Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.     

A discussion of the Pseudo-Weyl-type metrics

We discuss the pseudo-Weyl vacuum field for which the two metric functions, and , depend on the three coordinates, z, andt. It is shown that no solutions exist that depend on all three coordinates. Consideration is given to the time-dependent metric of Einstein and Rosen and the same (null) result is shown to hold for that case. Thus, the most general solutions to the Weyl-type metric appear to be those already found by Weyl and by Einstein and Rosen.     

Cantor spectrum and singular continuity for a hierarchical Hamiltonian

We study the spectrum of the HamiltonianH onl ₂() given by (H)(n)=(n+1)+(n–1)+V(n)(n) with the hierarchical (ultrametric) potentialV(2 ^m (2l+1))=(1–R ^m )/(1–R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0<R<1,R=1 andR>1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese _n (1)<e _n (2)<...<e _n (2 ⁿ –1) of the Dirichlet problemH=E, (0)=(2 ⁿ )=0,n1. In the gap opening ate _n (k) the integrated density of states takes on the valuek/2 ⁿ . The spectrum is purely singular continuous forR1 when the potential is unbounded, and the Lyapunov exponent vanishes in the spectrum. The spectrum is purely continuous forR<1 in (H)[–2, 2] and =0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.Work supported by the Fonds National Suisse de la Recherche Scientifique, Grant No. 2.042-0.86 (H.K. and R.L.) and 2.483-0.87 (A.S.)On leave from the Dipartimento di Fisica, Università degli Studi di Firenze, Largo E. Fermi 2, I-50125 Firenze, Italy     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

The Positive Action conjecture and asymptotically Euclidean metrics in quantum gravity

The Positive Action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics onR ⁴ and a large class of more complicated topologies and for self-dual metrics. We show that ifR 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under anSU(2) orSO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric onK3 — the only simply connected compact manifold which admits a self-dual metric.     

The velocity autocorrelation function of a finite model system

We investigate in detail the dependence of the velocity autocorrelation function of a one-dimensional system of hard, point particles with a simple velocity distribution function (all particles have velocities ±c) on the size of the system. In the thermodynamic limit, when both the number of particlesN and the length of the boxL approach infinity andN/L , the velocity autocorrelation function(t) is given simply by c² exp(–2ct@#@). For a finite system, the function _N(t) is periodic with period 2L/c. We also show that for more general velocity distribution functions (particles can have velocities ±c_i,i = 1,...), _N(t) is an almost periodic function oft. These examples illustrate the role of the thermodynamic limit in nonequilibrium phenomena: We must keept fixed while letting the size of the system become infinite to obtain an auto-correlation function, such as(t), which decays for all times and can be integrated to obtain transport coefficients. For any finite system, our _N (t) will be very close to(t) as long ast is small compared to the effective size of the system, which is 2L/c for the first model.Supported in part by the AFOSR under Contract No. F44620-71-C-0013.     

Asymptotic solutions of continuous-time random walks

The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distribution(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior of(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments of(t) exist; the other occurs when(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.This work was partially supported by NSF Grant No. 28501.     

Space-times admitting a three-dimensional conformal group

Perfect fluid space-times admitting a three-dimensional Lie group of conformal motions containing a two-dimensional Abelian Lie subgroup of isometries are studied. Demanding that the conformal Killing vector be proper (i.e., not homothetic nor Killing), all such space-times are classified according to the structure of their corresponding three-dimensional conformal Lie group and the nature of their corresponding orbits (that are assumed to be non-null). Each metric is then explicitly displayed in coordinates adapted to the symmetry vectors. Attention is then restricted to the diagonal case, and exact perfect fluid solutions are obtained in both the cases in which the fluid four-velocity is tangential or orthogonal to the conformal orbits, as well as in the more general tilting case.     

Quantum mechanics without the position operator

The formula for the differential scattering cross section in quantum mechanics is derived without the usual assumption that the square of the -function is a position probability density for particles. It is argued that position, like time, may be basically a macroscopic parameter rather than a random variable for microparticles.This research was partly supported by a grant from the NSERC.     

Random Analytic Chaotic Eigenstates

The statistical properties of random analytic functions (z) are investigated as a phase-space model for eigenfunctions of fully chaotic systems. We generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of (z) previously demonstrated for a spherical phase space [SU(2) polynomials]. For systems with time-reversal symmetry, the number of real roots is computed for the three geometries. In the semiclassical regime, the local correlation functions are shown to be universal, independent of the system considered or the geometry of phase space. In particular, the autocorrelation function of is given by a Gaussian function. The connections between this model and the Gaussian random function hypothesis as well as the random matrix theory are discussed.     

Power law growth for the resistance in the Fibonacci model

Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(n)=(n+1)+(n–1)+v(n) (n),n,l ²(),, wherev(n)=[(n+1)]–[n],[x] denoting the integer part ofx and the golden mean , give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.     

The collapse of the wave function

Probability distributions are seen to be observer dependent. The probability function can be put into an observer-dependent form. This eliminates the acausal behavior of the collapse of the wave function.     

