Higher order field equations II; Limit properties of green's functions

In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions G^M_R(x) and G^M_A(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for |M| → ∞ the Green's functions G^M_R(x) and G^M_A(x) approach the Green's functions Δ_R(x) and Δ_A(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of G^M_A(x) and G^M_A(x) is the same as of Δ_R(x) and Δ_A(x) - and also the same as for D_R(x) and D_A(x) for t→ ± ∞, where D_R and D_A are the Green n's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense.     

Magnetische Monopole und Quarks

We discuss the concept of Schwinger, which starts with the hypothesis of the existence of magnetical monopoles and results in a baryon model with magnetically charged constituents. Especially we analyse the mathematical consistency of such a theory. which admits a connection between some magnetically charged “quarks” and the homogeneous Maxwell-equations ?_v*F^μv(x^u) = 0, which, displaying a lack of symmetry with respect to the inhomogeneous one, ?_vF^μv(x^u) = 4?j^μ, are replaced by ?*j^μv. Here *j^μ(x^μ) means a conserved magnetic current which provides a monopole source for the magnetic field.     

Determination of the effective surface region thickness and of Begrenzungs effect

The thickness, x_s, of the effective surface region, defined as the region in which an electron travelling inside a material experiences surface excitations, as well as Begrenzungs effect, i.e. the variation of the bulk inelastic cross section in the surface region are theoretically determined for aluminium and silicon using the QUEELS-ε(k,ω)-REELS software. This software allows to determine the energy-differential inelastic electron scattering cross sections for reflection-electron-energy-loss spectroscopy (REELS) within the dielectric response theory. This study has been carried out for electron energies between 300 eV and 5000 eV. We find that the bulk inelastic cross section decreases exponentially with the distance to the surface, that x_s is practically independent of electron incidence and exit angles and that x_s=[2(v_B)^0.4/ω_s]v^0.6, where ω_s is the surface plasmon frequency, v the electron velocity and v_B the Bohr velocity ().     

QCD estimates for associated charm production by weak neutral currents

We give quantitative estimates for associated charm production in neutrino and antineutrino induced neutral current interactions, based on (a) quantum chromodynamics (QCD) and (b) the quark parton model (QPM) and a phenomenological generalization thereof. We emphasize the need for a precise measurement of the ratio σ(v_μN→v_μe⁺X)/σ(v_μN→μ^?X) and the corresponding ratio for antineutrinos, as these can provide clean tests of certain characteristic features of QCD. Bounds are obtained for single charm production by charm-changing neutral currents.     

High vibrational levels of the X state of ICl,and the electronic-coriolis coupling of the X and A states

Ro-vibrational levels of the ground state of ICl have been measured in the range v_x = 35–73 and J_x < 55 using three-stage polarization-labeling spectroscopy. When merged with established microwave and fluorescence data for the lower levels, these results lead to a self-consistent set of spectroscopic constants G_v, B_v, D_v and H_v for all levels v_x ≤ 73. The highest G_v and G_v + F_J term values recorded are, respectively, 52.7 and 1.37 cm^?1 below the dissociation limit. Coefficients of the long-range interaction I + Cl are determined by analysis of the outer wings of the RKR potentials for the X(0⁺) and A′(2) states. Electronic rotational interaction between X and the well-known A(1) state of ICl is shown to account for at least a major part of the $\text{Ω-doubling}$ splitting observed in A: the electronic matrix element characterizing this interaction is shown to be strongly r-dependent, and reasons for this are reviewed.     

The static potential in quantum chromodynamics

The effective interaction between a static quark-antiquark pair is computed within the framework of 〈G_μv^aG_a^μv〉 ≠ 0. Due to the static approximation the interaction takes the form of a potential, which is in striking agreement with phenomenological potentials.     

Space charge regions at silver halide surfaces: Experimental results for undoped AgCl

Vibrating capacitor measurements have been made of the variation with temperature of the surface potential of AgCl single crystals of surface orientation (100), (210), (110) and (111). The magnitude and temperature variation of the surface potential was found to be strongly dependent on the orientation of the free surface. The data are interpreted in terms of a model which attributes the existence of surface space charge potentials to differences in the binding energies of silver ions at surface sites relative to normal interior lattice sites. The results are summarized in terms of the Gibbs free energies ΔG_v^(hkl) for cation vacancy formation and give ΔG_v⁽¹¹⁰⁾ = + 0.01 eV + 5 kT, ΔG_v⁽¹⁰⁰⁾ = ? 0.2eV + 10 kT, ΔG_v⁽²¹⁰⁾ = = ? 0.8eV + 23kT, ΔG_v⁽¹¹¹⁾ = + 0.8 eV ? 20kT. These numbers have large absolute uncertainties but the relative values for the different orientations are reliable. The values of ΔG_v^hkl can be used to calculate the magnitude and sign of the surface space charge fields and allow some predictions on the formation of photographic latent images near free surfaces.     

