Toward a Collective Treatment of Quantum Gravitational Interactions

In this paper we study the gravitational effects induced by the quantum fluctuations of the energy–momentum tensor of scalar fields. Our treatment is based on the two-point correlation function of this operator. In a large N limit, this treatment constitutes the next contribution after the semiclassical treatment. The specific example we study are the gravitational interactions between outgoing configurations giving rise to Hawking radiation and in-falling configurations. Even when the latter are in vacuum state, the interactions grow boundlessly upon approaching the horizon. Their main effect is to wash out the trans-Planckian correlations which existed in a given background geometry. When evaluated in the lowest order, these interactions express themselves in terms of a stochastic ensemble of metric fluctuations. The propagation of Hawking radiation in this ensemble resembles that of sound propagation in a random medium. The analogies with acoustic black holes are manifest even though certain features differ.     

Semiclassical dynamics of Dirac particles interacting with a static gravitational field

The semiclassical limit for Dirac particles interacting with a static gravitational field is investigated. A Foldy–Wouthuysen transformation which diagonalizes at the semiclassical order the Dirac equation for an arbitrary static spacetime metric is realized. In this representation the Hamiltonian provides for a coupling between spin and gravity through the torsion of the gravitational field. In the specific case of a symmetric gravitational field we retrieve the Hamiltonian previously found by other authors. But our formalism provides for another effect, namely, the spin hall effect, which was not predicted before in this context.     

Semiclassical bosonic <Emphasis Type="Italic">D</Emphasis>-brane boundary states in curved spacetime

In this paper we discuss the existence of quantum D-brane states in the strong gravitational field and in the presence of a constant Kalb-Ramond field. A semiclassical string quantization method in which the spacetime metric g _AB and the constant antisymmetric Kalb-Ramond field b _AB are treated exactly is employed. In this framework, the semiclassical D-branes are defined at the first order perturbation around the trajectory of the center-of-mass of a string. The set of equations the semiclassical D-branes must satisfy in a general strong gravitational field are given. These equations are solved in the AdS background where it is shown that a D-brane coherent state exists if the operators that project the string fields onto the corresponding Neumann and Dirichlet directions satisfy a set of algebraic constraints. A second set of equations that should be satisfied by the projectors in order that the semiclassical state be compatible with the global structure of the D-brane are derived in the particle limit of a string in the torsionless AdS background.     

Stability of Semiclassical Gravity Solutions with Respect to Quantum Metric Fluctuations

We discuss the stability of semiclassical gravity solutions with respect to small quantum corrections by considering the quantum fluctuations of the metric perturbations around the semiclassical solution. We call the attention to the role played by the symmetrized 2-point quantum correlation function for the metric perturbations, which can be naturally decomposed into two separate contributions: intrinsic and induced fluctuations. We show that traditional criteria on the stability of semiclassical gravity are incomplete because these criteria based on the linearized semiclassical Einstein equation can only provide information on the expectation value and the intrinsic fluctuations of the metric perturbations. By contrast, the framework of stochastic semiclassical gravity provides a more complete and accurate criterion because it contains information on the induced fluctuations as well. The Einstein–Langevin equation therein contains a stochastic source characterized by the noise kernel (the symmetrized 2-point quantum correlation function of the stress tensor operator) and yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. These points are illustrated with the example of Minkowski space-time as a solution to the semiclassical Einstein equation, which is found to be stable under both intrinsic and induced fluctuations.     

Classical gravitational radiation from quasi-elastic particle scattering in models with extra dimensions

An equation for the emission of classical gravitational waves due to quasi-elastic N → N particle scattering is derived in a model with compactified extra dimensions. In addition to previous classical studies, additional terms that are suppressed by factors of one over the frequency time compactification radius are also calculated. From this, a single formula is given which predicts the energy loss into the gravitational radiation from elastic collisions as well in the low-and high-energy limit. The text was submitted by the authors in English.     

The quantum cosmology in the Brans-Dicke theory

In this paper, we try to establish the quantum cosmology based on the Brans-Dicke gravitational theory. The most remarkable property of it is that the scale factora(t) has a lower limit, where the cosmological wave function approaches zero.     

Raychaudhuri equation in quantum gravitational optics

The equation of Raychaudhuri is one of the key concepts in the formulation of the singularity theorems introduced by Penrose and Hawking. In the present article, taking into account QED vacuum polarization, we study the propagation of a bundle of rays in a background gravitational field through the perturbative deformation of Raychaudhuri’s equation. In a sense, this could be seen as another semiclassical study in which geometry is treated classically but matter (which means the photon here) is allowed to exhibit quantum characteristics that are encoded in its coupling to the background curvature.      

Probing the dark matter issue in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity via gravitational lensing

For a general class of analytic f(R)-gravity theories, we discuss the weak field limit in view of gravitational lensing. Though an additional Yukawa term in the gravitational potential modifies dynamics with respect to the standard Newtonian limit of General Relativity, the motion of massless particles results unaffected thanks to suitable cancellations in the post-Newtonian limit. Thus, all the lensing observables are equal to the ones known from General Relativity. Since f(R)-gravity is claimed, among other things, to be a possible solution to overcome for the need of dark matter in virialized systems, we discuss the impact of our results on the dynamical and gravitational lensing analyses. In this framework, dynamics could, in principle, be able to reproduce the astrophysical observations without recurring to dark matter, but in the case of gravitational lensing we find that dark matter is an unavoidable ingredient. Another important implication is that gravitational lensing, in the post-Newtonian limit, is not able to constrain these extended theories, since their predictions do not differ from General Relativity.     

