Elastic scattering and the associated anomalous dispersion in the energy range 8.63⩽E⩽42.75 keV from heavy atoms

Elastic scattering cross-sections for Pt, Au and Pb were measured using nearly monoenergetic unpolarized Kα x-ray photons in the energy range 8.63⩽E⩽42.75 keV at an angle of 90°. The experimental results were compared with theoretical elastic scattering cross-sections calculated using relativistic form factors (RFFs), relativistic modified form factors (RMFs), a combination of RFFs, RMFs with angle-independent anomalous scattering factors (ASFs) and relativistic numerical calculations based on the multipole expansion of the second-order S-matrix approach. The experimental results were in reasonable agreement with the S-matrix values and values based on an RMF+ASF approach compared with an RFF+ASF approach. © 1988 John Wiley & Sons, Ltd.     

On the connectedness structure of the CoulombS-matrix

The forward direction singularity of the non-relativistic CoulombS-matrix is examined and discussed. The relativistic CoulombS-matrix to order α is shown to have a similar singularity.     

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.     

Quantum field model with unrenormalizable interaction

The unitary relativistic model of quantum field theory with rapidly increasing spectral function (i.e. it grows faster than any finite power of momentum) is investigated. It is shown that there exist nontrivial Lagrangians, leading to this kind of spectral functions and allowing to construct the local theory without the ultraviolet divergences on their basis. In this theory theS-matrix is unitary and not equal identically to unity.     

Collision theory for waves in two dimensions and a characterization of models with trivialS-matrix

Within the framework of local relativistic quantum theory in two space-time dimensions, we develop a collision theory for waves (the set of vectors corresponding to the eigenvalue zero of the mass operator). Since among these vectors there need not be one-particle states, the asymptotic Hilbert spaces do not in general have Fock structure. However, the definition and “physical interpretation” of anS-matrix is still possible. We show that thisS-matrix is trivial if the correlations between localized operators vanish at large timelike distances.     

Three-body scattering in Poincaré-invariant quantum mechanics

The relativistic three-nucleon problem is formulated by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. Two-body interactions are included that preserve the Poincaré symmetry, lead to the same invariant two-body S-matrix as the corresponding non-relativistic problem, and result in a three-body S-matrix satisfying cluster properties. The resulting Faddeev equations are solved by direct integration, without partial waves for both elastic and breakup reactions at laboratory energies up to 2?GeV.     

Scattering formalism for non-localizable fields

We consider the theory of a non-localizable relativistic quantum field. Nonlocalizability means that the field is not a tempered distribution, but increases strongly for large momenta. Local commutativity can then not be satisfied. Instead we assume the existence of Green's functions with the usual analyticity properties. We show that in such a theory theS-matrix can be defined, and its elements can be expressed in terms of the fields by the usual reduction formulae.     

On the Connection between Particles and Poles of the S-Matrix

In axiomatic S-matrix theory it is usually assumed that stable particles give rise to simple poles of the S-matrix for real negative energies while unstable particles give rise to poles close to the real axis on an unphysical sheet of the energy Riemann surface. The stable particle — pole association has been known for a long time not to be always true. For example in potential scattering what is relevant in this case in fact is not the S-matrix but the Jost function. The zeroes of this function for real negative energies are in fact in one-to-one correspondence with the bound states, while the correspondence may break down for the poles of the S-matrix. On the other hand it has recently been pointed out that there also is in general no connection between unstable particles and poles of the S-matrix.     

Relativistic effects in the 3N continuum and the A y puzzle

The three-nucleon (3N) Faddeev equation is solved in a Poincaré-invariant model of the three-nucleon system. Two-body interactions are generated so that when they are added to the two-nucleon invariant mass operator (rest energy) the two-nucleon S-matrix is identical to the non-relativistic S-matrix with a CD Bonn interaction. Cluster properties of the three-nucleon S-matrix determine how these two-nucleon interactions are embedded in the three-nucleon mass operator. Differences in the predictions of the relativistic and corresponding non-relativistic models for elastic and breakup processes are investigated. Of special interest are the lowering of the A _y maximum in elastic nucleon-deuteron (Nd) scattering below ≈25?MeV caused by the Wigner spin rotations and the significant changes of the breakup cross sections in certain regions of the phase space.     

