Showing papers on "Quantum geometry published in 1994"
TL;DR: The treatment of gravity is described as a quantum effective field theory that allows a natural separation of the low energy quantum effects from the high energy contributions, and the leading quantum corrections to the gravitational interaction of two heavy masses are calculated.
Abstract: I describe the treatment of gravity as a quantum effective field theory. This allows a natural separation of the (known) low energy quantum effects from the (unknown) high energy contributions. Within this framework, gravity is a well-behaved quantum field theory at ordinary energies. In studying the class of quantum corrections at low energy, the dominant effects at large distance can be isolated, as these are due to the propagation of the massless particles ( including gravitons) of the theory and are manifested in the nonlocal and/or nonanalytic contributions to vertex functions and propagators. These leading quantum corrections are parameter-free and represent necessary consequences of quantum gravity. The methodology is illustrated by a calculation of the leading quantum corrections to the gravitational interaction of two heavy masses.
1,150 citations
01 Apr 1994
TL;DR: P-adic numbers padic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory padic strings as discussed by the authors.
Abstract: P-adic numbers p-adic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory p-adic strings.
1,147 citations
TL;DR: An overview of the integer quantum Hall effect is given in this paper, where a mathematical framework using non-ommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
Abstract: An overview of the integer quantum Hall effect is given. A mathematical framework using nonommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
626 citations
TL;DR: It is argued that the leading quantum corrections, in powers of the energy or inverse power of the distance, may be computed in quantum gravity through knowledge of only the low-energy structure of the theory.
Abstract: I argue that the leading quantum corrections, in powers of the energy or inverse powers of the distance, may be computed in quantum gravity through knowledge of only the low-energy structure of the theory. As an example, I calculate the leading quantum corrections to the Newtonian gravitational potential.
537 citations
TL;DR: In this article, the authors apply simultaneously the principles of quantum mechanics and general relativity to find an intrinsic limitation to quantum measurements of space-time distances and show that the intrinsic uncertainty of a length is proportional to the one third power of the length itself.
Abstract: Applying simultaneously the principles of quantum mechanics and general relativity we find an intrinsic limitation to quantum measurements of space-time distances. The intrinsic uncertainty of a length is shown to be proportional to the one third power of the length itself. This uncertainty in space-time measurements implies an intrinsic uncertainty of the space-time metric and yields quantum decoherence for particles heavier than the Planck mass.
227 citations
01 Jan 1994
196 citations
TL;DR: In this article, a systematic procedure to define and/or classify local transmission and reflection times for the passage of a quantum particle through a static potential barrier is described, and previously defined times and new quantities arise as particular cases of the general formalism.
Abstract: A systematic procedure to define and/or classify local transmission and reflection times for the passage of a quantum particle through a static potential barrier is described. Previously defined times and new quantities arise as particular cases of the general formalism. Generalizations for multidimensional and multichannel scattering systems are presented. The one-dimensional results are applied in detail to the rectangular potential. Other nonlocal approaches based on the current density are also examined, and the ``Hartman effect'' is quantitatively characterized for wave packets.
80 citations
TL;DR: In this article, a self-contained framework for quantum mechanics based on its path-integral or "sum-over-histories" formulation is proposed, which is very close to that for classical stochastic processes like Brownian motion, and its interpretation requires neither measurement nor state vector as a basic notion.
Abstract: I sketch a self-contained framework for quantum mechanics based on its path-integral or “sum-over-histories” formulation. The framework is very close to that for classical stochastic processes like Brownian motion, and its interpretation requires neither “measurement” nor “state-vector” as a basic notion. The rules for forming probabilities are nonclassical in two ways: they use complex amplitudes, and they (apparently unavoidably) require one to truncate the histories at a “collapse time,” which can be chosen arbitrarily far into the future. Adapting this framework to gravity yields a formulation of quantum gravity with a fully “spacetime” character, thereby overcoming the “frozen nature” of the canonical formalism. Within the proposed adaptation, the value of the “collapse time” is identified with total “elapsed” spacetime four-volume. Interestingly, this turns the cosmological constant into an essentially classical constant of integration, removing the need for microscopic “fine tuning” to obtain an experimentally viable value for it. Some implications of the “V = T” rule for quantum cosmology are also discussed.
