On the nonrelativistic limit of the Dirac theory

The relation between a “nonrelativistic” Hamiltonian of the formH ^∞=(A+B)²+C and a corresponding family of “Dirac-Hamiltonians”H(c) in the limitc → ∞ is investigated. It is shown that the resolvent (z ?H(c))^?1 and the relativistic perturbation of isolated eigenvalues ofH ^∞ are analytic in 1/c for sufficiently large |c|.     

On the criterion of the crystal-liquid phase transition

A localization criterion is proposed for the crystal-liquid phase transition (PT). According to this criterion, the PT begins when the E _d/k _b T ratio reaches a boundary value E _d(s)/k _b T _m such that a solid phase is present above it and a liquid phase is present below it in a phase diagram. Here, E _d is the energy of atom delocalization, k _b is the Boltzmann constant, T is the temperature, and E _d(s) is the delocalization energy for a solid phase at melting point T _m. This criterion is shown to generalize the Lindemann criterion of melting to the case of crystallization and the Löven criterion of crystallization to the case of melting. This localization criterion is found to be applicable for both normally melting substances and substances that melt with a decrease in the specific volume upon the transition into a liquid phase. The relation of the localization criterion to the vacancy and diffusional criteria of the crystal-liquid PT has been studied. The inequality T _N < T _m, where T _N is the temperature of the onset of crystallization, is explained using the localization criterion. The calculated values of the T _N /T _m ratio coincide well with the experimental estimates. The maximum value of T _N /T _m is likely to be most probable in crystals with a bcc structure and a small value of the Grüneisen parameter. The T _N /T _m ratio is analyzed at the points in the PT where no change in the specific volume occurs and an entropy jump is nonzero.     

Effects of the velocity dependence of the collision frequency on the Dicke line narrowing

The effect of the velocity (v) dependence of the transport collision frequency ν_trv on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance (M) and buffer (M _b) particles, β = M _b/M: it is at a minimum for β ? 1 and reaches a maximum for β ? 3. A power-law particle interaction potential, U(r) ∝ r ^?n , is used as an example to show that, compared to ν_trv (v) = const (n = 4), the line narrows if ν_trv (v) decreases with increasing v (n < 4) and broadens if ν _trv (v) increases with v(n > 4). At β ? 3, the line width can increase [compared to ν_trv (v) = const] by 5 and 12% for the potentials with n = 6 and n ? 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I(Ω) has been found to be weakly sensitive to ν_trv (v) at some detuning Ω_c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.     

Breakdown of renormalizability in the Kondo problem in the high-temperature expansion of the free energy

It is shown that there are two energy scales in the Kondo problem: T _k and T ₀, one of which (T _k) is exponentially small in the coupling constant g. The second scale T ₀is proportional to the squared coupling constant. Perturbation theory is valid only in the region T? T ₀. The point T ₀ is apparently the crossover from weak to strong coupling. The first indications of the breakdown of the hypothesis of only one energy scale in the Kondo problem appear in fourth order of perturbation theory.     

Field effect on the dynamics of the Ising spin glass near the percolation threshold

The effect of a uniform field H on the dynamics of the Ising spin glass Fe_xZn_1-xF₂, x=0.25, is studied through Monte Carlo simulations. The correlation function data are consistent with the droplets picture, with a single scaling variable t/t_w, where t_w is the waiting time under H, activation over logarithmic energy barriers near the percolation threshold, x_p=0.24, and considerable change of the glassy phase for intermediate and high H.     

Prediction of the one pion exchange contribution to the processπ − p→π − pω with the help of the Veneziano model

The one pion exchange contribution to the differentialπ ^? p→π ^? pω cross-section is calculated assuming that theπ→πω amplitude associated with one of the vertices is described by the Veneziano formula, and that only a trivial dependence of the latter on the mass of the exchanged pion needs to be taken into account. The remaining unknown overall constant is then fixed by considering the decayω→3π. A rough comparison with preliminary data from a 16 GeV/cπ ^? p experiment shows that the one pion exchange model could possibly describe the reactionπ ^? p→π ^? pω forπω masses above 1.4 GeV.     

On the renormalization of the quantum field theory of point-like monopoles and charges

The quantum field theory of point-like monopoles and charges is first formulated on a euclidean lattice for a convenient regularization. The regularization preserves the peculiar features of the theory, namely those related to the invariance and to the quantization condition. The partition function is expressed as a path integral over the particle's closed paths and the action is constructed in terms of arbitrary surfaces having those paths as boundaries. The possible divergences of the continuum limit are discussed, in particular the vacuum polarization ones. It is found that, although both the electric charge Q and the magnetic charge G are renormalized as Q = Z_QQ_R and G = Z_GG_R, the quantization condition is preserved by the renormalization i.e. Z_QZ_G = 1 so that QG = Q_RG_R = 2πn. Due to the dual symmetry of the theory, then, for Q = G we get Z_Q = Z_G = 1.     

Electroweak one-loop corrections to the decay of the charged vector boson

The electroweak radiative corrections to the decay widths of theW-boson, Γ \((W \to l\bar v,\bar ud,\bar cs)\) , have been calculated in the standard theory. The results are presented in terms of an electroweak form factorρ ^W and their dependence onm _t , andM _H (masses oft-quark and higgs boson) is studied. Typically,ρ ^W ?1 is of an order of one percent. The differenceρ _lv ^W ?ρ _qq′ ^W is negligible, 0.046%. The calculational scheme used is described in detail.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

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.     

