Showing papers on "Quantum geometry published in 1997"

Quantum theory of geometry: I. Area operators

Abstract: A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.

Non-pointlike particles in harmonic oscillators

Abstract: Quantum mechanics usually describes particles as being pointlike in the sense that, in principle, the uncertainty, , can be made arbitrarily small. Studies on string theory and quantum gravity motivate correction terms to the uncertainty relations which induce a finite lower bound to spatial localization. This structure is implemented into quantum mechanics through small correction terms to the canonical commutation relations. We calculate the perturbations to the energy levels of a particle which is non-pointlike in this sense in isotropic harmonic oscillators, where we find a characteristic splitting of the usually degenerate energy levels. Possible applications are outlined.

Quantum Geometry: A Statistical Field Theory Approach

Abstract: Preface 1. Introduction 2. Random walks 3. Random surfaces 4. Two-dimensional gravity 5. Monte Carlo simulations 6. Gravity in higher dimensions 7. Topological quantum field theories References Index.

String Theory on Calabi-Yau Manifolds

Abstract: These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality.

Forks in the road, on the way to quantum gravity

Abstract: In seeking to arrive at a theory of “quantum gravity,” one faces several choices among alternative approaches. I list some of these “forks in the road” and offer reasons for taking one alternative over the other. In particular, I advocate the following: the sum-over-histories framework for quantum dynamics over the “observable and state-vector” framework; relative probabilities over absolute ones; spacetime over space as the gravitational “substance” (4 over 3+1); a Lorentzian metric over a Riemannian (“Euclidean”) one; a dynamical topology over an absolute one; degenerate metrics over closed timelike curves to mediate topology change; “unimodular gravity” over the unrestricted functional integral; and taking a discrete underlying structure (the causal set) rather than the differentiable manifold as the basis of the theory. In connection with these choices, I also mention some results from unimodular quantum cosmology, sketch an account of the origin of black hole entropy, summarize an argument that the quantum mechanical measurement scheme breaks down for quantum field theory, and offer a reason why the cosmological constant of the present epoch might have a magnitude of around 10−120 in natural units.

Euclidean Quantum Gravity

Abstract: This chapter studies the linearized gravitational field in the presence of boundaries. For this purpose, zeta-function regularization is used to perform the mode-by-mode evaluation of Faddeev-Popov amplitudes in the case of flat Euclidean four-space bounded by two concentric three-spheres, or just one three-sphere. On choosing the de Donder gauge-averaging term, the resulting ζ(0) value is found to agree with the space-time covariant calculation of the same amplitudes, which relies on the recently corrected geometric formulae for the asymptotic heat kernel in the case of mixed boundary conditions. Two sets of mixed boundary conditions for Euclidean quantum gravity are then compared in detail. The analysis proves that one cannot restrict the path-integral measure to transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are only obtained on considering from the beginning all perturbative modes of the gravitational field, jointly with ghost modes. Unlike the mixed boundary conditions involving (complementary) projectors, one knows from chapter six that boundary conditions completely invariant under infinitesimal diffeomorphisms involve both normal and tangential derivatives of metric perturbations. The corresponding ζ(0) value is obtained, and the proof of symmetry of the Laplace operator in such a case is obtained. Mixed boundary conditions are also considered which lead to Robin conditions on spatial metric perturbations, and Dirichlet conditions on normal metric perturbations. Last, a review of Hawking’s proposal to consider smooth simply connected four-manifolds as the building blocks of Euclidean quantum gravity is presented. This makes it necessary to study physical processes in S 2 x S 2 , K3 and CP 2 geometries. Yet another open problem is a consistent formulation of quantum supergravity on manifolds with boundary.

Geometry of Quantum Principal Bundles II

Abstract: A general non-commutative-geometric theory of principal bundles is developed. Quantum groups play the role of structure groups and general quantum spaces play the role of base manifolds. A general conceptual framework for the study of differential structures on quantum principal bundles is presented. Algebras of horizontal, verticalized and "horizontally vertically" decomposed differential forms on the bundle are introduced and investigated. Constructive approaches to differential calculi on quantum principal bundles are discussed. The formalism of connections is developed further. The corresponding operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. In particular the analogs of the basic classical algebraic identities are derived. A quantum generalization of classical Weil's theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Interesting examples are presented.

The future of spin networks

Abstract: The roles that spin networks play in gauge theories, quantum gravity and topological quantum field theory are briefly described, with an emphasis on the question of the relationships among them It is argued that spin networks and their generalizations provide a language which may lead to a unification of the different approaches to quantum gravity and quantum geometry This leads to a set of conjectures about the form of a future theory that may be simultaneously an extension of the non-perturbative quantization of general relativity and a non-perturbative formulation of string theory

Quantum differentials and the q-monopole revisited

Abstract: The q-monopole bundle introduced previously is extended to a general construction for quantum group bundles with non-universal differential calculi. We show that the theory applies to several other classes of bundles as well, including bicrossproduct quantum groups, the quantum double and combinatorial bundles associated to covers of compact manifolds.

Loss of quantum coherence through scattering off virtual black holes

Abstract: In quantum gravity, fields may lose quantum coherence by scattering off vacuum fluctuations in which virtual black hole pairs appear and disappear. Although it is not possible to properly compute the scattering off such fluctuations, we argue that one can get useful qualitative results, which provide a guide to the possible effects of such scattering, by considering a quantum field on the $C$ metric, which has the same topology as a virtual black hole pair. We study a scalar field on the Lorentzian $C$ metric background, with the scalar field in the analytically continued Euclidean vacuum state. We find that there are a finite number of particles at infinity in this state, contrary to recent claims made by Yi. Thus, this state is not determined by data at infinity, and there is loss of quantum coherence in this semiclassical calculation.