Construction of kernels of the nonlinear collision integral in the Boltzmann equation using laplace transformation

It is shown that the relation between kernels L _l (v, v ₁) of the linear collision integral and kernels G _l,0 ^l (v, v ₁, v ₂) of the nonlinear collision integral can be reduced to the Laplace transformation. Analytic expressions for nonlinear kernels G _0,0 ⁺⁰ (v, v ₁, v ₂) and G _1,0 ⁺¹ (v, v ₁, v ₂) are determined for hard spheres and pseudo-Maxwellian molecules.     

Geometric study of the connection between the Lagrangian and Hamiltonian constraints

The second order differential equation character of the solutions of the dynamical equation i(Γ)ω_L = dE_L for a singular Lagrangian L, as well as the conditions for the existence of such a solution, are studied. We also introduce a couple of maps R (L)_v : T _FL(v)(T¹Q) → T_v(TQ) and T(L)_v : T_FL(v)(T¹Q) → T_FL(v)(T¹Q), with v ϵ TQ, which are shown to be very useful for establishing the connection between the constraints arising in the Lagrangian and Hamiltonian formulations.     

Influence of mobile space charges on electrical properties of ferroelectric Bi3.5La0.5Ti3O12 thin films

Lanthanum-substituted bismuth titanate, Bi_3.5La_0.5Ti₃O₁₂ (i.e., x=0.5 in Bi_4−xLa_xTi₃O₁₂), thin films have been grown on Pt/Ti/SiO₂/Si substrates using pulsed laser deposition. The frequency dependence of the real part ε′(ω) and the imaginary part ε″(ω) of the dielectric constant has been studied. The ε′(ω) does not show any sudden change within the frequency range of 10²-10⁶ Hz. In contrast, the ε″(ω) shows a large dispersion as frequency decreases. The observed relaxation behavior in ε″(ω) can be explained in terms of a migration of oxygen vacancies in (Bi₂O₂)²⁺ layers, not in Bi₂Ti₃O₁₀ perovskite layers.     

Ultra-high energy neutrino-nucleon and neutrino-electron cross-sections in the standard model,in supersymmetric and superstring models

The neutrino-nucleon cross-section is calculated for energies up to 10¹⁵ GeV in the framework of the standard model assuming that the nucleon structure functionF ₂(x, Q ²) goes like ln²(a/x) asx→0. The cross-sections forv _μ e ^?(v _e e ^?)-interactions rise linearly with energy up toE _ν～10⁷ GeV and amount at this energy to ～10% of the neutrino-nucleon cross-section. The νN-cross-sections with production of supersymmetric particles in the framework of a supersymmetric model with minimal particle content are estimated. Photino-nucleon and photino-electron interactions are considered for masses of the squarks and sleptons in the range of 80–300 GeV. In superstring inspired unified models an estimation is made of the νN-interaction with an exchange of light leptoquarks.     

Laser-induced fluorescence spectrum of nitrogen dioxide in the neighborhood of 6125 Å

The resonance fluorescence spectrum of nitrogen dioxide has been excited by a tunable, cw dye laser in the neighborhood of 6125 Å. The rotational constants of the ²B₂ upper electronic state are determined as follows, in units of cm^?1: A_v′ = 7.2 ± 0.6; $\text{B}\text{?}{}_{\mathrm{v}}\text{′ = 0.454 ± 0.015}$; B_v′ = 0.496 ± 0.046; C_v′ = 0.412 ± 0.019. The band origin is at 16 325.1 ± 1.8. Quoted error limits are standard deviations obtained from the fit of the data. The vibrational assignment of the upper state is (0, 5, 0), and by combination with the data of other workers, we estimate for its vibrational constants, in cm^?1: ω₁′ + x₁₁′ = 1425.7; ω₂′ = 876.6; x₂₂′ = ?0.83; x₁₂′ = ?8.1. The molecular geometry in the upper state is briefly discussed.     

Spin-dependent parton distributions in polarized deep inelastic lepton nucleon scattering

We analyze the recently measured first moment Г ^p (Q ²) of the structure functiong ₁ ^p (x,Q ²) obtained in deep inelastic scattering of polarized electrons from polarized protons. It is shown that the recent experimental result Г ^p (〈Q ²〉?10 GeV²)? 0.113 implies a polarized strange quark sea whose total polarization Δs(μ²)??0.22 is comparable to Δμ(μ²)?0.74 and Δd(μ²)??0.50 already at a typically hadronic scale μ=0(1 GeV). Most remarkably, the total polarization due to quark distributions Δμ(μ²)+Δd(μ²)+Δs(μ²) practically vanishes which implies that the spin of the proton is carried by the gluons, ΔG(μ²), and/or the angular momentum of the partons. This surprising conjecture can be experimentally tested with heavy quark production in longitudinally polarized lepton-nucleon scattering. In particular the (real) photon-gluon fusion process γg→Q \(\bar Q\) provides a rather unique measure of ΔG(x,Q ²).     

