Showing papers on "Quantum geometry published in 2001"
TL;DR: It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry and the correct semiclassical behavior is obtained.
Abstract: It is shown that the cosmological singularity in isotropic minisuperspaces is naturally removed by quantum geometry. Already at the kinematical level, this is indicated by the fact that the inverse scale factor is represented by a bounded operator even though the classical quantity diverges at the initial singularity. The full demonstration comes from an analysis of quantum dynamics. Because of quantum geometry, the quantum evolution occurs in discrete time steps and does not break down when the volume becomes zero. Instead, space-time can be extended to a branch preceding the classical singularity independently of the matter coupled to the model. For large volume the correct semiclassical behavior is obtained.
717 citations
TL;DR: An experimentally feasible scheme to achieve quantum computation based solely on geometric manipulations of a quantum system by driving the quantum system to undergo appropriate adiabatic cyclic evolutions is proposed.
Abstract: We propose an experimentally feasible scheme to achieve quantum computation based solely on geometric manipulations of a quantum system. The desired geometric operations are obtained by driving the quantum system to undergo appropriate adiabatic cyclic evolutions. Our implementation of the all-geometric quantum computation is based on laser manipulation of a set of trapped ions. An all-geometric approach, apart from its fundamental interest, offers a possible method for robust quantum computation.
636 citations
TL;DR: The physical resources available to access and manipulate the degrees of freedom of a quantum system define the set A of operationally relevant observables, and the notion of compoundness for quantum systems is accordingly relativized.
Abstract: The physical resources available to access and manipulate the degrees of freedom of a quantum system define the set A of operationally relevant observables. The algebraic structure of A selects a preferred tensor product structure, i.e., a partition into subsystems. The notion of compoundness for quantum systems is accordingly relativized. Universal control over virtual subsystems can be achieved by using quantum noncommutative holonomies
242 citations
TL;DR: In this article, the inverse scale factor is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity.
Abstract: The inverse scale factor, which in classical cosmological models diverges at the singularity, is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity. This procedure results in a bounded operator which is diagonalizable simultaneously with the volume operator and whose eigenvalues are determined explicitly. For large scale factors (in fact, up to a scale factor slightly above the Planck length) the eigenvalues are close to the classical expectation, whereas below the Planck length there are large deviations leading to a nondiverging behavior of the inverse scale factor even though the scale factor has vanishing eigenvalues. This is a first indication that the classical singularity is better behaved in loop quantum cosmology.
218 citations
TL;DR: In this article, the authors introduce spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity and topological quantum field theories.
Abstract: This is an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice field theory, matrix models, category theory and statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian cases. We focus in particular on the Barrett-Crane model for four-dimensional quantum gravity.
214 citations
TL;DR: In this paper, a dynamical interpretation of the Hamiltonian constraint equation as evolution equation is presented, which involves a transformation from the connection to a dreibein representation and the selection of an internal time variable.
Abstract: Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity, time also turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed within this framework, which enables an investigation of physical spectra of geometrical quantities. In particular, the physical volume spectrum is proven to equal the discrete kinematical volume spectrum in loop quantum cosmology.
131 citations
TL;DR: In this paper, a model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes is given which generalizes similar, previous approaches.
Abstract: A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial.
104 citations
TL;DR: Using algebraic geometry codes, this work gives a polynomial construction of quantum codes with asymptotically non-zero rate and relative distance.
Abstract: Using algebraic geometry codes we give a polynomial construction of quantum codes with asymptotically nonzero rate and relative distance.
103 citations
TL;DR: In this article, the role of the Barbero-Immirzi γ parameter in controlling spacetime discreteness is discussed, with an emphasis on the role played by the barbero-immirzi parameter.
Abstract: The continuum and semiclassical limits of isotropic, spatially flat loop quantum cosmology are discussed, with an emphasis on the role played by the Barbero-Immirzi γ parameter in controlling spacetime discreteness. In this way, standard quantum cosmology is shown to be the simultaneous limit γ→0, j→∞ of loop quantum cosmology. Here, j is a label of the volume eigenvalues, and the simultaneous limit is technically the same as the classical limit →0, l→∞ of angular momentum in quantum mechanics. Possible lessons for semiclassical states at the dynamical level in the full theory of quantum geometry are mentioned.
97 citations
01 Jan 2001
TL;DR: In this article, the authors argue that reconciling general relativity with the Standard Model requires a background-free quantum theory with local degrees of freedom propagating causally, which is to be expected in any background free quantum theory.
Abstract: This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' and `state', and similarly those of `spacetime' and `process', which we argue is to be expected in any background-free quantum theory. We sketch how higher-dimensional algebra provides the mathematical tools to make this link precise. Finally, we comment on attempts to formulate a theory of quantum gravity in 4-dimensional spacetime using `spin networks' and `spin foams'.
97 citations
TL;DR: In this article, the existence of fuzzy S-2 and S-4 models for the quantum geometry on the corresponding spheres in AdS/CFT correspondence was shown to fit nicely the data from the dipole mechanism for the stringy exclusion principle and a wrapped fractional membrane can be used to count for the large ground state degeneracy.