70 citations
TL;DR: In this article, the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method, and examples in mathematical physics are given within the language of quantum spaces.
Abstract: It is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. Within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the endq-deformed instanton models are introduced for every integral index.
65 citations
60 citations
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TL;DR: Fermi as discussed by the authors gave a lecture at the International School of Physics in Varenna, Italy, June 28-July 7, 1994 with Enrico Fermi.
Abstract: Lectures given at International School of Physics ``Enrico Fermi'', Varenna, Villa Monastero, June 28-July 7 1994
Journal Article•
TL;DR: In this article, the authors critically review recent developments in quantum statistical mechanics and in the quantum dynamics of the vortex system in high temperature superconductors, and present a detailed review of these developments.
Abstract: We critically review recent developments in the quantum statistical mechanics and in the quantum dynamics of the vortex system in high temperature superconductors.
TL;DR: In this article, a noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations, and it is shown with the example of a quadratically ultraviolet divergent graph in $\phi^4$ theory that nonzero minimal uncertainties in positions do have the power to regularise.
Abstract: A noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations. There appear nonzero minimal uncertainties in positions and in momenta. As the main result it is shown with the example of a quadratically ultraviolet divergent graph in $\phi^4$ theory that nonzero minimal uncertainties in positions do have the power to regularise. These studies are motivated with the ansatz that nonzero minimal uncertainties in positions and in momenta arise from gravity. Algebraic techniques are used that have been developed in the field of quantum groups.
TL;DR: The vortex structure of the steady-state probability flow for a general one-particle system in quantum mechanics is introduced and its relationship to the inverse-square potential is discussed.
Abstract: The vortex structure of the steady-state probability flow for a general one-particle system in quantum mechanics is introduced and its relationship to the inverse-square potential is discussed. The relationship is made clear by classifying the corresponding solutions of the related classical inverse-square-potential problem.
TL;DR: In this article, the authors present the results of a high statistics study of a model for four dimensional euclidean quantum gravity based on summing over triangulations, which they call summing-over-triangulations (SOT).
Abstract: We present the results of a high statistics study of a model for four dimensional euclidean quantum gravity based on summing over triangulations.
TL;DR: In this paper, the quantum Z3-graded space is discussed and its differential calculus is investigated and the quantum matrices in quantum Z 3-gated space are obtained, and the results of the differential calculus are obtained.
Abstract: In this paper the quantum Z3‐graded space is discussed and its differential calculus is investigated and the quantum matrices in quantum Z3‐graded space are obtained.
TL;DR: New solutions to the Wheeler-DeWitt equation are presented that reinforce the conjecture that the Jones Polynomial is a state of nonperturbative quantum gravity.
Abstract: We propose a new representation for gauge theories and quantum gravity. Alternatively, it can be viewed as a new framework for doing computations in the loop representation. It is based on the use of a novel mathematical structure that extends the group of loops into a Lie group. This extension allows the use of functional methods to solve the diffeomorphism and Hamiltonian constraint equations. It puts in a precise framework some of the regularization problems of the loop representation. It has practical advantages in the search for quantum states. Making use of it we are able to find a new solution to the Wheeler-DeWitt equation that reinforces the conjecture that the Jones polynomial is a quantum state of nonperturbative quantum gravity.
TL;DR: The de Broglie--Bohm interpretation of quantum mechanics (and field theory) is explored as a way of spontaneously breaking general covariance, and thereby give meaning to the equations that many authors have been using to analyze black hole evaporation.
Abstract: We discuss the derivation of the so-called semiclassical equations for both minisuperspace and dilaton gravity. We find that there is no systematic derivation of a semiclassical theory in which quantum mechanics is formulated in a space-time that is a solution of Einstein's equations, with the expectation value of the matter stress tensor on the right-hand side. The issues involved are related to the well-known problems associated with the interpretation of the Wheeler-DeWitt equation in quantum gravity, including the problem of time. We explore the de Broglie--Bohm interpretation of quantum mechanics (and field theory) as a way of spontaneously breaking general covariance, and thereby give meaning to the equations that many authors have been using to analyze black hole evaporation. We comment on the implications for the information loss'' problem.
