Showing papers on "Quantum geometry published in 1998"

Quantum geometry and black hole entropy

Abstract: A ``black hole sector'' of nonperturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large nonrotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}A/4{\ensuremath{\ell}}_{P}^{2}$. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.

Loop Quantum Gravity

Abstract: The problem of describing the quantum behavior of gravity, and thus understanding quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this problem. It is a mathematically well-defined background-independent quantization of general relativity, with its conventional matter couplings. Today research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant results obtained so far are: (i) The computation of the spectra of geometrical quantities such as area and volume, which yield tentative quantitative predictions for Planck-scale physics. (ii) A physical picture of the microstructure of quantum spacetime, characterized by Planck-scale discreteness. Discreteness emerges as a standard quantum effect from the discrete spectra, and provides a mathematical realization of Wheeler’s “spacetime foam” intuition. (iii) Control of spacetime singularities, such as those in the interior of black holes and the cosmological one. This, in particular, has opened up the possibility of a theoretical investigation into the very early universe and the spacetime regions beyond the Big Bang. (iv) A derivation of the Bekenstein-Hawking black-hole entropy. (v) Low-energy calculations, yielding n-point functions well defined in a background-independent context. The theory is at the roots of, or strictly related to, a number of formalisms that have been developed for describing background-independent quantum field theory, such as spin foams, group field theory, causal spin networks, and others. I give here a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.

Relativistic spin networks and quantum gravity

Abstract: Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a four-dimensional state sum model for quantum gravity based on relativistic spin networks that parallels the construction of three-dimensional quantum gravity from ordinary spin networks.

Quantum Gravity in 2+1 Dimensions

Abstract: 1. Why (2+1)-dimensional gravity? 2. Classical general relativity in 2+1 dimensions 3. A field guide to the (2+1)-dimensional spacetimes 4. Geometric structures and Chern-Simons theory 5. Canonical quantization in reduced phase space 6. The connection representation 7. Operator algebras and loops 8. The Wheeler-DeWitt equation 9. Lorentzian path integrals 10. Euclidian path integrals and quantum cosmology 11. Lattice methods 12. The (2+1)-dimensional black hole 13. Next steps Appendix A. The topology of manifolds Appendix B. Lorentzian metrics and causal structure Appendix C. Differential geometry and fiber bundles References Index.

Discrete Approaches to Quantum Gravity in Four Dimensions

Abstract: The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.

Quantum spin dynamics (QSD): IV. ? Euclidean quantum gravity as a model to test ? Lorentzian quantum gravity

Abstract: The quantization of Lorentzian or Euclidean 2 + 1 gravity by canonical methods is a well studied problem. However, the constraints of 2 + 1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3 + 1 constraints. In this paper we canonically quantize Euclidean 2 + 1 gravity for an arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3 + 1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3 + 1 case, SU(2) rather than . We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3 + 1 Lorentzian quantum field theory for gravity. The space of solutions to all constraints turns out to be much larger than that obtained by traditional approaches, however, it is fully included. Thus, by a suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining `spurious solutions' is discussed.

A general survey of molecular quantum similarity

Abstract: A comprehensive survey of the theoretical foundations and definitions associated with quantum similarity is given. In this task care has been taken to determine the primary mathematical structure which can be associated with quantum similarity measures. Due to this, the concept of a tagged set is defined to demonstrate how molecular sets can be described systematically. The definition of quantum object, a notion introduced by our laboratory and employed for a long time in quantum similarity studies, is clarified by means of a blend involving quantum theory and the tagged set structure formalism, and used afterwards as the cornerstone of the subsequent development of the theory. In the definition of quantum objects, density functions play a fundamental role. To formally construct the quantum similarity measure, it is very interesting to study the main algorithmic ideas, which may serve to compute approximate density forms, accurate enough to be employed in the practical calculation of nuclear, atomic and molecular quantum similarity measures. Thus, the atomic shell approximation is defined accompanied by all the implied computational constraints and the consequences they have in the whole theory development as well as to the physical interpretation of the results. A wide and complex field appears from all these ideas, where convex sets play a fundamental role, and a new definition emerges: one associated with vector semispaces, where the main numerical formalism of quantum similarity seems perfectly adapted. Applications of this development embrace quantum taxonomy, visual representation of molecular sets, QSAR and QSPR, topological indices, molecular alignment, etc., and among this range of procedures and fields, there appears with distinct importance the discrete representation of molecular structures.

Quantum geometry with intrinsic local causality

Abstract: The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group ${G}_{q}$ over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.

