Geometry Fluctuations Research Articles

Finite quantum electrodynamics with a gravitationally smeared propagator

On the basis of qualitative considerations of quantum fluctuations in the space-time geometry, a modification of the Feynman propagator is proposed involving a smearing of the propagator over a space-time region of an extent of about a Planck length, or $\ensuremath{\simeq}{10}^{\ensuremath{-}33}\mathrm{cm},$ surrounding the light cone. The smeared propagator is a weighted average of modified Bessel functions both in coordinate space and in momentum space. In coordinate space, the smeared propagator has a milder singularity on the light cone than the Feynman propagator, and in momentum space it provides an effective cutoff at the Planck mass. Radiative corrections in QED can be easily calculated with this propagator, and the results are finite. The renormalizations of the electron charge and mass are of the order of a few percent.

Optimal self-avoiding paths in dilute random medium

The combined effects of bond-energy disorder and random-bond exclusion on optimal undirected self-avoiding paths are studied by an original finite-size scaling method in two dimensions. For concentrations of accessible bonds between the undirected and directed percolation thresholds, overhangs do not seem to change the standard self-affine scaling regime characteristic of directed paths. At the undirected threshold the path becomes fractal, with a fractal dimension equal to that of the minimal length path on the infinite cluster backbone. At this point the optimal energy variance scales with time t as ${\mathrm{t}}^{{\mathrm{\ensuremath{\omega}}}_{\mathrm{c}}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathrm{c}}$=1.02\ifmmode\pm\else\textpm\fi{}0.05). Furthermore, ${\mathrm{\ensuremath{\omega}}}_{\mathrm{c}}$ turns out to be exclusively determined by fluctuations in backbone geometry and not by disorder in bond energies. This scenario is qualitatively confirmed and extended by renormalization-group calculations on hierarchical lattices.

Black hole thermodynamics from quantum gravity

The semiclassical approximation is studied on hypersurfaces approaching the union of future null infinity and the event horizon on a large class of four-dimensional black hole backgrounds. Quantum fluctuations in the background geometry are shown to lead to a breakdown of the semiclassical approximation in these models. The boundary of the region where the semiclassical approximation breaks down is used to define a ‘stretched horizon’. It is shown that the same effect that brings about the breakdown in semiclassical evolution associates a temperature and an entropy to the region behind the stretched horizon, and identifies the microstates that underlie the thermodynamical properties. The temperature defined in this way is equal to that of the black hole and the entropy is equal to the Bekenstein entropy up to a factor of order one.

Towards optimal measurement of power spectra - I. Minimum variance pair weighting and the Fisher matrix

This is the first of a pair of papers which address the problem of measuring the unredshifted power spectrum in optimal fashion from a survey of galaxies, with arbitrary geometry, for Gaussian or non-Gaussian fluctuations, in real or redshift space. In this first paper, that pair weighting is derived which formally minimizes the expected variance of the unredshifted power spectrum windowed over some arbitrary kernel. The inverse of the covariance matrix of minimum variance estimators of windowed power spectra is the Fisher information matrix, which plays a central role in establishing optimal estimators. Actually computing the minimum variance pair window and the Fisher matrix in a real survey still presents a formidable numerical problem, so here a perturbation series solution is developed. The properties of the Fisher matrix evaluated according to the approximate method suggested here are investigated in more detail in the second paper.

Gravity coupled with matter and the foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D −1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.

Note on the semiclassical approximation in quantum gravity.

We re-examine the semiclassical approximation to quantum gravity in the canonical formulation, focusing on the definition of a quasiclassical state for the gravitational field. It is shown that a state with classical correlations must be a superposition of states of the form $e^{iS}$. In terms of a reduced phase space formalism, this type of state can be expressed as a coherent superposition of eigenstates of operators that commute with the constraints and so correspond to constants of the motion. Contact is made with the usual semiclassical approximation by showing that a superposition of this kind can be approximated by a WKB state with an appropriately localised prefactor. A qualitative analysis is given of the effects of geometry fluctuations, and the possibility of a breakdown of the semiclassical approximation due to interference between neighbouring classical trajectories is discussed. It is shown that a breakdown in the semiclassical approximation can be a coordinate dependent phenomenon, as has been argued to be the case close to a black hole horizon.

Quantum gravity effects at a black hole horizon

Quantum fluctuations in the background geometry of a black hole are shown to affect the propagation of matter states falling into the black hole in a foliation that corresponds to observations purely outside the horizon. A state that starts as a Minkowski vacuum at past null infinity gets entangled with the gravity sector, so that close to the horizon it can be represented by a statistical ensemble of orthogonal states. We construct an operator connecting the different states and comment on the possible physical meaning of the above construction. The induced energy-momentum tensor of these states is computed in the neighbourhood of the horizon, and it is found that energy-momentum fluctuations become large in the region where the bulk of the Hawking radiation is produced. The background spacetime as seen by an outside observer may be drastically altered in this region, and an outside observer should see significant interactions between the infalling matter and the outgoing Hawking radiation. The boundary of the region of strong quantum gravitational effects is given by a time-like hypersurface of constant Schwarzschild radius r one Planck unit away from the horizon. This boundary hypersurface is an example of a stretched horizon.

