Study of Anderson localization by a continued-fraction method

The electron localization is studied for Anderson's tight-binding model of a disordered two-dimensional square lattice. For a large system of 10⁴ sites the averaged squared modulus $\text{| G}{}_{00}\text{|}{}^2$ of the Green's function is evaluated by a continued-fraction method. From this quantity, following Anderson's criterion, the energy of the mobility edge is found as a function of the degree of disorder. Also the Anderson transition is recognized.     

Derivation of Regge's action from Einstein's theory of general relativity

Regge's action for a discrete lattice is derived from Einstein's continuum formula $\text{?√ |g| R}\text{d}{}^{\mathrm{D}}\text{x}$ in general relativity.     

An eikonal perturbation theory having applications in hadron dynamics. II. Massive quantum electrodynamics

An eikonal perturbation theory (EPT), derived in previous work for a superrenormalizable coupling, is here developed for massive quantum electrodynamics (MQED) involving scalar or spinor matter fields minimally coupled to neutral massive vector gluons. After summarizing the functional method, we present the EPT for the external field problem. In agreement with results known within ordinary perturbation theory (OPT) in the eikonal approximation (EA), from an exact eikonal equation derived here we show that the EPT for the external field problem provides an excellent approximation method for Green's functions at large momenta. We then discuss some general features of the EPT for MQED, and show that it leads to a renormalizable approximation method. Our approach is then illustrated by deriving explicit expressions for various renormalized Green's functions in lowest order EPT. We also discuss some asymptotic properties of such Green's functions and indicate how to proceed with calculations in higher orders. As in our previous work, we again find that the renormalization procedure in EPT bears close resemblance to the one for OPT. Contrary to what happens with the EA, the inclusion of self-interactions and of other field-theoretic effects does not spoil the virtues of the EPT as a far better high-momenta approximation than the OPT. As a typical example, if s is an energy parameter and g the coupling constant with $\text{g}{}^2\text{4π}\text{< 1}$, OPT to order g²ⁿ often fails to be a good approximation as soon as $\text{(}\text{g}{}^2\text{4π}\text{)}\text{ln}\text{s ～ 1}$, while in such cases EPT to order g²ⁿ is still a good approximation as long as $\text{(}\text{g}{}^2\text{4π}\text{)}{}^{\mathrm{n+1}}\text{ln}\text{s < 1}$. We also find that the EPT is superior to the EA in that, contrary to the EA, it provides a step-by-step rigorous and renormalizable iterative approximation method which can account for self-interactions and other field-theoretic effects. We emphasize that the EPT is much simpler and more general than other explicit approximate summation methods of classes of OPT Feynman graphs.In field theory, we consider the use of the EPT as a generalization of the EA for discussing, e.g. high-energy behaviors in MQED as well as infrared divergence and bound-state problems in the limit of massless gluons. It is also suggested that, in view of its nice field-theoretic and high-energy properties, the EPT for MQED might provide a useful laboratory where ideas and problems in hadron dynamics could be meaningfully investigated within a Lagrangian field theory.     

Casimir self-stress on a perfectly conducting cylindrical shell

We compute the Casimir stress on a perfectly conducting cylindrical shell, due to quantum field fluctuations (zero-point energy) in both the interior and exterior regions, using a Green's dyadic formulation for the field strengths. To obtain a finite answer, a frequency cutoff must be inserted, but the result is independent of that cutoff. The Casimir stress is found to be attractive, the Casimir energy per unit length for a cylinder of radius a being $\text{E}$ = $\text{?0.014}\text{a}{}^2$.     

Inclusive production of K̄1(890) and K̄1(1420) in K̄−p interactions at 10 and 16 GeV/c

The inclusive production of K?¹(890) and K?¹(1420) is studied in K?^?p interactions at 10 and 16 GeV/c. At 10 GeV/c an enhancement in the ($\text{K}\text{?}{}^0\text{π}{}^{\mathrm{?}}$) mass distribution is found at 1.74 GeV, but no clear signal is seen at 16 GeV/c. The fraction of $\text{K}{}^0\text{'}\text{s}$ coming from decay of the $\text{K}{}^1\text{(890)}\text{or}\text{K}{}^1\text{(1420)}$ is large, being (50 ± 6)% and (45 ± 5)% at 10 and 16 GeV/c, respectively. The inclusive cross sections for K^1?(890) and $\text{K}{}^{10}\text{(890)}$ production are almost constant with energy from 8 to 32 GeV/c with values of 3.5 and 3.3 mb, respectively. The $\text{K}{}^1\text{(890)}$ production cross section is studied as a function of transverse and longitudinal variables and found to derive mainly from fragmentation of the incident K^? meson. The spectra of $\text{K}{}^0\text{'}\text{s}$ resulting from the decay of $\text{K}{}^1\text{(890)}$ are studied.     