TL;DR: In this article, a new entropy bound was derived for space-times with nonrotating isolated horizons from the quantum geometry approach, in which the horizon is described by the boundary degrees of freedom of a three dimensional Chern-Simons theory.
Abstract: A new entropy bound, tighter than the standard holographic bound due to Bekenstein, is derived for space-times with nonrotating isolated horizons from the quantum geometry approach, in which the horizon is described by the boundary degrees of freedom of a three dimensional Chern-Simons theory.
01 Mar 2001
TL;DR: In this paper, a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries is described.
Abstract: Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number of interesting and unexpected results that have been obtained for these dynamically triangulated models in two and three dimensions make discrete Lorentzian gravity a promising candidate for a non-trivial theory of quantum gravity.
TL;DR: In this paper, the authors explore the quantum aspects of an elastic bar supported at both ends and subject to compression, and calculate the quantum fluctuations about the classical value as a function of strain.
Abstract: We explore the quantum aspects of an elastic bar supported at both ends and subject to compression. If strain rather than stress is held fixed, the system remains stable beyond the buckling instability, supporting two potential minima. The classical equilibrium transverse displacement is analogous to a Ginsburg-Landau order parameter, with strain playing the role of temperature. We calculate the quantum fluctuations about the classical value as a function of strain. Excitation energies and quantum fluctuation amplitudes are compared for silicon beams and carbon nanotubes.
TL;DR: Les Houches as mentioned in this paper reviewed the formfactors technique and its extension to massless quantum field theories and discussed the calculation of correlators in integrable quantum impurity problems, with special emphasis on point contact tunneling in the fractional quantum Hall effect and the two-state problem of dissipative quantum mechanics.
Abstract: These are notes of lectures given at The NATO Advanced Study Institute/EC Summer School on “New Theoretical Approaches to Strongly Correlated Systems”, (Newton Institute, April 2000). They are a sequel to the notes I wrote two years ago for the Summer School, “Topological Aspects of Low Dimensional Systems”, (Les Houches, July 1998). In this second part, I review the form-factors technique and its extension to massless quantum field theories. I then discuss the calculation of correlators in integrable quantum impurity problems, with special emphasis on point contact tunneling in the fractional quantum Hall effect, and the two-state problem of dissipative quantum mechanics.
TL;DR: In this article, the energy spectrum and corresponding wave functions of a flat quantum dot with elliptic symmetry were obtained exactly and a detailed study of the effect of ellipticity on the energy levels and the corresponding wave function was made.
Abstract: The energy spectrum and corresponding wave functions of a flat quantum dot with elliptic symmetry are obtained exactly. A detailed study is made of the effect of ellipticity on the energy levels and the corresponding wave functions. The analytical behavior of the energy levels in certain limiting cases is obtained.
TL;DR: In this article, a model describing exactly a thin spherically symmetric shell of matter with zero rest mass is discussed, where the variables are embeddings, their conjugate momenta, and Dirac observables.
Abstract: We discuss a model describing exactly a thin spherically symmetric shell of matter with zero rest mass. We derive the reduced formulation of this system in which the variables are embeddings, their conjugate momenta, and Dirac observables. A non-perturbative quantum theory of this model is then constructed, leading to a unitary dynamics. As a consequence of unitarity, the classical singularity is fully avoided in the quantum theory.
TL;DR: In this article, the authors identify a scale of around 100 TeV as an upper limit on the domain of validity of a quantum field theory description of Nature and discuss the phenomenological consequences of such a scale.
Abstract: Quantum gravity arguments and the entropy bound for effective field theories proposed in PRL 82, 4971 (1999) lead to consider two correlated scales which parametrize departures from relativistic quantum field theory at low and high energies. A simple estimate of their possible phenomenological implications leads to identify a scale of around 100 TeV as an upper limit on the domain of validity of a quantum field theory description of Nature. This fact agrees with recent theoretical developments in large extra dimensions. Phenomenological consequences in the beta-decay spectrum and cosmic ray physics associated to possible Lorentz invariance violations induced by the infrared scale are discussed. It is also suggested that this scale might produce new unexpected effects at the quantum level.
TL;DR: In this paper, the first steps in the construction of semiclassical states for nonperturbative canonical quantum gravity using ideas from classical, Riemannian statistical geometry and results from quantum geometry of spin network states are described.
Abstract: I describe the first steps in the construction of semiclassical states for non-perturbative canonical quantum gravity using ideas from classical, Riemannian statistical geometry and results from quantum geometry of spin network states. In particular, I concentrate on how those techniques are applied to the construction of random spin networks, and the calculation of their contribution to areas and volumes.
TL;DR: In this article, the technique of dimensional regularization first developed in the context of Q.F.T. by 't Hooft and Veltmann is revisited from the E (∞) theory view point.