A discussion of the Barbier plot

At room temperature the constant C in Barbier's relation lg[S_v|_H=Hc]=-C+lgH_c, S_v is the magnetic viscosity parameter and H_c the coercivity, is approximately the same for a wide range of systems. We take into account the frequency dependence of H_c and derive an expression for C which is both system and measurement dependent. At low measurement frequencies and high temperatures, however, the system dependence of C is suppressed. A similar discussion is carried out for the activation volume v|_Hc.     

The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμ_λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm-Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ_m,n and the energies E_m,n of the bound states are exactly obtained in both the sphere S² and the hyperbolic plane H².     

On the validity of the scaling law for partial-wave amplitudes

It is shown that the scaling law for partial-wave amplitudes Im f^l(s) holds for lγ (√s/4μ) (α(4μ²)?1) log s, if the leading Regge singularity α(t) in the t-channel is bounded by

$\text{Re α(t) ? α(4μ}{}^2\text{+}\text{α(4μ}{}^2\text{)?1}\text{2μ}\text{(}\text{t}\text{?2μ)}$

and is smooth near t = 4μ² in the sense of eq. (4).     

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

Small Values of the Maximum for the Integral of Fractional Brownian Motion

We consider the integral of fractional Brownian motion (IFBM) and its functionals ξ _T on the intervals (0,T) and (?T,T) of the following types: the maximum M _T , the position of the maximum, the occupation time above zero etc. We show how the asymptotics of P(ξ _T <1)=p _T ,T→∞, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for p _T . The data do not reject the hypothesis that the exponent θ of the power law is related to the similarity parameter H of fractional Brownian motion as follows: θ=?(1?H) for the interval (?T,T) and θ=?H(1?H) for (0,T). The point 0 is special in that IFBM and its derivative both vanish there.     

The resonance decay function method in the determination of the pre-factor of the Néel relaxation time of single-domain nanoparticles

In its simple form, the relaxation time of the Néel relaxation process of the magnetic moment of single-domain particles is given by τ_N=τ_0Nexp(σ), σ being the ratio of anisotropy energy to thermal energy. The pre-factor, τ_0N, is normally given a value of 10⁻⁹ s, but values ranging from 10⁻⁸ to 10⁻¹² s have been reported in literature. Here, by means of the field and frequency dependence of the complex magnetic susceptibility, χ(ω,H)=χ′(ω,H)−iχ″(ω,H), of a magnetic fluid sample, in the MHz-GHz range, in conjunction with the determination of the sample decay function, b(t), the pre-factor τ_0N is determined. b(t) is readily obtained through the inverse Fourier transformation relationship, which exists between b(t) and χ″(ω).     

On the asymptotic behaviour of the mass operator near the Fermi energy

By taking due account of momentum conservation, it is shown that, when ω is near the Fermi energy ω_F, the imaginary part of the mass operator $\text{M(}\text{k}\text{, ω)}$ for an infinite Fermi system behaves like (ω ? ω_F)^p(k) where the exponent p(k) ? 2 depends on the interval in which $\text{|}\text{k}\text{|}$ is lying. In particular, the commonly asserted quadratic behaviour (ω ? ω_F² is shown to be true only for $\text{|}\text{k}\text{| ? 3k}{}_{\mathrm{<mtext>F</mtext>}}$. It is explicity assumed that the Fermi system admits a perturbative type treatment.     

Factorization and the couplings of the vacuum trajectory

The decoupling theorems associated with an isolated factorizable pomeron pole of unit intercept are re-examined. It is found that the coupling of three such poles, Γ(t, t, 0), need not vanish, precisely at the point t = 0. This is demonstrated by summing only over states in the appropriate unitarity sum, and sum rule, which are consistent with the M², s/M² → ∞ limit. The triple-Regge region then makes a constant contribution to σ_total, insteadsb of the ln lns result obtained if the isolated pole is assumed to couple also to states such that s/M² = constant. The physical implications regarding factorization and the pole-cut relationship are discussed. The relationship between higher order optical theorems (Mueller discontinuities) and particular terms in the unitarity sum for the two → two absorptive part A₂₂ is exploited. Consistent contributions to the triple-Regge region contribute constant vertex corrections to pure pole behaviour in A₂₂. There is no cut contribution and the magnitude of the vertex corrections reflects the relative amount of diffractive production. The analysis is extended to multiple fireball production where pure multipole structures emerge. The series naturally terminates if the diffractive component is sufficiently small. The implications for the behaviour of the total cross section at machine energies are discussed.     

Beyond the pulse-area theorem: Role of the absorption and the dispersion in the propagation of weak ultrashort resonant pulses

We present here some fundamental but yet underlooked features of the propagation of weak ultrashort pulses (with Δ_dτ << 1, where Δ_d is the Doppler width and τ is the pulse duration) in resonant atomic media. We show that the pulse area behaviour and the pulse spectrum at resonance are governed by the usual optical depth (α₀L, where α₀ is the absorption coefficient at resonance and L the length of the medium), whereas the pertinent parameter that governs the severity of the dispersion effects and the distortion of the pulse is the dispersion parameter e_disp = (α₀L)Δ_dτ that we introduce. Paradoxical effect such as distortionless propagation (e.g. e_disp << 1) with vanishing pulse area (when α₀L >> 1) can then explained within this formalism.