Yang-Mills instantons and the S-matrix

We present a scheme for calculating gauge-invariant S-matrix elements in the presence of instantons. We exploit the conformal invariance of the zero-mass field equations. The asymptotic in and out states are defined by their values on null infinity $\text{J}$. We use this method to calculate to lowest-order S-matrix elements for scalar particles and fermions in a dilute gas of SU(2) instantons and anti-instantons. The scalar particles acquire an effective mass and an effective interaction of the form $\text{exp}\text{(?(?}{}^2\text{/16π) ?}\text{?}\text{)}$, where ? is the scale of the instanton, plus other interactions which cannot be presented by a local effective lagrangian. The fermions acquire the effective lagrangian obtained by 't Hooft. In the case of a single flavour of fermions, this corresponds to a mass term.     

Equations for subsystems

The exp(S) equations of Coester and Kümmel are rederived from the Schrödinger equations for n-body subsystems which are specified by the amplitudes of n fermions being at certain positions in r-space and the rest moving in shell model states. This derivation needs no formal exp(S) ansatz and allows us to give physical significance to a certain method of truncating the equations. Approximations for systems with short- and long-range forces are distinguished and the application to the electron gas is briefly discussed. It is found that the mutual occurrence of long- and short-range difficulties makes it hard to truncate the equations, and a crude but simple approximation for this case is proposed. In relation to Brueckner-Bethe theory, additional self-consistency is found to be the most important difference. Its importance is stressed and discussed in relation to the concepts of a particle potential and occupation probabilities. The equations are finally studied for a solvable model which simulates long-range forces. It is seen that for this example the truncated equations are highly efficient in getting near to the exact ground state as long as the system shows no phase-transition-like phenomena.     

Amplitude of photostructural changes in chalcogenide vitreous semiconductors

It is shown that a position of an optical absorption edge (OAE) of amorphous AsSe and As₂S₃ films irradiated by light up to saturation is independent on T_exp the temperature at which the sample is exposed, and the amplitude of a reversible photoinduced shift of OAE ΔE is determined by its thermal shift ΔE as it is heated from T_exp up to the glass-transition temperature T_g. So, in order to obtain maximum photoinduced changes we need to use materials with maximum thermal variations of the forbidden zone width, and to tend to a greater difference between T_g and T_exp. The obtained results are well explained within the scope of the local heating model.     

Systematic behavior ofB(E2) values in the yrast bands of doubly even nuclei

The experimental information onB(E2) transition rates in the yrast bands of doubly even nuclei (126≦A≦184) is systematized. The strength functionS _exp≡B(E2,I→I?2)×E(I→I?2) is found to reveal characteristic behavior significant for structure studies of yrast bands. The energy-weightedB(E2,I→I?2) values (S _exp) and 2?/?²(?: moment of inertia) are plotted versus the rotational frequency squared ?²ω² for each nucleus. In strongly deformed nuclei (N≧90), theS _exp curves smoothly increase for low rotational frequencies suggesting that up to spin valuesI≈8 the ratioQ ₀ ² ? is nearly constant (Q ₀: quadrupole moment). This is not the case in nuclei with a soft core (N≦88). In the relevant discussion, the hydrodynamical model as well as the CAP effect are considered. The results in the backbending region are qualitatively discussed in terms of the two-band crossing model. Evidence is found supporting the prediction of an oscillating behavior of the yrast-yrare interaction.     

A simple relationship between chandrasekha's S and T functions

Chandrasekhar's equations for the functions S and T which represent the diffuse scattering and the diffuse transmission, respectively, of a beam of radiation by a planar atmosphere are invariant under the transformations S(τ₁; μ, φ - μ₀), φ₀) exp (-τ₁/μ₀) = T(τ₁; μ, φ; μ₀, φ₀) and T(τ₁; μ, φ; -μ₀, φ₀) exp(-τ_t/μ₀) = S(τ₁; μ, φ; μ₀, φ₀). This relationship reflects the physical symmetry of radiative transfer in a planar atmosphere. It offers a simple method for obtaining the expression for S (T) when an analytical expression for T (S) exists. However, it does not appear to ofter any simplification of numerical solutions.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems

The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Green's functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order $\text{m ? n}$ and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.     

Solving the relativistic inverse scattering problem with allowance for inelasticity effects on the basis of <Emphasis Type="Italic">N/D</Emphasis> equations and application of the resulting solution to an analysis of nucleon-nucleon interaction

A manifestly Poincaré-invariant approach to solving the inverse scattering problem is developed with allowance for inelasticity effects. The equations of the N/D method are used as dynamical equations in this approach. Two versions of the approach are considered. In the first version (method A), the required equations are constructed on the basis of the maximal-analyticity principle, which constitutes the basis of dynamical S-matrix theory. In formulating the second version of equations (method B), it is assumed that a partial-wave scattering amplitude may develop dynamical singularities that violate the requirement of maximal analyticity. The dynamics of interaction components that violate maximal analyticity is described within the model of a nonlocal separable potential. The method is used to analyze nucleon-nucleon interaction in the ¹S₀ and ³S₁ states. The results obtained by solving the inverse scattering problem for potential functions are compared with the predictions of the one-boson-exchange model.     

