Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.     

Embedded elliptic curves and the Yang-Baxter equations

The completely $\text{Z}$_n symmetric S-matrix defined by Belavin is shown to satisfy the Yang-Baxter equations. In the projective space of Boltzmann weights, the curves on which there exist commuting transfer matrices are shown to be embedded elliptic curves. Explicit polynomial equations for these curves are given. For n=2 these results reduce to the results of Baxter for the symmetric eight-vertex model.     

Vertex-IRF correspondence and factorizedL-operators for an ellipticR-operator

As for an ellipticR-operator which satisfies the Yang-Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex-IRF correspondence for the ellipticR-operator is obtained. The Boltzmann weights of the corresponding IRF model satisfy the star-traingle relation. By means of these intertwining vectors, the factorizedL-operators for the ellipticR-operator are also constructed. The vertex-IRF correspondence and the factorizedL-operators for Belavin'sR-matrix are reproduced from those of the ellipticR-operator.     

Hamiltonian structure of isospectral deformation equation and semi-classical approximation to factorizedS-matrices

We consider semi-classical approximation to factorizedS-matrices. We show that this new class of matrices, calleds-matrices, defines Hamiltonian structures for isospectral deformation equations. Concrete examples of factorizeds-matrices are constructed and they are used to define Hamiltonian structure for general two-dimensional isospectral deformation systems.     

Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1-Dimensions

The quantumS-matrix theory of straight-strings (infinite one-dimensional objects like straight domain walls) in 2+1-dimensions is considered. TheS-matrix is supposed to be purely elastic and factorized. The tetrahedron equations (which are the factorization conditions) are investigated for the special two-colour model. The relativistic three-stringS-matrix, which apparently satisfies this tetrahedron equation, is proposed.     

Characterization of completely X-symmetric factorized S-matrices for a special type of interaction applications to multicomponent field theories

We describe all the solutions of the factorization equations for the S-matrix for “pair-two pairs” interaction. These S-matrices are expressed in terms of elementary functions. They correspond to the multicomponent nonlinear Schrödinger equation and the multicomponent Heisenberg spin system.     

The Classical <Emphasis Type="Italic">R</Emphasis>-Matrix of AdS/CFT and its Lie Dialgebra Structure

The classical integrable structure of \mathbbZ₄{\mathbb{Z}_4}-graded supercoset σ-models, arising in the AdS/CFT correspondence, is formulated within the R-matrix approach. The central object in this construction is the standard R-matrix of the \mathbbZ₄{\mathbb{Z}_4}-twisted loop algebra. However, in order to correctly describe the Lax matrix within this formalism, the standard inner product on this twisted loop algebra requires a further twist induced by the Zhukovsky map, which also plays a key role in the AdS/CFT correspondence. The non-ultralocality of the σ-model can be understood as stemming from this latter twist since it leads to a non-skew-symmetric R-matrix.     

The local potential approximation for the Brueckner G-matrix and a simple model of the scalar-isoscalar Landau-Migdal amplitude

The Brueckner G-matrix for a slab of nuclear matter is analyzed in the singlet ^1S and triplet ³ S + ³ D channels. The complete Hilbert space is split into two domains, the model subspace S₀, in which the two-particle propagator is calculated explicitly, and the complementary one, S', in which the local potential approximation is used. This kind of local approximation was previously found to be quite accurate for the ^1S pairing problem. A set of model spaces S ₀(E ₀) with different values of the energy E₀ is considered, E₀ being the upper limit for the single-particle energies of the states belonging to S₀. The independence of the G-matrix on E₀ is assumed as a criterion for the validity of the local potential approximation. It turns out that such an independence holds within few percents for E ₀ = 10-20 MeV, for both channels under consideration. The G-matrix within the local potential approximation is used for justifying a simple microscopic model for the coordinate-dependent scalar-isoscalar component f (r) of the Landau-Migdal amplitude in terms of the free T-matrix. Received: 2 November 2001 / Accepted: 4 January 2002     

Factorizable Z(N) models

The factorizable S-matrix with Z(N) symmetry is constructed. It is speculated that the field theory belonging to this S-matrix matrix is related to the scaling limit of Z(N) generalizations of the Ising model.     

Bethe ansatz equations for the brokenZ N -symmetric model

We obtain the Bethe ansatz equations for the brokenZ _N -symmetric model by constructing a functional relation of the transfer matrix ofL-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.     

Contractions of bialgebras and super-bialgebras

The contraction scheme of Z ₂ graded super-bialgebras is discused. The existence of the contraction limit and the classical r-matrix with and without the coproduct rescaling is investigated. As an example the contraction of D = 4 Poincaré N = 2 super-bialgebra to D = 2 Poincaré N = 2 super-bialgebra is given.     