TL;DR: The reduction algorithms for functional determinants of differential operators on spacetime manifolds of different topological types are presented, which were recently used for the calculation of the no-boundary wavefunction and the partition function of tunnelling geometries in quantum gravity and cosmology.
Abstract: The reduction algorithms for functional determinants of differential operators on spacetime manifolds of different topological types are presented, which were recently used for the calculation of the no-boundary wavefunction and the partition function of tunnelling geometries in quantum gravity and cosmology.
TL;DR: In this article, the authors describe several models in the continuum with single-valued equations of motion in classical physics, but with multiple-valued Hamiltonians and their time displacements in quantum theory are therefore obliged to be discrete.
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TL;DR: A general noncommutative-geometric theory of principal bundles is presented in this paper, where quantum groups play the role of structure groups and quantum analogs of infinitesimal gauge transformations are studied.
Abstract: A general noncommutative-geometric theory of principal bundles is presented. Quantum groups play the role of structure groups. General quantum spaces play the role of base manifolds. A differential calculus on quantum principal bundles is studied. In particular, algebras of horizontal and verticalized differential forms on the bundle are introduced and investigated. The formalism of connections is developed. Operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. A quantum generalization of classical Weil's theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Illustrative examples and constructions are presented.
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TL;DR: The moduli space of N = 4,4 string theories with a K3 target space is determined in this article, and it is shown that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20).
Abstract: The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces.
TL;DR: It turns out that observables of the system possess finite expectation values although the Einstein-Hilbert action is unbounded, and this well-defined phase is found to be stable for a one-parameter family of measures.
Abstract: We analyze simplicial quantum gravity in four dimensions using the Regge approach. The existence of an entropy dominated phase with small negative curvature is investigated in detail. It turns out that observables of the system possess finite expectation values although the Einstein-Hilbert action is unbounded. This well-defined phase is found to be stable for a one-parameter family of measures. A preliminary study indicates that the influence of the lattice size on the average curvature is small. We compare our results with those obtained by dynamical triangulation and find qualitative correspondence.
01 Nov 1994
TL;DR: In this article, the authors present a model for two-dimensional quantum gravity with random surfaces and strings, and the mystery of $c > 1$ and Euclidean quantum gravity in $d > 2$.
Abstract: Contents:
1. Introduction
2. Bosonic propagators and random paths
3. Random surfaces and strings
4. Matrix models and two-dimensional quantum gravity
5. The mystery of $c > 1$
6. Euclidean quantum gravity in $d > 2$
7. Discussion
TL;DR: In this paper, the EPR argument is applied to one particle in a plane and contextuality emerges as a result, i.e., although the two orthogonal coordinates characterizing the position of a particle commute, there are states in which these coordinates are not independent, in the sense that the measurement or observation of one of them must have an effect on the other.
Abstract: The logic of the EPR argument is applied, not to two particles in a line but to one particle in a plane. Contextuality (defined here as a generalization of the notion of nonseparability) among commuting observables is shown to emerge as a result, i.e., although the two orthogonal coordinates characterizing the position of a particle commute, nonetheless there are states in which these coordinates are not independent, in the sense that the measurement or observation of one of them must have an effect on the other.
01 Jan 1994
TL;DR: A survey of supersymmetric: quantum mechanics and its applications to solvable quantum mechanical potentials can be found in this article, where the degeneracy of the energy levels of isospectral potentials is interpreted as supersymmetry.
Abstract: The introduction of supersymmetric quantum mechanics has generated renewed interest in solvable problems of non-relativistic quantum mechanics. This approach offers an elegant way to describe different, but isospectral potentials by interpreting the degeneracy of their energy levels in terms of supersymmetry. The original ideas of supersymmetric quantum mechanics have been developed further in many respects in the past ten years, and have been applied to a large variety of physical problems. The purpose of this contribution is to give a survey of supersymmetric: quantum mechanics and its applications to solvable quantum mechanical potentials. Its relation to other models describing isospectral potentials is also discussed here briefly, as well as some of its practical applications in various branches of physics.