Quantum smearing of spacetime singularity

We propose a quantum mechanical generalization of the affine length of a curve which gives a criterion of whether a classical singularity is smeared out by a quantum fluctuation of the spacetime geometry. An explicit example is demonstrated in the (2 + 1)-dimensional black-hole model.

Measurements of beam geometry fluctuations of typical argon-ion and Nd:YAG lasers with relevance to laser interferometer gravitational wave detectors

The best sensitivity of a laser interferometric gravitational wave detector can only be achieved after considerable reduction is made in fluctuations of the geometry of the laser beam. Beam geometry fluctuations can be divided into four main types; lateral beam movement, tilt of the beam, variations in the beam width, and fluctuations in the beam curvature. Here we describe new techniques of measurement. The results for two types of laser beam geometry fluctuation—lateral movement and width fluctuation—for an argon-ion laser and for a diode pumped monolithic ring Nd:YAG laser are presented.

Breakdown of the semiclassical approximation at the black hole horizon.

The definition of matter states on spacelike hypersurfaces of a 1+1 dimensional black hole spacetime is considered. The effect of small quantum fluctuations of the mass of the black hole due to the quantum nature of the infalling matter is taken into account. It is then shown that the usual approximation of treating the gravitational field as a classical background on which matter is quantized, breaks down near the black hole horizon. Specifically, on any hypersurface that captures both infalling matter near the horizon and Hawking radiation, quantum fluctuations in the background geometry become important, and a semiclassical calculation is inconsistent. An estimate of the size of correlations between the matter and gravity states shows that they are so strong that a fluctuation in the black hole mass of order exp[-M/M_{Planck}] produces a macroscopic change in the matter state.

Rational Scalar Field Theories on Quantum Spheres

Quantum manifolds can be used to describe strong quantum fluctuations of geometry which eliminate the ultraviolet divergences of the fields at microscopic level. It is shown that the numbers of scalar wave functions on quantum spheres in 2, 3 and 4 dimensions are finite when the deformation parameter is a root of unity. The deformation parameter is determined by the radius of the universe and the scale of the ultraviolet cutoff. Moreover, it is pointed out that the non-commutative nature of the wave functions depends on their scales. The wave functions and their energies are almost classical below the intermediate scale. If we set the scale far beyond our scale, the effect is almost invisible for our scale. A possible scenario for the quantum evolution of the universe is also discussed

Quantum Fluctuations of Black Hole Geometry

By using the minisuperspace model for the interior metric of static black holes, we solve the Wheeler·DeWitt equation to study quantum mechanics of the horizon geometry. Our basic idea is to introduce the gravitational mass and th,e expansions of null rays as quantum operators. Then, the exact wave function is found as a mass eigenstate, and the radius of the apparent horizon is quantum·mechanically defined. In the evolution of the metric variables, the wave function changes from a WKB solution giving the classical trajectories to a tunneling solution. By virtue of the quantum fluctuations of the metric evolution beyond the WKB approximation, we can observe a static black hole state with the apparent horizon separating from the event horizon. Since the discovery of the Hawking radiation, much work has been devoted to the analysis of the quantum evaporation of black holes. In particular, the problem of final fates of evaporating black holes has been much debated. In Hawking'S semi­ classical calculation, the emitted radiation was found to be exactly thermal. Then, if a black hole evaporates completely, an initially pure quantum state must evolve to a mixed state. This is known as the information loss paradox. As was emphasized by Preskill, I) it is very difficult to resolve the serious puzzle in quantum mechanics and general relativity. Before reaching the final resolution, we must develop quantum theories of the black hole geometry. A possible way of treating the horizon as a quantum system is to apply the Wheeler-DeWitt equation to spherically symmetric spacetimes. In the superspace canonical formulation, the Hamiltonian and momentum constraints work as quantum equations for the physical state IJf which is a functional of the metric variables. To make the calculation tractable, Rodrigues et al. 2 ) proposed a black hole minisuper­ space model and derived the simplified Hamiltonian constraint. Unfortunately their model was found to be incompatible with the momentum constraint. 3 ) The compati­ bility between the two constraints can be recovered if we consider a local minisuper­ space model valid near the apparent horizon. 4 ) From the wave function dependent on the local metric near the apparent horizon, we can derive the mass-loss rate due to the back-reaction of Hawking radiation and show the breakdown of the semi-classical result at the final stage of complete evaporation. The Wheeler-DeWitt approach will be viable as a quantum theory of the horizon. To advance this prospect, in this paper, we want to clarify another quantum feature of the horizon from the Wheeler­ DeWitt equation. We will consider static states of a spherically symmetric black hole instead of its evolutionary states. In classical relativity the apparent horizon is always located just on the event horizon. The degeneracy of the two horizons can be removed owing to quantum fluctuations of the metric. This was first pointed out by Y ork 5 ) under the

Quantum Fluctuations of Black Hole Geometry

By using the minisuperspace model for the interior metric ofstatic black holes, we solve the Wheeler-DeWitt equation to study quantum mechanics of the horizon geometry. Our basic idea is to introduce the gravitational mass and the expansions of null rays as quantum operators. Then, the exact wave function is found as a mass eigenstate, and the radius of the apparent horizon is quantum-mechanically defined. In the evolution of the metric variables, the wave function changes from a WKB solution giving the classical trajectories to a tunneling solution. By virtue of the quantum fluctuations of the metric evolution beyond the WKB approximation, we can observe a static black hole state with the apparent horizon separating from the event horizon.