R-matrix plus potential analysis of unbound states of33S

Single channel single levelR-matrix plus potential analysis of the resonances observed in the³²S(n, n) reaction for the neutron energy lying from 20–1100 keV has been carried out to determine the properties of these resonances. This analysis is further supported by a single channel multilevelR-matrix analysis of the data. Spectroscopic factors for the resonances have also been calculated by ourR-matrix method. Its comparison with those obtained bydwba analysis of the (d, p) data is discussed.     

Local fields with the wrong connection between spin and statistics

It is proved that there exist free field operators which satisfy local commutativity and which transform according to certain unitary representations of the homogeneous Lorentz group. The fields satisfy axioms similar to the Wightman axioms, and give rise to local algebras of observables obeying postulates similar to those suggested byHaag. They describe a tower of particles with spins 1/2, 1 1/2, 2 1/2, ..., but commute at space-like separation, giving rise to Bose statistics for the particles. This shows that the well-known theorem on spin and statistics cannot be extended to general theories of local observables; it also shows that the assumptions made in S-matrix theory do not hold for theS-matrix of a theory of interacting infinite fields.The research reported in this document has been sponsored in part by the Air Force Office ofScientific Research OAR through the European Office Aerospace Research U.S. Air Force.     

Construction ofS-matrix from scattering data

A method of construction of the scatteringS-matrix from scattering data is proposed. The scattering matrix is expressed in the form of a rational fraction and takes fully into account the analytic properties of theS-matrix. The method generates a unique and stable analytic continuation of theS-matrix into the complex energy plane. The method is applied to calculation of energy and widths of several resonances in nuclear and atomic physics. Its efficiency is compared with some recently proposed methods.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

On quantization of non-linear relativistic field without recourse to perturbation theory

A new method of quantization of non-linear relativistic field based on separation of the concepts of energy and Hamiltonian is suggested. The proposed method allows the making of the vacuum a stationary state with zero eigenvalues of energy, momentum and charge, and to obtain the exact solutions without recourse to perturbation theory. Infinite integrals, if they appear, turn to be in the denominator and cause no difficulties. An application of this method is demonstrated by an example of a nontrivial relativistic model. Mass-spectrum andS-matrix have been obtained. Only exact solutions are used.     

General properties of physical region spinor amplitudes

The manifestly covariant formalism for dealing with physical S-matrix elements for processes between relativistic particles of arbitrary spin is reviewed. An emphasis is made on the use of homogenous polynomial techniques. Some regularity properties of the spinor amplitudes implied by general on-shell conditions (such as Lorentz invariance and boundedness of the S-operator) are discussed. Covariant decompositions for 2-particles scattering amplitudes are derived in the framework of distribution theory.     

Z 4-symmetric factorizedS-matrix in two space-time dimensions

The factorizedS-matrix with internal symmetryZ ₄ is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ ₄-symmetricS-matrix corresponds to the Baxter transfer matrix itself.     

Threshold singularities of theS-matrix and convergence of Haag-Ruelle approximations

We establish a one-to-one correspondence between the continuity properties of theS-matrix at the 2-particle threshold and the rate of convergence of the Haag-Ruelle approximations ψ(t) for asymptotic 2-particle states ψ with smooth wavefunctions. It turns out that the norm distance ∥ψ?ψ(t)∥ approaches 0 liktt ^?5/4 if theS-matrix has the normal threshold singularities and liket ^?3/4 in the exceptional case where the threshold has “absorbed” a bound state. These connections are valid both in relativistic quantum field theory and in non-relativistic models with short range interaction.     

Quantization of relativistic systems with constraints

The general dynamical system with constraints is quantized, and the S-matrix is constructed in the most general class of gauges including relativistic ones. In the case when constraints do not form a group a new type of additional diagrams arises securing unitarity of the theory: the four-fermion interaction of ghost fields.     

Constraint algebra for interacting quantum systems

We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix.