Showing papers on "Quantum geometry published in 2004"
TL;DR: In this paper, it was shown that the contribution of spins greater than 1/2 to the entropy is not negligible, and the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived.
Abstract: Quantum geometry (the modern loop quantum gravity involving graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater than 1/2 to the entropy is not negligible. Hence, the value of the Barbero–Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between quantum geometry and the black-hole quasi-normal modes should be understood again.
440 citations
TL;DR: In this paper, a simple algebraic mechanism for the emergence of deformations of Poincar? symmetries in the low-energy limit of quantum theories of gravity is presented.
Abstract: We present a simple algebraic mechanism for the emergence of deformations of Poincar? symmetries in the low-energy limit of quantum theories of gravity. The deformations, called ?-Poincar? algebras, are parametrized by a dimensional parameter proportional to the Planck mass, and imply modified energy?momentum relations of a type that may be observable in near future experiments. Our analysis assumes that the low energy limit of a quantum theory of gravity must also involve a limit in which the cosmological constant is taken very small with respect to the Planck scale, and makes use of the fact that in some quantum theories of gravity the cosmological constant results in the (anti)de Sitter symmetry algebra being quantum deformed. We show that deformed Poincar? symmetries inevitably emerge in the small-cosmological-constant limit of quantum gravity in 2 + 1 dimensions, where geometry does not have local degrees of freedom. In 3 + 1 dimensions we observe that, besides the quantum deformation of the (anti)de Sitter symmetry algebra, one must also take into account that there are local degrees of freedom leading to a renormalization of the generators for energy and momentum of the excitations. At the present level of development of quantum gravity in 3 + 1 dimensions, it is not yet possible to derive this renormalization from first principles, but we establish the conditions needed for the emergence of a deformed low energy limit symmetry algebra also in the case of 3 + 1 dimensions.
252 citations
TL;DR: In this paper, a line-by-line derivation of canonical quantum mechanics stemming from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics is presented.
Abstract: We present a line by line derivation of canonical quantum mechanics stemming from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This viewpoint can be naturally extended to provide a conceptually novel, non-perturbative formulation of quantum gravity. Possible observational implications of this new approach are briefly mentioned.
190 citations
15 Sep 2004
TL;DR: A didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity can be found in this paper, where the main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described.
Abstract: These notes are a didactic overview of the non perturbative and background independent approach to a quantum theory of gravity known as loop quantum gravity. The definition of real connection variables for general relativity, used as a starting point in the program, is described in a simple manner. The main ideas leading to the definition of the quantum theory are naturally introduced and the basic mathematics involved is described. The main predictions of the theory such as the discovery of Planck scale discreteness of geometry and the computation of black hole entropy are reviewed. The quantization and solution of the constraints is explained by drawing analogies with simpler systems. Difficulties associated with the quantization of the scalar constraint are this http URL a second part of the notes, the basic ideas behind the spin foam approach are presented in detail for the simple solvable case of 2+1 gravity. Some results and ideas for four dimensional spin foams are reviewed.
163 citations
TL;DR: The main concepts of E Infinity theory and some of the principle results have been explained in this article, and the problem of the mass spectrum of the standard model from various viewpoints, in particular, from that of gravitational instanton.
Abstract: The paper is an elementary introduction to the concepts of E Infinity of quantum physics. In the first two paragraphs the main concepts of E Infinity theory and some of the principle results have been explained. Subsequently we address the problem of the mass spectrum of the standard model from various viewpoints, in particular, from that of gravitational instanton. Particular attention is paid throughout the paper to giving an intuitive grasp for an extended theory of vacuum fluctuation based on Dirac's hole theory and E Infinity theory. It is further shown how the golden mean and logarithmic scalings can be used to understand quantum gravity and how the new transfinite Dirac's theory can explain certain anomalous positron production which was observed by several experimental groups worldwide.
160 citations
TL;DR: A method to compute two-electron integrals over arbitrary regions of space is introduced and particularized to the basins appearing in the quantum theory of atoms in molecules, showing that the approach is always convergent and computationally efficient.
Abstract: A method to compute two-electron integrals over arbitrary regions of space is introduced and particularized to the basins appearing in the quantum theory of atoms in molecules. The procedure generalizes the conventional multipolar approach to account for overlapping densities. We show that the approach is always convergent and computationally efficient, scaling as N4 in the worst, two-center case. Several numerical results supporting our claims are also presented.
150 citations
TL;DR: In this paper, the kinematical setting of spherically symmetric quantum geometry derived from the full theory of loop quantum gravity is developed, which extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise.