A least upper bound on the superconducting transition temperature

It is rigorously shown that the superconducting transition temperature of any material for which the Eliashberg theory is valid must satisfy k_BT_c ? 0.2309 A, where A is the area under its electron-phonon spectral function α²F(ω). This relation is a least upper bound, not just an upper bound, in the sense that there is an optimal situation in which the equality holds. This occurs when the Coulomb pseudopotential parameter $\text{μ}{}^1$ is zero and the spectral function is the Einstein spectrum Aδ(ω ? 1.750 A). These results are generalized in an approximate, but sufficiently accurate, way to the case $\text{μ}{}^1\text{≠ 0}$ to obtain the more useful least upper bound $\text{k}{}_{\mathrm{<mtext>B</mtext>}}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{? c(μ}{}^1\text{) A}$ and the corresponding optimal spectrum $\text{Aδ[ω ? d(μ}{}^1\text{)A]}$. Numerical results for the functions $\text{c(μ}{}^1\text{)}$ and d(μ¹) are presented for $\text{0 ? μ}{}^1\text{? 0.20}$. It is shown that the T_c's of many materials (including Nb₃Sn), for which experimental values of A and $\text{μ}{}^1$ are available, do not lie very far below the upper bound.     

Drell-Yan-West relation for deep-inelastic and elastic form factors

It is shown, in a scalar current model, that the Drell-Yan-West relations is valid in the form of an inequality. Namely, it is proved that the elastic form factor decreases asymptotically as, or more rapidly than, $\text{～|t|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext><mtext>l</mtext>}}$, provided that the deep-inelastic structure function behaves near threshold as ～(1 ? ω)^{l ? 1}.     

Quark masses

In quark gluon theory with very small bare masses, -$\text{ψ}$$\text{M}$ψ, spontaneous breakdown of chiral symmetry generates sizable masses M_u, M_d, M_s, … We find ($\text{M}{}_{\mathrm{<mtext>u</mtext>}}\text{+ M}{}_{\mathrm{<mtext>d</mtext>}}\text{) /2 ≈ m}\text{p}\text{/ √6 ≈ 312}\text{MeV}\text{,}\text{and}\text{M}{}_{\mathrm{<mtext>s</mtext>}}\text{≈ 432}\text{MeV}$. Scalar densities have well determined non-zero vaccum expectations $\text{〈0|u}{}^{\mathrm{a}}\text{|0〉 ≡ 〈0|ψ(x) (λ}{}^{\mathrm{a}}\text{/2)ψ(x)/0〉 ≈ ?}{}_{\mathrm{\pi }}{}^2\text{M}{}^{\mathrm{a}}\text{,}\text{i.e}\text{〈0? u}{}^{\mathrm{o}}\text{/vb0〉 ≈ 8 × 10}{}^{\mathrm{?3}}\text{(}\text{GeV}\text{)}{}^3$ at an SU(3) breaking of the vacuum c′ ≡ 〈0|u⁸|〉/〈0|u^o|0〉 ≈ ? 16%     

Phase transitions in ferroelectric ceramics of the type Sr0.5Ba0.5Nb2O6

The ferro-paraelectric transition in Sr_0.5Ba_0.5Nb₂O₆ ceramics has been studied by measuring the dielectric pennitivity as a function of temperature at different frequencies. Experimental results in the transition region over ?'_max fit the relation $\text{1}\text{?'}\text{?}\text{1}\text{?'}{}_{\max }\text{= A(T ?}\text{T}{}_0\text{)}{}^{\mathrm{\gamma }}$ with γ > 1, as is typical for dif transitions. In an interval of temperatures of 17°C over T₀ (=Tfor?'_max), γ = 2. Using the relaxors' model of Kirillov-Isupov an equation is inferred in order to compute the activation energy of the relaxors as a function of parameters directly obtained from the experimental results.     

Replica variables,loop expansion,and spectral rigidity of random-matrix ensembles

The replica trick of statistical mechanics is used to derive integral representations of n-point Green's functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the one-loop correction to the GOE one-point function, it is found that the density of states at the edge of the semicircle scales is $\text{～N}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}\text{?}\text{(N}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{δ}\text{)}$ where N is the dimension of the matrix ensemble. For the n-point functions with n ≥ 2, the existence of the microscopic limit to all orders in N^?1 is proved by decomposing the integration variables into massive (i.e., macroscopic) and massless (microscopic) components. Evaluation of the EGOE two-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but not-stationary.     