On Quantum Spacetime and the horizon problem

In the special case of a spherically symmetric solution of Einstein equations coupled to a scalar massless field, we examine the consequences on the exact solution imposed by a semiclassical treatment of gravitational interaction when the scalar field is quantized. In agreement with Doplicher et al. (1995)  [2], imposing the principle of gravitational stability against localization of events, we find that the region where an event is localized, or where initial conditions can be assigned, has a minimal extension, of the order of the Planck length. This conclusion, though limited to the case of spherical symmetry, is more general than that of  [2] since it does not require the use of the notion of energy through the Heisenberg Principle, nor of any approximation as the linearized Einstein equations.We shall then describe the influence of this minimal length scale in a cosmological model, namely a simple universe filled with radiation, which is effectively described by a conformally coupled scalar field in a conformal KMS state. Solving the backreaction, a power law inflation scenario appears close to the initial singularity. Furthermore, the initial singularity becomes light like and thus the standard horizon problem is avoided in this simple model. This indication goes in the same direction as those drawn at a heuristic level from a full use of the principle of gravitational stability against localization of events, which point to a background dependence of the effective Planck length, through which a-causal effects may be transmitted.     

Gravitational radiation by quantum systems

The limits of validity of the semiclassical theory in which gravity is unquantized are discussed. This is done by comparing the emission of classical gravitational waves in the semiclassical theory with graviton emission in quantum gravity theory. It is shown that these can be quite different even for macroscopic systems. Thus quantum gravitational effects can manifest themselves on a macroscopic scale. A hypothetical experiment to demonstrate the existence of gravitons by means of such effects is discussed.     

πNN coupling determined beyond the chiral limit

Within the conventional QCD sum rules, we calculate the πNN coupling constant, g _πN, beyond the chiral limit using two-point correlation function with a pion. For this purpose, we consider the Dirac structure, iγ₅, at m _π ² order in the expansion of the correlator in terms of the pion momentum. For a consistent treatment of the sum rule, we include the linear terms in quark mass as they constitute the same chiral order as m _π ². In this sum rule, we obtain g _πN= 13.3 ± 1.2, which is very close to the empirical πNN coupling. This demonstrates that going beyond the chiral limit is crucial in determining the coupling. Received: 8 July 1999 / Revised version: 20 August 1999     

Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice – III. From ab initio models to WKB for Schrödinger–Poisson

This work is concerned with the semiclassical approximation of the Schrödinger–Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a N-body quantum problem, then K-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for Pauli’s exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects.     

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. Received: 7 February 2000 / Accepted: 7 July 2000     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Schrödinger invariance and strongly anisotropic critical systems

The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent =z=2, the group of local scale transformation considered is the Schrödinger group, which can be obtained as the nonrelativistic limit of the conformal group. The requirement of Schrödinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either spacelike or timelike. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model, and critical dynamics of the spherical model with a nonconserved order parameter. For generic values of , evidence from higher-order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.     

Yangian Gelfand-Zetlin Bases, -Jack Polynomials and Computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model

We consider the -invariant Calogero-Sutherland Models with N= 1,2,3,...; in the framework of Symmetric Polynomials. In this framework it becomes apparent that all these models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters q and t. The Hamiltonian of the -invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both q and t approach an N _th elementary root of unity. This is a generalization of the well-known situation in the case of the Scalar Calogero-Sutherland Model (N /equals 1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the -invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call -Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The -Jack Polynomials describe the orthogonal eigenbasis of the -invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N /equals 1). For each known property of Macdonald Polynomials there is a corresponding property of -Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the -invariant Calogero-Sutherland Model at integer values of the coupling constant. Received: 1 April 1997 / Accepted: 1 June 1997     

Density of states in a mesoscopic SNS junction

The semiclassical theory of proximity effects predicts a gap E _g～?D/L ² in the excitation spectrum of a long diffusive superconductor/normal-metal/superconductor (SNS) junction. Mesoscopic fluctuations lead to anomalously localized states in the normal part of the junction.As a result, a nonzero, yet exponentially small, density of states (DOS) appears at energies below E _g. In the framework of the supermatrix nonlinear σ model, these prelocalized states are due to the instanton configurations with broken supersymmetry. The exact result for the DOS near the semiclassical threshold is found, provided the dimensionless conductance of the normal part G _N is large. The case of poorly transparent interfaces between the normal and superconductive regions is also considered. In this limit, the total number of subgap states may be large.     

On the way towards a nonsingular gravitational collapse-the properties of the Yang-Mills cosmos

Adding gravitational self-interaction to general relativity in an intrinsic way changes drastically the behavior of a physical system under gravitational collapse. In our analysis of this question for homogeneous and isotropic matter distributions we show that (i) theSO(1,3) gauge theory of gravity of the Yang-Mills type has the correct Newtonian limit for the late universe, (ii) it defines intrinsically a dynamical gravitational stressenergy-momentum tensorG T _ab , and (iii) negative self-energy always prevents homogeneous and isotropic matter from forming a big-bang singularity; if the present universe disposes of a positive self-energy, pair creation on the eve of the lepton era generates sufficient gravity to stop the fatal collapse.This essay received an honorable mention (1977) from the Gravity Research Foundation-Ed.Research fellow of Schweizerischer Nationalfonds.     

Evolution on a smooth landscape

We study in detail a recently proposed simple discrete model for evolution on smooth landscapes. An asymptotic solution of this model for long times is constructed. We find that the dynamics of the population is governed by correlation functions that although being formally down by powers ofN (the population size), nonetheless control the evolution process after a very short transient. The long-time behavior can be found analytically since only one of these higher order correlators (the two-point function) is relevant. We compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean-field theory supplemented with a cutoff at small population density. Finally, we relate our results to the recently studied case of mutation on a totally flat landscape.