Abstract: The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise. A comparison between a reduced quantization and a derivation of the model from the full theory is presented in detail, with an emphasis on the resulting quantum representation. Similar concepts for Einstein–Rosen waves are discussed briefly.
137 citations
TL;DR: In this paper, it was shown that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.
Abstract: We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.
106 citations
TL;DR: In this paper, the authors considered the possibility of abandoning the Archimedean axiom and introducing a fundamental physical limitation on the smallest length in quantum spacetime, and they arrived at the conclusion that maximising the Hawking-Bekenstein informational content of spacetime makes the existence of a transfinite geometry for physical “spacetime” plausible or even inevitable.
Abstract: In a recent paper entitled “Quantum gravity from descriptive set theory”, published in Chaos, Solitons & Fractals, we considered following the P-adic quantum theory, the possibility of abandoning the Archimedean axiom and introducing a fundamental physical limitation on the smallest length in quantum spacetime. Proceeding that way we arrived at the conclusion that maximising the Hawking–Bekenstein informational content of spacetime makes the existence of a transfinite geometry for physical “spacetime” plausible or even inevitable. Subsequently we introduced a mathematical description of a transfinite, non-Archimedean geometry using descriptive set theory where a similar conclusion regarding the transfiniteness of quantum spacetime may be drawn from the existence of the Unruh temperature. In particular we introduced a straight forward logarithmic gauge transformation linking, as far as we are aware for the first time, classical gravity with the electroweak via a version of informational entropy. That way we found using e(∞) and complexity theory that α G =(2) α ew −1 =1.7×10 38 where α G is the dimensionless Newton gravity constant and α ew =128 is the fine structure constant at the electroweak unification scale. The present work is concerned with more or less the same category of fundamental questions pertinent to quantum gravity. However we switch our mathematical apparatus to a combination of Clifford algebras and set theory. In doing that, the central and vital role of the work of D. Finkelstein becomes much more tangible and clearer than in most of our previous works.
105 citations
TL;DR: In this paper, a reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is proved and a new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pullbacks of line bundles from the target via different markings is used for the reconstruction result.
Abstract: A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K -theory is proved. A new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pull-backs of line bundles from the target via different markings is used for the reconstruction result. Examples of calculations in quantum cohomology and quantum K -theory are given.
02 Jun 2004
TL;DR: In this article, the authors investigated the low-energy limit of loop quantum gravity and showed that any successful integration theory for such spaces of connections with these gauge groups will of necessity be different in essential structure from the theory for compact and non-compact, Abelian groups.
Abstract: In this thesis we address two remaining open questions in loop quantum gravity The first deals with the low-energy limit of the theory We illustrate some of the conceptual difficulties and their resolution through the study of a toy model: the quantum mechanics of a point particle We then find that this model can also be applied to the quantum mechanics of spatially isotropic, homogeneous cosmology within the framework of loop quantum cosmology (LQC) This leads us to extend our results to investigate, for the quantum constraint in LQC, the effective classical dynamics of the quantum theory We find that we can calculate an effective Hamiltonian constraint, and we employ this to calculate the modifications to Friedmann’s equations for a dust filled, spatially flat, isotropic universe We then turn to a mathematical question, investigating the extension of integration theory on spaces of connections to connections with non-compact structure group For groups that are the direct product of a compact group with a non-compact Abelian group, we demonstrate a fully satisfactory theory based on the almost periodic compactification of the group This approach fails for other non-compact groups, and for the case of SL(2,R) and SL(2,C) we present a partial ‘no-go’ theorem that demonstrates that any successful integration theory for such spaces of connections with these gauge groups will of necessity be different in essential structure from the theory for compact and non-compact, Abelian groups
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01 Mar 2004
TL;DR: Gromov-Hausdorff distance for quantum metric spaces Bibliography Matrix algebras Converge to the sphere for quantum Gromov Hausdhof distance as mentioned in this paper.
Abstract: Gromov-Hausdorff distance for quantum metric spaces Bibliography Matrix algebras Converge to the sphere for quantum Gromov-Hausdorff distance Bibliography.
TL;DR: The chaotic behavior of the Bianchi IX model stops once quantum effects become important, consistent with the discrete structure of space predicted by loop quantum gravity.
Abstract: In classical general relativity, the generic approach to the initial singularity is very complicated as exemplified by the chaos of the Bianchi IX model which displays the generic local evolution close to a singularity. Quantum gravity effects can potentially change the behavior and lead to a simpler initial state. This is verified here in the context of loop quantum gravity, using methods of loop quantum cosmology: The chaotic behavior stops once quantum effects become important. This is consistent with the discrete structure of space predicted by loop quantum gravity.
