Showing papers on "Quantum geometry published in 1996"

Quantum Theory of Gravity 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 {\it 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 1-dimensional, rather like polymers, and the 3-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite dimensional sub-spaces 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 3-dimensional geometric operators, e.g., the ones corresponding to volumes of regions.

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 {\it 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 1-dimensional, rather like polymers, and the 3-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite dimensional sub-spaces 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 3-dimensional geometric operators, e.g., the ones corresponding to volumes of regions.

Loops, knots, gauge theories and quantum gravity

Abstract: 1. Holonomies and the group of loops 2. Loop coordinates and the extended group of loops 3. The loop representation 4. Maxwell theory 5. Yang-Mills theories 6. Lattice techniques 7. Quantum gravity 8. The loop representation of quantum theory 9. Loop representation: further developments 10. Knot theory and physical states of quantum gravity 11. The extended loop representation of quantum gravity 12. Conclusions: present status and outlook References Index.

Geometry eigenvalues and the scalar product from recoupling theory in loop quantum gravity.

Abstract: We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its graphical tangle-theoretic Temperley-Lieb formulation, provides a powerful calculation tool in this context. We describe its application to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and the quantum volume operators. We compute several volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthonormal.

Introduction To Quantum Groups

Abstract: Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of classical Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: classical relativistic space-time symmetries Poincare group as a "classical" deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

Limits on the Measurability of Space-time Distances in (the Semi-classical Approximation of) Quantum Gravity

Abstract: By taking into account both quantum mechanical and general relativistic effects, I derive an equation that describes some limitations on the measurability of space-time distances. I then discuss possible features of quantum gravity which are suggested by this equation.

The complete spectrum of the area from recoupling theory in loop quantum gravity

Abstract: We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the `degenerate' sector, and agrees with the recently computed spectrum of the connection-representation area operator.

New features in supersymmetry breakdown in quantum mechanics

Abstract: The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered.

Large Quantum Gravity Effects: Unforeseen Limitations of the Classical Theory

Abstract: Three-dimensional gravity coupled to Maxwell (or Klein-Gordon) fields is exactly soluble under the assumption of axisymmetry. The solution is used to probe several quantum gravity issues. In particular, it is found that if there is an electromagnetic wave of Planckian frequency even with such low amplitude that the curvature of the classical solution is small, the uncertainty in the quantum metric can be very large. More generally, the quantum fluctuations in the geometry are large unless the number and frequency of photons satisfy the inequality $N(\ensuremath{\Elzxh}G\ensuremath{\omega}{)}^{2}\ensuremath{\ll}1$. Results hold also for a sector of the four-dimensional theory (consisting of Einstein-Rosen gravitational waves).

The geometry of quantum spin networks

Abstract: The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation of the observable algebra. Operators for area and volume are extended to this theory and, partly, diagonalized. The eigenstates are expressed in terms of q-deformed spin networks. The q-deformation breaks some of the degeneracy of the volume operator so that trivalent spin networks have non-zero volume. These computations are facilitated by use of a technique based on the recoupling theory of , which simplifies the construction of these and other operators through diffeomorphism invariant regularization procedures.

Geometry of Quantum Statistical Inference.

Abstract: An efficient geometric formulation of the problem of parameter estimation is developed, based on Hilbert space geometry. This theory, which allows for a transparent transition between classical and quantum statistical inference, is then applied to the analysis of exponential families of distributions (of relevance to statistical mechanics) and quantum mechanical evolutions. The extension to quantum theory is achieved by the introduction of a complex structure on the given real Hilbert space. We find a set of higher order corrections to the parameter estimation variance lower bound, which are potentially important in quantum mechanics, where these corrections appear as modifications to Heisenberg uncertainty relations for the determination of the parameter. [S0031-9007(96)01153-2]

Physics with nonperturbative quantum gravity: Radiation from a quantum black hole

Abstract: We study quantum gravitational effects on black hole radiation, using loop quantum gravity. Bekenstein and Mukhanov have recently considered the modifications caused by quantum gravity on Hawking's thermal black-hole radiation. Using a simple ansatz for the eigenstates of the area, they have obtained the intriguing result that the quantum properties of geometry affect the radiation considerably, yielding a discrete spectrum, definitely non-thermal. Here, we replace the simple ansatz employed by Bekenstein and Mukhanov with the actual eigenstates of the area computed using loop quantum gravity. We derive the emission spectra, using a classic result in number theory by Hardy and Ramanujan. Disappointingly, we do not recover the Bekenstein-Mukhanov discrete spectrum, but — effectively — a continuum spectrum, consistent with Hawking's result. The Bekenstein-Mukhanov argument for the discreteness of the specrum is therefore likely to be an artifact of the ansatz, rather than a robust result (at least in its present kinematical version). The result is an example of concrete (although somewhat disappointing) application of nonperturbative quantum gravity.