Numerically stable solution of coupled channel equations: The local reflection matrix

The second order linear Schrödinger equation is transformed to a first order nonlinear differential equation for a quantityp=(iq ^–1 –)/(iq ^–1 +). In a coupled channel problem all quantities occurring in this equation includingq (the WKB wave number) are matrices andp may be calledlocal reflection matrix. This quantity is closely related to the logarithmic derivative of the Schrödinger function but has no singularities in the classically allowed region. In the asymptotic region where the potential is constant the local reflection matrix approaches the physical reflection matrix. In a pure reflection problem (with an infinite potential on one side) this is the fullS-matrix, in a transmission problem (with an activation barrier of finite height) unitarity of theS-matrix can be used to determine most quantities of physical interest fromp. While standard logarithmic derivative methods can become instable for transmission problems the solution with the local reflection matrix is completely stable both for reflection and transmission problem.     

Geodesics for the NUT metric and gravitational monopoles

In order to provide insight about the physical interpretation of the NUT parameter, we solve the geodesic equations for the NUT metric. We show that the properties of NUT geodesics are similar to the properties of trajectories for charged particles orbiting about a magnetic monopole. In summary, we show that (1) the orbits lie on the surface of a cone, (2) the conserved total angular momentum is the sum of the orbital angular momentum plus the angular momentum due to the monopole field, (3) the monopole field angular momentum is independent of the separation between the source of the gravitational field and the test particle, and (4) the geodesics are almost spherically symmetric. The strong similarities between the NUT geodesics and the electromagnetic monopole suggest that the NUT metric is an exact solution for a gravitational magnetic monopole. However, the subtle difference of being only almost spherically symmetric implies that the analogy is not perfect. The almost spherically symmetric nature of the NUT geodesics suggest that the energy of the Dirac string makes a contribution to the solution. We also construct exact solutions for special orbits, discuss a twin paradox, and speculate about the Dirac quantization condition for a gravitational magnetic monopole.     

Solitonic gravitational waves in Bianchi II cosmologies. 1. The general framework

The inverse scattering method is applied to a class of space-times belonging to the Bianchi types I–VII. Solitonic perturbations corresponding to one or two poles on an arbitrary cosmological background are described in detail. The fundamental matrix ₀ is explicitly calculated for a Bianchi II background, thus providing the first known example of a non-diagonal case.     

Scale invariance,killing vectors,and the size of the fifth dimension

An analysis is made of the classical five-dimensional sourceless Kaluza-Klein equations with the existence of the usual/ Killing vector not assumed, where is the coordinate of the fifth dimension. The physical distance around the fifth dimensionD ₅, needed for the calculation of the fine structure constant, is not calculable in the usual theory because the equations have a global scale invariance. In the present case, the Killing vector and the global scale invariance are not present, but it is found rather generally thatD ₅=0. This indicates that quantum gravity is a necessary ingredient if is to be calculated. It also provides an alternate explanation of why the universe appears four-dimensional.     

What Can We Learn from Nonminimally Coupled Scalar Field Cosmology?

A novel exploration of nonminimally coupled scalar field cosmology is proposedin the framework of spatially flat Friedmann—Robertson—Walker spaces forarbitrary scalar field potentials V() and values of the nonminimal couplingconstant . This approach is self-consistent in the sense that the equation of stateof the scalar field is not prescribed a priori, but is rather deduced together withthe solution of the field equations. The role of nonminimal coupling appears tobe essential. A dimensional reduction of the system of differential equations leadsto the result that chaos is absent in the dynamics of a spatially flat FRW universewith a single scalar field. The topology of the phase space is studied and revealsan unexpected involved structure: according to the form of the potential V()and the value of the nonminimal coupling constant , dynamically forbiddenregions may exist. Their boundaries play an important role in the topologicalorganization of the phase space of the dynamical system. New exact solutionssharing a universal character are presented; one of them describes a nonsingularuniverse that exhibits a graceful exit from, and entry into, inflation. This behaviordoes not require the presence of the cosmological constant. The relevance of thissolution and of the topological structure of the phase space with respect to anemergence of the universe from a primordial Minkowski vacuum, in an extendedsemiclassical context, is shown.     

A solution spectrum of the nonlinear schrödinger equation

The soliton solutions of the form=A/coshkx and=B tanhkx of the nonlinear Schrödinger equation have been considered with respect to many problems. In this paper, it is shown that the nonlinear Schrödinger equation also possesses a solution manifold that generalizes the above soliton functions and provides a discrete spectrum of eigenfunctions and eigenvalues. With the help of a slight modification of these eigenfunctions, it is possible to extend them to the relativistic case, where they become solutions of a nonlinear Klein-Gordon equation associated with a discrete mass spectrum.     

Plane-Symmetric Solitons of Spinor and Scalar Fields

We consider a system of nonlinear spinor and scalar fields with minimal coupling in general relativity. The nonlinearity in the spinor field Lagrangian is given by an arbitrary function of the invariants generated from the bilinear spinor forms S= and P=i⁵; the scalar Lagrangian is chosen as an arbitrary function of the scalar invariant = _,^,, that becomes linear at 0. The spinor and the scalar fields in question interact with each other by means of a gravitational field which is given by a plane-symmetric metric. Exact plane-symmetric solutions to the gravitational, spinor and scalar field equations have been obtained. Role of gravitational field in the formation of the field configurations with limited total energy, spin and charge has been investigated. Influence of the change of the sign of energy density of the spinor and scalar fields on the properties of the configurations obtained has been examined. It has been established that under the change of the sign of the scalar field energy density the system in question can be realized physically iff the scalar charge does not exceed some critical value. In case of spinor field no such restriction on its parameter occurs. In general it has been shown that the choice of spinor field nonlinearity can lead to the elimination of scalar field contribution to the metric functions, but leaving its contribution to the total energy unaltered.