TL;DR: In this article, the singularity resolution issue in quantum gravity was examined by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics, inspired by the loop quantum gravity program, and is based on an alternative to the Schr\"odinger representation normally used in metric variable quantum cosmology.
Abstract: We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity program, and is based on an alternative to the Schr\"odinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute ``singularity resolution'' in quantum gravity.
TL;DR: In this article, a non-commutative version of the Hamiltonian of a quantum mechanical particle on spacetime has been shown to have identical spectra, and the spatial coordinates of the particle have identical spatial coordinates.
Abstract: In this work quantum physics in noncommutative spacetime is developed. It is based on the work of Doplicher et al. which allows for time-space noncommutativity. The Moyal plane is treated in detail. In the context of noncommutative quantum mechanics, some important points are explored, such as the formal construction of the theory, symmetries, causality, simultaneity and observables. The dynamics generated by a noncommutative Schrodinger equation is studied. We prove in particular the following: suppose the Hamiltonian of a quantum mechanical particle on spacetime has no explicit time dependence, and the spatial coordinates commute in its noncommutative form (the only noncommutativity being between time and a space coordinate). Then the commutative and noncommutative versions of the Hamiltonian have identical spectra.
TL;DR: In this article, the authors propose a background independent quantum theory of gravity and matter based on the interplay between the symplectic form, dynamical metric and non-integrable almost complex structure of the space of quantum events.
TL;DR: This work uses quantum-correction factors to calculate approximately the quantum velocity time-correlation function (TCF) of supercritical Lennard-Jones argon from the classical TCF, and shows that the harmonic quantum correction factor works the best for this system.
Abstract: We use quantum-correction factors to calculate approximately the quantum velocity time-correlation function (TCF) of supercritical Lennard-Jones argon from the classical TCF. We find that for this quite classical system, several different quantum-correction schemes yield essentially identical results for the real and imaginary parts of the quantum TCF, and also agree well with the recent forward–backward semiclassical dynamics (FBSD) results of Wright and Makri [J. Chem. Phys. 119, 1634 (2003)]. We also consider a more quantum-mechanical fluid of lighter atoms (neon) at a lower temperature. In this case different quantum-correction schemes give different results. FBSD calculations show that the harmonic quantum correction factor works the best for this system.
TL;DR: In this paper, the scattering of He atoms from adsorbed CO molecules on the Pt(111) surface is described within the formalism of quantum trajectories provided by Bohmian mechanics.
Abstract: The scattering of He atoms from adsorbed CO molecules on the Pt(111) surface is described within the formalism of quantum trajectories provided by Bohmian mechanics. We show that the main mechanism leading to the formation of quantum rainbows and resonance enhanced trapping is quantum vortices.
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16 Jan 2004
TL;DR: In this paper, it was pointed out that the standard quantum geometry formalism is not consistent with 3/2 and favours 1/2, and that the area law with logarithmic corrections can be obtained with a coefficient 1 /2.
Abstract: Various approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient 1/2, and some involving 3/2. It is pointed out here that the standard quantum geometry formalism is not consistent with 3/2 and favours 1/2.
TL;DR: In this paper, a connection between supersymmetric quantum mechanics and ordinary quantum mechanics is established, and the connection between the two types of quantum mechanics can be found in this paper.
TL;DR: In this paper, the spherically symmetric volume operator is discussed and all its eigenstates and eigenvalues are computed, and the formulae of this derivation complete the derivation of an explicit calculus for SPV models, which is needed for future physical investigations.
Abstract: The spherically symmetric volume operator is discussed and all its eigenstates and eigenvalues are computed. Even though the operator is more complicated than its homogeneous analogue, the spectra are related in the sense that the larger spherically symmetric volume spectrum adds fine structure to the homogeneous spectrum. The formulae of this paper complete the derivation of an explicit calculus for spherically symmetric models which is needed for future physical investigations.
Posted Content•
08 Nov 2004
TL;DR: In this article, it was pointed out that the dominant term in the entropy is somewhat higher while the logarithmic correction is unaffected in their calculation, and they investigated the modification of the lower bound of [2] in view of this development.