Noncommutative Geometry of the h-deformed Quantum Plane

Abstract: The $h$-deformed quantum plane is a counterpart of the $q$-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the $h$-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a skew-symmetric non-degenerate bilinear map. The skew-symmetric map resembles a symplectic 2-form and induces a metric. It is also shown that the extended $h$-deformed quantum plane is a noncommutative version of the Poincar\'e half-plane, a surface of constant negative Gaussian curvature.

The Quantum Geometry of N=(2,2) Non-Linear Sigma-Models

Abstract: We consider a general N=(2,2) non-linear sigma-model in (2,2) superspace. Depending on the details of the complex structures involved, an off-shell description can be given in terms of chiral, twisted chiral and semi-chiral superfields. Using superspace techniques, we derive the conditions the potential has to satisfy in order to be ultra-violet finite at one loop. We pay particular attention to the effects due to the presence of semi-chiral superfields. A complete description of N=(2,2) strings follows from this.

Quantum fields and quantum space time

Abstract: Lectures: Non-communicative Gauge Fields from Quantum Groups A.Yu. Alekseev, V. Schomerus. Duality in N=2 SUSY SU(2) Yang-Mills Theory: A Pedagogical Introduction A. Bilal. Noncommunicative Differential Geometry and the Structure of Space Time A. Connes. Quantum Integrable Models on 1+1 Discrete Space Time L. Faddeev, A. Volkov. Supersymmetry and Non-Communicative Geometry J. Frohlich, et al. Turbulence under a Magnifying Glass K. Gawedzki. Quantization of Space and Time in 3 and in 4 Space-Time Dimensions G. 't Hooft. Pushing Einstein's Principles to the Extreme G. Mack. Integrable Classical and Quantum Gravity H. Nicolai, et al. Quantum Affine Algebras and Integrable Quantum Systems V. Chari, A. Pressley. Seminars: T-Duality and the Moment Map C. Klimcik, P. Severa. Symmetries of Dimensionally Reduced String Effective Action J. Maharana. Non Local Observables and Confinement in BF Formulation of Yang-Mills Theory F. Fucito, et al. Disorder Operators, Quantum Doubles, and Haag Duality in 1+1 Dimensions M. Muger. The Cohomology and Homology of Quantum Field Theory J.E. Roberts. 2 Additional Lectures. Index.

Supersymmetric quantum theory, non-commutative geometry, and gravitation. Lecture Notes Les Houches 1995

Abstract: This is an expanded version of the notes to a course taught by the first author at the 1995 Les Houches Summer School. Constraints on a tentative reconciliation of quantum theory and general relativity are reviewed. It is explained what supersymmetric quantum theory teaches us about differential topology and geometry. Non-commutative differential topology and geometry are developed in some detail. As an example, the non-commutative torus is studied. An introduction to string theory and $M$(atrix) models is provided, and it is outlined how tools of non-commutative geometry can be used to explore the geometry of string theory and conformal field theory.

Some Complex Quantum Manifolds and their Geometry

Abstract: Quantum spheres can be defined in any number of dimensions by normalizing a vector of quantum Euclidean space [1]. The differential calculus on quantum Euclidean space [2] induces a calculus on the quantum sphere. The case of two-spheres in three-space is special in that there are many more possibilities. These have been studied by P. Podleś [3, 4, 5, 6] who has defined quantum spheres as quantum spaces on which quantum SU q (2) coacts. He has also developed a noncommutative differential calculus on them. In these lectures we consider, following [7], a special case of Podles spheres which is one of those special to three space dimensions.

Weave States in Loop Quantum Gravity

Abstract: Weaves are eigenstates of geometrical operators in nonperturbative quantum gravity, which approximate flat space (or other smooth geometries) at large scales. We describe two such states, which diagonalize the area as well as the volume operators. The existence of such states shows that some earlier worries about the difficulty of realizing kinematical states with non-vanishing volume can be overcome. We also show that the Q operator used in earlier work for extracting geometrical information from quantum states does not capture more information than the area and volume operators.

The hidden measurement formalism: quantum mechanics as a consequence of fluctuations on the measurement.

Abstract: We expose a formalism, that we have called the’ hidden measurement formalism’, where the quantum structure is due to the presence of ‘fluctuations’ on the interaction between the system and the measurement apparatus. In this formalism the quantum mechanical probabilities are not ontological but arrise as a consequence of ‘lack of knowledge’ about this interaction. We study the quantum classical limit and the EPR problem in the light of this explanation.

Spin networks and recoupling in loop quantum gravity

Abstract: I discuss the role played by the spin-network basis and recoupling theory (in its graphical tangle-theoretic formulation) and their use in performing explicit calculations in loop quantum gravity. In particular, I show that recoupling theory allows the derivation of explicit expressions for the eigenvalues of the quantum volume operator. An important side result of these computations is the determination of a scalar product with respect to which area and volume operators are symmetric, and the spin network states orthonormal.

Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials canonical commutation relations, quantum algebras, and reduction to lattice quantum systems

Abstract: Some representation-theoretic aspects of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R2 are investigated. For each vector v in a set V(D)⊂R2\{0}, the projection Pv of the physical momentum operator P≔p−αA to the direction of v is defined by Pv≔v⋅P as an operator acting in L2(R2), where p=(−iDx,−iDy)[(x,y)∈R2] with Dx (resp., Dy) being the generalized partial differential operator in the variable x (resp., y) and α∈R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter unitary group {eitPv}t∈R generated by the self-adjoint operator Pv (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2\D are also considered. Conjugately to Pv and Pw [w∈V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a ...