Particle creation if a cosmic string snaps

We calculate the Bogolubov coefficients for aC ⁽⁰⁾ metric which describes the snapping of a cosmic string. In this background, we show that there are noregular solutions with particle interpretation, but we find ageneralized solution with integrable discontinuity, which exhibits particle creation. We also find a regular solution if we allow wave packets.     

Creation of Scalar and Dirac Particles in Asymptotically Flat Robertson-Walker Spacetimes

In the present article we obtain the exact solutions of the Klein-Gordon and Dirac equations for two models of Robertson-Walker spaces with asymptotically Minkowskian regions. Using the obtained exact solutions we calculate the density of scalar and Dirac particles created through Bogolubov transformations technique. For Dirac field it is shown that the creation rate of particles and anti particles are equal.     

Currents,stress tensor and generalized unitarity in conformal invariant quantum field theory

Matrix elements of internal symmetry currents and energy momentum density tensor are constructed in Migdal Polyakov conformal invariant bootstrap field theory. Their 3-point functions satisfy Bethe Salpeter equations which determine any free coefficients that may still occur in the conformal invariant Ansatz. Ward identities are verified for alln-point functions. They imply correct equal time current commutation relations. A proof of generalized unitarity is also given. Various equivalent forms of the propagator bootstrap are discussed. Our algebraic techniques also yield an eigenvalue equation for first order correction to the exactly conformal invariant theory, assuming the latter is Gell-Mann Low large momentum asymptote of a renormalizable finite mass theory.     

Long-range interaction between hydrogen atoms: estimate of third-order contributions

Third-order long-range coefficients C ₁₁, C ₁₃, C ₁₅ for two hydrogen atoms in their ground states are evaluated by perturbation-variation methods in terms of finite basis sets of Slater-type orbitals. Some tests for the accuracy of the variational solutions are proposed and discussed.     

\left\langle {:\bar \psi \psi :} \right\rangle and three-point functions in QCD

We calculate the coefficients of to leading order in _s in the operator product expansion of the fundamental three-point functions of QCD in the deep Euclidean region. We demonstrate that these coefficients satisfy the Ward identities.Supported by BMFT 0233 REB4     

The Andrews–Gordon Identities and q-Multinomial Coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form , with , and . The bosonic side of the identities involves q-deformations of the coefficients of x ^a in the expansion of . A combinatorial interpretation for these q-multinomial coefficients is given using Durfee dissection partitions. The fermionic side of the polynomial identities arises as the partition function of a one-dimensional lattice-gas of fermionic particles. In the limit , our identities reproduce the analytic form of Gordon's generalization of the Rogers–Ramanujan identities, as found by Andrews. Using the duality, identities are obtained for branching functions corresponding to cosets of type of fractional level . Received: 22 January 1996 /Accepted: 4 September 1996     

Causality in quantum field theory with nonlocal interaction

It is proved that theS-matrix satisfies the Bogolubov microcausality condition in each order in perturbation theory in a quantum field theory with nonlocal interaction, where the nonlocality is introduced with the help of form factors being entire analytical functions of the order 1/2.     

Generalized hydrodynamic equations in the hard-spheres model

Generalized linearized hydrodynamic equations in the hard-sphere model are considered on the basis of N.N. Bogolubov’s approximate approach to the analysis of collective interactions. The generalized matrix of transfer coefficients is shown not to be self-adjoint when the finiteness of the two-particle interaction region is taken into account.     

EXTENDED SSH MODEL FOR C58B2 AND C58N2

The extended Su-Schrieffer-Heeger model is used to study the electronic energy levels and molecular structures for two kinds of isomers of C₅₈X₂(X=B, or N). In each case we obtain two sets of stable solutions. For C₅₈X₂ (1, 2), both sets of solutions are found to be reasonable, indicating two different phases. For C₅₈X₂(1, 60 ), only one set of solutions which satisfy the central inversion symmetry is physically reasonable.     

EXTENDED SSH MODEL FOR C58B2 AND C58N2

The extended Su-Schrieffer-Heeger model is used to study the electronic energy levels and molecular structures for two kinds of isomers of C₅₈X₂(X=B, or N). In each case we obtain two sets of stable solutions. For C₅₈X₂ (1, 2), both sets of solutions are found to be reasonable, indicating two different phases. For C₅₈X₂(1, 60 ), only one set of solutions which satisfy the central inversion symmetry is physically reasonable.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

Eigenfunction expansions in the imaginary Lobachevsky space

Eigenfunctions of the Schrödinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of mutual expansions for some sets of solutions are found.     

Solution and ellipticity properties of the self-duality equations of Corriganet al. in eight dimensions

We study the two sets of self-dual Yang-Mills equations in eight dimensions proposed in 1983 by E. Corriganet at. and show that one of these sets forms an elliptic system under the Coulomb gauge condition, and the other (overdetermined) set can have solutions that depend at most onN arbitrary constants, whereN is the dimension of the gauge group, hence the global solutions of both systems are finite dimensional. We describe a subvarietyP ₈ of the skew-symmetric 8 x 8 matrices by an eigenvalue criterion and we show that the solutions of the elliptic equations of Corriganet al. are among the maximal linear submanifolds ofP ₈. We propose an eighth-order action for which the elliptic set is a maximum.     

On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.     

Ward Identities and Chiral Anomaly in the Luttinger Liquid

Systems of interacting non-relativistic fermions in d =1, as well as spin chains or interacting two dimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non-universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed.     

Invariant tensors inSU (3)

The construction of independentSU (3) tensors out of octets of fields is considered by investigating numerically invariantSU (3) tensors. A method of obtaining independent sets of these to any rank is discussed and also independent sets are explicitly displayed up to fifth rank. It is shown that this approach allows us to obtain relations among the invariant tensors, and useful new identities involving thed _ijk andf _ijk tensors are exhibited.     

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V ₁, V ₂ in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V ₁, V ₂. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V ₁ or V ₂ and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V ₁ and V ₂. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters. Received: 14 September 2001 / Accepted: 18 February 2002     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

A combination of Yang’s equations forSU(2) gauge fields and Charap’s equations for pion dynamics with exact solutions

Two sets of nonlinear partial differential equations originating from two different physical situations have been combined and a new set of nonlinear partial differential equations has been formed wherefrom the previous two sets can be obtained as particular cases. One of the two sets of equations was obtained by Yang [1] while discussing the condition of self-duality ofSU(2) gauge fields on Euclidean four-dimensional space. The second one was reported by Charap [2] for the chiral invariant model of pion dynamics under tangential parametrization. Using the same type of ansatz in each case De and Ray [16] and Ray [7] obtained physical solutions of the two sets of equations. Here exact solutions of the combined set of equations with particular values of the coupling constants have been obtained for a similar ansatz. These solutions too are physical in nature.