Low Temperature Behavior and Crossovers of the Square Lattice Quantum Heisenberg Antiferromagnet

Abstract: We present thermodynamic measurements of various physical observables of the two-dimensional $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ isotropic quantum Heisenberg antiferromagnet on a square lattice, obtained by quantum Monte Carlo methods. The results are in excellent agreement with field-theoretical predictions. The issue of the existence of a crossover from quantum critical to renormalized classical regime is clarified.

The spectral dimension of 2D quantum gravity

Abstract: We show that the spectral dimension ds of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1. The same arguments provide a simple proof of the known result ds = 4/3 for branched polymers.

Quantum geometry of algebra factorisations and coalgebra bundles

Abstract: We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct $q$-monopoles on all the Podle\'s quantum spheres $S^2_{q,s}$.

Quantum rectifiers from harmonic mixing

Abstract: We investigate dissipative quantum transport in extended periodic systems that are subjected to electric harmonic mixing fields Ehm(t) = E1cos (Ωt) + E2cos (2Ωt + ). Although such a drive possesses no net bias on average, the interplay of quantum dissipation and nonlinear response causes a finite directed current. We thus discover the paradigm of a dissipative quantum rectifier. The quantum current exhibits multiple reversals when driven in the nonadiabatic regime. As a function of temperature the quantum current displays a bell-shaped characteristic–constituting the benchmark for quantum stochastic resonance. Moreover, harmonic mixing also serves as a novel tool to selectively control quantum diffusion.

Loop quantum gravity and quanta of space: a primer

Abstract: We present a straightforward and self-contained introduction to the basics of the loop approach to quantum gravity, and a derivation of what is arguably its key result, namely the spectral analysis of the area operator. We also discuss the arguments supporting the physical prediction following this result: that physical geometrical quantities are quantized in a non-trivial, computable, fashion. These results are not new; we present them here in a simple form that avoids the many non-essential complications of the first derivations.

Linear quantum trajectories : applications to continuous projection measurements

Abstract: We present a method for obtaining evolution operators for linear quantum trajectories. We apply this to a number of physical examples of varying mathematical complexity, in which the quantum trajectories describe the continuous projection measurement of physical observables. Using this method we calculate the average conditional uncertainty for the measured observables, being a central quantity of interest in these measurement processes.

Penrose universe and Cantorian spacetime as a model for noncommutative quantum geometry

Abstract: Cantorian spacetime E ∞ , with the continuous dimensional expectation E ∞ >=4+φ ∞ is shown to consist of hierachical self affine Penrose universes with noncommutative properties constituting a geometrical realization of the noncommutative formalism of orthodox quantum mechanics.

Lack of strong ellipticity in Euclidean quantum gravity

Abstract: Recent work in Euclidean quantum gravity has studied boundary conditions which are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using the de Donder gauge-averaging functional, this scheme leads to both normal and tangential derivatives in the boundary conditions. In the present paper, it is proved that the corresponding boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary.

On the Geometrization of Bohmian Mechanics:. a New Approach to Quantum Gravity

Abstract: In this paper, a new approach to quantum gravity is presented in which the de-Broglie–Bohm quantum theory of motion is geometrized. This way of considering quantum gravity leads automatically to the fact that the quantum effects are contained in the conformal degree of freedom of the space–time metric. The present theory is then applied to the maximally symmetric space–time of cosmology, and it is observed that it is possible to avoid the initial singularity, while at large times the correct classical limit emerges.

Kinematical bounds on optimization of observables for quantum systems

Abstract: Upper and lower kinematical bounds for the expectation values of arbitrary observables of driven quantum systems in mixed states are derived, and criteria for their attainability established. The results are applied to the problem of maximizing the energy of a laser-driven four-level Morse oscillator model for HF, as well as a four-level harmonic-oscillator model.

Quartic Gauge Couplings from K3 Geometry

Abstract: We show how certain F^4 couplings in eight dimensions can be computed using the mirror map and K3 data. They perfectly match with the corresponding heterotic one-loop couplings, and therefore this amounts to a successful test of the conjectured duality between the heterotic string on T^2 and F-theory on K3. The underlying quantum geometry appears to be a 5-fold, consisting of a hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this fiber is the symmetric product Sym^2(K3). We are lead to this structure by analyzing the implications of higher powers of E_2 in the relevant Borcherds counting functions, and in particular the appropriate generalizations of the Picard-Fuchs equations for the K3.

Quantum Geometry and Black Holes

Abstract: Non-perturbative quantum general relativity provides a possible framework to analyze issues related to black hole thermodynamics from a fundamental perspective. A pedagogical account of the recent developments in this area is given. The emphasis is on the conceptual and structural issues rather than technical subtleties. The article is addressed to post-graduate students and beginning researchers.