Generation of squeezed radiation from vacuum in the cosmos and the laboratory.

The close analogy between the generation of gravitons from vacuum fluctuations of the gravitational field and the generation of photons from vacuum fluctuations of the electromagnetic field is studied. Gravitons produced in the cosmos and photons produced in the laboratory are governed by similar physical principles and mathematical equations. Both are described by the so-called squeezed vacuum quantum states. Squeezed vacuum optical radiation has been generated experimentally by separating the pump light from the squeezed fluctuations in an interferometric geometry. It may prove possible that some predictions of gravitation theory can be modeled, or even tested, in the laboratory by quantum optics experiments

Resistance fluctuations in a four-probe geometry with infinite leads.

The mean-square variation in the magnetoresistance of small metal wires is calculated in a four-probe geometry with infinite leads. Dephasing scattering is included in a simplified but current-conserving manner. As a result, a region of each lead approximately one dephasing length from the sample contributes to the fluctuations, which, however, then remain finite as the leads are made long, as one would expect on phyiscal grounds. The calculated symmetric and antisymmetric parts of the resistance fluctuations are in semiquantitative agreement with experiment.

Nonlinear gyrokinetic equations for tokamak microturbulence

A nonlinear electrostatic gyrokinetic Vlasov equation as well as a Poisson equation have been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov–Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits.

An erosion criterion for gravel‐bed rivers

AbstractA theoretically‐based erosion criterion is developed for gravel‐bed rivers which incorporates the effect of both grain geometry and turbulent velocity fluctuations. It is derived from a balance of instantaneous drag, lift, and gravity forces operating on individual grains and is calculated for spherical grains arranged in three distinct geometries. To accommodate the temporal variation in bed shear stress, the model includes a stochastic element based on the characteristics of turbulence derived from the flume evidence of McQuivey (1973a, b). In terms of the Shields parameter, results show reasonable agreement with the range of observations quoted from the field and with the experimental data of Fenton and Abbott (1977). Finally, the argument is generalized to cover applications in the wider context of field conditions including a range of grain sizes and flow conditions.

Noise in high performance thin-film longitudinal magnetic recording media

The problem of noise in thin-film longitudinal media is analyzed experimentally and theoretically. The physical mechanism for the noise is shown to be fluctuations in the geometry of the zig-zag transitions separating bit cells. The shifted-transition noise model is introduced as a means of quantifying the noise processes. Spatial, spectral, and autocorrelation properties are introduced. A calculation of the RMS noise voltage yields the characteristic noise versus density curves found experimentally, and clarifies their interpretation with respect to the signal-to-noise ratio. The corresponding experimental data for several plated and sputtered media are presented and analyzed in the light of the model predictions.

Ray path identification and acoustic tomography in the Straits of Florida

The Florida Current, which flows through the Straits of Florida, transports a significant fraction of the heat which must be transferred from low to high latitudes in order for the earth to remain in thermal balance. While ocean acoustic tomography is an atttactive candidate for monitoring this heat transport, its use is dependent on being able to identify individual ray paths. All published investigations of tomography have been concerned with deep‐ocean propagation where experience has shown that paths can be identified. Propagation in the Straits is quite different from deep‐ocean propagation, however. The sound speed is such that a ray path which travels modest distances necessarily experiences bottom bounces. It is not clear, a priori, that paths can be identified in the Straits. We report the results of an investigation of this question. Using a fast, specialized ray‐tracing program, we analyzed the fluctuations in ray path geometry resulting from small perturbations in the bathymetry. We found that the topographic features amplify these fluctuations to such a degree that identification of individual rays becomes impossible after only a few bottom bounces. Consequently, there is no adequate interpretive tool for relating an acoustic signal recorded more than a few tens of kilometers from a source to the heat transport, i.e., for carrying out the tomographic inversion. We propose a way of overcoming this identification problem based on the use of acoustic profilers.

Shape fluctuations of spherical micelles

The thermal excitation of capillary waves on the surface of a spherical micelle has been analysed in an approximate manner using a series expansion in spherical harmonics and restricting the treatment to small amplitudes. The potential energy governing the dynamics of these motions is computed as the surface tension, γ, of the micelle multiplied by its surface area, A. It is found that a closed, ordinary surfactant micelle in aqueous solution with γ≈ 25 mN m–1 exhibits a root-mean-square dispersion, σr/R, of its radius of several percent because of shape fluctuations on the time-scale 10–10s. The present study also indicates that (closed) micellar or microemulsion droplets with γ≲ 1 mN m–1 experience large fluctuations in geometry.