Showing papers on "Quantum geometry published in 2003"
TL;DR: In this article, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has been made explicit, thereby providing a firmer mathematical and conceptual foundation to the subject.
Abstract: Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
794 citations
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TL;DR: In this paper, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has been made explicit, thereby providing a firmer mathematical and conceptual foundation to the subject.
Abstract: Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
533 citations
TL;DR: The Loop Quantum Gravity (LQG) theory as mentioned in this paper is a mathematically rigorous candidate quantum field theory of the gravitational field, which has been shown to have background independence and minimality of structures.
Abstract: Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures.
380 citations
TL;DR: In this article, the authors illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle, which can also serve as a simple introduction to many of the ideas and constructions underlying quantum geometry.
Abstract: A programme was recently initiated to bridge the gap between the Planck scale physics described by loop quantum gravity and the familiar low energy world. We illustrate the conceptual problems and their solutions through a toy model: quantum mechanics of a point particle. The model can also serve as a simple introduction to many of the ideas and constructions underlying quantum geometry. Maxwell fields will be discussed in the second paper of this series which further develops the programme.
286 citations
TL;DR: In this paper, a discrete time description of usual quantum mechanics was obtained without any approximation or explicit discretization, which mimics features of the discrete time evolution of loop quantum cosmology.
Abstract: Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding the viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.
143 citations
TL;DR: In this paper, the problem of the treatment of scalar fields is addressed at the kinematic level by constructing the appropriate background-independent operator algebras and Hilbert spaces.
Abstract: In loop quantum gravity, matter fields can have support only on the 'polymer-like' excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states cannot refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address two issues already at the kinematic level. First, one has to construct the appropriate background-independent operator algebras and Hilbert spaces. Second, to make contact with low-energy physics, one has to relate this 'polymer description' of matter fields to the standard Fock description in Minkowski space. While this task has been completed for gauge fields, important gaps remained in the treatment of scalar fields. The purpose of this letter is to fill these gaps.
121 citations
TL;DR: In this paper, the relation of the dynamical initial conditions with boundary conditions such as the no-boundary or the tunneling proposal is investigated, and a discussion of inflation from quantum cosmology is discussed.
Abstract: Loop quantum cosmology of the closed isotropic model is studied with a special emphasis on a comparison with traditional results obtained in the Wheeler-DeWitt approach. This includes the relation of the dynamical initial conditions with boundary conditions such as the no-boundary or the tunneling proposal and a discussion of inflation from quantum cosmology.
106 citations
TL;DR: In this article, the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2 + 1 spacetime dimensions, was studied and it was shown that spacelike intervals are continuous whereas timelike interval is discrete.
Abstract: We study the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2 + 1 spacetime dimensions. We find that the spectrum of spacelike intervals is continuous, whereas the spectrum of timelike intervals is discrete. This result contradicts the expectation that spacelike intervals are always discrete. On the other hand, it is consistent with the results of the spin foam quantization of the same theory.
92 citations
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TL;DR: In this article, the authors propose an interpretation for the non-chiral correlation functions of Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.
Abstract: The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.
74 citations
TL;DR: The causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets as discussed by the authors.
74 citations
TL;DR: In this paper, the isolated horizon framework is extended to include non-minimally coupled scalar fields and entropy is no longer given just by (a quarter of) the horizon area but also depends on the scalar field.
Abstract: The isolated horizon framework is extended to include non-minimally coupled scalar fields. As expected from the analysis based on Killing horizons, entropy is no longer given just by (a quarter of) the horizon area but also depends on the scalar field. In a subsequent paper these results will serve as a point of departure for a statistical mechanical derivation of entropy using quantum geometry.
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TL;DR: In this paper, the authors describe the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is their main concern, on Barrett-Crane model.
Abstract: Spin foam models are a new approach to a formulation of quantum gravity which is fully background independent, non-perturbative, and covariant, in the spirit of path integral formulations of quantum field theory. In this thesis we describe in details the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach. We concentrate, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is our main concern, on the Barrett-Crane model. In particular, for the Barrett-Crane model: we describe the general ideas behind its construction, and review what has been achieved up to date, discuss in details its links with the classical formulations of gravity as constrained topological field theory; we show a derivation of the model from a lattice gauge theory perspective, in the general case of manifold with boundaries, presenting also a few possible variations of the procedure used, discussing the problems they present; we analyse in details the classical and quantum geometry; we also describe how, from the same perspective, a spin foam model that couples quantum gravity to any gauge theory may be constructed; finally, we describe a general scheme for causal spin foam models, how the Barrett-Crane model can be modified to implement causality and to fit in such a scheme, and the resulting link with the quantum causal set approach to quantum gravity.
