continuum limit in canonical loop quantum gravity Loop quantum gravity is formulated as a continuum theory, we therefore should clarify the need for a continuum limit in canonical loop quantum gravity. To this end we will shortly discuss how

Effects of quantum gravity on black holes

Abstract: In this review, we discuss effects of quantum gravity on black hole physics. After a brief review of the origin of the minimal observable length from various quantum gravity theories, we present the tunneling method. To incorporate quantum gravity effects, we modify the Klein-Gordon equation and Dirac equation by the modified fundamental commutation relations. Then we use the modified equations to discuss the tunneling radiation of scalar particles and fermions. The corrected Hawking temperatures are related to the quantum numbers of the emitted particles. Quantum gravity corrections slow down the increase of the temperatures. The remnants are observed as $M_{\hbox{Res}}\gtrsim \frac{M_p}{\sqrt{\beta_0}}$. The mass is quantized by the modified Wheeler-DeWitt equation and is proportional to $n$ in quantum gravity regime. The thermodynamical property of the black hole is studied by the influence of quantum gravity effects.

Loop quantum cosmology and the fate of cosmological singularities

Abstract: Singularities in general relativity such as the big bang and big crunch, and exotic singularities such as the big rip are the boundaries of the classical space- times. These events are marked by a divergence in the curvature invariants and the breakdown of the geodesic evolution. Recent progress on implementing techniques of loop quantum gravity to cosmological models reveals that such singularities may be generically resolved because of the quantum gravitational e ects. Due to the quantum geometry, which replaces the classical di erential geometry at the Planck scale, the big bang is replaced by a big bounce without any assumptions on the matter content or any fine tuning. In this manuscript, we discuss some of the main features of this approach and the results on the generic resolution of singularities for the isotropic as well as anisotropic models. Using e ective spacetime description of the quantum the- ory, we show the way quantum gravitational e ects lead to the universal bounds on the energy density, the Hubble rate and the anisotropic shear. We discuss the geodesic completeness in the e ective spacetime and the resolution of all of the strong singular- ities. It turns out that despite the bounds on energy density and the Hubble rate, there can be divergences in the curvature invariants. However such events are geodesically extendible, with tidal forces not strong enough to cause inevitable destruction of the in-falling objects.

Discreteness corrections and higher spatial derivatives in effective canonical quantum gravity

Abstract: Canonical quantum theories with discrete space may imply interesting effects. This paper presents a general effective description, paying due attention to the role of higher spatial derivatives in a local expansion and differences to higher time derivatives. In a concrete set of models, it is shown that spatial derivatives one order higher than the classical one are strongly restricted in spherically symmetric effective loop quantum gravity. Moreover, radial holonomy corrections alone cannot be anomaly free to this order.

Modified gravity from the quantum part of the metric

Abstract: It is shown that if a metric in quantum gravity can be decomposed as a sum of classical and quantum parts, then Einstein quantum gravity looks approximately like modified gravity with a nonminimal interaction between gravity and matter

Spectral dimension of quantum geometries

Abstract: The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them.

The Soccer-Ball Problem

Abstract: The idea that Lorentz-symmetry in momentum space could be modified but still remain observer-independent has received quite some attention in the recent years. This modified Lorentz-symmetry, which has been argued to arise in Loop Quantum Gravity, is being used as a phenomenological model to test possibly observable effects of quantum gravity. The most pressing problem in these models is the treatment of multi-particle states, known as the 'soccer-ball problem'. This article briefly reviews the problem and the status of existing solution attempts.

Solution to the Lorentzian quantum reality problem

Abstract: The quantum reality problem is that of finding a mathematically precise definition of a sample space of configurations of beables, events, histories, paths, or other mathematical objects, and a corresponding probability distribution, for any given closed quantum system. Given a solution, we can postulate that physical reality is described by one randomly chosen configuration drawn from the sample space. For a physically sensible solution, this postulate should imply quasiclassical physics in realistic models. In particular, it should imply the validity of Copenhagen quantum theory and classical dynamics in their respective domains. A Lorentzian solution applies to relativistic quantum theory or quantum field theory in Minkowski space and is defined in a way that respects Lorentz symmetry. We outline a new solution to the non-relativistic and Lorentzian quantum reality problems, and associated new generalizations of quantum theory.

Semiclassical dynamics of horizons in spherically symmetric collapse

Abstract: In this paper, we consider a semiclassical description of the spherically symmetric gravitational collapse with a massless scalar field. In particular, we employ an effective scenario provided by holonomy corrections from loop quantum gravity (LQG), to the homogeneous interior spacetime. The singularity that would arise at the final stage of the corresponding classical collapse, is resolved in this context and is replaced by a bounce. Our main purpose is to investigate the evolution of trapped surfaces during this semiclassical collapse. Within this setting, we obtain a threshold radius for the collapsing shells in order to have horizons formation. In addition, we study the final state of the collapse by employing a suitable matching at the boundary shell from which quantum gravity effects are carried to the exterior geometry.

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 under PSL$_2(\mathbb{C})$-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quantum Gravity.