Theory of the local field correction in an electron gas

The wave-vector- and frequency-dependent dielectric function ?(k,ω) of an electron gas can be expressed in terms of Lindhard's function and a complex local field correctionG(k,ω) which incorporates all the effects of dynamic exchange and correlation in the system. The general properties ofG(k,ω) are discussed, in particular the static and high-frequency limits. It is shown that for smallk, bothG(k, 0) andG(k, ∞) vary ask ², with different coefficients, but both determined by the average kinetic and potential energies per particle. For largek,G(k, ∞) varies again ask ² and it is argued that the same holds true forG(k, 0), with both coefficients (though different) determined by the average kinetic energy per particle. General formulas for the plasma dispersion relation and damping, involving, respectively, the real and imaginary parts ofG(k,ω), are given. The term in the plasma frequency which is proportional tok ² is given directly in terms of the average kinetic and potential energies per particle, a result true at all temperatures. A calculation of the frequency dependence ofG(k,ω), starting from the exact equation of motion for the particle-hole operator and employing a decoupling approximation introduced previously by Toigo and Woodruff, is presented. Explicit results forG(k,ω) are obtained for smallk and allω. The complete expressions forG(k, 0) andG(k, ∞) in this approximation have been obtained and are plotted.     

超导临界温度理论(Ⅰ)

本文求出了Eliashberg方程在T=T_c时的解,得到了下面的临界温度级数表示式:T_c=α₀(μ^*)(λ〈ω²〉)^1/2{1+1/λα₁(μ^*)〈ω⁴>/〈ω²>²+1/λ²(α₂₁(μ^*)〈ω⁶>/〈ω²>³+α₂₂(μ^*)〈ω⁴>²/〈ω²>⁴) +1/λ³(α₃₁(μ^*)〈ω⁸>/〈ω²>⁴+α₃₂(μ^*)(〈ω⁴>〈ω⁶>)/〈ω²>⁵)+α₃₃(μ^*)〈ω⁴>³/〈ω²>⁶+…},其中α₀(μ^*),α₁(μ^*)等仅是μ^*的函数。新的T_c公式表明了,T_c不仅依赖于λ、μ^*和〈ω²〉,而且依赖于有效声子谱α²F(ω)的各级矩〈ω²ⁿ〉。     

Measurement of the nuclear magnetic dipole moment of Au197 and hyperfine structure measurements in the ground states of Au197, Ag107, Ag109 and K39

Using an atomic beam magnetic resonance apparatus the nuclear magnetic dipole momentμ _I of the stable isotope Au¹⁹⁷ was measured directly with the doublet method. The result isμ _I(Au¹⁹⁷)=0.143491 (9)μ _n, uncorrected for atomic diamagnetism. Further hyperfine structure measurements were performed in the ground states of K³⁹, Ag¹⁰⁷, Ag¹⁰⁹ and Au¹⁹⁷ with the following results:Δv(K³⁹)=461.719723 (38) MHzΔv(Ag¹⁰⁷)=1712.512111 (18) MHzΔv(Ag¹⁰⁹)=1976.932075 (17) MHzΔv(Au¹⁹⁷)=6099.320184 (13) MHzg _J(Ag¹⁰⁷)/g _J(K39)=1.0000260 (20)g _J(Au¹⁹⁷)/g _J(K39)=1.0005076 (20).     

Influence of A D.C. electric field on the polarization of an intense laser beam in liquids

Analytical and numeric solutions of the nonlinear equation describing polarization changes of a strong laser beam propagating in liquid in the presence of a stationary electric field are derived. The polarization ellipse of the optical wave was found to oscillate along the propagation path. A new method of estimating the x^{(-ω,ω,ω,-ω)}_zyyz value is proposed.     

Quark confinement and the baker-ball-zachariasen approximation

The Yang-Mills effective action ?1/4∫dxxF _μv ^a (D ²/M ²)F _μv ^a recently proposed by Baker et al., and also much earlier by the author and R.L. Stuller, is considered. An extension of the effective action to include Fermions is constructed in which quarks can be confined. The Schwinger-Dyson equation for the quark propagator is studied, and conditions are derived for the quark mass to be driven selfconsistently to infinity.     

Causality on the world-sheet of the string

The (anti) commutator of the string coordinates x^μ(ω,θ) and S^Aa(ω,θ) is studied for the open and closed free strings. By observing that this commutator is constant on each face of a polygonal pattern of the string's world-sheet, a close connection between causality and topology is arrived at.