TL;DR: In this paper, the authors argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics and discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space.
Abstract: We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space. Specifically, we apply our scheme to the problem of a background independent formulation of Matrix Theory.
TL;DR: In this article, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity, in combination with a solution of the singularity problem.
Abstract: In physical theories, boundary or initial conditions play the role of selecting special situations which can be described by a theory with its general laws. Cosmology has long been suspected to be different in that its fundamental theory should explain the fact that we can observe only one particular realization. This is not realized, however, in the classical formulation and in its conventional quantization; the situation is even worse due to the singularity problem. In recent years, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity. Here, the dynamical law and initial conditions turn out to be linked intimately, in combination with a solution of the singularity problem.
TL;DR: In this article, the authors discuss applications of loop quantum gravity in the cosmological realm, including the basic formalism and recent results in loop quantum cosmology, and a computation of modified dispersion relations for quantum gravity phenomenology.
Abstract: After a brief introduction to classical and quantum gravity we discuss applications of loop quantum gravity in the cosmological realm. This includes the basic formalism and recent results of loop quantum cosmology, and a computation of modified dispersion relations for quantum gravity phenomenology. The presentation is held at a level which does not require much background knowledge in general relativity or mathematical techniques such as functional analysis, so as to make the article accessible to graduate students and researchers from other fields.
TL;DR: In this article, the black-hole entropy for type I isolated horizons, based on loop quantum gravity, is extended to include non-minimally coupled scalar fields, and the resulting expression of blackhole entropy now depends also on the scalar field precisely in the fashion predicted by the first law in the classical theory.
Abstract: The black-hole entropy calculation for type I isolated horizons, based on loop quantum gravity, is extended to include non-minimally coupled scalar fields. Although the non-minimal coupling significantly modifies quantum geometry, the highly non-trivial consistency checks for the emergence of a coherent description of the quantum horizon continue to be met. The resulting expression of black-hole entropy now depends also on the scalar field precisely in the fashion predicted by the first law in the classical theory (with the same value of the Barbero–Immirzi parameter as in the case of minimal coupling).
TL;DR: In this article, it was shown that the AIL-representation is irreducible, provided it is viewed as the representation of a C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.
Abstract: Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Isham-Lewandowski representation, has been constructed. Recently, several uniqueness results for this representation have been worked out. In the present article, we contribute to these efforts by showing that the AIL-representation is irreducible, provided it is viewed as the representation of a certain C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.
TL;DR: The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing high-level approaches to quantum information processing.
TL;DR: In this paper, the quantum mechanics of BMN operators with two scalar impurities and arbitrarily many traces, at one loop and all genus, were studied, and an operator identity which partially elucidates the structure of this quantum mechanics, provides some support for a conjectured formula for the free all genus two-point functions, and demonstrates that a single O (g 2 2 ) contact term arises in the Hamiltonian as a result of transforming from the natural gauge theory basis to the string basis.
TL;DR: In this paper, a model of spherically symmetric thin shell of light-like substance in its own gravitational field is presented. But the model is based on the exclusive use of gauge-invariant variables, the so-called Dirac observables, and on privileged dynamical symmetries such as the asymptotic time translation.
Abstract: These notes consist of three parts. The first one contains the review of previous work on a gauge-invariant Hamiltonian dynamics of generally covariant models. The method is based on the exclusive use of gauge-invariant variables, the so-called Dirac observables, and on privileged dynamical symmetries such as the asymptotic time translation. The second part applies the method to the model of spherically symmetric thin shell of light-like substance in its own gravitational field following a paper by C. Kiefer and myself. A natural set of Dirac observables is chosen and the Hamiltonian defined by the time translation symmetry is calculated. In the third part, my construction of a version of quantum mechanics for the model is reviewed. The quantum evolution is unitary in spite of the classical theory containing black and white holes and singularities. The wave packet describing the quantum shell contracts, bounces and reexpands. The state of the quantum horizon is a linear combination of the “white” and “black” (past and future) apparent horizons.
TL;DR: In this paper, it was shown that the effects of quantum gravity are linearly proportional to the ratio of the photon energy to the characteristic scale energy of QG, and that the polarization of photons with energies of about 0.1 MeV should be completely random, contrary to what is observed.
