Asymptotics of the S-matrix for perturbed Hill operators

We consider the Schrödinger operator with a periodic potential p plus a compactly supported potential q on the real line. We assume that both p and q have m ? 0 derivatives. For generic p, the essential spectrum of the operator has an infinite sequence of open gaps. We determine the asymptotics of the S-matrix at high energy.     

The breaking of exchange degeneracy in dual unitarisation

The splitting of trajectories with non-vacuum quantum numbers is discussed within the dual unitarisation scheme. The estimated difference in intercepts between vector-tensor trajectories of opposite signatures, such as ? ? A₂ and $\text{K}{}^1\text{?}\text{K}{}^7$, agrees both in sign and magnitude with that deduced from phenomenological analyses.     

Baryon exchange effects in dual unitarisation

The effects of baryon exchanges in the renormalisation of Regge trajectories are studied in the dual unitarisation scheme. The main results are that: (i) the pomeron is boosted above α = 1, giving rising total cross sections beyond baryon-antibaryon thresholds, and (ii) the ω trajectory remains approximately at α = 0.5 but acquires a sizeable admixture of the exotic $\text{q}\text{q}\text{qq}$ state, which enhances its coupling to baryons. There are in addition a number of other interesting predictions.     

The vertex bootstrap conditions in the dual unitarisation scheme

The bootstrap conditions for the triple- and double-Regge vertices are derived utilising the planar unitarity condition for six- and five-point amplitudes. The problem of symmetry breaking is considered within the one-dimensional approximation. It is shown that the vertex and propagator equations together predict the equal spacing rule for Regge trajectory intercepts and a well defined pattern of symmetry breaking for vertices.     

On the gauge independence of the S-matrix

The S-matrix is invariant with respect to the variation of any (global) parameter involved in the gauge-fixing conditions, if that variation is accompanied by a certain redefinition of the basis of polarization vectors. Renormalizability of the underlying gauge theory is not required. The proof is nonperturbative and, using the “extended” BRS transformation, quite simple. Received: 27 April 2001 / Published online: 6 July 2001     

S-matrix derivation of the Weinberg Model

By applying only the requirement of asymptotic softness (i.e., the requirement that amplitudes should not grow, asymptotically, faster than that in a renormalizable theory) to various Born amplitudes with four external lines and making some specific assumptions about the number of added particles to meet the requirement, we obtain all of the interaction Lagrangian and relations among the masses and coupling constants of the Weinberg Model.     

S-matrix theory and the density matrix formalism

A general equation governing the time development of the diagonal part of the density matrix is proposed for weakly interacting systems possessing no off-diagonal long range order. This equation, which involves generalized Møller operators of the type employed in S-matrix theory, is solved for two cases and leads to the generalized Pauli and Boltzmann equations.     

The Dirac S-matrix in the k-plane

Expressions of the S-matrix, for the Dirac case, which are suitable for analytic continuation over the complex k-plane for a large class of potentials, fulfilling conditions (1), are derived.     

S-matrix elements from T-duality

Recently it has been speculated that the S-matrix elements satisfy the Ward identity associated with the T-duality. This indicates that a group of S-matrix elements is invariant under the linear T-duality transformations on the external states. If one evaluates one component of such T-dual multiplet, then all other components may be found by the simple use of the linear T-duality. The assumption that fields must be independent of the Killing coordinate, however, may cause, in some cases, the T-dual multiplet not to be gauge invariant. In those cases, the S-matrix elements contain more than one T-dual multiplet which are intertwined by the gauge symmetry.     

The S-matrix in classical mechanics

In the Hilbert-space version of classical mechanics, scattering theory forN-particle systems is developed in close analogy to the quantum case. Asymptotic completeness is proved for forces of finite range. Infinite-range forces lead to the problem of stability of bound states and can be dealt with only in some simple cases.It is a pleasure to thank Prof.L. Motchane for his kind hospitality at the I.H.E.S., where most of this work was done, and where the author profited from discussions withD. Ruelle andO. E. Lanford.     

s-channel discontinuities of the torus insertion in the dual unitarisation scheme

The refined analysis of the s-channel discontinuities of the torus insertion in the topological expansion is performed. Assuming the dependence on reggeon masses of the fixed-pole residue similar to that given by the Veneziano model we obtain almost exact cancellations of the discontinuities which in the one-dimensional approximation were argued to be responsible for the exchange degeneracy breaking between ? and A₂ trajectories. Implications on different f-IP schemes are discussed.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

Diagonal structure of the S-matrix in the lee model

The diagonal structure of the S-matrix in the Lee model is studied. The N2θ- and N3θ-type eigenvalues are shown to factor exactly into the products of the corresponding Nθ eigenvalues, and it follows that these eigenvalues are free of production thresholds. The (implicit) eigenvalue equation for the V2θ-type eigenvalue is given, and this eigenvalue is shown to factor asymptotically into the product of the corresponding Vθ eigenvalues. These properties are conjectured to hold for Njθ- and Vjθ-type eigenvalues. It is conjectured that the two-body type eigenphase operator gives the dominant contribution to the S-matrix at high energy.     

Feynman rules and S-matrix equivalence of the vilkovisky-DeWitt effective action

The reparametrization covariant background field method introduced by Honerkamp in the context of non-linear θ-models and the reparametrization invariant off-shell effective action due to Vilkovisky and DeWitt, which applies to gauge theories as well, are compared. For the latter the Feynman rules of the diagrammatic expansions beyond 1-loop order are made explicit, and the difference between the original definition of Vilkovisky and its modification by DeWitt is explained. Furthermore, within perturbation theory it is shown that the resulting S-matrix is equivalent to the usual one.     

Integrability of the S-matrix versus integrability of the Hamiltonian

This report reviews the relations between the integrability properties of the S-matrix and of the Hamiltonian. Particular emphasis is put on the situation where the Hamiltonian has a conserved quantity which is not compatible with the asymptotics and where correspondingly the integrability does not transfer to the S-matrix. As questions of integrability are more readily handled in classical dynamics, all developments are first performed classically. Several examples are discussed to illustrate the main points. The quantum mechanical discussion reveals that the eigenphase statistics of the S-matrix depends principally on the chaoticity of the scattering map while basis dependent quantities such as the distribution of matrix elements tend to have random matrix behaviour only in the presence of topological chaos. The relevance of these considerations to the evaluation of scattering data is discussed.     

Uniqueness and symmetry breaking in S-matrix theory

Assuming that the physical world is a solution of theS matrix equations, nonlinear functional analysis enables its uniqueness to be tested experimentally. As a first step, we develop such tests within the limits of partial wave dispersion relations,with crossing symmetry included. They are closely related to Levinson's theorem. We show that they give conditions for the validity of the bootstrap hypothesis, of the dynamical generation of symmetries, and of Dashen-Frautschi perturbation theory. They do not appear to be satisfied experimentally.     

Beta functions and S-matrix in string theory

The relationship between sigma model β-functions and string theory scattering amplitudes is proved. We derive our results for the closed bosonic string using the weak field expansion around the flat space. The equations of motion for all the background fields, including the heavy fields are considered. We show that the effective equations for the light fields are obtained by integrating out the heavy fields. It is shown that the contributions to the β-functions come from the boundary of moduli space on a punctured Riemann surface. String loop corrections to the equations of motion are also studied.