Abstract: The technique of dimensional regularization first developed in the context of Q.F.T. by 't Hooft and Veltmann is revisited from the E (∞) theory view point. It is concluded that this computational procedure is an “ontological” reflection of the physical reality of the underlying quantum geometry of micro space-time and not merely a cunning mathematical device for the avoidance of non-analyticity at a critical state, and extracting an approximate solution where other direct methods fail.
Posted Content•
TL;DR: A state sum model based on the group SU(1,1) is defined in this article and its geometry and asymptotics suggest it is a good candidate for modelling (2+1) Lorentzian quantum gravity.
Abstract: A state sum model based on the group SU(1,1) is defined Investigations of its geometry and asymptotics suggest it is a good candidate for modelling (2+1) Lorentzian quantum gravity
TL;DR: In this paper, the physical and conceptual need for a theory of quantum gravity is discussed, and some knowledge of general relativity and non-relativistic quantum mechanics is assumed.
Abstract: This paper provides a thorough introduction to the physical and conceptual need for a theory of quantum gravity; some knowledge of general relativity and nonrelativistic quantum mechanics is assume...
TL;DR: Recently, a number of fundamental physical issues were addressed in loop quantum gravity as discussed by the authors, including a statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons, a natural resolution of the big-bang singularity, spin-foam models which provide background independent path integral formulations of quantum gravity and ''finiteness proofs'' of some of these models.
Abstract: Over the last three years, a number of fundamental physical issues were addressed in loop quantum gravity. These include: A statistical mechanical derivation of the horizon entropy, encompassing astrophysically interesting black holes as well as cosmological horizons; a natural resolution of the big-bang singularity; the development of spin-foam models which provide background independent path integral formulations of quantum gravity and `finiteness proofs' of some of these models; and, the introduction of semi-classical techniques to make contact between the background independent, non-perturbative theory and the perturbative, low energy physics in Minkowski space. These developments spring from a detailed quantum theory of geometry that was systematically developed in the mid-nineties and have added a great deal of optimism and intellectual excitement to the field.
The goal of this article is to communicate these advances in general physical terms, accessible to researchers in all areas of gravitational physics represented in this conference.
TL;DR: In this paper, a detailed analysis of the boundary element formulation of the electronic states of quantum structures is presented, and techniques for minimizing computation time by reducing the number of boundary integrals, utilizing the repetitive nature of embedded multiple quantum structures, and eliminating boundary elements for modeling the effects of quantum wells that contain quantum structures are discussed.
Posted Content•
TL;DR: In this paper, a model of a self-referential universe is presented and a diagrammatic approach is used to discuss the quantum topology of the EPR paradox, particle decays and scattering processes.
Abstract: Some principles underpinning the running of the Universe are discussed. The most important, the machine principle, states that the Universe is a fully autonomous, self-organizing and self-testing quantum automaton. Continuous space and time, consciousness and the semi-classical observers of quantum mechanics are all emergent phenomena not operating at the fundamental level of the machine Universe. Quantum processes define the present, the interface between the future and the past, giving a time ordering to the running of the Universe which is non-integrable except on emergent scales. A diagrammatic approach is used to discuss the quantum topology of the EPR paradox, particle decays and scattering processes. A toy model of a self-referential universe is given.
TL;DR: The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations as discussed by the authors.
Abstract: The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. This is interpreted as a tunneling effect in the quantum geometry.
TL;DR: In this article, the quantum Hamiltonian for a gravitational system with boundary was derived and the resulting thermody-namics were used to fix the fundamental scale m. In this model, states of quantum geometry are represented by spin networks.
Abstract: Using the quantum Hamiltonian for a gravitational system with boundary, we find the partition function and derive the resulting thermody- namics. The Hamiltonian is the boundary term required by functional differ- entiability of the action for Lorentzian general relativity. In this model, states of quantum geometry are represented by spin networks. We show that the statistical mechanics of the model reduces to that of a simple non-interacting gas of particles with spin. Using both canonical and grand canonical descrip- tions, we investigate two temperature regimes determined by the fundamental constant in the theory, m. In the high temperature limit (kT ≫ m), the model is thermodynamically stable. For low temperatures (kT ≪ m) and for macroscopic areas of the bounding surface, the entropy is proportional to area (with logarithmic correction), providing a simple derivation of the Bekenstein- Hawking result. By comparing our results to known semiclassical relations we are able to fix the fundamental scale m. Also in the low temperature, macro- scopic limit, the quantum geometry on the boundary forms a 'condensate' in the lowest energy level (j = 1/2).
TL;DR: A quantum action with parameters different from the classical action is introduced and results for the harmonic oscillator with weak perturbation, the quartic potential, and the double well potential are presented.
Abstract: We suggest a closed form expression for the path integral of quantum transition amplitudes We introduce a quantum action with parameters different from the classical action We present numerical results for the harmonic oscillator with weak perturbation, the quartic potential, and the double well potential The quantum action is relevant for quantum chaos and quantum instantons
TL;DR: In this article, the superpotential and the two-component wave functions of the ground state of a two-dimensional physical system were found out using super-ymmetry in non-relativistic quantum mechanics.