Abstract: Gamma rays from the γ-ray burst (GRB) 021206 have been reported to be strongly linearly polarized1, with the estimated degree of polarization (80 ± 20%) being close to the absolute maximum of 100% — affording us the opportunity to constrain models of quantum gravity, which has had 1010 years to act on the photons as they travelled towards us. Here I show that if the effects of quantum gravity are linearly proportional to the ratio of the photon energy to the characteristic scale energy of quantum gravity, then the polarization of photons with energies of about 0.1 MeV should be completely random, contrary to what is observed. I conclude that, should the polarization measurement be confirmed, quantum gravity effects act with a power that is greater than linearity, or that loop quantum gravity is not viable. Compared with previous methods and results (see ref. 2, for example), testing of the linear polarization of cosmic γ-ray bursts may substantially extend the observational window on the theory of quantum gravity.
TL;DR: In this paper, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity, in combination with a solution of the singularity problem.
Abstract: In physical theories, boundary or initial conditions play the role of selecting special situations which can be described by a theory with its general laws. Cosmology has long been suspected to be different in that its fundamental theory should explain the fact that we can observe only one particular realization. This is not realized, however, in the classical formulation and in its conventional quantization; the situation is even worse due to the singularity problem. In recent years, a new formulation of quantum cosmology has been developed which is based on quantum geometry, a candidate for a theory of quantum gravity. Here, the dynamical law and initial conditions turn out to be linked intimately, in combination with a solution of the singularity problem.
TL;DR: In this article, the non-perturbative quantum geometry of the universal hypermultiplet (UH) was investigated in N = 2 supergravity, and the classical pre-potential corresponding to the standard (Ferrara-Sabharwal) metric of the UH arising in the Calabi-Yau compactification of type-II superstrings was calculated.
TL;DR: In this paper, a study of dynamics and dissipative tunneling in a symmetric quartic double well potential is provided, where the numerical solution for the position autocorrelation function obtained through the Wigner-Fokker-Planck equation is compared with numerically exact results of Stockburger and Mak.
Abstract: A study is provided of dynamics and dissipative tunneling in a symmetric quartic double well potential. The numerical solution for the position autocorrelation function obtained through the Wigner–Fokker–Planck equation is compared with numerically exact results of Stockburger and Mak [J. Chem. Phys. 110, 4983 (1999)]. We find that the Wigner–Fokker–Planck dynamics agree well with the numerically exact computations, they account for both quantum coherences as well as quantum tunneling phenomena. This, in contrast to the mixed quantum classical approximation, which does not perform as well.
TL;DR: In this paper, a decomposition of the stress tensor correlation function into three parts is discussed, and the physical implications of each part are discussed. And the operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity is discussed.
Abstract: The quantum fluctuations of the stress tensor of a quantum field are discussed, as are the resulting space-time metric fluctuations. Passive quantum gravity is an approximation in which gravity is not directly quantized, but fluctuations of the space-time geometry are driven by stress tensor fluctuations. We discuss a decomposition of the stress tensor correlation function into three parts, and consider the physical implications of each part. The operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity are discussed.
TL;DR: In this article, a non-perturbative path integral for Lorentzian space-times has been proposed, which solves the problems of having a well-defined Wick rotation, possessing a coordinate-invariant cutoff, and leading to convergent sums over geometries.
Abstract: In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a well-defined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to convergent sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d=2 and d=3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.
TL;DR: The quantum mechanical formalism for the position and momentum of a particle on a one-dimensional lattice has been developed in this paper, where some mathematical features characteristic of finite-dimensional Hilbert spaces are compared with the infinite-dimensional case.
Abstract: The quantum mechanical formalism for the position and momentum of a particle on a one-dimensional lattice is developed. Some mathematical features characteristic of finite-dimensional Hilbert spaces are compared with the infinite-dimensional case. The construction of an unbiased basis for state determination is discussed.
TL;DR: In this paper, a thermodynamic approach to introducing quantum corrections to the classical transport picture in semiconductor device simulation is presented, leading to a modified Boltzmann equation with a quantum corrected force term and to quantum corrected fluid, or quantum hydrodynamic models.
Abstract: We present a thermodynamic approach to introducing quantum corrections to the classical transport picture in semiconductor device simulation. This approach leads to a modified Boltzmann equation with a quantum corrected force term and to quantum corrected fluid, or quantum hydrodynamic models. We present the quantum interaction of electrons with a gate oxide barrier potential and quantum hydrodynamic simulations of a resonant tunneling diode as application examples.
TL;DR: In this article, it is shown that the measured quantum states are defined mainly by a mirror and the gravitational field, and possible false effects in the experiment on the measurement of quantum states of neutrons in the Earth's gravitational field are discussed.
Abstract: Possible false effects in the experiment on the measurement of quantum states of neutrons in the Earth's gravitational field are discussed. It is shown that the measured quantum states are defined mainly by a mirror and the gravitational field.