Quantum features of Brownian motors and stochastic resonance.

Abstract: We investigate quantum Brownian motion sustained transport in both, adiabatically rocked ratchet systems and quantum stochastic resonance (QSR). Above a characteristic crossover temperature T0 tunneling events are rare; yet they can considerably enhance the quantum-noise-driven particle current and the amplification of signal output in comparison to their classical counterparts. Below T0 tunneling prevails, thus yielding characteristic novel quantum transport phenomena. For example, upon approaching T=0 the quantum current in Brownian motors exhibits a tunneling-induced reversal, and tends to a finite limit, while the classical result approaches zero without such a change of sign. As a consequence, similar current inversions generated by quantum effects follow upon variation of the particle mass or of its friction coefficient. Likewise, in this latter regime of very low temperatures the tunneling dynamics becomes increasingly coherent, thus suppressing the semiclassically predicted QSR. Moreover, nonadiab...

Density Formalism for Quantum Theory

Abstract: A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a picture of an external reality. The new formalism, developed first for the single-particle case, has the advantage of generalizing immediately to quantum field theory and to the description of relativistic phenomena such as particle creation and annihilation.

Quantum Geometry and Diffusion

Abstract: We study the diffusion equation in two-dimensional quantum gravity, and show that the spectral dimension is two despite the fact that the intrinsic Hausdorff dimension of the ensemble of two-dimensional geometries is very different from two. We determine the scaling properties of the quantum gravity averaged diffusion kernel.

The area spectrum in quantum gravity

Abstract: We show that, apart from the usual area operator of non-perturbative quantum gravity, there exists another, closely related, operator that measures areas of surfaces. Both corresponding classical expressions yield the area. Quantum mechanically, however, the spectra of the two operators are different, coinciding only in the limit when the spins labelling the state are large. We argue that both operators are legitimate quantum operators, and the choice of which one to use depends on the context of the physical problem of interest. Thus, for example, we argue that it is the operator proposed here that is relevant for use in the context of black holes as measuring the area of the black-hole horizon. We show that the difference between the two operators is due to the non-commutativity that is known to arise in the quantum theory. We give a heuristic picture explaining the difference between the two area spectra in terms of quantum fluctuations of the surface whose area is being measured.

Midi-superspace quantization of non-compact toroidally symmetric gravity

Abstract: We consider the quantization of the midi-superspace associated with a class of spacetimes with toroidal isometries, but without the compact spatial hypersurfaces of the well-known Gowdy models. By a symmetry reduction, the phase space for the system at the classical level can be identified with that of a free massless scalar field on a fixed background spacetime, thereby providing a simple route to quantization. We are then able to study certain non-perturbative features of the quantum gravitational system. In particular, we examine the quantum geometry of the asymptotic regions of the spacetimes involved and find some surprisingly large dispersive effects of quantum gravity.

Model for the effect of finite phase-coherence length on resonant transmission and capture by quantum wells

Abstract: We study the effect of an imaginary potential and (separately) of a finite coherence length on the transmission, reflection, and capture fractions for a thermal distribution of carriers incident on a single quantum well. The formalism used is closely related to one used by Kuhn and Mahler for the same purpose. Closed-form expressions are obtained for the three transport fractions resulting from a single incident beam. Three independent fitting parameters are used in this formalism, namely, the size of the imaginary potential, the extent it penetrates into the barriers adjacent to the well, and the phase-coherence length. This last is a length scale associated with a correlation function that appears when the phase of the wave function is treated as a stochastic variable. We show that the parameters can be chosen so that the transport fractions agree with those calculated from first principles, and show how a shortening of the coherence length, e.g., by electron-electron interactions that have been left out of the first-principles calculation, destroys the resonant behavior of these fractions predicted by Brum and Bastard [Phys. Rev. B 33, 1420 (1986)].

Dimensionful deformations of poincaré symmetries for a quantum gravity without ideal observers

Abstract: Quantum mechanics is revisited as the appropriate theoretical framework for the description of the outcome of experiments that rely on the use of classical devices. In particular, it is emphasized that the limitations on the measurability of (pairs of conjugate) observables encoded in the formalism of quantum mechanics reproduce faithfully the "classical-device limit" of the corresponding limitations encountered in (real or gedanken) experimental setups. It is then argued that devices cannot behave classically in quantum gravity, and that this might raise serious problems for the search of a class of experiments described by theories obtained by "applying quantum mechanics to gravity." It is also observed that using heuristic/intuitive arguments based on the absence of classical devices one is led to consider some candidate quantum gravity phenomena involving dimensionful deformations of the Poincare symmetries.