Approximate solution of bound state problems through continued fractions

A method to solve ordinary linear differential equations through continued fractions is applied to several physical systems. In particular, results for the Schrödinger equation give a good accuracy for the eigenvalues of bound states in theS-wave Yukawa potential, and the lowest order approximations provide exact-values for the harmonic oscillator and Coulomb potential eigenvalues and eigenfunctions.     

Thermodynamic properties of magnetic bilayers

Two models of magnetic bilayers are presented both based on the Heisenberg model. In the first case of ferromagnetically ordered ferromagnetically coupled planes of S=1 the anisotropy is of easy plane/axis type, while in the study of antiferromagnetically ordered antiferromagnetically coupled planes of S=1/2, the anisotropy is of XXZ type. Both systems are treated by Green's function method, which consistently applied within Random Phase Approximation lead to excitation energies and the system of equations for order parameters which can be solved numerically and which satisfies both Mermin-Wagner and Goldston theorem in the corresponding limit and also agrees with the Mean Field results. The basic result is that the transition temperature for magnetic dipole order parameter is unique for both planes. The results are compared with previous theoretical and experimental results.     

Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

We consider U(n + 1) Yang–Mills instantons on the space Σ × S ², where Σ is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n + 1) instanton equations on Σ × S ² are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,?), where A is a gauge potential of the group U(n) and ? is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g > 1, when Σ × S ² becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.     

Trigonometric and Sn symmetric solutions of triangle equations with variables on the faces

General properties of triangle equations with variables on the faces have been considered. A simple method to search for the solutions of these equations is offered. Some examples of applying this method in the case of the general vertex model with the three colors as well as for S_n symmetric model are given.     

From phenomenological thermodynamics to the canonical ensemble

Given the generic canonical probability in phase φ=exp[β(Ψ-H)], contact is traditionally made with phenomenological thermodynamics by comparing the identity δ〈φ〉=0 with the relationTδS=δU+δW, δ indicating an arbitrary infinitesimal variation of the thermodynamic coordinates and angular brackets ensemble means. This paper is concerned with the inverse problem of finding both the generic form of the phase functionw such thatS=〈w〉 and the explicit form φ=αexp[(F-H)/kT] of the canonical distribution on the basis of the requirement that the consequences of the phenomenological laws must be safeguarded, i.e., the relations between the quantities which are their concomitants must also be satisfied by their statistical representatives. Given the generic statistical formalism and specifically thatU=〈H〉, δW=?〈δH〉, the problem is soluble, granted the following generic assumption: “the statistical representative of the entropy is the ensemble mean of a function which depends upon the phase through φ alone.”     

Bottlenecking of indirect g shifts of local moments in metals

The indirect g shift in the electron spin resonance of local moments in metals is examined using the Bloch-Hasegawa equations. The g shift is found to have three components. First the usual Hasegawa term that decreases as the bottleneck increases, and second, a temperature dependent indirect shift that is unaffected by the bottleneck. The third term becomes unobservable in the unbottlenecked limit; it is proportional to the ratio of the non S-state local moment susceptibility to the S-state local moment susceptibility. The width of the resonance is found to be proportional to T — θ, where θ is the ordering temperature, and not to temperature T, thereby providing an explanation for the “negative” residual linewidth frequently observed in magnetically concentrated systems.     

Simple method for determining of the isokinetic temperature value for the SNAr reactions in solution

The changes of the free energy of activation δ?G^≠_exp and the activation entropy δ?S^≠ in the framework of the isokinetic relationship δ?G^≠_exp versus (T_iso ? T_exp) δ?S^≠ were explored quantitatively to predict the isokinetic temperature T_iso for the aromatic nucleophilic substitution reactions in solution.     

Infinite Ergodic Theory and Non-extensive Entropies

We recapitulate results from the infinite ergodic theory that are relevant to the theory of non-extensive entropies. In particular, we recall that the Lyapunov exponent of the corresponding systems is zero and that the deviation between neighboring trajectories does not necessarily grow polynomially. Nonetheless, as we show, no single quantity can describe this subexponential growth, the generalized q-exponential exp _q being, in particular, ruled out. We also revisit a number of dynamical systems preserving nonfinite ergodic measure.     

Microscopic calculation of low-energy deuteron-deuteron scattering on the basis of the cluster-reduction method

The cluster-reduction method is used to solve numerically the differential equations for the s-wave Yakubovsky components characterizing the nnpp system in the S=2 spin state. Elastic deuteron-deuteron scattering is analyzed for the case where nucleon-nucleon interaction is simulated by the MT I–III potentials. Effective equations describing the relative motion of clusters is derived. The scattering length and phase shifts for low-energy deuteron-deuteron scattering are calculated.     

New inversion techniques for the triton photoeffect

The photoeffect cross section for a particle in a model potential has been calculated analytically along with five energy-weighted moments. Two sets of orthonormalized polynomials R_n and S_n are developed with the weighting functions $\text{exp}\text{(?x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{exp}\text{(?x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. respectively. These polynomials are used to invert the calculated moments to get the cross section corresponding to 5, 4, 3 and 2 moments.The cross sections found by S-inversion are in good agreement with the model; those using R-inversion give somewhat poorer agreement.Clare and Lally calculated triton moments and used Laguerre inversion to find the triton cross section. We use their moments with R-inversion and S-inversion. Our resulting cross section agrees reasonably well with Gorbunov's and Faul's measurements of the ³He photoeffect. We find better agreement than that found by Clare and Lally.