Line reversal in baryon exchange reactions and the energy dependence of dip locations

An empirical investigation of $\text{p}\text{p}\text{→ππ}\text{and}\text{π}\text{p}\text{→π}\text{p}$ data provides evidence for a dynamical variable connected with the boundary of the physical region. Absence of a fixed u = ?0.2 dip in recent $\text{p}\text{p}\text{→π}{}^{\mathrm{?}}\text{π}{}^{\mathrm{+}}$ high-energy data disagrees with nucleon-exchange (N) line-reversal expectations from π⁺p→pπ⁺ backward scattering. We notice however that the π⁺p→pπ⁺ dip is not located at fixed u but remains with a surprising accuracy at a fixed distance u' = u ? u_min from the physical boundary. Conjecturing that dips associated with particle exchanges occur at fixed u', we predict that the N-exchange dip in $\text{p}\text{p}\text{→π}{}^{\mathrm{?}}\text{π}{}^{\mathrm{+}}$ will be considerably displaced at non-asymptotic energies. Our conjecture places the fixed u' = ?0.2 dip near $\text{u ? ?0.4}\text{in}\text{p}\text{p}\text{→π}{}^{\mathrm{?}}\text{π}{}^{\mathrm{+}}$ for momenta around 2 GeV/c. Folded $\text{p}\text{p}\text{→ ππ}$ distributions exhibit a dip structure at this location. These observations suggest that the failure of line reversal at low energies is due to direct-channel effects, since fixed u' dips occur naturally in geometrical and direct-channel models.     

Remarks on Mc Bryan's convergence proof for (α-4(cos(αϕ)−1+α2ϕ2/2))2-quantum fields

With the help of Mc Bryan's method we prove the existence of F(?)₂-fields, where $\text{F(x) ?}\text{∑}\text{n=0}\text{∞}\text{a}{}_{\mathrm{2n}}\text{x}{}^{\mathrm{2n}}$ is a power series with fast decreasing coefficients. Hypercontractive estimates show the existence of fields with a superposition of exponential or cosine functions as interaction, similar to Mc Bryan's interaction.     

Quantum theory of longitudinal dielectric response properties of a two-dimensional plasma in a magnetic field

An analysis of dynamic and nonlocal longitudinal dielectric response properties of a two-dimensional Landau-quantized plasma is carried out, using a thermodynamic Green's function formulation of the RPA with a two-dimensional thermal Green's function for electron propagation in a magnetic field developed in closed form. The longitudinal-electrostatic plasmon dispersion relation is discussed in the low wavenumber regime with nonlocal corrections, and Bernstein mode structure is studied for arbitrary wavenumber. All regimes of magnetic field strength and statistics are investigated. The class of integrals treated here should have broad applicability in other two-dimensional and finite slab plasma studies.The two-dimensional static shielding law in a magnetic field is analyzed for low wavenumber, and for large distances we find $\text{V(}\text{r}\text{) ～}\text{Q}\text{k}{}_0{}^2\text{r}{}^3$. The inverse screening length $\text{k}{}_0\text{=}\text{2πe}{}^2\text{??}\text{?ξ}$ (? = density, ξ = chemical potential) is evaluated in all regimes of magnetic field strength and all statistical regimes. k₀ exhibits violent DHVA oscillatory behavior in the degenerate zero-temperature case at higher field strengths, and the shielding is complete when $\text{ξ = r′}\text{l}\text{z.shtsls;}\text{ω}$, but there is no shielding when $\text{ξ ≠ r′}\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}$. A careful analysis confirms that there is no shielding at large distances in the degenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> ξ}$. Since shielding does persist in the nondegenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> KT}$, there should be a pronounced change in physical properties that depend on shielding if the system is driven through a high field statistical transition. (It should be noted that the static shielding law of semiclassical and classical models has no dependence on magnetic field in two dimensions, as in three dimensions.) Finally, we find that the zero field two-dimensional Freidel-Kohn “wiggle” static shielding phenomenon is destroyed by the dispersal of the zero field continuum of electron states into the discrete set of Landau-quantized orbitals due to the imposition of the magnetic field.     