Three-nucleon calculations without the explicit use of two-body potentials

Using as two-nucleon interaction input the ³S₁-³D₁ and ¹S₀ partial waves, the Faddeev equations are solved for the three-nucleon bound state. The ³S₁³D₁T-matrix is calculated from the Reid potential. Avoiding the usual potential fit, the ¹S₀T-matrix is directly continued off-shell and is constructed consistent with the ¹S₀ phase shift of elastic two-nucleon scattering. The off-shell part of the ¹S₀T-matrix is parametrized and with this parametrization the dependence of the three-nucleon bound-state properties is studied. As a result it is found that the binding energy varies only between 6.2 MeV and 6.8 MeV, while the minimum in the charge form factor for electron scattering from ³He lies between 12.9 fm^?2 and 18.7 fm^?2. The larger (smaller) ³He binding energy is accompanied by a ³He charge form factor whose minimum is at larger (smaller) momentum transfers.     

Metalloradical EPR Signals from the YZ·S-State Intermediates in Photosystem II

The redox-active tyrosine residue (Y_Z) plays a crucial role in the mechanism of the water oxidation. Metalloradical electron paramagnetic resonance (EPR) signals reflecting the light-induced Y_Z· in magnetic interaction with the CaMn₄-cluster in the particular S-state, Y_Z·S_X intermediates, have been found in intact photosystem II. These so-called split EPR signals are induced by illumination at cryogenic temperatures and provide means to both study the otherwise transient Y_Z· and to probe the S-states with EPR spectroscopy. The illumination used for signal induction grouped the observed split EPR signals in two categories: (i) Y_Z in the lower S-states was oxidized by P680⁺ formed via charge separation, while (ii) Y_Z in the higher S-states was oxidized by an excited, highly oxidizing Mn species. Applied mechanistic studies of the Y_Z·S_X intermediates in the different S-states are reviewed and compared to investigations in photosystem II at physiological temperature. Addition of methanol induced S-state characteristic changes in the split signals’ formation which reflect changes in the magnetic coupling within the CaMn₄-cluster due to methanol binding. The pH titration of the split EPR signals, on the other hand, could probe the proton-coupled electron transfer properties of the Y_Z oxidation. The apparent pK _as found for decreased split signal induction were interpreted in the fate of the phenol proton.     

XIII. Exact S-matrices and form factors in 1 + 1 dimensional field theoretic models with soliton behaviour

A review is given of the derivation of exact S-matrices in field theoretic models with soliton behaviour, that means models obeying infinitely many conservation laws which imply the factorization of the S-matrix. Form factors of various operators are calculated exactly by means of Watson's theorem. The exact value of the finite Sine-Gordon wave function renormalization constant is determined.     

Gravitational waves as string vacua II

Classical propagation of (super)strings through gravitational shock waves is analyzed. The exact classical solutions are used for quantization and for the identification of the exact quantumS-matrix describing string scattering by the wave. ThisS-matrix coincides with theS-matrix of the string-string scattering in theflat space-time for particular profile of the shock wave! This is interpreted as the generation of curved geometry from the flat space-time string theory. The quantum consistence of (super)string motion in gravitational plane wave backgrounds is then studied. It turns out that for the standard dimensionsD=26 (D=10) the vanishing of the Ricci tensor for the plane wave is sufficient condition for vanishing of the Weyl (superWeyl) anomaly. Thus, plane wave solutions of the Einstein equations are automatically the classical (super)string vacua. For particular plane waves the anomaly can be evaluated even nonperturbatively.This is the second part of the review based on the PhD thesis of the author defended in 1989 at SISSA, Trieste.     

Conserved currents and symmetry transformations in local scattering theory

The relation between conserved currents and symmetries of theS-matrix is investigated within the framework of Wightman field theory. Assuming a complete particle interpretation with no massless particles, it is shown that every conserved current yields a self-adjoint charge operator which acts additively onn-particle states and commutes with theS-matrix. For currents satisfying current algebra relations of a groupG, the corresponding charges generate a unitary representation ofG.     

Method for Lippmann-Schwinger equations

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind is extended to singular equations of the Lippmann-Schwinger type. Numerical calculations of on- and off-shell t-matrix elements are carried out at positive and negative energies, for the ¹S₀ Reid soft-core nucleon-nucleon potential. Satisfactory convergence is achieved, even with the simplest version of the s     

Low-energy three-body scattering in a hyperspherical adiabatic basis

We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particles in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exactS-matrix analytically, in an expansion in powers of the wave number. We were able to associate the definite powers ofq in the expansion of theS-matrix to the corresponding inverse powers of in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the usual approximations found in the literature (extreme and uncoupled adiabatic approximations), when calculating the diagonal and off-diagonalS-matrix elements. Finally, we show that the coupled adiabatic equations uncouple as the energy goes to zero.