Abstract: The corrected counting of states for black holes in the quantum geometry approach shows that the dominant configurations are distributions of spins that include spins exceeding one-half at the punctures. This alters the value of the Immirzi parameter and the black hole entropy. A framework for the calculation of black hole entropy in the quantum geometry approach was formulated in [1]. In that paper a lower bound for the entropy was worked out on the basis of the association of spin one-half to each puncture and found to be proportional to the area of the horizon. The proportionality constant involves what is known as the Immirzi parameter, which can be chosen so that the entropy becomes a quarter of the area. Recently, this lower bound was sharpened in [2] to include a logarithmic correction − 1 2 lnA. Subsequently, it was pointed out in [3] that the dominant term in the entropy is somewhat higher while the logarithmic correction is unaffected in their calculation. In the present note we investigate the modification of the lower bound of [2] in view of this development. A general configuration has sj punctures with spin j. Note that 2 ∑
TL;DR: Two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium, are derived.
Abstract: We formally derive two nonlinear Ginzburg-Landau type models starting from the Wigner-Fokker-Planck system, which rules the evolution of a quantum electron gas interacting with a heat bath in thermodynamic equilibrium. These models mainly consist of a quantum, dissipative O(Planck 3) hydrodynamic/O(Planck 4) stochastic correction to the frictional (Caldeira-Leggett-)Schrodinger equation. The main ingredient lies in the use of the hydrodynamic/stochastic fluid model approach associated with the quantum Fokker-Planck equation and the identification of the associated pressure field. Then, Madelung transformations set the problem in the Schrodinger picture of dissipative quantum mechanics. We also describe the stationary dynamics associated with both systems.
TL;DR: In this article, a Hilbert space representation of a contextual Kolmogorov model is constructed based on two fundamental observables, i.e., position and momentum observables.
Abstract: We constructed a Hilbert space representation of a contextual Kolmogorov model. This representation is based on two fundamental observables—in the standard quantum model these are the position and momentum observables. This representation has all distinguishing features of the quantum model. Our representation is not standard model with hidden variables. In particular, this is not a reduction of the quantum model to the classical one.
TL;DR: In this paper, the authors show that a recent proposal for the quantization of gravity based on discrete spacetime implies a modification of standard quantum mechanics that naturally leads to a loss of coherence in quantum states of the type discussed by Milburn.
Abstract: We show that a recent proposal for the quantization of gravity based on discrete spacetime implies a modification of standard quantum mechanics that naturally leads to a loss of coherence in quantum states of the type discussed by Milburn. The proposal overcomes the energy conservation problem of previously proposed decoherence mechanisms stemming from quantum gravity. Mesoscopic quantum systems (as Bose–Einstein condensates) appear as the most promising testing grounds for an experimental verification of the mechanism.
TL;DR: In this paper, it was shown that the contribution of spins greater than 1/2 to the black hole entropy is not negligible and that the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived.
Abstract: Quantum Geometry (the modern Loop Quantum Gravity using graphs and spin-networks instead of the loops) provides microscopic degrees of freedom that account for the black-hole entropy. However, the procedure for state counting used in the literature contains an error and the number of the relevant horizon states is underestimated. In our paper a correct method of counting is presented. Our results lead to a revision of the literature of the subject. It turns out that the contribution of spins greater then 1/2 to the entropy is not negligible. Hence, the value of the Barbero-Immirzi parameter involved in the spectra of all the geometric and physical operators in this theory is different than previously derived. Also, the conjectured relation between Quantum Geometry and the black hole quasi-normal modes should be understood again.
TL;DR: Loop quantum gravity as mentioned in this paper is a background independent approach to quantum gravity, which is based on the basic physical principles and how one deduces predictions from them, at a level suitable for physicsts in other areas such as string theory, cosmology, particle physics, and condensed matter physics.
Abstract: We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one deduces predictions from them, at a level suitable for physicsts in other areas such as string theory, cosmology, particle physics, astrophysics and condensed matter physics. No details are given, but references are provided to guide the interested reader to the literature. The present state of knowledge is summarized in a list of 35 key results on topics including the hamiltonian and path integral quantizations, coupling to matter, extensions to supergravity and higher dimensional theories, as well as applications to black holes, cosmology and Plank scale phenomenology. We describe the near term prospects for observational tests of quantum theories of gravity and the expectations that loop quantum gravity may provide predictions for their outcomes. Finally, we provide answers to frequently asked questions and a list of key open problems.
TL;DR: In this paper, the authors give a modern account of the construction and structure of the space of generalized connections, an extension of the spaces of connections that plays a central role in loop quantum gravity.
Abstract: We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
TL;DR: In this article, the equivalence principle is invoked to require both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities, and the observed numerical smallness of the cosmological constant can be rationalized in this approach.
Abstract: Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum theory to be fully dynamical quantities. Our approach sheds new light on such basic issues of quantum gravity as the nature of observables, the problem of time, and the physics of the vacuum. In particular, the observed numerical smallness of the cosmological constant can be rationalized in this approach.