A measurement of e+e− → pp for (1975 ⩽ s ⩽ 2250) MeV

The $\text{e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→}\text{p}\text{p}$ cross section has been measured in the energy interval (1975 ? 2E ? 2250) MeV for |cosθ| < 0.7. The measurement is based on ～ 100 events, thus improving by a factor 3 on the previous existing statistics in this energy interval. The form factor |G|² is given as a function of energy under the assumption |G_E| = |G_M|. We also give the first measurement of the differential cross section, averaged over the energy interval, and estimate the ratio G_M|/|G_E| from it.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

Order relations for lattice sums from order relations for theta functions

Let {τ} and {γ} denote mutually reciprocal unit Bravais lattices in an n-dimensional Euclidean space, and consider the Theta Functions (TF's) $\text{V}{}_{\mathrm{\tau }}\text{(t) = t}{}^{\mathrm{<mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}\text{exp}\text{(?πtτ}{}^2\text{)}$ for all 0 < t < ∞. By showing how to evaluate a larger class of sums $\text{Z}{}_{\mathrm{\tau }}{}^{\mathrm{(K)}}\text{(t)  π}{}^{\mathrm{k}}\text{t}{}^{\mathrm{<mtext>k\; +\; </mtext><mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}{}_{\mathrm{r}}{}^{\mathrm{2k}}\text{exp (?πtτ}{}^2\text{)}$, k a nonnegative integer, we are able to evaluate any derivative of the V-functions. With this information we find order relations for the TF's on the cubic lattices in three dimensions. Coupling these relations with Ewald's Theta Function method, we secure order relations for Lennard-Jones, Chaba-Pathria, and other lattice sums on cubic lattices. We also sketch extensions to non-Bravais lattices and give an order relation for TF's on the non-Bravais hexagonal closepacked and the Bravais facecentered cubic.     

Calculation of the γγ → ϱ0ϱ0 cross section at high energies

The processes with the cross sections not decreasing with energy become important at high energies. The simplest processes of this kind are γγ → V_i⁰V_j⁰ where V⁰ = ?⁰, ω, ?, ….. We calculate their cross sections in the high-energy small angle region s ? |t| ? μ². The cross section γγ → ?⁰?⁰ at high energies (s ? 10 GeV²) exceeds those of γγ → ππ, ?⁺?^? considerably. At $\text{s ? 10}{}^4\text{GeV}{}^2$ (this is the characteristic energy for the VLEPP and SLC colliders) and $\text{|t| ? 2}\text{GeV}{}^2$, the ratio $\text{(}\text{d}\text{σ/}\text{d}\text{t)(γγ → ?}{}^0\text{?}{}^0\text{)/(}\text{d}\text{σ/}\text{d}\text{t)(γγ → μ}{}^{\mathrm{+}}\text{μ}{}^{\mathrm{?}}\text{) ? 70}$.     

The effects of density-dependence and exchange on the equivalent sharp radius of the real proton optical potential

The real part of the proton optical potential is calculated using the folding model with Green's strongly density-dependent nucleon-nucleon interaction. The results for a range of nuclei give, approximately, for the equivalent sharp radius for proton optical potential $\text{R}{}_{\mathrm{<mtext>v</mtext>}}\text{= 1.13 A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0.55}$ with exchange. The additive constant is interpreted as the range of the hard core of the nucleon-nucleon interaction.     

Problems of a perpetually oscillating universe

A model is constructed where a Friedmann universe, when collapsing, passes the region of Planek's dimensions near the classical singularity in the De Sitter state. The model assumes that the condition of the one-loop approximation $\text{R}{}_{\mathrm{\mu \nu \lambda \delta }}\text{· R}{}^{\mathrm{\mu \nu \lambda \delta }}\text{≤}\text{1}\text{l}{}_{\mathrm{<mtext>P1</mtext>}}{}^4$ is a universal law of nature. In the case of a dust-like matter the law restricts the mass density (?) to Planck's density $\text{?}{}_{\mathrm{<mtext>P1</mtext>}}\text{=}\text{c}{}^5\text{khx}{}^2$; $\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}{}^2\text{≤ 1}$. In the dust-like model it is assumed that the gravitation constant χ depends on the density as $\text{χ = χ}{}_0\text{φ(}\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}\text{?1)}$, the function φ vanishes at ? = ?_P1 so that the matter tensor in the right-hand side of the Einstein equation χT_μ^ν disappears in this limit, and the Friedmann universe becomes a De Sitter universe whose Λ¹ term is written in the form $\text{Λ}{}^1\text{θ(}\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}{}^2\text{)}$ and at ? → ?_P1, θ → 1. As kh → 0, the theory becomes classical. Some difficulties of a perpetually oscillating model, namely, entropy increase, mass increase due to particle production, and increase of metric perturbations (appearance of gravitational waves) in the process of collapse, are considered in the framework of the model. Various possibilities of the mathematical apparatus of the theory that naturally involve limitations on the curvature value and, in particular, nonlinear Born-type lagrangians are discussed.