Band structure and quantum Poincaré sections of a classically chaotic quantum rippled channel

Abstract: We obtain the energy band spectra, eigenfunctions, and quantum Poincar\'e sections of a free particle moving in a two-dimensional channel bounded by a periodically varying (ripple) wall and a flat wall. Classical Poincar\'e sections show a generic transition from regular to chaotic motion as the size of the ripple is increased. The energy band structure is obtained for two representative geometries corresponding to a wide and a narrow channel. The comparison of numerical results with the level-splitting predictions of low-order quantum degenerate perturbation theory elucidate some aspects of the classical-quantum correspondence. For larger ripple amplitudes the conduction bands for narrow channels become flat and nearly equidistant at low energies. Quantum-classical correspondence is discussed with the aid of quantum Poincar\'e (Husimi) plots.

Time symmetry breaking, duality and cantorian space-time

Abstract: Connections are drawn between the global thermodynamical interpretation of quantum mechanics and the reductionist Cantorian-fractal space-time approach. The objective is to show the influence of the thermodynamical approach on development in both fields and to explain how Cantorian space can serve as a geometrical model for a space-time support of the thermodynamical approach to quantum mechanics. Seen through both theories, quantum mechanics could appear to be the result of a turbulent but homogeneous diffusion process in a transfinite non-smooth micro space-time with an area-like quantum ‘path’. Time symmetry breaking is then a consequence of the transfinite information barrier of Cantorian space-time. An important result found here is that the four dimensionality of micro space-time is a consequence of a discrete Maxwell-Boltzmann distribution of the elementary Cantor sets forming this space. In fact, it will be shown here that many of the paradoxes of quantum mechanics can be traced back to the contraintuitive character of the underlying unstable and nonsmooth Cantorian geometry of micro space-time.

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.

Quantum homogeneous spaces as quantum quotient spaces

Abstract: It is shown that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. A construction of quantum fibre bundles on such spaces is proposed. The quantum plane and the general quantum two‐spheres are discussed in detail.

Quantum phase in interferometry

Abstract: Quantum estimation of a phase shift is formulated within quantum mechanics and information theory. Using Bayes' estimation, the phase can be attributed even to a single interfering particle. The method is demonstrated for neutron interferometry operating near the quantum limit.

Closure conditions for classical and quantum moment hierarchies in the small-temperature limit

Abstract: We analyze closure conditions in the small initial temperature limit for classical and quantum mechanical moment hierarchies of corresponding collisionless kinetic equations. Euler equations with a nondiagonal pressure tensor are obtained in the classical case. In the quantum case we consider the cases of fixed and small (scaled) Planck constant and derive Quantum Hydrodynamic equations.

Loop quantum gravity and black hole physics

Abstract: I summarize the basic ideas and formalism of loop quantum gravity. I illustrate the results on the discrete aspects of quantum geometry and two applications of these results to black hole physics. In particular, I discuss in detail a derivation of the Bekenstein-Hawking formula for the entropy of a black hole from first principles.

Loop quantum gravity and black hole physics

Abstract: I summarize the basic ideas and formalism of loop quantum gravity. I illustrate the results on the discrete aspects of quantum geometry and two applications of these results to black hole physics. In particular, I discuss in detail a derivation of the Bekenstein-Hawking formula for the entropy of a black hole from first principles.

Quantum integrability for the one-dimensional Hubbard open chain

Abstract: An extended formalism is presented to treat quantum integrable systems on a finite interval with independent boundary conditions on each end. It is shown that this formalism provides a natural description for the quantum integrability of the one-dimensional Hubbard open chain. \textcopyright{} 1996 The American Physical Society.