Nonsingular AdS-dS transitions in a landscape scenario

Abstract: Understanding transitions between different vacua of a multiverse allowing eternal inflation is an open problem whose resolution is important to gain insights on the global structure of the spacetime as well as the problem of measure. In the classical theory, transitions from the anti--de Sitter to de Sitter vacua are forbidden due to the big-crunch singularity. In this paper, we consider toy landscape potentials: a double well and a triple well potential allowing anti--de Sitter and de Sitter vacua, in the effective dynamics of loop quantum cosmology for the $k=\ensuremath{-}1$ FRW model. We show that due to the nonperturbative quantum gravity effects as understood in loop quantum cosmology, nonsingular anti--de Sitter to de Sitter transitions are possible. In the future evolution, an anti--de Sitter bubble universe does not encounter a big-crunch singularity but undergoes a big bounce occurring at a scale determined by the underlying quantum geometry. These nonsingular transitions provide a mechanism through which a probe or a ``watcher,'' used to define a local measure, can safely evolve through the bounce and geodesics can be smoothly extended from anti--de Sitter to de Sitter vacua.

Kantowski-Sachs spacetime in loop quantum cosmology: bounds on expansion and shear scalars and the viability of quantization

Abstract: Using effective dynamics, we investigate the behavior of expansion and shear scalars in different proposed quantizations of the Kantowski-Sachs spacetime with matter in loop quantum cosmology. We find that out of the various proposed choices, there is only one known prescription which leads to the generic bounded behavior of these scalars. The bounds turn out to be universal and are determined by the underlying quantum geometry. This quantization is analogous to the so called 'improved dynamics' in the isotropic loop quantum cosmology, which is also the only one to respect the freedom of the rescaling of the fiducial cell at the level of effective spacetime description. Other proposed quantization prescriptions yield expansion and shear scalars which may not be bounded for certain initial conditions within the validity of effective spacetime description. These prescriptions also have a limitation that the "quantum geometric effects" can occur at an arbitrary scale. We show that the `improved dynamics' of Kantowski-Sachs spacetime turns out to be a unique choice in a general class of possible quantization prescriptions, in the sense of leading to generic bounds on expansion and shear scalars and the associated physics being free from fiducial cell dependence. The behavior of the energy density in the `improved dynamics' reveals some interesting features. Even without considering any details of the dynamical evolution, it is possible to rule out pancake singularities in this spacetime. The energy density is found to be dynamically bounded. These results show that the Planck scale physics of the loop quantized Kantowski-Sachs spacetime has key features common with the loop quantization of isotropic and Bianchi-I spacetimes.

The Foundations of Physical Law

Abstract: Introduction to Foundational Physics Mathematical Ideas and Methods The Most Primitive Concepts A Fundamental Symmetry Nilpotent Quantum Mechanics I-II Nilpotent Quantum Field Theory Gravity Particles Return to Symmetries

On Precanonical Quantization of Gravity

Abstract: Precanonical quantization is based on mathematical structures of the De Donder–Weyl Hamiltonization of field theories. The resulting formulation of quantum gravity describes the quantum geometry of space-time in terms of operator-valued distances and the transition amplitudes between the values of spin connection at different points of space-time, which

Geometric uncertainty relation for mixed quantum states

Abstract: In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schrodinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrodinger, and vice versa.

Spectra of geometric operators in three-dimensional loop quantum gravity: From discrete to continuous

Abstract: We study and compare the spectra of geometric operators (length and area) in the quantum kinematics of two formulations of three-dimensional Lorentzian loop quantum gravity. In the $\mathrm{SU}(2)$ Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $\ensuremath{\gamma}$ exactly like in the four-dimensional case. However, we show that when working with the self-dual variables and imposing the reality conditions the spectra become continuous and $\ensuremath{\gamma}$ independent.

Quantum groups and functional relations for lower rank

Abstract: A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_2))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a degression of the corresponding consideration for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ with an extensions to the higher spin case.

Aspects of Nonlocality in Quantum Field Theory, Quantum Gravity and Cosmology

Abstract: This paper contains a collection of essays on nonlocal phenomena in quantum field theory, gravity and cosmology. Mechanisms of nonlocal contributions to the quantum effective action are discussed within the covariant perturbation expansion in field strengths and spacetime curvatures and the nonperturbative method based on the late time asymptotics of the heat kernel. Euclidean version of the Schwinger-Keldysh technique for quantum expectation values is presented as a special rule of obtaining the nonlocal effective equations of motion for the mean quantum field from the Euclidean effective action. This rule is applied to a new model of ghost free nonlocal cosmology which can generate the de Sitter stage of cosmological evolution at an arbitrary value of $\varLambda$ -- a model of dark energy with its scale played by the dynamical variable that can be fixed by a kind of a scaling symmetry breaking mechanism. This model is shown to interpolate between the superhorizon phase of gravity theory mediated by a scalar mode and the short distance general relativistic limit in a special frame which is related by a nonlocal conformal transformation to the original metric. The role of compactness and regularity of spacetime in the Euclidean version of the Schwinger-Keldysh technique is discussed.