Interaction hierarchy: string and quantum gravity

Abstract: We have found that the functional integral for quantum gravity can be represented as a superposition of less complicated theory of random surfaces with Euler character as an action. We propose an alternative linear action A(M4) for quantum gravity. On the lattice we constructed spin system with local interaction, which has the equivalent partition function. The scaling limit is discussed.

Small Volumes in Compactified String Theory

Abstract: We discuss some of the classical and quantum geometry associated to the degeneration of cycles within a Calabi-Yau compactification. In particular, we focus on the definition and properties of quantum volume, especially as it applies to identifying the physics associated to loci in moduli space where nonperturbative effects become manifest. We discuss some unusual features of quantum volume relative to its classical counterpart.

Geometry of Pdes and Mechanics

Abstract: Algebraic geometry differential equations (PDEs) mechanics continuum mechanics quantum field theory geometry of quantum PDEs.

Polymer geometry at Planck scale and quantum Einstein equations

Abstract: It is well known that quantum general relativity is perturbatively non-renormalizable. Particletheorists often take this to be a sufficient reason to abandon general relativity and seek an alternative which has a better ultraviolet behavior in perturbation theory. However, one is by no means forced to this route. For, there do exist a number of field theories which are perturbatively non-renormalizable but are exactly soluble. An outstanding example is the Gross-Neveau model in 3 dimensions, (GN)3, which was recently shown to be exactly soluble rigorously [1]. Furthermore, the model does not exhibit any mathematical pathologies. For example, it was at first conjectured that the Wightman functions of a non-renormalizable theory would have a worse mathematical behavior. The solution to (GN)3 showed that this is not the case; as in familiar renormalizable theories, they are tempered distributions. Thus, one can argue that, from a structural viewpoint, perturbative renormalizability is a luxury even in Minkowskian quantum field theories. Of course, it serves as a powerful guiding principle for selecting physically interesting theories since it ensures that the predictions of the theory at a certain length scale are independent of the potential complications at much smaller scales. But it is not a consistency check on the mathematical viability of a theory. Furthermore, in quantum gravity, one is interested precisely in the physics of the Planck scale; the short-distance complications are now the issues of primary interest. Therefore, it seems inappropriate to elevate perturbative renormalizability to a viability criterion.

Correlations in quantum plasmas. II. Algebraic tails.

Abstract: For a system of point charges that interact through the three-dimensional electrostatic Coulomb potential ~without any regularization ! and obey the laws of nonrelativistic quantum mechanics with Bose or Fermi statistics, the static correlations between particles are shown to have a 1/ r 6 t il, at least at distances that are large with respect to the length of exponential screening. After a review of previous work, a term-by-term diagrammatic proof is given by using the formalism of paper I, where the quantum particle-particle correlations are expressed in terms of classical-loop distribution functions. The integrable graphs of the resummed Mayerlike diagrammatics for the loop distributions contain bonds between loops that decay either exponentially or algebraically, with a 1/ r 3 leading term analogous to a dipole-dipole interaction. This reflects the fact that the charge-charge or multipole-charge interactions between clusters of particles surrounded by their polarization clouds are exponentially screened, as at a classical level, whereas the multipole-multipole interactions are only partially screened. The correlation between loops decays as 1/ r , but the spherical symmetry of the quantum fluctuations makes this power law fall to 1/ r , and the harmonicity of the Coulomb potential eventually enforces the correlations between quantum particles to decay only as 1/ r . The coefficient of the 1/ r 6 tail at low density is planned to be given in a subsequent paper. Moreover, because of Coulomb screening, the induced charge density, which describes the response to an externalinfinitesimal charge, is shown to fall off as 1/ r , while the charge-charge correlation in the medium decreases as 1/ r . However, in spite of the departure of the quantum microscopicorrelations from the classical exponential clustering, the total induced charge is still essentially determined by the exponentially screened charge-charge interactions, as in clas ical macroscopic electrostatics. @S1063-651X~96!05205-1#

Glueing operation for R-matrices, quantum groups and link-invariants of Hecke type

Abstract: We introduce an associative glueing operation $\oplus_q$ on the space of solutions of the Quantum Yang-Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves $2\times 2$ quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard $su(n)$ solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.